Landing Gear Integration in Aircraft Conceptual Design

Chapter I Introduction

1.1. Introduction The design of the landing gear, which is considered "the essential intermediary between the aeroplane and catastrophe" [1], is one of the more fundamental aspects of aircraft design. The design and integration process encompasses numerous engineering disciplines, e.g., structures, weights, runway design, and economics, and has become sophisticated in the last few decades.

The landing gear design process is well-documented by Conway [ 1] and more recently by Currey [2] and is experience-based and graphically-oriented in nature. As such, it is a key responsibility of the configuration designer during initial concept studies. However, as industry and government try to incorporate multidisciplinary design optimization (MDO) methods in the conceptual design phase, the need for a more systematic procedure has become apparent. Accordingly, NASA Ames provided Virginia Tech with a two-year research grant to develop a landing gear design methodology that can be implemented within an MDO environment, with a special emphasis on design considerations for advanced large subsonic transports. The result of this research project, known as Landing Gear Integration in Aircraft Conceptual Design, is the topic of this report.

1.2. Overview Several design considerations that must be addressed are briefly discussed to illustrate the complexity involved in the development of such a methodology. The list is made up of an ever-increasing, and sometimes conflicting, number of requirements, e.g., component maximum strength, minimum weight, high reliability, low cost, overall aircraft integration, airfield compatibility, etc., and truly reflect the multidisciplinary nature of the task.

The weight of the landing gear, which typically ranges from three to six percent of the maximum aircraft takeoff weight, is also a design consideration. With advances in flight science technologies, which result in reduced structural and mission fuel weights, the landing gear may become an increasingly large weight fraction in future large aircraft.

Figure 1.1 illustrates the issue. Several typical weight estimating equations are compared

with datatabulated by Roskam[3]. The figure alsocontainsrecentestimates from Boeing

andAirbus for their proposednew largeaircraftdesigns.In particular,note the difference

in trendsbetweenthe ACSYNT andDouglasandTorenbeek equations for weightsabovea

million pounds.It is particularlyinterestingto note that the Airbus estimateagreeswith

both the ACSYNT and Douglasequations.However,which trendis correct?Because the

curvescrossat this point, it is impossibleto tell which estimateis appropriate. Thus,one

design objective of the study is to be able to estimate the weight of the landing gear early in the design phase using a first principals analysis. A major reduction in the landing gear weight may be hard to realize because landing gears are one of the few non-redundant load- paths in an aircraft, and any reduction in reliability from current fail-safe standard is not acceptable [4].

-- ACSYNT gear wt/togw • DC-10-10 Wg/Wto v A-300-132 WgAVto -- - - GD Gear wt/togw • DC-10-30 Wg/Wto [] DC-8 Wg/Wto -- -- - Torenbeek Gear wt/togw [] B 737-200 Wg/Wto • C-5A Wg/Wto ..... Douglas Eqn. [] B 727-100 Wg/Wto [] A-3XX Wg/Wto • DC-9-30 Wg/Wto • B 747-100 Wg/Wto _-') Boeing NLA Wg/Wto MD-80 Wg/Wto O B 707-320 Wg/Wto 0.0600 , , , , , Torenbeek Eqn.

0.0500 ACSYNT • _ " 0.0400

-

Gear Wt/ • .-- " "" No aircraft da!

TOGW _,,,-_o °° 0.0300 ____. __ Do._uglas Eqn i° check trend GD Eqn. J (data source: Roskam's Aircraft Design Weights Volume.._) i i i i I 0.0200 100 103 TOGW 1.0 106 Figure 1-1. Initial comparison of weight equations with aircraft weights data.

The location of the aircraft center of gravity (cg) is critical in the design and location of the landing gear. The nose and main assemblies must be located within specific distances from the aircraft cg, in both the longitudinal and lateral directions, such that the aircraft is in

no dangerof tipping back1 or turning over on its side over the full range of cg locations

under static or dynamic conditions. Another issue to be considered is the distribution of the aircraft weight, which is dependent on the distances between the aircraft cg and the nose and main assembly. Between 85 and 92 percent of the MTOW must be maintained on the main assemblies such that the brakes can provide sufficient energy to slow down the aircraft within a given runway length [5].

Airfield compatibility has become one of the primary considerations in the design of landing gears due to the high cost associated with infrastructure modification, e.g., pavement reinforcement and runway and taxiway expansion[6]. 2 Pavement bearing strength, which varies from one airport to another due to variations in subgrade materials, dictates the number and arrangement of tires needed to produce the required flotation characteristics. Flotation is defined as the capability of the runway pavement and other surfaces, e.g., taxiway and apron, to support the aircraft. In addition, the disposition of the landing gear is constrained by the runway and taxiway geometry as found at the airports to be served. Since the ground track is dependent on the dimensions of the wheelbase and track, an increase in these dimensions could bring the aircraft over the edge of the pavement during certain maneuvers, e.g., a 180-degree turn and centerline-tracing taxiing, and cause the aircraft to bog down in soft soil [7].

Some aircraft have tail props to ensure that the aircraft does not tip back while parked at the gate.

From the AIAA case study series on the 727: "When the first National Airlines 727-200 rolled to a stop as it was delivered in Miami, the pilot touched the brakes, the airplane nose went down and then recoiled up, lifting the nose gear off the concrete about 6 to 8 inches. The gasp in the crowd were heard 3,000 miles away in Seattle .... As far as we know, no 727-200 has ever sat on its tail and maybe we overreacted to the National incident, but that's why you will nearly always see a 727 with its rear airstairs down when parked. There are some rare cases where we attach lead to the radome bulkhead for extreme loading conditions." Note that the cg range of the 727-200 ranges from 8 to 42% of the mac.

2The prototype B-36 had single large main wheels, 110 inches in diameter. They were the largest aircraft wheels ever made. They required a 22 1/2 inch thick runway, thus limiting the prototype to three specially strengthened runways, those at Fort Worth, Eglin AFB and Fairfield-Suisan AFB (later Travis AFB also). A multi-wheel gear could not be obtained until adequate brakes could be designed.

Finally, a four-wheel gear using 56 inch diameter tires was perfected for the B-36A. A 13 1/2 inch thick runway was needed, and 22 primary and a further 22 alternate air fields could handle the production bombers. (source: Meyers K. Jacobson and Ray Wagner, B-36 in action, squadron/signal publications Aircraft No. 42, 1980, the initial TOGW of the B-36 was 265,000 lbs, and grew to 360,000 lbs.).

The soundness of a landing gearconceptdependson the efficacy of overall system

integration. Ground clearance,particularly between the engine nacelleand the static

groundline, plays a key role in determining the length of the landing gear and the

permissible takeoffrotationangle.Insufficientallowancecanresultin costly modifications, e.g., lengthening of the strut with concomitant stowage constraints or complicated strut shrinkage mechanisms, or repositioning of the under-wing engines, that effectively rule out future growth options." The landing gear stowage issue must also be addressed as the number of main assembly struts increases with the increase in aircraft weight [8]. Trade- off studies concerning space availability, structural integrity, and weight penalties resulting from local structural reinforcements are needed to arrive at an optimum design.

With the financial challenges arising from the deregulation of the air-travel industry, airlines need to reduce operating costs to remain competitive. As a result, airlines are demanding that aircraft manufacturers produce new designs with high reliability and low maintenance requirements. Recent technologies, e.g., carbon-carbon heat sinks, radial tires, and high-strength steel, are being introduced. In addition, simplified design and improved manufacturing techniques, e.g., die-forging and three-dimensional machining [9], are being used to reduce the part-count associated with the landing gear system.

1.3. Objectives The development of an MDO-capable design methodology is focused on providing the conceptual designer with tools to help automate the disciplinary analyses, i.e., geometry, kinematics, flotation, and weight. Documented design procedures and analyses as found and referenced by Curry [2] and Torenbeek [3] were examined to determine their applicability, and to ensure compliance with current practices and regulations. Although in most cases the documented analyses were developed for a specific type of aircraft, the essential fundamentals remain unchanged for any type of aircraft. Thus, using the latest information as obtained from industry during an initial industry survey [App. A], the analyses were developed to accommodate the design criteria associated with possible advanced large subsonic transports. Algorithms were then developed based on the updated analysis procedures as a package to be incorporated into existing MDO codes.

"This was the case with the Boeing 727.

Chapter 2 Aircraft Center of Gravity

2.1. Introduction The precise location of the aircraft cg is essential in the positioning of the landing gear, as well as for other MDO applications, e.g., flight mechanics, stability and control, and performance. Primarily, the aircraft cg location is needed to position the landing gear such that ground stability, maneuverability, and clearance requirements are met. Given the fact that none of the existing conceptual design-level cg estimation procedures has the degree of responsiveness and accuracy required for MDO applications, a new approach is formulated to provide a reliable range of cg locations that is better suited for MDO applications.

The connection between the landing gear and the cg has become even more critical with the adoption of advanced control systems. As pointed out by Holloway[10] in 1971, and illustrated here in Fig. 2.1, once the aft cg limit is no longer based on stability but on the ability to generate the required nose down pitching moment, the wing tends to move forward relative to the cg and the landing gear may "fall off" the wing. Thus, the tip-back angle may become an important consideration in determining the aft cg limit. Sliwa identified this issue in his aircraft design studies. [ 11 ] 2.2. Current Capabilities Although not expected to determine the location of the aircraft cg, current aircraft sizing programs, as typified by Jayaram et al. [12] and McCullers [13], do provide some rudimentary estimates. These codes use estimated component weights obtained from statistical weight equations, and either user-specified or default component cg locations to arrive at the overall aircraft cg location. However, as demonstrated by Chai et al. [14], the lack of responsiveness and accuracy have rendered current approaches inadequate for MDO application.

required

back

Tail cg range

limit

Size I

Minimuml,,._I

Tail Size F[

_sedown limit

noseup

limit

allowable

region

fwd aft

cg location Figure 2.1 Typical tail sizing chart with tip back limit becoming the aft cg limit for relaxed static stability aircraft (after Holloway, et al., [10]).

The lack of responsiveness is attributed to the fact that each aircraft component is assigned a specific location within the airframe. Typically, these approaches do not estimate the operational range of cg locations. The cg location is a complicated function of the configuration, loading, and fuel state, with an allowable range limited by a number of operational factors [ 15]. Although a range of cg locations can be established by varying the configuration, equipment arrangement, and payload and fuel states individually, the process is difficult. The accuracy limitations arise because the codes assume that the user has the experience and knowledge required to make adjustments to the component weight and cg estimates. Unfortunately, this approach is not suitable for use in automated procedures required in MDO.

Evidently, what is needed is a new approach which is capable of establishing a maximum permissible cg range for a given configuration. This available cg range can then be compared with the desired operational cg range obtained from performance, control, and operational requirements. If the desired cg range is within the available cg range, the

conceptis viable andcanbe balanced. If not,theconfigurationmustbe changed, eitherby

the designeror an MDO procedure if anautomated process is beingused.

2.3. Alternate Method Component location flexibility at the conceptual design phase is actively exploited as a means to improve the responsiveness and accuracy of current cg estimation procedures.

In the proposed procedure, aircraft components are assigned a range of cg locations based on the geometry, as well as physical and functional considerations, associated with each component. By arranging the cg of the components at their fore- and aft-most limits, the maximum permissible cg range of a particular layout can be established. This cg range can then be used by an MDO procedure to determine the forward and aft aircraft cg limits required to meet performance and stability and control considerations. Adjusted for uncertainty, this maximum permissible cg range can be used as a constraint for the operational cg range during the optimization.

2.3.1. Establishment of Component CG Range The assignment of component cg range is based on the geometry, planform, and the type of components involved. In the case of the primary components, e.g., fuselage, wing, and empennage, the location of these items remains relatively unchanged once the concept is frozen. Consequently, the cg range is expected to be centered near the volumetric center of the component and is unlikely to shift too much. For ease of identification, the primary components will be referred to as the constrained items.

As for secondary components, e.g., equipment and operational items, the location of each component varies from one aircraft concept to another, depending on the philosophy and preference of the airframe manufacturer. Note that as long as the stowage and functionality constraints are not violated, these components can be assigned to any available space throughout the aircraft due to their compactness. Consequently, the

corresponding cg range is defined by the forward and aft boundaries of the stowage space

within which the item is located. Accordingly, these components are termed the unconstrained items.

Although the payload and passenger amenity, i.e., furnishings and services, are confmed within the cargo holds and cabin, operational experience has shown that the cg location of these items varies according to the loading condition and cabin layout as specified by the airlines, respectively. Similarly, the cg location of the fuel varies as a function of time as the fuel is being consumed during the duration of the mission. Given the added freedom in terms of the loading pattern, these components are also classified as unconstrained items.

2.3.2. Generic Component Layout The proposed aircraft component cg ranges are listed in Table 2.1 and represented graphically in Fig. 2.2. The ranges are based on the layout of existing commercial transports [16 and 17] and can be modified to accommodate any unique layout of the aircraft concept under consideration.

The locations of the front and rear spar for the wing and empennage are dictated by space required for housing the control surfaces and the associated actuation systems, where values of 15 and 65 percent chord, respectively, are typically used. As in the conventional cantilever wing and empennage construction, the majority of the structure, i.e., bulkheads, ribs, and fuel tanks, are located between the front and rear spars. Thus, it can be expected that the cg of the wing is most likely to be located between the two, along the respective mean aerodynamic chords (mac). In addition, given the physical arrangement of the fuel tanks, the cg of the fuel and the fuel system can be expected to be located near the same vicinity.

Table2.1

Genericcomponentlocationfor conventionalcivil transports

Corn p onent Typ e Component cg range Wing Constrained Between fore and aft spars along wing mac Constrained 40 to 50 percent fuselage length

Fuselage

Horizontaltail Constrained

Between fore and aft spars along horizontal tail mac Vertical tail Constrained Between fore and aft spars along vertical tail mac Constrained 45 to 60 percent engine length Engines/Nacelles Constrained Between fore and aft wheelwell bulkheads Nose gear Constrained Between fore and aft wheelwell bulkheads Main gear Unconstrained Fuel system Between fore and aft spars along wing mac Unconstrained Between fore and aft wing spars along aircraft Hydraulics centerline; Between aft pressure bulkhead and tip of tailcone Unconstrained Electrical system Between forward pressure bulkhead and nose wheelwell; Between fore and aft wing spars along aircraft centerline Avionics Unconstrained Between forward pressure bulkhead and nose wheelwell Instrumentation Unconstrained Between forward pressure bulkhead and nose wheelwell Environmental Unconstrained Between fore and aft wing spars along aircraft centerline Unconstrained Between aft spar and trailing-edge along surface Flight control mac Unconstrained Between aft pressure bulkhead and tip of Auxiliary power tailcone Unconstrained 45 to 60 percent cabin length Furnishings Services Unconstrained 45 to 60 percent cabin length Unconstrained 45to 60 percent cabin length Passengers Unconstrained 45 to 55 percent forward and aft cargo holds Cargo Fuel Constrained Between fore and aft spars along wing mac; Between fore and aft wing spars along aircraft centerline

The cg of the fuselage depends on the structural arrangement of the pressure

bulkheads, frames, and the aft-body taper ratio. Other factors include local structural reinforcement around the landing gear wheelwells, cargo holds, and the layout of the cabin, e.g., a forward upper-deck as found on the Boeing Model 747 or a double-decker as found on the proposed ultra-high-capacity transports. Taking these factors into consideration, the proposed procedure assumes that the cg of the fuselage is most likely to be located between 40 and 50 percent of the fuselage length.

_ent_l ,.,.__ Electrical system ! _ Hydraufics _" Engines/N_:elles Electrical system _ _ Main gear Auxifiary power ( Hydraulics _vio Instrumentation nics Nose gear tail Furnishings Passengers rvices B Cargo l Cargo Figure 2.2 Ranges of available component cg locations The cg of the engine group varies according to the dimensions of the engine, nacelle, and engine pylon. To account for weight-affecti_,_ factors such as compressor fan diameter, the shape of the nacelle, thrust reverser and pylon structure arrangement, forward and aft cg limit of 45 and 60 percent of the length of the engine, respectively, were assigned.

Regardless of the configuration of the landing gear, the cg of the landing gear will be confined between the landing gear wheelwells in flight. Thus, the forward and aft cg limits of the landing gear are assumed to coincide with the forward and aft stowage volume boundaries of the nose and main assembly wheelwells.

Hydraulics is divided into the wing and empennage group, with the weight proportional to the ratio of the respective control surface area to the total control surface area. The wing group is assumed to be located beneath the wing torsion box, which results in a cg range that is defined by the fore and aft wing spars along the aircraft centerline. On the other hand, the cg range of the empennage group is limited to the space behind the aft pressure bulkhead. Besides providing the stowage volume for the empennage hydraulics, the tail cone space also houses the auxiliary power unit.

Similarly, flight controls are divided into the wing and empennage group, with the weight proportional to the ratio of the local control surface area to the total control surface area. The proposed procedure assumes that the weight of the leading-edge control surfaces is negligible and that the trailing-edge control surfaces are in the retracted position. Thus, the cg of the flight controls are bounded by the rear spar and the trailing edge of each surface, along the respective macs.

The electrical system is divided into the battery and generator groups, assuming that the weight is distributed evenly between the two. The battery group is to be located between the forward pressure bulkhead and the nose wheelwell, although it can also be located in the cavity between the nose wheelwell and the forward cargo hold. The generator group is to share the wing-body fairing cavity as being used to stow the wing hydraulics, i.e., under the wing torsion box. Due to functionality constraints, avionics and instrumentation are assumed to be located in the same compartment which houses the batteries. Similarly, environmental control packs are to share the wing-body fairing cavity with the electrical generator and wing hydraulic groups.

Given -hat the aircraft is fully loaded, the cg of the furnishings, services, and

passengers is limited to between 45 and 60 percent of the cabin length. This assumption takes into account the distribution of the passengers and the corresponding arrangement of the furnishings and passenger services in different cabin layouts. To accommodate the variable nature of the cargo loading operation, which is affected by the type and weight of the baggage and bulk materials, forward and aft cg limits of 45 and 55 percent, respectively, of both forward and rear cargo holds were assigned.

2.3.3. Validation of Analysis A simple spreadsheet software, where the component cg range data as presented in Table 2. I are stored and a macro is defined for calculation purposes, is created to establish the forward and aft limits of the permissible aircraft cg range. A detailed description of the spreadsheet can be found in Chapter Nine. The Boeing Models 737, 747, 767, and McDonnell Douglas DC-10 were used to validate the proposed cg estimation procedure as outlined above. Estimated component weights were obtained from ACSYNT(AirCraft SYNThesis) [12] and used for all four aircraft, while component cg ranges were determined using the generic layout as detailed in the previous section. Essentially, the four aircraft are treated as conceptual aircraft. The objective here is to determine if the maximum permissible cg range as established by the new approach can enclose the actual operational cg range. Actual [18] and estimated aircraft cg ranges determined using the spreadsheet are listed in Table 2.2, both sets of data are shown in Fig. 2.3 for ease of comparison.

Table 2.2 Aircraft cg range Aircraft Estimated, % mac Actual, % mac 12.0/30.0 B737 (forward/aft) 0.0/68.0 1.1.0/32.0 B767 (forward/aR) -4.0/67.0 8.0/18.0 DC 10 (forward/aft) -7.0/46.0 13.0/33.0 B747 (forward/aft) .................... 4.0/63.0 , , B737 , , B767 , , DC10 , , B747 l ] I I , , ] , _ , l 0.0 50.0 Aircraftcg range, °/mac I -'" Operationalcgrange ,--', Permissiblecg range I Figure 2.3 Actual and estimated aircraft cg range comparison As shown in Fig. 2.3, the new approach is capable of producing a permissible aircraft cg range that brackets in the actual operational cg range for all four aircraft. In addition, the estimated cg range offers a generous margin at either end-limit of the band representing the actual operational cg range. Since both the weight and location of the components are based on statistical information, the margin would ensure that the operational cg range remains within the obtainable range even when the uncertainty is included. Evidently, the proposed cg estimation procedure is able to meet the flexibility and reliability requirements that are essential for MDO applications.

Chapter 3 Landing Gear Concept Selection

3.1. Introduction The design and positioning of the landing gear are determined by the unique characteristics associated with each aircraft, i.e., geometry, weight, and mission requirements. Given the weight and cg range of the aircraft, suitable configurations are identified and reviewed to determine how well they match the airframe structure, flotation, and operational requirements. The essential features, e.g., the number and size of tires and wheels, brakes, and shock absorption mechanism, must be selected in accordance with industry and federal standards discussed in the following chapters before an aircraft design progresses past the concept formulation phase, after which it is often very difficult and expensive to change the design [ 19]. Three examples of significant changes made after the initial design include the DC-10-30, which added the third main gear to the fuselage, the Airbus A340, where the main gear center bogie increased from two to four wheels in the - 400 series, and the Airbus A-300, where the wheels were spread further apart on the bogie to meet LaGuardia Airport flotation limits for US operators.

Based on the design considerations as discussed in this chapter, algorithms were developed to establish constraint boundaries for use in positioning the landing gear, as well as to determine whether the design characteristics violate the specified requirements. The considerations include stability at takeoff/touchdown and during taxiing, braking and steering qualities, gear length, attachment scheme, and ground maneuvers.

3.2. Configuration Selection The nose wheel tricycle undercarriage has long been the preferred configuration for passenger transports. It leads to a nearly level fuselage and consequently the cabin floor when the aircraft is on the ground. The most attractive feature of this type of undercarriages is the improved stability during braking and ground maneuvers. Under normal landing attitude, the relative location of the main assembly to the aircraft cg produces a nose-down pitching moment upon touchdown. This moment helps to reduce the angle of attack of the aircraft and thus the lift generated by the wing. In addition, the braking forces, which act behind the aircraft cg, have a stabilizing effect and thus enable the pilot to make full use of the brakes. These factors all contribute to a shorter landing field length requirement.

The primary drawback of the nose wheel tricycle configuration is the restriction placed upon the location where the main landing gear can be attached. With the steady increase in the aircraft takeoff weight, the number of main assembly struts has grown from two to four to accommodate the number of tires required to distribute the weight over a greater area. However, stability and performance constraints as identified by Holloway et al. [10] and Sliwa [11] effectively eliminate all but a few locations where the main assembly can be attached. The attachment limitation phenomenon is known as the location stagnation [App.

A] and can become a major concern for future large aircraft, where additional tires and struts are required to alleviate the load being applied to the pavement. Typically, a large trailing-edge extension, i.e., the Yehudi, l is employed to alleviate at least in part the location stagnation problem. The Yehudi can result in weight and aerodynamic penalties due to local structural reinforcement and increased wetted area, respectively. However, the increased root chord also allows an increase in absolute root thickness for a given t/c. This advantage may outweigh other penalties.

3.3. Landing Gear Disposition The positioning of the landing gear is based primarily on stability considerations during taxiing, liftoff and touchdown, i.e., the aircraft should be in no danger of turning over on its side once it is on the ground. Compliance with this requirement can be determined by examining the takeoff/landing performance characteristics and the relationships between the locations of the landing gear and the aircraft cg.

3.3.1. Angles of Pitch and Roll During Takeoff and Landing The available pitch angle (0) at lifloff and touchdown must be equal, or preferably exceed, the requirements imposed by performance or flight characteristics. A geometric limitation to the pitch angle is detrimental to the liftoff speed and hence to the takeoff field length. Similarly, a geometric limitation to the roll angle (_) could result in undesirable operational limit under cross-wind landing condition.

tApparently known as a "Yehudi", this inboard trailing edge extension actually first appeared on the Boeing B-29 to solve a fuselage-nacelle interference problem. Douglas used it first on a swept wing transport on the DC-8, and it was not adopted by Boeing until the 707 design went to the -320 model.

The name was first used to describe the wind tunnel part that was made on the spot during the wind tunnel test. "Who's Yehudi" was a running gag on a popular radio show at the time, as well as the name of a popular violinist (letter from Bill Cook, retired Boeing engineer and author of The Road to the 707).

For a given aircraft geometry and gear height (hs), the limit for the takeoff/landing pitch angle follows directly from Fig. 3.1. The roll angle at which the tip of the wing just touches the ground is calculated using the expression [5, p. 350] 2h tan _ = tan F + ""g - tan Otan A (3.1) s--t In this case, 1-"is taken as the dihedral angle, s is the wing span, t is the wheel track, and A is the wing sweep. Similar conditions may be deduced for other parts of the aircraft, except that F, A and s in Eq. (3.1) must be replaced with appropriate values. For example, the permissible roll angle associated with nacelle-to-ground clearance is determined with the following values: F measured from the horizon to the bottom of the nacelle in the front view, A measured from the chosen landing gear location to the engine in the top view, and s the distance between the engines.

3.3.1.1. Pitch Angle Required for Liftoff The takeoff rotation angle is prescribed in preliminary design, and then estimated. The final values for 0 and _ are found as the detailed performance characteristics of the aircraft become available. The pitch angle at liftoff (OLOF) is calculated using the expression [5, p.

350]

(3.2)

+d_ 2ll +l.g./2 CLLOF i OLOF =O_LOF dt [ VLo F dC L/da where aWF is the highest angle of attack anticipated for normal operational use, Vwv is the liftoff speed, g is the gravitational acceleration C L wp is the lift coefficient, and dCL/dot is the lift-curve slope. As shown in Fig. 3.1, the dimension of l_ and l 2 are defined by the line connecting the tire-ground contact point upon touchdown and the location of the tail bumper, if one is present. For large transports, the typical value for the rate of rotation (dO dO is taken as four degrees per second [5].

F tip loss static groundlim a) Front view • Chosen gem" location b) Top view static groundline c) Side view Figure 3.1 Geometric definitions in relation to the pitch and roll angles [5]

The detailedaerodynamic datarequiredto useEq. (3.2) is not alwaysavailableat the

conceptual designstage.In most aircraftthe aft-bodyand/ortail bumper is designedsuch

that the aircraft cannotrotate by more than a specifiednumber of degreesat liftoff.

Typically, the value is between 12 and 15 degrees[2]. In addition to the tail scrape

problem,the aircraft cg cannot rotate over and aft of the location of the main assembly, a

phenomenon known as tail tipping and is critical during landing.

3.3.1.2. Pitch and Roll Angles During Landing With the flaps in the fully-deflected position, the critical angle of attack of the wing during landing is smaller than in takeoff. Consequently, the pitch angle during landing is generally less than that during takeoff. In the absence of detailed information, the pitch angle on touchdown (0to) may be assumed equal to OLo F. As for the roll angle upon touchdown, an upper limit of between five [20] and eight [5] degrees is generally applied to large transport aircraft.

3.3.2. Stability at Touchdown and During Taxiing Static stability of an aircraft at touchdown and during taxiing can be determined by examining the location of the applied forces and the triangle formed by connecting the attachment locations of the nose and main assemblies. Whenever the resultant of air and mass forces intersects the ground at a point outside this triangle, the ground will not be able to exert a reaction force which prevents the aircraft from falling over. As a result, the aircraft will cant over about the side of the triangle that is closest to the resultant force/ground intersect.

Assuming first that the location of the nose assembly is fixed, the lower limit of the track of the landing gear, identified as constraint I in Fig. 3.2, is defined by the line passing through the center of the nose assembly and tangential to the circle with a radius of 0.54 times the height of the aircraft cg (hog) from the static groundline, centered at the fore-most cg location [5]. The constant 0.54 is based on static and dynamic instability considerations at touchdown and during taxiing. Conversely, if the location of the main assembly is

assumed to befixed, the aft-mostlimit of the noseassemblymounting location,identified

asconstraint II in Fig. 3.2,is definedasthe intersection of the aircraftcenterlineandthe line

thatpasses throughthecenterof the mainassembly, tangentialto thecircle with a radiusof

0.54timesof theheightof aircraft cg.

/S I " '-

0 54hq _(

•

• Chosen gear location 1 Sideways turnover, fixed nosewheet location m Touchdown E Sideways turnover, fixed main wheel location Figure 3.2 Limits for the undercarriage disposition based on stability [5] 3.3.2.1. Condition at Touchdown The most unfavorable condition at touchdown would be a landing with the aircraft cg at its aft-most and highest location, which can lead to the tail scrape and tail tipping phenomenon mentioned previously. Assuming there are no retarding forces, i.e., spin-up load, a vertical force acting at a distance behind the aircraft cg is needed to produce a moment that will pitch the nose downward. Thus, the minimum allowable offset between the aft-most cg and the main assembly mounting locations, identified as constraint III in Fig. 3.2, is determined using the following expression [5, p. 352] lm >_ ( hcg + es)tan OTD (3.3) where e s is the total static deflection of the shock strut and tire, and 0to is the pitch angle at touchdown. Note that the offset distance is dependent on the value of the pitch angle, whose value is similar to the pitch angle at lifloff, i.e., between 12 and 15 degrees. For a low-wing passenger aircraft, hog can be approximated assuming a full load of passengers and no wing fuel [2]. This generally results in a vertical cg position at the main passenger- deck level.

3.3.2.2. Sideways Turnover Angle Forces acting sideways on the airplane in cross-wind landing condition or a high-speed turn during taxiing could cause the aircraft to turnover on its side. It is thus desirable to keep the turnover angle (g t) as small as possible. The angle is determined using the expression [2, p. 38] hcg tan_ - (3.4)

In sin S

where t tanS- 2(lr a +in) (3.5) and 5 is defined as the angle between the aircraft centerline and the line connecting the center of the nose and main assembly. The dimensions used in the above equations are given in Fig. 3.3. For land-based aircraft, either the maximum allowable overturn angle of 63 degrees [2] or the stability considerations at takeoff and touchdown and during taxiing, whichever is the most critical, determines the lower limit for the track of the main assembly.

3.3.3. Braking and Steering Qualities The nose assembly is located as far forward as possible to maximize the flotation and stability characteristics of the aircraft. However, a proper balance in terms of load distribution between the nose and main assembly must be maintained. When the load on the nose wheel is less than about eight percent of the maximum takeoff weight (MTOW),

controllabilityonthegroundwill becomemarginal,particularlyin cross-windconditions."

This valuealsoallowsfor fuselage lengthincrease with aircraftgrowth. On the otherhand,

when the staticload on the nosewheel exceeds about 15percentof the MTOW, braking

quality will suffer,the dynamicbrakingloadon the noseassembly may becomeexcessive,

anda greatereffort may be requiredfor steering[5]. Note that thesefigures should be

lookeduponasrecommendations insteadof requirements.

Figure3.3Turnoveranglecalculation[2]

3.3.4. Gear Length Landing gear struts should be of sufficient length such that adequate clearance between the runway and all other parts of the aircraft, e.g., the aft-body, wingtips, and engine nacelles, is maintained when the aircraft is on the ground. For a low-wing aircraft with wing-mounted engines, the above requirement proves to be one of the most challenging design issues in terms of permissible roll angle at touchdown. Although engine nacelle-to- ground clearance has not been explicitly defined, a similar requirement for propellers was "There are exceptions. The DC-9-50 has 3% on the nosewheel.

specifiedin FAR Part25 andcanbeusedasan absoluteminimum: a seven-inch clearance

between the propellers andthe groundin level takeoffor taxiing attitude,whicheveris most

critical. To date, the smallestoffset on jet transportsis found on the Boeing Model

747/GE90testbed,wherethe GE90 enginenacelleclearsthe ground by a mere 13-inch

clearance[21]. As for operationalaircraft,the Boeing Model 737-300, -400, and -500

exhibit a 15-inchnacelle-to-ground clearance [22]. The length of the nose wheel strut is

generallybasedon the requirement that the fuselageshouldbe horizontalor tilted slightly

nose-down whenthe aircraft is on the ground.

Besidesthe clearance considerations, allowancemust also be consideredfor future

stretchingof the aircraft,which generallyinvolves addingplugs forward and aft of the

wing spars.Providedthatthe attitudeof the aircraftwill remain the same,the increasein

the aft fuselagelengthwould thusreduce the maximum permissibletakeoffrotationangle, which canresultin costly modificationsandthuseffectivelyruleout future growthoptions.

Boeing abandoned further stretches of the Model 727 partially becauseof the difficulties

encountered while attemptingto maintainan adequate tail scrapeangle, whereasDouglas

wasableto reduce the requiredtail scrape angleon the MD-11 by only structuralchanges,

increasingthe wing incidenceby threedegrees over thatof its 22-foot shorterDC-10-30

forebear.

3.3.5. Landing Gear Attachment From considerations of surrounding structure, the nose and main assembly are located such that the landing and ground loads can be transmitted most effectively, while at the same time still comply with the stability and controllability considerations. For a wing- mounted assembly, the trunnion is generally attached to the rear wing spar and the landing gear beam and the loads are transmitted directly to the primary wing-fuselage bulkheads.

With the inclusion of fuselage-mounted assemblies in the multiple main-strut configurations, a secondary frame would then be added at a distance behind the rear wing- spar, where loads are transmitted forward to the primary wing-fuselage bulkhead through the keel and by shear in the fuselage skin. As for the nose assembly, structural considerations may be conclusive in deciding the mounting location, i.e., at the proximity

of forward cabin bulkhead to minimize weight penalty due to local structural

reinforcement.

3.4. Ground Operation Characteristics Besides ground stability and controllability considerations, the high costs associated with airside infrastructure improvements, e.g., runway and taxiway extensions and pavement reinforcements, have made airfield compatibility issues one of the primary considerations in the design of the landing gear [23]. In particular, the aircraft must be able to maneuver within a pre-defined space as it taxies between the runway and passenger terminal. For large aircraft, this requirement effectively places an upper limit on the dimension of the wheelbase and track.

3.4.1. Aircraft Turning Radii As shown in Fig. 3.4, turning radii are defined as the distances between the center of rotation and various parts of the aircraft. The center of rotation is located at the intersection of the lines extending from the axes of the nose and main assemblies. For aircraft with more than two main struts, the line extending from the main assembly group is located midway between the fore and aft gears. The turning radii are a function of nose gear steering angle (fl); the greater the angle, the smaller the radii. The upper limit for this angle is determined by the methods available to provide the steering action, which generally limits the angle to +60 degrees [2]."

The turning radius corresponding to an 180-degree tum (r18oo_,_) as identified in Fig.

3.4 is determined using the expression t rl8OOturn = btan(90- fl)+ _ (3.6) where b and t are the wheelbase and track, respectively. Given the aircraft design group classification as listed in Table 3.1, the minimum turning diameter, i.e., twice of the 180- degree turn radius, should be less than the corresponding runway pavement width.

' This value may be low. The B737 has +75 °, the DC-8 -1-741/2 °, the DC-9 +80 °, and the B767 +65 °.

Table3.1 FAA airplane design group classification for geometric design for airports [7]

Airplane design group Wingspan, ft Runway width, ft III 79.0 < s < 118.0 100.0 IV 118.0 < s < 171.0 150.0 V 171.0 < s < 197.0 150.0 VI 197.0 < s < 262.0 200.0 With the greater wheelbase and track dimensions as exhibited by large aircraft, the 180- degree turn maneuver can no longer be achieved with the conventional nose-steering scheme alone. As a result, combined nose and main assembly steering systems have been introduced on the newer large aircraft, e.g., Boeing Models 747 and 777, to reduce the turning radii." Other advantages provided by this feature include reduced tire wear and scuffing of the pavement surface in a sharp turn. Note that at the conceptual design phase of an aircraft, Eq. (3.6) is sufficient in producing a first-cut estimate. The resulting turning radii, which are based on nose-steering scheme, are slightly larger than the ones corresponding to combined nose and main assembly steering scheme, and thus provide a built-in safety margin.

• However, the first generation DC-8 also incorporated this feature.

• Chosen gear location t I 180 ° turning radius I I !

t / S t ## ## Figure 3.4 Aircraft turning radii [7] 3.4.2. Centerline-guidance Taxiing The size of the fillets at runway and taxiway intersections depend not only on wheelbase, radius of centerline curve, width of taxiway, and total change in direction, but also on the path that the aircraft follows. There are two options in which an aircraft can be maneuvered on a turn: one is to establish the centerline of the taxiway as the path of the nose gear; the other is to assume that the nose gear follows a path offset outward of the centerline during the tum. The former is selected as the critical design case since it is the most demanding of the two in terms of piloting skill, i.e., difficult to keep the nose wheel, which is below and behind the pilot's field of view, on the centerline while taxiing, and thus requires a greater area of pavement during the maneuver as safety margin.

As shown in Fig. 3.5, the maximum castor angle (tp), i.e., the angle formed between the tangent to the centerline and the longitudinal axis of the aircraft, will occur at the end of the turn, where the nose wheel is at the point of tangency. The angle is approximated by [7, p. 3181 b sin_p = -- (3.7) R where R is the radius of centerline curve. A re-check should be made at this point to make sure that the design castor angle is within the permissible range of the steering angle.

For a given wheelbase and track dimension, the required f'lllet radius (F) is calculated using the expression [5, p. 318] t S (3.8) F = 4R 2 + b 2 - 2Rbsinrp -_- where S is the minimum distance required between the edge of the outboard tire and the edge of the pavement. Given the aircraft design group classification number as determined from Table 3.1 and the corresponding FAA design values as presented in Table 3.2, the upper limit for the wheelbase and track of the aircraft can be determined using Eqs (3.7) and (3.8).

Table 3.2 FAA recommended taxiway exit geometry [7] Group VI Group III Group IV Group V 150.0 150.0 170.0 Centerline radius, ft 100.0 80.0 85 85.0 Fillet radius, ft 55.0 15.0 15.0 20.0 Safety margin, ft ........ 10.0 Fillet radius, F

, \

\

\\ Safety margin, S Radius of centerline curve, R

/

/ \\ _'\\ \ • Chosen gear location Figure 3.5 Taxiway fillet design [7] 3.5. Landing Gear Disposition Constraints Landing gear location constraints as discussed in the above sections are superimposed on the three-view of a notional aircraft for illustrative purposes. As shown in Fig. 3.6a, the main assembly must be located such that when the shock strut is at the fully-extended position, the tire-ground contact point is below constraints IT and IV in the vertical direction and outboard of constraint I in the lateral direction. In the top view as shown in Fig. 3.6b, the main assembly must also be located aft of constraint IV in the longitudinal direction and outboard of constraint V in the lateral direction. As for the nose assembly, it must be

locatedbetweenconstraints I andII in the longitudin.al direction.And finally, as shown in

Fig. 3.6c,the fully-extended tire-groundcontact point is below constraintII in the vertical directionandaft of constraint 1Iin thelateraldirection.

m ii Static grotmdline I Sideways turnover II/ Wingtip-to-grotmd clearance Fuselage tail clearance at lit_off II Nacelle-to-ground clearance IV a) Front view I II III V I Nosewheel load, 8% MTOW IV Touchdown Nosewheel load, 15% MTOW V Sideways turnover, fixed nosewheel location III Sideways turnover, toted main wheel Ioaetion b) Top view Figure 3.6 Landing gear attachment location constraints [5] Static groundline Touchdown II Fuselage tail clearance at liftoff c) Side view Figure 3.6 Landing gear attachment location constraints (cont'd)

Chapter 4 Twes, Wheels, and Brakes

4.1. Introduction The number of tires required for a given aircraft design gross weight is largely determined by the flotation characteristics, which will be discussed in detail in Chapter Seven. Assuming that the number and distribution pattern of the tires is already known, this chapter provides guidelines to the selection of the tires, wheels, and brakes that will meet the performance and safety requirements [2 and 20].

As a part of the landing gear configuration definition process, tires, wheels, and brakes selection algorithms were developed based on the procedure as discussed in this chapter.

Specified selection criterion, e.g., minimum size, weight, or pressure, are used to select suitable tires and wheels from manufacturer's catalog [24] and industry standards [25], while statistical database was used to size the brakes as required to meet the braking requirements.

4.2. Type, Size and Inflation Pressure of the Tire The tire selection process involves listing all candidates that meet the performance requirements. A list of tires and wheels used on commercial transports can be found in Appendix D. The primary consideration is the load-carrying capacity of the tire during the speed regime normally applicable for landing or takeoff cycles. In addition, the number of plys and type of construction, which determines the weight of the tire and its operational life, is important from an economic standpoint. Other considerations include the inflation pressure of the tire and the size of the wheel. The former must be chosen in accordance with the beating capacity of the airfield from which the aircraft is designed to operate from, whereas the latter must have sufficient space to house the brake assembly.

3O 4.2.1. Basic Tire Constructions Radial tires have gained growing acceptance since their introduction despite a somewhat cautious approach at the beginning, which is attributed to lack of applicable standards, concerns about the mixability with bias tires, and retreadability of refurbished tires. Intermixing of radial and bias tires, or even with radial tires of different construction, is possible only if the loading is no more uneven than currently encountered with mixing of bias tires only. As for retreadability, it should be noted that multiple retreating is not necessarily a benefit to the airlines; instead, it could be an indication of low tire performance in terms of tread wear. Thus, the concern here is not as much how often the tire can be retreated, but how to extend the average total carcass life.

In radial construction, shear stresses in the rubber matrix are minimized and loads are efficiently distributed throughout the tire. Even if the same basic materials used in bias tires are used in the radials, the amount of material required for a particular application can be reduced. As a result, weight savings of up to 20 percent have been realized [26]. In addition, minimized slippage between the tire and the contact surface and the near optimal tuning of belt stiffness that comes with the radial construction all contributed to improved wear performance. In fact, some radial tires currently achieve twice as many landings per tread as conventional bias tires [26].

Operational experience has also shown that radial tires offer a greater overload bearing capacity and withstand under-inflation better. An approximately 10-percent increase in the footprint area improves the flotation characteristics and reduces hydroplaning on wet runways [26]. In addition, radial tires do not fail as suddenly as bias tires do. Warning signs such as external deformation and out-of-roundness exhibited prior to catastrophic failure provide indications of a potential blowout to maintenance personnel, and thus enhance operational safety.

4.2.2. Size of the Tire The choice of the main wheel tires is made on the basis of the static loading case. The total main gear load (F,,) is calculated assuming that the aircraft is taxiing at low speed without braking. As shown in Figure 4.1, equilibrium gives [5, p. 356]

In

F m - W (4.1) lm + ln where W is the weight of the aircraft and l,, and In are the distance measured from the aircraft cg to the main and nose gear, respectively. The design condition occurs at MTOW with the aircraft cg at its aft limit. For single axle configurations, the total load on the strut is divided equally over the tires, whereas in tandem configurations, the load per wheel depends on the location of the pivot point; to reduce overloading of the front wheels during braking, the pivot is usually positioned such that the distance between it and the front and rear wheel axles is about 55 and 45 percent of the truck beam, respectively [5].

The choice of the nose wheel tires is based on the nose wheel load (Fn) during braking at maximum effort, i.e., the steady braked load. Using the symbols shown in Fig. 4.1, the total nose gear load under constant deceleration is calculated using [5, p. 358] Fn lm+ In lm +'"_n W- D+ T (4.2) where L is the lift, D is the drag, T is the thrust, and hog is the height of aircraft cg from the static groundline. Typical values for aJg on dry concrete vary from 0.35 for a simple brake system to 0.45 for an automatic brake pressure control system [5]. As both D and L are positive, the maximum nose gear load occurs at low speed. Reverse thrust decreases the nose gear load and hence the condition T= 0 results in the maximum value [5, p. 359] Fn =lm +hcg(ax/g)w (4.3) lm + ln The design condition occurs at MTOW with the aircraft cg at its forward limit.

L _Z Figure 4.1 Forces acting on the aircraft during a braked roll [5] To ensure that the rated loads will not be exceeded in the static and braking conditions, a seven percent safety factor is used in the calculation of the applied loads [2].

In addition, to avoid costly redesign as the aircraft weight fluctuates during the design phase, and to accommodate future weight increases due to anticipated aircraft growth, the calculated loads are factored upward by another 25 percent prior to tire selection [2].

4.2.3. Inflation Pressure Provided that the wheel load and configuration of the landing gear remain unchanged, the weight and volume of the tire will decrease with an increase in inflation pressure.

From the flotation standpoint, a decrease in the tire contact area will induce a higher bearing stress on the pavement, thus eliminates certain airports from the aircraft's operational bases. Braking will also become less effective due to a reduction in the frictional force between the tires and the ground. In addition, the decrease in the size of the tire, and hence the size of the wheel, could pose a problem if internal brakes are to be fitted inside the wheel rims. The arguments against higher pressure are of such a nature that commercial operators generally prefer the lower pressures in order to maximize tire life and minimize runway stress [26].

4.3. Wheel Design The design of the aircraft wheel is influenced primarily by its requirement to accommodate the selected tire, to be large enough to house the brake, and to accomplish the above tasks with minimum weight and maximum life. As shown in Figure 4.2, two basic configurations of wheel design are currently available: A-frame and bowl-type [27].

The former is structurally the most efficient and therefore the lightest that can be achieved. However, this design has a limited space for housing the brake as compared to the bowl-type design. Consequently, as the braking energy requirement increases with aircraft weight and hence the size of the heat sink required, it might be necessary to resort to a bowl-type design even though it has a weight penalty [27].

/l_,¢/Be.ad seat tad Ftnible ,---n M'F . .j. ,.

........ -I-- "I I

a) A-frame b) Bowl-type Figure 4.2 Basic configuration of wheel design [27] Continued heavy dependence on forged aluminum alloy wheels is foreseen by industry, whereas steel and magnesium alloy wheels are no longer given serious consideration due to weight and corrosion problems, respectively [28]. Although practicable, titanium wheels are still quite expensive. Most of the premium for titanium wheels results from the expense for the forging process, which could be 10 to 11 times

thoseof aluminumalloy [28] In addition,currenttitanium forging tolerances haveyet to

reachthe precision obtainablefor aluminummaterial,thus machiningof all surfacesis

requiredto controlweightandobtainthe desiredform.

Basedon statisticaldata,the wheelassembly weight is determinedasa function of the

ratedperwheelstaticload (F) andaverage tire outerdiameter(D) [2, p. 145]

FD fw - (4.4) Given the type of material to be used, the wheel assembly unit weight is obtained from Figure 4.3 with the weight factor 0rw) as determined from Eq. (4.4).

-- _ Casted AI | .,_"_,_ -------a------ F e | _/'7_/-

1°3 II--For edA'

_ 10 2 _ 101 10 o ........ 1 , _ , ,,,,_1 j ....... t 101 102 103 104 Weight factor, fw Figure 4.3 Aircraft wheel assembly weight [2] 4.4. Brake Design Besides the primary task of stopping the aircraft, brakes are used to control speed while taxiing, to steer the aircraft through differential action, and to hold the aircrat_ stationery when parked and during engine run-up. Since the heat sinks account for a significant fraction of total landing gear weight, there is a continual effort to reduce their weight through the application of advanced materials, namely, carbon [ 19].

4.4.1. Heat Sink Material Material characteristics of steel and carbon are compared in Table 4.1. As shown in the table, carbon's high specific heat and thermal conductivity make it highly desirable as a heat absorber. The former ensures a reduction in brake weight, while the latter ensures that the heat transfer throughout the heat sink occurs more uniformly and at a faster rate.

In addition, carbon retains much of its specific strength, which is defined as the ultimate tensile strength divided by density, at high temperature while steel loses almost all of its strength.

Table 4.1 Heat sink materials comparison [2] Property ........... Steel Carbon Desired Density, lb/in 3 0.283 0.061 High Specific heat at 500°F, Btu/lbo°F 0.13 0.31 High Thermal conductivity at 500°F, Btu/hoft2o°F 24.0 100.0 High Thermal expansion at 500°F, 1.0E-6 ino°F/in 8.4 1.5 Low Thermal shock resistance index, x105 5.5 141.0 High Temperature limit, °F 2,100 4,000 Hi_ Long service life and low maintenance requirements for carbon brakes prove to be another plus from an economic standpoint. It was estimated that carbon would permit up to five to six times more landings as compared to steel between refurbishment and would require fewer man-hours for overhaul [27]. To illustrate the economic advantage of using carbon brakes, it was estimated that a total weight saving of 1,200 pounds could be achieved on the Concorde using carbon brakes. This is equivalent to five percent of its estimated transatlantic payload [29].

The primary drawback of carbon brakes is that a greater volume is required to absorb the same amount of energy in comparison to steel brakes. Some problems with carbon brakes include sudden loss of strength due to oxidation of the carbon, temporary loss of braking due to moisture contamination, and high initial cost. However, these issues have

largelybeenresolvedin favor of the performanceand economicaspectsof carbonheat

sinks [16]. In fact, advanced transportssuchas the BoeingModel 777 and the emerging

ultra-high-capacity aircraftall featurecarbonbrakes.

4.4.2. Brake Sizing The primary consideration in brake development is the size and weight of the brake required to meet the kinetic energy generated under the design landing weight, maximum landing weight, and rejected takeoff (RTO) conditions. Brake capacity requirements for these braking conditions are listed in Table 4.2.

Table 4.2 FAA commercial transport brake capacity requirements [20] Specifications 1O0 stops at average of 10 f_JS 2 deceleration Design landing weight 5 stops at average of 10 ft/s 2 deceleration Maximum landing weight 1 stol9 at average of 6 ft/s 2 deceleration Reiected takeoff The total kinetic energy is determined using the expression [20] KE = 0.0443WV 2 (4.5) where V is the power-off stalling speed in knots. Assuming that the power-off stalling speed is 1.2 times of the stalling speed (V,), it can be approximated using the expression [5, p. 577] 2W (4.6) V = 1.2V s = 1.2 I'13pSCL,max where/9 is the standard sea-level air density, S is the reference wing area, and CL.,,ax is the maximum wing lift coefficient. The constant 1.13 takes into account the speed loss in the FAA stall maneuver [5]. As illustrated above, the kinetic energy absorption requirements increase as the square of the velocity and hence the landing speed is significant.

The procedure used to size a steel brake is given here for illustrative purposes. Similar data for carbon are not available, but scaling factors of 1.28 and 0.40 can be used to relate the steel volumes and weights, respectively, to those values for carbon [30]. Kinetic energy levels expected under the normal landing weight, maximum landing weight, and RTO conditions are first calculated using Eq. (4.5) and the appropriate aircraft weights.

Brake assembly weights (Wbrake) corresponding to each kinetic energy level are obtained from Figure 4.4 and averaged to arrive at a compromise value. The required heat sink volume (Vbrake) is then approximated using the expression Vbrak e = 3.3Wbrak e - 84.2 (8.7) where the constant coefficients are determined using linear regression analysis on statistical database[2].

500.0 _1 -------o-----100 Stops ----o-----5Stops -----o--_RTO I 400.0 ._ 300.0 E _ 200.0 100.0 - i _ I I l I I I i i I I I I r I I I L I 0.0 0.0 20.0 40.0 60.0 80.0 100.0 Braking kinetic energy, ft lb x 10.0e06 Figure 4.4 Brake assembly weight vs. kinetic energy level [2] Given the tire wheel diameter as determined during the tire selection process, heat sink inner and outer diameters and the volume per inch width constant are selected from Table 4.3. Dividing the total volume by the constant then gives the necessary heat sink width.

The envelope for the heatsink andtorqueplate carrieris established by adding0.75 inch

on theinsidediameter andthe endfacingthewheel centerline.Finally, the piston housing

envelopeis approximated by addingtwo incheson the actuationsideof the heat sink as

shownin Figure4.5 [2].

Table 4.3 Heat sink dimensions [2] i Outer dia., in Volume/inch width, in 2 Inner dia., in Rim dia., in 7.375 12.000 70.4 14.0 13.000 80.9 15.0 8.125 13.750 88.4 16.0 8.750 14.750 100.0 17.0 9.500 15.750 114.3 18.0 10.125 16.500 123.1 19.0 10.750 17.500 136.7 20.0 11.500 18.500 150.9 21.0 12.250 19.500 168.5 22.0 12.875 20.375 176.3 23.0 12.750 21.375 195.2 24.0 14.375 22.375 212.1 25.0 15.125 iiiiii I II III IIII e wheel and rotate with the wheel Brakes are primarily stators and rotors, forced together to generate friction, which is converted -,w-...- Stators: keyed to the central torque tube, to heat.

_nT-_--_ _io--ff--ary Key elements of carbon brakes. [See [2], pg 138 for more detailed view.]

Figure 4.5

Chapter 5 Shock Absorber Design

5.1. Introduction The basic function of the shock absorber is to absorb and dissipate the impact kinetic energy to the extent that accelerations imposed upon the airframe are reduced to a tolerable level [2 and 20]. Existing shock absorbers can be divided into two classes based on the type of the spring being used: those using a solid spring made of steel or rubber and those using a fluid spring with gas or oil, or a mixture of the two that is generally referred to as oleo-pneumatic. The high gear and weight efficiencies associated with the oleo-pneumatic shock absorber make it the preferred design for commercial transports [2].

Based on the analysis procedure as outlined in this chapter, algorithms were developed to determine the required stroke and piston length to meet the given design conditions, as well as the energy absorption capacity of the shock absorber.

5.2. Oleo-Pneumatic Shock Strut Design The basic weight support function of the oleo-pneumatic shock struts, which have a high efficiency under dynamic conditions both in terms of energy absorption and dissipation, is provided by a compressed cylinder of air and oil. A single-acting shock absorber, which is the most commonly used design for commercial transports, is shown in Fig. 5.1. This type of shock strut absorbs energy by f'trst forcing a chamber of oil against a chamber of dry air or nitrogen and then compressing the gas and oil. During the compression process, the oil and gas either remain separated or are mixed depending on the type of design. After the initial impact, energy is dissipated as the air pressure forces the oil back into its chamber through recoil orifices.

Although the compression orifice could be merely a hole in the orifice plate, most designs have a metering pin extending through it, and by varying the pin diameter the orifice area is varied. This variation is adjusted so that the strut load is fairly constant under dynamic loading. If this can be made constant, the gear efficiency would be 100 percent. In practice, this is never obtained and efficiencies of 80 to 90 percent are more usual [4]. Since only the efficiency factor is of interest in the conceptual design phase, no additional discussion on the design of the metering pin will be provided.

Valve

Chamber (Air)

Orifice Support Tube,,

UpperBearing

Lower Chamber

g Pin

Piston

LowerBearin

Figure5.1 Single-acting shockabsorber, after[4], with no attribution

5.2.1. Stroke Calculation The first step in calculating the stroke (S) is to select the design reaction factor (N), sometimes called the landing load factor. This factor should not be confused with the aircraft load factor, which results from maneuvers or atmospheric disturbances. For a transport-type aircraft the landing load factor varies from 0.7 to 1.5, with 1.2 being the most widely used value [2].

Sink speed (Vs) is usually legislated by the procuring authority and/or the regulations pertaining to a particular category of aircraft. The FAA requires that a transport-type aircraft be able to withstand the shock of landing at 10 ft/s at the design landing weight and 6 ft/s at maximum gross weight [ 19]. In practice, sink speeds of this magnitude rarely occur due to ground effects and flare-out of the aircraft prior to touchdown.

The total energy (E) of the aircraft at the instant of touchdown, which consists of kinetic and potential energy, is approximated using the expression [2, p. 35] WV 2 E- F(W-LXS+S,) (5.1) 2g where W is the aircraft weight, V is the sink speed, g is the gravitational acceleration, L is the wing lift, and S, is the tire deflection. S is shock absorber stroke, which is the value we are trying to find. Given that the kinetic energy capacity of the shock absorber and tire must be equal to the total energy, Eq. (5.1) becomes [2, p. 35] WV 2 rlsSNW + rltStNW = _+ (W- L)(S + St ) (5.2) 2g where r/$ and r/t are the shock absorber and tire absorber efficiency factors, respectively.

The former is generally assumed to be 0.47 and the latter 0.8 for an oleo-pneumatic strut [2]. To maintain an adequate safety margin, an extra one inch of stroke is usually added to the calculated stroke.

5.2.2. Compression Ratios Compression ratios are the ratios of the pressure under one condition divided by the pressure under another condition, e.g., fully compressed to static. Two compression ratios are normally considered: static to fully extended and fully compressed to static. For transport-type aircraft, where floor height variation is important, a ratio of 4:1 for the static to extended case and 3:1 for the compressed to static case would be satisfactory [2].

Assuming a static pressure (P2) of 1,500 psi, which enables standard compressors to be used for servicing and provides enough margin to allow for aircraft growth, pressures at the extended (P1) and compressed (P3) positions are calculated using the compression ratios given above. Note that the piston area (A), and subsequently the displacement volume (d), are both a function of the static pressure, that is F A = -- (5.3) and d= SA (5.4) where F is the maximum static load per strut.

5.2.3. The Load-stroke Curve The energy absorbed by the strut during its stroke is obtained by integrating the area beneath the load-stroke curve, which relates the magnitude of the applied ground loads to the stroke traversed. Standard notation for shock strut sizing uses the subscript 1 to denote the fully extended position, 2 to denote the static position, and 3 to denote the compressed position. To accommodate excess energy produced in a heavy or semi-crash landing, shock absorbers are designed such that the piston is not fully bottomed even at the compressed position, i.e., V 3 _ O. The reserve air volume, which is assumed to be 10 percent of the displacement [2], allows the shock strut at a predetermined load to move through extra travel, absorbing the excess energy by the work done. Hence, the air volume at the fully-extended position is approximated as [2, p. 100]

vl = v3 + d (5.5)

Pressures between the extended and static positions are defined by the isothermal compression curve, which is representative of normal ground handling activity [2, p. 100] P1V1 = PxVx = const (5.6) Given the relationships of Eqs (5.4) and (5.5), the pressure at stroke X is obtained using the expression [2, p. 100] Px - P1V1 - PI(V3 +d) Sextend < X < Sstatic (5.7)

Vx Vl - XA

Pressures obtained using Eq. (5.7) are then multiplied by the piston area to arrive at the design loads as shown on the load-stroke curve.

A polytropic, i.e., real-gas, compression curve should be considered for pressures between the static and compressed positions. It is representative of dynamic compression cases such as landing impact and bump traversal and is based upon PV" being constant [2], hence Px = P2 V1 --'XA Sstatic < X < Scompress (5.8) The constant n can either be 1.35 or 1.1; the former is used when the gas and oil are separated and the latter when they are mixed during compression. The distance from the

staticto the fully compressed positionis largelya matterof choice.Statisticaldataindicate

thattransport-type aircrafttypically havefurther compression beyondthe staticposition of

about 16 percent[2] of the total stroke,a figure which tends to give a hard fide while

taxiing. However, with the staticpositionbeing so far up the load-strokecurve, where a

large amountof energy is absorbedwith a relatively small stroke travel, aircraft weight

variations do not result in substantialgear deflections.That is, the built-in margin

minimizes the needof redesigningthe baselineshock strut for uses on future growth

versionsof the aircraft.Again, thepressures obtainedusingEq. (5.9) aremultiplied by the

piston areato arriveat the designloads.At this point the valuesof P1 and P3 should be

checked to ensure that the former is greater than 60 psi to avoid sticking due to friction between the piston and the cylinder wall, while the latter is less than 6,000 psi to prevent seal leakage [2].

5.2.4. Internal Cylinder Length As specified by MIL-L-8552, the distance between the outer ends of the bearings shall be not less than 2.75 times the internal cylinder/piston outside diameter (D). Thus the minimum piston length is given by [2, p. 111] (5.9) Zpist = S + 2.75D where D = _'_ (5.10) 5.2.5. Sample Calculation The load-stroke curve of a notional single-acting shock absorber is generated for illustrative purposes. Based on the design requirements as stated in Table 5.1, Eq. (5.3) gives a piston cross-sectional area of 33.3 in 2, while Eq. (5.4) places the total displacement at 666.7 in 3.

Using the 16 percent extension figure, the static position at which the gas law switches from isothermal to polytropic gas law is estimated to be 3.2 inches from the fully compressed position, i.e., X at 16.8 inches. Loads corresponding to isothermal and polytropic compression were determined using Eqs (5.7) and (5.8), respectively, and presented in Table 5.2. The corresponding load-stroke curves are shown in Fig. 5.2.

Table5.1

Shock absorber sizing parameters

Parameter

Design value

Total stroke 20.0 in

Staticposition 16 percent of total stroke

Staticload 50,000 lb

1,500 lb

Staticpressure

4: I static to extended

Compression ratio

3 i! compressed to static i iii i Table 5.2 Calculations of isothermal and polytropic compression X, in. V, in 3 Pt,., psi Pootv, psi Pco,.b, psi Fco,.b, lb 0.0 727.3 375.0 375.0 375.0 12500.0 2.0 660.6 412.8 427.0 412.8 13760.0 4.0 593.9 459.2 493.0 459.2 15306.7 6.0 527.3 517.2 578.9 517.2 17240.0 8.0 460.6 592.1 694.8 592.1 19736.7 10.0 393.9 692.4 858.2 692.4 23080.0 12.0 327.3 833.3 1102.0 833.3 27776.7 14.0 260.6 1046.5 1498.9 1046.5 34883.3 16.0 193.9 1406.5 2234.2 1406.5 46883.3 16.8 167.3 1630.2 2726.6 1630.2 54340.0 18.0 127.3 2142.5 3943.0 2214.6 73820.0 20.0 60.6 4500.6 10740.1 5645.3 188176.7 iiiiii 4 10 5 4 10 5 polytropic I I ----o----- isothermal

•------o--- combined I

3 10 5 __ 3 105 "_2 105 "_ 2 105 1 10 5 5 10 4 '_'-I, "l'-h , , J I L _ I J J 0 10 ° 10.0 15.0 20.0 0.0 5.0 Stroke, in Figure 5.2 The load-stroke curve

Chapter 6 Kinematics

6.1. Introduction Kinematics is the term applied to the design and analysis of those parts used to retract and extend the gear [2]. Particular attention is given to the determination of the geometry of the deployed and retracted positions of the landing gear, as well as the swept volume taken up during deployment/retraction. The objective is to develop a simple deployment/retraction scheme that takes up the least amount of stowage volume, while at the same time avoiding interference between the landing gear and surrounding structures.

The simplicity requirement arises primarily from economic considerations. As shown from operational experience, complexity, in the forms of increased part-count and maintenance down-time, drives up the overall cost faster than weight [5]. However, interference problems may lead to a more complex system to retract and store the gear within the allocated stowage volume.

Based on the analysis as outlined in this chapter, algorithms were developed to establish the alignment of the pivot axis which permits the deployment/retraction of the landing gear to be accomplished in the most effective manner, as well as to determine the retracted position of the assemblies such that stowage boundary violations and structure interference can be identified.

6.2. Retraction Scheme For safety reasons, a forward-retracting scheme is preferable for the fuselage-mounted assemblies. In a complete hydraulic failure situation, with the manual release of uplocks, the gravity and air drag would be utilized to deploy and down-lock the assembly and thus avoid a wheels-up landing [2]. As for wing-mounted assemblies, current practice calls for an inboard-retraction scheme which stows the assembly in the space directly behind the rear wing-spar. The bogie undercarriage may have an extra degree of freedom available in that the truck assembly can rotate about the bogie pivot point, thus requiring a minimum of space when retracted. As will be illustrated in the following section, deployed/retracted

positionof thelanding gear,as well as possibleinterference betweenthe landing gearand

surroundingstructures, caneasilybeidentifiedusingthe mathematical kinematicanalysis.

6.3. Mathematical Kinematic Analysis A mathematical kinematic analysis, which is more effective and accurate than the graphical technique, was selected to determine the axis of rotation that will, in one articulation, move the landing gear assembly from a given deployed position to a given retracted position. As shown in Fig. 6.1, a new coordinate system, termed the kinematic reference frame here, is defined such that the origin is located at the respective landing gear attachment locations with the axes aligned with the aircraft reference frame. The aircraft coordinate system-based origin permits constraints established in the kinematic reference frame, e.g., assembly clearance envelope, retraction path, and swept volume, be translated into the aircraft reference frame and checked for interference with surrounding structures.

6.3.1. The Pivot Ax& and Its Direction Cosines In the determination of the alignment of the landing gear pivot axis, it is assumed that the axle/piston centerline intersection is brought from its deployed position to a given location within the stowage volume. For wing-mounted assemblies, the retracted position of axle/piston centerline intersection is assumed to coincide with the center of the stowage volume. In the case of fuselage-mounted assemblies with a forward-retracting scheme, the retracted position is assumed to be at the center of the cross-sectional plane located at the forward third of the stowage length.* " Note: to reduce structural cut-away, many forward retracting gears have shrink mechanisms. In particular, it appears that the Airbus A 330 and A340 aircraft may have shrink struts on the main gear. This consideration is neglected in the current analysis, but probably should be considered.

z Aircraft reference frame Y X f z Kinematic reference frame . Y x X_ear Tires in the deployed position _" Figure 6.1 Relationships between the aircraft and kinematic reference flames 6. 3.1.1. The Fuselage-mounted Assembly For fuselage-mounted assemblies with a forward retracting-scheme, the pivot axis is defined by the cross product of the space vectors corresponding to the deployed and retracted position of a point location on the truck assembly. As shown in Fig. 6.2, the cross product of two vectors (V_ and V2) representing the deployed and retracted positions of a given point location, here taken as the axle/piston centefline intersection, is orthogonal to both vectors, i.e., in the direction of the pivot axis. Thus, V = V 1 × V2 (6.1) From standard vector operation, the direction cosines of the fuselage-mounted assembly is given as X Y Z /- m= n= (6.2)

qX 2 + 1,2+ Z 2 4X 2 + I'2 + Z 2 4X 2 + y2 + Z 2

andthe anglebetweenthe two vectors, i.e., the angle of retraction (t_fun) in this case, is

calculated using the expression coS_ful I =lll 2 + mlm 2 + nln 2 (6.3) where I i, m e, and n, are the respective direction cosines of the deployed and retracted space vectors.

Z V, pivot axis

x

_" _J V2 (_fu//_ VI Figure 6.2 Fuselage-mounted assembly pivot axis alignment 6. 3.1.2. The Wing-mounted Assembly The determination of the wing-mounted assembly pivot axis involves the deployed and retracted positions of two points on the assembly. Essentially, the problem consists of bringing the line segment between the two points from its deployed position to its retracted position [31]. For ease of visualization, a twin-wheel configuration is used here to illustrate the procedure involved in determining the alignment of the desired pivot axis. Identical procedure is used for other configurations as well.

As shown in Fig. 6.3, the axle/piston centerline intersection is selected as the first point (point A), while the second point (point B) is conveniently located at a unit distance along

the axle,inboardfrom the first point location,retractedpositions of the first and second

pointsaregiven aspoint A' andB', respectively.

V4 z

P' _ , Y x Poi_'.'. ,,_.

, 29 Figure 6.3 Vector representation of the wing-mounted landing gear Of the four point positions required in the analysis, the positions of point A and A' are readily determined from the geometry of the landing gear and the stowage volume, respectively. From simple vector algebra V2 = V 1 + .) (6.4) where subscripts 1 and 2 denote the space vector corresponding to the deployed positions of points A and B, respectively. Similarly, V4 = V 3 + U r (6.5) where subscript 3 and 4 denote the retracted positions of point A and B, respectively, and U r defines the orientation of the unit vector in its retracted position and is unknown.

To solvefor Ur, it is assumed thatno devicesareusedto shortenthe length of the strut

duringtheretractionprocess, i.e., that the magnitudes of V 2 and V 4 remain constant, X2+(YI+I)2 +Z2=(X3+Xu)2+(y3+Yu¢+(Z3+Zu) 2 (6.6) and that the magnitude of the retracted unit vector remains at unity X 2 + YU 2 + Z2U = 1 (6.7) The angle of inclination (0)ofUr in the yz-plane, which is one of the design variables that can be used to position the retracted truck assembly to fit into the available stowage space, is given as tanO = Yu (6.8)

zu

The vector components of U r, and subsequently V4, can then be determined by solving Eqs (6.6), (6.7), and (6.8) simultaneously.

As shown in Fig. 6.4, the pivot axis that will permit the achievement of the desired motion is defined by the cross product of the space vectors between the deployed and retracted positions of the two point locations, in this case points A and B, V = V B × V A (6.9) where (6.10) V A =(X 3 - Xl_ ^ + (Y3 -Y1).i + (Z3 - Zl)/_ and VB = (X4- X2)/^ +(Y4- Y2)J +(Z4 -- Z2)/_ (6.11) Thus, the direction cosines of the wing-mounted assembly and the angle of rotation can be determined using Eqs (6.2) and (6.3), respectively. Note that the subscripts in Eq. (6.3) will be 1 and 3 in this case, i.e., the vectors corresponding to the deployed and retracted positions of point A, respectively.

z V, pivot axis Y VB Poi_ Figure 6.4 Wing-mounted assembly pivot axis alignment 6.3.2. Retracted Position of a Given Point Location In addition to determining the required pivot axis and angle of retraction, the analytic method is used to establish the retraction path and the stowed position of the landing gear assembly. Note that the drag and side struts are excluded in the analysis since the retraction of these items involves additional articulation, e.g., folding and swiveling, that cannot be modeled by the analysis.

Define point A as an arbitrary point location on the landing gear assembly. Given the angle of rotation and the direction cosines of the pivot axis as determined above, the retracted position of point A, denoted here as A', can be determined by solving the following system of linear algebraic equations [2, pp. 193-194] XA' II(D(,A+mYA+nZA)-XA1 rmZA-nYA1 IgA1 (6.12)

Ya' [=cllm(O:a+mYa+nZa)-Yal+c2l a-lZa]+ Ya

ZA'.J Ln(IXA+mYA+nZA)-ZAJ LIYA-mXAJ LZAJ where (6.13) c 1 = 1- cos_ c 2 = sin _ 0 < _ < _full

Similarly,the retractionpathandsweptvolumeof the assembly, as shown in Fig. 6.5, can

be established by calculatingseveralintermediate transit positionsat a given interval of

degrees. The aboveinformationcan thenbe usedto identify possibleinterference between

the landinggearandsurrounding structures duringdeployment/retraction.

i Y _ x is ",, "_" _\" '¢___ Figure 6.5 Retraction path and swept volume of the landing gear 6.4. Integration and Stowage Considerations For future large aircraft, interference between the landing gear assembly and the surrounding structure is one of the more important considerations in the development of kinematics. With the large number of doors required to cover the stowage cavity on such aircraft, a complex deployment/retraction scheme for both the landing gear and doors is required to ensure that no interference will occur under all conditions. Additionally, the availability of stowage volume can become a major integration problem as the number of tires increases with aircraft takeoff weight. Given the conflicting objectives between maximizing the volume that can be allocated for revenue-generating cargoes and providing adequate landing gear stowage space, a trade-off study involving crucial design parameters, e.g., pivot axis alignment, angle of retraction, and bogie rotation, is needed to arrive at a satisfactory compromise with surround structures.

6.4.1. Truck Assembly Clearance Envelope Clearances are provided to prevent unintended contact between the tire and the adjacent parts of the aircraft during operation, particularly in the case when the tire is damaged and continues to spin when stowed. As shown in Fig. 6.6, the maximum grown outside diameter (De) and section width (WG) are determined using the expressions [25, p. 8] D G = D+ 2(I.115-O.074AR)H (6.14) and WG = 1.04W (6.15) where D is the specified rim diameter, H is the maximum section height, W is the maximum section width, and AR is the tire aspect ratio defined as H AR = m (6.16) D The values for the radial and lateral clearance, i.e., C R and C w, respectively, are calculated using the expressions [25, p. 9] - ° 0.073 250MPH 0.060 225MPH 210MPH (6.17)

cR- 0.047 WG + 0.4 at

0.037 190MPH 0.029 160MPH and CW = 0.019W G +0.23 (6.18) The constant coefficients found in Eqs (6.14), (6.15), and (6.16) are based on the maximum overall tire dimensions, plus growth allowance due to service and the increase in diameter due to centrifugal force.

USED WS NEW INFLATED INFLATED TIRE TIRE

A,x

ts t

H I 4 D Figure 6.6 Clearance envelope for aircraft tires [25] Based on the clearance as determined above, the minimum radial and lateral distance between the tire and surrounding structures are calculated as follows [25, p. 9] R x =DG+c R (6.18) Wx = WG + Cw (6.19)

Cw+ CR

S x - (6.20) Given the minimum allowable distances obtained using Eqs (6.18), (6.19), and (6.20), a clearance envelope is established around the truck assembly. Then, using the kinematic analysis as outlined in the previous section, the boundary of the envelope is re-established in the retracted position. Note that the envelope is represented in the kinematic coordinate system, while the boundaries of the landing gear wheelwell are in the aircraft coordinate system. Recall that the origin of the kinematic reference frame is defined in the aircraft coordinate system. Thus, simple algebraic manipulation would bring both sets of data under the same coordinate system, whether it be the airframe or the kinematic reference frame. Stowage boundary violations can then be identified by comparing both sets of data for discrepancies.

Chapter 7 Aircraft Flotation Analysis

7.1. Introduction The configuration of the landing gear has a direct impact on ground flotation, a term used to describe the capability of pavement and other surfaces to support an aircraft [32].

The number and arrangement of the wheels, along with the aircraft weight and its distribution between the nose and main assemblies, dictates the required pavement thickness for a particular aircraft. In addition, the type of the pavement found at the airports to be served by the aircraft also need to be considered. As shown in Fig. 7.1, existing runway and apron pavements can be grouped into two categories: flexible and rigid [7]. A flexible pavement, more commonly known as asphalt, may consist of one or more layers of bituminous materials and aggregate, i.e., surface, base, and subbase courses, resting on a prepared subgrade layer. On the other hand, rigid pavement may consist of a slab of portland cement concrete placed on a layer of prepared soil. The thickness of each of the layers must be adequate to ensure that the applied loads will not damage the surface or the underlying layers.

Asphalt Thickness Cement- Thickness Base concrete Subbase Subbase Subgrade Subgrade a) Flexible pavements b) Rigid pavements Figure 7.1 Theoretical pavement cross-sections [33] Based on the analyses as outlined in this chapter, a program was developed to determine the required flexible and rigid pavement thickness for a particular aircraft.

Results obtained from the program were validated with actual design data to ensure that a high degree of reliability can be placed upon the program itself.

7.2. Design Pavement Thickness Various flotation analyses have been developed over time in different countries and by different government agencies and organizations. Some agencies and organizations and the corresponding design methods are listed as follows [7]: the Federal Aviation Administration (FAA), the Portland Cement Association (PCA), the Waterways Experiment Station (S-77-1), and the British Air Ministry (LCN). The majority of these methods use the California bearing ratio (CBR) method of design for flexible pavements and Westergaard stress analysis for the rigid pavements [7].

7.2.1. Flexible Pavements For flexible pavements, CBR is the standard measurement used to classify the bearing strength of the subgrade. It is essentially the ratio of the bearing strength of a given soil sample to that of crushed limestone gravel. It is expressed as a percentage of the limestone figure, i.e., a CBR of ten means that the subgrade has a bearing strength of ten percent to that of crushed aggregate. The original design method, which was developed by the California Division of Highways in 1928, evaluates the pavement thickness requirements for a given load condition and soil strength, assuming that the load is carried on a single wheel with a circular footprint area.

Until the middle of the 1950s, the analysis developed for the B-29, which features a dual wheel configuration, was extended to develop thickness design relationships for new aircraft with twin-tandem configurations. However, it appears that the analysis tends to produce slightly unconservative thickness estimates. Subsequent reevaluation of the theoretical work, which is based on Boussinesq's theory [5], and test data showed that the slopes of pavement deflection versus wheel offset for the single wheel were equal to or steeper than for dual wheels at equal depths, as shown here in Fig. 7.2. A direct result of this study is the introduction of the concept of the equivalent single-wheel load (ESWL), which eventually became the foundation of the S-77-1 design method [34 and 35]. ESWL is essentially a fictitious load on a isolated wheel, having the same inflation pressure, and causing the same stresses in the runway material as those due to a group of wheels. This fictitious wheel load accounts for the fact that a given loading, spread over a number of contact areas, causes lower stresses in the runway material than would be the case when the same load is concentrated on a single wheel.

0.00 -- 0.01 e- 0.02 -m f- O 0.03 0.04 0.05 ----o---- Dual 006. L _ _ I , L , I J t , I , , , I , _ , I , , , I , , , I , _ , I -8.0 -6.0 -4.0 -2.0 0.0 2.0 4.0 6.0 8.0 Offset, ft Figure 7.2 Comparison of single- and dual-deflection profiles, 1.O-foot depth [7] Probable locations where maximum pavement bearing stress might occur, e.g., directly under and between the tire contact areas, are shown in Fig. 7.3, The offset distance between these points and the center of individual tire contact area, as well as the depths below the surface at which the ESWL is computed, which is treated as the thickness of the pavement in the analysis, are subsequently represented in terms of the radius of the footprint area (r) [7, p. 429] r= _ (7.1) and the tire-ground contact area (A) is defined as F A = -- (7.2) P where F is the vertical main assembly load (per strut) and P is the tire inflation pressure.

Given the offset distances and depths, curves such as the ones shown in Fig. 7.4 are used to determine the corresponding deflection factors. The principle of superposition is then used in calculating the multiple-wheel deflection factor (;0, which is equal to the summation of the deflection factors produced by each tire in the multiple-wheel assembly at the point of analysis.

ysis lo tio _ i °tpnnts

Figure 7.3 Relationship between the tire-contact areas and the analysis locations The ratio of load intensity of the single-wheel configuration to a single wheel of the multiple-wheel configuration is defined as the inverse of the ratio of the maximum deflection factors at a given depth, i.e., the pavement thickness, [7, p. 430]

fm

- (7.3)

Fm fs

where subscripts s and m denote single- and multiple-wheel configurations, respectively.

Once the ratio of load intensity is determined, the ESWL is calculated using the expression ESWL= FsF (7.4)

FmSw

where N w is the number of wheels per strut. To account for the loading effect caused by the number of annual aircraft operations, the design thickness (t) corresponding to a given CBR value is estimated using the expression [7, p. 433] ot ! ESWL A (7.5) t= i _ 8-_-_-- R -_ where _ is the load repetition factor as shown in Fig. 7.5. It is categorized by the number of tires used to calculate the ESWL and typically value corresponding to 10,000 passes are used in the calculation [33].

10.0 -------w--- 4.0r offset 5.or offset 6.0r offset 8.0r offset 1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 Depth, r Figure 7.4 Deflection factor curves for Poisson's ratio of 0.5 [34] 1.20 _l _ 2wheels "_ 1.10 _1 _ 4 wheels El _ 8 wheels -,_ _ 0.80 .o 0.70 _ 0.90 -_

-_ o.60

102 103 104 lO s 106 Passes Figure 7.5 Aircraft load repetition factor [7] 7.2.2. Rigid Pavements Stress in a concrete pavement is induced in four ways: tire loads, change of shape of slab due to differential in temperature and moisture between the top and the bottom of the slab, and the friction developed between slab and foundation when the slab expands/contracts. Since the primary consideration in the design of any pavement is the load which it is to carry, only the stresses induced by tire loads will be addressed.

The Westergaard stress analysis [36] assumes that the slab is a homogeneous, isotropic, and elastic solid in equilibrium. The reactions of the subgrade are assumed to be in the vertical direction only, and is proportional to the deflections of the slab. Additionally, the wheel load is assumed to be distributed over an elliptical footprint area. The stiffness of the slab relative to that of the subgrade is represented by the radius of relative stiffness of the concrete (/) [37, p. 56] (7.6) where E is the modulus of elasticity for the concrete, d is the thickness of the slab,/1 is the Poisson's ratio for the concrete, and k is the modulus of subgrade reaction. Typically, E is taken as 4,000,000 psi and/.t as 0.15 [7].

Critical bearing stresses for the interior and edge loading cases are examined. For the interior loading case, the load is applied at the interior of the slab at a considerable distance from any edge or joint. The maximum tensile stress (cr) at the bottom of the slab is given as [7, p. 441]

Ea3

(7.7) ain t = F.-_ 0.275(1 + #)IOglO + 0.293(1 - #) a--'_a - b k[(a+b)/2_

a- L

where F s is the single wheel load, d is the design thickness, and a and b are the semi-axes of the footprint area ellipse. Considering the edge loading case next, the load is applied adjacent to an edge that has no capacity for load transfer. The maximum tensile stress is given as [7, p. 442]

_220

_ext (3+ p)d" l°glO lOOk[(a+b)/2_ + (7.8) 3(1 + bt)F [1.84_ 4/1 + (1 + a-b 2(1 ab 1.18(1 + 2/.t)b] tr(3+/./)d 2' /./)a--_+ -/./)(a+b) 2 I- Although the edge loading case produces a maximum stress that is the more critical of the two cases, in reality the probability of occurrence of this type of loading is relatively small, i.e., the traffic tends to be channelized with the highest concentration in the vicinity of the runway and taxiway centerlines [7]. In addition, rigid pavement design charts as provided by PCA, which are used as reference data in the following section, are based on the interior loading case. Therefore, the interior loading condition is selected as the basis of the rigid pavement analysis.

7.3. Pavement Thickness Estimates Design pavement thickness and corresponding ACNs for the Boeing Models 737, 747, 767, and McDonnell Douglas DC10 were determined for four subgrade strength categories: ultra-low, low, medium, and high [33]. Each category is assigned a CBR value for the flexible pavements and a k value for the rigid pavements; numerical values of each category are listed in Table 7.1.

Table 7.1 Subgrade strength categories [33] i ii Category CBR k, ib/in 3 Ultra-low 3.0 75.0 Low 6.0 150.0 Medium 10.0 300.0 High 15.0 550.0 For flexible pavements, ESWLs were computed using Eq. (7.4) from the surface down in multiples of footprint area radius. At each analysis depth, a CBR value was calculated using Eq. (7.5) and the repetition factor corresponding to 10,000 aircraft passes [33]. The result of this calculation is a set of design thickness and CBRs. Linear interpolation is then used to determine the final design thickness corresponding to the subgrade strength CBR values.

For rigid pavements, ls were computed using Eq (7.6) from the surface down in

predetermined increments, i.e., the design thickness, for each of four subgrade categories.

At each design thickness and k value, a maximum tensile stress was calculated using Eq.

(7.7). The result of this calculation is four sets of design thickness and the corresponding stresses. Linear interpolation is then used to determine the final design thickness corresponding to a concrete working stress of 400 psi [2].

Actual [7, 22, 38, and 39] and estimated pavement thickness are compared to determine the reliability of both analyses. As shown in Fig. 7.6a, the S-77-1 method tends to underestimate the required pavement thickness at the lower end of the CBR range, while it tends to overestimate the required pavement thickness at the upper end of the CBR range.

Yet, the trend is consistent with the results obtained from a number of full-scale test tracks, i.e., for heavy wheel loads, the theoretical thickness appeared to be too low for lower CBR values, and too high for higher CBR values. An interesting trend is observed upon closer examination of the actual pavement thickness data. As the subgrade strength increases, the required pavement thickness for aircraft with dual-twin truck assembly configurations, i.e., B747, B767, and DC10, approach, if not fall below, the one required by aircraft with twin- wheel configuration, i.e., B737. This can be attributed to the fact that the load on the pavement is better distributed as the number of wheels per assembly increases.

A vastly different trend, as shown in Fig. 7.6b, is exhibited by the Westergaard stress analysis: it tends to underestimate the required pavement thickness by roughly 30 percent across the entire k range. The discrepancy can be attributed to the simplicity of the analysis itself. Primarily, the analysis did not consider the variations in the location and direction of maximum moment and stress in the concrete slab [37]. Essentially, the position of the maximum stress can be shifted and rotated depending on the magnitude of 1 and the configuration and dimension of the truck assembly. In addition, the analysis did not include detailed design parameters such as fatigue of concrete due to repeated loading and interactions between layers of materials.

70.0 - ---o--- B737, act _ DC10, act • B737, est • DC10, est 60.0 -----o----- B767, act _ B747, act ¢.. • B767, est • B747, est • - 50.0 = 40.0 .- 30.0 20.0 , , , I , , _ I _ , _ I , , , i , , , I _ i _ I _ _ _ I 10.0 4 6 8 10 12 14 16 Sub-grade strength, CBR a) Flexible pavements 25.0 - + B737, act _ DC10, act • B737, est • DC10, est -------o----- B767, act .------o---- B747, act 20.0 • B767, est • B747, est ._ = 15.0 10.0 • • • • • • X • ,i, .... I , I i i I i i , _ I , , , , I .... I , ,°, , I 5.0 100 200 300 400 500 600 Sub-grade strength, k b) Rigid pavements Figure 7.6 Actual and estimated pavement thickness comparison Linear regression analysis was used to calibrate the estimated pavement thickness (tes,) against actual data. At each subgrade strength category, an aircraft weight-based correction factor is calculated using the expression fc = ClW + c c (7.9) where c I and c 2 are constant coefficients as listed in Table 7.2. The estimated value and correction factor are then combined to arrive at the calibrated pavement thickness (teat), that is, tcal = test + fc (7.10) The objective of this effort is to ensure that the discrepancy between the actual and estimated values will remain within a tolerable range. This is important when both analyses are used to examine the flotation characteristics of aircraft that are outside the existing pavement thickness database, namely, the next-generation high capacity commercial transports. As shown in Fig. 7.7, the calibrated thickness compared reasonably with the actual data.

Table 7.2 Pavement thickness correction constants C 1 C I Flexible Ultra-low 0.000017 3.726 Low 0.000002 0.198 Medium -0.000002 - 1.630 High -0.000007 -0.008 Rigid Ultra-low 0.000003 4.002 Low 0.000003 3.420 Medium 0.000001 3.407 High 0.000000 3.325 70.0 -----¢----- B737, act _ DC10, act • B737, cal • DC10, cal 60.0 B767, act + B747, act • B767, cal • B747, cal •- 50.0 : 40.0 ,m E- 30.0 20.0 , ,, I,, , I , , i [ N i i 1 i L i I t i J I i i , I 10.0 4 6 8 10 12 14 16 Sub-grade strength, CBR a) Flexible pavements 25.0 I B737, act _ DC10, act | • B737, cal • DC10, cal B767, act _ B747, act 20.0 • B767, cal • B747, cal

I

.u " 15.0 10.0 , = = , I , ¢ r i [ , = , , [ , J = _ I J , , , I , , _ , I 5.0 100 200 300 400 500 600 Sub-grade strength, k b) Rigid pavements Figure 7.7 Actual and calibrated pavement thickness comparison 7.4. ACN-PCN Conversion In an effort to resolve the difference among various pavement design and evaluation methods, the International Civil Aviation Organization (ICAO) recommended universal adoption of the Aircraft-Pavement Classification Number (ACN-PCN) system [39] in 1983. The ACN-PCN system is not intended for the design or evaluation of pavements. It is, instead, a convenient and simple way of categorizing and reporting the pavement's capability to support aircraft on an unrestricted basis. The major appeal of the system is that it allows aircraft manufacturers to use any design/evaluation method of choice to determine the pavement thickness requirements of a particular aircraft. The design thickness is then converted to ACN and compared to PCNs of the airports to be served. If the ACN is equal to or less than the PCNs, the aircraft is cleared to operate out of the given airports subject to any limitation on the tire pressure.

The flexible pavement ACN is calculated using the expression [33, p. 3-11 ] ACN _2/lO00) = (7.10) (0.878/CBR-0.01249) where the design thickness t is expressed in terms of centimeters. As for the rigid pavements, ACN is obtained using the conversion chart as shown in Fig. 7.8.

High Med /v Low / - Z < 0 _ _ _ _ I _ , , _ I , , , _ I .... I .... I , 10.0 20.0 30.0 40.0 50.0 60.0 Thickness, cm Figure 7.8 Rigid pavement ACN conversion chart [33] 7.4.1. ACN Estimates Flexible and rigid pavement thickness reqwrements obtained earlier were converted to ACNs for conversion validation purposes. As shown in Fig. 7.9a, the estimated flexible pavement ACNs exhibit a trend similar to that of the thickness estimates, i.e., too low for lower CBR values and too high for higher CBR values. Apparently, the thickness calibration process did not eliminate the discrepancy introduced in the pavement thickness calculation entirely, and that the trend is carried over into the ACN conversion process. On the other hand, it appears that the calibration process for the rigid pavement has removed most of discrepancy that was introduced in the pavement thickness calculation. As shown in Fig. 7.9b, the conversion, in fact, overestimated the ACN for all aircraft across the entire k range.

120 - _1 * B767, est ''--°'_ B737, act _ DCI0, act • B737, est • DC10, est B767, act ------q------ B747, act 8O • B747, est Z < .

4O _ c 2O , , I , , , I , , , I , , , I , , , I , _ , I _ _ , I 0 ' 4 6 8 10 12 14 16 Sub-grade strength, CBR a) Flexible pavements Figure 7.9 Actual and estimated ACN comparison B737, act _ DC10, act • B737, est • DC10, est B767, act _ B747, act • B767, est • B747, est Z < o--------o. G o ,_ j,I_t,llh,_,lt,,_l .... I .... I 0 100 200 300 400 500 600 Sub-grade strength, k b) Rigid pavements Figure 7.9 Actual and estimated ACN comparison (concluded) Linear regression analysis was again used to calibrate the estimated ACN (ACNes ,) against actual data. At each subgrade strength category, an aircraft weight-based correction factor is calculated using Eq. (7.9), except in this case the constant coefficients are c 3 and c a as listed in Table 7.3, The estimated value and correction factor are then combined to arrive at the calibrated ACN (ACNcal) , that is, (7.11) ACNcal = ACNest + fc As shown in Fig. 7.10, the calibration process has successfully brought the estimated ACNs closer to the actual data and thus improved the reliability of the flotation analysis.

Table 7.3 ACN correction constants

g

Flexible Ultra-low 0.000008 0.5178 Low 0.000010 -6.326 Medium 0.000009 -6.769 0.000022 -16.182 High Rigid Ultra-low 0.000006 -8.245 Low 0.000002 -4.940 Medium 0.000009 -7.628 0.000008 -6.519 High I I + B737, act _ DC10, act I • B737, cal • DC10, cal 100 _ I ""'¢-'-- B767, act ..... B747, act __1 • B767, cal • B747, cal _< 60 ......

20 - -"---'--'_ _ o 0 , , , I , _ , I _ , , I , , , I , , , I , , , I t _ _ 2 4 6 8 10 12 14 16 Sub-grade s_ength, CBR a) Flexible pavements Figure 7.9 Actual and calibrated ACN comparison 120 - B737, act _ DC10, act • B737, cal • DC10, cal B767, act -----q------ B747, act • B767, cal • B747, cal Z < • 4, 100 200 300 400 500 600 Sub-grade strength, k b) Rigid pavements Figure 7.10 Actual and calibrated ACN comparison (concluded)

Chapter 8 Weight Estimation

8.1. Introduction Statistical weight equations, although capable of producing landing gear group weights quickly and generally accurately, do not respond to all the variations in landing gear design parameters. In addition, the equations are largely dependent on the database of existing aircraft. For future large aircraft, such weight data is virtually non-existent. Thus, it is desirable that an analytical weight estimation method which is more sensitive than statistical methods to variations in the design of the landing gear should be adopted. The objectives are to allow for parametric studies involving key design considerations that drive landing gear weight, and to establish crucial weight gradients to be used in the optimization process.

Based on the procedures described in this chapter, algorithms were developed to size and estimate the weight of the structural members of the landing gear. The weight of non- structural members were estimated using statistical weight equations. The two were then combined to arrive at the final group weight.

8.2. Current Capabilities The primary shortcoming of statistical methods is that only a limited number of weight-affecting parameters are considered, e.g., length of the strut, material ultimate strength, vertical load, and number of tires. As a result, it is extremely difficult to distinguish landing gears with different geometric arrangements using these parameters alone. Statistical weight equations are also constrained by what has been designed in the past, i.e., if an unconventional design or a new class of aircraft such as the proposed ultra- high-capacity transport is involved, there might not be sufficient data to develop a statistical base for the type of landing gear required.

The majority of existing equations calculate the landing gear weight purely as a function of aircraft takeoff gross weight. It is the simplest method for use in sizing analysis, and is adopted in ACSYNT as well as by Torenbeek [5] and General Dynamics, as given by Roskam [3]. The Douglas equation used in the blended-spanload concept [41] also falls into this category. Other weight equations, e.g., Raymer [42] and FLOPS (Hight Optimization System) [13], include the length of the landing gear in the calculation and

thusareableto produceestimateswhich reflect the effectof varying designparameters to

someextent.

Actual andestimatedlandinggear weight fractionsare presentedin Fig. 8.1. Figure

8.1aprovidescomparisions for estimateswhich only use MTOW. Figure 8.1b provides

comparisionswith methodswhich take into accountmore details, specificallythe gear

length.As shown in Fig. 8.l a, for an MTOW up to around 200,000lb, the estimated

values from ACSYNT and Torenbeekare nearly equal. However, as the MTOW

increases, completelydifferenttrendsareobserved for thetwo equations: an increasingand

then a decreasinglanding gearweight fraction is predictedby ACSYNT, whereas a

continualincreasingweight fraction is predictedby Torenbeek.As for the Douglas

equation,an increasing weight fraction is observedthroughoutthe entireMTOW range.

Uponcloserexaminationof the datapresented, it was found thatonly a small numberof

actuallandinggearweightcases areavailableto establishtrendsfor aircrafttakeoff weight

above 500,000pounds. In addition, even within the range where significant previous

experience is available,the datascatterbetweenactualandestimated valuesis too largeto

draw conclusionson the accuracyof existing weight equations.Evidently a systematic

procedureis neededto validatethe reliability of the statisticalequations,and provide

another level of estimation.

8.3. Analytical Structural Weight Estimation Analytical weight estimation methods are capable of handling varying configurations and geometry, in addition to design parameters used in the statistical methods. As typified by Kraus [43] and Wille [44], the procedure consists of five basic steps: definition of gear geometry, calculation of applied loads, resolution of the loads into each structural member, sizing of required member cross-sectional areas, and calculation of component and total structural weight. Although these studies provided an excellent guideline toward the development of an MDO-compatible analysis algorithm, detailed discussions in the area of load calculations and structural design criteria were not included in the papers. To fill the gap, simplified loading conditions were determined from Torenbeek and the FAA [20], and structural analyses were developed as part of this work. Loading conditions are presented in Section 8.3.2., and the structural analyses are presented in Sections 8.3.3. and 8.3.4. and Appendix B.

5.00 ACSYNT ..... Douglas 4.50 ......... Torenbeek A • oo.,, o B737 4.00 • DC9 ° =- o B727 ¢' " t 3.50 * B707 -Q° ° ° zx LIOll • DCIO 3.00 Z- v B747 • C5 2.50 1 i [ I I I I I [ [ I I L I I I I I 2.00 10 5 107 MTOW, lb a) Pure weight fraction equations 5.00 • Raymer [] FLOPS 4.50 © o B737 [.., V • DC9 4.00 o B727 =- * B707 O O a LIOll • DCIO t_ v B747 3.00 • C5 2.50

3.50 1 i

l' ,1, ],,.....

2.00 105 106 MTOW, lb b) Weight fraction equations with landing gear length Figure 8.1 Landing gear weights comparison 8.3.1 Generic Landing Gear Model A generic model consisting of axles, truck beam, piston, cylinder, drag and side struts, and trunnion is developed based on existing transport-type landing gears. Since most, if not all, of the above items can be found in both the nose and main gear, the model can easily be modified to accommodate both types of assembly without difficulty. Although the torsion links are presented for completeness, they are ignored in the analysis since their contributions to the final weight are minor.

The model shown in Fig. 8.2 represents a dual-twin-tandem configuration. The model can be modified to represent a triple-dual-tandem or a dual-twin configuration with relative ease, i.e., by including a center axle on the truck beam, or replacing the bogie with a single axle, respectively. The model assumes that all structural components are of circular tube construction except in the case of the drag and side struts, where an I-section can be used depending on the configuration. When used as a model for the nose gear, an additional side strut arranged symmetrically about the plane of symmetry is included.

Trunnion Side strut Cylinder Drag strut Torsion links Truck beam Axle Figure 8.2 Generic landing gear model For added flexibility in terms of modeling different structural arrangements, the landing gear geometry is represented by three-dimensional position vectors relative to the aircraft reference frame. Throughout the analysis, the xz-plane is chosen as the plane of symmetry with the x-axis directed aft and the z-axis upward. The locations of structural components are established by means of known length and/or point locations, and each point-to-point component is then defined as a space vector in the x, y, and z directions. Based on this approach, a mathematical representation of the landing gear model is created and is shown in Fig. 8.3.

D \= [ Vector Description BA Forward trunnion B C Aft trunnion x BE Cylinder _/ AE Drag strut EF Piston

= i os=

L FG, FJ Truck beam H_/F GH, GI, JK, JL Axles Figure 8.3 Mathematical representation of the landing gear model 8.3.2. Applied Loads External loads applied to the gear assemblies can be divided into dynamic and static loads: the former occurs under landing conditions while the latter occurs during ground operations. As listed in Table 8.1, seven basic loading conditions have been selected for analysis with the applied loads calculated as specified in FAR Part 25 [20]. These conditions are also illustrated in Fig. 8.4.

Table 8.1 Basic landing gear loading conditions [20] Static Dynamic Three-point level landing Turning One-wheel landing Pivoting Tail-down landing Lateral drift landing Braked roll The corresponding aircraft attitudes are shown in Fig. 8.4, where symbols D, S and V are the drag, side and vertical forces, respectively, n is the aircraft load factor, W is aircraft maximum takeoff or landing weight, T is the forward component of inertia force, and I is the inertial moment in pitch and roll conditions necessary for equilibrium. The subscripts m and n denote the main and nose gear, respectively.

0.8Fn IF _ a) Three-point level landing b) One-wheel landing Figure 8.4 Aircraft attitudes under dynamic and static loading conditions [20] 0.8F_' I F m c) Tail-down landing 0"8F'_IF_, d) Lateral drift landing "--'_'F,, 0"8F'lFm e) Braked roll W -..=--It ' • 0.5F,,' O_'_F, _ f) Turning Figure 8.4 Aircraft attitudes under dynamic and static loading conditions [20] (continued) t i F. F.

g) Pivoting Figure 8.4 Aircraft attitudes under dynamic and static loading conditions [20] (concluded) For the dynamic landing conditions listed in Table 8.1, the total vertical ground reaction (F) at the main assembly is obtained from the expression [43] F- cW (V2 +scoso_ (8.1) rlScos o_ _ g where c is the aircraft weight distribution factor, 7/is the gear efficiency factor, S is the total stroke length, _ is the angle of attack at touchdown, V s is the sink speed, and g is the gravitational acceleration. Although the vertical force generated in the gear is a direct function of the internal mechanics of the oleo, in the absence of more detailed information Eq. (8.1) provides a sufficiently accurate approximation.

The maximum vertical ground reaction at the nose gear, which occurs during low- speed constant deceleration, is calculated using the expression [5, p. 359] Fn lm +ax/g hcg = W (8.2) Im + ln For a description of variables and the corresponding values involved in Eq. (8.2), refer to Chapter Four, Section Two.

The ground loads are initially applied to the axle-wheel centerline intersection except for the side force. As illustrated in Fig. 8.5, the side force is placed at the tire-ground contact point and replaced by a statically equivalent lateral force in the y direction and a couple whose magnitude is the side force times the tire rolling radius.

2" Z Figure 8.5 Location of the applied ground loads To determine the forces and moments at the selected structural nodes listed in Table 8.2, the resisting force vector (Fr,s) is set equal and opposite to the applied force vector (F_pp) Fre s = -F ap p (8.3) whereas the resisting moment vector (Mr,s) is set equal and opposite to the sum of the applied moment vector (Mopp) and the cross product of the space vector (r) with Fap p Mre s = -(Map p +r × Fap p ) (8.4) Table 8.2 Selected structural nodes description Node Description Location (Figure 8.3) 1 Axle-beam centerline intersection G/J 2 Beam-piston centerline intersection F 3 Drag/side/shock strut connection E 4 Cylinder-trunnion cente!_!!!ae i_atersection .... B 8.3.3. Forces and Moment Resolution Three-dimensional equilibrium equations are used to calculate member end reactions.

Internal forces and moments are then determined from equilibrium by taking various cross-sectional cuts normal to the longitudinal axis of the member. To ensure that the information is presented in a concise manner, the methods used in the analysis are discussed only in general terms, while detailed derivations are compiled and presented in Appendix B.

8.3:3.1. Coordinate Transformation Given that the mathematical landing gear model and the external loads are represented in the aircraft reference frame, transformation of nodal force and moment vectors from the aircraft to body reference frames are required prior to the determination of member internal reactions and stresses. The body reference frames are defined such that the x3-axis is aligned with the component's axial centerline, and xz-plane is a plane of symmetry if there is one. The transformation is accomplished by multiplying the force and moment vectors represented in the aircraft reference frame by the transformation matrix LBA [45, p. 117] F B = LBAF A (8.5) M B = LBAMA (8.6) where subscripts A and B denote the aircraft and landing gear body reference frames, respectively. By inspection of the angles in Fig. 8.7, where subscripts 1, 2, and 3 denote the rotation sequence from the aircraft (x, y, and z) to the body (x 3, Y3, and z3) reference frame, the three localized transformation matrices are [45, p. 117] (8.7a) L 1 (_o 1 ) = 0 cos_o 1 sintPl 1 0 0 0 -sintPl cosq) 1 (8.7b) L2 (¢P2) = 1

o_Sio ]

L sin ¢P2 0 cos _o 2 ] (8.7c) [cosq_3 sinq_3 i] L3(_P3)=[ si_ q)3 c°sq)30 Thus, the matrix LBa is given as [45, p. 117] (8.8) LBA -- t3(tP3)L2(cP2 )LI(tPl) or -cos tpl sin tp2 coscp 3" sin _o1 sin(P2 cos cp3 + sintp 1 sintp 3 + cos (Pl sin tp3 cos q)l sin ¢P2sintP3 - sintPl sinq_ 2 sin fP3 (8.9)

LBA =

+sin tpl cosq) 3 +cos tpl cos q)3 - sinq_lcosq_ 2 cos q)l cos tp2 sintp 2 Z Zl X, X 1 a) About the x, xt-axis zl yl,y2 .gl X2 b) About the y_, y:axis y3 Z2, Z3 _Y2x3 X2 c) About the z 2, z:axis Figure 8.6 Orientation of the axes and the corresponding rotation angles 8.3.3.2. The Main Assembly The main assembly drag strut and side strut structure is modeled as a space truss consisting of ball-and-socket joints and two-force members. As shown in Fig. 8.7 the loads applied to the cylinder consist of the side strut forces (F, ide), drag strut force (Fdros), an applied force with components F x, Fy, and F_, and an applied couple with moment components Cx, Cy, and C:. Internal axial actions are obtained using the method of sections.

Equilibrium equations are then used to determine the magnitude of the internal axial forces in the isolated portion of the truss.

The shock strut cylinder, in addition to supporting the vertical load, also resists a moment due to asymmetric ground loads about the z-axis. This moment is transmitted from the truck beam assembly to the cylinder though the torsion links. Note that in the tandem configurations, the moment about the y-axis at the piston-beam centerline is ignored because of the pin-connection between the two. However, this moment must be considered in the dual-twin configuration, where the moment is resisted by the integrated axle/piston structure.

\1 _l Cylinder Figure 8.7 Idealized main assembly cylinder/drag/side struts arrangement 8.3.3.3. The Nose Assembly As mentioned in the geometric definition section, an additional side strut, arranged symmetrically about the xz-plane, is modeled for the nose assembly. The addition of the second side strut results in a structure that is statically indeterminate to the first degree as shown in Fig. 8.8. The reactions at the supports of the truss, and consequently the internal reactions, can be determined by Castigliano's theorem [46, p. 611 ] °aV - _ Fill °9Fi (8.10) UJ - o31_ i=l AiEoqPj where uj is the deflection at the point of application of the load Pj, E is the modulus of elasticity, and/, F, and A are the length, internal force, and cross-sectional area of each member, respectively. The theorem gives the generalized displacement corresponding to the redundant, Pj, which is set equal to a value compatible with the support condition. This permits the solution of the redundant, and consequently all remaining internal actions, via equilibrium. As detailed in Appendix B, Section Two, the procedure is to first designate one of the reactions as redundant, and then determine a statically admissible set of internal actions in terms of the applied loads and the redundant load. By assuming a rigid support which allows no deflection, Eq. (8.10) is set to zero and solved for Pj.

z __ Trunnion connection Cylinder F_,os I_ F_ c_ Figure 8.8 Idealized nose gear cylinder/drag/side struts arrangement 8.3.3.4. The Trunnion When the gear is in the down-and-locked position, the trunnion is modeled as a prismatic bar of length L with clamped ends. As shown in Fig. 8.9, the trunnion is subjected to a force with components F x, Fy, and F z, and a couple with components C_ and C_, at axial position x = l_, where 0 < l_ < L and 0 < x < L. Clamped end-conditions at x = 0 and x = L yield ten homogeneous conditions, five at each end. At the load point x = l_, there are five continuity conditions, i.e., u, v, w, v', and w', and five jump conditions corresponding to point-wise equilibrium of the internal actions and the external loads.

The linear elastic response of the trunnion is statically indeterminate, but can be readily solved by the superposition of an extension problem for the x-direction displacement component u(x), a bending problem in the xy-plane for the y-direction displacement v(x), and a bending problem in the xz-plane for the z-direction displacement w(x). Using classical bar theory, the governing ordinary differential equation (ODE) for u(x) is second order, while the goveming ODEs for v(x) and w(x) are each fourth order. The goveming equations are solved in the open intervals 0 < x < l_ and 11 < x < L, where the 20 constants of integration (ci) resulting from integration of the ODEs with respect to x are determined using the boundary and transition conditions as given above. Details of the solution are given in Appendix B, Section Three.

Z Y Figure 8.9 Trunnion modeled as a clamped-clamped bar 8.3.4. Member Cross-sectional Area Sizing With the resolution of various ground loads, each structural member is subjected to a number of sets of internal actions that are due to combinations of extension, general bending, and torsion of the member. To ensure that the landing gear will not fail under the design condition, each structural member is sized such that the maximum stresses at limit loads will not exceed the allowables of the material and that no permanent deformation is permitted.

A description of selected cuts near major component joints and supports is given in Table 8.3. Normal and shear stresses acting on the cross section due to the internal actions were calculated at these locations and used in the sizing of the required member cross- sectional area.

Table 8.3 Sections description Section Description Location (Figure 8.3) 1 Axle-beam centerline intersection G/J F 2 Beam-piston centerline intersection 3 Piston E 4 Cylinder/struts connection E 5 Cylinder/trunnion centerline intersection B A 6 Forward trunnion mounting C 7 Aft trunnion mounting A 8 Drag strut 9 Side strut D i i iii i ii iii 8.3.4.1. Normal and Shear Stresses In a Thin-walled Tube The normal stresses induced on the structural members are determined by combining the effects of axial load and combined bending, while the shear stresses are determined by combining the effects of torsion and shear forces due to bending [47].

The normal stress (t=) due to combined axial force and bending moments is given as N My Mz vx_ =--+ (8.11) A Iy---_z- Iz'-'_y where N is the maximum axial force, A is the cross-sectional area of the member, My and M z are the internal moment components, and lyy and I= are the second area moments about the y- and z-axis, respectively. As shown in Appendix B, Section Four, the extremum values of the normal stress on a circular-tube cross section under combined axial and bending actions are N+ 1 2 (8.12)

= _ xx ._ A -

or rain where r is the mean radius of the tube and t is the wall thickness. In the case of drag and side struts, the last two terms in Eq. (8.11) are zero since both members are modeled as pin-ended two-force members, thus, N -cax = m (8.13) A The shear stress('rxs) due to combined transverse shear forces and torque is given as "Cxs = q(s) + Qgxs )torque (8.14) t where q is the shear flow due to bending of a thin-walled tube, see Fig. 8.10. Given that tanOma x = - Vz (8.15)

Vy

where 0,,_ is the polar angle where the bending shear flow attains an extremum value and V, and V z are the shear forces components, Eq. (8.14) then becomes = 1._--..T +,IV,2 + V 2) (8.16) T xs,_ rcrt \ 2r v J or rain where T is the applied torque. Details of the solution are given in Appendix B, Section Four.

F x y Figure 8.10 Shear flow around a tube 8.3.4.2. Design Criteria Although aircraft structural design calls for multiple load paths to be provided to give fail-safe capability, the concept cannot be applied in the design of the landing gear structures. Accordingly, the gear must be designed such that the fatigue life of the gear parts can be safely predicted or that the growth of cracks is slow enough to permit detection at normal inspection intervals [4].

Von Mises yield criterion for ductile materials combined with a factor of safety is used to determine the stress limit state. The Mises equivalent stress is given as [46, p. 368] (8.17) Mises = 4Z 2 + 3Z2xs and the factor of safety is defined as the ratio of the yield stress of the material to the Mises equivalent stress, that is, F.S.- tYyield (8.18) tYMises If this value is less than the specified factor of safety, the cross-sectional area of the component is increased until the desired value is attained.

In addition to material limit state, the critical loads for column buckling of the drag and side struts axe considered because of the large slenderness ratio associated with these members. The slenderness ratio is defined as the length of the member (L) divided by the minimum radius of gyration (P,.i.). Assuming a perfectly aligned axial load, the critical buckling load for a pin-ended two-force member can be calculated using Euler's formula [46, p. 635] rc2EI Ncr = L"_ (8.19) where E is the modulus of elasticity. In the case of a member with circular cross section, the moment of inertia I of the cross section is the same about any centroidal axis, and the member is as likely to buckle in one plane as another. For other shapes of the cross section, the critical load is computed by replacing I in Eq. (8.19) with lmi,, the minimum second moment of the cross section (bending about the weak axis). Note that the Euler's formula only accounts for buckling in the long column mode and is valid for large Aluminum alloy. For slendemess ratio

slendemess ratio, e.g., L/pmi, > 80 for 606 l-T6

below this range, intermediate column buckling should be considered [48].

8.3.4.3. Sizing of the Cross-sectional Area For thin-walled circular tubes, the cross-sectional area of the member is given as A = trDt (8.20) where the mean diameter (D) and design thickness (t) are both design variables. Instead of using these two variables in the analysis directly, the machinability factor (k), which is defined as the mean diameter divided by the wall thickness, is introduced to account for tooling constraints [49]. The factor is defined as D k = -- (8.21) t and has an upper limit of 40. For the thin-wall approximation to be valid in the structural analysis k > 20. Thus, the machinability factor is limited to 20 _< k _<40 (8.22) By replacing t in Eq. (8.20) with Eq. (8.21) and using D as a limiting design variable, the desired cross-sectional area can then be determined by iterating on k. Note that the lower limit of k given in Eq. (8.21) may be violated in some instances. For structural members such as the axles, the truck beam, and piston, which typically feature k values in the mid- teens, St. Venant's theory for torsion and flexure of thick-walled bars [50] should be used to calculate shear stresses. Essentially, the problem is broken down into torsion and bending problems and the shear stresses are calculated separately based on the linear theory of elasticity.

In general, the diameter of each cylindrical component is a function of either the piston or wheel dimension. In the case of shock strut, it is assumed that the internal pressure is evenly distributed across the entire cross-sectional area of the piston. That is, the piston area is a function of the internal oleo pressure (P2) and the maximum axial force, that is, N zrD2 A = m = _ (8.23) P2 4 where Dp is the outer diameter of the piston. Rearrangement of Eq. (8.23) gives

(8.24)

Assuminga perfectfit betweenthepistonlining andthe inner cylinder wall, the minimum

allowablemeandiameterof the cylinder is obtainedby addingthe wall thicknessof the

cylinder to the piston outer diameter.To reducethe level of complexity, the minimum

allowablemeandiameter of the trunnionis assumed to be identicalto thatof the cylinder.

Similar assumptionsaremadeconcerningthe axleandtruck beam, exceptthat the outer

diameterof the abovemembers is treatedasa functionof the diameter of the wheelhub. In

the caseof the axle,the maximumallowablemeandiameteris obtainedby subtracting the

axlewall thickness from the hubdiameter.

For the thin-walled I-section bar shown in Fig. 8.11, the cross-sectional area and

principal centroidal second area moments are A = t(2b + h) (8.25) (8.26) and b3t Izz 6 (8.27) where h is the web height and b is the width of the two flanges. Assume that I_y > I=, algebraic manipulations then result in h -- > _ (8.28) b and the z-axis is the weak axis in bending. The cross-sectional area is related to the second area moment by the minimum radius of gyration, that is, A Izz (8.29) =_'7/"- Pmin or for the I-section b

(8.30)

Pmin 412+6h/b I It h v y !

I b Figure 8.5 I-section truss bar Since only the cross-sectional area is used in the weight computation, it is not necessary to determine the actual dimensions of the sectional height and width. Instead, one of the dimensions, usually the height, is treated as a function of the piston diameter and the other is then calculated with a predetermined h/b ratio.

8.3.5. Structural Weight Calculation The final step of the analytical procedure is to calculate the weight of each member based on its cross-sectional area, length, and the material density. Recall that seven different loading conditions were examined in the analysis, which results in seven sets of cross- sectional areas for each member. To ensure that the component will not fail under any of the seven loading conditions, the maximum cross-sectional area from the sets is selected as the final design value. Component weights are then calculated by multiplying each of the cross-sectional areas by the corresponding length and material density. The summation of these calculations then becomes the structural weight of the idealized analytical model.

8.3.6. Validation of the Analysis For analysis validation purposes, the landing gears for the Boeing Models 707, 727, 737 and 747 were modeled and analyzed. The estimated structural weight, which includes the axle/truck, piston, cylinder, drag and side struts, and trunnion, accounts for roughly 75 percent of the total structural weight that can be represented in the model [43]. The remaining 25 percent of the gear structural weight is made up of the torsion links, fittings, miscellaneous hardware, and the internal oleo mechanism, e.g., the metering tube, seals,

oil, pins,andbearings. Notethatactualandestimated structuralweightspresented in Tables

8.4 and8.5 only accountfor the components thatweremodeledin the analysis.

Table8.4 Main assembly structuralweightcomparison

Aircraft Estimated, lb Actual, lb Est/Act

B737 784 768 1.02 B727 1396 1656 0.84 B707 2322 2538 0.91 B747 9788 11323 0.86 Table 8.5 Nose assembly structural weight comparison Aircraft Estimated, lb ...... Actual, lb Est/Act B737 107 145 0.74 B727 171 327 0.52 B707 159 222 0.72 B747 1010 1439 0.70 Differences between the actual and estimated structural weights can be attributed to several factors. First, the models analyzed are extremely simple, i.e., structural members were represented with simple geometric shapes and no considerations have been given to fillet radii, local structural reinforcement, beating surfaces, etc. As for the analysis itself, simplistic equations were used to calculate the applied static and dynamic loads, and idealized structural arrangements were used to determine the member internal reactions.

However, it should be noted that the results are consistent with Kraus' original analysis; where an average of 13 percent deviation was cited [43].

8.4. Landing Gear Group Weight Estimation Although proven to be far more responsive to variations in design parameters, it is unlikely that an analytical tool will replace statistical methods. In fact, both methods should be used as complements to one another. This is particularly true in the calculation of the landing gear group weight, where the analytical and statistical methods can be used to determine the structural and non-structural component weights, respectively.

For large transports, landing gear structural weight accounts for roughly 57 percent of the landing gear group weight. The remaining weight is made up by the rolling stock and controls; the former accounts for roughly 34 percent of the total weight, while the latter accounts for the last nine percent. Note that the weights of the tires, wheels and brakes that make up the rolling stock have already been determined in previous chapters and no

additional calculationsare required. As for the controls, i.e., actuation and steering

mechanisms, the items can be estimated statistically with sufficient accuracy and thus eliminates the need to resort to an analytical method [App. A]. A detailed weight breakdown is provided in Table 8.6; the values are presented in terms of percent total landing gear weight.

Table 8.6 Landing gear weight breakdown [2] iiii i iii Component Main assembly Nose assembly ......

Rolling stock 32.0 2.0 Wheels 6.0 1.0 Tires 10.0 1.0 Brakes 16.0 0.0 Miscellaneous 0.0 0.0 Structure 50.0 7.0 Shock strut 32.0 4.0 Braces 12.0 1.0 Fittings 5.0 1.0 Miscellaneous 1.0 1.0 Controls 7.0 2.0 Total 89.0 11.0 Nil Using the combined analytical and statistical approach presented here, the landing gear group weight for the Boeing Models 707, 727, 737, and 747 were calculated and compared with actual values. As presented in Table 8.7a, the analysis tends to underestimate the group weight as the aircraft takeoff weight increases. Linear regression analysis was used to calibrate the estimated group weights (West) so they agree with the actual values. Correction factors were calculated using the expression (8.31) fc = 0.005W - 525 where W is the aircraft weight. The correction factor is then combined with West to arrive at the calibrated landing gear group weight (Weal), that is, Weal = West + fc (8.32)

The objective of this effort is to ensure that the discrepancy between the actual and

estimated values will remain within a tolerable range. This is important when the analysis is used to examine the weight of landing gear for aircraft that are outside the existing pavement thickness database. The calibrated results are shown in Table 8.7b.

Table 8.7 Landing gear group weight comparison a) Estimated group weight ii inlnlllll i ii iiiii ii Aircraft Estimated, lb Actual, lb Est/Act B737 4479 4382 1.02 B727 5976 6133 0.97 B707 9510 11216 0.85 B747 27973 31108 0.90 b) Calibrated group weight I ii i I ii i i i iii i iii iii i iiii ii Aircraft Calibrated, lb Actual, lb Cal/Act B737 4499 4382 1.03 B727 6301 6133 1.03 B707 10545 11216 0.94 B747 31138 31108 1.00

Chapter 9 Analysis Package

9.1. Introduction Four FORTRAN programs and a spreadsheet based on the analyses as outlined in previous chapters were developed for eventual incorporation into existing MDO codes.

Programs CONFIG, LIMIT, PAVE, and GEARWEI can be used together in an iterative fashion to study the global effects of variations in the landing gear design parameters on configuration, system integration, airfield compatibility, and weight. In addition, the programs can be used individually to analyze a particular aspect of a given concept. In both cases, aircraft configuration characteristics have to be imported either from existing aircraft sizing codes or disciplinary analyses, while landing gear-related parameters must be specified by the user or set up as defaults. Within an optimization framework, these parameters would be treated as design variables whose optimum values would be computed by the optimizer to achieve a desired objective. However, the goal here is to demonstrate the algorithms, which can be used to help automate the landing gear design process.

In addition to the four programs as mentioned above, a simple Microsoft Excel-based spreadsheet was created to establish the maximum permissible cg range of a particular aircraft concept. The spreadsheet requires estimated component weights be imported from existing aircraft sizing code, while the corresponding component cg ranges can be specified by the user or set up as defaults.

9.2. Description of Programs A simple spreadsheet software is used to establish the forward and aft limits of the permissible aircraft cg range. Given the aircraft configuration characteristics and component weights, the spreadsheet uses the specified component cg range as detailed in Chapter Two to calculate the maximum permissible aircraft cg range.

The primary task for program CONFIG is to develop a landing gear model that can be used as the baseline configuration. Given the aircraft weight, configuration characteristics, and the number of struts and tires, the program determines the loads on the tires and the

total braking energyto be absorbed by the brakes.Suitabletires, wheels,and brakesare

eitherselected from manufacturers'catalogsor sizedstatisticallyas detailedin Chapters

Four andFive. The lengthof the structuralcomponents, e.g., axles, truck beam, piston, cylinder, and trunnion, are determined based on the attachment scheme and clearance requirements. As for the linkages, a generic attachment scheme derived from existing commercial transports is used to determine the arrangement and required length of the drag and side struts. Based on this information, the program establishes a mathematical model of the notional landing gear in three-dimensional space, which is to be used by the remaining programs for detailed analysis.

Program LIMIT is used to examine the design and kinematic characteristics of the landing gear. Given the configuration characteristics of the aircraft and the model of the notional landing gear, turnover angle, pitch and roll angles during takeoff and landing, ground clearance, and turning radii are calculated using procedures as detailed in Chapter Three. The calculated values are then compared with a list of specified requirements to identify possible constraint violations. From the dimension and arrangement of the landing gear and the allocated stowage space, pivot axis and retraction angle are determined using mathematical kinematic analysis as detailed in Chapter Six. In addition, retraction path, swept volume, and stowed position are established and compared with stowage boundaries for possible structural interference.

The flotation characteristics of the aircraft are determined by program PAVE. Flexible and rigid pavement bearing stresses associated with specified loading conditions are calculated using pavement design procedures as detailed in Chapter Seven. The required pavement thickness is converted to the standard pavement bearing strength reporting system and tabulated for comparison purposes.

The component and group weights of the landing gear are calculated by program GEARWEI. As detailed in Chapter Eight, the structural weight of the landing gear is determined analytically from the notional landing model, while the weight of the non- structural components is determined from a statistical database. These weights are combined to arrive at the landing gear group weight.

9.3. Organization of Analyses The programs are organized as shown in Figure 9.1 for use in an iterative fashion to study the global effects of variations in the landing gear design parameters. Aircraft weight and configuration characteristics, as well as a limited number of landing gear-related design parameters, enter the package through program CONFIG. The former set of data is obtained either from existing aircraft sizing codes or disciplinary analyses, e.g., ACSYNT and FLOPS, whereas the latter is user-specified or is set up as defaults. Using this information as a starting point, program CONFIG generates a notional landing gear model, as well as data sets to be used as inputs for programs PAVE, LIMIT, and GEARWEI.

The first two programs then assess flotation, operational stability, maneuverability, and stowage aspects of the aircraft/landing gear concept are examined. If all the design constraints are satisfied, landing gear weight is then estimated in program GEARWEI.

Note that if any of the design constraints cannot be satisfied by the current configuration, user-specified modifications to the model or design parameters will be needed to resolve the violations through an iterative process. The execution of all the programs is essentially instantaneous.

The current state of the analysis package is a compilation of a number of separate analysis codes. The package does not have the capability to generate the required landing gear-related parameters, e.g., the number of tires and struts, attachment location, and stowage space, based on imported aircraft configuration characteristics. Thus, starting values, or "guesstimates", must be provided for these design parameters. The parameters can then easily be varied by the user, or an optimizer, for parametric study purposes and the information used to select the optimum design.

Aircraf configuration characteristics [ Analysis package Program CONFIG landing gear configuration selection

-r

Program LIMIT flotation analysis constraint violations Program PAVE [ n moditication(s) User-specifed determination Program GEARWEI component group weight estimation Landing gear design characteristics [ Figure 9.1 Organization of analyses 9.3.1. Input Output Data Data required by the analysis package are listed in Table 9.1. The majority of this information consists of geometric and weight characteristics associated with the aircraft: wing area and span, quarter chord sweep, fuselage length and width, maximum takeoff/landing weight, aircraft cg location, etc. These design parameters are readily available from existing aircraft sizing codes and can easily be rean'anged into the "card- style" inputs used by the analyses. The remaining information consists of landing-gear related parameters, and as mentioned in the previous section, must be provided by the user or selected from defaults.

Table 9.1 Required input data Description Parameter Type Geometric characteristics; location Wing Imported Geometric characteristics Fuselage Imported Geometric characteristics; location Engine/Nacelle Imported Takeoff/landing weights; weight distribution; Weight Imported aircraft cg location Design/selection criteria; number of tires/struts; User-specified Landing gear or default location; clearance; stowage space A description of the results generated by individual analysis is given in Table 9.2. It should be pointed out that these data only represent part of information that is produced by the analyses. Intermediate results, e.g., constraint boundaries, landing gear loads and induced stresses, that might be of interest or importance to a particular discipline, are currently internal to the analyses. To access this information would require modification of the output section of the program(s) to extract these data. Sample input/output files for the four programs can be found in Appendix E.

Table 9.2 Analysis-generated output data i i Description Program CONFIG Selected tires/wheels data; strokes; load-stroke curve; mathematical landing gear model LIMIT Trunnion alignment; retracted landing gear position; stability/operational characteristics; constraint violations PAVE ESWLs; concrete bearing stresses; pavement thickness; ACNs GEARWEI Structural member dimensions; landing gear component/_roup weight 9.4. Aircraft CG Estimation Spreadsheet In addition to the four programs that made up the analysis package, a simple Microsoft Excel-based spreadsheet was created to establish the maximum permissible aircraft cg range for any given aircraft concept. Aircraft component identification was provided in the first column, while estimated component weights as obtained from existing sizing codes are entered into the second column. Given the aircraft geometric characteristics, the forward and aft component cg limits determined based on the generic aircraft layout developed in Chapter Three are entered into column three and four, respectively. The spreadsheet calculates the moments corresponding to the forward and aft component cg limits, and then divides the sums of the moments by the total component weight to arrive at the maximum forward and aft aircraft cg limits.

Chapter 10 Parametric Studies

10.1. Introduction The emergence of the next-generation high-capacity commercial transports [51 and 52] provides an excellent opportunity to demonstrate the capability of the landing gear analysis package as detailed in the previous chapter. Landing gear design variables were varied parametrically to show their effects on the weight, flotation, and stability characteristics.

Dependencies between the variables and characteristics established from the parametric analysis, as well as the magnitude of the effect, can be used as a guideline in selecting the most effective means to alter a particular aircraft-landing gear configuration so that the desired characteristics can be obtained.

10.2. The Ultra-High-Capacity Transports A conceptual ultra-high-capacity transport (UHCT) was established based on a study by Arcara et al. [53] and industry forecasts [54, 55 and 56]. Configuration characteristics of the aircraft are presented in Table 10.1. Note that the aircraft is classified as a Design Group VI aircraft according to its wingspan, which is slightly over the specified 262-foot upper limit [7]. To match the geometric model of the aircraft as found in ACSYNT, the wing is modeled as a simple trapezoid without an inboard trailing-edge extension, i.e., the Yehudi. As a result, the location of the wing mac and hence the aircraft cg location and the attachment position of the main assembly are slightly forward of where they would be in the actual design.

Twenty-four main assembly tires arranged in a triple-dual-tandem configuration, i.e., six tires per strut, are used as an initial design. Tire selection is based on the minimum weight criterion. Forged aluminum and carbon are selected as the construction materials for the wheels and brakes, respectively. For the landing gear structure, 300M high-strength steel is used. The attachment scheme calls for two main gear units mounted on the wing and two units on the fuselage: the wing-mounted units retract inboard, while the fuselage- mounted units retract forward into the fuselage. The ensuing wheelbase and track dimensions are approximately 102 and 39 feet, respectively. Given this information, the analysis package as described in Chapter Nine is used to determine the design characteristics associated with this particular aircraft-landing gear combination. As shown in Table 10.2, all design constraints are satisfied. The landing gear weighs about 56,900

poundsand accountsfor roughly 17.4percent of the aircraft structural weight, or 4.6

percent of the MTOW.

Table 10.1 Configuration characteristics of a conceptual UHCT Baseline Passenger capacity 800 Range, nmi 7,500 Fuselage length, ft 250.0 Fuselage width, ft 24.0 Wingspan, ft 264.0 Wing area, ft 2 8,324 Aspect ratio 8.4 MTOW, lb 1,230,000 Fuel, lb 550,000 Table 10.2 Baseline aircraft design characteristics iiii ii i iiiiiiii i iiiiiiiii i i ii ii iii i i iiiiiiiiii Calculated Constraint < 63.0 Sideways turnover angle, deg 40.7 < 8.0 Roll angle, deg 7.2 - 15.0 Available touchdown angle, deg 16.7 - 15.0 Available takeoff rotation angle, deg 15.4 > 7.0 Nacelle-to-ground clearance, in 10.0 < 60.0 Castor angle, deg 37.0 < 100.0 Turning radius, fl 78.4 Gear weight, lb 56,885 Weight fraction, %MTOW 4.63 The flotation characteristics are given in Table 10.3 along with actual data for the McDonnell Douglas DC10, which are highest among existing aircraft. As shown in Table 10.3, major runway reinforcements will be needed at airports with a combination of flexible pavements and a low beating strength subgrade. Costs associated with such an upgrade could be in the $100 million range [6], an investment that might not be acceptable to airport authorities. Consequently, some major international airports with flexible pavements might not be able to handle the UHCT unless design changes are made to the aircraft. Results in Table 10.3 indicate that airports with rigid pavements are better suited in handling this class of aircraft. Note that as the subgrade strength approaches its upper limit, the required flexible and rigid pavement thickness for the new aircraft are actually lower than the ones required by the DC10. This is consistent with the trend observed in Chapter

Seven, i.e., as the number of wheels per strut increases, the required pavement thickness

decreases with the increase in the subgrade strength.

Table 10.3 Baseline aircraft flotation characteristics ACN Subgrade strength Thickness, in (UHCT/DC10) (UHCT/DC10) Flexible Ultra-low 73.5/63.9 134/97 Low 39.1/37.8 80/70 Medium 25.5/26.9 60/59 High 16.0/20.2 47/53 Rigid Ultra-low 18.6/17.0 96/75 Low 16.4/15.2 79/64 Medium 13.3/13.0 62/53 High 11.5/11.8 50/44 10.3. Parametric Studies Given the baseline aircraft-landing gear combination as characterized in the previous section, landing gear design variables were varied parametrically to show their effects on the weight, flotation, and stability characteristics. Dependencies between the various control variables and resulting aircraft characteristics established from this study, as shown here in Fig. 10.1, can be used as a guideline in selecting the most effective means to alter a pza'ticular aircraft-landing gear configuration so that the desired characteristics may be obtained. Note that there are instances where flotation and stability characteristics remain unchanged despite variations in the design parameters. Thus, only the characteristics being affected will be discussed.

In order for the UHCT to be able to operate from current airports without extensive runway reinforcement, additional tires are required to redistribute the weight of the aircraft over a larger tire-ground contact area. Provided the number of main assembly struts remains unchanged at four, the number of tires were varied both above and below the baseline (24). As shown in Fig. 10.1a, landing gear weight fraction increases with the increase in the number of tires. Evidently, weight penalties associated with the dimension of the truck assembly as well as the increased part-count, easily outstrip weight savings obtained from lighter tire and wheel designs that come with reduced load-carrying requirements. As shown in Table 10.4, the increased tire-ground contact area leads to

reductions in requiredpavement thicknessandthe corresponding ACN when comparedto

the baseline figures.

4.8 - _ 4.8 •_ 4.7 _'_ _ 4.7 ---- 4.6 =- 4.6 =- O 4.5 • _ 4.5 4.4 4.4 4.3 - .E 4.3 I I I 4.2 : 4.2 I I I 20 tires 24 tires 30 tires 4 struts 5 struts 6 struts Number of tires Number of struts a) Number of tires, four-strut configuration b) Number of struts, 24-tire configuration 4.8 © 4.8 4.7 4.7 O- --<7"""""---'-O . 4.6 c- 4.6 o

L

• = 4.5 •_ 4.5 4.4 4.4 -_ 4.3 4.3 : I I I I I I I _: 4.2 4.2 Pressure Weight Size -10,000 lb Baseline-10,000 lb Tire selection criterion MTOW c) Tire selection criterion d) MTOW 4.8 __-- _ 4.8 4.7 _ 4.7 4.6 _ 4.6 • ,_ 4.5 • _ 4.5 4.4 4.4 4.3 _ 4.3 o_ I I I I : t I I I _ 4.2 -20.0 in Baseline +20.0 in -20.0 in Baseline +20.0 in Aircraft cg location Main assembly location, longitudinal e) Aircraft cg location f) Main assembly location, longitudinal Figure 10.1 Changes in landing gear weight fraction due to design parameter variations © 4.8 © 4.8 [.-, 4.7 ;_ 4.7 . 4.6 =. 4.6 o--"" • _ 4.5 •_, 4.5 u_ 4.4 4.4 -_ 4.3 • -_ 4.3 i I I ] = 1 I !

_. 4.2 _: 4.2 -20.0 in Baseline +20.0 in -3.0 in Baseline +3.0 in Main assembly location, lateral Strut length h) Strut length g) Main assembly location, lateral © 4.8 0 4.8 7" b- E- 4.7 4.7 d 4.6 =- 4.6 o •=_ 4.5 •= 4.5 t_ 4.4 ,._ 4.4 -_ 4.3 .E 4.3 I I I : I I k I I _: 4.2 4.2 -3.0 in Baseline +3.0 in -3.0 in Baseline +3.0 in Truck beam length Axle length i) Truck beam length j) Axle length Figure 10.1 Changes in landing gear weight fraction due to design parameter variations (concluded) Varying the number of main assembly struts is another option to be considered in producing the desired flotation characteristics. As shown in Fig. 10.1b, provided the number of tires remains unchanged at 24, a reduction in the landing gear weight fraction is realized with an increase in the number of main assembly struts. The reduction can be attributed to the decrease in the number of tires found on each strut, which effectively lowers the combined load on the structural members and therefore leads to a lighter structure. As shown in Table 10.5, a reduction in the required flexible pavement thickness is evident as the number of the struts increases. Recall that in multiple-wheel assemblies, the flexible pavement bearing stresses are directly proportional to the number of tires per strut involved in the calculation and hence the required pavement thickness. The rigid pavement thickness requirements remain unchanged since the stresses obtained from Westergaard' s analysis are independent of the number of main assembly struts.

Table 10.4 Number of main assembly tires, four-strut configuration iii ii IIIHlUl II In I III III IIII Subgrade strength Thickness, in ACN 20 tires 30 tires 20 tires 30 tires (Des./Base) (Des./Base) (Des./Base) (Des./Base) Flexible Ultra-low 71.1/73.5 68.5/73.5 127/134 118/134 39.0/29.1 35.7/29.1 80/80 68/80 Low Medium 24.6/25.5 22.6/25.5 56/60 48/60 15.6/16.0 13.6/16.0 45/47 37/47 High Rigid 19.6/18.6 17.6/18.6 106/96 86/96 Ultra-low 17.3/16.4 15.5/16.4 88/76 70/76 Low 14.1/13.3 12.6/13.3 69/62 55/62 Medium 12.2/11.5 10.9/11.5 56/50 45/50

.... H! h

i i iiiii iii ii iiiii iiiii i i i ii Table 10.5 Number of main struts, 24-tire configuration Subgrade strength Thickness, in ACN five struts six struts five struts six struts (Des./Base) (DesJBase) (Des./Base) (Des./Base) Flexible Ultra-low 73.5/73.5 67.4/73.5 135/134 115/134 Low 39.1/39.1 36.1/39.1 80/80 69/80 Medium 25.5/25.5 22.2/25.5 60/60 46/60 High 16.0/16.0 13.6/16.0 47/47 37/47 Rigid Ultra-low 18.6/18.6 18.6/18.6 96/96 96/96 Low 16.4/16.4 16.4/16.4 78/79 78/79 Medium 13.3/13.3 13.3/13.3 62/62 62/62 High 11.5/11.5 11.5/11.5 50/50 50/50 Besides increasing the number of main assembly tires and struts to bring about the desired reduction in the required pavement thickness, another option is to select a tire with a lower inflation pressure. As shown in Fig. 10. lc, the minimum inflation pressure candidate offers the lowest landing gear weight fraction of the three selection criteria. A reduced inflation pressure also means an increased tire-ground contact area, hence reduced

pavement loadsandpavement thicknessrequirements as shown in Table 10.6. It should

be noted that all but a select few of large tires availableare capableof meetingthe

performance requirements imposedby the UHCT. That is, the inflation pressure, size, and weight of the candidate tires are nearly identical. As a result, the effects due to such variations might not be as apparent as they would be for other types of aircraft, where the selection is based on a larger pool of candidate tires.

Variations in MTOW have an obvious impact on the configuration of the landing gear and the pavement thickness. As a minimum, the structural dimensions of the landing gear and hence the structural weight would vary as the design weight of the aircraft changes between different configurations. As shown in Fig 10. ld, the landing gear weight fraction decreases even though the actual landing gear weight increases with the MTOW. This can be attributed to the fact that the landing gear weight does not increase with the MTOW in a pound-for-pound manner, and therefore a decreasing weight fraction is observed.

Similarly, the landing gear weight decreases at a slower rate than the MTOW, yielding a higher weight fraction. The magnitude of the landing gear weight variation is similar to that provided by industry, where a 40-pound increase in the landing gear weight per 1,000 pounds increase in the MTOW is anticipated [App. A]. As reaffirmed in Table 10.7, an increase in the MTOW would require a thicker pavement to support the aircraft, and vice versa.

Table 10.6 Tire selection criteria, 24-tire configuration ACN Thickness, in Subgrade strength Min. press Min. size Min. press Min. size (Des./Base) (Des./Base) (Des./Base) (Des./Base) Flexible 133/134 135/134 73.1/73.5 73.5/73.5 Ultra-low 81/80 80/80 39.4/39.1 39.1/39.1 Low 55/60 60/60 Medium 24.3/25.5 25.5/25.5 44/47 47/47 15.3/16.0 16.0/16.0 High Rigid 92/96 96/96 18.3/18.6 18.6/18.6 Ultra-low 75/79 78/79 16.1/16.4 16.4/16.4 Low 58/62 62/62 Medium 12.9/13.3 13.3/13.3 45/50 50/20 10.9/11.5 11.5/11.5

Althoughthe locationof aircraft cg has always played a decisive role in the positioning

of the landing gear, instances are possible where design considerations become conclusive in deciding the mounting location, i.e., the landing gear has to be located at a specific location so that desired stability and maneuverability characteristics can be obtained. As shown in Fig. 10. le, for this particular aircraft-landing gear combination, provided that the location of the main assembly group is fixed, an optimum aircraft cg location exists at a short distance aft of the current position where the weight fraction of the landing gear is at its minimum. In such cases, the location of the aircraft cg must be maintained at a particular position during takeoff and landing conditions through a controlled loading scheme. Once airborne, the constraints can be relaxed by redistributing the fuel among various fuel tanks.

As shown in Fig. 10.1f, the repositioning of the main assembly group in the aft direction results in a landing gear weight fraction that is lower than the one corresponding to a shift in the forward direction. This trend can be attributed to the reduced load that follows directly from an increased offset between the main assembly group and the location of the aircraft cg, i.e., a longer moment arm to counteract the applied ground loads.

Note that when a highly-swept, high-aspect ratio wing is considered, a rearward movement of the main assembly group might be extremely difficult. Moving the gear aft could effect takeoff rotation speed and takeoff distance, which has to be checked. Also, brake weight may increase if the rotation speed increases, increasing the deceleration demands for the balanced field length requirement. Finally, the shift may not be feasible due to wing planform constraints, such as the size of the inboard trailing-edge extension (the Yehudi), required to provide suitable attachment location, as well as sufficient space to house the trailing-edge control surfaces and the associated actuation systems. The Yehudi also incurs drag and weight penalties that need to be considered.

The repositioning of the wing-mounted assemblies in the lateral direction affects primarily the stability and maneuverability characteristics of the aircraft. As shown in Table 10.8, an outboard movement of the wing-mounted assemblies produces a desired reduction in the sideways turnover angle; however, such a movement shifts the minimum 180-degree turn radius closer to the Class VI 100-foot upper limit [5]. As shown in Fig.

l O.lg, the increasing landing gear weight fraction can be associated with the outboard movement of the assemblies. This leads to an increase in the length of the side strut, as well as an increase in the drag and shock struts due to wing dihedral, and hence the

structuralweight of the landinggear.Conversely,an inboardmovementof the assemblies

exhibits a higher sidewaysturnover angle, a smaller turning radius, and a decreasing

landinggearweightfraction.

Table10.7 MTOW variations

ACN Thickness, in Subgrade strength -10,000 lb + 10,000 lb .10,000 lb + 10,000 lb (DesJBase) (DesJBase) (Des./Base) (Des.fBase) Flexible 73.2/73.5 73.8/73.5 134/134 136/134 Ultra-low Low 39.0/39.1 39.3/39.1 80/80 81/80 Medium 25.4/25.5 25.6/25.5 59/60 60/60 16.0/16.0 16.0/16.0 47/47 47/47 High Rigid Ultra-low 18.5/18.6 18.7/18.6 95/96 96/96 16.3/16.4 16.5/16.4 78/79 79/79 Low Medium 13.3/13.3 13.3/13.3 61/62 62/62 11.5/11.5 11.5/11.5 50/50 50/50 High Table 10.8 Wing-mounted assemblies location variations, lateral 20.0 in outboard 20.0 in inboard Design characteristics 43.2 Sideways turnover angle, deg 38.4 16.5 Available touchdown angle, deg 16.9 15.5 Available takeoff rotation angle, deg 15.3 76.7 ,,,Turning radius, ft ............................... 80.1 i iiiiii Changes in the stability characteristics and ground clearance due to variations in landing gear strut length are of primary interest when a growth version of the aircraft is considered.

Features typically associated with the growth options are a stretched fuselage obtained from the addition of plugs forward and aft of the wing, and upgraded power plants that come with a larger fan diameter. Both of the above features would require an extension of the strut length to maintain the desired operation angles and nacelle-to-ground clearance. As shown in Table 10.9, the growth-related modifications can result in an increased sideways turnover angle and a reduced permissible pitch angle during takeoff/landing operations. As can be expected and reaffirmed in Fig. 10.1h, an increase in strut length leads to an increase in structural weight, and therefore an increase in the landing gear weight fraction, as well as vice versa. The magnitude of the landing gear weight variation is again similar to the one

providedby industry,wherea 60-poundincrease in weightperstrutis anticipated for every

inch increase in strutlength[App. A].

Changes in thesize of the tires,wheels,andbrakesdueto varying designparameters, e.g., loading conditions and braking energy requirements, can alter the dimensions of the truck beam and axles. As can expected and reaffirmed by Figs 10.1i and 10.1j, an increase in the component length leads to a higher landing gear weight fraction, and vice versa. Data presented in Tables 10.10 and 10.11 show that an increase in either truck beam or axle length will result in a thicker pavement.

Table 10.9 Strut length variations -3.0 in +3.0 in Design characteristics 40.2 41.1 Sideways turnover angle, deg 16.9 16.5 Available touchdown angle, deg 15.3 15.5 ......... Avai!ab!e takeoff rotation an_le, de_ iiiii ii iiiii iiii ii i Table 10.10 Truck beam length variations III ACN Subgrade strength Thickness, in -3.0 in +3.0 in -3.0 in +3.0 in (Des./Base) (Des./Base) (Des./Base) (Des./Base) Flexible Ultra-low 73.1/73.5 73.7/73.5 133/134 135/134 Low 39.1/39.1 39.2/39.1 80/80 80/80 Medium 25.5/25.5 25.5/25.5 60/60 60/60 High 16.0/16.0 16.0/16.0 47/47 47/47 Rigid Ultra-low 18.6/18.6 18.6/18.6 96/96 96/96 Low 16.4/16.4 16.4/16.4 78/79 78/79 Medium 13.3/13.3 13.3/13.3 62/62 62/62 ............. High ..... 11.5/11.5 ...... 11.5/11.5 ...... 50/50 ................. 50/50 ...........

10.4. Derivatives of the Baseline Aircraft In today's highly competitive environment, flexibility in being able to meet the vastly different requirements from various airline customers, e.g., a longer range and an extended payload capacity, has become one of the primary considerations in the design and marketing of a new aircraft. To ensure that a customer will have a list of options to select

from when it comes time to placean order, derivativesare consideredearly on in the

conceptual design phase, and more than likely, pursued in parallel with the baseline aircraft.

Table 10.11 Axle length variations Subgrade strength Thickness, in ACN -3.0 in +3.0 in -3.0 in +3.0 in (DesJBase) (DesJBase) (DesJBase) (DesJBase) Flexible Ultra-low 73.4/73.5 73.6/73.5 134/134 135/134 Low 38.7/39.1 39.5/39.1 79/80 82/80 Medium 25.1/25.5 25.8/25.5 58/60 61/60 High 15.7/16.0 16.3/16.0 46/47 48/47 Rigid Ultra-low 18.6/18.6 18.6/18.6 96/96 96/96 Low 16.4/16.4 16.4/16.4 78/79 78/79 Medium 13.3/13.3 13.3/13.3 62/62 62/62 Hi[gh 11.5/11.5 11.5/11.5 50/50 50/50 Two derivatives were envisioned for the baseline UHCT: advanced (high aspect ratio) wing and extended range (8,000 nmi); corresponding configuration characteristics are shown in Table 10.12. Although the wing planform of the advanced wing derivative is slightly different from the baseline and the extended range version, it is assumed that the configuration of the landing gear on all three aircraft are identical, i.e., 24 main assembly tires on four struts. Note that this assumption does not imply that the weights of all three landing gear are identical.

Table 10.12 Derivative configuration characteristics Extended range Advanced wing Passenger capacity 800 800 Range, nmi 8,000 7,500 Fuselage length, ft 250.0 250.0 Fuselage width, ft 24.0 24.0 Wing span, ft 264.0 261.0 Wing area, ft 2 8,324 7,423 Aspect ratio 8.4 9.2 MTOW, lb 1,350,000 1,140,000 Fuel, Ib 640,000 460,000 I I I II IIIIII As shown in Figure 10.2, the advanced wing derivative has the highest landing gear weight fraction of the three configurations, whereas the extended range derivative has the

lowest of the three. For identicalmission requirementsbetweenthe baselineand the

advanced wing derivative,the baseline aircraftwill be the preferredchoiceif the deciding

factor is basedon landing gearweight fraction, its lower landing gear weight fraction

impliesthata greater fractionof the total aircraftweight is madeup by revenue-generating

payloads. However,if the decidingfactoris somethingother thanthe landinggearweight

faction, e.g., operating cost or runway upgrade cost, the advanced wing configuration will

be the preferred choice due to its lower mission fuel requirements and lighter MTOW, respectively. As for the extended range derivative, although the landing gear weight fraction is lower than the other two aircraft, the required pavement thickness as shown in Table 10.13 can result in a prohibitive runway upgrade cost. However, the desired flotation characteristics can be obtained by replacing the conventional wing design with the one found on the advanced wing derivative. The reduction in mission fuel weight associated with higher performance due to the advanced wing design would then lower the MTOW of the extended range derivative and hence the required pavement thickness.

_ 4.8 4.7 4.6 - ff • "_,-, 4.5 4.4 "_ 4.3 : "_ _: 4.2 I I I Adv wing BaselineExt range Aircraft configuration Figure 10.2 Changes in landing gear weight fraction due to aircraft configuration variations 10.5. Landing Gear Weight Trend for Large Aircraft The baseline aircraft along with its derivatives are used to provide some analytically-based landing gear weight estimates that can be used to help calibrate existing statistical weight equations. Although statistical weight equations are capable of producing quick and fairly accurate group weights within the range where significant previous experience is available, their reliability is questionable at best for aircraft with takeoff weight beyond one million pounds, i.e., they are constrained by what has been designed in the past. The uncertainty is

madeevidentby the two possibleweight trendsavailable: a decreasing trendas predicted

by ACSYNT andan increasing trend aspredictedby DouglasandTorenbeek. As shown

in Fig. 10.3,landing gear weight fractionscorrespondingto the baselineaircraft and its

derivativessuggest thatthe weightequationusedby ACSYNT is likely to producea more

accurate trendthantheonesusedby DouglasandTorenbeek. In addition,anincreasein the

numberof main assemblystrutsfrom four to six did not resultin a step increasein the

weightfractionasexpected. Again, this canbe attributedto the decrease in the number of

tires found on eachstrut, which effectively loweredthe combinedload on the structural

membersandthereforeled to a lighter structure.Note that additionalaircraft within the

UHCT classmustbe modeled to extendthe database sothat the weight trendsas observed

heremay beconfirmed.

Table10.13 Aircraft configuration variations

Subgradestrength Thickness,in ACN

Ext. range Adv. wing Ext. range Adv. wing

(Des./Base) (Des./Base) (Des./Base) (Des./Base)

Flexible

Ultra-low 77.1/73.5 70.0/73.5 148/134 90/134

Low 40.8/39.1 37.9/39.1 88/80 75/80

Medium 25.5/25.5 24.6/25.5 61/60 60/60

High 15.6/16.0 15.6/16.0 48/47 50/47

Rigid

Ultra-low 19.3/18.6 18.2/18.6 104/96 122/96

Low 16.9/16.4 16.1/16.4 84/79 75/79

Medium 13.6/13.3 13.2/13.3 65/62 55/62

High 11.6/11.5 11.6/11.5 52/50 43/50

J 5.00 -- ACSYNT nQ.°. °" 4.50 ...

..... Douglas ......... Torenbeek

./........ • DC10

4.00 • B747 • ° - ° ° - e,,,, o Base, 4 struts •-q 3.50 [] Adv, 4 struts o ER, 4 struts 3.00 • Base, 6 struts • Adv, 6 struts • ER, 6 struts _: 2.5o , , K , , _ I , , i i , L A , I 2.00 106 107 MTOW, lb Figure 10.3 Landing gear weight fraction beyond one million pounds MTOW

Chapter 11 Costs

1 I. 1. Introduction The manufacturing cost of the landing gear cannot be treated simply as a function of weight or strut length. Instead, cost estimation must take into account the costs of development, certification, marketing, life-cycle, spares, etc. A typical program cost is roughly in the range of $10 to $12 million dollars, based on industry survey [App. A].

However, detailed information is considered proprietary and is difficult to obtain from the manufacturers. Thus, the cost issue will only be discussed in qualitative terms, while actual unit costs will be provided whenever available.

11.2. Maintenance and Overhaul The maintenance costs associated with the landing gear represent a considerable item in the total maintenance bill [3]. The cost of the tires, wheels and brakes will remain relatively unchanged for new programs. The limiting factor is the size of the tire that can be constructed and tested without a major new investment in manufacturing and testing facilities. Current hardware limits the maximum diameter to 56 inches for the bias-ply tire and 58 inches for the radial-ply tire [App. A]. Dimensions and costs of several tires found on existing large aircraft are listed in Table 11.1. For the aluminum wheel and carbon- carbon heat sink found on the Boeing Model 747-400, the unit price is valued at $70,000.

Table 11.1 Description of selected aircraft tires [App. A] Tire Type Aircraft Application Cost, $ H49x 19.0-22, 32-ply Bias Boeing Model 747 Main/Nose 2,100 42x17.0-18, 28-ply Radial Boeing Model 777 Nose 2,100 50x20.0-22, 32-p1_, Radial, Boein_ Model 777 Main 2,900 The landing gear overhaul interval varies between 33,000 to 42,000 flight hours, or roughly within six years [App. A]. Generally, the parts of a landing gear are given an ultimate 'safe life' beyond which they would, if still in service, be scrapped [57]. A justification of this approach is that deterioration in service can go unseen since corrosion

and other process can occur in concealed areas which are only revealed when the assembly

is completely stripped down.

The preferred method is to overhaul the entire set at the same time to minimize the down-time; however, it might be necessary to overhaul the set separately due to schedule, parts and facility constraints. Components may require extensive rework in the shops and thus it is difficult to quote a total throughput time. Given a supply of serviceable components to replace those sent shop-to-shop, it is possible to turn around a B747 assembly within five weeks [57]. Due to the length of time required to rectify each constituent part of a particular assembly, a unit nearly always loses its identity as such, and the end product may contain only a few parts of the original assembly. However, it is noted that when refurbished, the assembly may be better than a new one since it embodies modifications designed to increase the subsequent overhaul life [57]. For the B747 type landing gear, the overhaul cost is estimated at $400,000 [App. A]. Replacement of the carbon heat sink occurs every 1,200 to 1,500 landings, while only 300 landings are allowed for the wheel before replacement. The overhaul cost for the wheel and heat sink is pre- negotiated with the contractors and is known as cost-per-landing. Quoting the B747 figures, the cost for the wheel, including tire, is estimated at $5 per landing, while the cost for the carbon-carbon heat sink is estimated at $10 per landing [App. A].

11.3. Cost Reduction With the financial challenges arising from the deregulation of the air-travel industry, the airlines are faced with the challenge of reducing operating costs to remain competitive.

As a result, the airlines have demanded that the aircraft manufacturers produce new designs with high reliability and low maintenance requirements. In basic design, costs associated with the landing gear may be reduced by aiming at simplicity, compactness, and minimum weight and maintenance requirements. Simplified design and improved manufacturing techniques, e.g., die-forging and three-dimensional machining [9], are being used to reduce the part-count associated with the landing gear system. In addition, recent technologies, e.g., carbon-carbon heat sinks, radial tires, and high-strength steel, are being introduced.

Potential savings associated with the application of these technologies have already been mentioned in Chapter Four.

Chapter 12 Future Considerations

Although an initial validation of the methodology was done, further validation and is required. Since all readily available data was used in this study, the methods should be checked against more thorough analysis done by industry on actual new products, such as the new Boeing 747 derivative, as they appear. In addition, several areas where refinements could be made have been identified as a result of the experience to date. However, we are not able to quantify the benefits that these extensions in the methodology would produce.

Refinement of the landing gear analysis package should include the improvement of the pavement thickness and landing gear weight predictions. A method to calculate the rigid pavement bearing stress that includes location and direction of maximum moment considerations [36] would improve the reliability of the estimated rigid pavement thickness and the corresponding ACN. The experimental test program being conducted by the FAA and Boeing to determine the exact flotation requirements for the B777 may provide useful information for this extension. The accuracy of the landing gear structural weight can be improved by extending the analysis to include intermediate column buckling analysis [48] for structural members with large slenderness ratio, e.g., drag and side struts, and St.

Venant's theory for torsion and flexure of thick-walled bars [50] for structural members with low machinability factors, e.g., axle and truck beam.

Finally, the full potential of the analysis package would emerge if a graphical front- end and the Dynamic Integration System (DIS)-based wrapping technique [58] were incorporated. The former would enable the user to interactively prepare input for the analysis and interpret the output, while the latter would provide a common interface such that coordinated execution of disciplinary analyses as found in ACSYNT can be achieved.

Chapter 13 Conclusions

The design of the landing gear is one of the more fundamental aspects of aircraft design. The design and integration process encompasses numerous engineering disciplines, e.g., structure, weights, runway design, and economics, and has become extremely sophisticated in the last few decades. These considerations were incorporated in an MDO procedure for use in the conceptual design of large transport aircraft. Accomplishments include: • Aircraft cg estimation methods were studied and a new approach to cg estimation in conceptual design was demonstrated.

• An automated landing gear modeling algorithm for large transport aircraft was developed, and conformance with typical FAR requirements was assessed automatically.

• Airfield compatibility considerations associated with pavement thickness and runway and taxiway dimensions were automated.

• An analytical structural weight estimation procedure was developed to complement existing statistical landing gear weight estimation methods.

• A multidisciplinary analyses computer program package for landing gear design and was created for use in large MDO aircraft design programs.

• Results obtained from the analysis package were presented, illustrating the trade-off studies and parametric results available for incorporation into a complete MDO design procedure.

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AppendixA Industry Survey

To try to ensure that our work was current and relevant, we made many contacts with industrial and government engineers that work with landing gears. The approach was to conduct a survey. In this section we present the material given to them, and report on what we found. In general, we got the best information in telephone interviews. The questions initiated discussions that were often broader and less focused than the questions themselves. Thus, the discussion of results presented in the following sections follows the broader areas, and does not explicitly summarize individual answers to the questions. In general, the company contacts were not able to give us detailed written material because they considered their expertise proprietary.

The issues we identified that needed to be addressed were: runway compatibility, landing gear integration, landing gear configuration, landing gear weight, advanced technologies, and cost. A list of questions was developed covering these considerations to ask engineers associated with landing gear systems. Using a few suggestions from contacts in industry and government, we started making calls. In some cases, we sent a fax of our questions. Often, we were directed to contact someone else in the organization, or, someone at another company. Eventually, the survey included major airframers, landing gear manufacturers, airlines, and government agencies and technical societies.

The list of questions was circulated among the manufacturers for comments and suggestions, while airlines were contacted to obtain operating and maintenance cost information.

A. 1. General script for our phone interviews The landing gear integration issue for advanced aircraft is being investigated under a NASA Ames research grant to Virginia Tech. The project objective is the

formulation of a methodologyto include landing gear considerationsexplicitly in the

conceptual design stage. In particular, the project addressesthe special design

considerations associated with the next-generation high-capacitytransportwith a TOGW

exceedingone million pounds. Our landing gear design and integration related issues

were defined during the initial background research with heavy reliance on N. S.

Currey's Aircraft Landing Gear Design: Principles and Practices. We have questions

concerning landing gear configuration, aircraft-landing gear integration, runway compatibility, advanced technologies, weight, maintenance, and cost.

A.2 The questions • What are the design parameters given to the landing gear designer? What is the design envelope you have to work with? Which is the primary design goal, minimum weight, stowage space, or complexity?

• What are the major problems encountered concerning the integration of the landing gear for the ultra high capacity type aircraft currently under study? What kind of special design considerations are required?

• What are some advanced technologies that will change the landing gear configuration of the ultra high capacity type aircraft dramatically in the next decade or two? How will they change the configuration? What kind of weight reduction can be expected with these technologies?

• What method is used to calculated the landing gear ground and landing loads?

Which specification is used? Is there a set of equations that can be readily used?

• What method is used to calculate the aircraft flotation requirements? How do you account for multiple main strut configurations? What kind of constraint in gear configuration is imposed by the flotation requirements?

• For a takeoff gross weight outside the experience base are there some "first principles" that can be followed for landing gear weight estimation?

• What will be the most likely landing gear configuration for the ultra high capacity type aircraft? How many main struts can be expected for a takeoff weight exceeding one million pounds? How would you arrange the main assembly if you have six main struts? What is the major advantage/disadvantage of increasing the number of main struts?

• What method is used to produce the initial landing gear weight estimation? What would be the scaling factor if we are to estimate the weight by scaling up current configurations to meet the demand? Can we obtain geometry and weight information on existing landing gears to be used as a design database?

• What methodis usedfor the initial landing gearcostestimate?What arethe major

costdriversandthe corresponding sensitivities?

A.3. The Contacts A list of survey participants and their telephone numbers are presented in Table A. 1.

A. 4. Findings A. 4.1 Runway Compatibility Due to economic considerations, the ultra high capacity transport, UHCT, must be able to operate out of Class V airports, e.g., the Boeing Model 747 class airports, without requiring extensive runway reinforcement and modification. Flotation requirements can be obtained using the PCA methods for rigid pavement and the CBR method for flexible pavements. Effects of multiple-strut/multiple-wheel landing gear configurations on the pavement bearing strength have yet to be addressed fully by industry. However, preliminary finite element analyses suggest interaction among wheels can be neglected outside a radius of ten footprint radii from the point where the flotation analysis is performed. Based on this information, the number of wheels, i.e., the equivalent number of wheels per strut (ENWS), used to select the proper repetition factor curve (this is the factor that accounts for the number of landings per year on the pavement) becomes the number of wheels found within the circle of ten foot-print radii centered at the strut-truck joint. With current tire inflation pressures, a 20-wheel main assembly is required for a TOGW between 1 and 1.2 million pounds, while a 24-wheel main assembly is required for a TOGW between 1.3 and 1.6 million pounds to produce the desired flotation characteristics. Both numbers include a 20 percent future growth factor.

Table A.1. Industry/Government Landing Gear Contact List Phone Fax Federal Aviation Administration John Rice Airport Standards and Safety (202) 267-8745 (202) 267-5383 Niel Schalekanp Aircraft Certification (206) 227-2112 Bill Perrella Aircraft Certification (206) 227-2116 (206) 227-1320 Wright-Patterson Air Force Base Paul Ulrich Vehicle Equipment (513) 255-2663 Henry Pollack (513) 255-4158 SAE A-5 Committee Richard Vandame (412) 776-4841 (412) 776-0002 Waterways Experiment Station Carlo Gonzalez (601) 634-2203 The Tire & Rim Association, Inc.

Joe Pacuit (216) 666-8121 Boeing Commercial Airplane Matt Travis Landing Gear System (206) 237-7744 John Potter Landing Gear System (206) 237-7745 Jerry Kileer Landing Gear System (206) 965-9775 Edward Gervais Airport Technology (206) 237-0175 Dave Nielson Configuration (206) 342-7577 Scott Perkins Structures (206) 266-7812 Bob Nielson Weights (206) 342-1522 McDonnell Douglas Aircraft Brian Lindley (310) 496-9129 AI Kemik (310) 593-7313 (310) 496-9244 Larry McBee (310) 496-9949 Cleveland Pneumatic Gene Stuczynski (216) 429-4213 (216) 883-7153 B.F. Goodrich Dave Moser (513) 440-2206 (513) 339-3813 Paul Snider (513) 440-2380 (513) 339-4556 Tom Kendall (513) 440-2205 (513) 339-6811 Dean Peters (513) 339-3811 (513) 339-6811 Menaseo Aerosystem Bill Luce (817) 685-3538 (817) 689-3852 Richard Luu (818) 847-9208 Michelin Marion DeWitt (704) 548-2483 Ron Olds Director of Sales (704) 548-2438 U.S Air Norman White Senior Airframe Engineer (412) 747-3425 (412) 747-3975 United James Gallivan Landing Gear, B747 (510) 382-8312 (510) 382-8302 Ed Pozzi Landing Gear, B757 (415) 634-6994 Northwest Jim Baumiller Landing Gear (612) 726-3885 Wheels & Brakes Steve Lydon (612) 726-7217 (612) 726-6844 A.4.2 Integration Aircraft-landing gear integration will be the primary concern for the next- generation high-capacity transports. The location dependency of the wing and the main gear assembly to the aircraft cg will play a major role in the integration issue. With the introduction of multiple-strut configurations, the envelope within which the landing gear has to be located to produce the ideal loading and stability characteristics may no longer be large enough to accommodate the increased number of main assembly struts. This phenomenon is known as location stagnation by the landing gear community.

Modification in design and flotation requirements must be made, if necessary, to accommodate kinematic and stowage constrains such that the landing gear can be deployed and stowed without interference with surrounding structures. A forward- retracting scheme for the fuselage struts is preferred, which allows the gears to be deployed using the slip-stream in case of a hydraulic failure. However, stowage limitations could result in an aft-retracting scheme for the center-line strut located between the wing-mounted struts in a five-plus struts configuration.

A.4.3 Configuration The number of wheels imposed by the flotation requirement can be accommodated with either a four-, five- or six-strut configuration. One of the centerline struts will be located abreast of the wing-mounted struts for the five-plus main gear struts configurations. With the introduction of the centerline strut(s), a double-keel layout is required, i.e., the stowage space is divided into three compartments with two identical keels placed parallel to each other. The centerline strut(s) will then be mounted and stowed between the keels. The fuselage width of the new aircraft, which will be 20 to 30 inches wider than that of the B747, should be able to accommodate the double-keel layout with relative ease. However, one of the drawbacks is that the structural weight associated with the keels will be doubled, since both keels have to withstand the same

bucklingload andthushaveto be similar in dimensionto the onefound in the single-keel

layout. Anotherdrawbackis that a complexdeploying/retractingschemefor the landing

gear doors must be developedto prevent interferencebetweenthe doors and the gear

itself.

The length of the strut will be dictated by the condition on aircraft ground

clearance requirements duringcross-windlandingsimposedby the large nacellediameter

of the advancedengines.The vertical spacingbetweenthe nacelleandthe wing, i.e., the

gully, will be reduced to a minimum, provided that desirable flow characteristics are maintained, before any extension in strut length is made. A main gear steering system will be needed to meet the ground operation requirements, with the most demanding maneuver being the 180-degree turn on existing runways. Options include the fuselage strut steering system found on the B747 and the forward-aft wheel steering system found on the B777.

The wheel truck dimensions of the dual-twin-tandem and triple-dual-tandem configurations will be similar to those of the B747 and B777, respectively. The longitudinal spacing between tires will be maintained at roughly six inches for ease of removal of the wheel plugs, while lateral spacing will be slightly wider in both configurations due to the increased brake size required for the new aircraft. Due to the limited stowage volume, the truck assembly might have to be rotated prior to retraction to minimize the stowage space required.

A.4.4 Loads The dynamic and ground loads are determined in accordance with FAR Part 25. It is unlikely that the new aircraft will be subjected to rough field operating requirements, and thus a single-acting shock absorber will be sufficient to handle the kinetic energy experienced during landing and taxiing. Based on preliminary analysis from industry, the

new aircraft will require a shockstrut with a 24-inch stroke at the minimum, a piston

diameterof 15inches,andinternaloleo pressures between1,500and 1,800psi. Canting

of the strut shouldbe avoided,if possible,due to the load path considerations. Active

struts will likely be used to provide improved and acceptableground ride quality.

Improvementswill probablybe internal, e.g., bearings,finishes, andrebounddamping, but little differencewill beseenin the externalconfiguration.

A.4.5 Weight The design of the new landing gear must be as simple as possible, since complexity drives up the cost faster than weight. However, weight also appears to be inversely proportional to the level of complexity. With the reduction in the complexity level, e.g., the number of supports, structural members are forced to withstand a higher load, which in term increases the structural weight due to an increase in cross-sectional area. Therefore, a balance must be reached between simplicity and weight, and this can only be accomplished through parametric studies of different landing gear configurations.

Note that a step increase in total landing gear weight occurs with each additional strut.

Therefore, the number of struts must be kept at a minimum while at the same time meeting the flotation and simplicity requirements. Existing data indicates that fuselage strut weight is roughly 25 to 40 percent less than that of the wing strut, and overall, total gear weight will remain at roughly five percent of the maximum take-off weight.

Structural weight estimation should be obtained using an analytical approach, while the following "rules of thumb" for sensitivities were provided by the industrial contacts. Weight scaling taken to a 1.1 power will give a reasonable estimation for sub- components, i.e., the steering system, up locks, down locks, fittings and miscellaneous items. A landing gear gross weight variation of 5 pounds per 1,000 pounds increase in TOGW for the nose gear was suggested, while a 40-pound variation per 1,000 pounds increase in TOGW for the main gear should be used. Weight variation of 40 pounds per inch increase in strut length per strut was also suggested. The wheel and tire weights will

be similar to that of the B747, i.e., 190 pounds and 290 pounds, respectively, while the

heat sink weight will be heavier, again due to the increased braking energy requirements.

A step increase in the landing gear group weight will occur with each additional strut; therefore, the number of struts should be kept at a minimum.

A.4.6 Advanced Technologies Advanced technologies will play a major role in reducing the weight of the UHCT type landing gears. A five to seven percent weight reduction can be obtained with the use of high strength steel for the landing gear strut and carbon for the brake. Radial-ply tires, although having a higher initial cost, offer a 20 percent weight reduction over bias-ply tires, while at the same time allowing more landings per life-cycle. Further weight reduction can be achieved by the use of a steer-by-wire concept in place of the conventional cable-and-pulley system. Electrical actuation units will be introduced as a way to reduce weight in secondary mechanisms, but the primary actuation method will remain hydraulic.

A 4. 7 Cost The manufacturing cost of the landing gear cannot be treated simply as a function of weight or strut length. Instead, cost estimation must take into account the costs of development, material and processes, certification, marketing, overhaul, refurbishment, and spares. Typical program cost is roughly in the range of $10 to $12 million dollars.

The cost of the tire, wheel and brake will remain relatively unchanged. The limiting factor is the size of the tire that can be constructed and tested without a major new investment in the manufacturing and testing facilities. Current hardware limits the maximum diameter to 56 inches for the bias-ply tire and 58 inches for the radial-ply tire.

The H49x 19.0-22, a 32-bias-ply tire found on both the nose and main gear of the B747, is valued at $2,100. This can be compared to the radial, the 50x20.0-22, which is found on the main gear of the B777 with a 32-ply rating, which is valued at $2,900, and the

42x17.0-18,which is found on the nosegear of the B777 with a 28-ply rating, andis

valued at $2,100.Due to its light weight and the increasednumber of landingsallowed

per life-cycle, the radial tire hasbecomethe preferredchoiceby airlines eventhough it

costsmore. As for the aluminumwheel andcarbon-carbon heatsink found on the B747, the unit priceis valuedat$70,000.

The landing gearoverhaulinterval variesbetween33,000to 42,000 flight hours.

The preferredmethodis to overhaulthe entiresetat the sametime to minimize the down-

time. However,it might benecessary to overhaulthe setseparatelydueto schedule,parts

andfacility constraints.For the B747type landing gear,the overhaulcost is estimatedat

$400,000.Replacementof the carbon heat sink occurs every 1,200to 1,500landings,

while only 300landingsareallowedfor the wheelbeforereplacement. The overhaulcost

for the wheelandbrakeis pre-negotiatedwith the contractorsandis known ascost-per-

landing.Quoting the B747 figures, the costfor the carbon-carbonbrakeis estimatedat

$10 per landing, while the cost for the wheel, including tire, is estimatedat $5 per

landing.

To conclude,dueto the competitionamongthe airframeandlanding gearmanufacturers,

landing geardesignprocedures, andweight andcostdataareconsideredto be company-

proprietary. As a result, the majority of the survey participants were only willing to

address the issues in general terms. However, the survey results did provide some useful insights to the design of the landing gear, and reaffirmed design and analysis procedures as previously documented.

Appendix B Structural Analysis Derivations

B. 1. Introduction Detailed derivations of selected structure analyses as introduced in Chapter Eight are compiled and presented in the following sections. The sections outline the procedures used to determine the internal actions for the nose assembly and the trunnion, both of which involve statically-indeterminate structures, as well as stress calculation for thin- walled circular tubes. These items involve derivations of the basic equations that cannot be presented in a concise manner within the main text.

B.2. The Nose Assembly The reactions at the supports of the truss that represents the nose gear cylinder/drag/side struts structure, and consequently the internal actions, can be determined by Castigliano's theorem [46, p. 611], that is, (B.1) where uj is the deflection at the point of application of the load P/, E is the modulus of elasticity, and l, F, and A are the length, internal force, and cross-sectional area of each member, respectively. The above theorem gives the generalized displacement corresponding to the redundant, Pj, which is set equal to a value compatible with the support condition. This permits the solution of the redundant, and consequently all remaining internal actions, via equilibrium.

As shown in Figure B.1, the port side strut* is designatedas redundantand

released from its support at point K. Using Eq. (B.1) the deflection at point K can be

written as FIolIo tgFlo + Fjoljo OFjo + FKOIKo OFKo (B.2) YK = A10E aR K Aj 0 _ aR K AKoE aR K F/o Cylinderj_ K side s S side strut Y 0s_d, _ x'S Cy O V-Fn,_ sin Oar_g Figure B. 1 Free-body diagram of the nose gear structure in the yz-plane From equilibrium, S (B.3a) FIO = RK + cOSOside (B.3b) Fjo = V - Fdrag sinO drag - ( FIo + RK ) sinO side (B.3c) FKO = R K " This is the strut on the right. You are looking aft in this figure.

where S and V are the applied side and vertical force, respectively, and Odrag and Oside are the angles between the axial centerlines of the drag and side struts** and the xy-plane, respectively. Differentiating Eqs (B.3a, b, and c) with respect to RK results in OFlo - 1 (B.4a) 3RK OFjo - sinOside (B.4b) 3RK °_FKo = 1 (B.4c)

aRK

To determine the reaction at point K and subsequently the internal force in each structural component, substitute the relationships of Eqs (B.3a, b, and c) and (B.4a, b, and c) back into Eq. (B.2), apply the no-deflection condition and then solve for Rx.

B.3. The Trunnion The trunnion model shown in Figure 8.9 is repeated here as Figure B.2. As shown in Figure B.2, the trunnion is subjected to a force with components Fx, Fy, and F_, and a couple with moment components Cy and Cz, at axial position x = Ii, where 0 < Ii < L and 0 < x < L. Clamped end-conditions at x = 0 and x = L yield ten homogeneous conditions, five at each end. At the load point x = Ii, there are five continuity conditions, i.e., u, v, w, v', and w', and five jump conditions corresponding point-wise equilibrium of the internal actions and the external loads. These twenty conditions are Ul(0) = u2(L) = 0 (B.5a) Vl(0) = Vl'(0) = vz(L) = vz'(L ) =0 (B.5b) "" A slight elaboration: the Oa,,s is the angle between the drag strut and the x-y plane. It, along with the drag strut, is not shown in Fig. B-1 because the attachment location of the drag strut is below the side strut attachment and the figure only represents the structural arrangement at a distance above that point.

(B.5c)

wl(O) = wl'(o)= wz(L) =.w2'(L)=0

(B.6a)

_ (ll)= U2(ll)

(B.6b)

Vlql)= v2(_1)

(B.6c) W1(11) = W2 (/1) dr1 (11) _ dv2 (/1) (B.6d) dx dr dwl (ll ) _ dw2 (ll ) (B.6e) dx dr (B.Va) -Nxl(l 1)+ Nx2(ll)+ Fx = 0 (B.7b) -Vyl (ll)+ Vy2(ll )+ Fy = O (B.7c) -Vzl(/1)+ Vz2(/1)+ Fz =0 (B.7d) -Mzl(ll)+ Mz2(II )+ Cz =0 (B.7e) -Myl(ll )+ My2ql)+ Cy =0 £ Figure B.2 Trunnion modeled as a clamped-clamped end bar

In the xz-plane, equilibrium gives

dMy

_- Vz = 0 (B.8) dr

and

dVz - 0 (B.9) dr Given the where My and V_ are the intemal moment and shear components, respectively.

equation of elastic curve as (B.10) where E is the modulus of elasticity and lyy is the second area moment about the y-axis, Eqs (B.8) and (B.9) become (B.11) and d4w (B.12)

yyl- l o

Integrating Eq. (B. 12) four times with respect to x results in w 1 =Clx3+C2x2+c3x+c 4 0_ x_ 11 (B.13a) 6 2 w 2= C5x3+C6x2+c7x+c 8 11 < x < L (B.13b) 6 2 To determine w 1 and w2 at either end of the trunnion, boundary conditions as given in Eqs (B.5b and c) and (B.6b, c, d, and e) were used to solve for the eight integration constants (c;) in Eqs (B.13a and b). Finally, substitute wl and w2 back into Eqs (B.10) and (B.11) and use the static boundary conditions as given in Eqs (B.7a, b, c, and d) to obtain the

intemal shearforce andbendingmoment, respectively.The sameprocedureis usedto

determine v(x) and the internal actions in the xy-plane.

In the longitudinal direction, equilibrium gives dN - 0 (B. 14) dr where N is the axial force. In addition, the material law gives du

N =EArn (B.15)

dr where A is the cross-sectional area. Since the axial force is spatially uniform, or piecewise constant, integrating Eq. (B. 15) once with respect to x results in u 1 = Nlx+c9 O<_x<_l 1 (B. 16a) EA W 2 = "'z X+Cl0 l 1 <x < L (B.16b) EA where the two integration constants c 9 and c10 are determined using the boundary conditions as given in Eq. (B.6a). Finally, substitute ul and u2 back into Eq. (B. 15) and sum the forces in the x direction at x = Ii to obtain the internal axial force.

B.4. Normal and Shear Stresses In a Thin-walled Tube The normal stresses induced on the structural members are determined by combining the effects of axial load and combined bending, while the shear stresses are determined by combining the effects of torsion and shear forces due to bending [47].

The normal stress ('r=,) due to combined axial force and bending moments is given as

N+MYz-MZ (B.17)

:A lyy

For a thin-walledcircular tube referredto polar coordinatesas shown in FigureB.3, the

principal centroidalsecond areamomentsaboutthe y- and z-axes are

(B.18) Iyy = Izz = _(rsin O)2trdO = zcr3 t where r is the mean radius and t is the wall thickness. Given the relationship of Eq.

(B. 18), Eq. (B. 17) becomes N 1 (B.19) V xx = --+ _r t (MA Y sinO - M z c°sO ) Differentiate Eq. (B. 19) with respect to 0 to get dTx_ 1 sin O)

(S.20)

"_ - -_t (MY cosO + M z and at the extremum, i.e., dz=/dO = 0, so that My tanO max =-_ (B.21a)

Mz

sinOma x = Mz (B.21b)

+

My (B.21c)

- +

Given the relationships ofEqs (B.21b and c), the extremum values of the bending normal stresses are determined using the expressions 1 2 _xx,bending(Omax)= - r--_t4M_ + M 2 (B.22a) and _ 1 2 Txx,bending(Oma x +7g)- _r t 4MY + M 2 (B.22b) Thus, the extremum values of the normal stress on a circular-tube cross section under combined axial and bending actions are N 1 2 (B.23) or rain Figure B.3 Annular section showing positive shear forces and bending moments The shear stress (zxs,) due to combined transverse shear force and torque is given as "cxs = q(s) + (.Cx s )torque (B.24) t where q is the shear flow due to bending. As indicated by in Figure B.4, the shear flow from some arbitrary origin to any point round the cross-section of a circular tube for axial equilibrium is dF - m (B.25) q=qo dx where dF d s 0 d.ca x (B.26) dx - dr '_0 zx_ tds = JO dx trdO F x y Figure B.4 Shear flow around a closed tube Given that dN = 0 (B.27) dx - Vz (B.28) dx and dMz dx - -Vy (B.29) rearrangement of Eq. (B.26) results in r dF _ r2 _ V z (B.30) Llyy (1-c°so)+ Vy sinO1 Izz ] From the relationships of Eqs (B.25) and (B.30), the integral of q round the cross section is (B.31) qrds = _ qords - _ _ rds Since only bending is considered in this case, the left-hand side of Eq. (B.31) is zero, that is, qo_dO= _d-_-._ dO (B.32) and the integration results in qo = r2t Vz (B.33) lyy Given the relationships of Eqs (B. 18), (B.30) and (B.33), the magnitude of the shear flow is determined using the expression (B.34) 1 E sin O) q=_rr(zC°S0-Vy Differentiating Eq. (B.34) with respect to 0 gives dodq - l(vycosO+ VzsinO) (B.35) and at the extremum, i.e., dq/dO = 0, so that tanO max = _ Vy (B.36a)

Vz

sin Omax - Vz (B.36b) 4Vy 2 +Vz 2

cosOmax = Vy (B.36c)

4Vy 2 +Vz 2

Given the relationships of Eqs (B.36b and c), the minimum and maximum values of the shear flow are determined using the expressions 1 2 (B.37a) q(Omax)=--_rt_JVy +Vz 2 and (B.37b) q(0 max + tr) = -_'4Vy2 + Vz2

The shearstressdueto torqueis given as

Zy ('C xs )torque = -7" (B.38)

where o r is the polar area moment

J = Sr3tdO = 2_rr3t (B.39) So, for the thin-wall approximation the maximum stresses will occur on the contour of the circular tube, consistent with bending analysis. Thus, given the relationships of Eqs (B.37a and b) and (B _9), Eq. (B.24) becomes "rxs - -- _Vy 2 + V 2 (B.40) - tcrtk,2r-

Appendix C Bibliography

This appendix provides a summary of the material used in doing the work. Not all of the citations included here were explicitly cited in the report, but contain material of interest to people studying the landing gear problem.

C.1. Textbooks Ashford, N., and Wright, P.H., Airport Engineering, 3rd Ed., John Wiley & Sons, Inc., New York, 1992.

Conway, H. G., Landing Gear Design, Chapman and Hall, London, 1958.

Currey, N. S., Aircraft Landing Gear Design: Principles and Practices, AIAA Education Series, Washington, 1988.

Horonjeff, R. and McKelvey, F. X., Planning and Design of Airports, McGraw-Hill, New York, 1994.

Kirschbaum, N., Aircraft Design Handbook." Aircraft Design Aid and Layout Guide, VPI Aircraft Design Series, Virginia, 1993.

Niu, M. C. Y., Airframe Structural Design, Conmilit Press, Hong Kong, 1988.

Raymer, D. P., Aircraft Design: A Conceptual Approach, AIAA Education Series, Washington, 1989.

Roskam, J., Airplane Design Part IV." Layout Design of Landing Gear and System, Roskam Aviation and Engineering, Ottawa, Kansas, 1986.

Roskam, J., Airplane Design Part V: Component Weight Estimation, Roskam Aviation and Engineering, Ottawa, Kansas, 1985.

Torenbeek, E., Synthesis of Subsonic Airplane Design, Delft University Press, Delft, 1981.

C.2. AGARD Reports AGARD CP 299: Subsystem Testing and Flight Testing Instrumentation, October 1980.

(N81-29077) AGARD CP 484: Landing Gear Design Loads, October 1990. (N91-28150), see in particular: • Kempf, G., and Haines, G.H., "Development of Undercarriage Design Loads," N91-28153.

• Krauss, A.J., "Considerations on Optimality of Landing Gear Arrangement and Design," N91-28156.

• Ladda, V., and Struck, H., "Operational Loads on Landing Gears," N91-28158.

C.3. Government Industry Standards FAR Part 25 Airworthiness Standards: Transport Category Airplanes, Federal Aviation Administration, Washington, DC, October 1994.

"Standardized Method of Reporting Airport Pavement Strength," AC 150/5335-5, Federal Aviation Administration, Washington, DC, June 1983.

Aerodrome Design Manual, Part 3: Pavements, International Civil Aviation Organization, Doc. 9157-AN/901, 1983.

1994 Aircraft Year Book, The Tire and Rim Association, Inc., 1994.

Special Design Charts for Concrete Airport Pavement, Portland Cement Association, 1993.

Packard, R. G., Design of Concrete Airport Pavement, Portland Cement Association, 1973.

The following aerospace recommended practices have been developed by the Society of Automotive Engineers (SAE) A-5 Aerospace Landing Gear System Committee: ARP 597 Wheels and Brakes, Supplementary Criteria for Design Endurance, Civil Transport Aircraft, April 1991.

ARP 1311 Aircraft Landing Gear, January 1995.

ARP 1494 Verification of Landing Gear Design Strength, February 1978.

ARP 1598 Landing Gear System Development Plan, August 1985.

ARP 1821 Aircraft Flotation Analysis Methods, May 1988.

ARP 1934 Use of Carbon Heat Sink Brakes on Aircraft, April 1990.

ARP 4243 Landing Area Landing Gear Compatibility, April 1993.

C. 4. Technical Papers Reports C.4.1 Landing Gear Design O'Massey, R. C., "Introduction to Landing Gear Design," ASM Paper W70-18.1, March 1970. (extensive data on shock absorber struts, somewhat lesser data on tire scrubbing and materials) Attri, N.S., and Amberg, R.L., "Advances in Landing Gear Systems," Paper No. 13, AGARD CP 299: Subsystem Testing and Flight Testing Instrumentation, October 1980 Young, D.W., "Aircraft Landing Gears - The Past, ,Present and Future," Institution of Mechanical Engineers, Proceedings, Part G, Journal of Aerospace Engineering, Vol. 200, 1986, pp. 179-196. (This is an excellent survey for designers to read to get an overview of the problem.)

Veaux, J., "New Design Procedures Applied to Landing Gear Development," Journal of Aircraft, Vol. 25, No. 10, October 1988, pp. 904-910 (This paper describes the use of CAD tools in the landing gear design process, and is not directly germane to the current project.)

Jenkins, S. F. N., "Landing Gear Design and Development," Institution of Mechanical Engineers, Proceedings, Part G, Journal of Aerospace Engineering, Vol. 203, 1989, pp.

67-73.

Tanner, J. A. Ed., Aircraft Landing Gear Systems, PT-37, SAE A-5 Landing Gear Systems Committee, 1990.

Greenbank, S. J., "Landing Gear: The Aircraft Requirement," Institution of Mechanical Engineers, Proceedings, Part G, Journal of Aerospace Engineering, Vol. 205, 1991, pp.

27-34.

C.4.2 Weight Estimation Wille, R. H., "Analytical Weight Estimation of Unconventional Landing Gear Design," SAWE Paper 1905, May 1989.

Wille, R.H., "Landing Gear Weight Optimization Using Taguchi Analysis," SA WE Journal, Winter 1990, pp. 31-40.

Kraus, P. R., "An Analytical Approach to Landing Gear Weight Estimation," SAWE Paper No. 829, May 1970.

Liebeck, R. H., Page, M. A., Rawdon, B. K., Scott, P. W., and Wright, R. A., "Concepts for Advanced Subsonic Transports," NASA CR-4624, September 1994.

Marsh, D. P., "Theory of Transport Aircraft Weight Fractions," SAWE Paper 1452, 1982.

Elsaie, A. M. and Santillan, R. Jr., "Structural Optimization of Landing Gear Using STARSTRUC," SAE Technical Paper 871047, 1987.

Saelman, B., "Designing Cylinders and Struts for Maximum Strength," Machine Design, Vol. 25, No. 8, August 1953, pp. 133-138 C.4. 3 Center of Gravity and Load Balancing Chai, S., Crisafulli, P., and Mason, W. H., "Aircraft Center of Gravity Estimation in Conceptual Design," AIAA Paper 95-3882, September 1995.

Davis, E., "Center of Gravity Envelope Development, 747-400F," SAWE Paper 2200, May 1994. (definitive explanation of 747 cg envelope) Field, G.G.,"MD-11 Design--Evolution not Revolution,"AIAA Paper87-2928,1987.

Containsdetailsof the DC-10 andMD-11 cg range and tail sizing X-plots. Interesting

description of the process.

Packing Problem References Samuel Eilon and Nicos Christofides, "The Loading Problem," Management Science, Vol. 17, No, 5, Jan. 1971, pp. 259-268 Samir V. Amiouny, John J. Bartholdi III, John H. Vande Vate and Jixian Zhang, "Balanced Loading," Operations Research, Vol. 40, No. 2, March-April 1992, pp.238- 246.

This paper has several algorithms for aircraft cargo loading and attainment of a certain balance range.

Kathryn A. Dowsland and William B. Dowsland, "Packing problems," Invited Review, European Journal of Operations Research, 56 (1992) 2-14.

Louis A. Martin-Vega, "Aircraft Load Planning and the Computer: Description and Review," Computers and Industrial Engineering, Vol. 9, No. 4, pp.357-369, 1985.

Douglas D. Cochard and Kirk A. Yost, "Improving Utilization of Air Force Cargo Aircraft," INTERFACES, 15: Jan.-Feb. 1985 pp. 53-68) H. Gehring, K. Menschner, and M. Meyer, "A computer-based heuristic for packing pooled shipment containers," European Journal of Operations Research, Vol. 44, (1990) pp. 277-288.

Robert W. Haessler and F. Brian Talbot, "Load planning for shipments of low density products," European Journal of Operations Research, Vol. 44 (1990) pp. 289-299.

Harold Dyckhoff, "A typology of cutting and packing problems," European Journal of Operations Research, Vol. 44, (1990) pp. 145-159.

C. 4.4 Pavement Flotation Dawson, J. L. and Mills, R. L., "Undercarriage Effects on (a) Rigid Pavements (b) Flexible Pavements," Aircraft Pavement Design, Institution of Civil Engineers, Proceedings, November 1970.

Bonney, K. V., "Aircraft Development in Relation to Pavement Design," Aircraft Pavement Design, Institution of Civil Engineers, Proceedings, November 1970.

Griffis, Jr., M. F. H. and Gamon, M. A., "Aircraft-Pavement Compatibility Study," Report FAA-RD-73-206, Waterways Experiment Station, US Army Corps of Engineers, Vicksburg, MS, September 1974.

Ahlvin, R. G. "Developinga Setof CBR DesignCurves,"InstructionReport4, US Army

Corpsof Engineers,WaterwaysExperimentStation,Vicksburg,MS, November1959.

Westergaard, H. M., "New formulasfor Stresses in ConcretePavements of Airfields,"

Transactions of American Society of Civil Engineers, Vol. 73, May 1947, pp. 687-701.

"Landing Gear," Aircraft Engineering, Vol. 58, No. 7, July 1986, pp. 3-6.

Pereira, A. T., "Procedures for development of CBR Design Curves," Information Report S-77-1, Waterways Experiment Station, US Army Corps of Engineers, Vicksburg, MS, June 1977.

Cameron-Johnson, A., "The Undercarriage in Aeroplane Project Design," Aircraft Engineering, Vol. XLI, No. 2, February, 1969, pp. 6-11 C.4.5 Cost Dean, E. B. and Unal, R., "Designing for Cost," AACE Transactions, 1991 C. 4. 6 Vehicle Integration Arcara, Jr., P. C., Bartlett, D. W., McGraw, Jr., M. E., and Geiselhart, K. A., "Technology Benefits for Very Large Subsonic Transports," AIAA Paper 93-1178, February 1993.

Holloway, R. B., Bums, P. M. and Johannes, R. P., "Aircraft Performance Benefits from Modern Control Systems Technology," Journal of Aircraft, Vol. 7, No. 6, November 1970, pp. 550-553.

Jayaram, S., Myklebust, A., and Gelhausen, P., "ACSYNT - A Standards-Based System for Parametric Computer Aided Conceptual Design of Aircraft," AIAA Paper 92-1268, February 1992.

McCullers, L. A., "Aircraft Configuration Optimization Including Optimized Flight Profiles", Proceedings of Symposium on Recent Experiences in Multidisciplinary Analysis and Optimization, 1984, pp. 395-412 (NASA CP-2327).

Sliwa, S. M., "Economic Evaluation of Flying-Qualities Design Criteria for a Transport Configured With Relaxed Static Stability," NASA Technical Paper 1760, December 1980.

Sliwa, S. M., "Impact of Longitudinal Flying Qualities Upon the Design of a Transport with Active Controls," AIAA Paper 80-1570, 1980.

Garner, P. L., Meredith, P. T., and Stoner, R. C., "Areas for Future CFD Development as Illustrated by Transport Aircraft Applications," AIAA Paper 91-1527, 1991. (contains explicit sensitivity values for the aircraft as a function of gear length) Eckels,W.E., "Civil TransportDesignMethodology,"AIAA Paper83-2463,Oct. 1983.

(connects geartradeoff studiesto thetotal system).

Baker, W. R. andGonzalez,C. R., "Design Considerations for Multi-Wheel Aircraft", 22nd ASCE IntemationalAir TransportationConference, Proceedings, June1992.

C.5. Related Articles (arranged in chronological order) Brahney, J.A., "Toward the all-electric airplane: electric brakes," Aerospace Engineering, Jan. 1989, pp. 17-20.

Smith, B.A., "Decisions to Build the New MD-12X Wing Based on Future Needs," Aviation Week and Space Technology, Oct. 8, 1990, pp. 28-29.

"Aircraft Tires: Bias or Radials?" Aerospace Engineering, Vol. 11, No. 9, September 1991, pp. 13 -14.

"Technical Advances in Tyres, Wheels and Brakes: Dunlop Keeps Britain in the Forefront," Aircraft Engineering, Vol., 59, No. 11, November 1987, pp. 2-5.

Proctor, P., "Super-Jumbos Pose Design Challenges," Aviation Week and Space Technology, Feb. 21, 1994, pg. 70.

Mecham, M., "Airport Officials: Superjumbos Mean New Headaches," Aviation Week andSpace Technology, Vol. 141, No. 21, November 21, 1994, pp. 76-80.

Mecham, M. and McKenna, J. T., "Cost, Not Size, to Drive Success of Super Jumbo," Aviation Week & Space Technology, Vol. 141, No. 21, November 21, 1994, pp. 45-46.

Proctor, P., "Boeing Ref'mes Designs for 600-Seat NLA," Aviation Week & Space Technology, Vol. 141, No. 21, November 21, 1994, pp. 48-53.

Proctor, P., "Boeing Presses Need for Multinational VLCT," Aviation Week & Space Technology, Vol. 141, No. 21, November 21, 1994, p. 75.

Shifrin, C. A., "New Jumbos, SSTs Face Tough Hurdles," Aviation Week and Space Technology, Vol. 141, No. 21, November 21, 1994, pp. 42-43.

Smith, B. A., "Douglas Awaits Chance to Revive Jumbo Plans," Aviation Week & Space Technology, Vol. 141, No. 21, November 21, 1994, p. 57.

Sparaco, P., "Airbus Weights Four A3XX Versions," Aviation Week & Space Technology, Vol. 141, No. 21, November 21, 1994, p. 54.

Kandebo, S. W., and Dornheim, M. A., "Operability Verified In GE90 Flight Tests," Aviation Week andSpace Technology, Vol. 142, No. 13, March 27, 1995, pp. 52-54 Sparaco, P., "Airbus Paves Way for A3XX Megajet," Aviation Week & Space Technology, Vol. 142, No. 21, May 22, 1995, pp. 26-27.

Proctor,P., "Boeing HomesIn on Future747Design," Aviation Week and Space

Technology, Vol. 144, No. 6, February 5, 1996, pp. 32-33.

Fiornio, F., Comp., "Too Big For Your Britches," Aviation Week and Space Technology, Vol. 144, No. 21, May 20, 1996, p. 19.

Sparaco, P., "No Money As Yet In Airbus Jumbo Plan," Aviation Week and Space Technology, Vol. 144, No. 25, June 17, 1996, pp. 27-28.

Proctor, P., "Boeing Poised to Offer Stretched 747 Versions," Aviation Week and Space Technology, Vol. 144, No. 27, July 1, 1996, pp. 20-21.

Smith, B., "Douglas Explores Design Options for Large, Long-Range MD-XXs," Aviation Week and Space Technology, Vol. 144, No. 27, July 1, 1996, pp. 43-44.

Sparaco, P., "Market Upturn Spurs Airbus To Plan New Aircraft Versions," Aviation Week and Space Technology, Vol. 144, No. 27, July 1, 1996, pg. 45.

C. 6. Aircraft Data Aircraft Type Certification Data Sheets and Specification, Vol. 3: Large Multiengine Aircraft, Federal Aviation Administration, Washington, DC, 1985.

Aircraft Data for Pavement Design, American Concrete Pavement Association, 1993.

Special Design Chart for Concrete Airport Pavement: Boeing 727, Portland Cement Association, 1994.

Airplane Characteristics for Airport Planning, 73 7-300, 73 7-400, 73 7-500, Boeing Document D6-58325-2, Boeing Commercial Airplanes, Seattle, Washington, September 1988.

Airplane Characteristics for Airport Planning, 747-400, Boeing Document D6-58326-1, Boeing Commercial Airplane Company, Seattle, Washington, March 1990.

Airplane Characteristics for Airport Planning, 777, Boeing Document D6-58329, Boeing Commercial Airplane Company, Seattle, Washington, February 1992.

Boeing B 73 7-130 Weight and Balance Control and Loading Manual, Boeing Commercial Airplane Company, Seattle, Washington, 1985.

Boeing Group Weight Statements, Boeing Commercial Airplane Company, Seattle, Washington.

Anon., MD-80 Component Weight Breakdown, McDonnell Douglas, 1987.

Green, W., Swanborough, G., and Mowinski, J., Modern Commercial Aircraft, Portland House, New York, 1978.

Lambert,M., Ed., Jane's All The World's Aircraft, Sentinel House, United Kingdom, 1972-1994.

C. 7 Simple Landing Gear Dynamics Models for Insight.

Fliigge, W., "Landing-Gear Impact," NACA TN 2743, October 1952. This is a classic for insight. It provides a simple dynamic model of the landing gear as a spring-mass-damper.

I actually think the equations are a tad oversimplified since today they would be solved using computational methods. An example analysis of the metering pin is included. The report also includes the analysis of the landing gear hitting a step in the pavement.

Curiously, the design community does not use this first principals analysis to do the design. Probably the analysis is not complete enough for actual design.

Abramson, H. Norman, An Introduction to the Dynamics of Airplanes, Dover Publications, New York, 1971. pp. 134-139. This is the book that led me to the Flfigge reference. Some other references are also cited.

von Mises, Richard, Theory of Flight, Dover Publications, New York, 1959, pp. 483-488.

This is another, slightly different, basic analysis of the landing impact problem.

C.8 Historical (pre 1970) and Miscellaneous Liming, R. A., "Analytic Definition of a Retractable Landing Gear Axis of Rotation," Journal of the Aeronautical Sciences, January 1947, pp. 19-23.

McBrearty, J. F., "A Critical Study of Aircraft Landing Gears," Journal of the Aeronautical Sciences, 16th IAS Annual Meeting, Proceedings, May 1948, pp. 263-280.

McBrearty, J. F. and Hill, D. C., "Landing Gear Strength Envelopes," Journal of the Aeronautical Sciences, IAS Los Angeles Meeting, Proceedings, May 1948, pp. 229-234.

Russell, A.G., "Some Factors Affecting Large Transport Aeroplanes with Turboprop Engines," Journal of Aeronautical Sciences, Vol. 17, No. 2, Feb. 1950, pp. 67-122.

Carter, K.S., "The Landing Gear of the Lockheed SST," SAE Paper 650224, 1965.

Stanton, G., "New Design for Commercial Aircraft Wheels and Brakes," AIAA Paper 67- 104, June 1967.

Ridha, R. A., "Minimum Weight Design of Aircraft Landing Gear Reinforcement Rings," AIAA Paper 68-328, March 1968.

Collins, R. L. and Black, R. J., "Tire Parameters for Landing Gear Shimmy Studies," AIAA Paper 68-311, April 1968.

Firebaugh, J. M., "Estimation of Taxi Load Exceedances Using Power Spectral Methods," Journal of Aircraft, Vol. 5 No. 5, September 1968, pp. 507-509.

Appendix D Aircraft Tire Database

This appendix contains the information on the tires and wheels required in the landing gear analysis. Table D. 1 contains the tire information, and Table D.2 contains the wheel information.

Table D. 1, Aircraft tire data, contains the following for 100 tires: (from Ref. 23.)

1. Item number 2.

Size (Outside diameter x section width -- rim diameter) 3.

Ply. This is an index of the tire strength, and does not necessarily represent the number of cord plies in the tire.

4. Speed in mph. The maximum speed to which the tire is qualified 5. Load in pounds. The maximum load for the ply rating of the tire.

6. Max braking in pounds. The maximum steady braking which may be applied to a tire during landing.

7. Inflation pressure in psf. This is the inflation pressure required to support the rated load.

8. Tire weight in pounds. This is the calculated weight of approved construction, not the maximum weight.

9. Maximum inflated outside diameter in inches 10. Maximum inflated width in inches 11. Aspect ratio, the ratio of the tire section height to the tire section width Table D.2, Aircraft Wheel data, contains the following for 100 wheels: 1. Item number 2. Size 3. Width in inches 4. Diameter in inches m_ G_ .... ° ° ° ° ° ° ° ° ° ° ° ° ...... 0 ° ° ° =_ 0m 0m o < oooooooooooooooooooooooooo I | ! I I I I I ! | I I I I I I I I I I I I i I N _,=., _ _ _ N _ _ _ _ _ • _ _ _ N _ N N N _ N _ N N _ N N

e,,lc,l _

.... ° .....

.... ° ° ° t< t< t--: t< r< t< t< t< ..................

t"q t.,,I ¢",1 ¢"4 e'4 ¢',1 ¢",1 m oo oo o_ oo oo oo oo oo oo oo oo oo o_ oo oo oo oo o_ oo t",_ _ _ _ _ _

?

ooooiiiiiiiiiiiiiiiiiiiiii

on N

o._

,,,.q _ o6 o6 o6 o6 o6 o_ oo _ oe oe oe o_ _',

il °

.N r._

TableD.2 Aircraft wheeldata[23]

Item Size Width Dia. Item Size Width Dia.

(in) (in) (in) (in) 1 32x11.5-15 9.0 15.0 27 24x7.7 5.5 10.0 2 32x11.5-15 9.0 15.0 28 24x7.7 10.0 5.5 3 32x11.5-15 9.0 15.0 29 24x7.7 5.5 10.0 4 36xli 9.0 16.0 30 24x7.7 5.5 10.0 5 37x14.0-14 11.0 14.0 31 24x7.7 5.5 10.0 6 37x14.0-14 11.0 14.0 32 24x7.7 5.5 10.0 7 40x15.5-16 10.0 16.0 33 24x7.7 5.5 10.0 8 40x15.5-16 10.0 16.0 34 24x7.7 5.5 10.0 9 44x16 13.3 18.0 35 34x9.9 8.0 16.0 10 44x16 13.3 18.0 36 34x9.9 8.0 16.0 11 49x17 13.3 20.0 37 36xll 9.0 16.0 12 49x17 13.3 20.0 38 36xl 1 9.0 16.0 13 50x20.0-20 16.3 20.0 39 36xl 1 9.0 16.0 14 50x20.0-20 16.3 20.0 40 39x13 10.0 16.0 15 50x20.0-20 16.3 20.0 41 39x13 10.0 16.0 16 50x20.0-20 16.3 20.0 42 39x13 10.0 16.0 17 50x20.0-20 16.3 20.0 43 39x13 10.0 16.0 18 50x20.0-20 16.3 20.0 44 39x13 10.0 16.0 19 50x20.5-23 13.0 23.0 45 39x13 10.0 16.0 20 50x20.5-23 13.0 23.0 46 39x13 10.0 16.0 21 50x20.5-23 13.0 23.0 47 39x13 10.0 16.0 22 50x20.5-23 13.0 23.0 48 39x13 10.0 16.0 23 50x20.5-23 13.0 23.0 49 39x13 10.0 16.0 24 50x20.5-23 13.0 23.0 50 39x13 10.0 16.0 25 49x17 13.3 20.0 51 39x13 10.0 16.0 26 49x17 13.3 20.0 52 39x13 10.0 16.0 IIIIIINN I I I NIIIUIIII I I Table D.2 Aircraft wheel data [23] (cont'd) Dia.

Width Width Dia. Item Size Item Size (in) (in) (in) (in) 20.0 13.3 10.0 16.0 77 46x16 53 39x13 20.0 13.3 10.0 16.0 78 46x16 54 39x13 20.0 13.3 10.0 16.0 79 46x16 55 39x13 13.3 20.0 56 39x13 10.0 16.0 80 46x16 13.3 20.0 40x14 11.0 16.0 81 46x16 20.0 13.3 58 40x14 11.0 16.0 82 46x16 13.3 20.0 40x14 11.0 16.0 83 46x16 20.0 13.3 60 40x14 11.0 16.0 84 46x16 20.0 13.3 40x14 11.0 16.0 85 46x16 20.0 13.3 40x14 11.0 16.0 86 46x16 20.0 13.3 11.0 16.0 87 49x17 63 40x14 13.3 20.0 11.0 16.0 88 49x17 64 40x14 13.3 20.0 65 40x14 11.0 16.0 89 49x17 13.3 20.0 66 40x14 11.0 16.0 90 49x17 20.0 13.3 40x14 11.0 16.0 91 49x17 20.0 13.3 40x14 11.0 16.0 92 49x17 13.3 20.0 40x14 11.0 16.0 93 49x17 13.3 20.0 70 40x14 11.0 16.0 94 49x17 13.3 20.0 71 40x14 11.0 16.0 95 49x17 13.3 20.0 11.0 16.0 96 49x17 72 40x14 13.3 20.0 13.3 20.0 97 49x17 73 46x16 13.3 20.0 13.3 20.0 98 49x17 74 46x16 20.0 13.3 13.3 20.0 99 49x17 75 46x16 20.0 14.3 13.3 20.0 100 50x18 76 46x16

Appendix E Analysis Package User's Manual

E. 1. Introduction

The package is intended to provide aircraft conceptual designers with tools to help

automate the landing gear design process.

E.2. Package Organization

The package consists of four executable files, configfor, limit.for, pave.for, and

gearwei.for. An input file with extension .inp is required for each program. Program

config.for currently acts as the front-end of the package and accepts all the data that is

input, even though some of the data may not be used by the program itself. Program

config.for then creates input files for the other three programs. Data files tire.dat and

pavecoefdat are required to provide database for programs config.for and pave.for, respectively.

The first line in each input file is a blank card, to be used as a case title card. All the rest of the input is formatted. We suggest that the sample input files be used as templates.

Typically, the character data is read in as alphanumeric format, the integer data is read in

as 3(10x, il0), and real data is read in as 3(10x, f10.2). The fields that are skipped are

intended for variable labels. Note that if the given aircraft does not exhibit a fuselage-

mounted landing gear, zeros should be entered in place of those input variables that are

related to the fuselage-mounted gear.

The codes produce minimal screen output, and do not write out anything until they

enter subroutine output, at the end of the computation. Config does provide some write

statements, to provide an indication of the progress during the calculation. If there are

problems with input data sets it will be at least slightly difficult to troubleshoot. The input

and output files names are hardwired, but could easily be changed to prompt the user for

file names. The most painful input appears to be the stowage volume definition. This input

in config can be fictitious and the program will still execute. Note that three sets of stowage

volumes are read in, whether a fuselage mounted main assembly is used or not. As

mentioned above, zeros should be entered when a fuselage mounted gear is not present. If

you don't input all three, the code will fail, giving an out of data error.

Process

Required aircraft/landing gear characteristics are arranged into card-style input file

"config.inp" to be read in by program "config.for". Selected tire/wheel characteristics and landing gear model, as well as initial data are arranged to form "limit.inp", "pave.inp", and "gearwei.inp" to be read in by "limit.for", "pave.for", and "gearwei.for", respectively.

Examination of the list of constraint violations as generated by "limit.for", e.g., sideways

turnover angle, takeoff rotation angle, turning radius, and stowage characteristics, and

pavement thickness requirement and ACN as generated by "pave.for" will provide insight

to what should be done to resolve these constraint violations. Some possible options

include relocating the landing gear, extending the strut length, modifying the aircraft cg

height off the ground, and an increase/decrease of clearance requirements. After all of the

design constraints are satisfied through an iterative modification process, the finalized

landing gear model is passed to "gearwei.for" for component/group weight estimation.

The following sections define the subroutines and calling tree for each program.

The program input and output is also summarized. Details are contained in Chapter 9.

config.for

Subroutines

datain

brsize

- brake sizing

select - tire/wheel selection

attach

- landing gear attachment scheme

pivaxi - pivot axis alignment

cropro - cross-product

output

Subroutine calling sequence:

datain

brsize

select

attach

pivaxi

cropro

output

Outputs:

Brake dimension and weight

Tire/wheel design characteristics: dimensions and weight

Stroke length

Load-stroke curve

Mathematical landing gear model: axle, truck beam, piston, cylinder, trunnion, drag

and side struts

limit.for

Subroutines

datain

layout - landing gear positioning constraints

runway - ground operation characteristics

stowag - stowage constraints

skewed

- skewed pivot axis alignment

output

Subroutine calling sequence

datain

layout

stowag

pivaxi

skewed

cropro

retrac

violat

output

Outputs

Takeoff/landing stability characteristics: pitch and roll angles

Ground stability characteristics: sideways turnover and tail-tipping

Ground clearance: nacelle-to-ground and wingtip-to-ground

Maneuverability characteristics: centerline-guidance tracking

and minimum turning radius

Kinematics: pivot axis alignment, retraction angle, landing gear retracted position

pave.for

Subroutines

datain

offset

- offset distance, analysis node to tire contact area

aceswl

- equivalent single wheel load

rigith

- rigid pavement thickness and ACN

flexth

- flexible pavement thickness and ACN

output

Subroutine calling sequence

dataJn

offset

aceswl

rigith

flexth

output

Ou_u_

Flexible and rigid pavement thickness and corresponding ACN

gearwei.for

This program computes an estimate of the landing gear weight.

Subroutines

datain

exload

- applied load

noreac

- structural nodal actions

crosec

- cross-sectional area sizing

weiest

- weight estimation

cropro - cross-product

cotran

- coordinate transformation

matinv

- matrix inverse

rowpiv

- row pivoting

ccross

- cylindrical cross section sizing

cirstr

- circular tube stresses

icross

- i-bar cross section sizing

select

- design cross section selection

output

Subroutine calling sequence

datain

exload

cropro

noreac

cropro

crosec

cotran

matinv

rowpiv

ccross

cirstr

icross

select

weiest

output

Outputs

Component dimensions

Component/group weight estimation

E.3.

Program Input Variables

aircraft Aircraft identification

brake Brake material

1 steel

2 carbon

wheel Wheel material

1 forged aluminum

2 cast aluminum

3 titanium

4 steel

Wheel selection criterion

objec

1 minimum pressure

2 minimum weight

3 minimum size

metal

Landing gear structure material

1 4340 steel

2 300M steel

mtow

MTOW, lb

mldw

Maximum landing weight, lb

fuel

Fuel weight, lb

ClTIaX Maximum main assembly load, percent MTOW

cmin

Minimum main assembly load, percent MTOW

warea Wing area, ft z

Wing span, in

wspan

Quarter chord sweep, deg

qswep

dihed

Dihedral, deg

croot Root chord, in

taper Taper ratio

clmax

Clmax, landing

nnls Number of main struts

Number of main wheels

nnlw

Number of nose wheels

nnw

Number of wheels per strut, main assembly

wpsm

Number of wheels per strut, nose assembly

wpsn

dyna Landing gear load factor

alpha Angle of attack, touchdown, deg

wbeta

Truck beam rotation angle, wing-mounted assembly, deg

fbeta

Truck beam rotation angle, fuselage-mounted assembly, deg

incl

Axle incline from the vertical, deg

Tail scrape angle, deg

scrap

dnace

Nacelle diameter, in

clear

Nacelle-to-ground clearance, in

Aircraft cg location, aircraft reference frame, in

cg(i)

wing(i) Wing root leading edge location, aircraft reference frame, in

i = 1.... 3 (x, y, and z coordinate, airframe)

engi(i) Inboard engine location, aircraft reference frame, in

i = 1.... 3 (x, y, and z coordinate, airframe)

tcon(i) Tail bumper location, aircraft reference frame, in

i = 1.... 3 (x, y, and z coordinate, airframe)

gear(i,j) Landing gear assembly location, aircraft reference frame, in

in the order: main, nose, body

well(id,k) Landing gear stowage volume, aircraft reference frame, in

in the ordermain, nose, body

Thenumberat the endof thevariabledenotes the comersof

therectangular-shaped stowage volume:

upperstarboard comer,forward

upper port comer, forward

lower starboard comer,forward

upper port comer, forward

upper starboard comer,aft

upper port comer, aft

lower starboard comer,aft

upper port comer, aft

E. 4. Sample Input Files

747conf inp

c landing gear layout/configuration input file aircraft: b747 brake = i, wheel = I, objec = 2 metal = 1 mtow = 738000.00, mldw = 564000.00, fuel = 316307.00 cmax = 0.96, cmin = 0.88, warea = 5500.00 wspan = 2348.00, qswep = 37.70, dihed = 7.00 croot = 642.00, taper = 0.25, clmax = 2.55 nms= 4.00, nmw = 16.00, nnw = 2.00 wpsm = 4.00, wpsn = 2.00, dyna = 1.20 alpha = 4.00, wbeta = 60.00, fbeta = 0.00 incl = 10.00, scrap = 12.00, dnace = 110.00 clear = 12.00 c component location xcg = 1260.00, ycg = 0.00, zcg = -24.00 xwing = 870.00, ywing = 0.00, zwing = -88.00 xengi = 1050.00, yengi = -465.00, zengi = -95.00 xtcon = 2375.00, ytcon = 0.00, ztcon = 0.00 xmain = 1254.00, ymain = -216.00, zmain = -62.00 xnose = 290.00, ynose = 0.00, znose = -106.00 xbody = 1375.00, ybody = -75.00, zbody = -118.00 c wing-mounted main assembly stowage -38.00 -17.00, zml = _nl = 1164 00, yml = -38.00 -115.00, zm2 = xm2 = 1164 00, ym2 = -136.00 -115.00, zm3 = _3 = 1164 O0 ym3 = -136.00 -17.00, zm4 = x_4 = 1164 O0 ym4 = -38.00 -17.00, zm5 = _n5 = 1260 O0 ym5 = -38.00 -115.00, zm6 = xm6 = 1260 00 ym6 = -136.00 -115.00, zm7 = xm7 = 1260.00 ym7 = -136.00 -17.00, zm8 = xm8 = 1260.00 ym8 = c nose assembly stowage -38.00 32.00, znl = xnl = 150.00 ynl = -38.00 -32.00, zn2 = xn2 = 150.00 yn2 = -88.00 -32.00, zn3 = xn3 = 150.00 yn3 = -88.00 32.00, zn4 = xn4 = 150.00 yn4 = -38.00 32.00, zn5 = xn5 = 290.00 yn5 = -38.00 -32.00, zn6 = xn6 = 290.00 yn6 = -i12.00 -32.00, zn7 = xn7 = 290.00, yn7 = -112.00 32.00, zn8 = xn8 = 290.00, yn8 = c fuselage-mounted main assembly stowage -38.00 xbl = 1260.00, ybl = -8.00, zbl = -38.00 xb2 = 1260.00, yb2 = -115.00, zb2 = -136.00 xb3 = 1200.00, yb3 = -115.00, zb3 = -136.00 xb4 = 1200.00, yb4 = -8.00, zb4 = -38.00 xb5 = 1390.00, yb5 = -8.00, zb5 = -38.00 xb6 = 1390.00, yb6 = -115.00, zb6 = -136.00 xb7 = 1390.00, yb7 = -115.00, zb7 = -136.00 xb8 = 1390.00, yb8 = -8.00, zb8 =

74 7lim i. inp

c landing gear layout/stowage constraints input file aircraft: b747 181.00 cmax = 0.96, cmin = 0.88, hcg = 7.00 wspan = 2348.00, qswep = 37.70, dihed = croot = 642.00, taper = 0.25 2.00 nms= 4.00, wpsm = 4.00, wpsn = 12.00 scrap = 12.00, dnace = 110.00, clear = 6.00 wbeta = 60.00, fbeta = 0.00, incl = 27.65 smain = 27.65, snose = 27.97, sfuse = c component location -24 O0 xcg = 1260.00, ycg = 0.00, zcg = -88 O0 xwing = 870.00, ywing = 0.00, zwing = -95 O0 xengi = 1050.00, yengi = -465.00, zengi = 0 O0 xtcon = 2375.00, ytcon = 0.00, ztcon = -64 O0 xmain = 1253.50, ymain = -215.00, zmain = -106 O0 xnose = 290.00, ynose = 0.00, znose = -118 O0 xfuse = 1375.00, yfuse = -75.00, zfuse = c wing-mounted main assembly stowage -38 O0 xml = -17.00, zml = 1164.00 yml = -38 O0 >un2 = -132.00, zm2 = 1164.00 ym2 = -136 O0 ma3 = -132.00, zm3 = 1164.00 ym3 = -136 O0 >un4 = -17.00, zm4 = 1164.00 ym4 = -38 O0 xm5 = 1260.00 ym5 = -17.00, zm5 = xm6 = -38 O0 1260.00 ym6 = -132.00, zm6 = _n7 = -136 O0 1260.00 ym7 = -132.00, zm7 = -136 O0 xm8 = 1260.00 ym8 = -17.00, zm8 = c nose assembly stowage -38 O0 xnl = 150.00 ynl = 32.00, znl = -38 O0 xn2 = -32.00, zn2 = 150.00 yn2 = -88 O0 xn3 = 150.00 yn3 = -32.00, zn3 = -88 O0 xn4 = 32.00, zn4 = 150.00 yn4 = -38 O0 xn5 = 290.00 yn5 = 32.00, zn5 = -38 00 xn6 = -32.00, zn6 = 290.00 yn6 = -112 00 xn7 = -32.00, zn7 = 290.00, yn7 = -112 O0 xn8 = 290.00, yn8 = 32.00, zn8 = c fuselage-mounted main assembly stowage -30.00 xfl = 1260.00, yfl = -17.00, zfl = -30.00 xf2 = 1260.00, yf2 = -115.00, zf2 = -136.00 -115 00, zf3 = xf3 = 1200.00, yf3 = -136.00 -17 00, zf4 = xf4 = 1200.00, yf4 = -30.00 -17 00, zf5 = xf5 = 1390.00, yf5 = -30.00 -115 00, zf6 = xf6 = 1390.00, yf6 = -136.00 -115 00, zf7 = xf7 = 1390.00, yf7 = -136.00 -17 00, zf8 = xf8 = 1390.00, yf8 = c selected tire data criterion: minimum weight type size ply speed load infl brake wei dia wid (mph) (ib) (psi) (ib) (ib) (in) (in) wing 49x17 32.0 235.0 50400.0 210.0 75600.0 243.3 48.8 17.3 nose 46x16 28.0 210.0 41800.0 210.0 62700.0 185.8 45.3 16.0 fuselage 49x17 32.0 235.0 50400.0 210.0 75600.0 243.3 48.8 17.3 c selected wheel data material: aluminum, forging type size dia wid hub wei (in) (in) (in) (ib) wing 49x17 13.3 20.0 I0.0 86.2 nose 46x16 13.3 20.0 10.0 105.3 fuselage 49x17 13.3 20.0 i0.0 86.2 c mathmatical model wing component xO yO zO xl yl zl (in) (in) (in) (in) (in) (in) tire 0.00 0.00 0.00 0.00 13.25 -20.20 axle 0.00 22.00 0.00 0.00 -22.00 0.00 truck beam -29.00 0.00 0.00 29.00 0.00 0.00 piston 0.00 0.00 0.00 0.00 0.00 -50.00 cylinder 0.00 0.00 0.00 0.00 0.00 -112.00 drag strut 0.00 0.00 0.00 -42.00 -4.00 -101.00 side strut 0.00 0.00 0.00 0.00 -84.00 -88.00 forward trunnion 0.00 0.00 0.00 36.00 12.00 0.00 aft trunnion 0.00 0.00 0.00 12.00 12.00 0.00 nose component xO yO zO xl yl zl (in) (in) (in) (in) (in) (in) tire 0.00 0.00 0.00 0.00 13.25 -20 20 axle 0.00 18.00 0.00 0.00 -18.00 0 O0 truck beam 0.00 0.00 0.00 0.00 0.00 0 00 piston 0.00 0.00 0.00 0.00 0.00 -34 O0 cylinder 0.00 0.00 0.00 0.00 0.00 -75 60 drag strut 0.00 0.00 0.00 -32.40 0.00 -88 O0 side strut 0.00 0.00 0.00 0.00 18.00 -32.40 forward trunnion 0.00 0.00 0.00 0.00 -18.00 0.00 aft trunnion 0.00 0.00 0.00 0.00 18.00 0.00 fuselage component xO yO zO zl xl yl (in) (in) (in) (in) (in) (in) tire 0.00 0.00 000 0.00 13.25 -20.20 axle 0.00 22.00 0.00 0.00 -22.00 0.00 truck beam -29.00 0.00 0.00 29.00 0.00 0 O0 piston 0.00 0.00 0.00 0.00 0.00 -50 00 cylinder 0.00 0.00 0.00 0.00 0.00 -60 O0 drag strut 0.00 0.00 0.00 92.00 -56.00 -40 O0 side strut 0.00 0.00 0.00 0.00 -48.00 -36 O0 forward trunnion 0.00 0.00 0.00 0.00 56.00 0 O0 aft trunnion 0.00 0.00 0.00 0.00 -8.00 0 O0

747pave.inp

c aircraft flotation input file aircraft: b747 mtow = 738000.00, mldw = 564000.00 cmax = 0.96 nmw = 16.00, wpsm = 4.00 c selected tire data criterion: minimum weight type size ply speed load infl brake wei dia wid (mph) (ib) (psi) (ib) (ib) (in) (in) wing 49x17 32.0 235.0 50400.0 210.0 75600.0 243.3 48.8 17.3 nose 46x16 28.0 210.0 41800.0 210.0 62700.0 185.8 45.3 16.0 fuselage 49x17 32.0 235.0 50400.0 210.0 75600.0 243.3 48.8 17.3 c selected wheel data material: aluminum, forging type size dia wid hub wei (in) (in) (in) (ib) wing 49x17 13.3 20.0 I0.0 86.2 nose 46x16 13.3 20.0 i0.0 105.3 fuselage 49x17 13.3 20.0 i0.0 86.2 c mathmatical model wing component xO yO zO xl yl zl (in) (in) (in) (in) (in) (in) 13.25 -20.20 tire 0.00 0.00 0.00 0.00 axle 0.00 22.00 0.00 0.00 -22.00 0.00 0.00 0.00 truck beam -29.00 0.00 0.00 29.00 0.00 -50.00 piston 0.00 0.00 0.00 0.00 0.00 -112.00 cylinder 0.00 0.00 0.00 0.00 4.00 -I01.00 drag strut 0.00 0.00 0.00 42.00 -84.00 -88.00 side strut 0.00 0.00 0.00 0.00 forward trunnion 0.00 0.00 0.00 16.00 4.00 0.00 aft trunnion 16.00 4.00 0.00 56.00 18.00 0.00 nose component xO yO zO xl yl zl (in) (in) (in) (in) (in) (in) tire 0.00 0.00 0.00 0.00 13.25 -20.20 axle 0.00 17.00 0.00 0.00 -17.00 0.00 truck beam 0.00 0.00 0.00 0.00 0.00 0.00 piston 0.00 0.00 0.00 0.00 0.00 -34.00 cylinder 0.00 0.00 0.00 0.00 0.00 -78.00 drag strut 0.00 0.00 0.00 41.00 0.00 -82.00 side strut 0.00 0.00 0.00 0.00 19.00 38.00 forward trunnion 0.00 24.00 0.00 0.00 0.00 0.00 aft trunnion 0.00 0.00 0.00 0.00 -24.00 0.00 fuselage component xO yO zO xl yl zl (in) (in) (in) (in) (in) (in) tire 0.00 0.00 0.00 0.00 13.25 -20.20 axle 0.00 22.00 0.00 0.00 -22.00 0.00

17!

truck beam -29.00 0.00 0.00 29.00 0.00 0.00 piston 0.00 0.00 0.00 0.00 0.00 -50.00 0.00 0.00 0.00 0.00 0.00 -64.00 cylinder 0.00 0.00 0.00 84.00 -56.00 -40.00 drag strut side strut 0.00 0.00 0.00 0.00 -48.00 -36.00 forward trunnion 0.00 0.00 0.00 0.00 -62.00 0.00 aft trunnion 0.00 -62.00 0.00 0.00 -72.00 0.00

74 7weig. inp

c landing gear weight estimation input file aircraft: b747 metal = 1 564000.00 mtow = 738000.00, mldw = cmax = 0.96, cmin = 0.88, hcg = 181.00 16.00, nnw = 2.00 nnas = 4.00, rumw = wpsm = 4.00, wpsn = 2.00, alpha = 4.00 dyna = 1.20, inpr = 1500.00, bwei = 262.11 26.97, sfuse = 26.65 smain = 26.65, snose = c component location -216.00, zmain = -62.00 xmain = 1254.00, ymain = 0.00, znose = -106.00 xnose = 290.00, ynose = -75.00, zfuse = -118.00 xfuse = 1375.00, yfuse = c selected tire data criterion: minimum weight type size ply speed load infl brake wei dia wid (mph) (ib) (psi) (ib) (ib) (in) (in) wing 49x17 32.0 235.0 50400.0 210.0 75600.0 243.3 48.8 17.3 nose 46x16 28.0 210.0 41800.0 210.0 62700.0 185.8 45.3 16.0 fuselage 49x17 32.0 235.0 50400.0 210.0 75600.0 243.3 48.8 17.3 c selected wheel data material: aluminum, forging type size dia wid hub wei (in) (in) (in) (ib) wing 49x17 13.3 20.0 10.0 86.2 nose 46x16 13.3 20.0 i0.0 105.3 fuselage 49x17 13.3 20.0 i0.0 86.2 c mathmatical model wing yl zl component xO yO zO xl (in) (in) (in) (in) (in) (in) 13.25 -20.20 tire 0.00 0.00 0.00 0.00 -22.00 0.00 axle 0.00 22.00 0.00 0.00 truck beam -29.00 0.00 0.00 29.00 0.00 0.00 0.00 -50.00 piston 0.00 0.00 0.00 0.00 0.00 -112.00 cylinder 0.00 0.00 0.00 0.00 4.00 -101.00 drag strut 0.00 0.00 0.00 42.00 -84.00 -88.00 side strut 0.00 0.00 0.00 0.00 4.00 0.00 forward trunnion 0.00 0.00 0.00 16.00 18.00 0.00 aft trunnion 16.00 4.00 0.00 56.00 nose component xO yO zO xl yl zl (in) (in) (in) (in) (in) (in) 13.25 -20.20 tire 0.00 0.00 0.00 0.00 -17.00 0.00 axle 0.00 17.00 0.00 0.00 0.00 0.00 truck beam 0.00 0.00 0.00 0.00 0.00 -34.00 piston 0.00 0.00 0.00 0.00 0.00 -78.OO cylinder 0.00 0.00 0.00 0.00 0.00 -82.00 drag strut 0.00 0.00 0.00 41.00 19.00 38.00 side strut 0.00 0.00 0.00 0.00

forward trunnion 0.00 24.00 0.00 0.00 0.00

aft trunnion

0.00 0.00 0.00 0.00 0.00

fuselage

component xO zO xl zl

yO yl (in) (in) (in) (in) (in) (in)

tire

0.00 0.00 0.00 0.00 13.25 -20 20

axle

0.00 22.00 0.00 0.00 -22.00 0 O0

truck beam

-29.00 0.00 0.00 29.00 0.00 0 O0

piston 0.00 0.00 0.00 0.00 0.00 -50 O0

cylinder 0.00 0.00 0.00 0.00 0.00 -64 00

0.00

drag strut 0.00 0.00 84.00 -56.00 -40 O0

side strut

0.00 0.00 0.00 0.00 -48.00 -36 O0

forward trunnion

0.00 0.00 0.00 0.00 -62.00 0 O0

aft trunnion

0.00 -62.00 0.00 0.00 -72.00 0 O0