Appendix A
Appendix A Development of Equations of Motion Fundamental relationships between forces acting on the landing gear and the fluid pressures inside the landing gear are developed in the following sections.
Pneumatic Pressure Equation The pressure terms that contributed to the forces F 1 and F 2 in equation (3) need to be related to position Xwg and X s or their derivatives. Nitrogen pressure changes have been described (ref. 13) by the polytrop_c gas law for a closed system as
(v°f
Pu = Pnil_nni) (ml) where V°i is the initial nitrogen volume, Vni is the actual nitrogen volume, Phi is the initial nitrogen charge pressure, and y is the polytropic gas constant. This representation of the pressure change is assumed to happen as a quasi-equilibrium process. In most situations the polytropic gas constant is actu- ally not constant and is usually calculated from pressure-stroke data. However, reference 13 showed the experimental estimation of a single value for the polytropic constant. Equation (A1) is defined in such a manner that Pu will become very large for small values of Vni; that is, the gear is nearly collapsed.
This equation is a suitable representation of the process, with only the polytropic gas constant y as an unknown.
Before developing expressions for hydraulic fluid motion between chambers, a fundamental assumption is that the upper chamber hydraulic fluid pressure equals the nitrogen pressure. An expres- sion for the nitrogen volume Vni is now sought. The total volume available in the system is defined by using the dimensions shown in figure A1 as (A2) V T = AL(L u-X s)+ArX s assuming A L = A u . This volume houses both hydraulic fluid and nitrogen in the system. Since the nitrogen level is given by Xni, with volume computed as Vni = AuXni, the corresponding volume of hydraulic fluid is given by (A3) Vii q = V T - Vni = ALL u - (A L - Ar)X s - ALXni and the rate of change of the volume of hydraulic fluid is dVliq - (A L- Ar)X s- ALXni (A4) dt In a closed system, the amount of hydraulic fluid Vii q is constant; therefore, the rate of change is zero. Using this fact with equation (A4) yields an expression for the nitrogen level as Xni - (AL- Ar) Xs (A5) A L
Ni_ogen
t u A Piston Figure A 1. Hydraulic fluid-nitrogen volumetric diagram.
If the system is not closed; for example, if hydraulic fluid is entering or leaving the chamber at a rate Qc, the rate of change of the volume of hydraulic fluid is given as dVliq dt - Qc = -(AL - AR)Xs - ALXni (A6) Solving this equation for the nitrogen level yields ac )_ni = - (AL - AR))_s AL AL (A7) This differential equation, along with the initial value for the nitrogen level X ° hi' describes changes in nitrogen level as a function of time. Canceling the area terms in equation (A1), the upper chamber pres- sure is now written as Pu = Pni (A8) Note that equations (A7) and (A8) relate the stroke to volumetric changes in hydraulic fluid. We will now develop the pressure equations for the system.
Lower Chamber Hydraulic Pressure Equation Hydraulic fluid pressure in the lower chamber and the snubber chamber is related to the flow rates of the hydraulic fluid into and out of those regions. The volumetric flow rates through the orifice plate hole Qo and the snubber orifices Qs can be determined by combining the continuity equation and Bernoulli's equation for fluids. Flow is always from high pressure to low pressure. Bernoulli's equation for an incompressible fluid states that along a streamline P 1 2 + _gV + z = c, where P is the pressure at some point, g is the gravitational acceleration, v is the velocity of the flow, O is the specific weight of the fluid, z is the height difference from some zero reference, and c is a constant. This equation assumes that the viscous effects within the fluid are negligible, the flow is steady and incompressible, and only points along a streamline were considered. Examine points 1 and 2 along a streamline; the mass conti- nuity for incompressible fluid states that the flow rates A lV 1 = A2v 2 . Assuming that the pressure at point 2 is P2 > PI and that the flow is from point 2 to 1, one can solve for the velocity v I from the con- tinuity equation as D 2 Vl =_v 2 D 1 where the area in the continuity equation is now expressed in terms of diameters. Substituting this velocity into the Bernoulli equation yields
-+ /
(A9) 9 1- 1 The ideal volumetric flow rate (Qideal) for an incompressible fluid can be expressed as Qideal = Av. In a realistic flow situation though, there is a flow rate loss due to flow restrictions in the orifice. This loss is empirically quantified by a discharge coefficient Cd, which represents the percentage of the ideal flow that actually occurs. This coefficient, when multiplied by the ideal flow, yields Qreal = CdQidea! = ACdV (AI0) Substituting the velocity in equation (A9) into equation (A 10) For the landing gear shown in figure 1, there are three flows that are of concern: the flow through the orifice plate Qo, flow in and out of the snubber chamber Qs, and if the gear is actuated externally, there is an additional flow Qc. The Qc is supplied by an external high- and low-pressure reservoir, which adds or removes hydraulic fluid from the system.
Defining a control volume as shown by the dashed line in figure A2, compression occurs when )/'s > 0, and extension occurs when Xs < 0. Flow is assumed to be positive leaving the control volume and negative entering it. For an incompressible fluid, continuity yields Qc + Qo + Qs = ALXs (A12) Equation (AI 2) defines the general form of the continuity equation. Because flow directions have to be taken into account explicitly to determine pressures in the various chambers, one needs to consider all Qo Orifice Lower chamber Supply reservoir X$ Piston Figure A2. Control volume between piston and orifice plate.
possibilities. Flow in or out of the snubber chamber always equals the volume displaced during expan- sion or compression Qs = ARJ(s" Substituting this value into equation (A12) yields Qc + Qo = (AL - AR))fs (A13) Substituting the appropriate pressures, areas, and diameters into equation (A 11 ), the flow rate through the orifice plate during the compression )(s > 0 can be written as Qo = P_L-Pu , PL > Pu (A14) A°Cd " _(Do14 ]
l[ 2
p 1 t DL ) ] where D o is the effective diameter of the main orifice, D L is the diameter of the lower chamber, C d is the discharge coefficient of the main orifice, and A o is the effective area of the main orifice. To sim- plify equation (AI 4), the nonpressure terms were grouped as E 1 = A oC d ¢Dol41
l;[2
1-t, DL) J
The flow through the orifice is now written as (A15) Qo = El _L- Pu for PL > Pu Obviously, if theupperchamber pressure is higherthanthe lowerchamber pressure, the order of the pressure terms must be reversed.
To actuate the piston, one can add or remove hydraulic fluid from the landing gear. Considering a system with infinitely large high- and low-pressure reservoirs, depicted in figure A2, an external flow into the lower chamber can be written as Oic=-CcXcJPHigh-P L xc<O (A16) and flow out of the lower chamber is given by QO = CcXc_/PL_PLo w xc>O (A17) where x c is a command variable, and the parameter C c combines empirically determined values for the discharge coefficient, orifice area, and flow gain per unit command input. Superscripts i or o are used to distinguish inflow from outflow. To evaluate the pressures, one needs to examine the different flow directions in the control volume.
Figure A3 shows three potential flow directions for Qc and Qo for the compression case. The cor- responding continuity equations for Xs > 0 are Qo-Oic = El P/-_-L-Pu+CcXc_/Pttigh-PL= (AL-AR))_s; xc<O, Qo>O Qo+a°=El_L_Pu+Ccxc_/PL_PLow=(aL_aR)Yfs; xc>O, Qo>O (A18) -Qo+Q ° =-EI_u-P L +CcXc_/PL-eLow = (AL-AR)Xs; Xc>O, Qo<O Similarly, the extension case has three potential flow directions shown in figure A4 with corresponding flow equations for )_s < 0 : i -Qo - Qc = E1 _u - PL + CcxcJPHigh - PL = (AL - AR))fs; Xc < O, Qo < 0 -Qo + Q°c = -El _u- PL + CcXcJPL - PLow = (AL - aR))_s; Xc > O, ao < 0 (A19) Qo + QI. = Et _L - Pu + CcXcJPHigh - PL = (AL - aR))fs; Xc < O, ao > 0 1 3 2 ao ao T Qo Figure A3. Potential flow directions for compression case Xs > 0.
ao ao
Q/
li!ii!!ii!
iiiiiiiiiii!iii!iii!iiii!iii!iiiiiiiiiii! i!ii!ii !iiii ii ii iiii!
Figure A4. Potential flow directions for extension case Xs < 0.
Equations (A8), (A 18), and (A19) need to be solved simultaneously to determine the pressures PL and Pu" Note that equations (A7) and (A8) couple the upper pressure to the flow rate Qc" Appendix B shows an explicit solution for the lower pressure PL, obtained by using the Mathematica Symbolic manipulation program (The Mathematica Book, third ed., Wolfram Media, Cambridge University Press), reference 15.
A case not considered thus far is when the system is locked due to friction; that is, )_s = 0 (lower and upper chambers move together). When no external hydraulic fluid source is in use, x c = 0 ; there is no hydraulic fluid transfer between chambers. However, when an external source is in use, hydraulic fluid transfer is determined by continuity Qo = Qc" Two hydraulic fluid flow cases are possible depending upon the control variable x c value; they are i -Qo = -El _ Pu = Qc = --CcXcJP High -- P L ' Xc < O, Qo < 0 (A20) Qo = EI_u-PL = QO = CcXcJPL_PLow ,xc>O, Qo>O Solving equation (A20) for the lower chamber pressure yields CcXc ) + PHigh PL = ( El ]2 f°r Xc<0 _ Ccxc j + 1 (A21) Ccxc ) Pu + PLow PL= (El 12 f°r xc>0 _Ccxc) + 1 Most of the pressures needed to solve equation (3) have been developed, except for the snubber cham- ber pressure. In the following section, these expressions are developed.
Snubber Chamber Hydraulic Pressure Equation The analysis in this section is similar to the lower chamber analysis just described. To develop pres- sure relationships for the snubber chamber, consider a control volume as shown by the dashed line in figure A5. The variables A R and D R in figure A5 are the snubber chamber annular area and effective diameter, respectively; Ps is the pressure in the snubber chamber, and D s is the diameter of the snub- ber orifices.
In general, the continuity equation relates the flow in or out of the snubber to the volumetric change as follows: Qs = AR)_s (A22) Flow into the control volume of the snubber area during a compression mode _'s > 0 is given by _L- Ps , P L > Ps Qs = AsCds (Dsl41 (A23)
il 2
1-t° J ]
= E2_L-P s where CdS is the discharge coefficient of the snubber orifices, and A s is the effective area of the snub- ber orifice. For the extension mode _'s < 0, the flow is into the control volume and equation (A23) yields P_s- PL ' Ps > PL Qs =-AsCds 1 (Dsl4] (A24)
I[2
-t .j ]
= -E3_s-PL / Cylinder O S wall / Piston Figure A5. Control volume for snubber chamber.
where D R is the effective diameter of the annular snubber chamber. Often the snubber orifice diameter D s is different in extension and compression. For those cases, the appropriate diameters need to be used in this equation for each case. Equations (A23) and (A24) can be used to determine the snubber pressure, given a flow rate Qs" Substituting the flow rate Qs in equations (A23) and (A24) into equa- tion (A22) yields for -_s > 0 E2,f-_L- Ps = ARXs (A25) for Xs < 0 -E3_s- PL = ARXs Solving for the snubber pressures ('A R') 2 .2 Ps =PL-1_z) Xs for Xs>0 (A26) Ps = PL +_E3) Xs for Xs<0 Equations (A8), (A18), (A 19), and (A26) provide the fundamental relationships between the pressures, displacements, and commanded input x c . To evaluate the time responses for all these quantities, equa- tions (3) and (A7) must be integrated in time with appropriate initial conditions for all variables. Fric- tion models will now be described.
Friction Models The only unknown term left in equation (3) is friction. Friction in this landing gear comes mainly from two sources: friction due to tightness of the seal and friction due to the offset wheel (moment). The seal friction has a maximum value when the system is locked and decreases as the system begins to move. The functional relationship between frictional force level and velocity is determined through test- ing. The friction due to the offset wheel is the result of the moment produced by the nonaxially loaded piston within the cylinder.
Figure A6 shows that the force between the piston head and the cylinder N is a result of the tire force F t applied at a distance I from the centerline of the piston. The frictional force due to the offset wheel is Fow = pN (A27) where N is the normal force on the cylinder wall resisting motion of the piston head, and p is the coeffi- cient of friction between the two parts. The normal force N is computed by adding the moments about point O; [3 is defined as one-half the thickness of the lower bearing. Solving for the normal force N and substituting into equation (A27) yields (A28) Cylinder N x_ Piston Figure A6. Schematic of landing gear friction model development.
The total friction force in the landing gearfis given by f = Fseal + Fow (A29) Friction effects were included in the numerical simulations by using the approach described in reference 16.
Note that this model takes into account only vertical loads on the strut. Furthermore, the tire is mod- eled as a nonlinear spring and damper combination and does not take into account radial stiffening due to centripetal forces. Also, all structural members were assumed to be rigid, each having only a vertical degree of freedom. These assumptions are adequate for taxiing over runway profiles and for landing impact (spin-up drag on the tire does not significantly affect the vertical loads on the strut). Any braking or turning maneuvers were not covered in the development. The equations developed here are the basis for a "rollout" simulation.
Appendix B
Appendix B
Analytical Solution of Pressure Equation
All possible sign combinations in equations (A18) and (A19) can be considered by using a generic form such as Cl ff_L- Pu + c2 P,_-_L-Pr = c3 (BI) Cl P_---_u- P L + c2 P,_L- Pr = c 3 (B2) Cl_u-P L +c2 Pff_r-PL:C 3 (B3) Solutions for these three equations have the following form: 2 2 4 2 2-c2 (Pr Pu)+ClPu+Cl c3-c2(r+Pu) c2c3+c2Pr+-2ClC2C3 3- Cl - PL = (B4)
224 jI2 21 2 4 2E2 2 c2c3+c2Pr+2ClC2C3 Cl +c2 (Pu- Pr) - c3 + ClPu+cl -c3 +c2(Pr+Pu)
PL = (B5) -c2c3+c2Pr+-2ClC2C3 3 + Cl-C 2 (Pr-Pu)+ClPu-c I c3-c2(Pr+P u) (B6) PL = These are potential solutions for equations (B1), (B2), and (B3), respectively. It is important to realize that physically realistic solutions may not exist for certain values of the parameters, c I ,c2,c3, and the pressures.
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2. Milwitzky, Benjamin; and Cook, Francis E.: Analysis of Landing-Gear Behavior. NACA Rept. No. 1154, 1953.
3. Ottens, H. H.: Predicted and Measured Landing Gear Loads for the NF-5 Aircraft Taxiing Over Bumpy Run- way. Aircraft Dynamic Response to Damage and Repaired Runways, AGARD CP-326, 1982.
4. Payne, B. W.; Dudman, A. E.; Morris, B. R.; and Hockenhull, M.: Development of a Cost Effective Approach to Modeling Aircraft Response to Repaired Runways. Aircraft Dynamic Response to Damaged and Repaired Runways, AGARD-CP-326, 1982.
5. Freymann, R.: An Experimental-Analytical Routine for the Dynamic Qualification of Aircraft Operating on Rough Runway Surfaces. AGARD R-731, 1987.
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7. Freymann, Raymond; and Johnson, William P.: Simulation of Aircraft Taxi Testing on the Agile Shaker Test Facility. Second International Symposium on Aeroelasticity and Structural Dynamics, DGLRfDFVLR/NLR/ ONERA, 1985, pp. 468-476.
8. Shepherd, Alan; Catt, Tyrone; and Cowling, David: An Aircraft Landing Gear Simulation Parametric Leg Model. SDL Rept. No. 234, Stirling Dynamics Limited, 1993.
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13. Daniels, James N.: A Method for Landing Gear Modeling and Simulation With Experimental Validation.
NASA CR-201601, 1996.
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3. REPORT TYPE AND DATES COVERED 1. AGENCY USE ONLY (Leave blank) 12. REPORT DATE Technical Publication May 1999 4. TITLE AND SUBTITLE 5. FUNDING NUMBERS Controlled Modeling and Validation of a Navy A6-lntruder Actively WU 522-18-11-04 Landing Gear System 6. AUTHOR(S) Lucas G. Horta, Robert H. Daugherty, and Veloria J. Martinson 7. PERFORMING ORGANIZATION NAME(S) AND ADDRESS(ES) 8. PERFORMING ORGANIZATION REPORT NUMBER NASA Langley Research Center L-17817 Hampton, VA 23681-2199 10. SPONSORING/MONrrORING 9. SPONSORING/MONITORING AGENCY NAME(S) AND ADDRESS(ES) AGENCY REPORT NUMBER National Aeronautics and Space Administration NASA/TP- 1999-209124 Washington, DC 20546-0001 11, SUPPLEMENTARY NOTES 12a. DISTRIBUTION/AVAILABILITY STATEMENT 12b. DISTRIBUTION CODE Unclassified-Unlimited Subject Category 05 Distribution: Standard Availability: NASA CASI (301) 621-0390 13. ABSTRACT (Maximum 200 words) Concepts for long-range air travel are characterized by airframe designs with long, slender, relatively flexible fuse- lages. One aspect often overlooked is ground-induced vibration of these aircraft. This paper presents an analytical and experimental study of reducing ground-induced aircraft vibration loads by using actively controlled landing gear. A facility has been developed to test various active landing gear control concepts and their performance. The facility uses a Navy A6 Intruder landing gear fitted with an auxiliary hydraulic supply electronically controlled by servo valves. An analytical model of the gear is presented, including modifications to actuate the gear externally, and test data are used to validate the model. The control design is described and closed-loop test and analysis com- parisons are presented.
14. SUBJECT TERMS 15. NUMBER OF PAGES Vibration; Active control; Landing gear; Aircraft 16. PRICE CODE A03 17. SECURITY CLASSIFICATION 18. SECURITY CLASSIFICATION 19. SECURITY CLASSIFICATION 20. LIMITATION OF REPORT OF THIS PAGE OF ABSTRACT OF ABSTRACT Unclassified Unclassified Unclassified UL NSN 7540-01-280-5500 Standard Form 298 (Rev. 2-89) Prescribed by ANSI Std. Z39-18 298-102