Theoretical dynamic analysis of the landing loads on a vehicle with a tricycle landing gear

APPENDIX A ANALYSIS O F VERTICAL LOADS ON THE X-15 LANDING GEAR Main Gear The X-15 main landing gear is of the articulated type, as illustrated in the sketch below. (See reference 1 for a detailed discussion. )

t

F V The ground reaction Fv is transmitted through a skid, landing-gear-leg, and bell-crank arrangement to an oleopneumatic shock strut inside the fuselage. Prior to shock-strut displacement, the vertical load is a function of the vertical skid a s displacement due to leg bending. The bending load can be expressed where K is the stiffness coefficient of the main-gear leg which relates vertical load to vertical deflection. This coefficient can be described analytically but is obtained more The vertical-bending deflection 6~ is equivalent to the practically by static tests.

overall gear deflection described by equation (4) or equations (20) o r ( 2 4 ~ ) .

After shock-strut motion, the vertical load is expressed a s in equation (3). Be- cause of the articulated nature of the main gear, the vertical load at the ground and the shock-strut load are not equivalent; however, they can be related through the gear geometry. Equating moments about the gear pivot (see preceding sketch) results in B F v = - F A s

APPENDIX A B

The gear vertical load is a function of the ratio A, which varies as the gear

rotates about the pivot, and of the shock-strut load Fs. The load Fs is given by where the quantities are defined as for equation (3). For the X-15 airplane, the friction forces Ffs are assumed to be negligible. The shock-strut airspring load

FA^ is determined experimentally for the X-15 but can be obtained analytically (see

ref. 7). The hydraulic load F H ~ is related t o the shock-strut velocity ds by

where C s is an experimentally defined hydraulic coefficient. The load does not have a direction sign as in reference 7 , since the X-15 shock strut produces hydraulic load only in compression. During the expansion stroke, hydraulic pressure is relieved This condition is through orifices so that the load is the result of the airspring only.

approximated by The vertical and shock-strut loads were related by equation (A2). Using the same technique is the rate of vertical deflection of the main gear due to the rigid rotation where 6~ R of the gear about the pivot.

and (A6) into equation (A2) results in Substituting equations (A3), (A4),

F * + C d 2

Fv = MR where

APPENDIX A

The X-15 main-gear shock strut also incorporates a pressure-relief valve which, -

upon opening, prevents a further increase in hydraulic pressure. Until the valve opens, the strut is velocity-sensitive, but after the orificing the strut load is the air-

-

spring load with the pressure-limited non-velocity-sensitive hydraulic load added to it.

Therefore, a s long as the relief valve is open, equation (A4) becomes is the maximum attainable hydraulic load.

where (E"s)

max The skid load of equation (A?) is then written

where @Ismax = A Hs )

The overall vertical-deflection rate 6 , given by equation (4) o r by equations (20) o r (24c) is the sum of the rates of vertical deflection o r The rate of bending deflection is now obtained by considering the main-gear leg as a single-degree-of-freedom damped spring mass. Then where g is the gravitational acceleration, C1 is the damping coefficient, and mM is the mass of the main-gear leg.

Nose Gear The X-15 nose gear is a conventional-type gear (ref. l), utilizing dual co- rotating wheels and tires and an oleopneumatic shock strut. The vertical load F, prior to shock-strut displacement is a function of the tire deflection 6, or

APPENDIX A where Ft is the vertical load on the tire. The relationship between F, and 6, is I obtained experimentally for the X-15. The tire deflection is equivalent to the overall nose-gear deflection given by equation (4) o r equations (20) or (24c).

.

After shock-strut motion, the vertical load is described by equation (3). Since gear bending has been neglected, the methods of reference 7 result in

Fv = FA + CNbS2 4

1% I

and the hydraulic coefficient CN are experimentally where the airspring load FA determined functions of the nose-gear shock-strut deflection 6 , .

The shock-strut deflection is related to the overall gear-deflection rate by The rate of the tire deflection is obtained by considering the wheel and tire as a single -degree+of-freedom damped spring mass. Then

.. Fv - Ft

6, = g + "N where mN is the mass of the tire and wheel.

APPENDIX B RELATIONS FOR DROP-TEST SIMULATION Drop tests of the X-15 landing gear were made to determine the actual gear characteristics. Each gear w a s tested individually rather than as a complete landing- gear system. Weight w a s added to the basic landing gear to simulate inertia loads due to vertical motion. Thus, equation (24a) for vertical motion becomes for each gear mZo = W - L - Fv (B1) where L represents a simulated aerodynamic force. Equations (24c) are then re- placed by 0 .

6 = zo where initially io = Vv.

The equations of appendix A are still applicable for the gear-drop tests. The mass m and weight W are the effective masses and weights computed for the critical landing conditions.

. REFERENCES

1. McKay, James M. ; and Scott, Betty J. : Landing-Gear Behavior lXlring Touchdown and Runout for 17 Landings of the X-15 Research Airplane. NASA TM X-518, 1961.

2. McKay, James M. ; and Kordes, Eldon E. : Landing Loads and Dynamics of the X-15 Airplane. NASA TM X-639, 1962.

3. Noll, Richard B. ; Jarvis, Calvin R. ; Pembo, Chris; Lock, Wilton P. ; and Scott, Betty J. : Aerodynamic and Control-System Contributions to the X-15 Airplane Landing-Gear Loads. NASA TN D-2090, 1963.

4. McKay, James M. ; and Noll, Richard B. : A Summary of the X-15 Landing Loads.

NASA TN D-3263, 1966.

5. Mechtly, E. A. : The International System of Units - Physical Constants and

SP-7012, 1964.

Conversion Factors. NASA Theoretical Investigation of the Slideout 6. Noll, Richard B. ; and Halasey, Robert L. : of a Vehicle Equipped With a Tricycle Skid-Type Landing-Gear System.

Dynamics NASA TN D-1828, 1963.

7. Milwitzky, Benjamin; and Cook, Francis E. : Analysis of Landing-Gear Behavior.

NACA Rep. 1154, 1953.

NASA-Langley, 1967 - 2 H-477

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