Document
AN ANALYTICAL PROCEDURE FOR
PREDICTING THE TWO-DIMENSIONAL IMPACT DYNAMICS OF SPACECRAFT LANDING GEAR By J a m e s T. Howlett Langley Research Center Langley Station, Hampton, Va.
NATIONAL AERONAUTICS AND SPACE ADMINISTRATION For s a l e by the Clearinghouse for Federal Scientific and T e c h n i c a l Information Springfield, Virginia 22151 - CFSTl price $3.00 ANANALYTICALPROCEDUREFOR PREDICTING THE TWO-DIMENSIONAL IMPACT DYNAMICS OF SPACECRAFT LANDING GEAR By James T. Howlett Langley Research Center SUMMARY This paper presents a method for analyzing the dynamic behavior of two- dimensional impacting struts and trusses of the type presently used in the construction of landing gear f o r lunar landing vehicles. The struts and trusses may contain shock absorbers. The method uses lumped masses to represent the system, and two- dimensional motion is assumed. The equations of motion a r e numerically integrated to obtain response time histories of each of the mass points, and to allow detailed analysis of the dynamic behavior of the system. Application of the method to a particular strut and truss indicates that axial elasticity is much more important than lateral elasticity in the dynamic behavior. The present limitation of the method is the large amount of computer time required for problems in which the time period of interest is large com- pared with the period of the highest natural frequency of the system.
INTRODUCTION Current concepts for space vehicles for soft landing on lunar and planetary sur- faces include landing gears consisting of legs which are trusses constructed of tubular struts. The t r u s s members may contain some type of shock absorber. In order to assure proper functioning of such landing gears, it is essential to understand the dynamic behavior upon impact with a landing surface. Especially important are the buckling behavior of the landing gear and the effects of landing-gear elasticity on the landing dynamics of the vehicle.
Previous analyses of landing dynamics (for example, refs. 1 and 2) have included axial but not lateral vibrations of the landing gear. Drop tests of a model of an early gear undergoes lateral vibra- version of the lunar module have shown that the landing tions during impact. The effect of lateral vibrations was not clearly understood.
Accordingly, the study reported herein was carried out. It consists of a mathematical analysis of the dynamic behavior of impacting trusses containing shock absorbers with lateral oscillations included as a possible deformation. The analysis, which is restricted to two-dimensional motion, represents the truss by a finite-element lumped-mass system, uses Euler's method of numerical integration to obtain response time histories of each m a s s point, and thus allows detailed examination of the dynamic behavior of the truss.
The analysis is applied to a specific strut and t r u s s configuration to serve as an example.
SYMBOLS
distances between vehicle center of mass and masses k l and k l + 1
axial stiffness flexural stiffness force moment of inertia length of beam-column element m a s s moment axial load time axes of inertial coordinate system axes distance along slope in inertial coordinate system total shortening of beam-column element shortening due to bending defined by equation (14) shortening due to compression defined by equation (26) lateral deflection in local coordinate system
r(5)
7 7 1 = 17@)
e slope of free end of beam-column element in local coordinate system
axes of local coordinate system
5,r
radius of gyration of c r o s s section P Subscripts: m a s s on main strut to which secondary strut is joined m a s s at end of secondary strut which attaches to main strut vehicle center of m a s s jth m a s s point jth mass point referred to first local coordinate system jth m a s s point referred to second local coordinate system m a s s point at vehicle end of main strut m a s s point at vehicle end of secondary strut components of forces total external Matrix notations:
0 column matrix
11 square matrix
Dots over a symbol denote differentiation with respect to time. P r i m e s denote differentiation in the local coordinate systems with respect to 5 . A bar over a symbol denotes a nondimensional quantity.
ANALYSIS The basic method of analysis is first developed for a strut. In a later section the method is generalized to include a t r u s s containing shock absorbers.
A strut is represented by a number of point masses connected by massless beam- column elements which may undergo both axial and lateral deformations. (See fig. 1.)
Each of these elements is uniform, but the length and stiffness parameters may differ from one element to the next. Only two-dimensional motion is allowed.
Coordinate Systems A s shown in figure 1, three different types of coordinate systems a r e used in the analysis. The axes of the inertial coordinate system are denoted by X and Y. In addition to the inertial coordinate system, there are two local coordinate systems asso- ciated with each m a s s point. The local coordinate systems are attached to the strut and move with it. The axes of the first local coordinate system associated with the jth m a s s a r e denoted by [j,j-l and qj,j-l. The origin of the first local coordinate system is located at m a s s j - 1. The C;j,j-l-axis is tangent to the strut at m a s s j - 1 and posi- tive in the direction of the jth mass. The positive direction of the qj,j-l-axis is 90° counterclockwise from the positive <j,j-l-axis.
The axes of the second local coordinate system associated with the jth m a s s are denoted by tj .+I and qj,j+l. The origin of the second local coordinate system is , J located at m a s s j + 1. The cj,j+l-aXiS is tangent to the strut at m a s s j + 1 and posi- tive in the direction of the jth mass. The positive direction of the qj,j+l-axis is 90° counterclockwise from the positive tj, j + l - a i S .
Equations of Motion Newton's equations of motion for the system take the following form for j = 1, 2, . . ., kl: where jth m a s s m j moment of inertia associated with jth m a s s 1 j F forces and the moment acting on jth m a s s from beam F Xj,j-l’ yj,j-l,Faj,j-l column j , j - 1 forces and the moment acting on jth m a s s from beam
FXj , j + 1 9 FYj, j+ 1 9 Fa j , j+ 1
column j , j + l components of external forces acting on jth m a s s Fe, xj 9 Fe, Yj Quantities with nonpositive subscripts or subscripts greater than k l do not appear. The procedure f o r computing the quantities FXj, j- 1, F yj, j- 1, F, j, j- 1, Fxj, j+l, Fyj, j+l, is given in the following section. Since the equations which a r e used to and Fa j,j+1 compute these quantities a r e nonlinear, the values are computed numerically for each time step. Once the values a r e determined, Euler’s method is used to integrate equa- tions (1) one time step with the assumption that Fx. 3 , j-1, FYj, j-1, Faj, j - 1 9 FXj,j+l’ and Faj,j+l remain constant during the interval.
Force-Displacement Equations To determine the quantities Fx. FYj,j-l, a d F,j,j-l, the elastic forces 1 , j - l ’
exerted on the jth mass by the beam column connecting masses j - 1 and j must be
determined, These forces can be determined by standard procedures. Classical beam- column theory gives equations relating the forces and displacements of the two ends of the beam column. These equations are more conveniently expressed in the first local coordinate system. The position of and slope at the jth m a s s in the first local coor- dinate system a r e given by the following formulas (fig. 1):
[ j , j - i = pj - xj-l)cos aj-1 + (yj - yj-1)Sin O j - 1 ( 2 4
qj,j-l = -(xj - xj-l)sin Q! j - 1 + ( Y j - Y j - l ) C O S aj-1 (2b)
Also define
where Zj,j-1 is the undeformed length of the beam column connecting masses j - 1
and j . The quantity 6 j , j - 1 is called the total shortening of the beam column con- necting masses j - 1 and j.
For notational convenience, the subscripts a r e dropped with the understanding that the quantities are in the first local coordinate system. Note that in the first local coor-
dinate system, the beam column connecting masses j - 1 and j may be viewed as a
cantilevered beam column (isolated section of fig. 1). In general, three types of loads may be needed at the f r e e end of this element to hold it statically in i t s instantaneous deformed configuration: a lateral force F, an applied moment M, and an end load P.
The following procedure is a well-known method of relating the forces to the displacements.
The equation for the lateral deflection of the cantilevered beam column shown in the isolated section of figure 1 is where lateral deflection E1 f lexur a1 stiffness total bending moment M t Note that
v@) = V Z = V j , j - 1
where qj,j-1 is given by equation (2b).
The boundary conditions a r e Two diffe ent cases may aris for case 1, P = 0 fo ca 2, P # 0.
Case 1; P = 0.- In this case, equation (4) becomes
~ = L[M + F(Z - tjl
d t 2 E1 I The solution of equation (7) subject to boundary conditions (6) is At the free end o r in matrix form Inverting equation (10) gives the relationship between the forces and displacements A convenient dimensionless form of equation (11) is where
- FZ2
F = - E1
-
M2 M = - E1
- vz
= v2 The longitudinal displacement of the f r e e end of the beam due to bending of the beam is (ref. 3): The quantity is called the shortening of the beam due to bending.
By using q(5) from equation (8), is zero, there is no shortening due to compression. Hence, for Since the end load P the present case, 6 = 61.
In dimensionless form, equation (15) becomes 1 1 - - 2 1 - -
5, = -i[3(~ + F) - - 4 F(M + F) + i@j
where
- 61
61 = 7
Case 2; P f 0.- In this case, equation (4) takes the form The solution of equation (18) with boundary conditions (6) is It follows that In matrix form equations (20) and (21) a r e ~ Inversion of equation (22) gives
-
--
D1
rl
where
Z\IP/EI - Z ~ P / E I 4
- e
- -
+ -
+ e The shortening due to bending is
F (" + F2 '?? + - - F F ) ( e -Z/= - 1 )
+ -
P P P 2 P In this case, however, 61 is not the only contribution to the shortening of the 62, which is beam. The end load produces an additional shortening (or lengthening) computed by the formula for compression of a bar.
6 2 = - = P Z The total shortening (or lengthening) is then
6 = 61 + 62 (27)
6 , and 6 are known.
Equations (23) and (27) determine F, M, and P i f qz, However, P is determined only implicitly; thus, an iterative numerical solution of the equations is required. Because of the latter point, it is convenient to consider separately the two possibilities which can occur: (a) P < 0; (b) P > 0.
Case 2(a); P < 0.- In this case equations (23) and (27) can be put in the following dimensionless form: sin M 1 k l - 1-- cos kl cos kl 1 sin kl 1
---
1-- cos kl kl cos kl
I I
.
+ F + (kl) qz (COS kl -
"-I
where
-
4 2 D 1 = k (EI) D1 and D1 is defined by equation (24).
c ~~ Case 2(b); P > 0.- In this case equations (23) and (27) can be put in the following dimensionless form: - -\ sinh kZ 1 kl c o s h k l cosh kl 1 sinh k l 1 - 1 1 - - cosh k l kl cosh kl '(sinh 2kZ 2kl
Z e d
(33) EA is given by equation (30).
where k It remains to be determined whether equations (12) and (16), (28) and (29), or (32) and (33) should be used to compute F, M, and kl for an arbitrary set of values of 171,
e, and E . In other words, if y l , 8, and are known, it must be determined whether
P = 0, P < 0, o r P > 0. Once this fact is known, the proper set of equations can be - solved for F, M, and kl. To this end, consider 62 as given in equation (26). If 62 < 0, then P > 0. But by 62 = 0, then P = 0; if 62 > 0, then P < 0; and if equation (27), &i2 = 6 - 61 Hence, 6 = 61 implies P = 0; 6 > 61 implies P < 0; and 6 < 61 implies P > 0.
Denote the value of 6 1 for P = 0 by 61(0) and denote the value of 61 f o r P # 0 by 61(P). By equations (11) and (15), 61(0) depends only on ql and 8. For the purposes of this test only, let 61 = 61(0) (34) This procedure amounts to neglecting the difference between G1(0) and 61(P). By using this assumption,
6 = 61(o) - P = 0
(3 5 4
6 > 61(o) - P < 0
(3 5b) where the arrow denotes an implication. There are heuristic arguments which indicate that the test may be rigorously valid, but no proofs have been found.
Note that the procedure does actually determine values of F, M, and P which will produce the correct values of qz, 8, and 6.
Once F, M, and P are determined, the forces which the beam column j, j - 1
is exerting on the jth mass a r e given by -F, -M, and -P. The components of these forces in the inertial coordinate system are the desired quantities (fig. 1):
F Xj,j-I = -P cos aj-l + F sin aj-1
( 3 6 4 The same type of procedure is used to compute F xj,j+l' FYj,j+l? and Fa j , j + l ' In the second local coordinate system, the position and slope of the jth mass are (374
t j , j + l = -(xj - xj+l)cos aj+l - ( y j - Yj+l)sin aj+l
The total shortening is where Zj -+I is the undeformed length of the beam column connecting masses j and 9 1
j + 1. With the understanding that all the quantities refer to the second local coordinate
system, the equations for F, M, and P are exactly as before. Hence, equations (35) are used to test 6 and the proper set of equations are used to obtain F, M, and P.
The forces which the beam column j, j + 1 are exerting on the jth m a s s are -F, -M, and -P. The components of these forces in the inertial coordinate system are (fig. 1): (394 F Xj, j + l = P cos aj+l - F sin aj+l (39b)
F Yj,j+l = P sin aj+l + F cos aj+l
I Boundary Conditions
The boundary conditions a r e handled in a straightforward manner. For example, if one end of the strut is pinned, then the X- and Y-coordinates of that end a r e held constant while the slope of that end is allowed to vary as determined by equation (IC). In analyzing a buckling test of a single strut, the upper end of the strut is forced to descend with con- stant velocity while the slope of that end is determined from equation (IC). In drop tests, o r t r u s s is considered to be pinned as long as it is in contact with the foot of the strut the ground. The instant that the foot t r i e s to leave the ground, the pin is released.
Analysis of Landing Gear Truss A typical two-dimensional truss with shock absorbers is shown in figure 2. This t r u s s is representative of a leg of a lunar landing vehicle for an impact in which the leg is subjected to no out-of-plane loads. The two members of the truss a r e joined by a pin l and their upper ends are pinned to the main vehicle. The two members of the truss may be joined at the foot, or in the section of the main strut below the shock absorber, or, as shown in figure 2, in the section of the main strut above the shock absorber.
The analytical model of the t r u s s can have up to 25 masses, including one m a s s f o r the center of m a s s of the vehicle. A shock absorber can be located between any two adjacent masses of the main strut and a second shock absorber can be located between The following definitions a r e needed any two consecutive masses of the secondary strut.
in connection with the analytical model: subscript indicating point on main strut to which secondary strut is joined ~ c1 I subscript indicating point at vehicle end of main strut k l k i f 1 subscript indicating point at vehicle end of secondary strut subscript indicating point at end of secondary strut which attaches to main c2 strut A typical analytical model of the t r u s s using 11 m a s s points is shown in figure 3.
The procedures discussed previously for a strut have been incorporated into a i procedure for dynamic analysis of the two-dimensional truss. The method is first
I
extended to a t r u s s without shock absorbers; generalization to include shock absorbers is done subsequently.
Consider, first, the pinned joint connecting the two struts.
The equations of motion I I for the joint a r e as follows: I + F + FYcl,cl-l e,YC 1 1 ; = c1 c1 F"cl, c 1- 1 + F"cl,cl+l xc2 = xc, Equations (40a) and (4Ob) imply that the position, velocity, and acceleration of the joint are determined by adding the forces from the three beam columns connected to the joint and integrating the resulting equations. Equations (40d) and (40e) express the fact that point c2 on the secondary strut coincides with point c1 on the main strut insofar as the x and y motion is concerned. Equations (40c) and (40f) indicate the fact that the moments on the main and secondary struts do not couple because of the pin.
The motion of the center of mass of the vehicle is determined by the forces and torques exerted by the two pinned joints connecting the struts to the main vehicle. The two upper ends of the struts and the vehicle center of m a s s are considered to be rigidly connected. The motion of the center of m a s s of the vehicle is governed by the following equations (fig. 4):
+ F t,Ykl+l (Xkl+l - ".rn)
where excluding the impacting leg mcm m a s s of vehicle coordinates of center of m a s s of vehicle xcm,ycm moment of inertia of vehicle Icm components of external force at vehicle center of m a s s Fe,&m7Fe,Ycm components of total force (elastic plus external) at mass j F t , ~ j ’ Ft,yj Note that equations (4 1) are coupled with equations (1).
The coordinates of masses k l and k l + 1 are determined from the vehicle cen-
ter of m a s s coordinates by the following equations for the vehicle orientation shown in figure 4:
Xkl = Xcm + d l C O S C Y c m
(424 where distance between vehicle center of m a s s and m a s s kl dl
distance between vehicle center of m a s s and m a s s k l + 1
d2
angle subtended at vehicle center of mass by masses k l and kl + 1
ecm Procedure for Incorporating Shock Absorbers F o r many applications, it is desirable to include a load-limiting shock absorber in a strut. A typical shock absorber of this type contains cartridges of crushable aluminum honeycomb. These shock absorbers are assumed to produce a constant force while stroking in either direction. The inclusion of such a shock absorber in the present anal- y s i s is accomplished as follows. The shock-absorber force is computed as a reaction force to the end load P. If the compressive load in the section of strut containing the shock absorber is less than the crush load of the cartridges, the shock-absorber force is set equal to the compressive load, and stroking is not considered to have occurred.
If the compressive load is computed to be greater than o r equal to the crush load, the shock-absorber force is set equal to the crush load, stroking occurs, and the reference length of the shock-absorber section is readjusted to a new shortened length. Once the shock absorber has shortened and then started to open, a small frictional force is applied until the shock absorber opens out to its original length. Then, if the shock absorber continues to open, the extensional shock-absorber force is applied in a manner analogous to the compressive shock-absorber force.
Note that there are two end loads associated with each section of the strut. For example, in figure 3, consider the section of strut connecting masses 2 and 3. There acting at mass 2 as seen in the local coordinate system will be an end load P 2 , 3 attached to m a s s 3; and there will be an end load P acting at mass 3 as seen in the local coordinate system attached to m a s s 2. Stroking is considered to occur only if both of these forces are greater than o r equal to the crush load.
In any attempt to analyze the response of a specific system, some modification and generalization of these shock absorbers will undoubtedly be necessary. The present shock absorber was considered in order to demonstrate the capability of this type of analysis to include shock absorbers of the load-limiting type.
NUMERICAL SOLUTIONS Solution of Force Equations earlier, for P f 0, equations (23) and (27) must be solved numerically As stated for F, M, and P . The solution can be obtained on a digital computer by using standard numerical techniques. However, these techniques, which require several iterations to achieve the desired accuracy, result in large amounts of computer time. Therefore, the following procedure has been used to determine the solution. This scheme determines values of F, M, and P which are well within the required accuracy and can reduce computer time by a factor of 10 o r more. From equations (26) and (27), 6 = 6 1 - - P1 EA Hence, By equation (30), Now El, which is the shortening of the beam due to bending, changes very little from one time increment to the next.
Therefore, as a first approximation to kl , where 61(x) is the value of 61 from the previous time increment. For case 2(a), 61(x) is determined from equation (29); for case 2(b), 61(x) is determined by equa- tion (33). The value of M as determined in this manner could be used in the right-hand side of equation (43) to compute a second approximation, and so on until the desired accu-
racy is achieved. However, because 8, changes only slightly from one time increment
to the next, results have shown that the first value of k 2 is sufficiently accurate and no
-
further iterations a r e necessary. Once kl is determined, F and a r e computed by equations (28) for case 2(a) (P < 0) or by equations (32) for case 2(b) (P > 0). Then F, a r e determined from equations (13a), (13b), and (30), where the sign of P M, and P must be chosen consistent with the appropriate case. The values of F, M, and P as thus determined do, in fact, produce the proper values of 0, and 6 to a satisfactory ql, degree of approximation.
Integration Scheme The numerical integration of equations (1) is carried out by using Euler's method.
This very simple scheme has been used effectively on other problems of this same type (ref. 1). It has the advantages of being self-starting and of requiring only one evaluation of the derivatives. Since the evaluation of the derivatives requires a considerable amount of computing time, this latter point is believed to outweigh any advantages of more sophis- ticated schemes, such as Runge-Kutta, which require several evaluations of the deriva- tives on each step. The inclusion of higher order differences in the present scheme (Adams method) produces a slight increase in accuracy. The equations for Euler's method are
x(t + At) = x(t) + At k(t)
$(t + At) = k(t) + A t x(t)
Equation (IC) requires special treatment for accurate numerical integration.
The parameter Ij which appears in equation (IC) is the rotary inertia of the jth mass.
Rotary inertia was included not because it was important, but because equations (1) were the simplest way to arrive at a consistent set of equations of motion. Equation (IC) could be completely eliminated from the analysis. However, the elimination of this equation would require an iterative method to determine the a j values.
When compared with motion in the x- and y-directions, the a-motion has very little inertial lag. The a-acceleration is much larger than the accelerations in the x- and y-directions. Also, the a-coordinate of a mass point oscillates at a much higher fre- quency than the x,y-coordinates. These high-frequency oscillations require an extremely small time increment for accurate numerical integration and result in unduly long com- puter runs. To reduce the computer time, an impulsive damper was introduced into the a!-motion. The action of the impulsive damper is explained in the following section.
Impulsive Damper 5 both with and without the impulsive damper is shown in figure 5 The behavior of for small values of the rotary inertia Ij. Without the impulsive damper, a! diverges very rapidly because of numerical instability of the integration scheme. If a time incre- ..
ment about one fiftieth as big were used, a ! would oscillate nearly harmonically with a maximum amplitude about equal to the first negative peak of the broken line.
The solid line shows the influence of the impulsive damper. A s the solution begins, a ! is most generally not in its instantaneous equilibrium position and an acceleration moves a ! toward its instantaneous equilibrium position. At this stage of the problem, two curves are identical. Since a ! is moving toward its instantaneous equilibrium the position, the computation is allowed to proceed normally. In the process of moving to i t s instantaneous equilibrium position, a ! acquires a finite velocity increment. Therefore, once a ! reaches its instantaneous equilibrium position, this velocity will cause a, to overshoot and begin to oscillate. In order to damp out the oscillation, the a ! velocity is set equal to zero once a ! reaches its instantaneous equilibrium position. This procedure amounts to adding a ficticious impulsive moment to the system which is just enough to reduce the a ! velocity to zero. Thus, a ! is prevented from overshooting and is actually stopped in its instantaneous equilibrium position. A s the solution pro- gresses, the instantaneous equilibrium value of a ! changes. When this happens, a ! is accelerated again and the entire process is repeated. As shown in figure 5, a, can have very high accelerations even with the impulsive damper. The high accelerations indicate that the instantaneous equilibrium value of a ! is changing rapidly and a, is lagging behind. Once a ! reaches its new instantaneous equilibrium position, the high accelera- tions a r e damped out. It is essential, of course, that a ! remains as near as possible to its instantaneous equilibrium position since, for small values of Ij, a ! reaches its new position almost instantaneously. For this reason, periodic checks must be made to assure that a ! is not lagging too far behind. In the cases so far considered, a ! has been found to follow along satisfactorily for sufficiently small values of Ij. As Ij increases in magnitude, the damping effect becomes more pronounced. An example is shown in figure 6. The upper plot is a time history of the deflection of the center of a vertical strut with shock absorber which is oscillating in the lateral direction. The lower plot is a time history of the a!-coordinate of the lower end of the strut.
One additional point should be mentioned. The time increment used in the numeri- cal integration must be many times smaller than the period of the highest natural fre- quency of the x- and y-motion. This statement reinains true even i f more sophisticated I numerical integration procedures are employed. Hence, if the period of the highest natural frequency of the t r u s s under consideration is short compared with the time period of interest then a large amount of computer time may be necessary. This condition can occur even when the period of interest is as short as 0.1 second. F o r a system with ten masses representing the truss and one mass for the vehicle, the computer program requires approximately 0.075 second p e r iteration on an IBM 7094 computer. This fact appears to be the most restrictive limitation of the method.
I I APPLICATIONS Buckling of Beam Column
I
To check the validity of the method presented in this paper, a comparison was made to results obtained by Hoff in reference 4. The accuracy of Hoff's results has been dis- ~ cussed by Sevin in reference 5. The system which was analyzed is a beam column pinned at both ends as shown in figure 7. The piston at the upper end of the beam column is forced to descend at a constant velocity of 0.256 inch p e r second (0.00650 m/sec). The case analyzed represents a very rapid loading of a slender beam column. In the work of Hoff and Sevin, the beam column had a small initial curvature. Since the present analysis does not account for initial deviations from straightness, the beam column w a s considered to be initially straight and it was given a small lateral velocity at the center. F o r this reason, the comparison of results shown in figure 8 indicate the same general type of behavior but not the same numerical magnitudes. A three-mass approximation of the beam column was used. The comparisons a r e presented in t e r m s of the nondimensional parameters used by Hoff.
The dashed curves in figure 8 show the results of the present analysis for an initial lateral velocity of 0.5 inch per second (0.0127 m/sec). Although different values of the initial lateral velocity produce different curves, the results shown a r e typical. In the ini- tial phase of the problem, the inertia of the beam column causes the lateral deflection of the center of the column to lag behind the static values, Eventually, this inertial lag is overcome and the final part of the solution consists of an oscillation about the static curve. Note that the axial load i n the beam column reaches the Euler buckling load at a dimensionless time of 1.0. Hence, failure of an actual beam column should occur very close to this time.
It should be pointed out that equations (28) are singular if the end load P is equal to the Euler buckling load. However, this difficulty occurs only i f P is exactly at the at least extremely close to it.
critical value, o r In actual practice, the value of P was never close enough to the exact critical value to cause any difficulties. The equations were simply integrated right through the singularity. In actual physical problems the I singularity should not cause any difficulties because the axial load in a beam column will ordinarily be kept well below the buckling load.
The next example illustrates the onset of dynamic buckling. A beam column is said to buckle dynamically if the amplitude of the vibrations tend to grow without limit (ref. 4). Behavior of this type is shown in figure 9 for three different cases. The num- ber of lateral oscillations which the column undergoes before buckling is dependent upon the velocity of the descending piston. The midpoint lateral deflections of the center of the beam column for three different values of the velocity of the descending piston are shown in figure 9. P r i o r to the onset of dynamic buckling, the number of oscillations which are predicted from the formula given by Hoff are approximately as follows: 1/3 oscillation for a velocity of the descending piston of 0.256 inch per second for a velocity of the descending piston of 0.128 inch (0.00650 m/sec); 2/3 oscillation per second (0.00325 m/sec); l3 oscillations for a velocity of the descending piston of 0.064 inch per second (0.001625 m/sec). In figure 9, the horizontal dashed lines indicate the region of normal vibrations. As the data in figure 9 indicate, the present analysis is in agreement with the values predicted by Hoff. For a velocity of the descending piston of 0.256 in./sec (0.00650 m/sec) buckling occurs before the midpoint of the beam column begins to swing back toward its initial position; for a velocity of the descending piston of 0.128 in./sec (0.00325 m/sec), the midpoint of the beam column begins to swing back and then buckles out; for a velocity of the descending piston of 0.064 in./sec (0.001625 m/sec), the midpoint swings out, comes back, and finally buckles out on the opposite side.
Buckling of Impacting Strut Figure 10 shows a strut with a shock absorber and its analytical model. The strut is of the general type presently being considered for the landing gear of the lunar module.
The strut is made of aluminum, is 12.3 f t (3.749 m) long and 6 inches (0.1524 m) in diameter, and has a wall thickness of about 0.03 inch (0.000762 m). At the upper end of the strut there is a heavy mass which weighs approximately 16 percent of the Euler buckling load. The shock absorber is a capsule of aluminum honeycomb with a crush Figure 11 shows a typical load-deflection curve which load of 3800 pounds (16903.24 N).
results from the analytical model of the shock absorber as described in a previous sec- tion. Note that the assumption of small deflections is valid because the length of the shock-absorber section is readjusted to a new shortened length as the shock absorber crushes.
Figure 12 shows the lateral displacements of the center of the strut without shock absorbers for various values of the impact velocity.
The strut is dropped straight down and the bottom of the strut is stopped instantaneously when it touches the ground. The center masses of the strut a r e given a small lateral velocity in order to induce bending.
At t = 0, the bottom of the strut first contacts the ground.
For an impact velocity of 1 ft/sec (0.3048 m/sec) the maximum axial load in the load and there is no pronounced effect on the lateral strut is well below the buckling vibrations. With an impact velocity of 2 ft/sec (0.6096 m/sec) the maximum axial load slightly exceeds the Euler buckling load. Before buckling begins, however, the downward velocity of the heavy mass at the upper end of the strut is stopped at approximately 0.02 second. As the strut begins to rebound, the upward displacement of the heavy mass at the upper end of the strut begins to relieve the compressive load. For impact veloc- ities greater than 2 ft/sec (0.6096 m/sec), the effect on the lateral displacements is clearly evident in figure 12.
Figure 13 shows the lateral vibrations of the strut with shock absorbers impacting a t 2 ft/sec (0.6096 m/sec). The center of the strut reaches its maximum lateral dis- placement on the first vibration and then oscillates with decreasing amplitude. The down- Buckling is pre- ward motion of the large mass is not stopped until about 0.076 second.
vented by the shock absorber, which limits the axial load on the strut. The shock absorber begins stroking at approximately 0.004 second and stops stroking at approxi- The lateral deflection of the strut without shock absorbers mately 0.076 second.
impacting at 1 ft/sec (0.3048 m/sec) is shown by the dashed line in figure 13 for com- parison. For the strut without shock absorbers impacting at 1 ft/sec (0.3048 m/sec), the downward motion of the heavy mass is stopped at approximately 0,021 second.
Effect of Lateral Elasticity on Impacting Strut As shown by the tests and supporting analysis of a 1/6-scale model of a lunar- module-type spacecraft described in reference 2, stored elastic energy can significantly affect the landing dynamics of lunar-landing vehicles. However, the analysis reported in reference 2 did not include lateral elasticity.
To determine the effect of lateral elasticity, the strut described in the previous section was impacted at 2 ft/sec (0.6096 m/sec) with varying amounts of axial and lateral elasticity. The initial lateral velocities of the center of the strut were varied from case to case in order to produce different amounts of maximum lateral displacement of the center of the strut. A maximum lateral deflection of 1 inch (2.54 cm) corresponds to an axial stress which exceeds the manufacturer's recommendation for the ultimate s t r e s s .
For the strut considered, the results indicate that axial elasticity is much more important than lateral elasticity in the dynamic behavior of the impacting strut. In fig- u r e 14, the maximum upward momentum of the rebounding heavy mass at the upper end of the strut is plotted against the maximum lateral amplitude of the center of the strut for different values of axial and lateral stiffness. As figure 14 shows, reducing E1 by a factor of 2 or 3 has very little effect on the maximum upward momentum. However, reducing EA by a factor of 2 increases the maximum upward momentum by almost 2 1 ateral vibrations have only a very slight 40 percent. The figure also indicates that effect on the maximum upward momentum.
In figure 15, vertical-acceleration time histories a r e shown for four of the extreme conditions presented in figure 14. Here also it can be seen that the reduction in E1 produces only a small change in the vertical-acceleration time history whereas the reduction in EA produces a much larger change in the curve.
Impact of Vehicle With Truss- Type Landing Gear Figure 16 shows a two-dimensional vehicle with a truss-type leg. The t r u s s con- s i s t s of two nearly equal struts joined at the foot. The struts a r e essentially the same as the strut considered above. The vehicle mass is approximately that of the lunar module at the moment of impact on the lunar surface. Lunar gravity is used. The vehicle impacts on one leg only. Calculation is stopped prior to impact of another leg.
An impact velocity of 1 ft/sec (0.3048 m/sec) is used. This velocity is just enough to build up the loads in the struts to the crush loads of the shock absorbers (3800 lb o r 16 903.24 N).
In figure 17, the vertical-acceleration time history of the center of m a s s of the is shown for three different cases.
vehicle For the first case, the full values of E1 and EA a r e used for both struts. The second case shows the effect of reducing the E1 of both struts by a factor of two while keeping EA at i t s full value. As the figure shows, these two curves are practically identical. The third case shows the vertical- EA of both struts is acceleration time history of the vehicle center of mass when the reduced by a factor of two and E1 is maintained at its full value. In this case, the curve is considerably changed and exhibits the same characteristics as for a single strut dropped straight down. The horizontal-acceleration response time histories show effects very similar to the vertical acceleration.
These results show that f o r the vehicle considered, a change in the bending stiff- ness by a factor of two produces essentially no change in the acceleration time history of the center of mass, and thus indicates that the motion of the vehicle is not substan- tially affected by the lateral vibrations of the landing gear. However, a change of the same amount in the longitudinal stiffness results in a much different acceleration time history of the vehicle center of mass. This result indicates that for the case analyzed, the longitudinal elasticity of the landing gear is much more important in the impact dynamics of the vehicle than the lateral elasticity.
CONCLUDING REMARKS A method has been developed for analyzing the dynamic behavior of two- dimensional impacting struts and trusses which may contain load-limiting shock absorbers. Lumped masses and finite elastic elements are used to represent the sys- tem and response time histories a r e obtained for each mass point. Application of the method to an impacting strut of the general type presently being considered for the landing gear of the lunar module indicates that the impact dynamics is essentially unaf- lateral elasticity. However, the impact dynamics is affected by fected by variations in variations in axial elasticity. Thus, axial elasticity is much more important than lateral A similar conclusion is elasticity in the dynamic behavior of the impacting strut.
obtained for a vehicle with a specific truss-type landing gear constructed of these struts.
The present limitation of the method is the large amount of computer time required for systems which have a high natural frequency with a period which is very short compared with the time period of interest.
Langley Research Center, National Aeronautics and Space Administration, Langley Station, Hampton, Va., November 28, 1967, 124-08-04-13-23.
REFERENCES 1. Walton, William C., Jr.; and Durling, Barbara J.: A Procedure for Computing the Motion of a Lunar Landing Vehicle During the Landing Impact. NASA TN D-4216, 1967.
2. Herr, Robert W.; and Leonard, H. Wayne: Dynamic Model Investigation of Touchdown Stability of Lunar-Landing Vehicles. NASA TN D-4215, 1967.
3. Timoshenko, Stephen P.; and Gere, James M.: Theory of Elastic Stability. Second ed., McGraw-Hill Book Co., Inc., c.1961.
4. Hoff, N. J.: The Dynamics of the Buckling of Elastic Columns. J. Appl. Mech., 18, no. 1, Mar. 1951, pp. 68-74.
vol.
5. Sevin, Eugene: On the Elastic Bending of Columns Due to Dynamic Axial Forces Including Effects of Axial Inertia. Trans. ASME, Ser. E, J. Appl. Mech., vol. 27, no. 1, Mar. 1960, pp. 125-131.
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