Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems

R. A. Wehage

E.J. Haug

Materials Division, College of Engineering, The University of Iowa,

Iowa City, Iowa 52242

Generalized Coordinate Partitioning for Dimension Reduction in Analysis of Constrained Dynamic Systems1

This paper presents a computer-based method for formulation and efficient solution of nonlinear, constrained differential equations of motion for mechanical systems. Nonlinear holonomic constraint equations and differential equations of motion are written in terms of a maximal set of Cartesian generalized coordinates, to facilitate the general formulation of constraints and forcing functions. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent variables, and constructs an influence coefficient matix relating variations in dependent and independent variables. This information is employed to numerically construct a reduced system of differential equations of motion whose solution yields the total system dynamic response. A numerical integration algorithm with positive-error control, employing a predictor-corrector algorithm with variable order and step size, is developed that integrates for only the independent variables, yet effectively determines dependent variables. Numerical results are presented for planar motion of two tracked vehicular systems with 13 and 24 degrees of freedom. Computational efficiency of the algorithm is shown to be an order of magnitude better than previously employed algorithms.

1. Introduction A method of formulating and solving differential equations

of motion for general mechanical systems subject to holonomic constraints has recently been presented [1,2,3], For planar systems, a three vector of Cartesian generalized coordinates is defined for each rigid body of the system.. Standard and user supplied constraints are formulated, yielding algebraic relations involving generalized coordinates common only to the bodies connected by each joint. A Lagrange multiplier technique allows coupling of the algebraic constraint equations and differential equations, yielding a large system of differential and algebraic equations which are solved iteratively using implicit numerical integration methods [1,2,3]. Because of weak constraint equation coupling, the resulting Jacobian matrix associated with the implicit method is sparse and sparse matrix algorithms are effectively employed to gain computational efficiency [4]. However, implicit numerical integration methods applied to mixed systems of differential and algebraic equations lead to numerical difficulties associated with badly conditioned matrices and artifically stiff problems [5].

Numerical difficulties associated with stiffness of equations Numerical analysis research supported by the U.S. Army

Research Office Project No. DAAG29-79-C-0221 and vehicle applications supported by the U. S. Army Tank and Automotive R & D Command Project No. DAAK30-78-C-0096-M2.

Contributed by the Mechanisms Committee and presented at the Design Engineering Technical Conference, Beverly Hills, Calif., Sept 28-Oct. 1, 1980 of the AMERICAN SOCIETY OF MECHANICAL ENGINEERS. Manuscript received at

ASME Headquarters June, 1980. Paper No. 80-DET-106.

have to some extent been circumvented in other methods of equation formulation by replacing the Cartesian coordinates with a smaller set of Lagrangian coordinates, equal in number to the constrained system degrees of freedom [6,7,8]. Loop closure or related methods are used to formulate constraint relations, yielding a smaller number of highly coupled differential equations. This is desirable from a numerical integration standpoint and leads to effective computer programs. However, depending upon the set of Lagrangian coordinates selected and the method of constraint formulation, it is difficult to incorporate nonstandard user supplied constraints. Provision for user supplied constraints thus leads to increased complexity in general purpose computer programs. It is apparent that there is a trade-off between program generality in the first approach and program efficiency in the second.

The method presented in this paper retains desirable features of both of the above methods: (1) the simple constraint formulation of the first in terms of Cartesian coordinates, and (2) the minimal system of differential equations of motion of the second, equal in number to the constrained system degrees of freedom. Excess coordinates are eliminated numerically, rather than analytically in the second method above. Gaussian elimination with full pivoting and subsequent L-U factorization [9] is applied to the Jacobian matrix of the constraint equations to determine: (1) the number of system degrees of freedom [10,11], (2) partitioning of generalized coordinates into independent and dependent sets and (3) construction of an influence

Journal of Mechanical Design JANUARY 1982, Vol. 104/247 Copyright © 1982 by ASME

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Fig. 1 Rigid bodies connected by a vector rp and rotational joints.

coefficient matrix expressing dependent velocities in terms of independent velocities [6,12], This method incorporates the desirable numerical stability properties of Gaussian elimination with full pivoting that is associated with the resulting L and [/matrices [13]. The numerical advantages of this approach have been demonstrated from a kinematic point of view in references [14] and [15].

2. Equations of Constrained Planar Motion For planar mechanical systems treated here, standard

constraints between rigid bodies are taken as friction free (workless) translational and rotational joints. User supplied constraint equations are also allowed, in order to make provision for nonstandard constraints such as cams and prescribed functional relations. Arbitrary (conservative or nonconservative) internal and external forcing functions are incorporated, including spring-damper-actuator elements connecting any pair of points on different bodies of the system. Allowance is also made for arbitrary user supplied forcing functions.

In order to determine the configuration or state of a planar mechanical system, it is first necessary to define generalized coordinates that specify the location of each body in the system. As shown in Fig. 1, let the x-y coordinate system be a fixed inertial reference frame. Define a body-fixed coordinate system £, - TJ, embedded in a typical body /. The location of body / is specified by the vector R, or global coordinates (xh y,) of the origin of its reference frame and the angle </>,• of rotation of the body fixed reference frame relative to the global coordinates.

The center of mass of body /' is located by a vector xmi in the body fixed coordinate system, with £, - ?j, components £„„• and j)mi. In terms of the generalized coordinates xh yh and 0, and the parameters £m, and i\mi, the center of mass is located globally by

xm, =Xj + amj cos </>,•-77m. sin <j>t

248/Vol. 104, JANUARY 1982

^ Ci

Fig. 2 Rigid bodies connected by a translational joint.

ym^yi + kmi Sin 0 ,+7) m . COS (/),• (1)

Thus, the kinetic energy of body / is

T< = i - m, (xl. +yj„.) + y J,tf (2)

where mt is the mass of body /' and 7, is its centroidal polar moment of inertia.

Fig. 1 further depicts an adjacent body j , with body-fixed coordinate system located by the vector R,. Let arbitrary points Pij on body /' and pM on body j be located by vectors r,7 and tjj, specified in the body fixed coordinate systems by coordinates £„, r/,y, £,-,-, and r/;,-. These points are in turn connected by a vector rp.

Tp = Ri+ru-Rj-Tji (3)

The vector condition for a rotational joint at points /?,-,• and Pji is simply xp = 0. The scalar equations of a rotational joint are thus

Xj + tjj cos #,• - riy sin fa - Xj - £,-,• cos </>,• + r/,v sin ty = 0

yt + £u sin </>,- + riy cos </>,- -y}- £,-,• sin ty - ijy7 cos 4>j = 0 (4)

If tp in Fig. 1 is taken as a nonzero vector of fixed length rp, a "massless link" of length rp connects points p,j and pir A single scalar equation for this constraint can then be written as

(X, + £,jj cos 4>j - riij sin </>, -Xj - £,,- cos ty

+ riji sin tj>j)2 + O,- + £,y sin «£,• + rj,7 cos </>,-

-yj ~ tji sin 4>j - riji cos(j>j)2-rl=0 (5)

For a translational joint shown in Fig. 2, let points pu and p^ lie on some line parallel to the path of relative motion between the two bodies, located such that r,7 and ry7 are perpendicular to this line and of nonzero magnitude. Equation (3) is also valid for this joint and the dot product of the vector r̂ , with ru yields one scalar equation

Transactions of the ASME


(gy cos <j>i - r\u sin <£,•)(*,• + £,-,- cos fr

- fly sin </>,• -xj - iji cos </>y + i)jj sin <£,•)

+ (Zu sin 4>j + riu cos (/>,) 0>, +£v- sin </>;

+ riy cos </>,• - ^ - £,-,- sin ^ - t/y7 cos <fy) = 0 (6)

A second scalar equation is obtained by noting that in three dimensions ry- x ry, = 0, whose z component gives

{hj c o s $> ~ ^ s i n * ' ) ( ^ ' s i n */ + 1/' c o s */)

- (£,-,- cos <£,- -?/;,• sinfy) (£y. sin4>, + rjy cos</>,)= 0 (7)

Other constraints may be formulated by a similar process. Denote by q, = [xh yh </>,]r the vector of generalized coor-

T T T

dinates of body / and by q = [q[ , q2 , . . .,q" ]T the composite vector of all system generalized coordinates. In this notation, the holonomic constraints of equations (4)-(7) and other (perhaps time dependent) holonomic constraints can be written in vector function form as

* ( * 0 = 0 (8)

where $(q,t) = [*i (q,t), . . . , $,„(q,t)]T are assumed independent. Virtual displacements dq that are consistent with constraints (i.e., with time fixed) satisfy

where

* =

%bq = Q

a * ~dq~ '

(9)

dqj

using subscript notation for differentiation with respect to a vector. Differentiation of equation (8) with respect to time gives the velocity relation

*,<7 + *, = 0 (10)

Internal forces acting between bodies may be obtained by a process similar to the constraint equation development. For example, since springs, dampers and actuators generally appear together, as shown in Fig. 3, they are incorporated into a single set of equations. The equation for spring-damper-actuator force is

1 ••\ku(lu-l0 ) +ciJvij+F,]TR (11)

where F„ is the resultant force vector FY..\ + Fv. in the element, R,.. is the vector ly cos ai + /,-,- sin aj between points Sy and Sjh ky is an elastic spring coefficient that may depend on generalized coordinates and time, Cy is a damping coefficient that may depend on generalized coordinates and time, /0. is the undeformed spring length, ly is the deformed spring length, fy is the time derivative of/y, and F0.. is an actuator force applied along the element that may depend upon generalized coordinates and time. The unit vectors i and j are parallel to the x and y axes, respectively.

Contributions of forces acting on body ;' to the system equations of motion are determined using the principle of virtual work. The virtual work of externally applied forces and spring-damper-actuator forces acting on body / is written as

bW=Qx (q,q,t)8xi + Qiy (q,q,t)Syi + Q^ (q,q,t)84>i (12)

The vector Q' = [Qx, Q'y, Q'lt>]r of generalized forces on body

/is thus defined and Q = [Ql , Q2 , . . ., Q" ]T is the vector of system generalized forces. Typical generalized forces are

Fig. 3 Rigid bodies connected by a spring-damper-actuator combination.

calculated to illustrate the procedure. Point Sy on body / is located by the vector R, + rs . A virtual change in the location of point Sy is given by

8 (R; + r ) = — (R, + r .) fix,- + — (R,- + r ) by, y aXj 'J ay, IJ

o<t>i ' (13)

Forming the dot product of Fy with 5 (R, + rs..) determines contributions to the generalized force expressions for body / as

Q^s =FXjj ( - Zsy sin <t>i - Vsy cos </>,•)

+ Fy.. (£SiJ cos </>,• - rjs.. sin <j>,)

Generalized force contributions are summed for all externally and internally applied forces acting on body /, to arrive at Q.

The variational form of Lagrange's equations of motion with workless constraints is [17], using subscript notation for differentiation with respect to a vector,

[jt(T,)-Tq-QT]bq = 0 (15)

which must hold for all bq satisfying equation (9). By Farkas' Lemma [18] there exists a vector \eRm of multipliers such that

Journal of Mechanical Design JANUARY 1982, Vol. 104/249


d

~dt (Tq)

T- TI-Q- *JX = 0 (16)

which with equation (8) form the constrained equations of motion of the system. For planar systems treated in this paper, equations (1) and (2) yield

d

dt

where

(T V ) ' - (T V )T=Mi{qi) qf -A' (J, q<) (17)

Since 4>„ is nonsingular one may use equation (24) to eliminate A, yielding equation (26) in the form

(M»v +DTM»" )v+ (AT" +DTM"U ) ii=A"

+ Q"+DT(A"+Q") (26)

Differentiating Eq. 10 with respect to time, and partitioning as above, one obtains

u=Dv-$u l[0M)?<y + 2*„7<7 + *„] (27)

M'

and

m, 0 - m, (£,„. sin </>,• + ijm . cos <£,)

0 m, - w,- ( - £,„. cos </>,- + i\m. sin 0,)

- ™; (£m, sin </>,• + r/m. cos 0,-) - m, ( - £„,. cos 0,- +i?„,. sin 0,) • / , + « , ( £?„,. + ^ )

(18)

/4' =

/«, (£,„. cos 0, -)?,„. sin fa) 4>}

w,(£,„ . sin 0,-+ ?;,„. cos (/>,-) (/>?

0

Using matrix notat ion A/ = diag(M' , A / 2 , . . ., M") 2 T T T

A = [A1 ,A2 ,. . .,A" ]Tequation(16)becomes

M(q)q = A(q,q)+Q(q,q,t) ~^(q,t)\

(19)

and

(20)

Initial conditions for system motion are given as

<7(0) = q°

q(0) = q° (21)

where q° and q° are consistent with constraints. That is q° satisfies equation (8) and q° satisfies equation (10).

3 . Reduced System of Differential Equations of Motion

Equations (20) and (8) constitute a system of 3« second order differential equations and m algebraic equations. Implicit methods have been employed to solve systems of equations of this form, however it is generally concluded that the combined system of algebraic and differential equations is inherently stiff and their direct solution is computationally expensive [2,3,5].

The matrix $q(q,t) has full row rank if the constraints are independent, so there is always at least one nonsingular submatrix of $ ? of rank m. Gauss-Jordan reduction of the matrix 4>9 with double pivoting defines a parti t ioning of q =

[uT,vT]T such that $„ is the submatrix of $ ? whose columns correspond to elements u of q and $„ is the submatrix of $ 9

whose columns correspond to elements v of q. Furthermore, the matrix $„ has ideal numerical properties associated with double pivoting. Equation (10) can be written in partitioned form as

$„ M + <J>„ f) = — *,

or since *„ is nonsingular

u= - $ , 7 ' * „ y — $ " ' $ , =Z?i)-$,7 '*,

where

£ > = - * - ' * „ .

Equation (20) may be partitioned as

M'" ii + M"v v = (A" + Q") - *^X

Mvuii+Mv"v= (Av+Q>)-$1\

(22)

(23)

(24)

(25)

Equation (26) can now be written as a second order differential equation explicitly in terms of the independent generalized coordinates v. In order to integrate the equations of motion, however, it is helpful to write them in first order form. To do this, define the vector s of independent velocities as

v=s (28)

and write the total vector q of generalized velocities in terms of s, using equation (22), as

w(s)=q(s) Ds - * , ; ' * ,

5 (29)

with this notat ion and equation (27), equation (26) becomes

[(M™ +DTM"U) + (AT" +DTM"U )D]s

= (Av + Q") +DT {Au + Q")

+ (M"u +DTM"")<i>;1[($gw)qw + 2<i>lqw + <i>l,] (30)

Equations (28) and (30) now form a set of 6n - 2m first order differential equations for the independent generalized coordinates v and velocites s. All terms in equation (10) are determined as functions of v and s through the constraint equations of equation (8) and equation (29). Solution of this system yields the complete system state, including Lagrange multipliers if constraint reaction forces are desired. Integration of equations (28) and (30) with initial conditions of equation (21) determines v and s. Equat ion (8) determines u, equation (22) determines «, equation (27) determines ii and equation (24) determines X.

The following observation is made concerning evaluation of the above equations. The matrix inverse * ,7 1 is never calculated. The matrix * ? is permuted with row and column pivoting into matrices Luu, U"u, and If" [9] such that # , -L"" • [UUU:UUV], *„ ~ L"" • U"", *„ ~ Lm • W" and D = _JJUU-\_ yuv w n e r e L"" is a nonsingular lower triangular matr ix, U"" is a nonsingular upper triangular matrix, and U"" is a rectangular matr ix.

Many practical statically indeterminant mechanical and vehicular systems are started from a state of static equilibrium or from a state of partial equilibrium, in which certain generalized coordinates are held fixed at prescribed initial values, while the remainder of the system is allowed to seek equilibrium consistent with constraints. These coordinates are then released to initiate transient analysis. Static equilibrium of the constrained system is achieved when v = s = 0 at t = t°, in which case equation (30) reduces to

250 / Vol. 104, JANUARY 1982 Transactions of the ASWlE


R" = (Av + Qv)+DT(Au+Q")=0 (31)

and equation 8 is satisfied. Equations (31) and (8) may be solved iteratively, using the linearized relations

A « = - * - ' * (32)

and

Av=-(-£R") ' RV (33)

where the matrix dR"/dv is obtained by numerical differencing, using du = Ddv.

4. Computational Algorithm Obtaining suitable initial conditions for transient analysis

poses problems when mechanical systems have many degrees of freedom. The following seven steps briefly outline an algorithm for achieving suitable initial conditions, from a starting configuration that is consistent with constraints or from a partial or full equilibrium configuration:

Step 1. Specify initial estimates q° of the constrained system configuration. These estimates should be near the expected final values to insure convergence. If static equilibrium is to be obtained, the estimates should be near the desired equilibrium configuration to avoid convergence to alternate configurations, or divergence. Set q = q°.

Step 2. If some elements of q are to remain fixed during initial assembly of the constrained system or static equilibrium, append to equation (8), additional algebraic constraints of the form q} — qj = 0, where j identifies the variables to be held fixed. These constraints will be removed prior to step 6.

Step 3. Evaluate * 9 (q) (including constraints of step 2) and factor to determine a submatrix $„ of maximal rank and to determine a partitioning of q into u and v (dependent and independent sets, respectively). If ^^ does not have full row rank then too many constraints or inconsistent constraints were specified. Therefore some equations are not independent and the dependent equations can be temporarily ignored in the iterative procedure of Step 4.

Step 4. Hold v fixed and iterate using equation (32)'to solve the independent constraint equations of step 2, until 11*11 <e, where e is some specified closure tolerance. If there are dependent constraint equations, check them for violation. If some equations are violated, then inconsistent constraint conditions have been specified and a meaningless solution may result, in which case stop. If static equilibrium is not desired, goto step 6.

Step 5. Determine a new estimate for v from equation (33) that reduces \\R" ||. If \\R" \\> 5, a specified force unbalance tolerance, return to step 3. If the iteration in steps 3 to 5 diverges, stop.

Step 6. Specify initial estimates for velocites q° and set q = q°. The velocities q must be adjusted to satisfy equation (10), evaluated at q from the previous steps. If some elements of q are to be held fixed at q°, append to equation (10) additional equations of the form qk — q° = 0, where k identifies the variables held fixed. These constraints will be removed prior to step 1 of transient analysis.

Step 7. Factor the modified matrix of equation (10) to determine a partitioning of q into it and v (not necessarily the same partitioning as in step 3). Calculate it using the above partitioning in equation (22). If the above matrix lacks full row rank, some equations are dependent and must be checked for consistency. If any dependent equations are not satisfied, a meaningless solution may result, therefore stop. Otherwise a consistent set of initial conditions is available for transient analysis.

The selection of numerical integration algorithms for

dynamic analysis should be based on the type of differential equations to be solved. Since most mechanical systems are non-stiff, an explicit predictor/implicit corrector algorithm DE/STEP, INTRP is employed [17]. It is chosen since predictor/corrector algorithms require fewer function evaluations than Runga—Kutta methods and allow interpolation between solutions with no additional function evaluations. This is important, since the computational overhead in evaluating the reduced system of differential equations is significant. The integration algorithm DE estimates relative stiffness and terminates execution if the system becomes too stiff.

A numerical integration algorithm based on the development in previous sections and the explicit/implicit integration method is now presented. The following steps summarize the numerical integration algorithm.

Let ;', initially 0, be an indicator for the current time step; i.e., /' = 0 implies t = t°. Steps 1 to 5 are satisfied when starting from results of the preceeding algorithm, so start with step 6.

Step 1. Check ||* || < e. If satisfied, the extrapolated dependent variables u are satisfactory so go to step 4. (A polynomial extrapolator is maintained for u by the integration algorithm at little additional overhead. Since this extrapolator may be of any order up to 12, ${u,v,t) will usually satisfy the above closure test, thus avoiding the following steps 2 and 3. In addition the partitioning of q into sets u and v depends upon q and t because $ 9 depends upon q and t. Since the extrapolator is maintained for u and the predictor for v, it is not necessary to interrupt the integration to account for different sets u and v. Error control is maintained only on v since steps 1 to 3 are equivalent to maintaining error control on u.)

Step 2. Evaluate * ? and perform L-U factorization to determine $„, $„, D, and the partitioning of q into u and v.

Step 3. Iterate to determine u using equation (32) until 11*11 < e.

Step 4. Evaluate * ? , factor and calculate u from equation (22). (The partitioning of q into it and s s ii depends upon q and /, because $ ? is a function q and t. Since all elements of q are candidates for s, a polynomial extrapolator for it and a predictor for s are maintained by the numerical integration algorithm, with error control only on s. This facilitates smooth transition of elements of q into and out of s within the integration algorithm, with no interruption of the integration process. Error control is maintained only on s since equation (22) is equivalent to maintaining error control on it.)

Step 5. Calculate s from equation (30) (if required, calculate ii from equation 27 and X from equation (24), where q = [tiT,sT]T and ii = s. These calculations are not required for the integration steps that follow, however ii may be used to improve the extrapolator of it.)

Step 6. Using the explicit/implicit Adams PECE algorithm, predict q and q at the ((' + l)st time-step. That is, predict v' +', s'+', and extrapolate « '+ ' , it'+'.

Step 7. Execute steps 1 to 5 to evaluate ii, v, s, and ii. Step 8. Using it, s, and ii from step 7, correct u, v, s, and

u (correcting u and ii is done only to improve the ex-trapolators of u and it for the next step which will reduce predictor error if some elements of u and it are subsequently picked up by v and s at the next step.)

Step 9. Estimate integration error on v and s. Adjust the current time step and integration order to suit integration error requirements. If integration error is too large, go back to step 6, otherwise, step 10.

Step 10. Execute steps 1 to 5 again to obtain updated values for q, q, and X, report the solution if at or past the desired reporting times. Increment / and return to step 6 or stop, if the final time is reached.



S7 S8 S 9 S10

Fig. 4 A computer model of the M551.

Sl2 S13 S I4 SIS SIB SI9 S20 S2>

Fig. 5 A computer model of the coupled M113's.

Fig. 6 Control system diagram of the coupled M113's.

5. Example Problems The analysis method presented here is demonstrated by

analyzing two highly nonlinear, multiple degree-of-freedom vehicular systems. Figure 4 illustrates a planar model of a 13 degree-of-freedom tracked vehicle with five roadwheels per side [19]. The transient response of the vehicle to a large weapon firing impulse is desired. Figure 5 illustrates a simplified planar model of a 24 degree-of-freedom system of two tracked vehicles connected by a rotational joint and a hydraulic actuator [20, 21]. A nonlinear electro-mechanical-hydraulic system with positional feedback control, shown in Fig. 6, controls the relative angular pitch rate between the two vehicles through the hydraulic actuator. The transient response of the coupled vehicles to pitch-up and pitch-down command signals is desired.

Since the two models have many similar characteristics, they are described simultaneously to avoid repetition. All of the kinematic constraint relations and many of the internal and external forces applied to the systems are supplied through the standard formulation of section 2. Additional generalized forces are added to incorporate such effects as suspension characteristics, track supports, wheel-ground reaction, and frictional forces.

The first example model consists of 8 rigid bodies. Bodies Bl to B5 represent roadwheels, where mass and inertia characteristics of roadwheels on opposite sides of the vehicle are combined. Body B6 includes the chassis mass and inertia. Bodies B7 and B8 represent massless road arms connecting the front and rear roadwheels to the chassis with revolute joints. These bodies are introduced for the convenience of connecting shock absorbers between road arms and chassis. The

^ JX:

FO

RC

E

REBOUND^

100

80

60

4 0

20

-

-

WHEEL NO.

1 2 and 3

4 and 5

« - " i ^ - *

JOUNCE

! 1 1

1

O

I

1

1 J

^ 1

1 J

1 _ J _ 5 0 5 10 15 2 0 25

DISPLACEMENT (cm) Fig. 7 Suspension spring characteristic for the M551.

remaining road wheels are attached using massless link constraints.

The vehicle has a torsion bar suspension. Spring-damper-actuators SI to S5 are employed to incorporate the nonlinear suspension characteristics, including jounce stops. Figure 7 illustrates typical force-displacement suspension characteristics. This force displacement relation as well as other nonlinear functions are incorporated into the model from discrete data points, using a cubic spline curve fitting algorithm. Actuators S12 and SI3 implement the damping characteristics of Fig. 8. Springs S6 to Sl l shown in Fig. 4 simulate a pretensioned track connecting the drive sprocket and roadwheels. These nonlinear springs do not support compression.

Additional generalized forces are introduced to represent roadwheel-ground interaction. These nonlinear functions allow the wheels to leave the ground and include damping. Frictional forces proportional to the normal component of the wheel-ground reaction oppose tangential displacement at the point of contact.

The vehicle of Fig. 4 is allowed to achieve an equilibrium configuration prior to initiating transient analysis. Then a force with an impulse of 16900 N»s (604000 N lasting .0280 s), that simulates recoil due to firing the main weapon, is applied to the chassis.

The simulation was carried out for 1.5 seconds, at which time the vehicle had essentially returned to rest. The problem had been earlier solved using a stiff, implicit method of integration that simultaneously solves differential and algebraic equations. Double precision was required to obtain a solution. It is solved here using the equation reduction method of sections 3 and 4 in single precision. An Itel AS6 computer was used for both simulations. Figs. 9 and 10 compare displacement of the centers of the drive and idler sprockets (marked A and B respectively on Fig. 4) with exisitng field test data. Table 1 compares execution costs by the previous and present methods.

The coupled tracked vehicle system of Fig. 5 consists of 12 rigid bodies. Bodies Bl to B10 constitute combined roadwheels, as in the previous example. Bodies Bl l and B12 are the front and rear sprung chassis, respectively. Roadwheels are connected to the chassis by massless links and the two chassis are connected by a revolute joint. Springs SI to S10 provide for torsion bar suspension characteristics using curves similar to Fig. 7. Springs Sl l to S16 and S17 to S22 simulate the pretensioned track on the front and rear units, respec-

252/Vol. 104, JANUARY 1982 Transactions of the AS ME


1 1 1

80 60 40

BLOW OFF

z

OR

CE

(k

u_

l

2 0

3 0

2 0

10

0 /

An

2 0

— /

' i

2 0

PISTON

-BLOW

z o

. CO

CO

M-

PR

ES

4 0

OFF

i l

60 80

VELOCITY (cm/sec)

OU

ND

m UJ

5 -r-

Fig. 8 Suspension damping characteristic for the M551.

MEASURED

SIMULATED

REARWARD——"

5 10 15

DISPLACEMENT (cm)

Fig. 9 Displacement of front idler sprocket for M551.

tively. As before, they do not support compression. Road-wheel support and friction forces similar to the previous model are included. Because of the large pitch displacements, the front and rear sprockets identified by letters A to D in Fig. 5 may contact ground. Provisions are made to prevent these sprockets from penetrating ground and frictional effects are included.

An additional set of differential equations is formulated for the electro-mechanical-hydraulic control system shown in Fig. 6. These equations are solved along with the mechanical system differential equations. The two systems are coupled together by spring-damper-actuator number S23, as illustrated in Figs. 5 and 6. Hydraulic pressure is converted to

\ REARWARD • \

5 10

DISPLACEMENT (cm)

15

— MEASURED

- SIMULATED

Fig. 10 Displacement of rear drive sprocket for M551.

Table 1 Comparison of execution costs for M551. Implicit method*

Explicit coordinate* reduction method

5 to 10 minuted execution time for 1 to 2 seconds simulation time

.5 to 1.5 minutes execution time for 1 to 2 seconds simulation time

"Programs were run on an Itel AS61 computer. Execution times depend on the amplitude and frequency of oscillations in the model. The above time ranges are due to varying model spring rates, and damping and friction coefficients.

Table 2 Comparison of execution costs for M113's. Implicit method*

Explicit coordinate* reduction method

8 to 20 minutes execution time for 4 seconds simulation time

2.5 to 5 minutes execution time for 4 seconds simulation time

"Programs were run on an Itel AS6 computer. Execution times depend on the amplitude and frequency of oscillations in the model. The ahove time ranges are due to varying model spring rates, and damping and friction coefficients.

an actuator force that is transmitted to the vehicles by the actuator. Positional feedback is provided by monitoring actuator length /23 and hydraulic fluid flowrate is determined by monitoring v2i, the time rate of change of the actuator length.

The pitching motion of the vehicles is controlled as the driver moves a joy stick forward or rearward. Positional feedback allows proportional pitch contol. A 100 percent pitch down command amounts to the driver pushing the stick forward to a mechanical stop and a full rearward displacement gives a 100 percent pitch up command. With the vehicle initially in a static, zero pitch attitude on level ground, the transient response of the system to 100 percent pitch-up and pitch-down commands is desired.

The differential equations were again solved by the stiff implicit and explicit algorithms on an Itel AS6 computer. Figs. 11 and 12 compare pump yoke displacement, actuator displacement, and actuator-hydraulic pressure (for the pitch-up and pitch-down commands respectively) with existing field



J— i'f^s^ I 1 \ ^ - s ^ PUMP YOKE DISPLACEMENT

J l NN ^ \ . PITCH UP

"H V v ^^-*-^

l | HEASimm "-•*.. .

[ | SIMULATED ~~

' 1 i i I

/

I MEASURED

SIMULATED

I

"-T^v

1

PUMP YOKE DISPLACEMENT

PITCH DOWN

^ * ^ .

1 _ _ j

/

ACTUATOR PRESSURE PISTON END

PITCH DOWN

_ _ _ _ — . MEASURED

SIMULATED

1 I

J-̂ *̂ -—'/

1 J

Fig. 11 Closed loop transient response of M113's-100-percent command up.

Fig. 12 Closed loop transient response ol M113's-100-percent command down.

test data. Fig. 13 compares simulated and actual vehicle pitch angles. Table 2 compares simulation costs of the solution methods.

References 1 Orlandea, N., Chace, M. A., and Calahan, D. A., "A Sparsity-Oriented

Approach to the Dynamic Analysis and Design of Mechanical Systems, Parts I and I I , " ASME, Journal ofEngineering for Industry, Vol. 99, Aug. 1977, pp. 773-84.

2 Orlandea, N., Calahan, D. A., "Description of a Program for the Analysis and Optimal Design of Mechanical Systems," Report No. AFOSR-TR-73-2026, University of Michigan, Nov, 15, 1973.

3 Haug, E. J., Wehage, R. A., Barman, N. C , "Dynamic Analysis and Design of Constrained Mechanical Systems," to appear in The Journal of Mechanical Design.

4 Duff, I. S„ "MA29-A Set of FORTRAN Subroutines for Sparse Un-symmetric Linear Equations," Report No. AERE-R.8730, Computer Science and Systems Division, AERE Harwell, Oxfordshire, July, 1977.

5 Gear, C. W., "Simultaneous Numerical Solution of Differential-Algebraic Equations," IEEE Transactions on Circuit Theory, Vol. CT-18, Jan. 1971, pp. 89-95.

6 Paul, B., "Analytical Dynamics of Mechanisms—A Computer Oriented Overview," Mechanism and Machine Theory, 1975, Vol. 10, pp. 481-507.

7 Chace, M. A., and Sherh, P. M., "Adaptation of Computer Techniques to the Design of Mechanical Dynamic Machinery," ASME Paper 73-DET-58, 1973.

8 Paul, B., "Dynamic Analysis of Machinery Via Program DYMAC,' ASME Paper 770049, 1977.

9 System/360 Scientific Subroutine Package (360A-CM-03X), Programmer's Manual (H20-0205-4), 1970.

10 Paul, B., "A Unified Criterion for the Degree of Constraint of Plane

254/ Vol. 104, JANUARY 1982 Transactions of the ASME


- 15 -L-

Fig. 13 Pitch angles of M113 'S .

Kinematic Chains," ASME, Journal of Applied Mechanics, Vol. 27, Series E., Vol.82, 1960, pp. 196-200.

11 Freudenstein, F., "On the Variety of Motions Generated by Mechanisms," ASME, Journal of Engineering for Industry, Vol. 84, Ser. B, 1962, pp.156-160.

12 Paul, B., and Krajcinovic, D., "Computer Analysis of Machines with Planar Motion, Part 1-Kinematics, Part 2-Dynamics," ASME, Journal of Applied Mechanics, Vol. 37, No. 3, Trans. ASME, Vol. 92, Series E, Sept. 1970, pp. 697-712.

13 Forsythe, G. E., and Moler, C. B., Computer Solution of Linear Algebraic Systems, Prentice-Hall, Englewood Cliffs, N.J., 1967.

14 Sheth, P. N., "Improved Iterative Techniques for the (4 X 4) Matrix Method of Kinematic Analysis," M.S. Thesis, Mechanical Engineering, University of Wisconsin, 1968.

15 Sheth, P. N., "A Digital Computer Based Simulation Procedure for Multiple Degree of Freedom Mechanical Systems with Geometrical Con

straints," Ph.D. Thesis, Mechanical Engineering, University of Wisconsin, 1972.

16 Shampine, L. F., and Gordon, M. K., Computer Solution of Ordinary Differential Equations: The Initial Value Problem, W. J. Freeman, San Francisco, Calif., 1975

17 Greenwood, D. T., Principles of Dynamics, Prentice-Hall, Englewood Cliffs, N.J . , 1965.

18 Haug, E. J., Arora, J. S., Applied Optimal Design, John Wiley & Sons, New York, NY., 1979.

19 Parent, R., "Integrated Engineering Service Test of XM551," TECOM Report #DPS-1897, March, 1966.

20 Beck, R. R., and Kamm, I. O., "Cybernetically Coupled Research Vehicle," Society of Automotive Engineers, Rep. 75-217, Feb. 1975.

21 Beck, R. R., and Wehage, R. A., "The Modeling and Simulation of Two Coupled M-113 Armored Personnel Carriers," Proceedings of the Tenth Annual Pittsburgh Conference on Modeling and Simulation, Vol. 10, Part 2, 1979, pp.353-59.



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