INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING VOL. 2 1 , 45 1-464 ( 1985)

GENERALIZED CO-ORDINATE PARTITIONING IN STATIC EQUILIBRIUM ANALYSIS

OF LARGE-SCALE MECHANICAL SYSTEMS?

PARVIZ E. NIKRAVESH' AND MANOHAR SKINIVASAN+

Aerospace and Mechanical Engineering Department, University of Arizona, Tucson, A%, U.S.A. Department o j Mechanical Engineering, Manhattun College, Riverdale, N r; U.S.A.

SUMMARY This paper presents a computer-based method for automatic formulation and efficient numerical solution of static equilibrium equations for nonlinear constrained mechanical systems with conservative forces. Nonlinear holonomic constraint equations and a potential energy function are written in terms of a maximal set of Cartesian generalized co-ordinates. A stable static equilibrium configuration is found by minimizing the potential energy of the system, subject to the kinematic constraint equations, i.e. constrained optimization. A Gaussian elimination algorithm with full pivoting decomposes the constraint Jacobian matrix, identifies dependent and independent co-ordinates and construct an influence coefficient matrix that relates variations in dependent and independent co-ordinates. This information is employed to convert the constrained optimization problem to an unconstrained optimization problem. A simple example is presented to illustrate the method. An algorithm that may be used in analysis of large-scale systems is presented.

INTRODUCTION

A method of formulating and solving differential equations of motion for planar mechanical systems, subject to holonomic constraints, was recently presented.' For planar systems, a vector of three Cartesian generalized co-ordinates is defined for each rigid body of the system. Standard and user supplied constraints are formulated, yielding algebraic relations among the generalized co- ordinates common 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. Gaussian elimination with full pivoting and subsequent L,-U factorization is applied to the Jacobian matrix of the constraint equations to: (1) determine the number of system degrees-of-freedom, (2) partition the generalized co-ordinates into independent and dependent sets, and (3) construct an influence coefficient matrix expressing dependent velocities in terms of independent velocities. A predictor/corrector numerical integration algorithm is employed to solve for independent generalized co-ordinates and velocities, i.e. a minimal system of differential equations of motion of second order, equal in number to the constrained system degrees-of-freedom. This method is formulated in the form of a general- purpose computer program called the Dynamic Analysis and Design System (DADS-2D).

In a later development, another version of the preceding code, called DADS-3D, for dynamic analysis of mechanical systems in spatial motion was presented.2 In this formulation, Euler paramters are used as rotational generalized co-ordinates. Euler parameters, in contrast to Euler

' Rescarch supported by U S . National Science Foundation, Project N. MEA 81-12947. 'Associate Professor of Mechanical Engineering. $Assistant Professor of Mechanical Engineering.

0029-598 1 /85/03 045 1 -- 14$01.40 0 1985 by John Wiley & Sons, Ltd.

Received 17 November 1983 Revised 28 February I984

452 P. E. NIKRAVESH AND M. SRINIVASAN

angles or any other set of three rotational co-ordinates, have no critical singular cases3 and are computationally more efficient.

Many practical statically indeterminate mechanical and vehicular systems have been analysed starting from a state of static equilibrium, or from a state of partial equilibrium in which certain generalized co-ordinates are held fixed at prescribed initial values, while the remainder of the system is allowed to seek equilibrium, consistent with constraints. These co-ordinates are then released to initiate transient analysis. Aithough it may not be apparent, for most large-scale systems, determination of a stable static equilibrium configuration can be a much more complicated problem than transient dynamic analysis of the same mechanism. A method for static equilibrium analysis was recently presented' and has been used in the DADS codes. In this method, static equilibrium of the constrained system is achieved by setting velocity and acceleration vectors to zero in the equations of motions, in which case equations of motions reduce to nonlinear algebraic equations that are solved iteratively with the kinematic constraint equations. This method possesses several drawbacks: (1) since the resultant algebraic equations are highly nonlinear and initial estimates of some of the unknown, such as Lagrange multipliers, are not readily available, iterative procedures may not find a solution, and (2) even if a solution is found, it may represent an unstable equilibrium configuration.

The method for static equilibrium analysis presented in this paper minimizes the potential energy of the system. As long as all forces contributing to the potential energy of the system are conservative, this method can be applied. The system may include nonconservative forces that do not affect the potential energy of the system; e.g., damping forces that are only a function of velocity. However, for nonconservative forces that are functions of initial (starting) positions, such as frictional forces, this method can not be applied. For such systems, a realistic configuration for static equilibrium can be found only by solving the transient dynamic problem and allowing sufficient settling time for equilibrium to be achieved.

In this paper, a general formulation for kinematics of mechanical systems is reviewed. The formulation is applicable to both planar and spatial motion. Generalized forces and equations of rigid body dynamics are discussed. Then, a potential energy function of some common forces and forcing elements is derived. A formulation and numerical algorithm for minimizing the potential energy function is presented and a simple example problem is given to illustrate the process more clearly. Finally, a general algorithm for static equilibrium analysis is presented.

MECHANICAL SYSTEM REPRESENTATION

Generalized co-ordinates

In order to specify the position of a rigid body in an inertial (global) x-y-z co-ordinate system, it is sufficient to specify the spatial location of the origin and the angular orientation of a 5-y-C co- ordinate system that is rigidly attached to the body. Let the Si-qi-Ci axes be attached to the ith body of the system, as shown in Figure l(b), where the origin Oi is located at the centre of mass of the body. A point P on body i is located in the inertial co-ordinate system by

rP = ri + AisiP

where sip = [tP, qp, C']T are the vector of co-ordinates of P in the co-ordinate system, ri E [x, y, z]? are the vector of co-ordinates of Oi in the x-y-z co-ordinate system, and Ai is the rotational transformation matrix of body i. The matrix Ai is a 3 x 3 matrix that can be expressed in terms of any set of rotational generalized co-ordinates for the body; e.g., direction cosines, Euler angles, or Euler parameters, etc. For planar motion, as shown in Figure l(a), only two translational

GENERALIZED CO-ORDINATE PARTITIONING 453

Y

Figure 1. Body-fixcd and global co-ordinate systems: (a) planar, (b) spatial

co-ordinates and one rotational co-ordinate are needed. In this case, equation (1) represents two algebraic equations and Ai is a 2 x 2 matrix.

Constraint equations

Standard constraints between rigid bodies are taken as friction free (workless) joints. Formulations of two spatial constraints are presented here. Detailed derivations for these and many more planar and spatial kinematic constraints can be found in References 1 and 2, respectively.

Spherical joint. Figure 2 shows two adjacent bodies i and j connected by a spherical joint (ball joint). A vector equation defining the joint can be written as

(2) r. + s!' - SP - r . = 0 I r J J

Using equation (l) , this equation can be rewritten in the form

ri + A&' - ri - Ajs[i' = 0

where the centre P of the ball and shocket is located by the body-fixed vectors sip and sip in bodies i

X

Figure 2. Spherical joint

454 P. E. NIKRAVESH AND M. SRINIVASAN

1

Figure 3. Universal joint

and j, respectively. Equation (2) consists of three algebraic equations that represent a spherical joint between bodies i andj .

Universal joint. A universal joint between bodies i and j is shown in Figure 3. Since point P is common to both bodies, equation (2) applies to this joint. The additional constraint is that the two vectors gi and g j remain perpendicular, thus their dot product is set to zero; ie.,

(3) g;gj = 0

Equations (2) and ( 3 ) yield a total of four algebraic equations that represent a universal joint between bodies i and j.

Force of springs Internal forces acting between bodies due to the action of springs may be obtained by a process

similar to constraint equation development.',2 For example, Figure 4 shows a translational (point- to-point) spring between bodies i and j , with attachment points Pi and Pi. Vectors I, from Pi to P , can be written as

I = r . + s . - r. - s. l J L 1

J J J = r . + A s'p - ri - A&"

and the magnitude 1 of vector I can be found from

(4)

J X

Figure 4. Translation spring

GENERALIZED CO-ORDINATE PARTITIONING 455

For a spring with linear characteristics, the spring force is

where k and 1" are the spring constant and undeformed spring length, respectively. If the spring has nonlinear characteristics, f(., can be determined from other expressions or curves in terms of A1 = 1 - 1'. The spring force f, acting at point Pi on body i can be expressed as

where 1/1 defines a unit vector from Pi to Pj. The spring force fj acting at point P, on body,j is the opposite direction of fi; i.e.,

Force determination for other types of force elements; e.g., torsional springs, is similar to that for translational springs.

EQUATIONS OF KIGID BODY DYNAMICS

For a mechanical system with b rigid bodies, the vector of generalized co-ordinatcs is denoted by

q = cqr,q:, . . . ,qTI' (9) where ql; i = 1,. . . , h, is the vector of generalized co-ordinates of body i. In planar motion, q, has three components two translational and onc rotational- -and for spatial motion q, has six components three translational and three rotational. However, if Euler parameters are employed to describe spatial angular orientation of a body, then q, will have seven components.2 The dimension on q, i.e. the total number of generalized co-ordinates in the system, is designated as n.

It is assumed that there are m holonomic constraints in the system written as

@(q, t ) = 0 (10) where t denotes the time variable. These constraints contain the kinematic joint equations or any other algebraic relationships between the generalized co-ordinates that may exist.

Lagrange's equations of motion for an unconstrained body is written as4

where Ti is the kinematic energy of body i, qi is the vector of velocities and g, is the generalized force vector. The kinetic energy of body i is expressed as

T . I 2 i I l = ' m i.'!f. + 'u<';l!w! 2 1 1 1 (121

where mi is the mass, JI is the inertia tensor and 0; is the vector ofangular velocities of the body. For a constrained body, subject to some or all of the constraints of equation (lo), equation (1 1) is modified to

where il is a vector of Lagrange multipliers. Substitution of equation (12) into equation (13), after defining proper angular co-ordinates, yields the equation of motion for body i as

~ ~ i j ~ + aq a = g, (14)

456 P. E. NIKRAVESH A N D M. SRINIVASAN

where Mi is the generalized mass matrix, qi is the vector of accelerations, and aq, = a<D/dqi. The total system equations of motion for b rigid bodies is then

Mq + @:A = g (15) where M = diag.[M,, M,, . . . , M,,], q = [q:, qz,. . . , q3’ and g = [g:, g;,. . . , gZlT. The term @:A in equation (15) provides the reaction forces at the joints and vector g contains all of the external forces and moments acting on the bodies, such as spring forces, gravitational forces, etc.

POTENTIAL ENERGY FUNCTION

For a conservative system, the generalized force g can be derived from a potential energy function I/, such that4

The potential energy I/ depends only on generalized co-ordinates q and not on time or velocities. The most common factors contributing to the system potential energy are gravitational field, externally applied forces and springs.

Gravitational fields

Assume that rigid body i can move in a global x-y-z reference frame, as shown in Figure 5. Furthermore, assume that the gravitational field is acting on the system in the negative z direction. If the gravitational constant is g and the mass of the body is mi, then equation (16) can be written as

where Vi(8) denotes potential energy of body i due to gravity. Equation (17) yields

Y . t(8) = - mig(zi - zo) (1 8) where zi is the z-co-ordinate of the mass centre of body i and z,, is a constant. For simplification, the potential energy T/Cg, may be taken as zero at the x-y plane. Then, equation (18) becomes

I/. r(9) = - mizig (19) Note that the gravitational constant g is a negative quantity, e.g. g = - 9-81 m/s2.

X ry Figure 5. Body i in a gravitational field

GENERALIZED CO-ORDINATE PARTITIONING 457

External forces

written as If a constant external force fk acts on body i, the potential energy corresponding to fk can be

v$) = - ( f : x ) x i +f:y)yi +f:z)zi) (20)

where ftX,, f & and f & are the constant x-y-z components off'. In equation (20), similar to the gravitational force, the x-y-z co-ordinate system is considered as the reference frame in which the potential energy due to f & , is zero in the co-ordinate planes, i.e. in the x-y, y-z and z-x planes.

Translational springs

shown in Figure 4, may be written as Potential (strain) energy stored in the kth translational spring that connects bodies i and j, as

1

qS) = 1 f d l (21) 10

If the spring characteristic is linear, then f ( s , = k(E - lo) and equation (21) yields

vlt, = $k(l - l0)2 (22) If the spring has nonlinear characteristics and the force f ts , is available as a nonlinear function of Al, then equation (21) may be integrated to obtain the potential energy.

Contribution to the potential energy of a system due to other sources, such as torsional springs or external moments, can be determined as in the preceding derivations.

Total potential energy

The total potential energy of a mechanical system may be defined as the sum of the potential energies of the springs, externally applied forces, etc. If there are b rigid bodies in the system the total potential energy due to the gravitational field, using equation (19), is

b

v(s) = C vi(s) (23) i = 1

For p constant externally applied forces, equation (20) yields

Similarly, if there are z translational springs in the system, equations (21) or (22) yield the total potential energy of the spring as

The system total potential energy is thus

I/= Vk7) + v,n + V(S) (26) If there are any other force elements in the system that are not discussed here, their contribution to the total potential energy must be added to equation (26).

458 P. E. NIKRAVESH A N D M. SRINIVASAN

STATIC EQUILIBRIUM ANALYSIS

The static equilibrium configuration of a mechanical system may be determined by evaluating the position for which the potential energy function is at its minimum. For the potential energy function V= V(q) of equation (26), the minimization problem can be stated as

minimize I/= V(q) subject to constraints @(q) = 0

Equation (27) represent a constrained minimization problem i.e. the kinematic constraints @(q) = 0 must be satisfied for all feasible q.

Many constrained optimization algorithms are available and can be applied to the problem of equation (27). However, equation (27) can be transformed to an unconstrained minimization problem. The method is based on the generalized co-ordinate partitioning technique of Reference 1.

For a mechanical system with a vector q of n generalized co-ordinates and m independent constraint equations @(q) = 0, the vector q may be partitioned into q z [uT, vTIT, where u is an m vector of dependent co-ordinates and v is an n-m vector of independent co-ordinates. Accordingly, the Jacobian matrix a,, is partitioned into a,, f [a,,,@,]. The total derivative of thc constraint equations @(q) = 0 is written as

dcf, = cf,, dq = 0 (28)

d@=cf,,,du+cD,dv=O (29)

or

Since m constraint equations are assumed to be independent, the submatrix a,, of aq is selected to be nonsingular; therefore equation (29) can be rewritten in the form

du = - a,,- 'av dv (30)

H = - cf,,,'cf,v (31)

d u = H d v (32)

The matrix H, called the influence coefficient matrix, is defined as

Then, equation (30) becomes

Partitioning of the generalized co-ordinates q into vector u and v may be achieved by employing L-U factorization with full pivotingo5 to decompose the constraint Jacobian matrix into and aY matrices. L-U factorization of the Jacobian with full pivoting permutes the rows and thc columns of the matrix. The variables corresponding to the columns of the nonsingular matrix are considered as dependent co-ordinates u. The variables corresponding to the columns of matrix a, arc considered as independent co-ordinates v. The influence coefficient matrix H is calculatcd according to equation (31) by most of the available L- U factorization algorithms. Construction of H is thus a byproduct of L-U factorization and is evaluated efficiently.

The total differential of the potential energy function V may be written as

dV= V,du + V,dv (33) Substitution of equation (32) into equation (33) yields

- = VuH + V, dV dV (34)

GENERALIZED CO-ORDINATE PARTITIONlNG 459

Hence, the minimization problem can be restated as

minimize V = V(v) dV

where - = V,H + V, dv (35)

Equation (35) is an unconstrained optimization problem in n-m variables. The gradient d V/dv is an n-m vector. This vector is needed for most of the commonly used optimization algorithms, e.g. steepest descent, conjugate gradient, etc.

The gradient vector d V/dv can be determined easily at a feasible position q, i.e. where @(q) = 0 is satisfied. The partial derivative V, is the negative of the generalized force vector gT at the position q. If g is partitioned, according to q = [uT, vTIT, into g = [g;,, gT,,IT, then

V,= -81'

V = - g T ( u ) (36) (U)

Substituting equation (36) into equation (34) yields the desired result:

dV dv - = - gT,p - g;, (37)

GENERAL COMPUTATIONAL ALGORITHM

The following steps outline an algorithm, based on minimization of potential energy, for finding a stable equilibrium configuration. A main routine partitions the generalized co-ordinates into dependent and independent co-ordinates, prior to entering an unconstrained minimization routine. Any well-developed minimization algorithm such as the conjugate gradient or Fletcher- Powell6 method can be used.

Algorithm

(a) Provide an initial estimate for the generalized co-ordinates q. (b) Partition q into u and v. (c) Enter the minimization routine. Minimization routine (general steps):

(d) Knowing v, call the function routine to determine the potential energy function V (equation 26) and the gradient vector dV/dv (equation 37). Function routine: (e.1) Knowing v solve equation (10) for u (iteratively).

If u cannot be found, set Flag = 1 and return to the minimization routine. Otherwise continue to step (c.2).

(e.2) Knowing u and v, determine thc potential energy V from equation (26) and the generalized force vector g from equation (1 6).

(e.3) Partition the Jacobian matrix a, into matrices a,, and and determine matrix H from equation (31).

(e.4) Partition the generalized force vector g into g,,, and g,,, vectors. (12.5) Determine the gradient vector dV/dv from equation (37). (e.6) Return to the minimization routine with Flag = 0.

(f) If Flag = 0, then continue the minimization process. If Flag = 1, i.e. a feasible solution for

460 P. E. NIKRAVESH AND M. SRINIVASAN

the kinematic constraint equations for the assigned values of v does not exist, then one of the following two conditions may exist: (1) The step taken by the minimization routine is too large. Make the step smaller, e.g.

cut it by half, update v and return to (d) to repeat the process. (2) The step taken by the minimization routine is too small. This may be an indication

that the set of co-ordinates considered as independent is not an adequate set. In order to define a new set of independent co-ordinates return to (b).

This algorithm has been implemented in the Dynamic Analysis and Design System (DADS) computer program. The Fletcher-Powell6 minimization routine has been employed in the algorithm. The method has proved to be efficient in determining stable equilibrium configuration for large-scale mechanical systems.

EXAMPLES

Pendulum

A simple example is presented here to assist in understanding the static equilibrium analysis method presented in preceding section. The mechanism under consideration is a single pendulum connected to a spring with linear characteristics. The problem is formulated in planar co-ordinates in order to reduce the number of variables, as shown in Figure 6, assuming d = 1 m, pendulum mass = 2 kg, pendulum polar moment of inertia = 2/3 kg.m2, undeformed spring length = 2 m, and k = 3 N/m.

A body-fixed 5-q co-ordinate system is embedded in the pendulum with its origin at C, the centre of mass of the body. The global co-ordinates x and y of point C and the angle 4 between the ( and x axes are the generalized co-ordinates of the pendulum, i.e. q = [x, y, +IT.

The constraint equations for the revolute joint are

.=[ X - C O S 4 . ]=o y-sinq5

The Jacobian matrix for equation (38) is

I 1.0 0.0 s in4 0.0 1.0 -cos&

Figurc 6. Simple pendulum-spring mechanism

GENERALIZED CO-ORDINATE PARTlTlONING 46 1

At the initial configuration, the vector qo = C0.707, - 0.707, - n/4IT satisfies the constraint equations. Substituting qo in Q4 and performing an G U factorization with full pivoting yields the dependent and independent co-ordinates and the influence coefficient matrix. However, since this is a very simple example and the Jacobian matrix of equation (39) has a simple form, the Jacobian can be partitioned in an explicit form, rather than in terms of the numerical values of its elements. With full pivoting, the Jacobian is partitioned as

@.=[ 1.0 0.0 1 , a"=[ s i n 4 1 0.0 1.0 - cos 4

where u = CX,YIT, v = C41T

For this simple problem, the form of the Jacobian suggests the above partitioning even without factorization. Since the form of and 0" are simple, H can be expressed explicitly as

Thus, from equation (32),

The total potential energy function can be written as the sum of gravitational potential and the energy stored in the spring, i.e.

V = - (2)( - 9.81)~ + 53)(1- 2)2 (43) here 1 = ((x' + 1)2 + (y' + 2)2)1'2, x' = x + cos 4, and yp = y + sin 4. The generalized force vector, from g = - VT, is

- 3(1 - 2)( - (x' + 1) sin+ + (y' + 2) cos $)/1

- 3(t - 2)(x' + l)/l 1 3(E - 2)(yp + 2)/Z ~ 19-62

which may be partitioned as

1 - 3(1- 2)(XP + 1)/1 gw = c - 3(1- 2)(yp + 2)/1- 19.62

(44)

(45)

g(") = [ - 3 1 - 2)( -- (xp + 1) sin + + (y' + 2) cos$)/l]

Hence, the gradient vector dV/dv of equation (35) is written as

I" - ::s$] 4v=[ 3(1- 2)(XP + 1 ) / 1 dv 3(1-- 2)(yp + 2)/1- 19.62

+ [3(1- 2)( - (x' + l)(yP - y) + (yP + 2)(XP - x))/l] which in this problem is a scalar.

Any minimization algorithm may be employed to solve this problem, since the function I/ and its gradient dV/dv is available. Table I shows the reduction in the potential energy from the initial position to the static equilibrium position. Equilibrium is achieved in only one iteration.

462 P. E. NIKRAVESH A N D M. SRINIVASAN

Table I. Numerical results for pendulum model

Iteration X Y 9 V

0 0'707 -0.707 - 0.785 - 13.52 1 0.188 -0.982 - 1.381 - 18.68

Connecting Rod

Slider

Figure 7. Slider-crank mechanism: (a) initial configuration, (b) intermediate configuration, (c) equilibrium configuration

Slider crank

The above algorithm was used to detcrmine the static equilibrium of a slider-crank mechanism, as shown in Figure 7(a). The initial crank angle is 45 degrees. The algorithm selected the rotational co-ordinate of the crank as the independent generalized co-ordinates throughout the simulation and found the cquilibrium in two iterations, as shown in Table 11. SI units were employcd. For this simulation.

A second simulation of the same mechanism was made with non-SI (but consistent) units. Since the entries of the Jacobian matrix are scaled differently depending upon the choice of the units, therefore, the selected independent generalized co-ordinates by the algorithm are affected. In the second simulation, the algorithm initially selected the x-co-ordinate of the slider as independent gcncralized co-ordinate (Figure 7(a)). In one iteration, the mechanism found the orientation shown in Figure 7(b). At this orientation, the algorithm switched to the y-co-ordinate of the connecting rod as the independent generalized co-ordinate. Two more iterations led to the equilibrium state shown in Figure 7(c).

Table 11. Numerical result for slider-crack model

Iteration Crank angle V

0 45" 55.475 1 - 87.2445 - 78.362 2 - 89.998 - 78.453

GENERALIZED CO-ORDINATE PARTITIONING 463

Figure 8. Tractor-trailer

Tractor-trailer

As a large-scale example, a tractor-trailer is shown in Figure 8. The front suspension of the tractor is a solid axle, supported from the chasis by two leaf springs. Two shock absorbers arc attached between the chassis and front axle. The rear suspension of the tractor consists of two driven tandem axles. Leaf springs supporting the axles are pivoted about the chassis on each side. The ends of the leaf spring rest in slippers on each of two tandem axles. Kinematics of the trailer bogie suspensions are essentially the same as the tandem axles of the tractor. The fifth wheel hardware coupling the tractor and trailer is configured to provide articulation between the tractor and trailer. Vertical tyre flexibility effect is included in the model for static equilibrium analysis. This system, with 25 degrees-of-freedom, was modelled by 17 moving bodies and 77 independent kinematic constraint equations representing the connectivities between the bodies. Translational spring elements were used to model the suspension system with nonlinear characteristics (equation 21).

Table I11 shows the change in the potential energy of the system versus iteration number. The static equilibrium is achieved in only four iterations. Note that, for this system, the initial estimates on the co-ordinates were given as close to the equilibrium as possible, e.g. the vertical position of the centre of gravity (e.g.) of the trailer was assigned an estimate 0.01 m away from the equilibrium. For this example, if the initial estimates on the co-ordinates are too far from the equilibrium, since the forces become extremely large and there are a large number of highly nonlinear equations to be considered, the search for equilibrium may diverge.

Table 111. Numerical result for tractor-trailer model

Vertical position of the Iteration c.g. of the trailer (SI) V

0.910 0.909 0.907 0.900 0-899

215,741.3 213,214.3 21 3,213.0 213,212.9 213,212.3

464 P. E. NIKRAVESH AND M. SRINIVASAN

DISCUSSION

A major advantage in using Cartesian co-ordinates to describe configuration of bodies in a mechanical system, instead of Lagrangian co-ordinates, is the simplicity of the form of the governing equations of motion for systematic generation by a computer program. However, a disadvantage associated with the Cartesian co-ordinates is the total number of co-ordinates involved in analysis. The method presented in this paper circumvents this drawback by partitioning the vector of co-ordinates into dependent and independent co-ordinates, where the number of independent co-ordinates is equal to the number of degrees-of-freedom of the system. An influence matrix relates variations of dependent co-ordinates to the variations of independent co-ordinates. The influence matrix is also used to convert the minimization process of the potential energy of a conservative system, from a constrained minimization formulation to an unconstrained minimiz- ation formulation.

Another method for finding the static equilibrium configuration of a mechanical system can be formulated by letting the velocity and acceleration vectors, q and q, in equation (1 5 ) to zero (g can be a function of q as well as 9). These n equations appended to m constraint equations of equation (10) provide a set of n + m nonlinear algebraic equations in n + m unknowns q and A. Iterative methods, such as the Newton-Raphson method, may be applied to solve these equations. A major difficulty involved with this process is to provide adequate initial estimates for the unknowns. Even for simple systems, numerical values for Lagrange multipliers A close to the correct values cannot be estimated. In contrast, the potential energy minimization method does not require computation of A; therefore, in this respect it is advantageous.

REFERENCES

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