Impulsive motion of non-holonomic deformable multibody systems Part I: Kinematic and dynamic equations

Journal

ofSound and

IMPULSIVE

(1988) 127(2), 193-204

Vibration

MOTION

OF NON-HOLONOMIC

MULTIBODY PART I: KINEMATIC J. Department

of Mechanical

(Received

Engineering,

9 December

SYSTEMS

AND

RISMANTAB-SANY

DEFORMABLE

AND

DYNAMIC

EQUATIONS

A. A. SHABANA

University of Illinois at Chicago, P.O. Box 4348, Chicago, Illinois 60680, U.S. A. 1986, and in revised form 5 November

1987)

In this investigation, a computer-based technique for the spatial kinematic and dynamic analysis of non-holonomic multibody systems consisting of interconnected rigid and deformable bodies is developed. Non-holonomic kinematic conditions, which can be expressed in terms of a coupled set of rigid body and modal elastic co-ordinates and velocities, restrict the infinitesimal change of system generalized co-ordinates. These kinematic constraints, however, impose no restrictions on the system reference or elastic modal co-ordinates and, accordingly, the number of independent velocities is smaller than the number of independent co-ordinates. In Part I, deformable components are discretized using the finite element method and the system configuration is identified by using a mixed set of reference and elastic co-ordinates. By using the generalized co-ordinate partitioning of the system constraint Jacobian matrix, holonomic constraint equations are used to identify a set of independent reference and elastic co-ordinates, while both holonomic and non-holonomic constraint equations are used to identify the independent reference and elastic velocities. In Part II, a method for the impact analysis of non-holonomic deformable multibody systems is presented. This method can be used to predict the jump discontinuity of the non-holonomic constraint forces based on new generalized impulsemomentum equations.

1.

INTRODUCTION

Constraint relationships in mechanical systems are categorized as holonomic, or integrable, and non-holonomic, which cannot be integrated. The constraint equations can be in the form of inequalities (or unilateral) that restrict the domain of motion and in the form of equalities (or bilateral) that restrict the infinitesimal change of co-ordinates. In this investigation only the equality constraints are considered. Among holonomic and non-holonomic constraints one should distinguish those that depend explicitly on time (rheonomic) and those that do not (scleronomic). In order to distinguish between the holonomic and non-holonomic constraints, let the motion of a system, with n generalized co-ordinates, be interpreted as the motion of a single point P in an n-dimensional space called the configuration space. The holonomic constraints are those that restrict the motion to a subspace of the configuration space, while non-holonomic constraints impose no restriction on the dimension of this space. It is believed that Lagrange, in his Mtkanique Analytique [l, p. 41, did not suspect the existence of such non-holonomic constraints. Therefore he assumed that arbitrary variations of independent co-ordinates could be chosen for any system once allowance had been made for the conditions imposed by the nature of the system [l, p. 51. The difference between holonomic and non-holonomic systems was first introduced by Hertz in 1894 [ 1, p. 51. Chaplygin, Voronec, Appell, Boltzmann, Hamel, Volterra, Maggi 193 0022-460X/88/230193+

12 %03.00/O

@ 1988 Academic Press Limited

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and others have also studied non-holonomic systems and each made a significant contribution to this branch of mechanics. Chaplygin in 1895 derived his equations of motion for a class of non-holonomic systems, called Chaplygin systems. This class includes conservative non-holonomic systems in which the variations of a number of co-ordinates equal to the number of degrees of freedom of the system may be assumed to be independent. The remaining co-ordinates which are equal in number to the number of kinematic constraints occur neither in the Lagrangian, nor in the coefficients of the equations of the non-integrable kinematic constraints. As pointed out in reference [ 1, p. 1011, Chaplygin’s systems are remarkable in that their dynamical equations governing the motion can be isolated from the non-integrable constraint equations. Voronec in 1901 relaxed some of the assumptions made by Chaplygin and derived equations of motion for a larger class of non-holonomic systems using the Hamilton-Ostrogradskii variational principle. He also derived the dynamic equations of non-holonomic systems in terms of the quasi-coordinates [ 1, p. 1141. However, when a system happens to be a Chaplygin system, it can be easily shown that Voronec’s equations are identical to Chaplygin’s equations [ 1, p. 1141. Appell in 1899 derived a simple form of the equations of motion, for holonomic and non-holonomic systems, in terms of a new scalar function S, later called the acceleration function, analogous to the kinetic energy in Lagrange’s equations. The technique required in using the Appell’s equations is fairly simple. One begins by determining the number m of degrees of freedom of the system, chooses m independent co-ordinates which may be Lagrangian co-ordinates or quasi-co-ordinates, whichever is more convenient and, next, expresses the acceleration function S in terms of generalized co-ordinates, generalized velocities, and only m generalized accelerations. However, in Appell’s equations the main drawback is that it is usually much harder to define the acceleration function for many systems than to define the kinetic energy. In all the dynamic formulations derived by Appell, Chaplygin, Voronec and others, and those considered in other relatively recent investigations [2,3,4,5, p. 2141, the attempts were mainly to eliminate the dependent variations and attain a set of equations of motion equal in number to the number of degrees of freedom of the non-holonomic systems. In fact the Lagrange form of d’Alembert’s principle, known as the basic postulates, is the starting point in deriving all these equations. Another method of treating systems with non-holonomic constraints, which is also based on the basic postulates, is the method of Lagrange multipliers. In this method, auxiliary parameters are introduced in order to retain all the generalized co-ordinates variations, whether they are dependent or independent. The number of these auxiliary parameters is equal to the number of dependent variations, and they are intimately related to the generalized reaction forces of holonomic and non-holonomic constraints. Although Lagrange may not have realized that the existence of non-holonomic constraints imposes no restrictions on the possible motion of the system, the principle of d’Alembert-Lagrange is valid for non-holonomic systems. A comprehensive treatment of the subject of nonholonomic system has been presented in reference [l], wherein different formulations are presented and compared. Very few investigations, however, have been devoted to the numerical aspect of the problem [6]. Furthermore, no attempts have been made to study the intermittent motion due to impacts of non-holonomic multibody systems that consist of interconnected rigid and deformable bodies. Accordingly, there are no numerical schemes available in the literature that can be used to predict the jump discontinuity in the non-holonomic constraint forces due to impacts hetween the system components. In these two companion papers, a method for the dynamic analysis of non-holonomic deformable multibody systems with intermittent motion is developed.

NON-HOLONOMIC

DEFORMABLE

SYSTEMS,

1

195

In Part I, the finite element method is applied to study the dynamics of deformable systems subjected to non-holonomic constraints. Deformable components in the system are discretized by using the finite element method. The configuration of each deformable body is then identified by using two coupled sets of generalized co-ordinates: reference and elastic co-ordinates [7,8]. The dynamic equations of motion are formulated by using Lagrange’s equation, and holonomic and non-holonomic constraints are introduced to the dynamic formulation by using the vector of Lagrange multipliers. In order to reduce the number of co-ordinates, the system differential and algebraic constraint equations are expressed in terms of a coupled set of reference and elastic modal co-ordinates [7,8]. Holonomic constraint equations are used to identify the independent reference and elastic co-ordinates, while holonomic and non-holonomic equations are used to identify the independent reference and modal velocities. The state equations associated with the independent co-ordinates and velocities are then identified and integrated forward in time, using a direct numerical integration method that has variable order and variable step size [9, p. 1091. The method developed in Part I can be applied to the dynamics and control of deformable multibody systems, in which the constraint (or control) equations are non-holonomic. An example of this type is given in Part II. In Part II, a method for the dynamic analysis of non-holonomic systems with intermittent motion is developed. As reported in reference [l, p. 1591, when the forces of constraints are eliminated from the dynamic formulation, non-holonomicity introduces nothing that is fundamentally new in the case of impulsive motion, since at the instant of impact, the jump discontinuity in the system velocities are determined by using a set of algebraic equations, and the system co-ordinates and time during the short-lived interval of impact are assumed fixed. Furthermore, the coefficients in the equations of holonomic and non-holonomic constraints, written in Pfaffian forms, are constant quantities. When the forces of non-holonomic constraints are retained in the dynamic formulation by using the method of Lagrange multipliers, the algebraic generalized impulse momentum equations have to be modified in order to account for the non-holonomic constraint equations. In Part II, new generalized impulse momentum equations are developed for the impact analysis of non-holonomic systems. These algebraic equations are used to determine the jump discontinuity in the system velocity vector as well as the jump discontinuity in the holonomic and non-holonomic constraint reaction forces. Planar and spatial numerical examples are presented.

2.

NON-HOLONOMIC KINEMATIC CONSTRAINTS

A mechanical system with non-integrable kinematic constraints that cannot be reduced to geometric constraints is called a non-holonomic system. The most distinguishing property of this kind of mechanical system is that not all variations of its generalized co-ordinates, regardless of the way they are chosen, correspond to a motion of the system which does not violate its constraints. It must be emphasized that, according to this statement, the existence of a single non-integrable constraint does not necessarily mean that a system is non-holonomic, since this constraint may prove to be integrable by virtue of the remaining constraint equations [l, p. 61. The type of non-holonomic constraints encountered in mechanical systems can usually be expressed in a form which is linear in the generalized velocities, that is G(q,, q2,. . ., qn,t)4+8(4,,4r,...,qn,t)=O

(1)

where q = [q, q2 - . * qnlT is the vector of system co-ordinates and 4 is the system velocity

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vector (a list of nomenclature is given in the Appendix). Equation (1) is called the linear, first-order non-holonomic constraint equation that must be satisfied throughout the motion of the multibody system.

3. NOTATIONS

AND

SPACE

DISCRETIZATION

The configuration of a body or substructure is determined by introducing a set of generalized co-ordinates that define the location of every point on the body with respect to a global reference frame. Let the XYZ Cartesian co-ordinate system represent an inertial frame of reference and the X’Y’Z’ represent a Cartesian body co-ordinate system. The origin of the co-ordinate system X’Y’Z’ does not have to be rigidly attached to the body. In fact a moving frame of reference is suggested for deformable bodies that undergo large angular rotational displacement. The rigid body motion with respect to the body reference can, however, be eliminated by using a set of reference conditions that define a unique displacement field [7]. The location of an arbitrary point on the body can then be defined by using two sets of generalized co-ordinates: reference and elastic generalized co-ordinates. The reference generalized co-ordinates q: define the global position and orientation of the body co-ordinate system X’Y’Z’, while the elastic generalized coordinates 4,; describe the deformation of the body with respect to the selected body reference. In terms of these generalized co-ordinates the position vector rb of an arbitrary point pi on body i, as shown in Figure 1, can be expressed as

r;= ri+Aiui

where

,.i=[Xi

yi

zi]T (293)

is the global position vector of the origin of the body co-ordinate system X’Y’Z’, u’ is the local position vector of point pi, and A’ is the transformation matrix from the ith body co-ordinate system to the inertial co-ordinate system. The transformation matrix A’ can be expressed as a function of a unit vector vi = [ ui u5 u:lT, along the instantaneous axis of rotation, and the angle of rotation 4’ about this axis as [lo] A’= I+ Vi sin (+i)+2V’Zsin2

(c5’/2)

(4)

ri

r

Figure

1. The global position of an arbitrary

point p’ on body i.

NON-HOLONOMIC

where

I is a 3 x 3 identity

matrix,

DEFORMABLE

and

SYSTEMS.

Vi is a skew symmetric 0

V’=

-v3

213 [ -vz

19’7

I

matrix

1

defined

as

vz i

0

-v,

v,

0

.

(5)

Alternatively, one may express A’ as the product of semi-transformation E”‘, which depend on the four Euler parameters, as

matrices

E’ and

A’ = E’E”‘T

(6)

where the superscript T denotes the transpose of a vector or a matrix, transformation matrices that depend linearly on the four Euler parameters 6; are given by -0,

60

-83

-e2 -e3

e3 -e2

e. 8,

02 i , -8, e. 1

-8, E*i=

-0 1

[ -e3

e.

e3

-0 3

b

e2

and 0’ = [ 0; 0; 0: &]’ is the vector of the four Euler parameters parameters satisfy the identity o”e’= The time and space dependent p’, can be written as

vector

and the semioh, 0:) O:, and

-e2

-8,

1 (7a, b)

0,

e. 1

of body

i. These

1

u’, which

four

(8) describes

the local position

ul= u;+ u;,

of point

(9)

where ub is the vector of the undeformed position of point p’ and u;. is the vector of the elastic deformation. The deformation vector, when using the finite element method, can be written in terms of the elastic generalized co-ordinates as ~1.zz N’q; in which the space-dependent matrix the vector of elastic co-ordinates.

(10)

N’ is an appropriate

4. KINEMATIC

shape function

[9] and q; is

CONSTRAINTS

As was pointed out earlier, constraints can be classified as holonomic and nonholonomic. In a multibody system consisting of N interconnected rigid and deformable bodies the holonomic kinematic relationships that represent any type of geometric constraint, such as spherical joints, revolute joints, etc., may be expressed as a system of non-linear algebraic equations C%L

4/, t) =o,

(11)

where qr and q./- are, respectively, the vectors of the system generalized reference and elastic co-ordinates, t denotes time, and Ch is the vector of mh linearly independent constraint equations that are twice differentiable. In fact, in most physical and mechanical applications which do not involve a variation in the kinematic structure [ 111, the vector function Ch is analytic. As an example of an holonomic relation, one may consider the constraint equations for a spherical joint between bodies i and j. These relations can be represented by using the loop equation r’+A’u’-r’-AJu’=O

7

(12)

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A. A. SHABANA

where rk = [X”

Yk

ZklT,

uk=ugk+&

(13914)

where ui is the local position vector of the joint definition point with respect to the kth body co-ordinate system in the undeformed state, and u; is the deformation vector of the joint definition point with respect to the kth body co-ordinate system. By using the finite element method, the deformation vector can be written as u; = N;q; where N,” is an appropriate write equation (12) as

(15)

shape function evaluated at point p of body k. One can then

ri +A’(&+

Nbq.)) - rj-Aj(ui+

N’,qf’) = 0.

(16)

Considering virtual changes in the system generalized reference and elastic co-ordinates that are consistent with the constraints of equation (1 l), one obtains C:,Q+C$q.f=o,

(17)

or, alternatively,

Cl)JW

[Ck

Sq/TIT=O,

(184

which can be written in a compact form as cpq

=o,

(18b)

where c; = [aCh/aq,

ach/aq,l

(19)

is the holonomic constraint Jacobian matrix which is assumed to be a differentiable function of the system generalized co-ordinates and time, and Q = [&IT

h.TIT

(20)

is the virtual change of the total vector of system generalized co-ordinates. the constraint equations of the spherical joint (16) lead to W+

Bi68’+AiN~Sq,f~--Sd-

For instance,

B’S@-A’N~Sq$=O,

(21)

or, in a matrix form, [I

B’

-I

-B’][6riT

MiT

8riT

68’T]T+[AiNL

-AjNi][SqF

GdTIT=O,

(22)

in which the matrix B’ is defined as B’ = a/M’(Aiui)

(23)

where the 8’ is the set of rotational co-ordinates that define the orientation of the body reference. Suppose that, in addition to the above holonomic constraints, the system is subjected to m, non-integrable linear, first order differential equations of the form Cnh(q,, q,,.,41,4.r, t) = G(q,, qr, W:

4.:lT+g(q,, qf, t) = 0

(24)

where 4, and 4, are, respectively, the vectors of the generalized reference and elastic velocities, and G is an m, x n matrix, and g is an m,-dimensional vector, which may depend on the system reference and elastic co-ordinates and possibly on time. It is assumed that the m, equations of equation (24) are independent, or G is a matrix of full

NON-HOLONOMIC

row rank, sequently, as

DEFORMABLE

SYSTEMS,

199

I

and both G and g are functions of the system co-ordinates and time. Con the Jacobian matrix of the non-holonomic constraint equations can be written c;;].

c;” = [c;: Since equation

(24) is assumed

to be linear

(25)

in the generalized

velocities,

one obtains

C:”= G(q,, 41, t), One may then define the system Jacobian

(26)

as [6] (27)

Differentiating

equation

(11) with respect

4;]‘+

c;,][d:

[C:,

to time yields c: = 0,

(28)

where C: is the partial derivative of the vector of holonomic constraint with respect to time t. From equations (24) and (26) one obtains [Ci,” Now combining

equations

c;:][q:

equation

yields

[c’”c’if] [;;]+[‘,‘]=o. 5. INDEPENDENT

REFERENCE

AND

Ch (29)

d:]‘+g=o.

(28) and (29) in one matrix

functions

ELASTIC

(30)

VARIABLES

Since non-holonomic constraints impose no restriction on the finite change of system co-ordinates, the number of independent reference and modal elastic co-ordinates of the non-holonomic deformable multibody system is not the same as the number of indepen dent reference and modal velocities. The holonomic kinematic constraints, if they exist, can be used to identify a set of independent co-ordinates by using the generalized co-ordinate partitioning of the constraint Jacobian matrix. Let the vectors of the system qr and qr be written as co-ordinates

PTI’,

% = [PI-

9, = [P:

km’,

(31,32)

where & and pj are, respectively, the dependent reference and elastic co-ordinates and p; and & are, respectively, the independent reference and elastic co-ordinates. Equation (17) can be then written as (33’)

c~,sp,+c~,sg,+c~,s~,+cp,sp;=o, or, alternatively,

as done by Chaplygin

[Ci,

and Voronec,

$’ =-[c;,

C&l [

which can be written

in a compact

/

1

in a matrix

$1

g [

9

-1 I

form as (34)

form as c;sp

= -c;sp;

(35)

where Ci and Cj are, respectively, the partitions of the Jacobian matrices associated with the dependent and independent co-ordinates, and /I and p are, respectively, the vectors of the dependent and independent co-ordinates defined as P = [P:

#a’?

p=[pJ

p7]?

(36,37)

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A. A. SHABANA

vector p is an mh-dimensional vector, while p is an (n - m,)-dimensional vector, where n is the total number of system co-ordinates. The m,, x m,, Jacobian matrix associated with dependent co-ordinates has a full row rank since the holonomic constraints are assumed to be linearly independent._Accordingly, the virtual changes of the dependent co-ordinates can be written in terms of the virtual changes of the independent co-ordinates. Since non-holonomic constraint equations impose restrictions on the variations of the co-ordinates, the components of the velocity vector 6 are not completely independent. This vector can be written as The

PJl’=h;T

P=tE

?T ti:

$I’,

(38)

in which the vector + = [yf +Tf’,,is an (n-m,,-m,)-dimensional vector, the components of which are the independent velocities, while the vectors 4 = [ 7jT GT]’ and B which have, respectively, dimensions m, and III,,,are the vectors of dependent velocities. Thus equation (30) can be written as

C =--

_-_

(39)

.g The vectors /I and [b’ 7jTlT are, respectively, the dependent co-ordinates and dependent velocities, while the vectors 7~and y are the independent generalized co-ordinates and the vector + is the independent generalized velocities. The dependent velocities can be written in terms of the independent ones by using equation (39) as follows (40) One may also write equation (11) as (41)

Ch(P, 77,x t) = 0, and upon considering leads to

a virtual change in the system total co-ordinates [c;

c”,

C”,][sp’

snT

equation (41)

6yTlT= 0,

(42)

GyTIT.

(43)

or c;sp Since Ci is non-singular,

= -[C”,

c;][sTJ’

one has sp = -[c;]-‘[C”,

Ch,][GnT

GyTIT.

(44)

Furthermore, equation (41) can be solved for j3 by using a Newton-Raphson algorithm once the vectors n and y are defined. After having determined the generalized co-ordinates, equation (40) can then be solved uniquely for the dependent velocities fi and + provided that i, and [p’ nT yTIT are known.

NON-HOLONOMIC

DEFORMABLE

SYSTEMS,

201

I

In order to get the constraint relationship, that is imposed on the system generalized accelerations by the holonomic and non-holonomic constraints, one begins by differentiating equations (28) and (29) with respect to time, which yields

Define

[ ir+cc:

I

[c;,

C&l ij,

[c;;

c;;][;;]+c$j+c:“=o.

fj)#j+2c&j+cI:=o,

(45,46)

(Y= [(Y”’ anhIlT, (y=-

(47)

where CYcan be determined once the co-ordinates outlined above. Finally, upon using equations become [z:/

Si,][,:

ii:]‘=

and velocities are found (27) and (47), equations

or, in a compact

[In:], [J][q]

6. DIFFERENTIAL

by the method (45) and (46)

form,

= a.

(48a, b)

EQUATIONS

A virtual displacement [8qT Sqr]’ of the total consistent with the kinematic constraints if

OF MOTION

system

generalized

co-ordinates

is

(49) The D’Alembert-Lagrange

equation d

aT’ _g,g_ zy aqi

LO iii

for body

i can also be written 0”

aq’

I

sq’ = 0,

as [I, p. 911 (50)

where T’ and U’ are, respectively, the kinetic and strain energy of body i and Q’ is the vector of generalized applied, potential, and non-potential forces associated with the ith body generalized co-ordinates. It is important, perhaps, to mention at this point that the non-holonomic equality constraints can be embedded [12, p. 2431 by using equation (49) to obtain a reduced system of non-linear differential equations that can be integrated forward in time by a direct numerical integration technique. The embedding of non-holonomic equality constraints has been discussed in detail in reference [ 121. The embedding technique, however, has some computational disadvantages, which are the consequence of losing the sparsity of the matrices that appear in the formulation in addition to the need to evaluate the inverse of some subjacobian matrices. In order to circumvent some of these computational difficulties, Lagrange multipliers can be used as adjoint variables. By using the method of Lagrange multipliers, the Jacobian of the constraint equations can be adjoined to the system differential equations of motion [l, p. 931, leading to daT’ (-) a$

&

i i -$+$+AzTJi=

QiT,

(51)

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A. A. SHABANA

where A’ is the vector of Lagrange multipliers and J’ is the Jacobian matrix associated with the generalized co-ordinates of body i and defined as

(52) It has been shown [13] that the kinetic and strain energy of the deformable body i in a multibody system can be written as

(534

where subscripts r and f refer, respectively, to reference and elastic co-ordinates, and M’ and K’ are, respectively, the ith body symmetric mass and stiffness matrices. By substituting equation (53) into equation (51) and following the procedure described in reference [13], the system differential equations of motion of the non-holonomic multibody system consisting of interconnected rigid and deformable bodies can be written as

(54) where the three coefficient matrices on the left are the system mass matrix M, the system stiffness matrix K, and the system Jacobian matrix J, respectively, and Q is the vector of generalized applied (potential and non-potential) forces, F is a quadratic velocity vector that contains the Coriolis or gyroscopic force components [8] and Ah, Anh are, respectively the vector of Lagrange multipliers associated with the holonomic and non-holonomic constraints, The system differential equations of motion of equations (54) and (48) can be combined in one matrix equation as follows:

This matrix equation can be solved for the acceleration vector 4 and the vector of Lagrange multipliers A = [AhT AnhTIT.

7. SUMMARY

AND

CONCLUSIONS

A method for the dynamic analysis of non-holonomic multibody systems consisting of interconnected rigid and deformable bodies has been presented. Non-holonomic constraints are assumed to depend linearly on the system velocities. While holonomic constraints are used to identify the system independent reference and elastic co-ordinates, non-holonomic as well as holonomic constraint equations are used to identify the system

NON-HOLONOMIC

DEFORMABLE

SYSTEMS, I

20.3

independent reference and elastic velocities. Deformable components in the system are discretized by using the finite element method, and component mode synthesis techniques can be employed to reduce the number of elastic co-ordinates. The system differential as well as holonomic and non-holonomic constraint equations are written in terms of a coupled set of reference and modal elastic co-ordinates. Since non-holonomic constraints impose restrictions on the infinitesimal change of the system co-ordinates, the number of independent generalized co-ordinates is larger than the number of independent generalized velocities by the number of non-holonomic constraint equations. The state vector can then be formed by using the independent co-ordinates, and independent velocities and the associated state equations can be identified and integrated forward in time by using a direct numerical integration method [9, p. 751. The solution of these state equations defines the independent variables. The dependent co-ordinates are determined by using the holonomic constraint equations, while the dependent velocities are determined by using both holonomic and non-holonomic constraint equations. The computational algorithm for this formulation is outlined in Part II, wherein a method for the impact analysis of non-holonomic multibody system is presented. In this method generalized impulse momentum equations that can be used to predict the jump discontinuities in the non-holonomic

reaction

forces

are developed.

ACKNOWLEDGMENT This research was supported by the U.S. Army Research office, Research Triangle Park, N.C. 27709.

REFERENCES 1. Ju. I. NEIMARK and N. A. FUFAEV 1967 Dynamics of Non-holonomic Systems. translations of Mathematical Monographs 33. Translated from the Russian by J. R. Barbour, American Mathematical Society, Providence, Rhode Island, 1972. 2. T. R. KANE 1961 American Society of Mechanical Engineers Journal of Applied Mechanics 28, 574-578. Dynamics of nonholonomic systems. 3. H. HEMAMI and F. C. WEIMER 1981 American Society of Mechanical Engineers Journal of Applied Mechanics 48, 177-182. Modeling of nonholonomic dynamic systems with applications. 4. C. E. PASSERELLO and R. L. HUSTON 1973 American Society of Mechanical Engineers Journal systems. of Applied Mechanics 40, 101-104. Another look at nonholonomic 5. E. T. WHITTAKER 1937 A Treatise on the Analytical Dynamics of Particles and Rigid Bodies. Cambridge University Press, fourth edition. 6. P. E. NIKRAVESH and E. J. HAUG 1983 American Society of Mechanical Engineers Journal of Mechanisms, Transmission and Automation in Design 105, 379-384. Generalized coordinate partitioning for analysis of mechanical systems with nonholonomic constraints. 7. 0. P. AGRAWAL and A. A. SHABANA 1985 International Journal of Computers and Structures 21(6) 1303-1312. Dynamic analysis of multibody systems using component modes. 8. A. A. SHABANA 1986 Computer Methods in Applied Mechanics and Engineering 54, 75-91. Transient analysis of flexible multibody systems, Part I: Dynamics of flexible bodies. 9. L. SHAMPINE and M. GORDEN 1975 Computer Solution of ODE; The Initial Value Problem. San Francisco: W. S. Freeman. 10. A. A. SHABANA 1989 Dynamics of Multibody Systems. John Wiley (to appear). 11. Y. KHULIEF and A. A. SHABANA 1986 American Society of Mechanical Engineers Journal of Mechanics, Transmissions, and Automation in Design 108, 167-175. Dynamic of multibody systems with variable kinematic structure. 12. R. M. ROSENBERG 1977 Analytical Dynamics of Discrete Systems. Plenum Press. 13. A. A. SHABANA and R. A. WEHAGE 1983 Journal of Structural Mechanics 11, 401-431. A coordinate reduction technique for dynamic analysis of spatial substructures with large angular rotations.

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APPENDIX:

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NOMENCLATURE

A’ the transformation matrix of body i Ch the vector of holonomic constraints rd, C the vector of non-holonomic constraints

E' pi

J K' M' mh

m,, N N’ N; n

0’ 4

4’ % 9,

d 4: TP r’

T’

u’

ui

ub

u; V’

8’ A

a semi-transformation matrix a semi-transformation matrix the system Jacobian matrix of holonomic and non-holonomic constraints the stiffness matrix of body i the mass matrix of body i the number of holonomic constraints the number of non-holonomic constraints the total number of bodies in the system the matrix of shape functions of body i the matrix of shape functions of body i evaluated at point p the total number of co-ordinates the vector of generalized forces the vector of system generalized co-ordinates the vector of generalized co-ordinates of body i the vector of system elastic generalized co-ordinates the vector of system reference generalized co-ordinates the vector of elastic co-ordinates of body i the vector of reference co-ordinates of body i the global position vector of an arbitrary point p on body i the global position vector of the origin of body i co-ordinate system kinetic energy of body i strain energy of body i the local position vector of an arbitrary point on body i the local undeformed position vector of an arbitrary point on body i the local deformation vector of an arbitrary point on body i a unit vector along the instantaneous axis of rotation of body i the set of the Euler parameters of body i the vector of Lagrange multipliers