The sensitivity of multibody systems with respect to a design variable matrix

Mechanics Research Communications, Vol. 21, No. 3, pp. 223-230, 1994 Copyright • 1994 Elsevier Science Printed in the USA. All rights reserved 0093-6413/94 $6.00 + .00

Pergamon

THE SENSITIVITY OF MULTIBODY SYSTEMS WITH RESPECT TO A DESIGN VARIABLE MATRIX YIMIN ZHANG" SUHU~N CHEN "° QIAOLING LIU "°° Mechanics Department, Jilin University of Technology Changchun,

130025, P. R. China

(Received 1 September 1993; acceptedfor print 24 January 1994)

Introduction

Calculation of derivatives serves as the foundation of optimun design and is necessary for machine any s e n s i t i v i t y analysis design and optimization.

During

the

in

above process,

finding

effective response sensitivity with respect to design

variables

is very important, and is especially difficult when the number of

variables is larger than one. For the case of large displacement mechanical systems,

there are many references (for example [I]-

[2]). The present paper studies the design variable with respect to n matrix of

the multibody systems

the sensitivity analysis of multibody systems

addition, when the direct method for

can the

be

in

separating

the

matrix

generalized.

there product

differentiation. This problem is also solved here.

* Ph.D. Student, ** Professor,

*** Lecturer 223

deduced from

sensitivity

of multibody system motion equations is used, difficulty

and extends

to matrix-valued

functions using the direct method. The formulae matrix calculus and Kronecker algebra

sensitivity

In

analysis arises during

a

224

Y. ZHANG, S. CHEN and Q. LIU

Mathematical Preliminaries [3J-[4]

[n

this

section

we

attempt to

summarize,

for

ease

of

reference, the methods of matrix calculus and Kronecker algebra. The matrix calculus proposed in [3]-[4] is employed. I. The elementary matrix The n × n unit matrix is vector which is "l"

denoted

by In. The

q- dimensional

in the kth element and zero elsewhere

is

called the unit vector and is denoted by ekCq~. The elementary matrix

has dimensions (p×q), and takes

~I"

in the i-k

element,

and

zero elsewhere. 2. The Kronecker product The Kronecker product of A(pXq)

and B(s×t)

is denoted by A ~ B

and is a ps×qt matrix defined by aziB a1=B

...

axqB-

(2)

A ® B=I

1

L aplB . . . . . .

ap~B

3. The permutation matrix Define the permutation matrix by P

q

(3)

i= lk=l This matrix is square (pq× pq) and has p r e c i s e l y a single in each row and in each column.

4. The column s t r i n g and row string of matrix We define the column string cs~) of A(p×q)

by

=1

"

225

SENSITIVITY ANALYSIS OF MULTIBODY SYSTEMS

q (4) the row s t r i g rs(A) of A is defined by P rs(A) x×p~-_j~leJ:p>A = (e sT< p ~ I q )

(5)

5. The basic d i f f e r e n t i a t i o n theorems The derivative structure of a matrix-valued function A(B)p× q with respect to a matrix B , × ,

is a matrix of dimension

ps× qt

defied by OA _- ( aA ) OB Oblj Suppose C ( r × l ) ,

(6)

D(q×u)

are

matrices

with

dimensions

as

indicated. The following i d e n t i t i e s can be established e(AD) = eB

~

(A@ C) ~B

Furthermore,

eA ( I t s D)+(I.~A) eB

aD eB

(7)

aA ec ~C + ( l . e U p × . ) ( ~A) ( l , e U 1 × ~ ) aB eB

if A=A(C) and C=C(B) so that

A= A ( C ( B ) ) ,

(8) the

following chain rule applies eA ; ( [ . e OA ) ( -e (r - sC) ~ T Iv) eB a (rsC) OB -(

eB

- I,) ( I , e

a (esC)

)

(9)

The Dynamic Governing Equation of Multibody System

Multibody dynamics is an uses

different

branches

interdisciplinary of

mechanics,

subject,

including

which analytic

mechanics, continuum mechanics, s t r u c t u r a l dynamics, etc. In the past 20 years, multibody systems dynamics has rapidly developed (for example [5]-[8]).

Basically,

the s t r u c t u r a l forms

of

the

Y. ZHANG, S. CHEN ~ d Q. LIU

226

governing equations have two categories. The first assumes the form: -

(lo) q (0) --qo where A is a n X n

generalized mass matrix,

q is a n-dismensional

vector that stands for generalized corrdinates,

a

columm

matrix of n order, n is the number of degrees of freedom,

and a

superscipt dot respresents

is

the time(t) derivative.

The second category is based upon method,

B

the

lagrangian

multiplier

and assumes the form: M~" +Kq+C T ~.--Q+F

-~ I

J,

C:O

2

where M stands for a n × n a n×n

(11)

stiffness matrix,

generalized mass matrix, K stands for k--(Xl,

holonomic constraint equations.

....

k~) z

Lastly, Cq

Jacobian constraint matrix and the

respresents stands

superscript

T

for

m

the

stands

for

transposition. Since these bodies make complex motions in the system, appears a high nonlinear phenomenon of the system. As a result, -varying mass matrices,

in the

governing

there exist sub- blocks

a fect which will bring

difficulty to the multibody system sensitivity

an

there

equations with

time

inevitabale

analysis

using

the direct method.

Sensitivity Matrix of Multibody Systems 1. Sensitivity Analysis of Multi-Rigid Body Systems The multi- rigid

body systems governing

equation

in

the

previous section can be expressed by the following mathematical

SENSITIVITY ANALYSIS OF MULTIBODY SYSTEMS

227

model

~ ' = f ~ , ~ ,q, t)

i

(12)

q(O)=qo where V = ( v , j ) , × t respresents a design variable matrix, f=A-~B,

here A-* is inverse matrix of A, Obviously, the mull-rigid systems governing equation is vector-valued

body

and matrix- valued

functional equation. Differentiating Eq. (12),

we get

the

following sensitivity

equation D~" DV

- -

Dq DV

= -

~f aV

of o q" T

+(I.~

D~ DV

)

÷(I.,6

ef o qT

)

Dq DV

(0)= 0

1

(13)

where D()/DV stands for the sensitivity matrix. For the governing equation of multibody system of eV sf af aqT

=

eA-~it ~ B) + (i,,® A_I) eB oV ev -

(14)

-

=A_1 aB

(15)

aA-~ aqT ( I . ® B ) + A

- -

aB aqT

- x ~

(16)

Substituting Eq. (14), (15) and (16) into the Eq. (13) ,

we get

the sensitivity equation for multi-rigid body systems. Hence it can be concluded that, solving the responses q, ~, ~" from the governing equation,

and then putting

the s e n s i t i v i t y equation, we obtain matrices

Dq

° ,

DV

!

DV

DV

the

response

them into sensitivity

228

Y. ZHANG, S. CHEN and Q. LIU

2. S e n s i t i v i t y The

Analysis of Flexible Multibody Systems

flexible

multibody

system

governing

previous section can be expressed in

terms

equation of

the

in

the

following

mathematical model f(V, ~', q' ,q, X, t)--0

C(V,q, t)--O

Obviously,

1

(17)

the flexible multibody system governing equation

is

vector-valued and matrix-valued functional equation. Differentiating Eq. (17), one obtains the following sensitivity equation: of

+(I

of

@

0V

)

O{I*T

of

+ (I,,~

OqT

)

Dq DV

D~I" + ( I . ~

of

DV

0q" T

+(I.~

of o ~. T

)

D ), DV

) Dq" DV

=0 (18)

oC oV

--.+(I.

~

oC OqT

)

Dq :0 DV

For the governing equation of flexible multibody systems of

( I , ~ ' ) + OK (I,~ q)+ oC~ (I,~ ~) oV oV oV

~ aM

oV of --

0~ "T

of

oq'T

eQ

OF

eV

aV

=M _-_

(19)

(2O) OQ

(21)

. OF

oq'T

Oq'T

o__~f -_ o__.M_M( I . ~ ' } + eq T eq T

K +

oCT ( I . ~ OqT

X}

OQ OqT

of -- c~ o~, T

Substituting

OF eq T

(22)

(2a)

Eqs. (19)-(23)

into

Eq. ( 1 8 )

, the

equation for f l e x i b l e multibody systems is obtained.

sensitivity

SENSITIVITY ANALYSIS OF THE MULTIBODY SYSTEMS

Thus, by solving for the responses q,

~ , ~'

229

from the

goYerning equation, and then putting them into the sensitivity equation, the response s e n s i t i v i t y matrices Dq , DV

Dq" DV

D~" are obtained. DV

Since the sensitivity equations contain the matrix,

it

is

clear

that

design

variable

generalized formulae have

developed using techniques from matrix calculus algebra. The governing equation and

been

and Kronecker

sensitivity

equation

can

all be solved by the stiffness method. Certainly, the algebraic equations

and d i f f e r e n t i a l

different

methods.

equations

Practical

can

be

solved using

applications and

algorithmic

coding of the formulae derived will be the subject of a

future

paper.

Conclusion

There arises a further need for

parameter

selection

in

the

dynamic analysis and design process for the mechanics model and mathematical model of high speed, large and precise systems, astronautic systems

and robots.

There

mechanical arises

also

problems of how to apply active control and optimization or to

change

the

structure

design

according

to

how

practical

requirements (such as displacement and velocity, responses etc.). This sensitivity

analysis

plays

an

important

problems. And what has been just shown above is matrix- valued

functions

role

in

such

work done

for

of multibody systems

along

this

direction. It can be clearly seen from our results,

that

this

method can be effectively used in the s e n s i t i v i t y matrix-

valued

functions

of

praticabillty and universality,

multibody

analysis

systems.

this method serves as a

for

Having solid

230

Y. ZHANG, S. CHEN ~ d Q. L1U

foundation for succeeding design

controlling

and

optimization

study, and is the basis of a univeral and analytic program.

References

I E. J. Haug, R. Wehage and N. C. Barman,

Design

Sensitivity

Analysis of Planar Mechanism and Dynamics, ASME,

Joural

of

Mechanical Design, Vol. 103, July, 1981, p. 560-570 2 V. Sohoni and E. J. Haug, A State Space Method for Optimization of Mechanisms and Machines,

ASME,

Kinematic Joural

of

Mechanical Design, Vol. 104, January, 1982, p. 101-106 3 W. J. Vetter, Expansions, 4

Matrix

Calculus

Operations

and

Taylor

SlAM Review, Vol. 15, 1973, p, 352-369

J. W. Brewer, Kronecker System Theory,

Products

and

IEEE Transactions on

Matrix

Circuits

Calculus and

in

Systems,

CAS-25, No. 9, 1978, p. 772-781 5

J. Wittenberg,

Dynamics of Systems of Rigid Bodies,

Teubner, Stuttgart,

B.

G.

1977

6 T. R. Kane, P. W. Likins and D.

A.

Levinson,

Spacecraft

Dynamics, McGraw Hill, 1983 7 W. S. Yoo and E. J. Haug, Dynamics of Articulated Structures, Part I: Theory,

J. Struct. Mech., Vol. 14, No.l, 198S,

p. 105

-126

8 h. A. Shabana,

Transient

Analysis

of

Flexible

Multibody

Systems, Part I: Dynamics of Flexible Bodies, Computer Me~hs hppl. Mech. Engrg., Voi. 54, 1986, p. 75-91