Ins t itut f t.i r
BI. :
Dynamik der Flugsysteme
IB.Nr.: 515 - 84/2
A Computer- Oriented Method for Reducing
Linearized Multibody System Equations
by Incorporating Constraints
__j
Freigabe:
Die Bearbeiter :
C. Führer
o.
Unterseh r i ften :
s -A-.~
~~
~val lr app
Der Abteilungsl ei ter
Dr.
w.
Korturn
Der stellv . Institut sdirektor :
Der Institutsdirektor:
Dieser Ber1cht enthält ·
11
_ _ _ Blatt davon
1 Bi lder
_ __
_ _ _ Diagramme
Or t ·
Obe~faffenhofen
Datum : 15 . 01 . 84
Bearbei t w,:
.FOhrer; wallrapp
CONTENTS
Summary
ii
1 . Introduction
2. Numerical Method
3
3. Condensation of the Dynamic Equations and Elimination of t he
Constraint Forces
5
4. Evaluation of the Constraint Forces
6
5. Example
7
5 .1.1
References
8
Summary
Consider a spati a l multibody system with rigid and elastic bodies . The bodies
are
linked by rigid interconnections (e . g . revolute joints) causing con -
straints, as well as by flexible interconnections (e.g. springs) caus ing
applied forces . Small motions of the system with respect to a given nominal
configuration can be described by linearized dynamic equations and kinematic
constraint equations .
We present a computer oriented procedure which allows
to develop a minimum nurober of these equations . There are three problems.
First: algorithmic selection of position coordinates ; second: condensation
of the dynamic equations; third: evaluation of the constraint forces. To
dernarrstrate the procedure, a closed loop multibody system is used as an exam ple.
- 1 -
1. INTRODUCTION
The · linearized equations of motion can be written in the following form:
M
( 1)
p
+ K p
+ D p•
=
g
Here M, K , D, a r e system matrices develop ed in [1, 2, 3 ]; p is the vector of
the n position coordinates of the unconstr aint sys tem and g the vector of
p
the internaland external f or ces. Any n
z holonomic constraints on the motions
of the system can be represented in linearized form as
Czt p(t ) = Z(t )
(2)
where CZ is a constant n x n - matrix.
p
z
Equation (2) r estricts the solution space of (1) to the set {a(t)} + ker( Czt )
where a(t ) is a parti cu l ar solution of (2) and where ker ( Czt )
t
space of Cz .
is the null -
Without lass of genera lity it can be assumed t hat a ( t) is an
element of the orthogonal complement of ker( Czt) which will be denoted by
t
+
t
ker(Cz ) . The dimension of ker(Cz ) corresponds to the number ny of degrees
of freedom. In the case of a syst em with a fu ll rank constraint matrix we have
ny = np - nz . Otherwise a r educt ion of Cz to a matrix with full rank must be
performed to evaluate n . This can be done in a numerically stable manner by
y
the singular value decompos ition [4]. We assume for the fo l lowing, that this
has been previously d one .
To describe a solution of the system ( 1) , ( 2 ) with n y ( independent) reduced
postion coordinates y it is necessary to choose a basis of the solution sp'ace
ker( Cz t). The bas i s vectors will be denoted by the matrix Jy .
basis J z of ker(Czt)+ is chosen with the feat ur e
(3)
a (t ) = Jz z(t )
Equivalently a
- 2 -
Thesedefinitionsare equi valent to the following matrix products :
t
=0
(4 )"
cz
(5 )
c zt J z = I
(6)
J tJ
y
Jy
z
=0
where I deno te s t he identity ma t rix. p can be spl i t t ed into a d irect sum of
two vectors with the following represent ation :
(7)
p
= Jy y
+ Jz z .
For simple constrained systems one may have so much ins i ght i n the structu re
that it ls po~sible to establish Jy by hand.
For more complex systems an automated approachwill be necessary . A method to
select the basis Jy has been proposed in [5 , 6] using the zero - eigenvalue
t heorem, which is equivalent to the singular value decompostion [4 ] . The
coordinates obtained by this method form an orthogonal basis of ker( Czt) ,
which is from the numerical point of view optimal. But the reduced position
vector y represented in an orthogonal basis has in general no more physically
interpretable components and a backtransformation of the vector y to the vec tor p will be nec es sary, where the numerical advantage can be lost again .
This disadvantage is avoided by the method pr esented here . This method also
automatically selects a basis Jy of the solution space, but the basis vectors
are chosen so that the components of y are identical to some components of p .
This choice keeps the physical interpr etability of y . I n addition one can
p rescribe some linear combination of the canonical basis vectors by giving a
re l ation
(8)
- 3 -
t
t o be taken as basis vectors of ker ( Cz ) . lf stands f or sorne cornponents of t he
reduced pos it ion vector y .
2. NUMERICAL METROD
For sirnplicit y it is requir ed that
it
c, has full r ank . To avoid contradictions
i s necessary to assurne that the colurnns of Cz a r e linearly i ndependent
frorn those of c,. Otherwise l inear dependency between colurnns of Cz and C~
would indicate , that coordinates n h ave been chosen, which cannot be used to
describe t he rnotion of the rnu l tibody systern (c. f . e xarnple).
There are t wo steps to perform in order to obtain the basis vectors : Evalua t i on of Jy by solving equation (4 ) with t he rest riction (8 ) and evaluation
of Jz by solving equation (5) and (6). The l ast step can be ornitted if z
and i f the cornputation of the constr aint forces is not required.
If we choose J z so, t hat
(9)
c'l
t
Jz = 0
holds,
(10)
follows ou t of equation (7). Equation (4) and (10) can be cornbined to
( 11)
=
=0
- 4 -
Using orthogonal Hauseholdertransformations and co lumn pivoting [4 ] , Ct can
be decomposed into a product of an orthogonal matr ix Q, an upper triangular
matrix R, with its d iagonal elements ordered in a sequence of decreasing
absolute values, and a p ermutation matrix P:
(12)
Ct=QRP
As Ct has full rank, R can be partitioned as
( 1 3)
R =
[ Rl' S]
where R
is quadratic with no zerodiagonal elements . Thus the dimension of
1
R 1 equa l s the rank of C. Jy is con sidered tobe of the form
(14)
Jy
=p
[
Jol J2IJ
where J is a (n + n ) x n - matrix, J 2 is a (n + n ) x (n - n ) - matrix .
1
z
Tl
Tl
z
Tl
p
Tl
With these definitions equation (1 1 ) reads
Denoting the first n 2 rows of Q by Qz and the remaining
the follow ing solut ion of equat ion (15)
R
( 16)
=p
-1 Q
1
[
I
1\
0
<·
n~
by
~
we obtain
- 5 -
Herein the orthogonal ity of
Q
has been us ed .
The column vectors of Jy are
indeed the basis vectors of ker ( Czt) and as a consequence of the identity
matrix in the lower right corner of Jy the components of y generated by this
method are really components of the unreduced position vector p.
The step establishing Jz is easier. Equation (5) and (6) can be combined to
the fol lowing system of linear equations
=
(17)
which can be solved by standard methods .
This algorithm has been integrated and tested in a FORTRAN - program which
gene r ates the linearized equations of motion f or genP.ral mult i body systems
[7 ,8] . For the orthogonal decomposition, subroutines fro m t he LINPACK pack age [9 ] have been used. Methods using orthogona l matrix decomposit ion are
numerical l y stable ; t hey are preferable to t hose using Gauss r elated algo rithms for determining the rank of a matrix .
3. CONDENSATION OF THE DYNAMIC EQUATIONS AND ELIMINATION OF
THE CONSTRAINT FORCES
With the matrices Jy and Jz we can define the gener alized applied f orces fy
and the generalized constraint forces fz as the projection of t he total force
g in the spaces ker( Czt) and ker(Czt) +:
fy
= Jytg
f
= J z tg
(18)
z
- 6 -
Correspondingl y
g
= gy +
g
can
be
splitted
into
a
dir ect
sum
of
two
forces
gz; with the applied force gy, defined by
(19)
and the constraint force gz , defined by
(20)
For simplicity we assume for the following the case z = 0 . The more general
equations can be easily obtained from this case. Introducing (7) in (1) and
premultiplying (1) with Jyt gives
By this procedure the number of equations of motion has been reduced to the
minimum and the constraint forces have been eliminated, which follows from
equation (19). Note that in order to obtain these results, the principle of
d'Alembert was not needed; the elimination of the constraint forces was a
consequence of geometrical considerations only.
4. EVALUATION OF THE CONSTRAINT FORCES
It fol lows from (20) that
(22)
holds.
Premultiplying (1) with Jzt instead of Jyt gives an expression for
the generation of the genera lized constraint forces:
- 7 -
From these generalized constraint forces the forces gz can be easi ly obtained
by using the relation
This relation can be verified by premultiplying it with Jzt and by using
(4),(5) and (18):
(25)
(26)
J
y
tc z rz = 0
5 . EXAMPLE
The kinematically closed chain treated in [1 0, page 182] has been us ed as an
example to demonstrate the method. Small motions about the nomina l triangular
configuration, shown in figure 1, have been studied. There are n
p
= 30
posi -
tion coordinates and n = 30 constraint equations. The l atter are linearly
z
dependent and the system has one degree of freedom. The computational results
of this example are shown in table 1 . As one can see, the for malism chooses
"y 1 = rotation of body 5 about axis 2" as reduced coordinate of the system in
triangular - shape configuration. This choice and in consequence the matrix
Jy are in general dependent on the nomina l configuration of the sys t em. The
method applied to the cube - shape configuration of the chain does not lead
to the same coordinate as found in the present case . The coordinate chosen by
the algorithm in this case cannot be used to describe the mot i on of t he chain
in the cube shape configuration .
- 8 -
5 .1.1
1]
Ref eren ces
MUELLER, P.C.; SCHIEHLEN, W. O., Lineare Schwingungen, Akad. Verlagsges.
Wiesbaden, 1976
[ 2]
WALLRAPP, 0 . , Die linearen Bewegungsgleichungen von elastischen
Mehrkoerperfahrzeugen, DFVLR inter nal report IB 552 - 80/4, Oberpfaffenhofen, 1980
[ 3]
DUFFEK, W.; KORTUEM, W.; WALLRAPP, 0., A general purpose program
for the sirnulation of vehicle - guideway interaction dynarn ics. In:
SLIBAR, A., SPRINGER, H., (eds . ), The Dynamics of Vehicles on Roads
and on Tracks , Pro c . 5th VSD - 2nd IUTAM Syrnp . , Vienna, Sept . 1977;
Swets & Zeitlinger, Arnsterdarn , 1978, p . 104 ff .
[ 4]
STEWART,
G.
W.,
Introduction to rnatrix computations , Academic
Press, New York/San Francisco/London , 1973
[ 5]
\~ALTON ,
[ 6]
HUSTON, R. L . ; KAMMAN , J. W., A discussion on cons traint equat i ons
in rnul tibody dynarnics, Mec . Res . Com . , Vol. 9 ( 4), 25 1 - 256 , 1982
[ 7]
WALLRAPP, 0 . , MEDUSA - ein allgerneines Rechenprogramm zur Analyse
und Auslegung der Dynamik von Rad/Schiene -Fahrzeugen. Forschungs information Bahntechnik, Deutsche Eisenbahn Consulting GmbH, Dor nier System GmbH ETR 32 (1983 ) Seite 365 - 366 , 1 Bild .
[ 8]
WALLRAPP , 0 . ; KORTUEM, W. ;
W. C., J r . , STEEVES, E. C., A new matrix theor em and its
application fo r establishing independent coordinates for complex
dynarnica l systerns with constraints, NASA technical report TR - R326 ,
1969
~fEDUSA -
ein allgerneines Rechenprogramm
zur Analyse und Auslegung der Dynamik von spurgefueh rten Fahrzeugen , VDI - Bericht , to appear.
- 9 -
[ 9]
DONGARRA , J . J.; MOLER, C. B. ; BUNCH, J. R. ; STEWART, G. W. , LIN PACK user's guide, SIAM, Philadelphia, 1979
[10]
WITTENBURG, J., Dynamics of Rigid Bodies, Teubner, Stuttgar t, 1977
- 10 -
body fixed frame 1
3
0.2- - - t
Figure 1.
1
- - - ----_j
The kinematically closed chain in triangular conf igur ation . Body
1 to 5 are free bodies, body 6 is fixed .
n : 30 postion coordinates , n : 30 cons t raints.
p
z
\
- 11 -
Number of degrees of freedom:
ny =
Definition of the reduced
position var iable
Matrix
Jy (30, 1 )
y1 = rotation of body 5 about axis 2
Position variables
(defined in the body fixed frame)
type
axis
body
tr anslation in
0.0
0.0
"
"
2
0.0
"
"
3
rotation about
0.0
"
"
0.0
0.0
"
2
"
3
translation in
0.0
"
"
0.0
0.0
1
1
2
"
2
2
"
3
2
rotation about
0.0
2
- 1. 0
"
"
2
2
0.0
"
"
3
2
translation in
- 0.1
3
0.0
"
"
2
3
0 .0
II
II
3
3
1
3
2
3
3
3
1
4
2
4
3
4
rot ation about
0.0
0.5
II
0.866
II
0.05
translation in
0.0
"
0 .0
II
0.0
"
"
"
"
rotat ion about
4
- 0.5
"
"
2
4
0.0
II
II
3
4
1
5
0.1
trans lation in
0.0
II
"
2
5
0.0
II
"
3
5
0.0
rotation about
1.0
"
II
0.0
"
"
Table 1:
1
5
2
5
3
5
Numerical res ults for the kinematically closed chain .