1 INTRODUCTION AND MOTIVATION

This paper was written, though not published, in November 2007, so it is more than three years

old. It is based in part on a talk given by the second author on “Some Problems Associated with

the Collaborative Simulation of Active Vibration Regulation Systems” at the 2007

MSC.Software Virtual Product Development (VPD) Conference in Frankfurt am Main, Ger-

many. It has its origins, however, almost ten years ago when a former German automobile

manufacturer, Wilhelm Karmann GmbH, set up projects to evaluate the use of control theoretic

techniques to yield improvements in ride and comfort in the development of modern “converti-

ble”-type automobiles, often (as here) simply referred to as convertibles. Each project was as-

signed the objective of identifying and evaluating promising existing strategies towards sup-

pressing low frequency vibrations which directly affect both the ride and comfort of a typical

convertible, but without significantly compromising other attributes, for example vehicle han-

dling and weight. As new techniques or problem areas developed new projects were set up.

Generally, the interaction of tyres and road irregularities, oscillations induced at the en-

gine/chassis contacts, and subjective human response are the chief causes of most of the com-

mon forms of low frequency (typically below 30 Hz) vibrations that are transmitted into the

elastic vehicle body, and they constantly act on and affect one another. When a convertible is

subjected to such considerable vibrational phenomena, the reduced rigidity (in torsion, as will

be seen shortly) of the body, due primarily to the absence of a solid roof, causes vibrations at

the body’s first natural frequency. Typically these manifest themselves as vibrations of the

A method for simplifying complicated multibody models for use

in experimental control

Reinhard Schmidt University of Applied Sciences Osnabrück, Faculty of Engineering and Computer Science, Osnabrück,

Germany

Mahyar Mahinzaeim Newcastle University, School of Mechanical and Systems Engineering, Newcastle upon Tyne, United

Kingdom

ABSTRACT: There are many ways to model large scale, flexible mechanical systems, and the

problem of deciding which one to use is of central importance in engineering. In recent years,

due mainly to the simultaneous improvement and reduction in cost of computers, commercial

multibody software packages have become widespread in industry. These are capable of model-

ling highly complex and nonlinear systems having a large number of “inputs” and “outputs”.

Unfortunately, from a control engineering standpoint, the order of these models is usually too

high, making it extremely difficult for modal control schemes based on these to be imple-

mented. This is a tutorial paper on the application of a combination of experimental, computa-

tional and analytical methods for the modelling, analysis and post-processing of general multi-

body systems. The purpose of this paper is to undertake in outline a systematic study of the

concepts involved in developing a model of a complex vehicle system which is well suited for

modal control design.

2 IOMAC'11 – 4

th International Operational Modal Analysis Conference

windscreen frame (easily visible to the human eye), and uncomfortable tactile steering wheel

vibrations. These will cause ride and comfort problems.

Many different approaches have been tried in the automobile industry over the years to im-

prove the rigidity – both in torsion and bending – of convertible bodies, with varying degrees of

success. For the torsional rigidity in particular, the adding or building-in of a torsionally rigid

frame within the vehicle floor or chassis is a measure widely used to achieve a considerable in-

crease in torsional rigidity. However, this will almost inevitably mean that the convertible will

be heavier than the corresponding saloon or coupé.

In this paper we are concerned with the application of a combination of experimental, com-

putational and analytical methods to the modelling, analysis and post-processing of general

multibody systems. In particular, we shall be concerned with the modelling capabilities of a

specific multibody software package, called ADAMS (developed by MSC.Software), in con-

structing a model of a complex convertible which is well suited for control design. Even though

there are many commercial programs to choose from, including not only ADAMS but also

other software packages such as MADYMO and SIMPACK, the descriptions in this paper are

based on ADAMS, because the two authors have between them many years of experience

working with the software. We will also have occasion to make some very brief remarks on the

role played by the PATRAN/NASTRAN finite element software package (also by

MSC.Software) in the development of the vehicle model.

Ideally, since much experimentation has been done in engineering laboratories on analysis,

by various techniques, of a number of systems related to automobiles, one could simply cite

those studies. Unfortunately, apart from the fact that studies that are explicitly concerned with

convertibles are scarce (e.g. Kalinke et al. (2001), Kalinke and Gnauert (2002), and Hiscutt et

al. (2008)) it is frequently difficult for a control theorist to obtain data in this way in a form

which he or she will find usable. A perusal of the summary of such studies provided by Pot-

tinger et al. (1986) will allow the reader to see what is meant here. It is for these reasons that

the second author, with the support of the Arbeitsgruppe Innovative Projekte (AGIP), has un-

dertaken a number of studies on his own, using the equipment available in the Laborbereich

Fahrzeugtechnik at the Hochschule Osnabrück (then called Fachhochschule Osnabrück) and,

occasionally, at Wilhelm Karmann GmbH in Germany. As they help to understand the issues

and the implications of our experimental work and, in part, also to understand the motivation

behind our work in this paper, we can summarise the results of this laboratory experience,

which is described more completely by the authors in Mahinzaeim et al. (2006) (see also

Mahinzaeim et al. (2007a, b, c)), as follows:

i. In the case of a typical open convertible, a model of the bare body shell of which

is illustrated in Fig. 1, it is found that, because of the reduced torsional rigidity,

the body experiences serious torsional motion in the low frequency range, admit-

ting a distinct torsional mode of vibration which is not present in the correspond-

ing coupé; its associated natural frequency is indicated in Fig. 2 by the peak at

roughly 14 Hz.

Figure 1: Body shell model of a typical convertible.

3

The directly measured observations used in producing the plots in Fig. 2 were the

lateral (side to side) accelerations of the test vehicles (open convertible and coupé)

at the top of each of the driver’s side A-columns, and they were made with accel-

erometers. (When looking at the side of a vehicle, the A-column or “A-pillar”, as it

is frequently called, is the post that attaches to the windscreen.) Examination of

the data from which Fig. 2 was obtained and comparison with data collected at

other points on the body further revealed that close to the top of the A-column

(where the measurements were taken) the lateral accelerometer signals were al-

most exclusively caused by torsional motions. Another notable observation in Fig.

2 is the highly damped peak, corresponding to the open convertible, just to the

right of the 25 Hz point.

Figure 2: Lateral accelerations of the top of the driver’s side A-columns: open

convertible ( ) and coupé ( ).

There is some evidence, supported by a finite element type of analysis of the mul-

tibody model (described later) of the convertible, that a group of natural frequen-

cies corresponding to coupled torsional/bending vibration modes occurs, not too

distinctly, from 20 Hz to 30 Hz, approximately. The peak to the right of the 25 Hz

point in Fig. 2 most likely corresponds to this group.

ii. In the case of a typical closed convertible, it is found that the folding textile roof

does not contribute significantly to the torsional rigidity of the body, as evidenced

in Fig. 3.

Figure 3: Lateral accelerations of the top of the convertible’s driver’s side A-column:

open ( ) and closed ( ).

Interestingly, however, the peak near the 25 Hz point, first seen in Fig. 2, no

longer appears in the frequency band of interest for the case when the convertible

is closed.

4 IOMAC'11 – 4

th International Operational Modal Analysis Conference

iii. Supporting experimental work to this study carried out by Wilhelm Karmann

GmbH, in cooperation with ERAS GmbH in Germany, on the convertible consid-

ered the incorporation of hydraulic actuators in the rear cross members of the ve-

hicle floor, which are illustrated in a portion of the body shell model (cf. Fig. 1)

depicted in Fig. 4, taking into account the transmission path for the torsional mo-

tion. The use of these actuators to carry torsional motion transferring forces into

the body was found to be an effective means, not necessarily of recovering the tor-

sional rigidity, but of influencing the body’s response and successfully dampening

any torsional vibrations that might arise.

Figure 4: Illustration of the rear cross members in the floor of a typical convertible.

Based on several approximating models of various orders and structures, feedback

control forces acting along the rear cross members were computed for the compos-

ite system consisting of the convertible, the hydraulic actuators and the acceler-

ometer placed at the top of the A-column. The results, for the same excitation as

used to generate Figs. 2 and 3, are shown in Figs. 5-1 and 5-2, superimposed on

the plots from the results in Fig. 3, for two specific and well known (see Anderson

and Moore (2007), e.g.) feedback control designs, the proportional plus integral

plus derivative (PID) controller and the linear quadratic Gaussian (LQG) control-

ler. What is striking in Fig. 5-1 is the fact that the feedback control forces, while

found to reduce the peak level at about 14 Hz, tend to emphasise the peak to the

right of the 25 Hz point, corresponding to the natural frequencies associated with

the clustered vibration modes in the group which we have briefly mentioned.

Figure 5-1: Lateral accelerations of the top of the open convertible’s driver’s side A-

column: no control ( ), PID control ( ), and LQG control (· · · · · ·).

5

Figure 5-2: Lateral accelerations of the top of the closed convertible’s driver’s side A-

column: no control ( ), PID control ( ), and LQG control (· · · · · ·).

This anomalous behaviour is thought to arise because the feedback controllers

were designed on the basis of finite dimensional approximations of the physical

system, by neglecting vibration modes higher than the first, although these effects

have been discussed (see Balas (1978)).

It is not at all clear, for example, if, for the very composite system sketched out above, in-

creasing the order of the approximating models, irrespective of how they were obtained, yields,

in the limit, an LQG controller which is also optimal (in the usual textbook sense, say) for the

real physical system. Nor is it clear whether this can be used to advantage to permit control

and/or observation of the first vibration mode without excessive interference from the second

(or higher) mode. To pursue any of these questions would take us away from our chief purpose

in this note.

2 ADAMS MODEL OF A TYPICAL CONVERTIBLE AND STATE SPACE REPRESENTATION

A multibody model of the convertible was built in ADAMS to capture the true physics of the

vehicle by matching the experimental results obtained as closely as possible, particularly in

view of the elastic body’s response to excitations of various kinds. This model forms part of the

system described here.

In general, the approach used to model a vehicle in ADAMS will first and foremost depend

on the type of analysis to be performed as well as the particular type of behaviour of the vehicle

that must be reproduced. Therefore, in this study, whether the vehicle body is incorporated

within the multibody model as a single rigid body, or as a finite element representation of it, is

clearly of special importance.

Three consecutive phases of modelling are involved, as shown conceptually in Fig. 6. The

first is to construct, in PATRAN, a finite element model of the bare body shell, or body-in-

white, as it is often called in the industry, which includes a range of “masses” being lumped (at

the appropriate mesh nodes) with the body-in-white to represent the engine, vehicle interior and

other payload. What is also included in this phase, and rather important here, is the specifica-

tion of the mesh nodes where the measurements are to be taken and of those where the forces

are to be applied later in the multibody model’s development. (An example of a body-in-white

model of a typical convertible was shown in Fig. 1.)

6 IOMAC'11 – 4

th International Operational Modal Analysis Conference

Figure 6: Overview of the vehicle modelling process.

This finite element model definition is converted in the second phase to a bulk data format

(BDF) which is then evaluated by NASTRAN, a comprehensive finite element analysis code, to

create a modal neutral file (MNF). ADAMS then, in phase three, evaluates the MNF and repro-

duces the elastic body model, which then is subsequently assembled to various vehicle subsys-

tems representing the suspensions, tyres and the steering system of the vehicle. This final phase

is perhaps the most difficult and challenging of the three. For more details, particularly on

phase three, the reader is referred to the book by Blundell and Harty (2004).

Figure 7: Lateral acceleration of the top of the driver’s side A-column of the ADAMS vehicle model.

7

Results from the ADAMS model, for lateral acceleration measurements taken at the top of

the driver’s side A-column, are presented in Fig. 7. These are compared with the measured ex-

perimental data from the real physical system in Fig. 8, where relatively good agreement is ap-

parent.

Figure 8: Lateral accelerations of the top of the driver’s side A-columns. Comparison of experimental

measurement ( ) and ADAMS simulation ( ).

It is now possible, using a simple ADAMS code statement (allowed in combination with cer-

tain types of analyses only!), to extract the linearised state space representation for the ADAMS

model sketched above in a format suitable for input to, for example, MATLAB/Simulink for

control design. Here the linearised state space representation of our model is given (formally)

by

( ),

.

t t t t

t t

z Az Bu Dw

y Cz (1)

In the above, 2t u and 4t w , which denote the control and the disturbance respec-

tively, represent the actuation forces in the rear cross members of the vehicle floor and the exci-

tation forces at the tyres. The observation 12t y represents measurements taken at several

points on the vehicle body. The state, which is given by 214t z , represents a subset of the

coordinates of the ADAMS model that are used to build the linearised state space representa-

tion above. However, the size of the state space 214 is too large, making it extremely difficult

for modal control in the state space to be implemented. A simple yet effective way to decrease

the dimension of the state space is to use modal truncation.

3 MODAL TRUNCATION AND CONTROL

Here we restrict ourselves to systems which are described by linear ordinary differential equa-

tions with a given initial state. We consider a special case of Eq. (1) where we assume, for con-

venience and without loss of generality for this discussion, that w 0 . Our notation and termi-

nology are fairly standard and are drawn largely from Kreyszig (1978).

3.1 The asymmetric matrix eigenvalue problem

Consider the standard state space representation of a constant linear system

0

, 0,

0 ,

t t t t

z Az Bu

z z (2)

8 IOMAC'11 – 4

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under an arbitrary control mt u , where 0 ,nt z z , A is a, not necessarily symmetric,

n n real matrix, and B is an n m real matrix. Eq. (2) is an initial-value problem on the state

space n and has, under suitable conditions on u , the unique solution

0

0

, 0.

tt stt e e s ds t

AA

z z Bu (3)

Along with (2) we consider the observation

, 0,t t t y Cz (4)

where C is a real q n matrix, so qt y . We assume, as is usually the case in practical

situations, that 0 m n and that 0 q n .

Let us consider the eigenvalue problem associated with A in Eq. (2). Suppose initially that

, 1,2, ,j k n and assume that A possesses n linearly independent eigenvectors

,

n

r j x 0 corresponding to n distinct eigenvalues j of A . It can be shown without

difficulty that then AT also possesses the property of possessing n linearly independent eigen-

vectors, call them ,

n

l j x 0 , associated with the eigenvalues j . We may thus write

, ,r j j r jAx x (5)

and, similarly,

, , ,l k k l kA x xT (6)

(Note that by elementary properties of the characteristic determinants of AT and A , we always

have that A and AT have the same spectrum.) If we now employ the usual inner product in

n , we obtain from Eq. (5)

, , , ,, ,r j l k j r j l kAx x x x

and, therefore, in view of Eq. (6),

, , , , , ,, , , .T

r j l k k r j l k j r j l k x A x x x x x

Hence

, ,, 0j k r j l k x x

and, if j k , we have

, ,, 0.r j l k x x (7)

That is, the two sets of vectors , , 1,2,r j j nx and , , 1,2,l j j nx , satisfying (7) for

j k , are orthogonal, or biorthogonal. Furthermore, define , ,:r j j r jcφ x and , ,:l j j l jcφ x .

Making use of a familiar procedure, these vectors may be normalised, by setting, for each

1,2, ,j n ,

, ,

1,

,j

r j l j

c x x

so that they satisfy the biorthonormality conditions

9

, ,, ,r j l k jkφ φ

where jk is Kronecker’s delta.

Now let , 1,2, , 2j k n , where n is an even integer. Write 2 1 ij j ju v and

2 2 1 ij j j ju v , where the constants ju and jv are both real, and, for simplicity, 0jv .

Write, further, ,2 1 , ,ir j r j r j φ α β and ,2 , ,ir j r j r j φ α β as well as ,2 1 , ,il j l j l j φ α β and

,2 , ,il j l j l j φ α β , where , , , ,, , , n

r j r j l j l j α β α β . Then it is routine to verify that

, , , , ,l j l j r k r k jk jα β α β TT

where 1 01

:0 12

j

T , and that

, , , , ,l j l j r k r k jk j jα β A α β T ΛT

where :j j

j

j j

u v

v u

Λ . We can now define the n n real, nonsingular matrices rΦ and lΦ

by

,1 ,1 ,2 ,2 , 2 , 2:r r r r r r n r nΦ α β α β α β

and

,1 ,1 ,2 ,2 , 2 , 2: ,l l l l l l n l nΦ α β α β α β

respectively. Clearly then

1

2

2

,l r

n

T 0

TΦ Φ

0 T

T (8)

and

1 1

2 2

2 2

.l r

n n

TΛ 0

T ΛΦ AΦ

0 T Λ

T (9)

3.2 Modal decomposition and truncation

Let us return to our system given by Eqs. (2) and (4). We put rz Φ q and rz Φ q to obtain,

from Eq. (2),

0

, 0,

0 .

r r

r

t t t t

Φ q AΦ q Bu

Φ q z

Premultiplication by lΦT yields, using (8) and (9),

10 IOMAC'11 – 4

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1 1 1

2 2 2

2 2 2

1

2

0

2

, 0,

0 ,

l

n n n

l

n

t t t t

T 0 TΛ 0

T T Λq q Φ Bu

0 T 0 T Λ

T 0

Tq Φ z

0 T

T

T

which may be written as

1

1 1

2 2

2 2

0

, 0,

0 ,

l

n n

t t t t

Λ 0 T 0

Λ Tq q Φ Bu

0 Λ 0 T

q q

T

where

1

1

2

0 0

2

: l

n

T 0

Tq Φ z

0 T

T . So we see that Eq. (2) is transformed into a set of un-

coupled initial-value problems each of which describes the dynamics of one vibration mode.

Also we have that the coordinate transformation rz Φ q carries Eq. (4) into

, 0.rt t t y CΦ q

Now define

1

1 1

2 2

2 2

: , : , : ,l r

n n

Λ 0 T 0

Λ TA B Φ B C CΦ

0 Λ 0 T

T

and distinguish between quantities reflecting, say, the 2n p controlled and 2p uncon-

trolled, or residual, vibration modes ( 0 p n ) by writing N

R

qq

q and

N

R

yy

y as well as

r rN rRΦ Φ Φ and l lN lRΦ Φ Φ , such that

1

1

2

0 0

2

:N lN

n p

T 0

Tq Φ z

0 T

T

and

, , ,NN

N R

RR

BA 0A B C C C

B0 A

11

where NA , NB , given by

1

1

2

2

,N lN

n p

T 0

TB Φ B

0 T

T

and NC , given by N rNC CΦ , are n p n p , n p m and k n p real matrices,

respectively. We obtain

0

, 0,

0 ,

N N N N

N N

t t t t

q A q B u

q q (10)

or (for suitable u ; cf. Eq. (3)), equivalently,

0

0

, 0,NN

tt st

N N Nt e e s ds t

AA

q q B u

together with

, 0.N N Nt t t y C q (11)

(It goes, of course, without saying that N and R stand for, respectively, the number of con-

trolled vibration modes and the number of residual vibration modes.)

One may now use, for example, the frequency band of interest as a criterion for selecting the

number of vibration modes to be controlled. A rather more interesting criterion is based on the

balanced realisation of the system, according to Moore (1981). For example, returning with

these results to the special case (cf. Eq. (1)) where, for each 0t , 2t u , t w 0 , and

1y t (representing acceleration measurements taken at the top of the driver’s side A-

column), it is perhaps not surprising, then, that the size of the state space can be reduced from

214 to 2.

3.3 Modal control

For ease of implementation, the traditionally favoured means for controlling and/or stabilising

the system described by Eqs. (10) and (11) is to use a linear feedback relation ˆN Nu K q ,

wherein at each time 0t the modal control tu is determined as a linear function of the es-

timated modal state ˆN tq , generated, for example, by a Luenberger observer. The problem

then becomes that of choosing the m n p feedback matrix NK in such a way that the cou-

pled system realised by substituting ˆN Nu K q into Eq. (10) and the relevant observer equa-

tions (see Anderson and Moore (2007)) is modified in order to achieve a desired behaviour.

Once this has been done, the control will be used for the full system given by Eqs. (2) and (4),

or (not equivalently) for the multibody system in ADAMS through a simulation involving par-

allel operation of ADAMS and MATLAB/Simulink (called “co-simulation”). It should be clear,

however, that in that case one has to face the question, at the methodological level, as noted in

the Introduction, of the relation between the controls ˆN NK q and ˆKq , with K being the m n

(full modal state) feedback matrix.

12 IOMAC'11 – 4

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4 SUMMARY

This paper has been written in an attempt to summarise, very briefly, the authors’ previous

works on the subject of modelling and control of modern convertibles and, at the same time, to

show how a combination of experimental, computational and analytical methods can lead to an

integrated approach to the subject.

ACKNOWLEDGEMENT

The studies reported in this paper were conducted during the second author’s visits to the Fac-

ulty of Engineering and Computer Science of the University of Applied Sciences Osnabrück

and were supported in part by AGIP, under research grant 2005.708 with the University of Ap-

plied Sciences Osnabrück, and in part by Wilhelm Karmann GmbH.

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Hiscutt, P., Ishikawa, S. and Croot, C. 2008. Reduction of whole body shake on a luxury sports converti-ble. Noise and Vibration, 2008, p. 85–92.

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