modeling multibody dynamic systems with uncertainties

8/10/2019 Modeling Multibody Dynamic Systems With Uncertainties

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Modeling Multibody Dynamic Systems With Uncertainties.Part II: Numerical Applications

Adrian Sandu*, Corina Sandu, and Mehdi Ahmadian

Virginia Polytechnic Institute and State University

*Computer Science Department, [email protected] Engineering Department, {csandu,ahmadian}@vt.edu

Abstract

This study applies generalized polynomial chaos theory to model complexnonlinear multibody dynamic systems operating in the presence of parametricand external uncertainty. Theoretical and computational aspects of thismethodology are discussed in the companion paper Modeling MultibodyDynamic Systems With Uncertainties. Part I: Theoretical and Computational

Aspects.In this paper we illustrate the methodology on selected test cases. Thecombined effects of parametric and forcing uncertainties are studied for a quartercar model. The uncertainty distributions in the system response in both time andfrequency domains are validated against Monte-Carlo simulations. Resultsindicate that polynomial chaos is more efficient than Monte Carlo and moreaccurate than statistical linearization. The results of the direct collocationapproach are similar to the ones obtained with the Galerkin approach. Astochastic terrain model is constructed using a truncated Karhunen-Loeveexpansion. The application of polynomial chaos to differential-algebraic systemsis illustrated using the constrained pendulum problem. Limitations of thepolynomial chaos approach are studied on two different test problems, one withmultiple attractor points, and the second with a chaotic evolution and a nonlinearattractor set.

The overall conclusion is that, despite its limitations, generalizedpolynomial chaos is a powerful approach for the simulation of multibody dynamicsystems with uncertainties.

Keywords: uncertainty, stochastic process, polynomial chaos, statisticallinearization, Monte Carlo, Karhunen-Loeve expansion, chaotic dynamics.


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Introduction

This study investigates the efficient treatment of uncertainties that occur inmechanical systems. Such uncertainties result from poorly known or variableparameters (e.g., variation in suspension stiffness and damping characteristics),

from uncertain inputs (e.g., soil properties in vehicle-terrain interaction), or fromrapidly changing forcings that can be best described in a stochastic framework(e.g., rough terrain profile). In the companion paper Modeling MultibodyDynamic Systems with Uncertainties. Part I: Theoretical and Computational

Aspects [1] we present the methodology developed to account for several typesof uncertainties, with the goal of obtaining realistic predictions of the systembehavior using computationally efficient methods.

As described in detail in Part I, this study applies the generalizedpolynomial chaos theory to formally assess the uncertainty in multibody dynamicsystems. The polynomial chaos framework has been chosen because it offers anefficient computational approach for the large, nonlinear multibody models of

engineering systems of interest, where the number of uncertain parameters isrelatively small, while the magnitude of uncertainties can be very large (e.g.,vehicle-soil interaction). The proposed methodology allows the quantification ofuncertainty distributions in both time and frequency domains, and enables thesimulations of multibody systems to produce results with error bars, similar tothe way the experimental results are often presented.

The results obtained with the polynomial chaos approach are comparedwith Monte Carlo simulations and with statistical linearization. Monte Carlosimulations are costly, and the accuracy of the estimated statistical propertiesimproves with only the square root of the number of runs. Statistical linearizationcomputes several moments of the uncertainty distribution of the solution, but do

not capture essential features of the nonlinear dynamics (e.g., as revealed bypower spectral density).

This paper is organized as follows: The response of a two degree-of-freedom system that represents a vehicle suspension is analyzed for parametricuncertainties, stochastic forcings, and combined sources of uncertainty. Next, wepresent numerical results for modeling uncertain terrain. The polynomial chaosapproach is compared with the statistical linearization method. Aspects related tospecial dynamical behavior are studied with the help of two test cases: thedouble-well problem and the Lorenz problem. The treatment of uncertainties inmultibody dynamic systems with constraints (modeled by differential algebraicequations (DAEs)) is illustrated on a simple pendulum problem. Finally, the main

conclusions of this study are summarized.

Two Degree of Freedom Vehicle Suspension Model

The treatment of nonlinear multibody dynamic systems with the proposedstochastic methods that account for uncertainties in the system is illustrated bythe following case study. Recognizing that most dynamic systems can beexpressed in a lumped parameter form (as a series of mass, spring, and


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damper), we consider the two degree-of-freedom quarter-car system illustratedin Fig. 1 which models a vehicle suspension.

Figure 1.The quarter-car model

The sprung mass sM and the unsprung mass uM are connected by a

nonlinear spring kand a linear damperc . The forcing ( )tz applied to uM through

the linear spring of stiffness Tk represents the interaction with the terrain. The

state-space representation of the quarter-car model system is:

( ) ( )( ) ( ) ( )221

3

2132

21

3

2131

22

11

)( xtzkvvcxxkvM

vvcxxkvM

vx

vx

Tu

s

++=

=

=

=

&

&

&

&

(1)

where 1x , 2x are the displacements of the sprung and unsprung masses, and

1v , 2v their respective velocities. The terrain is of the form:

)2sin()( tfAtz = (2)

The quarter-car model is subject to parametric and forcing uncertainties.The forcing uncertainty comes from the terrain profile, i.e., the forcedisplacement. Its amplitude can take values within +/- 50% of the nominal value.The uncertainty in the displacement is modeled using a uniform distribution(u.d.):

( ) ( ) ( ) ( ) ( ) u.d.]1,1[,2sin1, 111 +=+= ftaAtztztz (3)The parametric uncertainty comes from the variable tire stiffness and

damping constant. The probability distribution of the tire stiffness values is:

[ ] u.d.1,1, 22 += TTT kkk (4)and the probability distribution of the damping coefficient is:

[ ] u.d.1,1, 33 += ccc (5)We assume that the uncertainties in the terrain amplitude, tire stiffness,

and the damping coefficient are not correlated, i.e., 1 , 2 , and 3 are

independent, uniformly distributed random variables.The state is expressed as:


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( ) ( ) 4,3,2,1,)(),(1

===

itxtx jS

j

j

ii (6)

where Sis the dimension of the polynomial chaos space. Thus, the stochasticsystem is formulated as

( ) ( ) ( )( ) ( ) ( ) ( ) ( )jkxzkjkcvvcvM

jkcvvcvM

vx

Sjvx

T

jj

T

jjj

u

jjj

s

jj

jj

DBjA

BjA

++++=

=

=

=

23212

3211

22

11 1allfor,

&

&

&

&

(7)

where:

( ) ( ) ( )

( ) ( )( )( ) ( )

( ) ( ) ( )jkExz

jplmExxxxxx

jkEvv

S

k

kk

S

l

S

m

S

p

ppmmll

j

S

k

kk

j

,,41

,,,1

,,31

1

3221

1 1 1

42121212

1

3212

=

= = =

=

=

=

=

jD

jB

jA

(8)

with:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( )

( ) ( )

+

+

+

+

+

+

+

+

+

=

=

=

1

1

1

1

1

1 321

2

321321

1

1

1

1

1

1 3213213213214

1

1

1

1

1

1 3213213213213213

,,

,,,,,,,,,,

,,,,,,,,,,

ddd

dddw

kjiE

dddwkjiE

kjj

lkji

kji

(9)

System Response under Parametric Uncertainty

Due to the nonlinearity of the quarter-car system and the non-Gaussiandistribution of the parametric uncertainty, the model output uncertaintydistribution is non-Gaussian and difficult to estimate.

Figure 2 shows the uncertainty in the sprung mass displacement and inthe power density due to uncertainty in the damping coefficient. Only slightdifferences can be noticed between the evolution of the sprung massdisplacement in the deterministic case and the mean value of the systemresponse in the stochastic case with polynomial chaos.


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1 1.2 1.4 1.6 1.8 20.15

0.1

0.05

0

0.05

0.1

0.15

Time [s]

Displacement[m]

Deterministic

Polyn. Chaos (mean 1)

0 1 2 3 40

30

60

90

120

150

180

Frequency [Hz]

PowerD

ensity[m2

s

4H

z

1]

Deterministic

Polyn. Chaos (mean 1 )

(a) Sprung mass displacement (b) Power density

Figure 2.The uncertainty in the sprung mass displacement and in the powerdensity due to the uncertainty in the damping coefficient

Figure 3 shows the uncertainty in the sprung mass displacement and inthe power density due to uncertainty in the tire stiffness. It can be seen that thistype of uncertainty influences the response of the system substantially. The

range of the system responses, as illustrated by the error bars in Fig. 3a, is alsolarger than that due to uncertainty in the damping characteristics.

1 1.2 1.4 1.6 1.8 20.15

0.1

0.05

0

0.05

0.1

0.15

Time [s]

Displacement[m]

Deterministic


0 1 2 3 40

30

60

90

120

150

180

Frequency [Hz]

PowerDensity[m2

s

4H

z

1] Deterministic

Polyn. Chaos (mean1 )


Figure 3.The uncertainty in the sprung mass displacement and in the powerdensity due to the uncertainty in the tire stiffness

Uncertainty due to Stochastic Forcing

The forcing function uncertainty is first considered of the form shown in Eq. (3),where the random variable has (a) a uniform, and (b) a beta distribution, with

mean 0 and support [ ]1,1 . Thus, the forcing amplitude can change between

( )aA 1 and ( )aA +1 , and has meanA . The system response is parameterizedusing polynomial chaos decomposition up to order 5 (with Legendre polynomialsfor the uniform distribution, and Jacobi polynomials for the beta distribution). Theevolution of the sprung mass position for one second of evolution is shown in Fig.4a for the uniform forcing, and in Fig. 4b for the beta forcing. Note that the resultsof the deterministic simulation (using the nominal value of the amplitude) aredifferent than the probabilistic mean due to nonlinearities, but remain within onestandard deviation of it. As expected, the results for the beta uncertainty show a


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slightly smaller variance, and a smaller difference between the stochastic meanand the deterministic solution.

0 0.2 0.4 0.6 0.8 1

0.3

0.2

0.1

0

0.1

0.2

0.3

Time [s]

Displacement[m

]

Deterministic


0 0.2 0.4 0.6 0.8 1

0.3

0.2

0.1

0

0.1

0.2

0.3

Time [s]

Displacement[m

]

Deterministic


(a) Amplitude uniform distributed (b) Amplitude beta distributed

Figure 4. Time evolution of sprung mass displacement for stochastic forcing withamplitude uniformly distributed and beta distributed

Uncertainty from Combined Sources

In this section we present results reflecting the combined effect of bothparametric uncertainty and stochastic forcing. We analyze the resultinguncertainty in both time domain and frequency domain for the quarter-car model.

Figure 5 presents the uncertainty in the sprung mass displacement, andthe power density due to the combined effect of all uncertainty sources: forcing,damping parameter, and tire stiffness. The deterministic run results are differentthan the probabilistic mean, but they remain within one standard deviation of it.

1 1.2 1.4 1.6 1.8 20.15

0.1

0.05

0

0.05

0.1

0.15

Time [s]

Displacement[m]

Deterministic


0 1 2 3 40

30

60

90

120

150

180

Frequency [Hz]

PowerDensity[m2s4Hz1]

Deterministic

Polyn. Chaos (mean 1 )


Figure 5.The uncertainty in the sprung mass displacement and in its powerspectrum due to the combined effect of all uncertainty sources: forcing, damping

parameter, and tire stiffness

Figure 6 presents the mean values (upper panels) and standarddeviations (lower panels) for the sprung mass (left panels) and unsprung mass(right panels) displacements. Shown are the moments computed from thepolynomial chaos simulation, and from a Monte Carlo simulation with 2000 runs.The results are indistinguishable for the means and very close for standard


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deviations. The cpu times (of Matlab implementations) are 20 seconds for thepolynomial chaos simulation and 360 seconds for the Monte Carlo.

0 1 2 3

0.25

0

0.25

0.5Sprung Mass

Time [s]

Displacement[m]

0 1 2 3

0.25

0

0.25

0.5Unsprung Mass

Time [s]

Displacement[m]

0 1 2 30

0.03

0.06

0.09

0.12

0.15

Time [s]

Displacem

ent[m]

0 1 2 30

0.03

0.06

0.09

0.12

0.15

Time [s]

Displacem

ent[m]

Monte Carlo Std. Dev.Polyn. Chaos Std. Dev.

Monte Carlo Std. Dev.Polyn. Chaos Std. Dev.

DeterministicMonte Carlo MeanPolyn. Chaos Mean

DeterministicMonte Carlo MeanPolyn. Chaos Mean

Figure 6. The mean values (upper panels) and standard deviations (lowerpanels) for the sprung mass (left panels) and unsprung mass (right panels)

displacements

The polynomial chaos simulation can produce the probability density

function (PDF) of the results. The PDFs of the of the sprung and unsprung massdisplacements after 3 seconds of simulation are shown in Fig. 7a and in Fig. 7b.To validate the approach we carried out a 2,000-run Monte Carlo simulation. TheMonte Carlo PDF is similar to the polynomial chaos PDF (ignoring the additionalpeak for the sprung mass), but is obtained in a much longer CPU time.

0.2 0.1 0 0.10

5

10

15

20Sprung Mass

Displacement [m]

PDF

Monte CarloPolyn. Chaos

0.2 0.1 0 0.10

5

10

15

20

Displacement [m]

PDF

Unsprung Mass


(a) Sprung mass (b) Unsprung mass

Figure 7.The probability density of the displacements of sprung mass andunsprung mass


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2670

1780

890

0

890

1780

2670

Figure 8 shows the power density of the sprung mass displacement, andthe probability density of the power spectral density. Both the deterministic andthe probabilistic mean of the power density peak at 3 Hz, as can be seen in Fig.8a. At lower frequencies, the peaks are at 1 Hz, respectively at 1.33 Hz. For 1.33Hz, the probability density increases rapidly and drops abruptly at a power

density of approximately Hzsm 42

30 , as shown in Fig. 8b. For the 3 Hz case, theprobability density increases slowly until the power density reaches Hzsm 4230 ,

it stays relatively constant until Hzsm 42150 , then decreases slowly, as

illustrated in Fig. 8b.

0 1 2 3 40

30

60

90

120

150

180

Frequency [Hz]

Powe

rDensity[m2s4Hz1] Deterministic

Polyn. Chaos

(mean 1 )

0 50 100 150 200 2500

0.01

0.02

0.03

0.04

Power Density [ m2s

4Hz

1]

ProbabilityDensity

PDF (3 Hz)PDF (1.33 Hz)

(a) Displacement power density (b) Probability density of the power density

Figure 8.(a) Uncertainty in the power density of the sprung mass displacement;(b) the probability density of the power spectral density

The Collocation Approach

To illustrate the collocation approach we consider a cubic spring and a nonlineardamping element which models a magneto-rheological (MR) damper. The MR

damper force depends on the velocity difference v between the sprung andunsprung masses, and on the applied current intensity I ,

( ) ( ) ( )vIvIF 3.14tanh1015453195.701 +++= (10)The MR response is shown in Fig. 9a. It simulates the real MR damper

characteristics shown in Fig. 9b. The current intensity is a random variable,uniformly distributed between A]2,0[ . The range of values of the damping force

for different velocities is illustrated in Fig. 9b.

1.5 0.75 0 0.75 1.56000

4000

2000

0

2000

4000

6000

Velocity [m/s]

Force[N]

MR Damper

I = 0 AI = 1 AI = 2 A

(a) MR damper response (b) Experimental damping characteristics for

Force

[N]

-0.64 -0.38 0.13 0 0.13 0.38 0.64Velocity [m/s]

0 Amps

0.2 Amps

0.4 Amps

0.6 Amps

0.8 Amps

1.0 Amp

1.2 Amps

1.4 Amps

1.6 Amps

1.8 Amps

NOTE: + Force = Extension

Note: Curves h ave been zeroed

as described in the text.

0 Amps

0.2 Amps

0.4 Amps

0.6 Amps

0.8 Amps

1.0 Amp

1.2 Amps

1.4 Amps

1.6 Amps

1.8 Amps

NOTE: + Force = Extension

Note: Curves h ave been zeroed

as described in the text.


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ML-430 MR damper [2,3]

Figure 9. MR damper response and experimental damping characteristics

Because of the hyperbolic tangent nonlinearity, the application of theGalerkin formulation of the polynomial chaos is difficult. Instead, we define the

stochastic system using the direct collocation approach. The basis functions areLegendre polynomials of order up to 3, and the collocation points are the roots ofthe fourth order Legendre polynomial.

The evolution of the sprung mass and unsprung mass displacements areshown in Fig. 10, when the vehicle negotiates over a 25 cm square bump. Notethat the deterministic and the stochastic mean results do not coincide.

Figure 10.The response of the nonlinear quarter-car with uncertain MR dampingcharacteristics when negotiating over a square bump

The probability density functions of the displacements after 6 seconds isshown in Fig. 11. The polynomial chaos results are in excellent agreement withMonte Carlo simulation results.

0.06 0.07 0.08 0.09 0.10

50

100

150

Displacement [m]

PDF

PDF of Sprung Mass Displacement

PDF of Unsprung Mass Displacement

Figure 11.The probability density functions of the response of the nonlinear

quarter car at t = 6 seconds

Modeling Uncertain Terrain

Accurate modeling of the soil characteristics and terrain geometry is one of themain challenges in modeling off-road vehicles [4-6]. Another non-trivial issue ismodeling unprepared terrain, due to the high degree of uncertainty in itsgeometry. The accuracy of the terrain model is directly related to the response of


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the vehicle analyzed, such as its mobility. It is clear that a deterministic approachis not suitable in this case [7], and the uncertainties in the terrain profile must betaken into account.

The proposed terrain model exemplifies the use of Karhunen-Loeve (KL)expansion to represent stochastic forcings. KL expansion is based on the

eigenfunctions and eigenvalues of the covariance function. As explained in thecompanion paper [1], the covariance function of the forcing can be determinedexperimentally, for example from power spectrum density data.

We consider a one-dimensional course with an uncertain profile. Theterrain height is periodic in space with a period D and is modeled as a stochasticprocess periodic in space.

( ) ( )

( )

integersand,2

,2

2sin

sincossincos

21

2

2

1

1

0

2

22

2

212

1

12

1

111

nnn

DLn

DL

D

xAxz

L

x

L

xA

L

x

L

xAxzxz

==

=

+

+

+

+=

(11)

In Eq. (11) ij are independent normal random variables of mean 0 and

standard deviation 1. The terrain height at each location x is a normal random

variable of mean )(xz and standard deviation 222

1 AA + . The correlation between

terrain heights at two spatial points is

( ) ( ) ( )[ ]

+

==

2

212

2

1

212

1221121 coscos)()()()(,L

xxA

L

xxAxzxzxzxzExxR (12)

We regard the stochastic process from Eq. (11) as the superposition of

two periodic stochastic processes with correlation lengths 1L and 2L respectively.

The correlation function )( 21 xxR is also periodic with period D . Lucor et. al. [8]

show that the Karhunen-Loeve representation of a periodic stochastic process is:

( )

=

+

+=

1

00 2sin

2cos

2,

n

sncnncD

xn

D

xn

DDxz

(13)

They also establish that the eigenvectors of the periodic covariance kernel

are )/cos( nLx , )/sin( nLx , where the correlation length of theth

n eigenvector is

)2/( nDLn = . The periodic correlation function in Eq. (12) has only two nonzero

eigenvalues,

2

22

2

112

,2

ADAD == (14)

From Eq. (13) and Eq. (14) we conclude that Eq. (11) is the Karhunen-Loeve representation of the periodic stochastic process with covariance given byEq. (12). The choice of the model Eq. (11) is motivated by the analysis of Lucoret al. [8]. The authors consider bilateral autoregressive processes of the form:


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( ) ( ) ( )

iii

i axxzxxz

bxz +++

=2

(15)

Intuitively, the terrain height at ix depends on the average of terrain

heights at locations situated both to its left and to its right, plus a random term. In[8] it is shown that, in the continuous limit 0x , 1b , and for particular

choices of the correlation lengths, the second order autoregressive process fromEq. (15) can be represented as in Eq. (11).

In the numerical experiments presented next we consider the followingnumerical values:

mD

LmD

LmAmAmAmD 3.214

,3.56

,1.0,2.0,5.0,100 21210 ======

(16)

Therefore the mean terrain profile is a sine wave of period m100 . The

uncertainty in terrain height is modeled as the sum of two stochastic processeswith normal profiles and spatial correlation lengths of m3.5 and m3.2

respectively. Three different realizations of this model are shown in Fig. 12.

We now consider driving the quarter-car at smv 10= over this uncertainterrain profile. We wish to assess the uncertainty in the displacements of thesprung and unsprung masses. A third order polynomial chaos expansion is used,with four independent Gaussian variables. The bases are Hermite polynomials.

0 20 40 60 80 1001

0.5

0

0.5

1

Terrainheight[m]

Distance [m]

0 20 40 60 801

0.5

0

0.5

1

Distance [m]

0 20 40 60 80 1001

0.5

0

0.5

1

Distance [m]

Figure 12.Three different realizations of the periodic terrain stochastic model.The mean (the deterministic part of the profile) is represented in dashed red line.

For comparison, Fig. 13 presents the displacements of the sprung massand of the unsprung mass obtained by polynomial chaos of order 3 and by a2000-run Monte-Carlo simulation. The results are very similar, with thepolynomial chaos method having the advantage of reduced computational time.

0 1 2 3 4 5 6 7 8 9 101

0.75

0.5

0.25

0

0.25

0.5

0.75

1

Time [s]

Displacement[m]

MC Sprung Mass (mean 1)

0 1 2 3 4 5 6 7 8 9 101

0.75

0.5

0.25

0

0.25

0.5

0.75

1

Time [s]

Displacement[m]

MC Unsprung Mass (mean 1)


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Figure 13.Time evolution of the displacements of sprung and unsprung massesobtained by polynomial chaos of order 3 and by a 2000-run Monte-Carlo.

Figure 14 illustrates the probability density functions at the final time forthe sprung and unsprung masses. The polynomial chaos and the 2000-runMonte-Carlo estimates give similar results.

1 0.5 0 0.5 10

0.5

1

1.5

2

Sprung Mass Displacement [m]

PDF

Polyn. ChaosMonte Carlo

1 0.5 0 0.5 10

0.5

1

1.5

2

Unsprung Mass Displacement [m]

PDF

Polyn. ChaosMonte Carlo

Figure 14.The probability density function at the final time for the sprung andunsprung masses.

Polynomial Chaos versus Statisti cal Linearization

Statistical linearization is a powerful approach to approximate the mean andvariance of the response of nonlinear dynamic systems to stochasticperturbations [9]. To compare polynomial chaos with statistical linearization,consider the single degree of freedom nonlinear oscillator:

( ) )(tfxgxcxm =++ &&& (17)where )(tf is a stochastic forcing and )(xg a nonlinear spring. The system in

Eq. (17) is approximated by a linear system [9]:)(tfkyycym =++ &&& (18)

with the equivalent stiffness coefficient k chosen such that the mean-square

response ][ 2yE of the linear system is an optimal estimate of the mean square

response ][ 2xE of the nonlinear system in Eq. (17). Based on this approximation,

and assuming the response of the system in Eq. (18) has zero mean, its varianceis given by:


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[ ] ( )( )

+=

d

cmk

SyE

f

2222

2 (19)

where )(fS is the spectral density of the stochastic excitation )(tf .

The standard equivalent linearization applied to the one-dimensional

system in Eq. (17) provides the following formula for the equivalent stiffnesscoefficient [9]:

( )[ ][ ]22 ][

)]([

yE

yygE

xE

xgxEk = (20)

The value of this coefficient can be derived only if the statistics of thenonlinear solution x are known. In practice these statistics are approximated by

the statistics of the linear response y , which is often assumed Gaussian.

Equations (19) and (20) are solved simultaneously for kand ][ 2yE .

We consider a nonlinear cubic spring with 3)( xhxg = . Under the Gaussian

assumption for the distribution of y we have that

( ) ][3][

][3

][

][ 22

22

2

4

yEhyE

yEh

yE

yhEk == (21)

Consider the stochastic forcing be the stochastic terrain surface in Eq.

(11) with zero mean, ( ) 0=xz . The power density of this forcing is:

( ) ( ) ( ))()(2

)()(2

1

2

1

2

2

2

1

1

1

1

2

1

+++++= LLALLASf

(22)

From Eq. (19) and Eq. (22) the variance of the response is obtained:

[ ]( ) ( ) 222

22

2

2

2

2

1

222

1

2

12 22

++

+=

LcmLk

A

LcmLk

AyE

(23)

Equations (21) and (23) are solved together for kand ][ 2yE .

In the numerical experiments presented below we use the followingnumerical values (in non-dimensional units):

[ ]200,0,2,4,1,1,10 21 ===== tAAhcm (24)The system starts from rest position with zero velocity. The results are

shown in Table 1. Since statistical linearization estimates the variance at steadystate we consider a long integration interval (200 time units) for Monte Carlo andpolynomial chaos integrations. Polynomial chaos and Monte Carlo give verysimilar estimates of the variance, while statistical linearization overestimates it.

Table 1. The variance and equivalent stiffness estimated by different methods

Method ][xE ][ 2xE k

Monte Carlo (3000 runs) 0.033 1.463 4.389

Polynomial Chaos (order 3) 0 1.452 4.356

Statistical Linearization 0 1.956 5.869


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We conclude this comparison by noting that the shorter the correlationtime/distance of the stochastic forcing is, the more terms are needed in theKarhunen-Loeve expansion for a given accuracy. The polynomial chaosapproach becomes impractical for weakly correlated forcings (and in particularfor white noise forcing).

Special Dynamical Behavior

We study the quality of the polynomial chaos approximation for two systems withvery particular dynamics. The first example is the double-well problem; the PDFof the solution becomes bi-modal and difficult to approximate by smoothpolynomials. The second example is the chaotic 3-variable Lorenz model; withproper initial values, the solutions collapse onto an attractive manifold. Weexpect the chaotic dynamics to challenge the polynomial chaos approximationsof the PDF.

The Double-Well Problem

The double-well problem is governed by the ordinary differential equations(ODEs):

( ) ( ) 02 0,20,1 xxtxxx ==& (25)There are two stable attractors at 1=x , and one unstable steady state at

0=x . For every negative initial condition 00x toward 1+=x . Thus any

probability distribution of the initial state (which spans both positive and negativevalues) will evolve toward a couple of Dirac distributions on the attractor states.The problem is challenging since the state probability density becomes lesssmooth as the system evolves, making it increasingly difficult to construct

spectral approximations.Consider the case where the initial state is uniformly distributed between

]375.0,625.0[ , as shown in Fig. 15a. Since this distribution is skewed to the left,

we expect the system to reach the final state 1=x with higher probability than1+=x . The resulting PDF was computed with Monte Carlo (10,000 runs) and

Legendre chaos of orders 3 and 7. The results are shown in Fig. 15b for 2=t and in Fig. 15c for 4=t . We notice that the polynomial chaos approximationcaptures the bimodal result distribution. For 2=t the PDF calculated bypolynomial chaos is reasonably accurate. As the real distribution approaches twoDirac functions, the quality of the polynomial chaos PDF decreases, as seen in

Fig. 15c. Nevertheless, the bimodal character is well captured, with theprobability of the left final state being larger than that of the right final state.


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1.5 1 0.5 0 0.5 1 1.50

0.5

1

1.5

State at T = 0

PDF

1.5 1 0.5 0 0.5 1 1.50

3

6

9

State at T = 2

PDF

1.5 1 0.5 0 0.5 1 1.50

10

20

30

State at T = 4

PDF

Monte CarloPolyn. Chaos 3Polyn. Chaos 7

(a) (b) (c)

Figure 16.Polynomial chaos approximations of the PDF for the double-wellproblem reproduce the two peaks of the final distribution.

The Lorenz Prob lem

We now consider the classical Lorenz 3-variable system [10] motivated by

meteorological applications. ( )

zyxz

zxyxy

xyx

=

=

=

&

&

&

(26)

This system has been extensively studied since it provides a simpleexample of a particular type of nonlinear dynamics. For certain values of theparameters the system has chaotic solutions: very small perturbations in theinitial conditions lead quickly to very different solutions. From the point of view ofthis study, small uncertainties in the initial conditions grow rapidly as the systemevolves. Trajectories fall onto the Lorenz attractor, a low-dimensional stablemanifold. Note that the concepts of chaotic dynamics and polynomial chaos arenot directly related despite the similar terminology.

We consider the initial values of the system to be uncertain, and uniformlydistributed in the cube

5.155.14,10,10 000 zyx (27)

First we study the behavior for the parameter values 1= , 10= , and

1= , and a time interval of 15 units. We use 3 uncertain variables, and

Lagrange chaos of order 5. There are 56 basis functions. The system evolvestoward the stable equilibrium point ,3== yx 9=z (there are several stable

equilibria, but the cube of initial values lies in the attractive basin of this particularone). Figure 16 presents a comparison of the mean and standard deviation of

each variable against a 1000 ensemble Monte Carlo run. The polynomial chaosapproximation of the PDF is very accurate, and becomes sharper in time,simulating the Dirac distribution at the equilibrium point, as can be seen from thevanishing variance in Fig. 16.


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0 5 10 150

1

2

3

4X mean

0 5 10 151

0

1

2

3

4

5

6Y mean

0 5 10 152

4

6

8

10

12

14

16Z mean

0 5 10 150

0.2

0.4

0.6

0.8

1

1.2

1.4X std

Time units0 5 10 15

0

0.5

1

1.5

2

2.5Y std

Time units0 5 10 15

0

1

2

3

4Z std

Time units


Figure 16.Lorenz system, the stable attractor case. The polynomial chaosapproximation of the PDF is very accurate.

Next, we consider the parameter values that were originally chosen by

Lorenz [10] to obtain chaotic dynamics: 10= , 28= , and 38= . A phase plot

of the Lorenz attractor can be seen in Fig. 17. The attractor has a complicatedgeometry, which makes it difficult for polynomial chaos to approximate it.

The cube of initial values lies near the attractor, as seen in Fig. 17 (bluedots). Even if the set of initial conditions is very localized, after 15 time units thetrajectories have diverged and span all the attractor, as seen in Fig. 17 (reddots). This behavior is typical for chaotic dynamics, and is what makes this test

case challenging for polynomial chaos approximations.Figure 18 compares the mean and standard deviations obtained bypolynomial chaos with a 1000 ensemble Monte Carlo run. The rapid increase instandard deviation in the beginning reflects the rapid amplification of uncertainty,in chaotic dynamics. We notice that polynomial chaos approximation is quiteaccurate in the very beginning (for about 3 time units). After that theapproximation of the Z mean, as well as the standard deviations, becomeinaccurate. The polynomial chaos PDF after 15 time units is shown in Fig. 19a. Itis not a very accurate approximation of the PDF estimated by Monte Carlo. Thephase plot of the polynomial chaos predicted distribution is shown in Fig. 19b.This phase plot reveals that an attractor is also predicted by the polynomial

chaos approach. Its shape, however, is only a rough approximation of the realattractor shown in Fig. 17.


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17

25 15 5 5 15 2525

15

5

5

15

25

X

Y

0 10 20 30 40 5025

15

5

5

15

25

Z

Y

25 15 5 5 15 250

10

20

30

40

50

X

Z

Figure 17.Lorenz system, the chaotic case. Phase plot of the Lorenz attractor.The initial ensemble (blue) is localized in a 1x1x1 cube. After 15 time units ofevolution the trajectories have fallen onto, and dispersed all over the Lorenz

attractor (red).

0 5 10 1515

10

5

0

5

10

15

20X mean

0 5 10 1520

10

0

10

20Y mean

0 5 10 155

10

15

20

25

30

35

40Z mean


0 5 10 150

2

4

6

8

10

12

14X std

Time units0 5 10 15

0

5

10

15Y std

Time units0 5 10 15

0

2

4

6

8

10

12Z std

Time units Figure 18.Lorenz system, the chaotic case. The approximations of the mean

and standard deviation of the three variables is accurate only for a short period(~3 time units) at the beginning of the simulation.


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18

20 0 200

0.05

0.1

0.15

PDF

Monte Carlo

25 0 250

0.05

0.1

0.15

0 25 500

0.05

0.1

0.15

20 0 200

0.05

0.1

X

PDF

Polyn. chaos

25 0 250

0.05

0.1

Y0 25 50

0

0.05

0.1

Z

25 15 5 5 15 2525

15

5

5

15

25

X

Y

0 10 20 30 40 5025

15

5

5

15

25

Z

Y

25 15 5 5 15 250

10

20

30

40

50

X

Z

(a) Component-wise PDF (b) Phase plot of polyn. chaos solution

Figure 19.(a) The PDF at T = 15 units predicted by polynomial chaos and byMonte Carlo. (b) The phase plot of the attractor predicted by polynomial chaos is

only a rough approximation of the real attractor.

DAE Problems

The simple pendulum offers a simple example of a constrained mechanicalsystem. In the ODE formulation:

( ) 0,4

,sin 00 ===

&&&L

g (28)

where L is the length of the pendulum, defines the angular position of the

pendulum, && is the pendulums angular acceleration, g is the gravitational

acceleration, and 00 , & are the initial position and the initial velocity conditions.

In Cartesian coordinates ( ) ( ) cos,sin LyLx == (29)the pendulum equations form an index-3 DAE system:

( )

( )

222

2

222

2

22

0 LyxL

gxyxyy

L

gxyyxxx

+=

++=

+=

&&&&

&&&&

(30)

This system is solved by differentiating the constraints twice andintegrating the resulting index-1 DAE. Each step is followed by a projection of thesolution onto the velocity constraint manifold.

We consider the length of the pendulum string to be uncertain, with a

uniform distribution between [ ]m2.1,8.0 .mLmLLLL 2.0,1,u.d.]1,1[, ==+= (31)

The stochastic DAE is formulated by direct stochastic collocation.Figure 20 shows the variation of the angle for a simple pendulum

represented as an ODE in the deterministic model, in the stochastic model withpolynomial chaos, and using a 2000-run Monte Carlo simulation.


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20

0 3 6 9 12 151

0.5

0

0.5

1

Time [ s ]

Displacement[m]

X Deterministic

X Polyn. Chaos (mean 1)

0 3 6 9 12 15

0.7

0.8

0.9

1

1.1

Time [ s ]

Y Deterministic

Y Polyn. Chaos (mean 1)

Figure 22. Variation in the pendulum x and y coordinates for the DAE

formulation, using stochastic modeling with polynomial chaos of order 7

Figure 23 shows the variation of the position in the x and y directions for

a simple pendulum modeled with DAE in the deterministic model, and in the2000-run Monte Carlo simulation.

0 3 6 9 12 151

0.5

0

0.5

1

Displacement[m]

Time [ s ]

X DeterministicX Monte Carlo

0 3 6 9 12 15

0.7

0.8

0.9

1

1.1

Time [ s ]

Y DeterministicY Monte Carlo

Figure 23. Variation in the pendulum x and y coordinates for the DAE

formulation, using a 2000-run Monte Carlo simulation

Figure 24a presents the position constraint for a simple pendulummodeled with the stochastic model with polynomial chaos of order 3, and Fig.24b for polynomial chaos of order 7. The order 3 approximation shows anincrease in variance after about 10 seconds, which confirms that theconvergence of the polynomial chaos approach for DAE problems is slow.

0 3 6 9 12 150

0.5

1

1.5

Time [ s ]

PositionConstraint (X2+Y

2)

0 3 6 9 12 150

0.2

0.4

0.6

0.8

1

1.2

1.4

Time [ s ]

PositionConstraint (X2+Y

2)

(a) Polynomial chaos of order 3 (b) Polynomial chaos of order 7


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Figure 24. The position constraint for the pendulum modeled with DAE

Conclusions

Many real life mechanical systems operate under uncertainty. Uncertaintiesresult from poorly known or variable parameters (e.g., variation in suspension

stiffness and damping characteristics), from uncertain inputs (e.g., soil propertiesin vehicle-terrain interaction), or from rapidly changing forcings that can be bestdescribed in a stochastic framework (e.g., rough terrain profile).

This study investigates the use of generalized polynomial chaos theory formodeling complex nonlinear multibody dynamic systems with uncertainty. Thetheoretical and computational aspects of the polynomial chaos approach arediscussed in the companion paper Modeling Multibody Dynamic Systems withUncertainties. Part I: Theoretical and Computational Aspects [1].

This paper continues the study with numerical investigations of a selectedset of test problems. A quarter-car model is subject to both parametric andexternal forcing uncertainties. Polynomial chaos allows the calculation of mean

and variance of the solution, and of its probability density distribution in both timeand frequency domains. For nonlinear systems of interest the deterministicresults with the most likely values of the parameters are different than thestochastic mean. The modeling of uncertain terrain profiles is illustrated using aperiodic, bilateral, autoregressive stochastic process. This stochastic process isdiscretized using a Karhunen-Loeve expansion, and the system response isobtained using polynomial chaos.

Numerical experiments performed with the simple pendulum illustrate themodeling of uncertainties in DAE systems. The statistics convergence is slow,and higher order polynomial chaos expansions are employed for the DAEformulation than for the ODE formulation. The reasons for this slow convergence

in the DAE formulation require further investigations.Limitations of the polynomial chaos approach are studied on two test

problems with special dynamical behavior. While the method correctlyapproximates multiple attractor points, its accuracy decreases after a short timeinterval when it follows a chaotic evolution.

The main advantages of the polynomial chaos approach to simulatemultibody dynamic systems with uncertainty are:

Allows an accurate treatment of nonlinear dynamics and of non-Gaussian probability densities;

Does not rely on the assumption that uncertainties are small; Is well suited for multibody dynamic systems of interest, which have a

small number of uncertain parameters with large levels of uncertainty; Is more efficient than Monte Carlo; Is more accurate than statistical linearization; Allows the quantification of uncertainty distribution in both time and

frequency domains;

Allows a consistent discretization of external stochastic forcingsthrough truncated Karhunen-Loeve expansions;

Can qualitatively capture the dynamics for multiple attractor states;


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Allows simulations for uncertain differential-algebraic equations.The main weaknesses of the polynomial chaos approach are:

The dimension of the stochastic space increases exponentially with thenumber of independent random variables. Consequently, the approachis efficient only for a relatively small number of sources of uncertainty;

The discretization of external stochastic forcings by truncatedKarhunen-Loeve expansions require many terms for processes withshort correlation lengths/times;

Treatment of non-polynomial nonlinearities in the Galerkin frameworkrequires the use of multidimensional quadrature rules. On the otherhand, a complete convergence theory for the less expensivecollocation approach is not available at this time;

For chaotic dynamics the approximation of the PDF is accurate only forshort simulation times;

The convergence of the method for the DAE (pendulum) example isnotably slower than for the ODE formulation of the same system.

The overall conclusion is that despite its limitations, polynomial chaos is apowerful and potentially very useful, approach for the simulation of multibodydynamic systems with uncertainty.

Acknowledgements

The work of A. Sandu was supported in part by NSF through the awardsCAREER ACI-0093139 and ITR AP&IM 0205198. The work of C. Sandu wassupported in part by the AdvanceVT faculty development grant 477201 fromVirginia Techs NSF ADVANCE Award 0244916.

References

[1] Sandu, A., Sandu, C., and Ahmadian, M., Modeling Multibody DynamicSystems with Uncertainties. Part I: Theoretical Development, submitted Sept.2004.

[2] Ahmadian, M., Appleton, R., and Norris, J. A., Designing Magneto-Rheological Recoil Dampers in a Fire-out-of Battery Recoil System, IEEETransactions on Magnetics, Vol. 37, No. 1, Jan. 2003, pp. 430 435.

[3] Ahmadian, M. and Poynor, J.C., Effective Test Procedure for EvaluatingForce Characteristics of Magneto-rheological Dampers, Proc. of ASME IMECE2003, IMECE2003-38153, Nov. 15 21,2003, Washington, D.C.

[4] Bekker, M.G., Theory of Land Locomotion, The U. of Michigan Press, AnnArbor, Michigan, 1956.

[5] Bekker, M.G., Off-the-Road Locomotion, The University of Michigan Press,Ann Arbor, 1960.

[6] Bekker, M.G., Introduction to Terrain-Vehicle Systems, The University ofMichigan Press, Ann Arbor, Michigan, 1969.


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[7] Harnisch, C., and Jakobs, R., Impact of Scattering Soil Strength andRoughness on Terrain Vehicle Dynamics, Proc. of the 13thInt. Conf. of ISTVS,Munich, Germany, 1999.

[8] Lucor, D., Su, C.-H., and Karniadakis, G.E., Karhunen-Loeve Representationof Periodic Second Order Autoregressive Processes, M. Bubak editor, ICCS

2004, LNCS 3038, pp. 827-834, Springer-Verlag, 2004.

[9] Crandall, S.H., Is StochasticLinearization a fundamentally flawedprocedure?,Probabilistic Engineering Mechanics, Vol. 16, pp. 169-176, 2001.

[10] Lorenz, E.M., Deterministic Nonperiodic Flow, J. Atmos. Sci., 20, 448-464,1963.

modeling multibody dynamic systems with uncertainties

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