Accurate Models for Bushings and Dampers using the Empirical Dynamics Methodweb.mscsoftware.com/support/library/conf/adams/euro/… · · 2002-12-091 Accurate Models for Bushings

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Accurate Models for Bushings and Dampers using the Empirical DynamicsMethod

Andrew J. Barber

Modeling Integration Products and ServicesMTS Systems Corporation

Abstract

Conventional models for complex nonlinear components (like bushings and dampers) are ofteninadequate to represent behavior over wide frequency ranges, and at large amplitudes. EmpiricalDynamics models provide a solution to this problem, via a nonlinear, dynamic, blackbox approach.These models accurately replicate both frequency and amplitude dependence effects, without thelimitations of conventional schemes. Examples of Empirical Dynamics models will be presented forautomotive shock absorbers and a biaxial rubber bushing. Benefits and limitations of the EmpiricalDynamics approach are discussed along with requirements for interfacing these models to the ADAMSvirtual prototyping environment.

Accurate Models of Bushings and Shock Absorbers using the Empirical Dynamics Method A.J.Barber/ MTS Systems

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1 Introduction

ADAMS virtual prototype models have shownimpressive results in the ability to simulatebehavior of complex engineering systems.They make it possible to study the effects of anytype of dynamic input, for a wide range ofsystems. They also permit manipulation ofsystem variables in a controlled manner, asrequired for optimizing system design. As aresult, these techniques are rapidlyrevolutionizing the way engineering systems aredesigned.

Despite the successes of the ADAMS approach,there are nevertheless limitations. Systems thatinclude friction, rubber or other elastomericmaterials, complex fluid dynamic behavior,biological systems, and others, can be difficultto model accurately in reasonable time. Thedifficulty arises for some systems because theyrequire a very large number of state variablesfor an adequate representation. For example,fluid dynamic systems may require simulation oflarge, complex flow fields, just to assess forcevs. velocity relations. For other systems,difficulties arise because important physicalproperties cannot be easily measured orunderstood. Another source of problems occursfor systems whose properties are notrepeatable, or whose properties changesignificantly with temperature.

Often, these complexities are handled by usingblackbox models. Here, experimental data isobtained for inputs and outputs of a complexsystem or component, and a relation betweenthe inputs and outputs is formed using amathematical curve fitting technique. Examplesinclude polynomial or spline curve fits fordamper force vs. velocity, and frequencydependent bushing models, which providestiffness coefficients as functions of frequency.Blackbox methods are effective because theylargely ignore internal operation of the physicalsystem. This is an obvious advantage, when thephysical principles are not well understood, orwhen the system model requires an enormousnumber of states.

The blackbox approach contrasts sharply withanalytical modeling, or 'first principles' modeling,where geometric parameters, mechanical

properties and physical principles are used togenerate differential equations, and these aresolved numerically. Examples of these includethe finite element method, computational fluiddynamics, and multi-body dynamics (in a limitedsense). Not surprisingly, these latter methodshave been called 'whitebox' methods. Ofcourse, the whitebox and blackbox methods arenot mutually exclusive. Curve fits representingcomplex component behaviors are easilyintegrated into the ADAMS 'whitebox'environment.

One important difference between blackbox andwhitebox models is their adjustability. A majordisadvantage for blackbox models is their lackof adjustability. Many of the important physicalor design parameters are hidden in theblackbox, and cannot be changed. Alternatively,a blackbox model has parameters (coefficientsfor a curve fit), but these often do not havemeaningful physical counterparts. This makesblackbox models less than desirable for designoptimization. In contrast, the parameters of awhitebox model often have immediate and clearinterpretations, and are very useful for designstudies.

Another important difference between blackboxand whitebox models is their computationalefficiency. Blackbox models are superior in thisregard. They are typically represented bystraightforward algebraic relations, which canbe solved by direct (non-iterative) calculation.Whitebox models, by contrast, require astepwise solution of complex differential andalgebraic equations (DAE's), often withconsiderable iteration required.

Clearly, there is a direct tradeoff betweenadjustability and efficiency. The choice ofblackbox vs. whitebox models must be basedon a clear understanding of the modelingobjective. For example, a vehicle model, built tostudy ride and handling characteristics, requiresa tire model. A whitebox tire model might beconstructed using finite elements, with nonlinearstiffness and damping coefficients for therubber. A blackbox model of the same tire couldbe constructed by fitting curves to experimentaldata (for example, lateral force vs. slip angle).


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The choice of blackbox vs. whitebox in this casemay depend on the focus of the model. If it is toimprove the tire design, then clearly thewhitebox model is best. If it is to improve asuspension design, then the blackbox modelmay be sufficient. As a general philosophy, itmakes sense to use blackbox models wheneverpossible, i.e., for elements that are not the focusof the design optimization.

Of course, the accuracy of the two methodsmust also be considered. Historically, accuracyhas been one of the most serious challenges forboth blackbox & whitebox modeling. Whiteboxmethods may be inadequate for certain types ofsystems, as mentioned above. Blackboxmethods have had shortcomings for systemsexhibiting characteristics such as:

• amplitude dependence (or nonlinearity)• frequency dependence (or memory)• arbitrary (or random) input• multiple inputs and outputs

Blackbox modeling is relatively straightforward,if the system has only amplitude dependence oronly frequency dependence. When these arepresent simultaneously, however, the necessarymathematical sophistication increasesdramatically. Limitations in the type of system,type of input, or number of inputs, then become

necessary to keep the blackbox methodsmanageable.

In recent years, blackbox techniques have beendeveloped which can handle joint amplitude andfrequency dependence, with very fewrestrictions on the class of system or type ofinputs. This paper will discuss one suchtechnique called Empirical Dynamics Modeling(ED modeling or EDM).

The presentation begins with some background,covering definitions and explaining whyblackbox modeling becomes difficult whenamplitude and frequency dependence arepresent simultaneously. The next sectionintroduces Empirical Dynamics models, with abrief overview of their principles of operation. Aseries of case studies follow, with examples ofEmpirical Dynamics models for complex vehiclesubsystems and components such as shockabsorbers and rubber bushings. In the nextsection, some general characteristics ofEmpirical Dynamics models are presented,including benefits, limitations, and methods tocircumvent those limitations. Finally, themethods to interface ED models to the ADAMSenvironment will be discussed


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2 Background____________________________________________________________________________________

This section provides a brief overview ofconventional blackbox modeling. It discussessystems having only amplitude dependence oronly frequency dependence, and how these aremodeled using blackbox methods. It thendescribes the complications involved when asystem has both amplitude and frequencydependence, especially how the conventionalmodels fail to provide an adequaterepresentation. Finally, a brief discussion ofsome alternative methods and workarounds isprovided.

Amplitude dependence is synonymous withnonlinearity, which refers to the situation wheresuperposition fails to hold. That is, if systeminput amplitude is scaled by some factor, theoutput amplitude doesn’t scale by the samefactor. Most physical systems display thisproperty to some extent. A simple example of anonlinear system is an automobile shockabsorber, whose force vs. velocity relation ismodeled by a simple curve or a two-part linearrelation (Figure 2.1). Note that linearity shouldnot be confused with proportionality.Proportional systems (i.e., output is a scalarmultiple of input) are linear, but the converseneed not be true. The correct way to understandlinearity is by the effect of scaling: if the input ismultiplied by some scale factor, the output willscale by the same factor, but the input andoutput may look completely different.

Figure 2.1 Simple Nonlinear Damper Models

Amplitude dependent behavior is often modeledusing simple input-output curves or algebraicfunctions, as shown for the shock absorberabove. This method applies when the systemhas a single input and single output (SISO).When such a system has multiple inputs, thesimple curve characterization must be replacedwith a higher dimensional equivalent. For twoinputs, the relation must be represented by afamily of curves in two dimensions, or by asurface in three dimensions (3D). For threeinputs, it becomes more complex: now a familyof surfaces in 3D is needed. With more inputs,more dimensions are needed, and the ability tovisualize the relationship is lost. Nevertheless,equations may be fitted to define a usefulmodel. This approach of using simple curves orsurfaces, or equivalent equations, is applicablewhen there are no frequency dependent effects.Under these conditions, the model can be usedfor any type of input signal: sinusoidal, step,sweep, random, or arbitrary.

Frequency dependence refers to a systemwhose behavior depends on the rate or timescale of input. For instance, a frequencydependent system subjected to a 1 Hzsinusoidal input may give a 1 Hz sine output,but with 30 degrees phase lag. When 10 Hz isapplied, the phase lag may be 45 degrees(Figure 2.2). The ratio of output/ inputamplitudes may differ as well, at the twofrequencies. Resonance is an example of sucha frequency dependent effect.

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Figure 2.2 Frequency Dependent Behavior

Frequency dependent systems are oftenmodeled via frequency response functions(FRFs), where the relation between input andoutput signals is characterized in terms ofmagnitude and phase vs. frequency. FRFs aretypically explained for the case of sinusoidalinput and output: The FRF magnitude gives theratio of sinusoidal output amplitude oversinusoidal input amplitude, and the FRF phaseis the difference between output phase andinput phase (Figure 2.3). Transfer functions,based on Laplace transforms, offer a similarcapability. Frequency dependent systems canalternatively be modeled in the time domain,using ARMA type models, adaptive digitalfilters, and others. The time domain model ischaracterized using the impulse response of thesystem, and the output at any time is evaluatedby convolution of the signal with the impulseresponse.

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Figure 2.3 Frequency Response Function (FRF)

Note that these techniques for modelingfrequency dependent systems are applicable, ina strict sense, only for linear systems (i.e., noamplitude dependence). Linear frequencydependent systems can often be identified bythe fact that a sinusoidal input gives asinusoidal output. If the output is not sinusoidal,this is a sure sign of nonlinearity. It’s importantto keep in mind, however, that linearity ornonlinearity is a property of the system,independent of input signal type. Note alsothat linearity does not mean the FRF magnitudeis flat. Resonant peaks and rolloff at high or lowfrequencies may (and typically do) occur forlinear systems. A flat FRF magnitude doeshave a special interpretation, however: it meansthe system has no memory. This property willbe explained shortly.

The methods for modeling frequency dependentsystems can also deal with arbitrary inputwaveforms. For example, FRFs can be used ina straightforward manner to model systems withrandom inputs. In this case, the random signalsare taken to be composed of a large number ofsinusoidal components. The amplitudes ofthese components in the signal arecharacterized in terms of power spectral density(PSD, or power spectrum, Figure 2.4). Whenrandom inputs are applied to a dynamic system,a random output is produced. The effect of thesystem is to modify the shape of the powerspectrum between input and output. The outputpower spectrum is just the input spectrum bymultiplied by the FRF magnitude squared(Figure 2.5).


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Figure 2.5 Obtaining Output PSD from Input PSD andFRF

The dynamic system essentially functions as afilter, to amplify sinusoidal components at somefrequencies, and to attenuate them at otherfrequencies. For example, a building structuresubjected to wind loading vibratespredominantly at the building's naturalfrequency. This is not because the windnecessarily flows at that frequency, but ratherthat the random wind may be considered tocontain a wide band of frequencies, and thebuilding is a filter that acts to focus the motion atone frequency.

When FRFs are used with power spectraldensity descriptions of random input, animportant limitation is that the input must beGaussian, i.e., its probability density function(pdf) must follow the normal (bell shaped)distribution. Statisticians often argue that manyrandom signals have such a pdf, owing to thecentral limit theorem. However, engineersresponsible for designing systems subject torandom loading (especially road vehicles)understand that non-Gaussian random input istypical (but not ‘normal’). Luckily, FRFs may beused in the case of non-Gaussian inputs aswell. The catch is that the input power spectrumdescription must then be replaced by an explicitdescription of input waveform, i.e., a timehistory. Moreover, the output can no longer becalculated by simple frequency domainmultiplication; instead, convolution operationsmust be performed. These complications arenot serious, since analytical tools for performingsuch operations are well established. Likewise,the many of the concepts for random inputsapply: the system can still be considered as afilter, the input may be understood as a sum ofsinusoidal components, etc. In the remainder ofthis discussion, the terms ‘random’ and‘arbitrary’ will be used interchangeably, althoughtheir strict definitions are quite different.

When the system has multiple inputs andoutputs (MIMO), there arises the possibility ofcross-coupling: there may be a differentfunctional relation for each input-output signalpair. For FRF models, these are often shownas a matrix (Figure 2.6). Each element of thematrix is a frequency response function, and therows and columns of the matrix correspond todifferent inputs and outputs, respectively. Anoutput signal is calculated using a combinationof matrix multiplication and convolution.


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Frequency dependence is synonymous withconcept of system memory [10]. A system hasmemory when its output at any time depends onthe input at that same time, but also on pastinputs and outputs. Equivalently, a system hasmemory if there is any phase lag between inputand output. In physical systems, memory maybe associated with certain types of energystorage elements. In whitebox modeling,memory is associated with integrators and statevariables. Classic linear state spaceformulations using A, B, C, D matrices fall intothis category. The concept of memory is usefulbecause it ties together the integrator, state,and energy storage perspectives. However, itmay be considered more broadly, to includesystems that don’t fit into these categories.Furthermore, memory focuses on therelationship between inputs and outputs, ratherthan on the internal operation of the system.This is particularly useful for blackbox modeling,where the concepts of state variables andenergy storage aren’t necessarily relevant.

When there are no memory effects, systemoutput at any time depends only on the input atthat same time. Such systems are called'memoryless', 'instantaneous', or 'static'. Figures2.7 & 2.8 demonstrate the relations betweeninput and output, for memoryless (static) andmemory (dynamic) systems. In this case, staticdoes not mean motionless, or ‘at equilibrium’; itmeans that the model equations for the staticsystem can be used, even when the input istime varying. In whitebox terms, the systemdynamics can be described without need forstate variables. Likewise, the term 'dynamic'may used to describe frequency dependentsystems. In this case, dynamic doesn’t onlymean 'varying in time’; it means ‘having

memory’, or, ‘requiring equations that use statevariables’.

Figures 2.7 & 2.8 Static and Dynamic Relationsbetween Input and Output Signals

Although these interpretations are somewhatesoteric, they are useful to describe systemsand models more concisely. Specifically, thefollowing substitutions can be used:

amplitude dependent,but notfrequency dependent

static nonlinear

frequency dependent,but notamplitude dependent

linear dynamic(or just linear)

amplitude dependentandfrequency dependent

nonlinear dynamic

Note that these terms may apply to bothsystems and blackbox models. For example, anonlinear dynamic system may be modeled


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using static nonlinear equations. Such a modelwould ignore the memory effects.

The following paragraphs describe some of thelimitations of static nonlinear and linear dynamicmodeling methods when applied to nonlineardynamic systems. This is generally a difficulttask, because nonlinear dynamic behavior isdifficult to describe. However, much can belearned by simple cross-application of thetechniques, i.e.,• static nonlinear models don’t work well for

linear dynamic systems• linear dynamic models don’t work well for

static nonlinear systems

Consider first using a static nonlinear model fora linear dynamic system. The static nonlinearmodel is just a simple input-output curve (for asingle input single output (SISO) system).However, when memory is present, the input-output curves may exhibit hysteresis (multiplevalues, or loops). For example, the input-outputcurves for sinusoidal excitation may yield anelliptical (Lissajous) pattern (Figure 2.9).

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Figures 2.9 & 2.10 Lissajous Patterns

According to this representation, there are twooutput values for each input, and the correctoutput value at any time depends on the path orhistory of input. Another problem is that thesize and shape of the elliptical plot may changewith frequency. Thus, to model a frequencydependent system, it would be necessary todefine a family of such curves. To make mattersworse, the elliptical shape occurs only forsinusoidal inputs. Complex input waveforms,composed of sums of sinusoids, can yieldextremely complex hysteresis patterns. Furthercomplication arises when multiple inputs areinvolved. Try to imagine a family of surfaces inthree dimensions, whose shape changes withfrequency. Clearly, the static nonlinear modelingapproach rapidly becomes impractical for lineardynamic systems.

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Figure 2.10 Family of FRFS (3 amplitudes)

Similarly, linear dynamic models are ofteninadequate for characterizing static nonlinearsystems. Consider the simple case of a SISOsystem, modeled using an FRF. The amplitudedependence of the system means that theremay be a different FRF for each input amplitude(Figure 2.10). Occasionally, one hears of

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Figure 2.11 Generation of Harmonics by Smooth and Abrupt Nonlinearities

proposals to build a family of FRFs, eachassociated with a different amplitude, and tointerpolate between them. Problems with thisapproach can be understood from theperspective of harmonic distortion. Whenever asinusoidal input is applied to a nonlinearsystem, the output waveform is periodic, but notsinusoidal. Such a waveform may beconsidered to consist of one sinusoidalcomponent having the same frequency as theinput, plus additional harmonic components(i.e., at multiples of that frequency). A systemwith a ‘smooth’ nonlinearity may generate onlyfew weak harmonic components in the output,whereas one with an abrupt nonlinearitygenerates strong harmonics over a widefrequency range (Figure 2.11).

Further problems arise when the input signal iscomposed of a few sinusoids at differentfrequencies. In this case, one effect of thenonlinearity is to generate frequencycomponents at the sum and differencefrequencies. So, if a sine-on-sine input having a5 Hz component and a 1 Hz component isapplied to an amplitude dependent system, theoutput waveform would have components at 1and 5 Hz, but also at 4 and 6 Hz, the sum anddifference frequencies. Harmonics and sum &difference frequencies (“cross-frequency”effects) are the reason FRFs are not viable fornonlinear systems: the output at 6 Hz dependsnot only on the input at 6 Hz, but also on theinput at 1 Hz and 5 Hz. Or, the output at 6 Hzmay be a harmonic for input at 1 Hz, 2 Hz, or 3

Hz. Even worse, the output component at 6 Hzcould be the result of 4+ 2 Hz, or 8 -2 Hz, 10-4,etc., or all of these together. These effectscannot be quantified with a family of FRFs,since the FRF assumes that output at a givenfrequency is caused only by input at thatfrequency.

Clearly, the situation is even more complex forrandom input signals, since they consist of alarge number of sinusoidal components. In thiscase, a family of FRFs (each membercorresponding to a different amplitude) cannotbe defined, because there is no singleamplitude that characterizes the random input.Instead, the family would have to include adifferent FRF for each input power spectrum,which is completely impractical.

The preceding paragraphs showed how cross-application of the models fails for simple SISOsystems. New problems arise when there aremultiple inputs. In particular, there may be“nonlinear cross-coupling” effects. For example,an output resulting from 2 inputs may depend insome way on the product of those inputs.There is no provision for such terms in the FRFmatrix framework.

Despite all these problems, many nonlineardynamic systems can be effectively modeled asblackboxes. Two basic ways to do this are:• Restrict the types of inputs that can be used• Restrict the types of systems that can be

modeled.


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Examples of restricted inputs include amplitudelimits and frequency limits. For example, lineardynamic methods are sometimes used fornonlinear dynamic systems, when amplitudesare small, say when determining vehicle NVHproperties. Similarly, static nonlinear modelsmay be used for nonlinear dynamic systemswhen large amplitudes are incurred, but theinput are limited to a narrow frequency band,where the FRF doesn’t vary much. Such is thecase for events like vehicle maneuvering andhandling, where frequencies are typically low.

Nonlinear dynamic systems can also bemodeled if the system is of a limited class. Forexample, if the amount of nonlinearity is small,an FRF may be used as a least squaresapproximation. Alternatively, some systemscan be characterized as a series combination oflinear dynamic and static nonlinear parts.Models for these are known as Wiener orHammerstein models [5], depending on how thetwo parts are connected (Figure 2.12). Volterraseries models offer another way to handlenonlinear dynamics by restricting the class ofsystems [10]. They use higher order frequencyresponse functions, where magnitude andphase are functions not just of frequency, but ofseveral frequencies. For instance, consider adynamic system with ‘quadratic’ nonlinearbehavior: a whitebox representation for thissystem would include squares and product oftwo terms. The Volterra series model for thissystem gives sinusoidal output amplitude as afunction of two frequencies (f1, f2). That is, thesystem generates sums and differences of twofrequency components at a time, so there mustbe two independent variables for frequency.The problem with such an approach is that fewnonlinear systems fit into the quadraticnonlinear category. A cubic nonlinearity may

be more useful but this requires three frequencyaxes, which doesn’t work well in practice.

This brief background has outlined some of thechallenges and limitations of blackbox systemmodeling. It’s important to remember that,despite these limitations, the linear dynamic andstatic nonlinear approaches cover a widespectrum of applications. They deserveappropriate recognition, and should be includedas part of any modelers toolkit. Variants ofthese conventional methods, to handlenonlinear dynamic systems, should be usedcautiously: they may severely compromise theaccuracy of the model, or they may beappropriate for only limited classes of systemsor inputs. Of course, whitebox methods mayprovide a workaround in these situations.Alternatively, the Empirical Dynamics modelingmethod may be useful, as it enables accurateblackbox modeling for a very wide class ofnonlinear dynamic systems.

Figure 2.12 Wiener and Hammerstein NonlinearSystems


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3 Empirical Dynamics Models____________________________________________________________________________________

Recent developments in dynamic systemmodeling methods now enable generation ofaccurate models from experimental data, withvery few restrictions on the type of system ortype of input [5,6,8]. These models canproperly replicate amplitude dependence andfrequency dependence together. Thecomplications of multiple inputs and outputs,and cross-coupling, are easily handled.Moreover, the advances in computationalhorsepower allow such models to be built inreasonable time.

This convergence of modeling capabilities maybe considered an important breakthrough.However, the nature of this breakthrough issubtle. It is the combination of multiple factors:• blackbox• amplitude dependence• frequency dependence• multiple input and output• arbitrary input waveform• applicability to a wide class of systemsthat define the technique. As described in theBackground section, conventional methodshave been able to encompass many or most ofthese characteristics, but not all of them. Adescriptive title for such a method shouldinclude all of these important aspects.However, even the most concise description,‘nonlinear multi-input dynamic blackbox forarbitrary input and system types’, is a mouthful.The name Empirical Dynamics Modeling (EDM)has been coined as a convenient substitute.

ED models are based on neural networktechnology. A neural network is acomputational paradigm (i.e., data structure andalgorithms) originally developed to model thephysiological function of brains [1,4]. Neuraltechniques have been primarily aimed atproblems in pattern recognition, but they havealso enabled improved solutions for nonlinearregression (curve fitting) problems, whichincludes nonlinear dynamic modeling. Followingis a brief introduction describing how neuralnetworks are used for this objective.

A neural network is constructed fromfundamental units called neurons. A neurontakes a series of varying inputs uk and multiplies

them by constant weights wk, sums these alongwith a constant bias term, and then applies theresult to a nonlinear 'activation' function, to givean output value y. This process is shown inFigures 3.1 and 3.2.. The first figure presentsthe process in a conventional block diagramform, the second shows the same process ascommonly depicted in the neural networkliterature. The activation function is usually asigmoid (S shaped) function; for example, ahyperbolic tangent (Figure 3.3) . A neuralnetwork is constructed by connecting multipleneurons to the same inputs to make a 'layer',and by using the outputs of one layer as inputsto another layer. This structure is known as amultilayer perceptron (MLP) (Figures 3.4-3.5).

Figure 3.1 Neuron, Block Diagram

Figure 3.2 Neuron, Simplified Representation

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Figure 3.3 Hyperbolic Tangent Sigmoid Function


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Figure 3.4 Neurons formed into a Layer

Figure 3.5 Multi-Layer Perceptron (MLP)

To model a dynamic system, inputs to theneural network are obtained from discrete timesignals via tapped delay lines (Figure 3.6) [6].The number of taps (i.e., number of current andpast inputs and outputs) is chosen based onsample rate and on the characteristics of thedynamic system being modeled. Within theneural network, the number of layers andneurons in each layer are chosen similarly.

Figure 3.6 Tapped Delay Structure for Neural NetInput

Following this 'design' phase, training canbegin. Training (or supervised learning) isperformed by submitting a set of input data

(signal samples) to the tapped delay inputs, andadjusting the neural network weights for eachlayer, until the neural net output approximatesthe corresponding output signals from thedynamic system, with minimum mean squareerror [4]. This adjustment process is calledbackpropagation, because the calculations areperformed beginning with the neural net outputand proceeding backward through the network.

The process of tapping the inputs and outputs,in conjunction with adjustment of neural networkweights, can be understood as a nonlinearregression (curve fitting) problem [1]. Since theoutput for a dynamic system at any timedepends on values of present and past inputsplus past outputs, a functional relationship ofthe following form is used [5,6,8]:

yk = f ( uk,u k-1,…, uk-M , yk-1,…, y k-N )

where k is a time index;u’s are inputsy’s are outputsf is a nonlinear function, to be

determined (the primary goal of theregression problem).

M is a parameter, = number of pastinputs to include in the model.Ideally, this is an infinite number, torepresent the fact that all pastinputs affect the current output;practically, this is truncated at somepoint where the effect of past inputsbecomes insignificant.

N is a parameter denoting the numberof past outputs to use, the result ofa truncation process similar to thatfor M.

The function f maps a space of M+Ndimensions to a space of one dimension. TheM+N dimensional space is populated by samplepoints, where one sample point represents acombination of M inputs and N past outputs,and has an associated function value equal toone value of system output. The goal ofregression is to find one function f(.) whichapproximately fits the sample values at thesepoints.

An important facet of this process is theseparation of the input and output signals intomultiple sets, for training and prediction (orevaluation) [1]. A training set is a subset of thesignal that is used for creating the model, as


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defined above. The prediction set is used todetermine how well the model predicts outputswhen supplied with 'new' inputs (i.e., not usedfor training). The use of these two sets ismotivated by the following reasoning: Thetraining process aims to fit a function to pointsfrom the training set, with minimum meansquare error. It's possible with sufficientnumber of neurons to fit the training dataperfectly (this property is called the universalapproximation property). A perfect fit is notdesired, however, for several reasons. One isthat more computation time is incurred whenmore neurons are used. More importantly, aperfect fit will accommodate any and allmeasurement noise.

Figure 3.7 demonstrates how a simple but noisydata set may be fit with various curves(models). The left panel shows a linear fit, theright panel shows a perfect fit, and the middlepanel shows an intermediate fit. Assuming thefit in the center panel is somehow optimal, theleft and right panels are described as ‘underfit’and ‘overfit’, respectively.

Figure 3.7 Multiple Fits to Noisy Data

Overfitting may lead to large errors when theneural net model is applied to a new data set(which is ultimately the goal of the modelingprocess). In the neural network jargon, thissituation is called 'memorization'. The preferredbehavior of the neural model is then describedas 'generalization', where the model predictsoutputs with low error when new inputs aresupplied. Generalization may be facilitated bylimiting the number of neurons to a smallnumber (e.g., less than 50). This causes theregression function f to track the general trendof the input points while averaging out thenoise. The ability of the neural network togeneralize can only be assessed by testing it ondata that was not used for model generation;this data is the prediction set.

Figure 3.8 Training vs. Prediction Error

Errors for the training set and prediction setboth change with the number of neurons, but indifferent ways (Figure 3.8). Training errorgenerally decreases monotonically with moreneurons, asymptotically approaching zero asthe number of neurons becomes large.Prediction set error normally decreases for lownumbers of neurons, then increases, so thatthere is an optimal number of neurons thatgives the lowest prediction error. The process offinding this minimum is known as regularization,model selection, or complexity control [1].Ideally, complexity control should be performedwhenever a model is built. Moreover, a thirddata set (the validation set) should be used toperform this process. Practically, however, thisoptimization process may be extremely timeconsuming, as ever larger models take longer togenerate, with diminishing returns in predictionerror. It is often the case that excellent modelsmay be obtained with a small but arbitrarynumber of neurons, and that optimal modelselection is then not necessary.

These ideas about training and prediction errorsand generalization are relevant for all blackboxmodeling methods, not just for neural networks.They apply whenever the model order can beadjusted. For example, when polynomials areused for modeling, the model order is thedegree of the polynomial. This can be adjustedto give optimal prediction error, just as thenumber of neurons can be chosen for a neuralnetwork.

Perhaps the most important lesson from thisdiscussion is the idea that model performanceshould always be evaluated using a predictiondata set; it should never be evaluated based ontraining data alone. All of the plots and otherresults presented in this report are based on aprediction data set.


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Its important to understand that EmpiricalDynamics method is more than just a neuralnetwork dynamic modeling (systemidentification) method. Other aspects ofmodeling may be considered integral to thetechnique, including test excitation, parameteradjustment, and interfacing to ADAMS, which

are described in later sections. In addition,special proprietary enhancements have beenemployed to give the results shown in thefollowing sections. The nature of theseenhancements may be disclosed pendingpatentization.

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4 Case Studies___________________________________________________________________________________

This section will present case studies involvingEmpirical Dynamics models. These includemodels for 2 different shock absorbers and for abiaxially loaded rubber bushing, from a latemodel passenger car. Neural networks havebeen used for modeling shock absorbers sincethe early 1990’s [2,3]. The results of the currentcase study confirm and extend these results.It’s not apparent whether neural approaches formodeling rubber bushings have beeninvestigated.

Case Study 1: Shock Absorber #1

Automobile shock absorbers (or dampers) areoften modeled by fitting a curve to a plot of forcevs. velocity. These plots are generated byapplying a sine sweep displacement as input,and measuring the resultant force as output.Velocity is calculated from the knowndisplacement history. The changing frequencyof the sweep gives rise to varying velocities.

The current case study takes a slightly differentapproach, by using a random displacementsignal as input. Of course, this is more realisticthan a sine sweep, as the displacement profilefor most roads is random, not sinusoidal.Another reason to use a random input is that itprovides a richer content than a sine sweep,i.e., a wider variety of events are present in thesignal. Training of the neural network model isbest performed under these conditions.

The random signal for shock absorber excitationwas generated as a discrete Gaussian timehistory, with power spectral density havingmagnitude proportional to 1/frequency2. Theexponent 2 gives a random signal with most ofits energy at low frequencies, similar to thepower spectrum of real road input. Moreover,the exponent 2 gives a flat velocity power

spectrum, which is arbitrary but a reasonablechoice for a velocity-based characterization.

The shock absorbers were tested in an MTSservohydraulic test rig. Use of hydraulic powerwith servo-control of displacement allowsaccurate application of the displacement profilesto the specimen, as the ‘independent variable’.Force was measured as the ‘dependentvariable’, using a strain gage load cell. Despitetheir accuracy, all servocontrollers have someresponse limitations. The actual displacement ofthe specimen may differ from the commandedvalue, in the form of small discrepancies inamplitude and phase at higher frequencies.These effects were circumvented in thisinvestigation, by using the actual shockabsorber displacement, measured with anLVDT, for subsequent model generation.Essentially, this procedure allows the dynamicsof the test rig to be excluded from those of theshock absorber.

Velocity was calculated from the displacementsignal, using a high order digital differentiator[7]. Such a differentiator reduces the highfrequency phase errors of conventional first orsecond finite difference differentiators.

The force-velocity plot resulting from therandom input shows a complex pattern (Figure4.1). The pattern has two importantcharacteristics: a curved ‘backbone’, and somehysteresis (width or thickness) around thebackbone. These two parts coincide roughlywith the presence of nonlinearity and memory,respectively.


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Figure 4.1 Shock Absorber #1 Force-Velocity,Original Lab Data

It is instructive to demonstrate how wellconventional blackbox models work to replicatethis pattern. For this purpose, a static nonlinearmodel was generated using a least squarespolynomial curve fit, and a linear dynamic modelwas generated as an FRF. Each of thesemodels was then used with velocity as input topredict the force. Results are plotted in Figures4.2-4.3.

Figure 4.2 Shock Absorber #1 Force-Velocity,Polynomial Model

The polynomial model yields a plot whichfaithfully replicates the ‘backbone’, but fails tocapture any of the original hysteresis. Thiscompromise is expected for a static nonlinearcharacterization.

The FRF model gives a force-velocity plot thatreplicates the hysteresis to some extent, butfails to capture the curved backbone nature ofthe original data. This is the expected result fora linear dynamic model.

Figure 4.3 Shock Absorber #1 Force-Velocity, FRFModel

The Empirical Dynamics model for this shockabsorber gives a force-velocity plot very close tothe original lab data (Figure 4.4). It replicatesboth the backbone (nonlinear part) and thehysteresis (dynamic or memory part) of theoriginal data.

Figure 4.4 Shock Absorber #1 Force-Velocity, EDModel

These plots give a quick overview of thecapabilities of the Empirical Dynamics method.Further detail of these capabilities is provided inthe next section.

Case Study 2: Shock Absorber #2

For this specimen, the laboratory test wasessentially the same as described above.However, in this case, displacement was usedrather than velocity, as the independent variablefor the model. This approach may seemunusual, since curves of force vs. displacementfor a shock absorber are usually large loops,


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with no apparent ‘curve fit’ (Figure 4.5). In otherwords, there is no way that a static nonlinearcurve (polynomial) could be used to model thisrelationship. The large hysteresis loops suggestthat an FRF may work; however, thenonlinearity of this specimen isn’t apparent fromthis plot, so blind trial and error is the only wayto determine if an FRF will work.

Figure 4.5 Shock Absorber #2, Force-Displacement,Original Lab Data

Use of displacement as input rather thanvelocity is motivated by a couple of ideas:• A system with displacement as input and

force as output is equivalent to systemcomposed of a differentiator, followed by asystem with velocity input and force output.A differentiator is just a linear dynamicsystem, so it doesn’t add any appreciablechallenge (i.e., nonlinearity) to the problem.If the differentiator is included in theblackbox, the model generation processshould be able to account for it.

• By relying on the model generation processto handle the differentiation, problems thattypically arise from numerical differentiationmay be avoided. For example,conventional ‘first finite difference’differentiation causes large phase shifts athigh frequencies. Also, the same schemeyields amplitude errors, at high frequencies,owing to measurement noise. Theseproblems could be overcome by using highorder differentiators [7], with corrections tothe gain function based on the noisespectrum (i.e., optimum filtering). However,a similar effect can be obtained by lettingthe model generation process take care ofthese problems automatically. This ispossible because the optimal filtering

approach and the FRF and ED models areall based on minimizing the mean squareerror.

Force-displacement plots for the 2nd shockabsorber are shown, for the original lab data,the FRF model, and the Empirical Dynamicsmodel (Figures 4.5-4.7). Clearly, the ED modelreplicates the original data most accurately.

Figure 4.6 Shock Absorber #2, Force-Displacement,FRF Model

Figure 4.7 Shock Absorber #2, Force-Displacement,Empirical Dynamics Model

Note that the Empirical Dynamics model doesn’tjust replicate these ‘on average’. Rather, everyevent is replicated closely. This can be seenmore directly on a plot of time histories (Figure4.9). A plot of original (laboratory) forceoverlaid with the model-predicted force showsthat the ED model is predicting the signal veryprecisely at every point in time. An additionalerror time history overlaid on the same plotshows that the discrepancies are small. Forcomparison, time histories of original and


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predicted force are also shown for the FRFmodel (Figure 4.8). These plots show that theconventional models may introduce substantialerrors in both amplitude and phase of thesignals.

Figure 4.8 Shock Absorber #2, Time Histories, FRFModel

Figure 4.9 Shock Absorber #2, Time Histories,Empirical Dynamics Model (Legend in Fig 4.8)

Differences between the models can also beevaluated in the frequency domain. FourierTransform (FFT) plots of load amplitude vs.frequency are plotted for the original lab dataand each model (Figures 4.10-4.11); however,the FFT’s of error (original – model) show mostclearly that the Empirical Dynamics model isaccurate over a wider frequency range. It’simportant to note again, not only the trend isrepresented, but the small detail as well. TheFFT plot for the Empirical Dynamics modelindicates that this model accurately predicts theforce for a bandwidth of DC-100Hz. Anotheritem to note is the spectrum of the error is veryflat. This indicates that the modeling process

has extracted essentially all of the informationfrom the data; the error that remains is white, orcompletely uncorrelated with the input (or anyother signal).

Figure 4.10 Shock Absorber #2 FFT’s, FRF Model

Figure 4.11 Shock Absorber #2 FFT’s, EmpiricalDynamics Model

The higher accuracy of the Empirical Dynamicsmodel suggests that the specimen may haveappreciable nonlinearity. However, there is nodirect indication of this from any of the plotsthus far. Therefore, velocity was calculatedusing a high order differentiator, and a force-velocity plot was generated (Figure 4.12). Thisplot is significant in that it indicates a verystrong level of nonlinearity, as indicated by theabrupt change. Again, it must be emphasizedthat this velocity was not used to generate themodel.


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Figure 4.12 Shock Absorber #2 Force-Velocity,Original

Case Study 3: Biaxial Bushing

This case study demonstrates that the EDtechnique can be effectively used for multipleinputs, and for a specimen whose nonlinearcharacter is different than that of a damper.

The specimen in this case is a rubber bushingfrom a passenger car front suspension, from thejoint where the strut-fork connects to the lowercontrol arm. The bushing consists of a rubberannulus sandwiched between two steelcylinders (Figure 4.13). The operating loadingexperienced by this bushing is primarilyuniaxial, along the axis of the strut. However,for this case study, biaxial motion was appliedalong two orthogonal radial axes (x, z). Theobjective was to provide demonstration data fora multiple input nonlinear model, not to simulatethe true bushing environment.

The bushing lab test was performed using anMTS multi-axial elastomer test rig (Figure 4.14).The inputs were displacements, and outputswere forces, for each of the orthogonaldirections. The actual displacements weremeasured with LVDT’s, and these were used asinputs for the models, to isolate the specimenfrom the servoloop dynamics (as explainedearlier for the shock absorbers).

Figure 4.13 Bushing Specimen

Figure 4.14 Bushing Adapter and Test Rig

The displacement signals were discreteGaussian time series, with a power spectrumwhose magnitude was proportional to1/frequency. This spectral shape was selectedto give most of the energy at low frequencies.Amplitudes were selected to give noticeablenonlinear behavior.

A special characteristic of the two randomsignal channels was that they were statisticallyorthogonal. This means the statisticalcorrelation between the signals is essentiallyzero. Excitation having this characteristic isuseful when generating FRF models: it enablesthe contributions of each input to be easilyidentified in either output.

The nonlinearity of the specimen can be seenfrom plots of force vs. displacement measuredalong matching axes (Figures 4.15, 4.16). Formeasurements along the z-axis, the overallpattern displays a dual-inflection shape, similarto a cubic function. Along the x-axis, the same


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tendency appears, although less pronounced.Both plots show some hysteresis. Note that thehysteresis here may be caused by cross-coupling, in addition to memory effects.

Figure 4.15 Bushing Force-Displacment, OriginalLab Data, z axis

Figure 4.16 Bushing Force-Displacement, OriginalLab Data, x axis

Using these signals, polynomial, linear and EDmodels were generated.

The linear model is a 2x2 matrix of FRFs,whose magnitude parts are plotted in Figure4.17. Columns correspond to inputs, and rowscorrespond to outputs (responses). The maindiagonal shows the direct coupling effects: thenearly constant FRFs indicate that the bushingis behaving like a spring, as expected. The factthat the upper left magnitude is larger than thelower right suggests some asymmetry ispresent in the bushing. The off-diagonalelements are non-zero, indicating that cross-coupling is present. This provides furtherevidence of asymmetry, since a radially

symmetric bushing should generate onlysymmetric (even order) cross terms, whichwould not appear in the FRFs. Visual inspectionof the specimen suggests the asymmetry mayoriginate at the inner sleeve, which has a seam.

Figure 4.17 Bushing FRF Matrix

The static nonlinear model was generated astwo independent polynomials (Figure 4.18),between force and displacement for matchedaxes. In other words, cross-coupling wasignored. This was done primarily for simplicity,as a model with cross-coupling would requirerepresentation by two independent surfaces in athree dimensional space (i.e., two axes are theinputs, and the third is one of the outputs). Thissimplification is justified by the fact that cross-coupling is small, as shown by the FRF.

Figure 4.18 Uncoupled Polynomial Model

The ED model for the bushing was generatedas a pair of independent multiple input, singleoutput (MISO) units. That is, each output maybe affected by both inputs, but the outputs maybe considered independent of each other.(Figure 4.19)


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Figure 4.19 Double MISO Structure for EmpiricalDynamics Model

The models were evaluated using the sametechniques as for the shock absorbers. Plots offorce vs. displacement, for the z-axis, show thatthe polynomial model predicts the 'backbone',but not the hysteresis (Figure 4.20), and that theFRF predicts the hysteresis but not thebackbone (Figure 4.21). (Plots for the x-axisshow similar results).

Figure 4.20 Bushing Polynomial Model, z axis

Figure 4.21 Bushing Force-Displacement, FRFModel, z axis

The ED model, however, gives a plot nearlyidentical to the original lab data (Figure 4.22).

Figure 4.22 Bushing Force-Displacement, ED Model,z axis

Figure 4.15 is replicated here for directcomparison with Figure 4.22:

Figure 4.15 Bushing Force-Displacement, OriginalLab Data, z axis

Time history and FFT plots likewise show that:the ED model has lowest errors in both domains(Figures 4.24-4.29; plots are shown for the z-axis only; the x-axis plots are similar).


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Figure 4.24 Bushing Time Histories, PolynomialModel, z axis

Figure 4.25 Bushing Time Histories, FRFModel, z axis

Figure 4.26 Bushing Time Histories, ED Model,z axis

Figure 4.27 Bushing FFT’s, Polynomial Model,z axis

Figure 4.28 Bushing FFT’s, FRF Model, z axis

Figure 4.29 Bushing FFT’s, ED Model, z axis


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The preceding case studies have shown thepower of the Empirical Dynamics technique formodeling vehicle components with one or twoinputs. Other case studies (to be reported)demonstrate that this technique can be appliedto 3 or more inputs. In one example, the lateralstrain at a vehicle suspension ball joint, which isinfluenced by vertical, lateral, and longitudinalroad inputs, was predicted using those threeinputs, with error reduction of 40-50% over FRFmodels. The most significant aspect of this casestudy was that the specimen included not onlythe ball joint, or the suspension, but also theentire vehicle along with 12 channels of multi-axial road simulation test equipment (Figure4.30).

From these examples, it seems reasonable thatthe technique may be extended to other typesof vehicle components, including tires, engine

mounts, exhaust system hangers, etc.Additional applications may include non-vehicleapplications, including biomechanical systems,electrodynamic systems, and fluid dynamicsystems.

Figure 4.30 3x1 Blackbox Model for Ball Joint Strain

5 EDM Benefits and Limitations____________________________________________________________________________________

The preceding Case Studies have shown byexample how the Empirical Dynamics methodyields models with higher accuracy thanconventional blackbox methods. This sectionprovides a summary of benefits and limitationsof the Empirical Dynamics approach

Benefits of EDM include:• Significant improvements in accuracy• Ability to model a wider range of systems

and components, with just one technique• Simpler and faster modelingEach of these is elaborated below.

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Benefit 1: Significant Improvements inAccuracy

Empirical Dynamics models are more accuratethan conventional blackbox models. In manycases, Empirical Dynamics models will faroutperform linear dynamic models and static-nonlinear models (e.g., polynomials). Thisdepends of course on the degree of nonlinearityand/or memory in the system. EmpiricalDynamics models are most beneficial whenthese effects are appreciable. (That is, if thesystem is already highly linear, EDM won'tprovide much improvement over an FRF).

Evidence for this benefit was provided fornumerous specimens in the Case Studies.

Empirical Dynamics models can be moreaccurate than whitebox models. Whiteboxmodels often oversimplify complex physicalphenomena. For example, a whitebox model fora 'friction element' may be represented in termsof a simple (static or dynamic) coefficient,obtained from a table in a design handbook. This value is subject to wide variability (relatedto surface finish, contamination, relativevelocity, etc) and doesn't account for transitionsfrom static to dynamic (e.g., stiction). Bycontrast, an Empirical Dynamics model of asystem with physical friction can include theseeffects, in proper proportion. The result is amore accurate model overall.

Empirical Dynamics models are 'self-validating'. The accuracy of an Empirical Dynamics modelis assessed as part of the model generationprocedure. By splitting the data into trainingand validation sets, the performance of themodel is easily assessed, without any need toreturn to the lab. Another important aspect ofvalidation: since Empirical Dynamics modelsare generated and validated using random data,they automatically encompass a wide range ofoperating conditions. Whitebox models are


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often validated using simple waveforms(sinusoidal, step); these results may be difficultto extrapolate to more general conditions. Ofcourse, whitebox models can also be validatedusing random signals, but this may requiresignificant computational resources. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Benefit 2: A Wider Range of Systems,Components, and Signal Types

Simple components or complex systems can behandled with the same method. The onlyrequirement is to have input and output signalsfrom the specimen being modeled. Considerations re: actual complexity of thespecimen, in terms of number of moving parts,degrees of freedom, etc., can be often beignored.

The Empirical Dynamics approach may be usedfor a wide range of applications. That is,Empirical Dynamics modeling may be appliednot just to mechanisms, but also to hydraulic,electrical, thermal, optical, digital/software, &other systems.

Empirical Dynamics models can be used whenwhitebox models aren't available. For manyphysical systems, there are no whiteboxmethods that can accurately predict dynamicbehavior. For example, components thatinclude elastomers/rubber bushings, fabric,biomaterials, etc., are sometimes modeledusing the finite element method (FEM). TheFEM models don't include provisions formodeling significant damping or hysteresisbehavior in these components. EmpiricalDynamics models may be the only choice in thiscase.

For other components, there may be specialwhitebox methods available, but only atprohibitive cost ($ or expertise). For example,fluid-filled or fluid-dynamic components may bemodeled using a computational fluid dynamics(CFD) modeling package. If this tool is usedonly infrequently, the total cost to build a modelmay far exceed its capital outlay.

The Empirical Dynamics method allows use ofalternative signal types: For example, dampers(shock absorbers) are often characterized interms of force vs. velocity. With EmpiricalDynamics modeling, accurate damper modelsmay be generated in terms of force vs.

displacement. Essentially, the EmpiricalDynamics model includes the differentiator thatconverts displacement to velocity. This helpsprevent errors that may be introduced bynumerical differentiation schemes, or fromvelocity transducers which may be noisesensitive. - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Benefit 3: Simpler and Faster Modeling

Empirical Dynamics modeling enables simplermodel construction. In particular, generation ofEmpirical Dynamics models may be automated,in many cases. This happens in part because ofthe generic nature of Empirical Dynamicsmodeling, i.e., only input and output signals arerequired; internal specimen details can beignored. Automation may be applied mostbeneficially in scenarios where all specimensare similar (for example, shock absorbers testedin an MTS shock absorber test rig).

Empirical Dynamics models may be constructedfaster. In situations where a test rig, specimen,and related instrumentation are readilyavailable, it may be more expedient to generatea model by the Empirical Dynamics approach,even if a whitebox method is available. Notethat the test equipment and specimen are oftenavailable, to validate whitebox models. Thebenefit of Empirical Dynamics modeling is toconsolidate validation and model generationinto one procedure.

At this stage it is informative to present sometiming benchmarks for the main EmpiricalDynamics processes.

Specimen Lab TestDuration

TrainingDuration

Shock absorber 1 minute 1 hourRubber bushing 2 minutes 2 hours

for a 2x2 model

Some conditions: all calculations are based onusing a 450 MHz Pentium II processor. Notethat the lab test times do not include installationof the specimen, servo-loop tuning, amplitudeadjustment, and other preliminary setup. Modelcalculation times do not include miscellaneouspreliminary processing, including signal channelrearrangement, offset removal, and others.Most importantly, the calculation presumes thedesign for the neural net (including number oflayers and number of neurons) has already


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been finalized. This last task may be the mosttime consuming part of the Empirical Dynamicsmodeling process; on the other hand, it mayseldom be necessary, for many test scenarios.For example, when a series of EmpiricalDynamics models is generated, for specimensof similar behavior, it will generally benecessary to design the network only once.Thus, network design may be pre-defined, forsome specimen types.

Furthermore, a neural net design may beremarkably robust. Perhaps one of the mostsurprising discoveries in the development of theEmpirical Dynamics method was that the neuralnetwork designed for use with the shockabsorbers could be used, without significantchange, to the rubber bushing. This issurprising because the two systems havesignificantly different mechanical behavior, asshown by their force-displacement plots.

Empirical Dynamics models executefaster. Empirical Dynamics models are usuallyrepresented as closed-form algebraicequations, which are quickly evaluated by'turning the crank'. By contrast, whiteboxmodels often yield differential and algebraicequations (DAE's), which are often algebraicallycomplex and may require considerable iterationto solve.

At this time, benchmarks for EDM execution arenot available, for operation with ADAMS.Standalone operation of EDM, where outputsignals are calculated directly from predefinedinputs, has been measured. An ED shockabsorber model, with a sampling rate of 204.8Hz (giving model bandwidth ~ 80 Hz) has mostrecently been measured at 250 times fasterthan real-time. In other words, 10 seconds ofinput signal, equivalent to 2048 sample points,executes in approximately 0.04 seconds (withthe CPU described above).

Note that for Empirical Dynamics models,generation time may in many cases be offset byexecution time. For the shock absorberexample, two hours of processing ( 1 forgeneration, 1 for execution) effectively buys 250hours of simulation. (A direct comparison withwhitebox methods cannot be made, as awhitebox shock absorber model has not beeninvestigated. Clearly, this will be a worthwhileinvestigation).

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Limitations of the Empirical Dynamics approachinclude:• Test rig & specimen are required• Specimen parameters cannot be adjusted.• The model requires a large number of

coefficients.• The number of inputs is limited.

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Limitation 1: A Test Rig & Specimen areRequired.

Being a blackbox technique, EmpiricalDynamics requires a specimen from whichsignals can be obtained. However, test rigs andspecimens are often available, as needed forvalidation of whitebox models.

A future application of the Empirical Dynamicsapproach includes "whitebox substitution",where a time consuming whitebox subsystemmodel is replaced by an Empirical Dynamicsequivalent. In this case, the EmpiricalDynamics model is generated directly from thewhitebox model, with no need for a test rig orspecimen.

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Limitation 2: Specimen Parameters Cannot BeReadily Adjusted.

Parameters refer to specimen variables thatmay be adjusted as part of the specimen designprocess. For example, parameters of a tiremight include:• geometry (diameter, width, sidewall height,

tread pattern)• mechanical properties (viscoelasticity of the

rubber, anisotropic flexibility of therubber/cord matrix)

• pressure Whitebox models are usually configured toallow adjustment of these parameters in themodel.

Blackbox models, by ignoring the internaldetails of the physical system, preclude theability to manipulate various parameters of thatsystem. Nevertheless, there are multiplecounterpoints that bear mention:


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Complete adjustability is often not required: Adynamic vehicle model may be needed to studybody loads incurred while traversing a roughroad. Tire models for this vehicle may bewhitebox or blackbox. For the purpose ofmodeling suspension-to-body loading, the tireparameters (listed above) are not needed. Instead, a single unchanging tire may be used;it can be modeled using a blackbox. Thebenefit of the blackbox is improved accuracyand execution speed.

Complete adjustability has serious economicramifications: A whitebox model with fulladjustability introduces excessive complexityinto the modeling process, and with it muchlonger computation time.

Limited parameter adjustments are possiblewith Empirical Dynamics models. Twoexamples of these include 'extrinsic conditions'and 'model stretching'.

Extrinsic conditions refer to the use of multiplemodels, each corresponding to a different testcondition, which is constant or slowly varying.For example, to account for thermally inducedsystem variation, a series of EmpiricalDynamics models is constructed, eachcorresponding to a separate temperature(Figure 5.1). These are combined into a singlefile structure (Figure 5.2). During execution,temperature is used as an input to this series ofmodels, to select the correct model.Interpolation between temperatures could beperformed with minimal difficulty (Figure 5.3).This same approach can be used for conditionssuch as pressure, preload, etc., but also fordesign variables. For example, a bushing couldbe measured using several diameters, rubbercompositions, etc., and interpolation could beused to provide a wide range of parameteradjustment.

Figure 5.1 Extrinsic Condition, Measurement ofMultiple Conditions

Figure 5.2 Extrinsic Conditions, Model Combination

Figure 5.3 Extrinsic Conditions, Selection orInterpolation

Model stretching refers to the application of gainfactors to input and output signals of a model,and to the time base as well. Whereas ameasured input-output plot shows actualsystem behavior, this plot can be 'stretched' in


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the horizontal or vertical directions to perform a'what if' evaluation. This may actually be morestraightforward than adjusting physical/whitebox parameters; for those systems: it's notalways clear which parameter provides astretching effect, and it's very likely that aphysical parameter adjustment one way causesunintended side effects elsewhere in thesystem.

Moreover, model stretching adjustments mayhave actual physical counterparts. A case inpoint is that of a coil spring, with nonlinearityappearing when the coils ‘bottom out’ at largecompressive displacement. As shown in Figure5.4, this system may be modeled simply as anonlinear force-displacement relation. Bystretching the model along the displacementaxis, the nonlinearity is encountered at adifferent displacement, but at the same force(Figure 5.5). This is equivalent to adding coils tothe spring. If the model is stretched along theforce axis, the nonlinear transition occurs at thesame displacement (Figure 5.6). This isequivalent to changing the coil diameter (andleaving the wire diameter intact).

Figure 5.4 Coil Spring

Figure 5.5 Coil Spring, Empirical Dynamics ModelStretching in Displacement

Figure 5.6 Coil Spring, Empirical Dynamics ModelStretching in Force

A similar capability is available by changing thetime base of the Empirical Dynamics model(i.e., the sample rate used for the input andoutput histories). For a resonant mass-spring-damper system, changing the time base isequivalent to changing the natural frequency ofthe system, without changing the output/inputgain. Note that in the whitebox realm, thiswould require modification of both the mass andthe stiffness, to achieve the same effect.Another example is that of a damper: changingthe time base is equivalent to changing thekinematic viscosity (assuming piston inertia isnegligible), which can be further connected tothermal changes.

Empirical Dynamics modeling may be applied todimensionless characterizations. An EmpiricalDynamics shock absorber model doesn’t allowdirect adjustment of viscosity, orifice size, springstiffness, etc., and so it isn't particularly usefulfor shock absorber design. However, even awhitebox damper model has limitations,particularly in the fluid dynamic coefficients,which are typically quasi-static 'cookbook'values. These may be inadequate for assessingthe high frequency phenomena associated withclosing valves. A possible improvement for thefuture is to use the Empirical Dynamicsapproach to model the only the fluid dynamicbehavior, rather than the entire shock absorber.For example, a pressure coefficient may bemodeled as a function of a 'dynamic' Reynoldsnumber and valve position, normalized to somereference length, e.g., orifice size. This wouldallow the unsteady fluid dynamics to be properlymodeled. Note that such a characterizationwould entail construction of a special fluiddynamic test rig. Similarly, an EmpiricalDynamics model may be constructed for just thefriction behavior. Once these ‘subcomponent’Empirical Dynamics models are obtained, they


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can be integrated with whitebox spring andinertia components to provide improvedaccuracy, while retaining the adjustabilityneeded for design optimization.- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Limitation 3. Empirical Dynamics Models MayRequire Many Coefficients

The number of coefficients represents themodel size, i.e., the amount of informationrequired to specify the model. Large model sizehas historically had economic ramifications(connected to disk & RAM capacities, forexample). Likewise, large models may meanmore complexity from a user perspective,especially if these numbers have to beinterpreted, or manually determined.

With large numbers of neurons, an accurateEmpirical Dynamics model may containhundreds of coefficients for each combination ofinput and output.

However, large is relative. Compared towhitebox parameterizations, EmpiricalDynamics models may seem large; comparedto modern disk storage and RAM capacities,they are minute.Similarly, large is misleading. The coefficients ofa nonlinear blackbox model do not translate toadditional complexity for a user (i.e., to identify,understand, or modify).- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -

Limitation 4. Empirical Dynamics Models areLimited to a Few Inputs

At this time, Empirical Dynamics models withthree or fewer inputs have been studied. Modelgeneration time for a three input system isexcessive (> 2 days). Research anddevelopment efforts to reduce this duration arecurrently underway.


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6 Interfacing Empirical Dynamics Models to the ADAMS Environment

The ADAMS Solver has been designed foroptimal solution of equations in state variableform. This conflicts with several aspects of theEmpirical Dynamics paradigm. Primarydifferences include:• Empirical Dynamics models use a fixed

sample rate, or time base. The ADAMSSolver typically uses variable time steps.

• Empirical Dynamics models requireinformation about past inputs and outputs.The ADAMS Solver does not normallyretain a history of such information; instead,it stores the necessary 'history' informationas state variables.

• The ADAMS Solver supplies inputs to acomponent one at a time, and these may betrial values, whose convergence status isindeterminate. If past inputs and outputsare retained (in a buffer) for use with anEmpirical Dynamics model, the trial valuesmust be identified and excluded from thebuffer.

To negotiate between these differentenvironments, a prototype interface betweenADAMS and Empirical Dynamics has beendeveloped. Called the ADAMS/EDM Socket,this interface works within the ADAMSenvironment to perform the following functions• sample rate conversion (via interpolation)• determination of convergence status for

past inputs

• buffering of past inputs and outputs.

Preliminary tests on the prototype socket havedemonstrated its basic feasibility, and furthertests are underway.

Future development plans for the Socketinclude provision for the following features:

Physical interfacing. This includes componentspecifications that aren't directly required forEmpirical Dynamics model generation. Forinstance, the length of a shock absorber isneeded within the ADAMS environment, but itisn't needed for Empirical Dynamics modelgeneration. A related issue is matching ofcoordinate systems: the X,Y,Z axes used for alab test must be properly identified and matchedin the ADAMS environment.

Adjustments. As defined in the earlier Section,adjustments via extrinsic conditions and modelstretching alleviate some of the limitations of theblackbox approach. These features bestimplemented at the socket level.

Limit checking: Empirical Dynamics models areonly valid for the range of input data used totrain and test them. If the ADAMS Solversupplies input outside of this range, it will beimportant to provide a warning to the User.

7 Summary

This paper has presented Empirical Dynamics modeling as an effective method to generate and useblackbox models for complex components. The complications of amplitude dependence, frequencydependence, arbitrary input, multiple inputs and outputs, and arbitrary system types, are allaccommodated within the Empirical Dynamics framework, via use of neural networks and tapped delaystructures for inputs and outputs. The capabilities of Empirical Dynamics modeling were demonstratedin case studies for shock absorbers and a rubber bushing from a passenger car. In general, EmpiricalDynamics models provide a way to model complex components simpler and faster. The integration ofEDM into the ADAMS environment is currently under development.


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8 References

[1] Vladimir Cherkassky and Filip Mulier, Learning from Data: Concepts, Theory, and Methods, JohnWiley & Sons, 1998.

[2] J.W. Fash, Modeling of Shock Absorber Behavior using Artificial Neural Networks, SAE TechnicalPaper 940248, 1994.

[3] Joseph Giacomin, Neural Network Simulation of an Automotive Shock Absorber, EngineeringApplications of Artificial Intelligence, Vol. 4, No.1, pp. 59-64, 1991.

[4] Martin T. Hagan, Howard B. Demuth, Mark Beale, Neural Network Design, PWS Publishing Co.,Boston, 1996.

[5] Lennart Ljung, System Identification:Theory for the User, 2nd Ed., PTR Prentice-Hall, Upper SaddleRiver, NJ, 1999.

[6] Kumpati S. Narendra and Kannan Parthasarathy, Identification and Control of Dynamical SystemsUsing Neural Networks, IEEE Trans. Neural Networks, Vol.1, No.1, March, 1990. pp. 4-27.

[7] Boaz Porat, A Course in Digital Signal Processing, John Wiley & Sons, 1996.

[8] Jonas Sjoberg and Lester S.H. Ngia, Neural Networks and Related Model Structures for NonlinearSystem Identification, in Nonlinear Modeling: Advanced Blackbox Techniques, Johan A.K. Suykens andJoos Vandewalle (Eds.), Kluwer Academic Publishing, 1998, pp 1-28.

[9] Donald D. Weiner, John E. Spina, Sinusoidal Analysis and Modeling of Weakly Nonlinear Circuits,Van Nostrand Reinhold, 1980.

[10] Rodger E. Ziemer, William H. Tranter, D. Ronald Fannin, Signals and Systems, Continuous andDiscrete, Macmillan Publishing, New York, 1983.

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