material design in physical modelling sound synthesis

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Material Design in Physical Modelling Sound SynthesisPirouz DjoharianPublished online: 09 Aug 2010.

To cite this article: Pirouz Djoharian (2001) Material Design in Physical Modelling Sound Synthesis, Journal of New MusicResearch, 30:3, 227-241

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Abstract

This paper deals with designing material parameters forphysical models. Using the linear theory of viscoelasticity,we show that the sound signature of a material is a frequency-damping characteristic function. Some qualitative features of the sound signature of linear hypothetical materials arestated. Physical modelling is then organised in two inde-pendent steps: shape modelling within the spring-massmodel paradigm and material modelling according to mate-rial sound signatures. Any conservative model can then becovered with a viscoelastic dress in order to represent an arbi-trary shaped material.

1. Introduction

Among various sound synthesis methods, Physical Model-ling has distinguished itself by its ability to produce lifelikerealistic sounds with outstanding rich transients. Unlikesignal modelling synthesis, it is now the actual instru-ment that is modelled. Synthesis parameters have then physical meaning and can be controlled in a natural way. In a word, the physical model can be manipulated as a realinstrument.

Due to the close interaction with musical acoustics, physical modelling is often understood as a tool for repro-ducing realistic sounds of existing instruments. However, inan artistic context, computer models and their real world ref-erences are related in a loosely way. Therefore, physical mod-elling can be seen as an open and challenging framework,that allows musicians to explore their own imaginations.

The sound produced by a sound body is affected by manyfactors including its size and shape, its material, as well as external factors such as the way it is played and the environment. To concentrate on intrinsic features of soundbodies, all external factors, such as radiation, reverberationor external damping are neglected throughout this work.

The control of the sound quality of a resonator by meansof shape and material features is all but an easy or a straight-forward task, in real instrument making as well as in com-puter modelling. However, we want to introduce shape andmaterial as natural parameters into the sound synthesisprocess. Recent works in Psychoacoustics tend to prove thatthe auditory system encodes audio information to recognizein some extent shape and material features (Lakatos et al.,1997). But, it is not clear in which extent, sound perceptionsof shape and material are independent? Anyhow, in thevirtual world, materials may take arbitrary shapes. This will open the way for unusual combinations of shape andmaterial, such as strings made of concrete or plates made ofhuman bone!

Leaving the above psycho-acoustical question open, theproblem will be treated here from a purely physical point ofview. The first step is to look for invariant features in theacoustical signals produced by resonators having differentshapes but made of a single material or conversely havingconstant shape and made of different materials. Free oscilla-tions of a linear resonator are entirely characterised by theset of its normal modes. The invariant of shape is the set ofthe normalised modal frequencies. We show further that theinvariant of the material is to be found in the linear theory of viscoelasticity. Indeed, in a homogeneous isotropic vis-coelastic material, modal frequencies and damping constantssatisfy a particular characteristic equation, which is inde-pendent of the shape and the boundary conditions. This frequency-damping relationship is what we refer to as thesound signature of the material.

Thanks to approximation techniques such as the finite dif-ference or the finite element methods, a resonator’s shape canbe represented by a spring-mass system. This conservativeskeleton is then enriched by a viscoelastic model, accordingto a particular sound signature. At the final step, the physi-cal model ends up as a mass-spring-dashpot network.

Accepted: 4 April, 2001

Correspondence: A.C.R.O.E., 46, Avenue Felix Viallet, 38000 Grenoble, France. E-mail: [email protected]

Material Design in Physical Modelling Sound Synthesis

Pirouz Djoharian

A.C.R.O.E., Grenoble, France

Journal of New Music Research 0929-8215/01/3003-227$16.002001, Vol. 30, No. 3, pp. 227–241 © Swets & Zeitlinger

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228 Pirouz Djoharian

The paper is organised as follows. In section 2, an intro-duction to mechanical properties of materials is brieflyreviewed. In section 3, it is shown how shape and materialparameters affect vibration properties of mechanical sys-tems. Simple rheological models with their correspondingsound signatures are discussed. Section 4 outlines a generalframework for Physical Modelling, including shape andmaterial design. The last section discusses sound synthesisand simulation techniques.

2. Mechanical properties of materials

An acoustical instrument is divided into two parts: the exciter (plectrum, hammer, bow, etc.), and the resonator(string, membrane, etc.). Even though the exciter is in turnan actual vibrating structure, shape and material featureshave their greatest relevance in the resonators. In most in-struments, resonator’s vibrations are of small amplitude.Therefore, the linearity hypothesis is assumed throughoutthis work.

2.1 Conservative models

Though each material has a particular density and a charac-teristic elastic moduli, free oscillations of conservativemodels do not show any invariant feature of the material. Infact, inertia and elasticity combine into a single coefficientc, which is the speed of sound in the material. However, theshape of a resonator determines its reduced spectrum, i.e.,the frequency spectrum up to a scaling factor.

2.1.1 Inertia and elasticity

In homogeneous materials, inertia is defined by a single coefficient m, the density of the material. Depending on thedimensionality of the resonator, m denotes mass per unitlength, unit area or unit volume respectively. Density ofsolids varies from hundreds to thousands Kg/m3, accordingto the density of atoms and the crystal packing arrangements.

Elasticity of a material is its ability to return to its origi-nal shape and size when the forces causing the deformationare removed.1 Elastic property is then expressed in form ofa constitutive equation, i.e., an equation that is independentof the geometry of the body and depends only on its material nature. Therefore, a pair of intensive and extensivedependent variables, stress and strain, is introduced. Stress,s = f/A, is force per unit area and strain, e = Dl/l, is the dimen-sionless change of size or shape (length, volume, angle, etc.).The simplest model of elastic behaviour is the Hooke’s lawwhich states that stress is proportional to strain, s = ke. Thetypical situation is the case of a uniaxial load and the resulted

extension of a rod. The ratio E = s/e is known as the Young’smodulus of elasticity. Depending on the deformation type,various other elastic moduli are defined: shear, bulk, etc.(Tschoegl, 1989).

Elastic properties of an anisotropic material are expressedby a larger number of elastic moduli, with help of tensoralgebra (Tschoegl, 1989). However, the generalised Hooke’slaw is still a linear relation between the strain and the stresstensors. Nevertheless, all real materials deviate from thesimple Hooke’s law in various ways: nonlinearity of con-stitutive laws and/or time and frequency dependence ofelastic moduli. Time and frequency dependence of elasticmoduli is known as the viscoelastic properties of materials.Linearity is assumed throughout this work, then linear vis-coelasticity will be considered as the fundamental behaviourof solids.

2.1.2 Invariant of shape

Let us consider first the case of an ideal flexible string oflength l. It is well known that the modal frequencies wn ofthe string are multiples of the fundamental frequency w1 =cp/l, where c is the sound velocity along the string (Fletcher& Rossing, 1991). Therefore, two strings having differentlengths l1 and l2 will produce the same frequencies providedthat c1

2l2 = c22l1. This can be realised by tuning the strings, i.e.,

by adjusting their respective tensions. This example showsthat the frequency spectrum is neither an invariant of thematerial, nor the size. However, all ideal strings have thesame reduced spectrum wn/w1 = n. Reduced spectrum is,indeed, invariant under material and size change.

The above example may be generalised to any conserva-tive resonator. For example, in the case of an acoustic tube(closed at one end), the reduced spectrum is wn/w1 = 2n - 1while for simply supported bars wn/w1 = n2 (Fletcher &Rossing, 1991). The dynamic equilibrium governing vibra-tions of an isotropic sound body has the following form

(2.1.1)

where L is the differential operator expressing the localstrain, k an elastic modulus and m the density of the material. For transverse vibrations of strings and membranesinvolving stretching tension, L is the Laplacian operator D;for beams and plates involving bending waves, L = D2. Theshape and the size of the resonator as well as the boundaryconditions determine the eigenvalues ln of L (see § 3.2.3).The frequency spectrum is then obtained by

w 2n = lnc

2 (2.1.2)

where c2 = k/m. Thus the material quality appears through the single scaling factor c. Resonators made of differentmaterials with homothetical shapes have then the samereduced frequency spectrum. It follows that the conservativemodel reflects much more the geometry of the resonator.

∂∂

=2

2

u

t

kLu

m

1 Deformations may be of various types: elongation, bending, shear-ing, torsion, etc.

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Material design in physical modelling sound synthesis 229

2.2 Dissipative models

Loss of mechanical energy results from numerous externalas well as internal factors. External damping consists intransferring energy to another mechanical system. Forinstance, in stringed instruments, part of the string vibra-tional energy is transferred to the soundboard via the bridge.In the same way, a part of the bridge’s energy is in turn sup-plied to the surrounding air by acoustic radiation. In spite ofthe critical influence of external damping, to focus on intrin-sic features of materials, we consider here only internalfactors linked to the very nature of the material.

2.2.1 Internal damping in materials

When a body of material is subjected to a deformation, itsmicroscopic structure may experience local activities. As arule, the more the microstructure is ordered, the less defor-mation produces local activities. In crystalline materials,point defects such as solute atoms or impurities produce localjumps between neighbouring atomic sites. In amorphouscrystals such as glasses or polymers, deformation entailsmolecular rearrangements. In composite materials, slip atvarious interfaces between constituents, causes mechanicaldamping. In porous materials (wood, bone), fluid flow (air orwater) is responsible of damping phenomenon (Lakes, 1999).

Some mechanisms, such as thermoelastic damping, occureven in perfect ideal crystals. The temperature of the parts of the body under compression being raised and of thoseunder tension being lowered. The temperature gradient will cause the heat to flow. The mechanical energy is then trans-formed into heat in an irreversible way. Thermoelastic mechanism is important in metals and any materials with ahigh conductivity.

In any real material, all microscopic rearrangements nec-essarily require a finite time during which material proper-ties will vary. Thus, constitutive equations must involve thetime variable, or equivalently, the mechanical properties ofthe material are frequency dependent.

2.2.2 Linear viscoelastic constitutive equations

The mathematical formulation of linear viscoelasticity con-siders a sample of material as a causal linear time-invariant(LTI) system, with stress/strain as input/output variables.Aging phenomenon is then neglected and temperature isassumed constant. A general LTI system can be characterisedby one of its response functions: impulse response, stepresponse, harmonic response, etc. Considering the strain e asthe input and the stress s as the output, constitutive equa-tions have the general form of a convolution equation:

s = kd * e (2.2.1)

in which kd is the impulse response of the material sample.The time dependent relaxation modulus kh(t) is the ratios(t)/e(t), where the strain e(t) is the Heaviside unit step. The

relaxance k(s) and the complex modulus k*(w) are respec-tively the Laplace and the Fourier transforms of the impulseresponse. The constitutive equation (2.2.1) expressed in fre-quency domain or in the Laplace transform plane, reduces tomultiplication by k*(w) and k(s) respectively. The complexmodulus k*(w) = k(iw) = k¢(w) + ik≤(w) is the ratio s/e whenthe material sample experiences harmonic oscillations at fre-quency w. The real part of k*(w) is proportional to the poten-tial energy stored, while the imaginary part is proportional tothe energy dissipated. The loss factor h(w) = k≤(w)/k¢(w) is a quantitative measure of the damping capacity of thematerial at the frequency w. Single crystal materials exhibitvery small h, followed by the most conventional metals, andhighest of all for woods and rubberlike polymers. Typicalvalues of the loss factor at room temperature, for w = 1Hzare shown below.

The most easily measurable viscoelastic functions are thestep and the harmonic responses, kh(t) and k*(w). Regardlessof the precise damping mechanism involved, some generalproperties of kh(t), k¢(w) and k≤(w) may be outlined. For apassive material, the relaxation modulus is a decreasing func-tion with a decreasing derivative, i.e., the curve of kh(t) isconcave up (Lakes, 1999). The storage modulus k¢ is anincreasing function of w. In fact, at high rate strain, fewerrelaxation processes have enough time to be completed. Eachrelaxation phenomenon requires time but also minimum acti-vation energy. Hence, the plot of k≤(w) may exhibit severalpeaks at various frequencies, each connected to a particularrelaxation phenomenon. Below a peak, energy is too smallto activate the process and beyond it, time is too short for therelaxation to be completed. Roughly speaking, the frequencyaxis can be divided intro three regions:

1. The rubbery or leathery2 region, where k¢ and k≤ are relatively low and have slow variations,

2. The transition region, where k¢ increases rapidly and k≤exhibits one or several peaks,

3. The glassy region, where k¢ reaches a high stationaryvalue and k≤ has again a small value.

The location and the strength of the relaxation peaks deter-mine the viscoelastic properties of a material. The upper limit

Table 2.1. Typical values of the loss factor at room temperaturefor w = 1Hz (Lakes, 1999).

Material Loss Factor h

Steel 0.0005Aluminium 0.001Glass 0.0043Wood 0.02Plexiglas 0.1

2 The terminology rubbery, glassy, . . . refers basically to rubberlikematerials exhibiting significant viscoelastic behaviour.

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of the storage modulus is known as the glassy or the instan-taneous modulus kg = k¢(•) = kh(0), whereas its lowest valueis referred to as the rubbery or the equilibrium modulus ke = k¢(0) = kh(•). The ratio c = kg/ke is a measure of theoverall viscoelastic strength.

It must be pointed out that since k¢ and k≤ are the real andimaginary parts of the transfer function of a causal system,they are a Hilbert transform pair3 (Papoulis, 1977). In prin-ciple, all viscoelastic functions may be obtained by any oneof them, via the transform calculus.

2.3 Mathematical models

Several mathematical formulations of the relaxation modulusof materials have been proposed: sums of decaying expo-nentials, stretched exponentials or power law behaviours(Tschoegl, 1989). Decaying exponentials arise from quitegeneral considerations (Lakes, 1999). Our discussion will belimited to models based on them. A general relaxation func-tion has then the form of a sum of exponentials, each con-nected to a particular internal phenomenon. According towhether this sum is finite or infinite, lumped or distributedmodels are to be considered.

2.3.1 Lumped models

By considering two idealised elements, the pure springand the pure dashpot and combining them in series-parallelassemblies, one obtains a variety of viscoelastic behaviours.Consider a pure spring with stiffness k and a pure dashpotwith viscosity constant z. The spring reacts to a strain eaccording to the Hooke’s law s = ke, while the dashpotimpedes the flow s = zde/dt. The corresponding relaxancesare then k(s) = k and k(s) = zs respectively. Maxwell andKelvin units are series and parallel combinations of these twoelements (Fig. 2.1, a–b).

The Maxwell model has no equilibrium elasticity (i.e., ke

= 0) while the Kelvin model (also known as Voigt) does notexhibit relaxation behaviour. By adding a parallel pure springke to the Maxwell unit, one obtains the Zener model, whichis the simplest model of linear viscoelastic solids (Fig. 2.1.c).The Zener model can be generalised to a Wiechert model

which is the parallel assembly of a pure spring ke and nMaxwell units (ki,zi) (Fig. 2.1.d). Each Maxwell unit can becharacterised by two parameters: its relaxation time ti = zi/ki

(resp. relaxation frequency zi = ki/zi), and the correspondingstrength ki. Elastic moduli add in parallel mounting. Therelaxance of a Wiechert model is then the sum of ke and then relaxances of the underlying Maxwell units. The relaxanceand the relaxation modulus are then expressed as

(2.3.1)

(2.3.2)

The relaxed modulus of a Wiechert model is obtained bysetting s = 0 in (2.3.1) or t = • in (2.3.2): k(0) = kh(•) = ke.The glassy modulus k(•) = kh(0), is the sum of the stiffnessconstants of all the springs, kg = ke + Ski.

If the relaxation frequencies zi are well separated, the plotof the loss modulus k≤(w) versus w has n peaks located at therelaxation frequencies zi (Fig. 2.2). We define the dimen-sionless moduli (denoted by underlined letters), by nor-malizing with respect to kg. For example, the normalizedrelaxance k(s), is defined by

(2.3.3)

Every lumped viscoelastic model can be represented as an n-order Wiechert model. The decomposition is derived fromthe partial fraction decomposition of k(s) = P(s)/Q(s). Themost important feature is that for lumped systems, the con-volution equation (2.2.1) may be replaced by the differentialequation

(2.3.4)

Wiechert models with n = 4 to 10 elements can representlinear viscoelastic properties of most solids with a goodapproximation. Relaxation peaks zi and correspondingstrengths ki are to be obtained by direct measurement of k≤(w). An alternative way is to measure the transient re-sponse kh(t). Viscoelastic parameters are then derived bymodel fitting methods so that ke + Ski exp(-zit) approachesan observed relaxation modulus (Tschoegl, 1989).

2.3.2 Continuous relaxation spectra

Wiechert models having a large number of Maxwell units aregeneralised in models having an infinite number of relaxationpeaks. The distribution of relaxation frequencies is given bya weighting function H(z ), known as the relaxation fre-quency spectrum. The relaxation modulus of an infinite orderWiechert model is then a generalisation of (2.3.2) to an infinite sum

ss

ee

+ = +==ÂÂa

d

dtk b

d

dti

i

i e i

i

ii

n

i

n

11

k sk s

kg

( ) =( )

k t k k e k k eh e it

e it

i

n

i

ni i( ) = + = +- -

==ÂÂ t z

11

k s kk z s

k z sk

k

sei i

i ig

i i

ii

n

i

n

( ) = ++

= -+==

ÂÂ zz11

(a)

k z

(b)

k

z

(c)

ke

(d)

ke

k1

z1

(e)

ke

Fig. 2.1. (a) Kelvin model, (b) Maxwell model, (c) Zener model,(d) 2-order Wiechert model, (e) continous spectrum model.

3 In particular, k¢ and k≤ are respectively even and odd functions ofw.

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(2.3.5)

Relaxed modulus ke and instantaneous modulus kg areobtained from (2.3.5) by setting t = 0

(2.3.6)

Note that kh(t) may be recognised as the Laplace transformof H(z)/z, with t as Laplace variable, restricted to real nonnegative values. Equation (2.3.1) for the relaxance becomes

(2.3.7)

in which the Hilbert transform of H(-z) can be recognised.Some particular spectra allow analytical expression of the

k s kH

sdg( ) = -

( )+

•

Úz

zz

0

k k

Hdg e= +

( )•

Úz

zz

0

k t kH

e dh et( ) = +

( )•-Ú

zz

zz

0

response functions. In particular, the wedge spectrum,defined by

(2.3.8)

where z0, k0 and q are parameters. In other words, the material is supposed to exhibit relaxation phenomenonbetween two frequencies z1 and z2, with a power law distri-bution of the relaxation strengths.

The particular case of q = 0 is known as the box spectrum.The viscoelastic moduli are easily integrated. For instance,

(2.3.9)

(2.3.10)k k≤ ( ) =-( )

+w

w z zw z z0

2 1

21 2

arctan

k s k ks

sg( ) = -++0

2

1

logzz

H k

H

z z z z z zz

q( ) = ( ) < <( ) =

ÏÌÓ

0 0 1 2

0

,

otherwise

100

101

102

103

104

0.5

0.6

0.7

0.8

0.9

1

FREQUENCY w (rad/s)

NO

RM

. ST

OR

. MO

D. k

'(w)

100

101

102

103

104

0

0.05

0.1

0.15

0.2

0.25

FREQUENCY w (rad/s)

NO

RM

. LO

SS. M

OD

. k"(

w)

10-4

10-3

10-2

10-1

100

0.5

0.6

0.7

0.8

0.9

1

TIME t (s)

NO

RM

. RE

LA

X. M

OD

. kh(t

)z

2-1

z

1-1

zmax-1

z1

zmax

z2

zmax

ZenerWiechert z1 = 10, z2 = 103

Box (q = 0) [z1,z2]

+ Wedge (q = 0.5) [z1,z2]

Fig. 2.2. Plots of the normalized relaxation (top), storage (middle) and loss (bottom) moduli for Zener, Wiechert, Box and Wedge spectrum(q = 0.5) models.

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The loss modulus has a peak at zmax = (z1z2)1/2 and if z2 is

greater than z1 by two ore more decades, k≤(w) will exhibita plateau region between the two frequencies (Fig. 2.2). Thebox spectrum is certainly unrealistic. It is only a tool formodelling very slow evolution of the loss modulus betweenmany decades of frequencies.

Note that two or several functions may be used in con-junction to model the entire relaxation spectrum. Each modelwould contribute in an additive manner to the response functions.

Figure 2.2 shows viscoelastic moduli of the Zener andWiechert models in comparison to the wedge spectrummodel for q = 0 and q = 0.5. All of the four viscoelasticmodels have the same overall viscoelastic strength c = 2. The relaxation frequency of the Zener model is set to z0 = (z1z2)

1/2.It should be noted that the case (z1,z2) = (0,+•), reduces

to the power law relaxation. Namely,

(2.3.11)

Equation (2.3.11) turns out to be the relaxation modulus of the Kelvin fractional calculus model, obtained from theKelvin model by replacing the pure dashpot by an hybrid viscoelastic element, the spring-pot (Koeller, 1984). Therelaxance of a spring-pot is k(s) = k0(s/z0)

q, 0 £ q £ 1, in which q is an interpolation parameter between the purespring (q = 0) and the pure dashpot (q = 1). Just as for thepure dashpot, the spring-pot may be combined with puresprings in series-parallel mountings, in order to representvery slow decaying relaxation modulus with a small numberof parameters.

3. Vibrating systems

We consider next vibrating systems. We shall see how vis-coelastic properties of materials affect vibrating behaviour of viscoelastic bodies. According to the viscoelastic model,we shall refer to as Kelvin (respectively Zener, Wiechert,continuous spectrum, etc.) oscillators.

3.1 Single degree of freedom

Let us consider first the case of a little masse m linked to theground by a viscoelastic link, characterised by the relaxancek(s). The dynamic equilibrium equation, expressed in theLaplace transform plane, is

(3.1.1)

where x(s) and f(s) are the Laplace transforms of the displacement and the external force. Storage modulus of aviscoelastic system is intrinsically frequency dependent andvaries between two bounds, which are kg and ke. Conse-quently, two distinguish natural frequencies may be defined:

ms k s x s f s2 + ( )( ) ( ) = ( )

k t k kt

h e( ) = +( )

( )0

0

G q

z q

the rubbery we = (ke/m)1/2 and the glassy wg = (kg/m)1/2. To beconcordant with the corresponding definition for the Kelvinoscillator,4 we shall refer to wg as the natural frequency. Inother words

(3.1.2)

Free oscillations of the system are characterised by nontrivial solutions of the characteristic equation

(3.1.3)

where k(s) is the normalised relaxance of the viscoelasticmodel. Solving (3.1.3) for complex solutions s = -a ± iwyields the damping constant a and the damped frequency w. For a lightly damped oscillator, i.e., c ª 1, one mightassume that the damped frequency w is close to the naturalfrequency w0 and that a is small in comparison to w. Itfollows that k(a + iw) ª k*(w). By separating real and imaginary parts, one obtains then an approximation for thedamping constant a

(3.1.4)

3.1.1 Wiechert oscillator

Examination of the pole-zero location, shows that equation(3.1.3) has at least n negative solutions -ai and at most onesingle pair of conjugate complex roots: -a ± iw. It followsthat free vibration of an n-order Wiechert oscillator is a com-bination of n decaying exponentials exp(-ait) (overdampedcomponents) and at most a single oscillating solution exp(-at)exp(±iwt).

Complex poles -a ± iw are acoustically of prime impor-tance. For reasons that become apparent hereafter, we areparticularly interested in the relation between a and w0.Although (3.1.3) cannot be solved analytically, some quali-tative properties can be stated. In the rubbery region, i.e., w0 << z1, w2 ª we

2 = ke/m and

(3.1.5)

where z1 is the lowest relaxation frequency. The above rela-tion should be compared with the corresponding relation fora Kelvin oscillator, in which case a = w0

2/2z, for all w0.5

Beyond the highest peak zn, i.e., zn << w0, w ª w0 and aapproaches an upper bound amax, given by

a wwz0

1 02

12( ) ª

k

a ww

w00

02( ) ª

≤ ( )k

s k s202 0+ ( ) =w

w 02 =

k

mg

4 This choice is finally motivated by the fact that damping isexpected to lower the “natural” frequency w0.5 For a viscous oscillator, a = z/2m = w0

2t/2 = w02/2z (see also

§ 4.2.2).

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(3.1.6)

If the relaxation peaks are separated enough, it can be shownthat, in the transition region, i.e., z1 << w0 << zn, a grows bystages. Each stage is a plateau region corresponding to a par-ticular peak zi. The stationary value of a along each plateauis obtained by a formula similar to (3.1.6), in which the sum-mation is restricted to j = 1 . . . i. This plateau region is thenfollowed by a new quadratic growth region associated to thenext peak zi+1 (Fig. 3.1).

As mentioned above, free vibration of a Wiechert oscilla-tor is a linear superposition of a set of decaying exponentialsand a single exponentially damped sinusoid

(3.1.7)

Coefficients C and Cj’s are to be determined by initial con-ditions: initial position, initial speed, acceleration, . . . Thetime domain evolution of the oscillator is then an n + 2 orderconstant coefficient linear differential equation

(3.1.8)

3.1.2 Continuous spectrum oscillator

If the relaxation spectrum is bounded, some asymptotic fea-tures of the Wiechert oscillator remain unchanged. Forinstance, if H(z) = 0 for z > z2, in the glassy region (i.e., w0 >> z2), w ª w0 and the damping constant reaches a stationary value amax given by

(3.1.9)a z z zz

max = ( )Ú1

20

2

H d

a x b fin i

in i

i

n

i

n+ -( ) -( )

==

+

= ÂÂ 2

00

2

X t Ce C ei tj

tj( ) = +- ±( ) -Âa w a

a zmax ==

Â1

2 1

k j jj

n In the same way, if the spectrum vanishes below z1, in therubbery region (i.e., w0 << z1), the damping constant is pro-portional to the natural frequency squared:

(3.1.10)

and w 2 ª ke/m. In the case of a wedge spectrum bounded atboth sides, the foregoing integrals reduce to

(3.1.11)

(3.1.12)

Within the transition region [z1,z2], the plot of Log aversus Logw is then close to a straight line with slope 1 + q(Fig. 3.2). In particular, for a box spectrum oscillator, sub-stitution of (2.3.10) into (3.1.4) gives the approximateformula

(3.1.13)

The above relation is a good approximation of a if w0 > z1.Within the box, i.e., if z1 << w0 << z2, arctan(z2/w0) ª p/2.Hence, equation (3.1.13) yields then the linear relation

(3.1.14)

Moreover, the relaxation function k(s) has two poles at -z1

and -z2. Then, if ke > 0, the wedge spectrum oscillator has anegative pole -a1 in the interval [-z1,0], corresponding to anoverdamped component.

a wp

w z00

0 12 2( ) ª -Ê

Ëˆ¯

k

a w wzw

z00

02

012

( ) ª ÊË

ˆ¯ -Ê

Ëˆ¯

karctan

a w wz z

q

q q

0 02 0 2

11

1

2 1( ) ª

--

- -k

az z

q

q q

max =-+

+ +k 0 21

11

2 1

a ww z

zz

z0

02

221

( ) ª( )•

ÚH

d

FREQUENCY w0 (rad/s)

10-1

100

101

102

103

104

105

10-6

10-4

10-2

100

102

(D

AM

PIN

G C

OE

FF.

aw 0)

(s-1

)

Wiechert z1

, , k1, k

2Zener z

1, k

1 = 0.25

Zener z2, k

2 = 0.25

z2 z

1

z2

a(w0) ª k1

2z1

w02

k2

2z2

k1

2z1

Fig. 3.1. Damping constant versus frequency curves for Zener and Wiechert oscillators.

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Note that several boxes or wedges may be placed side byside. Just as for the Wiechert models, the resulting a versusw0 curve may be approximated by the superposition of curvescorresponding to each box.

3.2 Multiple degrees of freedom

Let us consider now a network of p little masses intercon-nected by viscoelastic links. The viscoelastic properties ofthe (i, j) interconnection is defined by the relaxance kij(s). Theequilibrium equation in the Laplace transform plane is

(3.2.1)

where F is the external force vector, [M] the diagonal matrixof masses and [K(s)] the symmetric transfer matrix of thesystem. The coefficients of [K(s)] are the relaxance functionskij(s). Multiple degrees of freedom systems will be referredto as MDOF oscillators.

3.2.1 Homogeneous system

Consider first the homogeneous case. All the viscoelasticlinks represent then the same material. To distinguish shapeindependent features, homogeneity is to be taken in a weaksense: all the links must have the same normalised relaxancek(s). In particular, all the links must have the same relaxationpeaks and the same corresponding normalised strengths.Therefore, the transfer matrix [K(s)] may be factored into[K(s)] = k(s)[K ], where [K ] is a symmetric real matrix. Clas-sical modal analysis may be applied to the underlying con-

M s K s X s F s[ ] + ( )[ ]( ) ( ) = ( )2

servative mass-spring system defined by [M] and [K]. Modalfrequency w0 and corresponding mode shape q are definedby the reduced wave equation

(3.2.2)

In the coordinate system yj, defined by the mode shapevectors (Djoharian, 1993), equation (3.2.2) is converted intoa set of n uncoupled scalar equations

(3.2.3)

For every mode, (3.2.3) is the equilibrium equation of asingle degree of freedom (SDOF) oscillator, called the modaloscillator. It follows that a homogeneous MDOF oscillatormay be decomposed into SDOF oscillators defined by(3.2.3). The mode shape vectors are identical to those of theunderlying conservative system. Damped frequencies anddamping constants of the j-th mode are obtained from thecharacteristic equation (3.1.3) in which w0 is replaced by w0j.

3.2.2 Non homogeneous system

Unlike the homogeneous case, for inhomogeneous systems,there is no natural way to define undamped frequencies. So,no shape or material invariant features can be expected.However, for completeness, we sketch here the modal analy-sis of non homogeneous Wiechert oscillators. The coeffi-cients of [K(s)] are rational functions of the Laplace variables. Multiplication of the two members of (3.2.1) by a suitablepolynomial D(s) leads then to an equation containing onlymatrix polynomials:

s k s y s f sj j j2

02+ ( )[ ] ( ) = ( )w

K q q[ ] = w 02

10 102

FREQUENCY w0 (rad /s )

DA

MPI

NG

CO

EFF

ICIE

NT

a(s

-1)

10-2

1

10

102

10-1

103 104

Zener, z0q = 0 (box)q = 0.25q = 0.75

z1 z2z0

Fig. 3.2. Frequency dependence of the damping constant of wedge spectrum oscillators: z1 = 10, z2 = 1000 (rad/s), c = 2.

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3.3 Sound signature of a material

For a homogeneous/isotropic resonator, the underlying con-servative skeleton determines the natural frequencies w0. Allthe material information is contained in the normalised relax-ance k(s). Solving the characteristic equations (3.1.3) forcomplex frequencies leads to an implicit damping constantversus frequency function a(w0). On the assumption that thematerial viscoelastic properties are independent of shape andsize, the resulting implicit function a(w0) will be shape andsize independent too. This particular function is what werefer to as the sound signature of the material.

Being a microscopic scale phenomenon, viscoelasticity isnot much affected by the macroscopic shape of the materialsample. However, it should be noted that some relaxationphenomena are size dependent. For instance, thermoelasticrelaxation involving heat transfer between extended andcompressed regions, is size dependent. In fact, the corre-sponding relaxation peak depends then on the sample width:

(3.3.1)

where D is the coefficient of thermal diffusion and d thelength of the heat path (Lakes, 1999).

Qualitative features of the a(w0) function of isotropicmaterials are outlined below:

• a(w0) is increasing,• In the rubbery region a(w0) µ w0

2,• If the relaxation spectrum is bounded then a(w0) has an

asymptotic upper bound amax.

Moreover, in the case of a well separated line spectrum, thea(w0) curve passes through the transition region by a seriesof plateau and quadratic growth regions.

4. Physical modelling

As stated above, shape and material modelling may be organ-ised in two independent steps: shape modelling by designinga spring-mass skeleton, and material modelling by replacingthe pure springs by viscoelastic links.

4.1 Shape modelling

Shape models are created within the spring-mass model paradigm.

4.1.1 Spatial design

A conservative resonator is characterized by physical datasuch as density and elasticity as well as geometrical data: thedimensionality (line, surface, etc.) and the boundary condi-tions (fixed, free, etc.). All the shape information is containedin the differential operator L, its spatial region of definition,together with the corresponding boundary conditions.

z ªD

d 2

(3.2.4)

where [A(s)] = [M]s2D(s) + [C(s)]. The roots of D(s) are infact the poles of kij(s), namely the relaxation peaks of thewhole set of edges. Now, the trial function, X(t) = exp(lt)qis a free vibration solution of (3.2.1) if l and q are solutionsof the generalised eigenvalue problem

(3.2.5)

The eigenvalues l are then the roots of the characteristicequation A(l) = det[A(l)] = 0. The (n + 2)p roots of A(l) are either negative, l = -ai, or they occur in conjugatecomplex pairs s = -a ± iw. Each complex pair adds up to a single real solution. Moreover, if the matrix polyno-mial [A(s)] is of simple structure (Lancaster, 1985) then allfree vibration solutions of (3.2.4) may be expressed as a sumof elementary solutions qexp(lt), where q and l are definedby (3.2.5). Otherwise, the matrix polynomial is said to bedefective and solutions involving polynomials in t mayappear.

A sufficient condition for a system to be of simple struc-ture is that all eigenvalues l be distinct. This is indeed thecase in most real situations. Roots with multiplicity ordergreater than one appear in systems having a mathematicallyperfect symmetry. But, in real life, perfect symmetry isunlikely.

3.2.3 Continuous systems

Vibrating behaviour of the spatially continuous system isquite similar to that of the finite degrees of freedom case.Here, matrices and vectors have to be replaced by differen-tial operators and functions. For conciseness, we considerhere only the case of transverse vibrations of a one or twodimensional medium. Let u(x,t) be the displacement of apoint x at time t and u(x,s) its time domain Laplace trans-form. If the material is isotropic (i.e., material properties areinvariant under rotation), the dynamic equilibrium equationfor zero external force can be written as

(3.2.6)

where m is the density of the material, k(s) the normalisedrelaxance and L a (self-adjoint) linear differential operatorcontaining derivatives with respect to space coordinates (cf. § 2.1.2). A normal mode is a standing wave u(x,t) =q(x)exp(-at)exp(±iwt), satisfying (3.2.6). By separating timeand space functions, it follows that

(3.2.7)

which means that l and q(x) are a pair of (mass-normalised)eigenvalue and eigenfunction of L. The eigenfunctions are then identical to those of the underlying conservativesystem. Thus, complex frequencies s = -a ± iw are solutionsof the dispersion equation (3.1.3), where w0 is a solution of(3.2.7).

Lq x x q x( ) = - ( ) ( )w m02

ms u x s k s Lu x s2 , ,( ) = ( ) ( )

A ql( )[ ] = 0

A s X s D s F s( )[ ] ( ) = ( ) ( )

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Equation (3.2.7) has to be approximated by a discretemodel, expressed in matrix form as

(4.1.1)

where [K ] and [M ] are the stiffness and the mass matricesof a spring-mass network.

The first step is to partition the continuous domain of theresonator, in small regions: intervals for one dimensional andrectilinear/curvilinear polygons for 2-dimensional regions(Fig. 4.1 and 4.2). The main methods for deriving the [K ]and [M ] matrices are the finite difference (FDM) and thefinite element (FEM) methods.

We sketch here the finite difference method using meshregions (Varga, 2000). A one dimensional flexible resonator(string, thin bar or air column) involves L = ∂2/∂x2. At eachnode O, with left and right neighbours W and E, the meshregion RO is defined as the interval bounded by the middleof each mesh. A little mass mO equal to the mass of RO isassociated to the node O. To each edge is associated a springwith a stiffness constant proportional to the inverse of theedge length (Fig. 4.1).

For membranes, usual mesh shapes are triangular or rec-tangular polygons. The mesh region RO is now defined as thepolygon bounded by the perpendicular bisectors of the edgescontaining O (Fig. 4.2). Here again mO is the mass of the meshregion RO. To each edge OP is then associated a spring with astiffness constant proportional to the ratio WqWr/OP (see Fig.4.2). In the case of triangular meshes, this is equivalent to

(4.1.2)

The above method may be applied to any surface and any(rectilinear or curvilinear) meshes. Note that a rectangular

k k Q R

mQ

OP

OP

O

= +( )

=

ÏÌÔ

ÓÔ Âcot cot

cotm

82

K q M q[ ] = [ ]w 02

mesh is equivalent to two triangular meshes in which thehypotenuse spring vanishes (cot Q = cot R = cot p/2 = 0).

Finite element methods use continuous domains, but theapproximate solutions are found in form of piecewise poly-nomial functions, characterised by their values at controlnodes (Reddy, 1993). For linear elements, FEM producesstiffness matrix identical to the FDM above.

In the same way, similar spring-mass networks approxi-mating bars and plates, involving flexural stiffness may beobtained by both methods.

4.1.2 Topological Design

Some geometrical constructions may be extended to abstractoperations on spring-mass networks. The most prominentexample is the product of two networks (Djoharian, 1993).The basic idea is the separation of variables in equation(3.2.7) (Courant, 1953). It can be applied if the variables areseparated in the expression of L and in the boundary condi-tions as well. Then, eigenfunctions may be found in form ofa product, i.e., un,p(x,y) = fn(x)gp(y). The corresponding modalfrequency satisfies the hypotenuse rule

w 2n,p = w 2

n,x + w 2y,p (4.1.3)

where wn,x and wy,p are modal frequencies derived from theseparated equations. This is obviously the case for the rec-tangular and cylindrical membranes. In matrix form, theproduct operation is expressed with the help of the Kroneckertensor product of matrices (Lancaster, 1985)

hW hE

kOW = k/hW kOE = k/hE

mO = m(hW + hE)/ 2

W EO

Fig. 4.1. Approximation of a continuous flexible string by a mass-spring network.

OP

mO = mRO

Q

R

Wq

Wr

ROkOP = k

WqWr

OP

Fig. 4.2. Spring-mass network resulting from finite differenceapproximation over triangular meshes.

(a) (b)

(c) (d)

f2 / f1 = 1.90

f2 / f1 = 1.99

f2 / f1 = 1.58

f2 / f1 = 1.72

Fig. 4.3. Approximation of planar (a, b) and non planar drums (c, d). In each case, the reduced frequency and the mode shape ofthe second mode are displayed.

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(4.1.4)

The product operation may be defined for any pair of net-works, regardless of any geometrical interpretation. It maybe generalised in several ways. For example, the L-product(Djoharian, 1993) in which a liaison pattern scatters throughthe product. Strictly speaking, if L is a liaison between twocopies of a network Y, represented by an interconnectionmatrix [L], the L-product of X and Y is obtained by substi-tuting a copy of Y to each node of X and by replacing everyspring (of X ) with stiffness constant k with the liaison k[L].

The product is twisted if in the substitution process, some liaisons are subjected to a permutation x leaving Yunchanged (Fig. 4.4).

These abstract topological constructions have an acousti-cal pertinence. Indeed, in a large variety of cases, modal fre-quencies of the product may be obtained by an explicitformula. For instance, if the liaison L is orthogonal, i.e.,[Qy]

T[L][Qy] is a diagonal matrix diag{l1, l2, . . .}, and if thesum of the stiffness constants at any node of X and L are con-stants sx and sL, then the spectrum of the product is obtainedfrom the spectra of X, Y and L by a generalised hypotenuserule (Djoharian, 1993):

(4.1.5)

Twisted products are significant in the case of a periodic(looped) multiplier network. Elasticity matrix of a periodicnetwork X is a constant diagonal matrix (Lancaster, 1985).Thus, X may be prolonged in 2X, having twice more masses.As x leaves Y invariant, the mode shapes qy of Y are (or maybe chosen) symmetric or anti-symmetric, i.e., [x]qy = qy or[x ]qy = -qy. Then symmetric and anti-symmetric modes of Ycombine with even and odd numbered modes of 2X respec-tively, according to (4.1.3). Twisted products may be used tomodel exotic surfaces such as the Möbius strip or the Kleinbottle (Fig. 4.5).

w w w s sn p y p p x n x L pl l, , ,2 2 2= + + -( )

M M M

K M K K Mx y

x y x y

[ ] = [ ] ƒ [ ][ ] = [ ] ƒ [ ]+ [ ] ƒ [ ]

ÏÌÓ

It should be noted that the mode shapes of a periodicnetwork (i.e., with circulant stiffness matrix) are the columnsof the Fourier matrix. So the modal frequencies are obtainedas the discrete Fourier transform of the network stiffness constants. This can be used to generate, via inverse DFT, periodic networks having a given prescribed spectrum.

4.2 Material modeling

As stated in § 3.3, shape invariant material properties appearthrough the a(w0) function. Strictly, this is valid only forisotropic materials. However, in some extent, the productoperation can be used to model fibered materials.

4.2.1 Isotropic materials

The sound signature of an isotropic material is an a versusw0 function. According to the sound synthesis algorithm, thematerial can be modelled in the following ways:

A. Definition of the material viscoelastic model, namely therelaxation spectrum or any other viscoelastic function.

X

Y Product Twisted Product

L-Product Twisted L-Product

[L] =0 0 1

1 0 00 1 0

[L] =0 1 1

1 1 01 0 1

L

L

[x] =1 0 0

0 0 10 1 0

[x][L] =011

01

1

0

11

L

xL

[x][L] =1 0 0

0 0 10 1 0

Fig. 4.4. Examples of products: direct and twisted.

(a) (b)

(c) (d)

Fig. 4.5. Products: (a) cylinder = loop ¥ chapelet, (b) torus = loop¥ loop; twisted versions: (c) Möbius strip, (d) Klein Bottle.

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B. Explicit definition of the sound signature in the form ofan a(w0) function.

The A method is the full description of the viscoelasticbehaviour. All vibrational properties of the body of materialare so modelled. The B method ignores the negative roots of(3.1.3). However, as far as free oscillations are concerned, theoverdamped components do not have any acoustic relevance.Nevertheless, in modelling interactions between two soundbodies (e.g., in percussion instruments), overdamped com-ponents must be taken into account (cf. § 5.2).

Direct definition of the a(w0) function is well suited formodal synthesis algorithms, where the physical model isdesigned by direct access to spectral parameters (cf. § 5.2.2).One may then choose an a(w0) curve in a quite fancifulmanner, as long as the qualitative features stated in § 3.3 aresatisfied, e.g., a(w0) µ w0

2 in low frequencies and a(w0) ªamax, at high frequencies.

4.2.2 Proportional viscosity

The transfer matrix [K(s)] of a Kelvin MDOF oscillator maybe decomposed into [K(s)] = [K] + [Z]s, where [K ] and [Z ]are known as the stiffness and viscosity matrices. The homo-geneity property [K(s)] = k(s)[K], introduced in § 3.2.1, isequivalent to k(s) = 1 + ts and [Z] = t[K], where t is thecharacteristic time of all the Kelvin models. The characteris-tic equation (3.1.3) reduces then to the classical second orderequation

(4.2.1)

Solving for complex roots -a ± iw, yields the following frequency-damping relationship

(4.2.2)

Equation (4.2.2) complies with the qualitative features statedin § 3.1, only in the rubbery region, or in the region preced-ing an isolated relaxation peak z = t -1 (compare with 3.1.5).Hence, using proportional viscosity for the entire frequencyaxis is not physically founded. However, in some cases, rela-tion (4.2.2) can represent the frequency-damping relationshipwithin a limited region of frequency (Doutaut et al., 1998).Its main merit is definitely its great simplicity, especially thatthe corresponding time domain equation is a second orderordinary differential equation, without input derivatives (see § 5.2).

Proportional viscosity is often used in conjunction withexternal viscosity. Let us assume that all the nodes are con-nected to the ground by dashpots having viscosity constantsproportional to masses, i.e., [Z ] = 2a0[M ] + t[K ]. Then, itis easily seen that this adds a constant term to the formera(w0) function

(4.2.3)a w at

w0 0 02

2( ) = +

a wt

w0 02

2( ) =

s s202 1 0+ +( ) =w t

Representing the external damping by a single constant para-meter 2a0, can be considered as a first approximation tomodel air viscosity. It can be used in conjunction to any inter-nal damping model. The characteristic equation (3.1.3) isthen converted into

(4.2.4)

On the assumption of light damping, approximation (3.1.4)combined with (4.2.4) yields

(4.2.5)

This shows that adding an external viscosity is a means totranslate the signature curve along the a axis. Another wayto extend the proportional viscosity is to use a general poly-nomial relation between [Z ] and [K ]1/2:

(4.2.6)

This leads to a polynomial a(w0) function

(4.2.7)

This generalised relation can be used as a trick to generateany a(w0) signature by means of Kelvin models. In fact, bypolynomial curve fitting, one can approximate, in the audiorange, any arbitrary material signature by a polynomial of asuitable order. Therefore, the viscosity matrix [Z ] can beobtained from [M ] and [K ] by (4.2.6). It should be pointedthat, even in the case of a homogeneous conservative skele-ton, the resulting system would not be a homogeneous Kelvinoscillator.

4.2.3 Fibered materials

For anisotropic materials, no shape invariant acoustic featurecan be found. However, for a fibered material, the a(w0) func-tion can be obtained from the sound signatures of each of thecomponents. The idea is to use the product operation intro-duced in § 4.1.2. It is easily seen that if both factors arehomogeneous/isotropic networks, then the characteristicequation of the (n,p) mode is

(4.2.8)

where kx and ky are the normalised relaxance of the components. Solving this equation for complex solutions -a ± iw, does not define a single valued a(w0) function. In fact, in a square membrane, wn,p = wp,n but if kx π ky thena(wn, p) π a(wp,n) (Fig. 4.6). In other words, the a versus w0

relation is shape dependent.It should be noted that problems involving anisotropic

plates are not of the “product” type. In fact, plate problemsinvolve at least two elastic/viscoelastic parameters, e.g., the Young’s modulus and the Poisson ratio of the material(Fletcher and Rossing, 1991).

s k s k sn x x y p y2 2 2 0+ ( ) + ( ) =w w, ,

a w w w w0 0 1 0 2 02

0( ) = + + + +a a a app. . .

Z a M a K a Kp

p[ ] = [ ]+ [ ] + + [ ]0 1

1 2 2. . .

a w a ww

0 0 00

2( ) ª +

≤( )k

s s k s20 0

22 0+ + ( ) =a w

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5. Sound synthesis

Numerical simulation of viscoelastic oscillators can be per-formed in time or in frequency domains. Lumped models aregoverned by constant coefficient differential equations. Thisenables us to use numerical simulations based on differencemethods. For continuous spectrum oscillators, frequencydomain simulation is the suitable choice.

5.1 Single degree of freedom

The special feature of a viscoelastic system is the high order n + 2 of the time domain differential equation (3.1.8)and the occurrence of the input derivatives in the right handmember. By standard state variable techniques, equation(3.1.8) may be reduced to a first order equation (Takahashiet al., 1972),

(5.1.1)

X A X B F

Y C X D F

= [ ] + [ ]= [ ] + [ ]

ÏÌÓ

where X is the state variable, containing the displacementsand their n + 1 derivatives. All the system poles are simpleand real, except a single pair of complex conjugate poles,corresponding to the oscillating component. In the canonicalmodal realisation, the state transition matrix [A] is diagonal,with the real poles -ai and the complex pair -a ± iw as di-agonal coefficients. The [B] matrix is a column vector con-taining ones, [C ] is the row vector containing the residuesand [D] = 0. Complex arithmetic can be avoided by splittingthe conjugate pair -a ± iw into a 2 ¥ 2 diagonal block. Forinstance, for a Zener oscillator, [A] may take the followingform

(5.1.2)

Simulation of the state equation (5.1.1) can be performed by standard finite difference techniques, e.g., Eulerforward/backward, Runge-Kutta, . . . (Hildebrand, 1968). Itshould be pointed out that for stability criterion, the samplingfrequency Fs must be larger than the highest relaxation fre-quency zmax. Moreover, for high damping materials, negativepoles -ai may have very large magnitudes, making the dif-ference system stiff. Implicit methods such as Euler back-ward or the trapezoidal method are then the suitable choices.

An alternate method is to use digital filters. The analogfilter defined by (3.1.8) can be converted into a digital IIRfilter, by digital filter techniques such as impulse invariantdesign or zero order hold (Oppenheim and Schafer, 1989).

5.2 Multiple degrees of freedom

As for the SDOF case, state variable techniques may be used.However, repeated poles may appear if the viscoelasticnetwork involves a mathematically perfect symmetry.However, if the system is of simple structure, canonical real-isation of the state space model in modal form is still pos-sible. Nevertheless, for a non homogeneous MDOF system(i.e., large p and large n), state space simulation is a heavytask. An alternative method is to replace the viscoelasticsystem by a suitable Kelvin oscillator as suggested in § 4.2.2.

5.2.1 Second order systems

According to § 4.2.2, as far as only free oscillations are con-cerned, the negative poles may be ignored. Oscillating com-ponents can be generated by a standard mass-spring-dashpotmodel, involving only Kelvin viscoelastic models, i.e., vis-cous damping. In order to represent a particular material, theviscosity matrix [Z] has to be chosen by (4.2.6). Differentialequation governing oscillation of a Kelvin oscillator is

(5.2.1)

Various finite difference schemes as well as digital filter tech-niques may be used. The CORDIS-ANIMA system uses a

M X Z X K X F[ ] + [ ] + [ ] =˙ ˙

A[ ] =-

- --

È

Î

ÍÍÍ

˘

˚

˙˙˙

aa w

w a

1 0 0

0

0

wn,xwy,p

a(w

n,x,w

y,p)

Isotropic

101

102

103

104

105

101

102

103

104

105

10-2

10-1

100

101

102

wy,p wn,x

a(w

n,x,w

y,p)

Anisotropic 10

1

102

103

104

105

101

102

103

104

105

100

101

102

Fig. 4.6. Damping constant a versus (wn, x,wy, p) functions forisotropic (top) and anisotropic fibered material (bottom).

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two step finite difference scheme with backward derivativeand centred acceleration (Florens and Cadoz, 1991).

Figure 5.1 shows the difference of transient responses pro-duced by the Zener and its equivalent Kelvin SDOF oscilla-tor. Both systems have the same complex pole (a = 10s-1 andw = 10Hz). The negative pole of the Zener oscillator is setto a1 = 20s-1. The input force is a square pulse of 0.5 sec. In the top figure both oscillators are isolated while in thebottom, they are faced to an elastic wall.

5.2.2 Modal synthesis

Modal synthesis enable us to simulate a linear oscillatordefined in terms of its spectral features. In particular, directcontrol of damping parameters may be used for assigning dif-ferent material signatures a(w0), over a conservative model.Classical modal synthesis (Adrien, 1991) is based on Kelvinmodels. Since it neglects overdamped components and inputderivatives, it is equivalent to the preceding approach. Eachmode can be simulated by a two order finite differencescheme or by digital filter techniques.

For homogeneous MDOF systems, a special modal syn-thesis can be considered. According to § 3.2.1, a homoge-neous viscoelastic oscillator can be decomposed in highorder viscoelastic real modes. The synthesis algorithm is then

similar to the classical modal synthesis. However, here, thetime domain equation of each mode is a high order differ-ential equation with input derivatives

(5.2.2)

Each modal equation can be simulated by the same tech-niques as before, i.e., finite differences or IIR digital filters.

6. Conclusion

A general framework for introducing materials in physicalmodeling sound synthesis was presented. It was shown thatthe acoustic invariant of an isotropic material has to be foundin some specific damping constant versus frequency functiona(w0). Thus, a physical model may be designed in two steps:1) modelling the geometrical data by a mass-spring model,2) covering the conservative skeleton by a viscoelastic dressin order to represent a particular material.

References

Adrien, J.M. (1991). The missing link: Modal synthesis. In: G.De Poli, A. Picialli, & C. Roads (Eds.), Representation ofMusical Signals. Cambridge, Massachusetts: MIT Press.

a y a y a y b f b fjn

jn

n j jn

n j02

11

2 0+( ) +( )

+( )+ + + = + +. . . . . .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

-2

0

2

4

6

8

10x 10

-3 Free Oscillation (K = 0)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1

-0.5

0

0.5

1

1.5

2x 10

-3 Knocking against an elastic wall (K = 105)

TIME (sec)

Elastic obstacle

DIS

PLA

CE

ME

NT

(A

rbitr

ary

Uni

ts)

DIS

PLA

CE

ME

NT

(A

rbitr

ary

Uni

ts)

K K

Kelvin

Zener

Dead Zone Dead Zone

TIME (sec)

Fig. 5.1. Response to a square pulse: free oscillations (top), knocking against a wall (bottom).

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Barr, S. (1964). Experiments in Topology. New York: Harper &Row.

Courant, R. & Hilbert, D. (1953). Methods of MathematicalPhysics (Vol. I). New York: Interscience-Publishers.

Djoharian, P. (1993). Generating model for modal synthesis.Computer Music Journal, 17, 57–65.

Doutaut, V., Matignon, D., & Chaigne, A. (1998). Numericalsimulation of xylophones. II. Time-domain modeling of the resonator and of the radiated pressure, Journal of theAcoustical Society of America, 104, 1633–1647.

Fletcher, N.H. & Rossing, T.D. (1991). The Physics of MusicalInstruments. New York: Springer-Verlag.

Florens, J.L. & Cadoz, C. (1991). The physical model: Model-ing and simulating the instrumental universe. In: G. De Poli,A. Picialli, & C. Roads (Eds.), Representation of MusicalSignals. Cambridge, Massachusetts: MIT Press.

Hildebrand, F.B. (1968). Finite-Difference Equations and Simu-lations. Englewood Cliffs, New Jersey: Prentice-Hall.

Koeller, R.C. (1984). Applications of Fractional Calculus to the

Theory of Viscoelasticity. Journal of Applied Mechanics, 51,299–307.

Lakatos, S., McAdams, S., & Caussé, R. (1997). The represen-tation of auditory source characteristics: simple geometricform. Perception & Psychophysics, 59, 1180–1190.

Lakes, R.S. (1999). Viscoelastic Solids. Boca Raton, Florida:CRC Press.

Lancaster, P. (1985). Theory of Matrices (2nd ed.). New York:Academic Press.

Oppenheim, A.V. & Schafer, R.W. (1989). Discrete-time SignalProcessing. Englewood Cliffs, New Jersey: Prentice-Hall.

Reddy, J.N. (1993). An Introduction to Finite Element Method.Mc-Graw Hill.

Takahashi, Y., Rabins, M.J., & Auslander, D.M. (1972). Controland Dynamic Systems. Reading, Mass.: Addison-Wesley.

Tschoegl, N.W. (1989). The Phenomenological Theory of LinearViscoelastic Behavior. Berlin: Springer-Verlag.

Varga, R.S. (2000). Matrix Iterative Analysis. Berlin: Springer-Verlag.

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material design in physical modelling sound synthesis

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