Physical synthesis of six-string guitar plucks using the ...

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Physical synthesis of six-string guitar plucks using theUdwadia-Kalaba modal formulation

V Debut J Antunes

To cite this versionV Debut J Antunes Physical synthesis of six-string guitar plucks using the Udwadia-Kalaba modalformulation Journal of the Acoustical Society of America Acoustical Society of America 2020 148(2) pp575-587 101121100001635 hal-03604430

Physical synthesis of six-string guitar plucks usingthe Udwadia-Kalaba modal formulation

V Debut1 2 a) and J Antunes1 2 b1INET-md Departamento de Ciencias Musicais Faculdade de Ciencias Sociais e Humanas FCSH

Universidade Nova de Lisboa 1069-061 Lisbon Portugal2Centro de Ciencias e Tecnologias Nucleares Instituto Superior Tecnico Universidade de Lisboa

Estrada Nacional 10 Km 1397 2695-066 Bobadela LRS Portugal

(Dated May 14 2020)

Extending previous work by the authors this paper develops a time domain synthesismethod for classical guitar based on substructuring concepts and using the Udwadia-Kalaba(U-K) modeling strategy Adopting a modal description of the dynamics of the separateflexible subsystems in terms of their unconstrained modes and enforcing coupling constraintconditions for the assembly the result is an explicit dynamical modal formulation for thecoupled system that directly lends itself to time-stepping methods for simulation Theguitar model couples six strings through a body model of an actual instrument includestwo string polarizations and the string geometrical nonlinear effects as well as for thestringfret interaction as the instrument is played Details are given for putting all thevibrating components together in a satisfying manner and a specific strategy is exploredto allow for a non-rigid fret using flexible-dissipative-inertial constraints The reliability ofthe approach is demonstrated with simulation examples that confirm the features one wouldexpect regarding the dynamical behaviour of classical guitars Finally a pragmatic approachis made to calculate the radiated sound by convolution combining the computed bridgeforce with a measured vibro-acoustic impulse response of the instrument which proved togive satisfactory sounding results

ccopy2020 Acoustical Society of America [httpdxdoiorg(DOI number)]

[XYZ] Pages 1ndash13

I INTRODUCTION1

The sound produced by musical instruments is the2

result of complex vibratory events usually involving3

multiple interaction between structural components and4

the player Beyond describing the excitation process the5

geometry and the dynamical behavior of the separate6

components any physical model of musical instrument7

also requires to develop dedicated methods that enforce8

contact conditions between the various subsystems to9

make them vibrate consistently In practice this is a10

rather complex issue that involves considerable modeling11

and computational difficulties and which can be critical12

in view of computational efficiency and accuracy of13

simulation Among the methods developed in vibration14

analysis to assemble structural components we can15

distinguish predominantly two standard techniques each16

one with specific merits and drawbacks Penalty17

methods which approximate the contact condition in18

terms of an inter-penetration force are often easy to19

implement but may lead to large computational expense20

when impacting near-rigid obstacles since the penalty21

parameters then impose small computational time scales22

a)vincentdebutfcshunlptb)jantunesctntecnicoulisboapt

andor the need for iterative numerical schemes On the23

other hand Lagrange multipliers allow constraints to be24

applied in a mathematically exact manner but demand25

the introduction of extra variables thus increasing the26

order of the system of equations127

Recently the present authors23 demonstrated that28

reliable simulations of flexible systems can be carried29

out by formulating their dynamics using the Udwadia30

and Kalaba4 equations which were originally proposed31

for modeling constrained discrete dynamical systems32

The extension was achieved using a modal approach33

for the continuum in terms of the unconstrained34

modes of the components and the resulting modal U-K35

formulation was found successful for systems involving36

point-constraints either linear or intermittent thus37

enabling the dynamical computations of many systems38

of practical interest Regarding the very modeling39

challenging systems that are musical instruments it40

seems that the modal U-K approach could be used41

profitably for physics-based sound synthesis purposes42

The physics of guitar plucks has been covered by43

many authors notably Valette5 and Woodhouse6 and44

models of musical strings have now reached a fine45

level of detail including frequency-dependent losses46

flexural dispersion and nonlinear features For obtaining47

the actual solutions different formulations have been48

proposed for solving the partial differential equations49

J Acoust Soc Am May 14 2020 Guitar synthesis using Udwadia-Kalaba modal formulation 1

numerically using wave propagation and reflection50

functions7 finite difference8 or the modal approach9 and51

also deriving energy preserving schemes10 to guarantee52

stability of the numerical solutions Modal methods53

- which will be employed here - have formed the54

basis of successful modeling strategies both for linear55

and nonlinear string vibration and with or without56

constraints11ndash16 with the advantages of being simple to57

implement and of dealing in an exact manner with some58

complex physical phenomena ie string damping and59

dispersion Despite the good deal of work on string60

modeling most of the models of pluck instruments in the61

literature remain highly simplified cases of the underlying62

physics and rarely address the entire problem taking63

account the full set of strings that drives the instrument64

body and that in turn produces propagating pressure65

waves In this view the modeling work by Bader1766

that employs finite difference by Mansour et al18 that67

uses a wave-based approach and by Debut et al14 that68

makes use of a modal representation certainly pertain to69

the small group of studies that deal with most complete70

models of actual multi-stringed instruments71

Following the work by Antunes and Debut2 the task72

of this paper is to show how flexible and versatile the73

modal U-K approach can be for the physically based74

sound synthesis of stringed instruments by exploring75

the issues of assembling the structural components in76

a realistic manner and illustrating the ability of the77

technique to achieve reliable simulations While the78

technique was presented for a crude model of guitar this79

paper refines and extends our previous work in several80

respects First the model to be built here involves all the81

major components of a classical guitar that technically82

raises the issue of multiple-point coupling and makes the83

implementation more complex It includes six strings84

vibrating in both polarizations and coupled together at85

the bridge by the body motions one or more pressing86

fingers on the fingerboard to control the frequency of the87

played notes as well as other relevant features of vibrating88

string namely wave dispersion frequency-dependent89

damping and nonlinear geometrical effects The issue90

of coupling the subsystems is dealt in regard to the U-91

K formalism by means of additional constraining terms92

in the dynamical equation and hence contrasts with the93

more traditional approach that uses penalty formulation94

to model contact in musical instruments111219ndash21 In95

order to model realistic finger interaction a specific96

strategy is also developed and explored to account for97

the possible stiffness damping and inertial properties98

of the constraints thus bypassing the non-deformable99

nature of common kinematic constraints Finally to100

achieve the end goal of synthesis the resulting sound101

is computed using a simple and pragmatic approach102

Based on convolution it uses the computed bridge force103

and a measured impulse response that characterizes the104

vibro-acoustic behavior of the guitar body and hence105

gets around the problem of producing satisfactory sounds106

without performing complicated numerical calculation of107

sound radiation108

The outline of this paper is as follows The U-K109

theory and its extension to deal with flexible systems110

is briefly described Then we give details of the guitar111

model and develop the dynamical equations for the112

separate subsystems and their coupling The final section113

is devoted to present numerical results obtained by time-114

domain simulations with intention of illustrating the115

dynamical behavior of the fully coupled model in view116

of the physics of classical guitars117

II THEORETICAL FORMULATION118

A Basic U-K formulation119

In general mathematical terms Udwadia and Kalaba120

proposed the following standard form for the study of the121

dynamics of constrained discrete mechanical systems4122

Mx = F(x x t) + Fc(x x t) (1)

which expresses the system response x as the result123

of the application of the constraint-independent forces124

F (external and internal including the stiffness and125

damping force terms) acting on it and some additional126

forces Fc stemming from a set of constraints which127

both are considered as known functions of x x and t128

The U-K formulation then benefits from an alternative129

expression of the usual constraint equations ψi(x x t) =130

0 (i = 1 m) obtained by differentiation with respect131

to time written as a matrix-vector constraint system in132

terms of accelerations as133

A(x x t) x(t) = b(x x t) (2)

where A is the constraint matrix and b is a known vectorTheir main result is then to provide an explicit expressionfor the constrained dynamics x(t) and the constraintforce vector Fc(t) at each instant given by

x = xu + Mminus12B+(bminusAxu) (3a)

Fc(t) = M12B+(bminusAxu) (3b)

denoting B+ the Moore-Penrose pseudo-inverse of B =134

AMminus12 The vector xu represents the dynamical135

response of the unconstrainted system solution of136

xu = Mminus1F(x x t) (4)

while the second term in the right-side hand of Eq137

(3a) accounts for the influence of the constraints on138

the system since the unconstrained system must further139

comply with the physical constraints The superlative140

elegance of the U-K formulation lies in the fact that141

it expresses the dynamics of the constrained system142

through a single dynamical equation (3a) that takes143

constraints into account and allows if needed the144

knowledge of the constraining force through (3b) No145

additional variables such as Lagrange multipliers are146

needed thus avoiding difficulties in the computation147

of the solutions as for Differential-Algebraic system of148

equations Notably Eqs (3) may be applied to linear149

2 J Acoust Soc Am May 14 2020 Guitar synthesis using Udwadia-Kalaba modal formulation

or nonlinear conservative or dissipative systems and150

may also be efficiently solved using a suitable time-step151

integration scheme for a given excitation152

B Modal U-K formulation153

Extension to constrained continuous multibody154

systems defined through their unconstrained modes Φ155

was formulated in Ref 2 by transforming (3) to the156

modal space through x = Φqqq leading to the formulation157

158

qqq = qqqu +MMMminus12BBB+(bminusAAAqqqu) (5)

where qqqu are the modal accelerations of the159

unconstrained configuration MMM = ΦTMΦ is the160

modal mass matrix AAA = AΦc is the modal constraint161

matrix with Φc the matrix of all modeshapes where162

constraints are defined and BBB = AAAMMMminus12 Finally the163

modal constraining force FFF c can also be computed as164

FFF c = MMM12BBB+(bminusAAAqqqu) (6)

C Dynamically coupled subsystems165

The application of the U-K modal formulation to a166

set of subsystems coupled through a number of kinematic167

constraints is now considered In a modal description the168

dynamics of the subsystem ` (` = 1 2 L) subjected169

to an external force field can be classically written as a170

set of modal equations in terms of the vector of modal171

amplitudes qqq` and its derivatives172

MMM `qqq` +CCC`qqq` +KKK`qqq` +FFF `nl(qqq` qqq`) = FFF èxt +FFF `nl (7)

where MMM ` CCC` and KKK` are diagonal matrices of the173

modal parameters m`n c`n and k`n (n = 1 N `) defined174

according to the modeshapes φ`n and FFF èxt and FFF `nl are175

the modal forces obtained by projection of the external176

and nonlinear forces on the modal basis The modal177

accelerations of the unconstrained system to be used in178

Eq (5) are given by179

qqqù =(MMM `)minus1

FFF ` (8)

denoting FFF ` the vector of all the constraint-independent180

modal forces181

FFF `= FFF èxtminusCCC`qqq`minusKKK`qqq`minusFFF `nl(qqq` qqq`) (9)

for which it is assumed that the vectors of modal182

constrained displacements and velocities are known at183

each time-step Stacking the modal quantities of the184

unconstrained subsystems in compact vectors as185

qqqu =[qqq1uT qqqLuT

]T(10)

the unconstrained modal accelerations of the coupled186

system read finally as187

qqqu = MMMminus1

[FFFext minusCCC qqq minusKKKqqq minus FFFnl(qqq qqq)

](11)

denoting188

qqq =[qqq1T qqqLT

]T(12)

In view of Eq (10) the matrices MMM CCC and KKK areblock diagonal set up by the submatrices of the modalparameters of the subsystems assembled as

MMM = diag(MMM1 MMML) (13a)

CCC = diag(CCC1 CCCL) (13b)

KKK = diag(KKK1 KKKL) (13c)

and FFFext and FFFnl are modal vectors associated to theexternal and nonlinear interaction forces acting on thevarious subsystems written respectively as

FFFext =FFF 1

extT FFFLextTT

(14a)

FFFnl =FFF 1

nlT FFFLnlTT

(14b)

The second set of equations to be considered concerns the189

P coupling constraints at locations r`c In most practical190

situations these are amenable to linear relationships by191

appropriate differentiation with respect to time leading192

to the standard form193

AAA(qqq qqq t) qqq = b(qqq qqq t) (15)

where AAA = AΦc with Φc = diag(Φ1c Φ

Lc ) and where194

Φ`c contains the modeshape vectors of each subsystem195

at the constraint location r`c Notice that because the196

constraints are implemented at the acceleration level197

Eq (15) generally leads to error accumulation in the198

computed motion during numerical integration due to199

numerical approximations and round-off errors resulting200

in constraint drift phenomenon To compensate for these201

errors a number of constraint stabilization techniques202

have been proposed22 and can be implemented if203

necessary as recently done by the authors for systems204

with intermittent contacts3 However for the problem205

at hand there was no necessity for implementing such206

correction terms Indeed by monitoring the constraints207

as the numerical simulation proceeds we observed a208

constraint drift under 20times10minus13 m which is negligible209

III GUITAR MODEL210

The guitar model involves six strings interacting with211

a guitar body and includes further models for the pressing212

finger on the fingerboard The strings are modeled213

using a simplified nonlinear formulation considering both214

polarizations and the body behavior is expressed in terms215

of modes identified from bridge input measurements on216

a real-life guitar217

A Nonlinear string model218

The vibratory motion of the strings is modeled219

using the KirchhoffndashCarrier approach2324 which is a220

simplified manner to describe large amplitude string221

motions Essentially the model discards the dynamics222


of the longitudinal modes but retains the effect of the223

geometrical nonlinearity by means of a pseudo-external224

forcing term added to the string transverse equations225

In our context it has two advantages First the model226

is known to capture relevant nonlinear phenomena for227

musical strings1425 thus enabling the reproduction of228

typical audible nonlinear effects Second it serves as a229

convenient means of illustrating that the U-K framework230

can deal effectively with nonlinear terms231

1 Kirchhoff-Carrier simplified nonlinear string model232

We consider a stiff string of length L cross-sectionalarea S mass density ρ Youngrsquos modulus E momentof inertia I and coefficient of dissipation η subjectto an axial tension T0 vibrating in two perpendiculartransverse motions Y (x t) and Z(x t) According to theKirchhoff-Carrier assumptions a simplified model of thefree vibration of the string is given by

ρSpart2Y

partt2+ η

partY

partt+ EI

part4Y

partx4=[T0 + Tdyn(t)

]part2Ypartx2

(16a)

ρSpart2Z

partt2+ η

partZ

partt+ EI

part4Z


]part2Zpartx2

(16b)

which is the standard wave equations for vibrating string233

including both polarizations with an additional forcing234

term proportional to the net dynamical increase in string235

length expressed as236

Tdyn(t) =ES

2L

int L

0

[(partY (x t)

partx

)2

+

(partZ(x t)

partx

)2]dx (17)

Eq (17) shows that the Kirchhoff-Carrier model only237

requires the knowledge of the transverse motions and238

that is independent of the x coordinate As far as the239

numerical technique is concerned the form of Eq (17)240

is very convenient particularly for modal synthesis and241

contrasts with the delicate numerical resolution of the242

ldquogeometrically exactrdquo model26 by modal discretization243

due the spatio-temporal nature of the nonlinear forcing244

2 Kirchhoff-Carrier nonlinear modal terms245

Denoting X a generic direction of string motion (X246

stands for Y or Z) the modal forces associated to the247

string nonlinear behavior are obtained by integration of248

the additional force terms of Eq (16) as249

fXn (t) =

int L

0

[Tdyn(t)

part2X

partx2

]φXn (x)dx (18)

Considering a string pinned at the nut and free at the250

bridge the modeshapes φXn (x) are given by251

φXn (x) = sin

[(2nminus 1)πx

2L

](19)

Substituting Eq (19) in Eq (18) and using a modalexpansion for the string motion the nonlinear modal

terms are finally given by

fXn (t)=minusESπ4

128L3(2nminus 1)2qXn (t)(

NYsumm=1

(2mminus1)2[qYm(t)

]2+

NZsumm=1

(2mminus1)2[qZm(t)

]2)(20)

with NY and NZ the sizes of the two string modal basis252

B The instrument body253

In view of Eq (7) the dynamic behaviour of the254

body is described by its modal properties extracted255

from input admittance measurements In contrast256

to modal computations based on a full model of the257

instrument body this is a direct approach that avoids258

complications of modeling in particular the frequency-259

dependent dissipative effects which significantly affect260

the computed sounds but are very difficult to model261

properly and that readily lend the synthesis method262

to be applied to any stringed instruments Since both263

polarizations of string motions are accounted a correct264

modeling of the stringbody coupling would demand265

the knowledge of both the in-plane and out-of-plane266

body modeshapes at the bridge However a simpler267

model is taken here by considering that only string268

motions normal to the soundboard couple to the body269

motions and also by neglecting the in-plane motions of270

the soundboard This means that the string motions in271

the plane parallel to the soundboard are unable to radiate272

sound and that the sound radiation is produced only273

by the normal motions of the soundboard Of course274

none of these choices is entirely satisfactory for accurate275

synthesis because in-plane and out-of-plane motions are276

coupled and also because in-plane motions couple the277

top and back plates via the ribs However since278

radiation is dominated by the out-of-plane vibration279

there is no doubt that in-plane motions are of second-280

order significance In practice the main advantage of281

our assumptions is to make the body characterization282

considerably simpler involving only transfer function283

measurements in the direction normal to the soundboard284

and avoiding the delicate measurements of the cross285

terms of the transfer function matrix286

The test guitar was a high-quality concert guitar287

built by Friederich number 694 During the tests it288

was positioned in the vertical position clamped to a289

rigid support by the neck and softly restrained in the290

lower bout with all the tuned strings damped (see291

Fig 1) The excitation and vibratory response were292

measured using a miniature force sensor (Kistler type293

9211) and a light-weight accelerometer (BampK 4375)294

respectively attached to the tie block using a thin layer295

of bee-wax Modal data were collected at two bridge296

locations close to the attachment points of the lowest297

and highest strings and a simple linear interpolation298

between the two transfer functions was performed to299

provide the unmeasured transfer functions at the other300

points where the strings make contact with the bridge301


Figure 1 Set-up for transfer function measurements at the

guitar body bridge

Modal identification was then achieved using the set of302

impulse responses expressed in terms of velocity and303

implementing the Eigensystem Realization Algorithm27304

The modal parameters for the first-ten identified modes305

are listed in Table I Fig 2 shows an example of306

an impulse response and its transfer function measured307

at the bridge end of the lowest string together with308

the corresponding reconstructed functions showing that309

the estimation was generally reliable up to 700 Hz310

Fig 3 also shows the identified mode shapes of the311

body at the bridge which exhibit typical motions of312

guitar modes28 The modes show a small degree of313

complexity but the effect of the imaginary components314

on the transfer function is small and can be neglected315

for our purpose Note that modes have been normalized316

according to max(|real(φBn (rs))|) = 1 (s = 1 6)317

Table I Identified modal parameters of the guitar body

Mode n fn (Hz) ζn () mn (Kg)

1 894 15 0402 1826 182 0083 2299 064 0204 3102 089 0475 4035 055 1066 5114 075 0587 5442 053 1828 6139 061 0699 6698 069 03810 8484 059 081

C Constraining forces and modal constraints318

1 Stringbody constraints319

Assuming a rigid transmission at the points wherethe strings make contact with the bridge the modalequations of the constrained system require to satisfya set of constrained modal equations stemming fromthe condition that at the bridge the strings and thebridge undergo the same motion Because the bodyresponse is here described only in the direction normal

0 005 01 015

Time [s]

-20

0

20

h (

t )

[ (

ms

)N

]

100 200 300 400 500 600 700 800 900

Frequency [ Hz ]

10-2

100

| H

( f

) |

[ (

ms

)N

]

Figure 2 Measured (green) and reconstructed (red) impulse

responses and transfer functions close to the lowest string

Gray dotted lines stand for the identified modal frequencies

E A D G B E

-1

0

1

1(

xB

)

f1 = 8939 Hz

E A D G B E

-1

0

1

2(

xB

)

f2 = 18262 Hz

E A D G B E

-1

0

1

3(

xB

)

f3 = 22986 Hz

E A D G B E

-1

0

1

4(

xB

)

f4 = 31021 Hz

E A D G B E

-1

0

1

5(

xB

)

f5 = 40352 Hz

E A D G B E

-1

0

1

6(

xB

)

f6 = 51143 Hz

E A D G B E

-1

0

1

7(

xB

)

f7 = 54417 Hz

E A D G B E

-1

0

1

8(

xB

)

f8 = 61386 Hz

E A D G B E

-1

0

1

9(

xB

)

f9 = 66978 Hz

Figure 3 Identified modeshapes of the guitar body at the

stringbridge contact points Real (blue) and imaginary

(magenta) components

to the soundboard the string motion Ys(xb t) in thisplane must be the same as the body motion at thestring contact point Yb(rs t) while its motion Zs(xb t)in the direction parallel to the soundboard must be nilFormally for each string s (s = 1 S) this results inthe conditions

Ys(xb t)minus Yb(rs t) = 0 (21a)

Zs(xb t) = 0 (21b)

or in terms of modal amplitudesΦYs

b

TqqqYs(t)minus

ΦBrsTqqqB(t) = 0 (22a)

ΦZs

b

TqqqZs(t) = 0 (22b)


where the modeshape vectors are written as

ΦYs

b =φYs1 (xb) φ

Ys

NYs(xb)

T(23a)

ΦZs

b =φZs1 (xb) φ

Zs

NZs(xb)

T(23b)

ΦBrs =φB1 (rs) φ

BNB

(rs)T

(23c)

Defining a vector of modal coordinates qqq built by stacking320

the individual modal displacements of the subsystems ie321

the strings including both polarizations and the body322

qqq = qqqY qqqZ qqqBT (24)

where qqqY and qqqZ are partitioned into terms associatedwith the different strings as

qqqY =qqqY1T qqqY6T

T(25a)

qqqZ =qqqZ1T qqqZ6T

T(25b)

and323

qqqB = qB1 qBNBT (26)

the constraint equations fit the matrix form324

AAAb qqq = 000 (27)

with the stringbody constraint matrix AAAb written as325

AAAb =

[ΦΦΦYb 000 minusΦΦΦB

000 ΦΦΦZb 000

](28)

where ΦΦΦYb and ΦΦΦZb are matrices of the modeshape vectors326

of the strings taken at the bridge location xb built as327

ΦΦΦYb =

ΦY1

b

T000 000

000

ΦY2

b

T 000

000 000

ΦY6

b

T

(29)

and similarly for the perpendicular direction and ΦΦΦB328

stands for the modeshapes of the body at the stringbody329

contact points rs330

ΦΦΦB =

ΦBr1T

ΦBr6T (30)

2 Stringfinger coupling at the fingerboard331

Modeling a stopping string for playing different332

notes can be thought in several ways according to the333

type of sound to simulate as well as to the degree of334

simplicitycomplexity aimed for the model The simplest335

model was taken in Ref 2 imposing a rigid kinematical336

constraint at a single location If such model could affect337

the tuning of the string effectively it is nonphysical338

in that part of the vibrational energy flows beyond the339

contact point (see Ref 2 for discussion) This simple340

approach is here extended to allow for multiple-point341

rigid coupling or for a non-rigid interaction as shown in342

Fig 4343

a Multiple-point rigid coupling This first optiondirectly enters into the realm of the standard kinematicalconstraints common in multibody theory The stringmotions Ys and Zs at the multiple finger locations xFf

(f = 1 F ) should be nil at all times so that

Ys(xFf t)=0 rArr

ΦYs

Ff

TqqqYs(t) = 0 (31a)

Zs(xFf t)=0 rArr

ΦZs

Ff

TqqqZs(t) = 0 (31b)

where the modeshape vectors at the finger locations are

ΦYs

Ff=φYs1 (xFf

) φYs

NYs(xFf

)T

(32a)

ΦZs

Ff=φZs1 (xFf

) φZs

NYs(xFf

)T

(32b)

Eq (31) can be rearranged in a matrix form as344

AAAf qqq = 000 (33)

where the modal constrained matrix is345

AAAf =

[ΦΦΦYs

F 000

000 ΦΦΦZs

F

](34)

with346

ΦΦΦYs

F =

ΦYs

F1

T

ΦYs

FF

T ΦΦΦZs

F =

ΦZs

F1

T

ΦZs

FF

T (35)

b Non-rigid coupling To extend the model to347

allow for non-rigid contact requires a little more care348

since a flexible-dissipative-inertial model for coupling349

subsystems cannot be directly formulated as kinematic350

constraints Nevertheless they can be implemented351

by assigning one or more additional flexible-dissipative-352

inertial elementary subsystems which are constrained to353

follow the string motions at the finger location(s) exactly354

as the body modes are included This modeling strategy355

somehow creates a bridge between computationally356

efficient multibody modeling providing ldquorigidrdquo kinematic357

constraints and the versatile but computationally slower358

modeling strategy through penalty-based constraints359

Non-rigid connections on the fingerboard cantherefore be obtained by extending the guitar modelusing a set of mass-spring-damper oscillators at locationsxFf

(f = 1 F ) and enforcing coupling as

Ys(xFf t)minusY Ff

f (t)= 0 (36a)

Zs(xFf t)minusZFf

f (t)= 0 (36b)

where YFf

f and ZFf

f are the responses of the fingercontact points in the two perpendicular directionsReturning to Eq (11) the vector and matrices mustnow be partitioned as

qqqu =qqqYsu T qqqZs

u T qqqFsu T

T(37a)

MMM = diag(MMMYsMMMZsMMMFs

)(37b)

CCC = diag(CCCYsCCCZsCCCFs

)(37c)

KKK = diag(KKKYsKKKZsKKKFs

)(37d)


where the vectors and matrices for the strings are as360

beforeMMMFs CCCFs andKKKFs are block diagonal matrices of361

the inertia damping and stiffness properties associated362

with the ldquofingerrdquo subsystem acting on string s including363

both directions of string motions364

qqqFsu =

Y F1

f Y FF

f ZF1

f ZFF

f

T(38)

In terms of modal coordinates Eqs (36) becomeΦYs

Ff

TqqqYs(t)minus Y Ff

f (t) = 0 (39a)ΦZs

Ff

TqqqZs(t)minus ZFf

f (t) = 0 (39b)

which can be rearranged as365

AAAf qqq = 000 (40)

with the modal constrained matrix AAAf written as366

AAAf =

[ΦΦΦYF 000 minusI 000

000 ΦΦΦZF 000 minusI

](41)

Figure 4 Rigid kinematical constraints (left) and flexible-

dissipative-inertial constraints (right) at multiple locations

D Dynamical formulation of the fully coupled model367

The formulation (11) readily applies to the full guitar368

model including the non-rigid constraint model for the369

stringfret interaction by stacking the unconstrained370

modal displacements into a vector371

qqqu = qqqYu qqqZu qqqBu qqqFu T (42)

where qqqYu qqqZu and qqqBu are as before and372

qqqFu=qqqY1

ufT qqqY6

ufT qqqZ1

ufT qqqZ6

ufTT

(43)

In view of the definition of qqq the assembled inertia MMM373

stiffnessKKK and dampingCCC matrices which are uncoupled374

are given by375

MMM=

MMMY 000 000 000

000 MMMZ 000 000

000 000 MMMB 000

000 000 000 MMMF

CCC=

CCCY 000 000 000

000 CCCZ 000 000

000 000 CCCB 000

000 000 000 CCCF

KKK=

KKKY 000 000 000

000 KKKZ 000 000

000 000 KKKB 000

000 000 000 KKKF

(44)

with MMM ` KKK` and CCC` the inertia stiffness and dampingmatrices for each individual subsystem ` and the vectorsFFFext and FFFnl write

FFFext =FFFYextT FFFZextT000000

T(45a)

FFFnl =FFFYnlT FFFZnlT000000

T(45b)

where FFFYext and FFFZext correspond to the plucking action376

of the player expressed for both planes of polarization377

and FFFYnl and FFFZnl are the modal vectors associated to the378

string nonlinear effects with entries given by Eq (20)379

The form of the modal constraint equation is now380

AAAqqq = 000 (46)

with381

AAA =

[AAAbAAAf

]=

ΦΦΦYb 000 minusΦΦΦB 000 000 000

000 ΦΦΦZb 000 000 000 000

ΦYFΦYFΦYF 000 000 000 minusFFFY 000

000 ΦZFΦZFΦZF 000 000 000 minusFFFZ

(47)

where ΦYFΦYFΦYF and ΦZFΦZFΦZF are block matrices with entries given382

by Eq (35) and FFFY and FFFZ are block diagonal matrices383

built with either the identity or null matrix depending on384

the constraints enforced on each string The form of (47)385

shows that the interface forces and constraint conditions386

couple the Y -motions of the strings and the body as well387

as coupling between the finger and the strings in each388

direction of motions Notice that coupling between both389

polarizations of string motions is not apparent in (47)390

although it is formulated in the nonlinear forces (45b)391

IV ILLUSTRATIVE COMPUTATIONS392

A System parameters393

For given initial conditions the system of equations394

comprising (5) with the unconstrained accelerations (11)395

and the constraint matrix (15) and using the specific396

matrices given by Eqs (44) and (47) can be integrated in397

time to produce the transient response The time-domain398

simulations were performed based on the identified body399

modal data and using standard string properties of400

classical guitars Assuming non-ideal strings with fixed-401

free ends the modeshapes are given by Eq (19) and402

their modal frequencies can be approximated by403

fYn = fZn =cT2πpn

(1 +

B

2T0p2n

)(48)

where cT is the velocity of transverse waves B is the404

bending stiffness of the string and T0 its tensioning405

pn = (2n minus 1)π2L and n the mode number Modal406

damping values are introduced following the pragmatic407

formulation given in Ref 6408

ζYn = ζZn =1

2

T0(ηF + ηAωn

)+ηBBp

2n

T0 +Bp2n(49)


where ηF ηA and ηB are the loss coefficients related to409

ldquointernal frictionrdquo ldquoair viscous dampingrdquo and ldquobending410

dampingrdquo The length of the strings is L=065 m411

and their density bending stiffness tensioning and loss412

coefficients are taken from Ref 29 The size of the modal413

basis of each string is defined by the maximum frequency414

of the computations (10000 Hz) which proved a sensible415

compromise for convergence to realistic simulations416

resulting in 800 modes for the set of strings To417

obtain suitable values for the elasticity parameter ES418

we perform simple vibratory tests on string samples419

Stretching a string vertically and suspending a mass to420

its free end one can easily measure the longitudinal421

vibration of the string-mass oscillator and obtain its422

natural frequency Assuming the string acting as a423

spring with no mass and reminding that the natural424

frequency of the longitudinal vibration is given by f =425

12π

radicESML then an estimate of ES is obtained The426

simulation results were obtained by time-step integration427

of the ordinary modal equations using the velocity-428

Verlet numerical scheme30 an integrator that combines429

simplicity second order accuracy with good numerical430

stability - see implementation details in Ref 2 We431

adopted a time-step of 10minus5s and assumed all initial432

modal displacements and velocities nil initially433

B Computational results434

Pluck excitations were simulated by considering a435

point-wise external force at xe=09L with a linear force436

ramp applied to one or more strings during 10 ms and437

by assuming null excitation force at subsequent instants438

1 Dynamical behaviour of the fully coupled model439

Results in Fig 5 were obtained for the open string440

E2 excited in the direction normal to the soundboard441

first considering a rigidly mounted string then coupling442

one string and finally all six strings to the instrument443

body Assuming no dynamics for the body the response444

spectra of string motion at the bridge is virtually nil445

according to the pinned boundary condition When446

coupling the instrument body to the string the response447

becomes obviously different and shows the classical series448

of near harmonic string partials together with smaller449

peaks from the body resonances While the first modal450

frequency of the unconstrained string is 41 Hz a close451

look at Fig 5 shows that the constraint at the bridge452

yields the fundamental frequency for the tuned string453

with pinned ends (f = 82 Hz) thus confirming the454

correct implementation of the bridge coupling Coupling455

the other five strings results in new peaks in the456

spectrum extending over the frequency range of the body457

modes and with an amplitude difference of about 100 dB458

compared to the most excited modes which clearly relate459

to the non-excited strings As found in some stringed460

instruments this dynamical coupling of all the strings via461

the bridge can lead to sympathetic vibrations which can462

further results in slight interference beats in the sound463

due to the inharmonicity produced by the bridge coupling464

and of the string partials465

0 200 400 600 800 1000 1200 1400 1600 180010

-40

10-20

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 180010

-20

10-10

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 1800

Frequency [ Hz ]

10-20

10-10

100

| Y

6(f

) |

Figure 5 Response spectra of string motion at the bridge

Top rigidly mounted string middle single stringbody

coupling bottom six-stringbody coupled model Force

amplitude 5 N Linear model

0 02 04 06 08 1

Time [ s ]

0

05

1

15

2

Mo

dal en

erg

y [

J ]

10-3

STRINGSFRETBODYTOTAL

Figure 6 Instantaneous modal energies Oblique excitation

with amplitudes of 2 N and 1 N in the Y and Z directions

Linear six-stringbody coupled model Conservative model

Figs 6 and 7 then serve at checking other466

implementation details Simulations pertained to the467

string E2 constrained at xF = 02509L by a single ldquostiffrdquo468

fret modeled with mF=5 10minus3 Kg cF=1 Nsm and469

kF=106 Nm plucked in the oblique direction The470

modal energies E` of the subsystems ` shown in these471

plots were computed from the results of the simulations472

as follows in terms of the modal parameters and the473


0 02 04 06 08 10

05

1

15

[ J ]

10-3 STRINGS

Y-dir

Z-dir

0 02 04 06 08 10

05

1

15

[ J ]

10-5 FRET

Y-dir

Z-dir

0 02 04 06 08 10

05

1

[ J ]

10-4 BODY

0 02 04 06 08 1

Time [ s ]

0

05

1

15

10-3 FULL SYSTEM

Figure 7 Instantaneous modal energies Same plucking

conditions as in Fig 6 Linear six-stringbody coupled model

Non-conservative model

modal displacements and velocities474

E`(t) =

Nsumn=1

[1

2m`n(ω`n)2

[q`n(t)

]2+

1

2m`n

[q`n(t)

]2](50)

One important feature seen in Fig 6 is that the total475

energy remains roughly constant over a conservative476

simulation while energies flow back and forth between477

the subsystems thus giving confidence in the numerical478

model for simulations of reasonable duration For479

the non-conservative case shown in Fig 7 the large480

difference in energy decay for the two polarizations481

confirms the different stringbody coupling with the482

directions of string motion Note that both the body483

and the fret oscillator are dominated by periodic energy484

exchanges at the frequency of the played note as the wave485

travels back and forth between bridge and fret Finally486

looking closely at the modal energy of the fret one can487

see smaller superimposed periodic disturbances related488

to the reflections between fret and nut illustrating that489

energy still flows through the modeled fret490

Finally we compare in Fig 8 results by varying491

the excitation amplitude in order to illustrate the string492

nonlinearity The simulations were obtained for the493

string E2 using the experimentally determined value494

of 32 kN for the flexibility ES which is comparable495

to the value given by Chaigne31 Fig 8 shows the496

temporal variations of string tension with an estimate497

of the playing frequency computed from time domain498

evaluations of the zero crossing frequency The general499

features of large amplitude string motions are clearly500

visible with a quasi-static increase of tension and501

superimposed oscillations at twice the string fundamental502

frequency and a small frequency variation of the played503

note which compares well with measurements32504

1 2 3 4 5 6 7

716

718

72

722

724

T (

t )

[ N

] Fexc

= 1 N

Fexc

= 5 N

1 2 3 4 5 6 7

Time [ s ]

822

824

826

828

f (

t )

[ H

z ]

Figure 8 String tension (top) and instantaneous frequency of

string motion (bottom) Nonlinear rigid body model

2 Parameter study of the stringfinger coupling model505

We now illustrate the different strategies to account506

for a stopping finger on the fingerboard and explore the507

influence on the coupled system of the finger model508

In contrast with the approach in Ref 2 the strategy509

in terms of auxiliary oscillators rigidly coupled to the510

string seems more realistic and is undoubtedly more511

versatile allowing different types of contacts to be512

modeled by varying the oscillator parameters However513

one could expect the finger dynamics to perturb the514

coupled system with changes in frequency and damping515

for its modes similar to what observed for a multi-modal516

system coupled to a system exhibiting resonance To517

give some insights Figs 9-11 show the effects of varying518

the finger mechanical parameters Simulations pertain to519

the string E2 constrained at xF = 01857L by a finger of520

mass mf = 5 10minus3 Kg and ignore the body dynamics521

The first test consists in varying the stiffness of the522

constraint assuming no damping Fig 9 shows the523

frequency spectra of the coupling force at the rigid bridge524

where the peaks of the string modes can be seen clearly as525

well as the resonance frequency of the auxiliary oscillator526

which moves in frequency with the stiffness and slightly527


modulates the other peaks A close look shows that528

the spectra are affected much more sensitively below529

the resonance of the finger oscillator with amplitude530

variations analogous to a system driven in the stiffness-531

controlled region Decreasing the stiffness causes a rise532

in amplitude of the undesirable modes while increasing533

the stiffness lower their amplitudes As observed for the534

stringbody coupling33 inharmonicity is also created by535

the stiffness termination which in turn affects the tuning536

of the note With the increase of stiffness perturbations537

of the string modes are reduced and tuning is improved538

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

4 Nm

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

5 Nm

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

kf = 10

6 Nm

Figure 9 Spectra of the coupling force at the rigid bridge

Stiffness is varied F=1 c1 = 0 Nsm

Fig 10 then shows the effect of varying the damping539

keeping the stiffness constant Not surprisingly if540

damping increases the amplitude of the coupled541

modes close to the oscillator resonance decrease and542

the frequency region affected becomes wider This543

significantly affects the time envelope of the pluck note544

that becomes smoother and has a shorter-time decay545

Finally the effect of the number of contacts is studied546

using the rigid contact model which is convenient for547

this purpose Fig 11 gives a picture of the spatial548

distribution of the string energy by plotting the standard549

deviation of string motion as a function of its length550

As seen the ldquopassiverdquo region is affected with great551

sensitivity by the number of contacts and surprisingly552

increasing the contact points does not necessarily reduce553

the undesirable transfer of energy Computations show554

that the effectiveness of the multi-point rigid constraints555

approach is strongly problem-dependent and varies in a556

non-obvious way with several factors including the finger557

width the spacing between contact points the relative558

string length from the ends and the excitation spectrum559

Taken in combination these results and noticing560

that the perturbations induced by each finger oscillator561

are additive due to the linear coupling a robust and562

physically sound strategy to simulate the pressing finger563

would be to implement a set of flexible-dissipative564

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

cf = 1 Nsm

cf = 10 Nsm

0 1 2 3 4 5

Time [ s ]

-2

0

2

4

Y (

t )

10-3

Figure 10 Spectra of the coupling force at the rigid

bridge (up) and time history of the string motion (bottom)

Damping is varied F=1 k1 = 105 Nm

constraints imposing a first stiff constraint in order565

to ensure a precise tuning followed by several soft566

constraints that would control the waveform and spectral567

content of the decaying response depending on their568

number and damping properties For illustration569

the spectrograms of two plucks for different sets of570

parameters are presented in Fig 12571

0 01 02 03 04 05 06

x [ m ]

10-5

10-2

ST

D (

Y(t

) )

F=1

F=2

F=3

F=5

016 017 018 019 02

x [ m ]

10-5

ST

D (

Y(t

) )

Figure 11 Standard deviation of the string motion Number

of contacts is varied Vertical dotted lines represent the

constraints locations Finger width (0015m) is kept constant

C Simulated acoustic radiation572

As an alternative to computing sounds from full573

vibro-acoustic models that usually require the use of574

massive computational resources it is tempting to apply575


Figure 12 Spectrograms of the force at the rigid bridge Up

F = 3 bottom F = 5 Parameter values kF1=106 Nm

cF1=1 Nsm and kFj =104 Nm cFj =10 Nsm (j = 2 F )

Constraint at xF = 01857L (sound files available)

simpler techniques that could be efficient to generate576

convincing sounds from the vibratory responses The577

method used here is to treat the guitar body as a linear578

input-output system and to compute the pressure by579

convolution benefiting from the local excitation of the580

body by the strings and using a vibro-acoustic impulse581

response of the test guitar This approach does not by582

any means simulate the sound response everywhere in the583

pressure field but it has the merit of adding the influence584

of the radiating properties of the body at a very low585

computational cost and of providing sounds of greater586

perceptual quality than the commonly used vibratory587

signals at the bridge The pressure at location rp can be588

computed by the convolution of the time-history of the589

stringbridge interaction force FY (rb t) and the vibro-590

acoustic impulse response function hY (rb rp t) between591

the bridge and microphone by592

p(rp t) = hY (rb rp t) lowast FY (rb t) (51)

The extension to account for the effects of all the593

strings is readily obtained by linear superposition594

principle summing the contributions from each string595

Measurements of the vibro-acoustic impulse response596

were performed under the same experimental conditions597

as for transfer function measurements using an598

additional condenser microphone Impulsive excitation599

was applied in the perpendicular direction to the600

soundboard and the acoustic pressure was measured601

at a distance of 1 m from the central axis of the602

guitar in a quiet room with all strings damped (see603

Fig 1) The impulse response was computed by inverse604

Fourier transform of the complex transfer function built605

from the force and response signals described in the606

frequency domain Results in Fig 13 pertain to the607

simulation of a G major arpeggio obtained with the608

two lowest and the top strings constrained using five609

auxiliary flexible-inertial constraints on each string The610

effect of the radiation properties of the body on the611

computed vibratory signals is supported when comparing612

the spectrograms in Fig 13 which clearly shows the613

spectral filtering of the vibration signal614

Figure 13 Spectrograms of the bridge force (top) and sound

pressure (bottom) Nonlinear six-stringbody coupled model

Arpeggio of a G major chord with three strings stopped using

a set of auxiliary oscillators (sound files available)

V CONCLUSIONS615

Following our recent work showing the potential616

of the modal Udwadia-Kalaba equations to physically617

simulate musical instruments we presented in this paper618

a computational approach that allows efficient time-619

domain simulations of realistic guitar plucks In this620

work we further generalized our simple stringbody621

coupled model toward the development of a more622

realistic guitar model by including the complete set623

of strings the two polarizations of string motions the624

string geometric nonlinearity the stringfret interaction625

that controls the playing frequency and the body626


dynamics of an actual instrument The dynamics of627

the various subsystems were formulated in terms of628

their unconstrained modes and coupling were achieved629

by enforcing several constraint conditions One of the630

developments of the model was the introduction of631

flexible-dissipative-inertial supplementary equations as632

an alternative to the common kinematical constraints633

which here improve the role of the stringfingerfret634

interaction at the fingerboard This approach can635

also benefit the modeling of other complex dynamical636

systems outside the realm of physical instrument637

modeling To improve the realism of the synthesis638

we also included crudely the radiation properties of639

the instrument body into the computational model640

through an ad hoc procedure Sound pressure was641

computed adopting a hybrid approach by convoluting642

the computed bridge force signals with a vibro-acoustic643

impulse response measured on the test guitar Such644

vibro-acoustic response can be measured in an anechoic645

chamber or else include the acoustics of any specific646

room This approach appeared not only straightforward647

to compute realistic sounds of string instruments for648

which the radiating components are subjected to point-649

wise excitation but also very efficient compared to heavy650

computational sound radiation techniques651

ACKNOWLEDGMENTS652

The authors thanks Paulo Vaz de Carvalho for653

lending the guitar and interesting discussions This work654

is funded by national funds through the FCT ndash Fundacao655

para a Ciencia e a Tecnologia IP under the Norma656

Transitoria - DL 572016CP1453CT0007657

REFERENCES658

1JH Ginsberg Mechanical and Structural Vibrations Theory659

and Applications chapter 9 Wiley 2001660

2J Antunes and V Debut Dynamical computation of constrained661

flexible systems using a modal Udwadia-Kalaba formulation662

Application to musical instruments J Acoust Soc Am663

141764 2017664

3J Antunes V Debut L Borsoi X Delaune and P Piteau665

A modal Udwadia-Kalaba formulation for the vibro-impact666

modelling of continuous flexible systems with intermittent667

contacts Procedia Engineering 199322ndash329 2017668

4FE Udwadia and RE Kalaba Analytical dynamics - A new669

approach Cambridge University Press 1996670

5C Valette The mechanics of the vibrating string Mechanics of671

musical instruments Springer 1995672

6J Woodhouse On the synthesis of guitar plucks Acta Acust673

united Ac 90928ndash944 2004674

7M E McIntyre RT Schumacher and J Woodhouse On the675

oscillations of musical instruments Journal of the Acoustical676

Society of America 741325ndash1345 1983677

8A Chaigne On the use of finite differences for musical678

synthesis Application to plucked stringed instruments Journal679

drsquoAcoustique 5181ndash211 1992680

9JM Adrien The missing link Modal synthesis In G DePoli681

A Picalli and C Roads editors Representations of musical682

sounds MIT Press Cambridge 1991683

10S Bilbao Numerical Sound Synthesis Finite Difference684

Schemes and Simulation in Musical Acoustics Wiley Publishing685

2009686

11L Trautmann and R Rabenstein Multirate simulations687

of string vibrations includingnonlinear fret-string interactions688

usingthe functional transformation method EURASIP Journal689

on Applied Signal Processing 7949ndash963 2004690

12V Chatziioannou and M van Walstijn Energy conserving691

schemes for the simulation of musical instrument contact692

dynamics J Sound Vib 339262ndash279 2015693

13S Bilbao and A Torin Numerical modeling and sound synthesis694

for articulated stringfretboard interactions Acta Acust united695

Ac 63336ndash347 2015696

14V Debut M Carvalho M Marques and J Antunes Physics-697

based modeling techniques of a twelve-string portuguese guitar698

a three-dimensional non-linear time-domain computational699

approach for the multiple-stringsbridgesoundboard coupled700

dynamics Appl Acoust 1083ndash18 2016701

15M van Walstijn and J Bridges Simulation of distributed contact702

in string instruments A modal expansion approach In 24th703

European Signal Processing Conference (Eusipco 2016) pages704

1023ndash1027 2013705

16C Issanchou S Bilbao JL Le Carrou C Touze and O Doare706

A modal-based approach to the nonlinear vibration of strings707

against a unilateral obstacle simulations and experiments in the708

pointwise case J Sound Vib 393229ndash251 2017709

17R Bader Computational Mechanics of the Classical Guitar710

Springer-Verlag Berlin Heidelberg 2005711

18H Mansour J Woodhouse and G Scavone Enhanced wave-712

based modelling of musical strings Part 1 Plucked strings Acta713

Acust united Ac 1021094ndash1107 2016714

19O Inacio J Antunes and MCM Wright Computational715

modeling of string-body interaction for the violin family and716

simulation of wolf notes J Sound Vib 310260ndash286 2007717

20S Bilbao A Torin and V Chatziioannou Numerical modeling718

of collisions in musical instruments Acta Acust united Ac719

101155ndash173 2015720

21C Issanchou JL Le Carrou C Touze B Fabre and O Doare721

Stringfrets contacts in the electric bass sound Simulations and722

experiments App Acoust 129217ndash228 2018723

22D J Braun and M Goldfarb Eliminating constraint drift in the724

numerical simulation of constrained dynamical systems Comput725

Methods in Appl Mech Eng 198(37)3151 ndash 3160 2009726

23GF Carrier On the nonlinear vibration problem of the elastic727

string Q Appl Math 3157ndash165 1945728

24G Kirchhoff Vorlesungen uber Mechanik Teubner 1883729

25J Pakarinen V Valimaki and M Karjalainen Physics-based730

methods for modeling nonlinear vibrating strings Acta Acust731


26PMC Morse and KU Ingard Theoretical Acoustics733

International series in pure and applied physics Princeton734

University Press 1968735

27J Juang Applied System Identification PTR Prentice-Hall736

Inc New Jersey 1994737

28B E Richardson The acoustical development of the guitar J738

Catgut Acoust Soc 51ndash10 1994739

29J Woodhouse Plucked guitar transients Comparison of740

measurements and synthesis Acta Acust united Ac 90945ndash741

965 2004742

30M Tuckerman B J Berne and G J Martyna Reversible743

multiple time scale molecular dynamics J Chem Phys744

97(3)1990ndash2001 1992745

31A Chaigne Viscoelastic properties of nylon guitar strings J746

Catgut Acoust Soc Series II 121ndash27 1991747

32T Tolonen V Valimaki and M Karjalainen Modeling of748

tension modulation nonlinearity in plucked strings IEEE Trans749

Speech Audio Process 8(3)300ndash310 2000750

33A Chaigne and J Kergomard Acoustics of Musical Instruments751

Modern Acoustics and Signal Processing Springer New York752

2016753


754


Physical synthesis of six-string guitar plucks usingthe Udwadia-Kalaba modal formulation

V Debut1 2 a) and J Antunes1 2 b1INET-md Departamento de Ciencias Musicais Faculdade de Ciencias Sociais e Humanas FCSH

Universidade Nova de Lisboa 1069-061 Lisbon Portugal2Centro de Ciencias e Tecnologias Nucleares Instituto Superior Tecnico Universidade de Lisboa

Estrada Nacional 10 Km 1397 2695-066 Bobadela LRS Portugal

(Dated May 14 2020)

Extending previous work by the authors this paper develops a time domain synthesismethod for classical guitar based on substructuring concepts and using the Udwadia-Kalaba(U-K) modeling strategy Adopting a modal description of the dynamics of the separateflexible subsystems in terms of their unconstrained modes and enforcing coupling constraintconditions for the assembly the result is an explicit dynamical modal formulation for thecoupled system that directly lends itself to time-stepping methods for simulation Theguitar model couples six strings through a body model of an actual instrument includestwo string polarizations and the string geometrical nonlinear effects as well as for thestringfret interaction as the instrument is played Details are given for putting all thevibrating components together in a satisfying manner and a specific strategy is exploredto allow for a non-rigid fret using flexible-dissipative-inertial constraints The reliability ofthe approach is demonstrated with simulation examples that confirm the features one wouldexpect regarding the dynamical behaviour of classical guitars Finally a pragmatic approachis made to calculate the radiated sound by convolution combining the computed bridgeforce with a measured vibro-acoustic impulse response of the instrument which proved togive satisfactory sounding results

ccopy2020 Acoustical Society of America [httpdxdoiorg(DOI number)]

[XYZ] Pages 1ndash13

I INTRODUCTION1

The sound produced by musical instruments is the2

result of complex vibratory events usually involving3

multiple interaction between structural components and4

the player Beyond describing the excitation process the5

geometry and the dynamical behavior of the separate6

components any physical model of musical instrument7

also requires to develop dedicated methods that enforce8

contact conditions between the various subsystems to9

make them vibrate consistently In practice this is a10

rather complex issue that involves considerable modeling11

and computational difficulties and which can be critical12

in view of computational efficiency and accuracy of13

simulation Among the methods developed in vibration14

analysis to assemble structural components we can15

distinguish predominantly two standard techniques each16

one with specific merits and drawbacks Penalty17

methods which approximate the contact condition in18

terms of an inter-penetration force are often easy to19

implement but may lead to large computational expense20

when impacting near-rigid obstacles since the penalty21

parameters then impose small computational time scales22

a)vincentdebutfcshunlptb)jantunesctntecnicoulisboapt

andor the need for iterative numerical schemes On the23

other hand Lagrange multipliers allow constraints to be24

applied in a mathematically exact manner but demand25

the introduction of extra variables thus increasing the26

order of the system of equations127

Recently the present authors23 demonstrated that28

reliable simulations of flexible systems can be carried29

out by formulating their dynamics using the Udwadia30

and Kalaba4 equations which were originally proposed31

for modeling constrained discrete dynamical systems32

The extension was achieved using a modal approach33

for the continuum in terms of the unconstrained34

modes of the components and the resulting modal U-K35

formulation was found successful for systems involving36

point-constraints either linear or intermittent thus37

enabling the dynamical computations of many systems38

of practical interest Regarding the very modeling39

challenging systems that are musical instruments it40

seems that the modal U-K approach could be used41

profitably for physics-based sound synthesis purposes42

The physics of guitar plucks has been covered by43

many authors notably Valette5 and Woodhouse6 and44

models of musical strings have now reached a fine45

level of detail including frequency-dependent losses46

flexural dispersion and nonlinear features For obtaining47

the actual solutions different formulations have been48

proposed for solving the partial differential equations49




























































sound radiation108















Mx = F(x x t) + Fc(x x t) (1)












A(x x t) x(t) = b(x x t) (2)






























158


























FFF ` (8)


modal forces181







]T(10)



qqqu = MMMminus1


](11)

denoting188

qqq =[qqq1T qqqLT

]T(12)






FFFext =FFF 1

extT FFFLextTT

(14a)

FFFnl =FFF 1

nlT FFFLnlTT

(14b)








Lc ) and where194
















III GUITAR MODEL210

























ρSpart2Y

partt2+ η

partY

partt+ EI

part4Y


]part2Ypartx2

(16a)

ρSpart2Z

partt2+ η

partZ

partt+ EI

part4Z


]part2Zpartx2

(16b)





Tdyn(t) =ES

2L

int L

0

[(partY (x t)

partx

)2

+

(partZ(x t)

partx

)2]dx (17)














fXn (t) =

int L

0

[Tdyn(t)

part2X

partx2

]φXn (x)dx (18)



φXn (x) = sin

[(2nminus 1)πx

2L

](19)



fXn (t)=minusESπ4


NYsumm=1

(2mminus1)2[qYm(t)

]2+

NZsumm=1

(2mminus1)2[qZm(t)

]2)(20)





















































guitar body bridge



















1 894 15 0402 1826 182 0083 2299 064 0204 3102 089 0475 4035 055 1066 5114 075 0587 5442 053 1828 6139 061 0699 6698 069 03810 8484 059 081




0 005 01 015

Time [s]

-20

0

20

h (

t )

[ (

ms

)N

]

100 200 300 400 500 600 700 800 900

Frequency [ Hz ]

10-2

100

| H

( f

) |

[ (

ms

)N

]




E A D G B E

-1

0

1

1(

xB

)

f1 = 8939 Hz

E A D G B E

-1

0

1

2(

xB

)

f2 = 18262 Hz

E A D G B E

-1

0

1

3(

xB

)

f3 = 22986 Hz

E A D G B E

-1

0

1

4(

xB

)

f4 = 31021 Hz

E A D G B E

-1

0

1

5(

xB

)

f5 = 40352 Hz

E A D G B E

-1

0

1

6(

xB

)

f6 = 51143 Hz

E A D G B E

-1

0

1

7(

xB

)

f7 = 54417 Hz

E A D G B E

-1

0

1

8(

xB

)

f8 = 61386 Hz

E A D G B E

-1

0

1

9(

xB

)

f9 = 66978 Hz






Zs(xb t) = 0 (21b)


b

TqqqYs(t)minus


ΦZs

b

TqqqZs(t) = 0 (22b)



ΦYs

b =φYs1 (xb) φ

Ys

NYs(xb)

T(23a)

ΦZs

b =φZs1 (xb) φ

Zs

NZs(xb)

T(23b)

ΦBrs =φB1 (rs) φ

BNB

(rs)T

(23c)






qqqY =qqqY1T qqqY6T

T(25a)

qqqZ =qqqZ1T qqqZ6T

T(25b)

and323



AAAb qqq = 000 (27)


AAAb =


000 ΦΦΦZb 000

](28)



ΦΦΦYb =

ΦY1

b

T000 000

000

ΦY2

b

T 000

000 000

ΦY6

b

T

(29)




ΦΦΦB =

ΦBr1T

ΦBr6T (30)













Fig 4343



Ys(xFf t)=0 rArr

ΦYs

Ff

TqqqYs(t) = 0 (31a)

Zs(xFf t)=0 rArr

ΦZs

Ff

TqqqZs(t) = 0 (31b)


ΦYs

Ff=φYs1 (xFf

) φYs

NYs(xFf

)T

(32a)

ΦZs

Ff=φZs1 (xFf

) φZs

NYs(xFf

)T

(32b)


AAAf qqq = 000 (33)


AAAf =

[ΦΦΦYs

F 000

000 ΦΦΦZs

F

](34)

with346

ΦΦΦYs

F =

ΦYs

F1

T

ΦYs

FF

T ΦΦΦZs

F =

ΦZs

F1

T

ΦZs

FF

T (35)
















Ys(xFf t)minusY Ff

f (t)= 0 (36a)

Zs(xFf t)minusZFf

f (t)= 0 (36b)

where YFf

f and ZFf



u T qqqFsu T

T(37a)


)(37b)


)(37c)


)(37d)







qqqFsu =

Y F1

f Y FF

f ZF1

f ZFF

f

T(38)


Ff

TqqqYs(t)minus Y Ff

f (t) = 0 (39a)ΦZs

Ff

TqqqZs(t)minus ZFf

f (t) = 0 (39b)


AAAf qqq = 000 (40)


AAAf =



](41)










qqqFu=qqqY1

ufT qqqY6

ufT qqqZ1

ufT qqqZ6

ufTT

(43)



are given by375

MMM=

MMMY 000 000 000

000 MMMZ 000 000

000 000 MMMB 000

000 000 000 MMMF

CCC=

CCCY 000 000 000

000 CCCZ 000 000

000 000 CCCB 000

000 000 000 CCCF

KKK=

KKKY 000 000 000

000 KKKZ 000 000

000 000 KKKB 000

000 000 000 KKKF

(44)



T(45a)


T(45b)






AAAqqq = 000 (46)

with381

AAA =

[AAAbAAAf

]=


000 ΦΦΦZb 000 000 000 000



(47)























fYn = fZn =cT2πpn

(1 +

B

2T0p2n

)(48)






ζYn = ζZn =1

2

T0(ηF + ηAωn

)+ηBBp

2n

T0 +Bp2n(49)



















12π









































0 200 400 600 800 1000 1200 1400 1600 180010

-40

10-20

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 180010

-20

10-10

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 1800

Frequency [ Hz ]

10-20

10-10

100

| Y

6(f

) |





0 02 04 06 08 1

Time [ s ]

0

05

1

15

2

Mo

dal en

erg

y [

J ]

10-3














0 02 04 06 08 10

05

1

15

[ J ]

10-3 STRINGS

Y-dir

Z-dir

0 02 04 06 08 10

05

1

15

[ J ]

10-5 FRET

Y-dir

Z-dir

0 02 04 06 08 10

05

1

[ J ]

10-4 BODY

0 02 04 06 08 1

Time [ s ]

0

05

1

15

10-3 FULL SYSTEM





E`(t) =

Nsumn=1

[1

2m`n(ω`n)2

[q`n(t)

]2+

1

2m`n

[q`n(t)

]2](50)































1 2 3 4 5 6 7

716

718

72

722

724

T (

t )

[ N

] Fexc

= 1 N

Fexc

= 5 N

1 2 3 4 5 6 7

Time [ s ]

822

824

826

828

f (

t )

[ H

z ]






































0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

4 Nm

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

5 Nm

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

kf = 10

6 Nm





























0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

cf = 1 Nsm

cf = 10 Nsm

0 1 2 3 4 5

Time [ s ]

-2

0

2

4

Y (

t )

10-3











0 01 02 03 04 05 06

x [ m ]

10-5

10-2

ST

D (

Y(t

) )

F=1

F=2

F=3

F=5

016 017 018 019 02

x [ m ]

10-5

ST

D (

Y(t

) )

























































V CONCLUSIONS615






































ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754




























































sound radiation108















Mx = F(x x t) + Fc(x x t) (1)












A(x x t) x(t) = b(x x t) (2)






























158


























FFF ` (8)


modal forces181







]T(10)



qqqu = MMMminus1


](11)

denoting188

qqq =[qqq1T qqqLT

]T(12)






FFFext =FFF 1

extT FFFLextTT

(14a)

FFFnl =FFF 1

nlT FFFLnlTT

(14b)








Lc ) and where194
















III GUITAR MODEL210

























ρSpart2Y

partt2+ η

partY

partt+ EI

part4Y


]part2Ypartx2

(16a)

ρSpart2Z

partt2+ η

partZ

partt+ EI

part4Z


]part2Zpartx2

(16b)





Tdyn(t) =ES

2L

int L

0

[(partY (x t)

partx

)2

+

(partZ(x t)

partx

)2]dx (17)














fXn (t) =

int L

0

[Tdyn(t)

part2X

partx2

]φXn (x)dx (18)



φXn (x) = sin

[(2nminus 1)πx

2L

](19)



fXn (t)=minusESπ4


NYsumm=1

(2mminus1)2[qYm(t)

]2+

NZsumm=1

(2mminus1)2[qZm(t)

]2)(20)





















































guitar body bridge



















1 894 15 0402 1826 182 0083 2299 064 0204 3102 089 0475 4035 055 1066 5114 075 0587 5442 053 1828 6139 061 0699 6698 069 03810 8484 059 081




0 005 01 015

Time [s]

-20

0

20

h (

t )

[ (

ms

)N

]

100 200 300 400 500 600 700 800 900

Frequency [ Hz ]

10-2

100

| H

( f

) |

[ (

ms

)N

]




E A D G B E

-1

0

1

1(

xB

)

f1 = 8939 Hz

E A D G B E

-1

0

1

2(

xB

)

f2 = 18262 Hz

E A D G B E

-1

0

1

3(

xB

)

f3 = 22986 Hz

E A D G B E

-1

0

1

4(

xB

)

f4 = 31021 Hz

E A D G B E

-1

0

1

5(

xB

)

f5 = 40352 Hz

E A D G B E

-1

0

1

6(

xB

)

f6 = 51143 Hz

E A D G B E

-1

0

1

7(

xB

)

f7 = 54417 Hz

E A D G B E

-1

0

1

8(

xB

)

f8 = 61386 Hz

E A D G B E

-1

0

1

9(

xB

)

f9 = 66978 Hz






Zs(xb t) = 0 (21b)


b

TqqqYs(t)minus


ΦZs

b

TqqqZs(t) = 0 (22b)



ΦYs

b =φYs1 (xb) φ

Ys

NYs(xb)

T(23a)

ΦZs

b =φZs1 (xb) φ

Zs

NZs(xb)

T(23b)

ΦBrs =φB1 (rs) φ

BNB

(rs)T

(23c)






qqqY =qqqY1T qqqY6T

T(25a)

qqqZ =qqqZ1T qqqZ6T

T(25b)

and323



AAAb qqq = 000 (27)


AAAb =


000 ΦΦΦZb 000

](28)



ΦΦΦYb =

ΦY1

b

T000 000

000

ΦY2

b

T 000

000 000

ΦY6

b

T

(29)




ΦΦΦB =

ΦBr1T

ΦBr6T (30)













Fig 4343



Ys(xFf t)=0 rArr

ΦYs

Ff

TqqqYs(t) = 0 (31a)

Zs(xFf t)=0 rArr

ΦZs

Ff

TqqqZs(t) = 0 (31b)


ΦYs

Ff=φYs1 (xFf

) φYs

NYs(xFf

)T

(32a)

ΦZs

Ff=φZs1 (xFf

) φZs

NYs(xFf

)T

(32b)


AAAf qqq = 000 (33)


AAAf =

[ΦΦΦYs

F 000

000 ΦΦΦZs

F

](34)

with346

ΦΦΦYs

F =

ΦYs

F1

T

ΦYs

FF

T ΦΦΦZs

F =

ΦZs

F1

T

ΦZs

FF

T (35)
















Ys(xFf t)minusY Ff

f (t)= 0 (36a)

Zs(xFf t)minusZFf

f (t)= 0 (36b)

where YFf

f and ZFf



u T qqqFsu T

T(37a)


)(37b)


)(37c)


)(37d)







qqqFsu =

Y F1

f Y FF

f ZF1

f ZFF

f

T(38)


Ff

TqqqYs(t)minus Y Ff

f (t) = 0 (39a)ΦZs

Ff

TqqqZs(t)minus ZFf

f (t) = 0 (39b)


AAAf qqq = 000 (40)


AAAf =



](41)










qqqFu=qqqY1

ufT qqqY6

ufT qqqZ1

ufT qqqZ6

ufTT

(43)



are given by375

MMM=

MMMY 000 000 000

000 MMMZ 000 000

000 000 MMMB 000

000 000 000 MMMF

CCC=

CCCY 000 000 000

000 CCCZ 000 000

000 000 CCCB 000

000 000 000 CCCF

KKK=

KKKY 000 000 000

000 KKKZ 000 000

000 000 KKKB 000

000 000 000 KKKF

(44)



T(45a)


T(45b)






AAAqqq = 000 (46)

with381

AAA =

[AAAbAAAf

]=


000 ΦΦΦZb 000 000 000 000



(47)























fYn = fZn =cT2πpn

(1 +

B

2T0p2n

)(48)






ζYn = ζZn =1

2

T0(ηF + ηAωn

)+ηBBp

2n

T0 +Bp2n(49)



















12π









































0 200 400 600 800 1000 1200 1400 1600 180010

-40

10-20

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 180010

-20

10-10

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 1800

Frequency [ Hz ]

10-20

10-10

100

| Y

6(f

) |





0 02 04 06 08 1

Time [ s ]

0

05

1

15

2

Mo

dal en

erg

y [

J ]

10-3














0 02 04 06 08 10

05

1

15

[ J ]

10-3 STRINGS

Y-dir

Z-dir

0 02 04 06 08 10

05

1

15

[ J ]

10-5 FRET

Y-dir

Z-dir

0 02 04 06 08 10

05

1

[ J ]

10-4 BODY

0 02 04 06 08 1

Time [ s ]

0

05

1

15

10-3 FULL SYSTEM





E`(t) =

Nsumn=1

[1

2m`n(ω`n)2

[q`n(t)

]2+

1

2m`n

[q`n(t)

]2](50)































1 2 3 4 5 6 7

716

718

72

722

724

T (

t )

[ N

] Fexc

= 1 N

Fexc

= 5 N

1 2 3 4 5 6 7

Time [ s ]

822

824

826

828

f (

t )

[ H

z ]






































0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

4 Nm

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

5 Nm

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

kf = 10

6 Nm





























0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

cf = 1 Nsm

cf = 10 Nsm

0 1 2 3 4 5

Time [ s ]

-2

0

2

4

Y (

t )

10-3











0 01 02 03 04 05 06

x [ m ]

10-5

10-2

ST

D (

Y(t

) )

F=1

F=2

F=3

F=5

016 017 018 019 02

x [ m ]

10-5

ST

D (

Y(t

) )

























































V CONCLUSIONS615






































ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754










158


























FFF ` (8)


modal forces181







]T(10)



qqqu = MMMminus1


](11)

denoting188

qqq =[qqq1T qqqLT

]T(12)






FFFext =FFF 1

extT FFFLextTT

(14a)

FFFnl =FFF 1

nlT FFFLnlTT

(14b)








Lc ) and where194
















III GUITAR MODEL210

























ρSpart2Y

partt2+ η

partY

partt+ EI

part4Y


]part2Ypartx2

(16a)

ρSpart2Z

partt2+ η

partZ

partt+ EI

part4Z


]part2Zpartx2

(16b)





Tdyn(t) =ES

2L

int L

0

[(partY (x t)

partx

)2

+

(partZ(x t)

partx

)2]dx (17)














fXn (t) =

int L

0

[Tdyn(t)

part2X

partx2

]φXn (x)dx (18)



φXn (x) = sin

[(2nminus 1)πx

2L

](19)



fXn (t)=minusESπ4


NYsumm=1

(2mminus1)2[qYm(t)

]2+

NZsumm=1

(2mminus1)2[qZm(t)

]2)(20)





















































guitar body bridge



















1 894 15 0402 1826 182 0083 2299 064 0204 3102 089 0475 4035 055 1066 5114 075 0587 5442 053 1828 6139 061 0699 6698 069 03810 8484 059 081




0 005 01 015

Time [s]

-20

0

20

h (

t )

[ (

ms

)N

]

100 200 300 400 500 600 700 800 900

Frequency [ Hz ]

10-2

100

| H

( f

) |

[ (

ms

)N

]




E A D G B E

-1

0

1

1(

xB

)

f1 = 8939 Hz

E A D G B E

-1

0

1

2(

xB

)

f2 = 18262 Hz

E A D G B E

-1

0

1

3(

xB

)

f3 = 22986 Hz

E A D G B E

-1

0

1

4(

xB

)

f4 = 31021 Hz

E A D G B E

-1

0

1

5(

xB

)

f5 = 40352 Hz

E A D G B E

-1

0

1

6(

xB

)

f6 = 51143 Hz

E A D G B E

-1

0

1

7(

xB

)

f7 = 54417 Hz

E A D G B E

-1

0

1

8(

xB

)

f8 = 61386 Hz

E A D G B E

-1

0

1

9(

xB

)

f9 = 66978 Hz






Zs(xb t) = 0 (21b)


b

TqqqYs(t)minus


ΦZs

b

TqqqZs(t) = 0 (22b)



ΦYs

b =φYs1 (xb) φ

Ys

NYs(xb)

T(23a)

ΦZs

b =φZs1 (xb) φ

Zs

NZs(xb)

T(23b)

ΦBrs =φB1 (rs) φ

BNB

(rs)T

(23c)






qqqY =qqqY1T qqqY6T

T(25a)

qqqZ =qqqZ1T qqqZ6T

T(25b)

and323



AAAb qqq = 000 (27)


AAAb =


000 ΦΦΦZb 000

](28)



ΦΦΦYb =

ΦY1

b

T000 000

000

ΦY2

b

T 000

000 000

ΦY6

b

T

(29)




ΦΦΦB =

ΦBr1T

ΦBr6T (30)













Fig 4343



Ys(xFf t)=0 rArr

ΦYs

Ff

TqqqYs(t) = 0 (31a)

Zs(xFf t)=0 rArr

ΦZs

Ff

TqqqZs(t) = 0 (31b)


ΦYs

Ff=φYs1 (xFf

) φYs

NYs(xFf

)T

(32a)

ΦZs

Ff=φZs1 (xFf

) φZs

NYs(xFf

)T

(32b)


AAAf qqq = 000 (33)


AAAf =

[ΦΦΦYs

F 000

000 ΦΦΦZs

F

](34)

with346

ΦΦΦYs

F =

ΦYs

F1

T

ΦYs

FF

T ΦΦΦZs

F =

ΦZs

F1

T

ΦZs

FF

T (35)
















Ys(xFf t)minusY Ff

f (t)= 0 (36a)

Zs(xFf t)minusZFf

f (t)= 0 (36b)

where YFf

f and ZFf



u T qqqFsu T

T(37a)


)(37b)


)(37c)


)(37d)







qqqFsu =

Y F1

f Y FF

f ZF1

f ZFF

f

T(38)


Ff

TqqqYs(t)minus Y Ff

f (t) = 0 (39a)ΦZs

Ff

TqqqZs(t)minus ZFf

f (t) = 0 (39b)


AAAf qqq = 000 (40)


AAAf =



](41)










qqqFu=qqqY1

ufT qqqY6

ufT qqqZ1

ufT qqqZ6

ufTT

(43)



are given by375

MMM=

MMMY 000 000 000

000 MMMZ 000 000

000 000 MMMB 000

000 000 000 MMMF

CCC=

CCCY 000 000 000

000 CCCZ 000 000

000 000 CCCB 000

000 000 000 CCCF

KKK=

KKKY 000 000 000

000 KKKZ 000 000

000 000 KKKB 000

000 000 000 KKKF

(44)



T(45a)


T(45b)






AAAqqq = 000 (46)

with381

AAA =

[AAAbAAAf

]=


000 ΦΦΦZb 000 000 000 000



(47)























fYn = fZn =cT2πpn

(1 +

B

2T0p2n

)(48)






ζYn = ζZn =1

2

T0(ηF + ηAωn

)+ηBBp

2n

T0 +Bp2n(49)



















12π









































0 200 400 600 800 1000 1200 1400 1600 180010

-40

10-20

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 180010

-20

10-10

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 1800

Frequency [ Hz ]

10-20

10-10

100

| Y

6(f

) |





0 02 04 06 08 1

Time [ s ]

0

05

1

15

2

Mo

dal en

erg

y [

J ]

10-3














0 02 04 06 08 10

05

1

15

[ J ]

10-3 STRINGS

Y-dir

Z-dir

0 02 04 06 08 10

05

1

15

[ J ]

10-5 FRET

Y-dir

Z-dir

0 02 04 06 08 10

05

1

[ J ]

10-4 BODY

0 02 04 06 08 1

Time [ s ]

0

05

1

15

10-3 FULL SYSTEM





E`(t) =

Nsumn=1

[1

2m`n(ω`n)2

[q`n(t)

]2+

1

2m`n

[q`n(t)

]2](50)































1 2 3 4 5 6 7

716

718

72

722

724

T (

t )

[ N

] Fexc

= 1 N

Fexc

= 5 N

1 2 3 4 5 6 7

Time [ s ]

822

824

826

828

f (

t )

[ H

z ]






































0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

4 Nm

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

5 Nm

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

kf = 10

6 Nm





























0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

cf = 1 Nsm

cf = 10 Nsm

0 1 2 3 4 5

Time [ s ]

-2

0

2

4

Y (

t )

10-3











0 01 02 03 04 05 06

x [ m ]

10-5

10-2

ST

D (

Y(t

) )

F=1

F=2

F=3

F=5

016 017 018 019 02

x [ m ]

10-5

ST

D (

Y(t

) )

























































V CONCLUSIONS615






































ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754













ρSpart2Y

partt2+ η

partY

partt+ EI

part4Y


]part2Ypartx2

(16a)

ρSpart2Z

partt2+ η

partZ

partt+ EI

part4Z


]part2Zpartx2

(16b)





Tdyn(t) =ES

2L

int L

0

[(partY (x t)

partx

)2

+

(partZ(x t)

partx

)2]dx (17)














fXn (t) =

int L

0

[Tdyn(t)

part2X

partx2

]φXn (x)dx (18)



φXn (x) = sin

[(2nminus 1)πx

2L

](19)



fXn (t)=minusESπ4


NYsumm=1

(2mminus1)2[qYm(t)

]2+

NZsumm=1

(2mminus1)2[qZm(t)

]2)(20)





















































guitar body bridge



















1 894 15 0402 1826 182 0083 2299 064 0204 3102 089 0475 4035 055 1066 5114 075 0587 5442 053 1828 6139 061 0699 6698 069 03810 8484 059 081




0 005 01 015

Time [s]

-20

0

20

h (

t )

[ (

ms

)N

]

100 200 300 400 500 600 700 800 900

Frequency [ Hz ]

10-2

100

| H

( f

) |

[ (

ms

)N

]




E A D G B E

-1

0

1

1(

xB

)

f1 = 8939 Hz

E A D G B E

-1

0

1

2(

xB

)

f2 = 18262 Hz

E A D G B E

-1

0

1

3(

xB

)

f3 = 22986 Hz

E A D G B E

-1

0

1

4(

xB

)

f4 = 31021 Hz

E A D G B E

-1

0

1

5(

xB

)

f5 = 40352 Hz

E A D G B E

-1

0

1

6(

xB

)

f6 = 51143 Hz

E A D G B E

-1

0

1

7(

xB

)

f7 = 54417 Hz

E A D G B E

-1

0

1

8(

xB

)

f8 = 61386 Hz

E A D G B E

-1

0

1

9(

xB

)

f9 = 66978 Hz






Zs(xb t) = 0 (21b)


b

TqqqYs(t)minus


ΦZs

b

TqqqZs(t) = 0 (22b)



ΦYs

b =φYs1 (xb) φ

Ys

NYs(xb)

T(23a)

ΦZs

b =φZs1 (xb) φ

Zs

NZs(xb)

T(23b)

ΦBrs =φB1 (rs) φ

BNB

(rs)T

(23c)






qqqY =qqqY1T qqqY6T

T(25a)

qqqZ =qqqZ1T qqqZ6T

T(25b)

and323



AAAb qqq = 000 (27)


AAAb =


000 ΦΦΦZb 000

](28)



ΦΦΦYb =

ΦY1

b

T000 000

000

ΦY2

b

T 000

000 000

ΦY6

b

T

(29)




ΦΦΦB =

ΦBr1T

ΦBr6T (30)













Fig 4343



Ys(xFf t)=0 rArr

ΦYs

Ff

TqqqYs(t) = 0 (31a)

Zs(xFf t)=0 rArr

ΦZs

Ff

TqqqZs(t) = 0 (31b)


ΦYs

Ff=φYs1 (xFf

) φYs

NYs(xFf

)T

(32a)

ΦZs

Ff=φZs1 (xFf

) φZs

NYs(xFf

)T

(32b)


AAAf qqq = 000 (33)


AAAf =

[ΦΦΦYs

F 000

000 ΦΦΦZs

F

](34)

with346

ΦΦΦYs

F =

ΦYs

F1

T

ΦYs

FF

T ΦΦΦZs

F =

ΦZs

F1

T

ΦZs

FF

T (35)
















Ys(xFf t)minusY Ff

f (t)= 0 (36a)

Zs(xFf t)minusZFf

f (t)= 0 (36b)

where YFf

f and ZFf



u T qqqFsu T

T(37a)


)(37b)


)(37c)


)(37d)







qqqFsu =

Y F1

f Y FF

f ZF1

f ZFF

f

T(38)


Ff

TqqqYs(t)minus Y Ff

f (t) = 0 (39a)ΦZs

Ff

TqqqZs(t)minus ZFf

f (t) = 0 (39b)


AAAf qqq = 000 (40)


AAAf =



](41)










qqqFu=qqqY1

ufT qqqY6

ufT qqqZ1

ufT qqqZ6

ufTT

(43)



are given by375

MMM=

MMMY 000 000 000

000 MMMZ 000 000

000 000 MMMB 000

000 000 000 MMMF

CCC=

CCCY 000 000 000

000 CCCZ 000 000

000 000 CCCB 000

000 000 000 CCCF

KKK=

KKKY 000 000 000

000 KKKZ 000 000

000 000 KKKB 000

000 000 000 KKKF

(44)



T(45a)


T(45b)






AAAqqq = 000 (46)

with381

AAA =

[AAAbAAAf

]=


000 ΦΦΦZb 000 000 000 000



(47)























fYn = fZn =cT2πpn

(1 +

B

2T0p2n

)(48)






ζYn = ζZn =1

2

T0(ηF + ηAωn

)+ηBBp

2n

T0 +Bp2n(49)



















12π









































0 200 400 600 800 1000 1200 1400 1600 180010

-40

10-20

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 180010

-20

10-10

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 1800

Frequency [ Hz ]

10-20

10-10

100

| Y

6(f

) |





0 02 04 06 08 1

Time [ s ]

0

05

1

15

2

Mo

dal en

erg

y [

J ]

10-3














0 02 04 06 08 10

05

1

15

[ J ]

10-3 STRINGS

Y-dir

Z-dir

0 02 04 06 08 10

05

1

15

[ J ]

10-5 FRET

Y-dir

Z-dir

0 02 04 06 08 10

05

1

[ J ]

10-4 BODY

0 02 04 06 08 1

Time [ s ]

0

05

1

15

10-3 FULL SYSTEM





E`(t) =

Nsumn=1

[1

2m`n(ω`n)2

[q`n(t)

]2+

1

2m`n

[q`n(t)

]2](50)































1 2 3 4 5 6 7

716

718

72

722

724

T (

t )

[ N

] Fexc

= 1 N

Fexc

= 5 N

1 2 3 4 5 6 7

Time [ s ]

822

824

826

828

f (

t )

[ H

z ]






































0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

4 Nm

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

5 Nm

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

kf = 10

6 Nm





























0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

cf = 1 Nsm

cf = 10 Nsm

0 1 2 3 4 5

Time [ s ]

-2

0

2

4

Y (

t )

10-3











0 01 02 03 04 05 06

x [ m ]

10-5

10-2

ST

D (

Y(t

) )

F=1

F=2

F=3

F=5

016 017 018 019 02

x [ m ]

10-5

ST

D (

Y(t

) )

























































V CONCLUSIONS615






































ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754



guitar body bridge



















1 894 15 0402 1826 182 0083 2299 064 0204 3102 089 0475 4035 055 1066 5114 075 0587 5442 053 1828 6139 061 0699 6698 069 03810 8484 059 081




0 005 01 015

Time [s]

-20

0

20

h (

t )

[ (

ms

)N

]

100 200 300 400 500 600 700 800 900

Frequency [ Hz ]

10-2

100

| H

( f

) |

[ (

ms

)N

]




E A D G B E

-1

0

1

1(

xB

)

f1 = 8939 Hz

E A D G B E

-1

0

1

2(

xB

)

f2 = 18262 Hz

E A D G B E

-1

0

1

3(

xB

)

f3 = 22986 Hz

E A D G B E

-1

0

1

4(

xB

)

f4 = 31021 Hz

E A D G B E

-1

0

1

5(

xB

)

f5 = 40352 Hz

E A D G B E

-1

0

1

6(

xB

)

f6 = 51143 Hz

E A D G B E

-1

0

1

7(

xB

)

f7 = 54417 Hz

E A D G B E

-1

0

1

8(

xB

)

f8 = 61386 Hz

E A D G B E

-1

0

1

9(

xB

)

f9 = 66978 Hz






Zs(xb t) = 0 (21b)


b

TqqqYs(t)minus


ΦZs

b

TqqqZs(t) = 0 (22b)



ΦYs

b =φYs1 (xb) φ

Ys

NYs(xb)

T(23a)

ΦZs

b =φZs1 (xb) φ

Zs

NZs(xb)

T(23b)

ΦBrs =φB1 (rs) φ

BNB

(rs)T

(23c)






qqqY =qqqY1T qqqY6T

T(25a)

qqqZ =qqqZ1T qqqZ6T

T(25b)

and323



AAAb qqq = 000 (27)


AAAb =


000 ΦΦΦZb 000

](28)



ΦΦΦYb =

ΦY1

b

T000 000

000

ΦY2

b

T 000

000 000

ΦY6

b

T

(29)




ΦΦΦB =

ΦBr1T

ΦBr6T (30)













Fig 4343



Ys(xFf t)=0 rArr

ΦYs

Ff

TqqqYs(t) = 0 (31a)

Zs(xFf t)=0 rArr

ΦZs

Ff

TqqqZs(t) = 0 (31b)


ΦYs

Ff=φYs1 (xFf

) φYs

NYs(xFf

)T

(32a)

ΦZs

Ff=φZs1 (xFf

) φZs

NYs(xFf

)T

(32b)


AAAf qqq = 000 (33)


AAAf =

[ΦΦΦYs

F 000

000 ΦΦΦZs

F

](34)

with346

ΦΦΦYs

F =

ΦYs

F1

T

ΦYs

FF

T ΦΦΦZs

F =

ΦZs

F1

T

ΦZs

FF

T (35)
















Ys(xFf t)minusY Ff

f (t)= 0 (36a)

Zs(xFf t)minusZFf

f (t)= 0 (36b)

where YFf

f and ZFf



u T qqqFsu T

T(37a)


)(37b)


)(37c)


)(37d)







qqqFsu =

Y F1

f Y FF

f ZF1

f ZFF

f

T(38)


Ff

TqqqYs(t)minus Y Ff

f (t) = 0 (39a)ΦZs

Ff

TqqqZs(t)minus ZFf

f (t) = 0 (39b)


AAAf qqq = 000 (40)


AAAf =



](41)










qqqFu=qqqY1

ufT qqqY6

ufT qqqZ1

ufT qqqZ6

ufTT

(43)



are given by375

MMM=

MMMY 000 000 000

000 MMMZ 000 000

000 000 MMMB 000

000 000 000 MMMF

CCC=

CCCY 000 000 000

000 CCCZ 000 000

000 000 CCCB 000

000 000 000 CCCF

KKK=

KKKY 000 000 000

000 KKKZ 000 000

000 000 KKKB 000

000 000 000 KKKF

(44)



T(45a)


T(45b)






AAAqqq = 000 (46)

with381

AAA =

[AAAbAAAf

]=


000 ΦΦΦZb 000 000 000 000



(47)























fYn = fZn =cT2πpn

(1 +

B

2T0p2n

)(48)






ζYn = ζZn =1

2

T0(ηF + ηAωn

)+ηBBp

2n

T0 +Bp2n(49)



















12π









































0 200 400 600 800 1000 1200 1400 1600 180010

-40

10-20

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 180010

-20

10-10

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 1800

Frequency [ Hz ]

10-20

10-10

100

| Y

6(f

) |





0 02 04 06 08 1

Time [ s ]

0

05

1

15

2

Mo

dal en

erg

y [

J ]

10-3














0 02 04 06 08 10

05

1

15

[ J ]

10-3 STRINGS

Y-dir

Z-dir

0 02 04 06 08 10

05

1

15

[ J ]

10-5 FRET

Y-dir

Z-dir

0 02 04 06 08 10

05

1

[ J ]

10-4 BODY

0 02 04 06 08 1

Time [ s ]

0

05

1

15

10-3 FULL SYSTEM





E`(t) =

Nsumn=1

[1

2m`n(ω`n)2

[q`n(t)

]2+

1

2m`n

[q`n(t)

]2](50)































1 2 3 4 5 6 7

716

718

72

722

724

T (

t )

[ N

] Fexc

= 1 N

Fexc

= 5 N

1 2 3 4 5 6 7

Time [ s ]

822

824

826

828

f (

t )

[ H

z ]






































0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

4 Nm

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

5 Nm

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

kf = 10

6 Nm





























0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

cf = 1 Nsm

cf = 10 Nsm

0 1 2 3 4 5

Time [ s ]

-2

0

2

4

Y (

t )

10-3











0 01 02 03 04 05 06

x [ m ]

10-5

10-2

ST

D (

Y(t

) )

F=1

F=2

F=3

F=5

016 017 018 019 02

x [ m ]

10-5

ST

D (

Y(t

) )

























































V CONCLUSIONS615






































ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754



ΦYs

b =φYs1 (xb) φ

Ys

NYs(xb)

T(23a)

ΦZs

b =φZs1 (xb) φ

Zs

NZs(xb)

T(23b)

ΦBrs =φB1 (rs) φ

BNB

(rs)T

(23c)






qqqY =qqqY1T qqqY6T

T(25a)

qqqZ =qqqZ1T qqqZ6T

T(25b)

and323



AAAb qqq = 000 (27)


AAAb =


000 ΦΦΦZb 000

](28)



ΦΦΦYb =

ΦY1

b

T000 000

000

ΦY2

b

T 000

000 000

ΦY6

b

T

(29)




ΦΦΦB =

ΦBr1T

ΦBr6T (30)













Fig 4343



Ys(xFf t)=0 rArr

ΦYs

Ff

TqqqYs(t) = 0 (31a)

Zs(xFf t)=0 rArr

ΦZs

Ff

TqqqZs(t) = 0 (31b)


ΦYs

Ff=φYs1 (xFf

) φYs

NYs(xFf

)T

(32a)

ΦZs

Ff=φZs1 (xFf

) φZs

NYs(xFf

)T

(32b)


AAAf qqq = 000 (33)


AAAf =

[ΦΦΦYs

F 000

000 ΦΦΦZs

F

](34)

with346

ΦΦΦYs

F =

ΦYs

F1

T

ΦYs

FF

T ΦΦΦZs

F =

ΦZs

F1

T

ΦZs

FF

T (35)
















Ys(xFf t)minusY Ff

f (t)= 0 (36a)

Zs(xFf t)minusZFf

f (t)= 0 (36b)

where YFf

f and ZFf



u T qqqFsu T

T(37a)


)(37b)


)(37c)


)(37d)







qqqFsu =

Y F1

f Y FF

f ZF1

f ZFF

f

T(38)


Ff

TqqqYs(t)minus Y Ff

f (t) = 0 (39a)ΦZs

Ff

TqqqZs(t)minus ZFf

f (t) = 0 (39b)


AAAf qqq = 000 (40)


AAAf =



](41)










qqqFu=qqqY1

ufT qqqY6

ufT qqqZ1

ufT qqqZ6

ufTT

(43)



are given by375

MMM=

MMMY 000 000 000

000 MMMZ 000 000

000 000 MMMB 000

000 000 000 MMMF

CCC=

CCCY 000 000 000

000 CCCZ 000 000

000 000 CCCB 000

000 000 000 CCCF

KKK=

KKKY 000 000 000

000 KKKZ 000 000

000 000 KKKB 000

000 000 000 KKKF

(44)



T(45a)


T(45b)






AAAqqq = 000 (46)

with381

AAA =

[AAAbAAAf

]=


000 ΦΦΦZb 000 000 000 000



(47)























fYn = fZn =cT2πpn

(1 +

B

2T0p2n

)(48)






ζYn = ζZn =1

2

T0(ηF + ηAωn

)+ηBBp

2n

T0 +Bp2n(49)



















12π









































0 200 400 600 800 1000 1200 1400 1600 180010

-40

10-20

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 180010

-20

10-10

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 1800

Frequency [ Hz ]

10-20

10-10

100

| Y

6(f

) |





0 02 04 06 08 1

Time [ s ]

0

05

1

15

2

Mo

dal en

erg

y [

J ]

10-3














0 02 04 06 08 10

05

1

15

[ J ]

10-3 STRINGS

Y-dir

Z-dir

0 02 04 06 08 10

05

1

15

[ J ]

10-5 FRET

Y-dir

Z-dir

0 02 04 06 08 10

05

1

[ J ]

10-4 BODY

0 02 04 06 08 1

Time [ s ]

0

05

1

15

10-3 FULL SYSTEM





E`(t) =

Nsumn=1

[1

2m`n(ω`n)2

[q`n(t)

]2+

1

2m`n

[q`n(t)

]2](50)































1 2 3 4 5 6 7

716

718

72

722

724

T (

t )

[ N

] Fexc

= 1 N

Fexc

= 5 N

1 2 3 4 5 6 7

Time [ s ]

822

824

826

828

f (

t )

[ H

z ]






































0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

4 Nm

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

5 Nm

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

kf = 10

6 Nm





























0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

cf = 1 Nsm

cf = 10 Nsm

0 1 2 3 4 5

Time [ s ]

-2

0

2

4

Y (

t )

10-3











0 01 02 03 04 05 06

x [ m ]

10-5

10-2

ST

D (

Y(t

) )

F=1

F=2

F=3

F=5

016 017 018 019 02

x [ m ]

10-5

ST

D (

Y(t

) )

























































V CONCLUSIONS615






































ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754







qqqFsu =

Y F1

f Y FF

f ZF1

f ZFF

f

T(38)


Ff

TqqqYs(t)minus Y Ff

f (t) = 0 (39a)ΦZs

Ff

TqqqZs(t)minus ZFf

f (t) = 0 (39b)


AAAf qqq = 000 (40)


AAAf =



](41)










qqqFu=qqqY1

ufT qqqY6

ufT qqqZ1

ufT qqqZ6

ufTT

(43)



are given by375

MMM=

MMMY 000 000 000

000 MMMZ 000 000

000 000 MMMB 000

000 000 000 MMMF

CCC=

CCCY 000 000 000

000 CCCZ 000 000

000 000 CCCB 000

000 000 000 CCCF

KKK=

KKKY 000 000 000

000 KKKZ 000 000

000 000 KKKB 000

000 000 000 KKKF

(44)



T(45a)


T(45b)






AAAqqq = 000 (46)

with381

AAA =

[AAAbAAAf

]=


000 ΦΦΦZb 000 000 000 000



(47)























fYn = fZn =cT2πpn

(1 +

B

2T0p2n

)(48)






ζYn = ζZn =1

2

T0(ηF + ηAωn

)+ηBBp

2n

T0 +Bp2n(49)



















12π









































0 200 400 600 800 1000 1200 1400 1600 180010

-40

10-20

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 180010

-20

10-10

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 1800

Frequency [ Hz ]

10-20

10-10

100

| Y

6(f

) |





0 02 04 06 08 1

Time [ s ]

0

05

1

15

2

Mo

dal en

erg

y [

J ]

10-3














0 02 04 06 08 10

05

1

15

[ J ]

10-3 STRINGS

Y-dir

Z-dir

0 02 04 06 08 10

05

1

15

[ J ]

10-5 FRET

Y-dir

Z-dir

0 02 04 06 08 10

05

1

[ J ]

10-4 BODY

0 02 04 06 08 1

Time [ s ]

0

05

1

15

10-3 FULL SYSTEM





E`(t) =

Nsumn=1

[1

2m`n(ω`n)2

[q`n(t)

]2+

1

2m`n

[q`n(t)

]2](50)































1 2 3 4 5 6 7

716

718

72

722

724

T (

t )

[ N

] Fexc

= 1 N

Fexc

= 5 N

1 2 3 4 5 6 7

Time [ s ]

822

824

826

828

f (

t )

[ H

z ]






































0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

4 Nm

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

5 Nm

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

kf = 10

6 Nm





























0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

cf = 1 Nsm

cf = 10 Nsm

0 1 2 3 4 5

Time [ s ]

-2

0

2

4

Y (

t )

10-3











0 01 02 03 04 05 06

x [ m ]

10-5

10-2

ST

D (

Y(t

) )

F=1

F=2

F=3

F=5

016 017 018 019 02

x [ m ]

10-5

ST

D (

Y(t

) )

























































V CONCLUSIONS615






































ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754



















12π









































0 200 400 600 800 1000 1200 1400 1600 180010

-40

10-20

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 180010

-20

10-10

100

| Y

6(f

) |

0 200 400 600 800 1000 1200 1400 1600 1800

Frequency [ Hz ]

10-20

10-10

100

| Y

6(f

) |





0 02 04 06 08 1

Time [ s ]

0

05

1

15

2

Mo

dal en

erg

y [

J ]

10-3














0 02 04 06 08 10

05

1

15

[ J ]

10-3 STRINGS

Y-dir

Z-dir

0 02 04 06 08 10

05

1

15

[ J ]

10-5 FRET

Y-dir

Z-dir

0 02 04 06 08 10

05

1

[ J ]

10-4 BODY

0 02 04 06 08 1

Time [ s ]

0

05

1

15

10-3 FULL SYSTEM





E`(t) =

Nsumn=1

[1

2m`n(ω`n)2

[q`n(t)

]2+

1

2m`n

[q`n(t)

]2](50)































1 2 3 4 5 6 7

716

718

72

722

724

T (

t )

[ N

] Fexc

= 1 N

Fexc

= 5 N

1 2 3 4 5 6 7

Time [ s ]

822

824

826

828

f (

t )

[ H

z ]






































0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

4 Nm

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

5 Nm

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

kf = 10

6 Nm





























0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

cf = 1 Nsm

cf = 10 Nsm

0 1 2 3 4 5

Time [ s ]

-2

0

2

4

Y (

t )

10-3











0 01 02 03 04 05 06

x [ m ]

10-5

10-2

ST

D (

Y(t

) )

F=1

F=2

F=3

F=5

016 017 018 019 02

x [ m ]

10-5

ST

D (

Y(t

) )

























































V CONCLUSIONS615






































ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754


0 02 04 06 08 10

05

1

15

[ J ]

10-3 STRINGS

Y-dir

Z-dir

0 02 04 06 08 10

05

1

15

[ J ]

10-5 FRET

Y-dir

Z-dir

0 02 04 06 08 10

05

1

[ J ]

10-4 BODY

0 02 04 06 08 1

Time [ s ]

0

05

1

15

10-3 FULL SYSTEM





E`(t) =

Nsumn=1

[1

2m`n(ω`n)2

[q`n(t)

]2+

1

2m`n

[q`n(t)

]2](50)































1 2 3 4 5 6 7

716

718

72

722

724

T (

t )

[ N

] Fexc

= 1 N

Fexc

= 5 N

1 2 3 4 5 6 7

Time [ s ]

822

824

826

828

f (

t )

[ H

z ]






































0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

4 Nm

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

5 Nm

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

kf = 10

6 Nm





























0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

cf = 1 Nsm

cf = 10 Nsm

0 1 2 3 4 5

Time [ s ]

-2

0

2

4

Y (

t )

10-3











0 01 02 03 04 05 06

x [ m ]

10-5

10-2

ST

D (

Y(t

) )

F=1

F=2

F=3

F=5

016 017 018 019 02

x [ m ]

10-5

ST

D (

Y(t

) )

























































V CONCLUSIONS615






































ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754













0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

4 Nm

0 1000 2000 3000 4000 5000

100

1010

| F

SB

(f)

|

kf = 10

5 Nm

0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

kf = 10

6 Nm





























0 1000 2000 3000 4000 5000

Frequency [ Hz ]

100

1010

| F

SB

(f)

|

cf = 1 Nsm

cf = 10 Nsm

0 1 2 3 4 5

Time [ s ]

-2

0

2

4

Y (

t )

10-3











0 01 02 03 04 05 06

x [ m ]

10-5

10-2

ST

D (

Y(t

) )

F=1

F=2

F=3

F=5

016 017 018 019 02

x [ m ]

10-5

ST

D (

Y(t

) )

























































V CONCLUSIONS615






































ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754


















































V CONCLUSIONS615






































ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754



























ACKNOWLEDGMENTS652






REFERENCES658






141764 2017664






















2009686










Ac 63336ndash347 2015696









1023ndash1027 2013705















101155ndash173 2015720






















965 2004742



97(3)1990ndash2001 1992745








2016753


754


754


Physical synthesis of six-string guitar plucks using the ...

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Transcript of Physical synthesis of six-string guitar plucks using the ...