Computation of the Violin Front Response to an Excitation Generated

by String Vibrations

1

By Claire Saint-Pierre

CE 291: Control and Optimization of DistributedSystems and Partial Differential Equations

Professor Alexandre Bayen

University of California, Berkeley December 4th, 2007

• Introduction

• Definition of the Studied System

• Splitting Method

Outlines

2

• Calculation of Eigensolutions

• Experimental results

• Next steps

• References

Front page image: http://neureither-luthier.com/instruments.htm

String vibration

Bridge

Bass bar Sound post

Introduction

3

Bass bar Sound post

Front Back

Vibration of the body

Sound we hear Source: http://library.thinkquest.org/27178/en/section/2/2.html

y

Bass Bar Simplified Problem

1 2

Front Table Vibration

4

x

L1 L2

l

Source: http://www.chismviolins.com/photos.htm

Wave equation u(x,y,t) through the front table:

02 =−∆ ttuua

Definition of the System

5

With a the speed of propagation of the wave (in wood)

Boundary Conditions: Initial Conditions:

0),,(

0),0,(

==

tlxu

txu 0)0,,( =yxu

SOV:

The equation becomes: 22 " ω−==∆

T

T

U

Ua

)().,(),,( tTyxUtyxu =

Separation of Variables

6

Solution of the time component:

from the initial condition

B = 0 with ω ω ω ω the oscillation frequency

)cos()sin()( tBtAtT ωω +=

)sin()( tAtT ω=

),(),(),(),( 210 yxUyxUyxUyxU ++=

1Domain

0=U 01 =U 02 =U00 =U

Splitting Method

7

= + +

0=U

),( yxh

U =∆

01 =U

01 =∆U 02 =∆U

02 =U00 =U

),(

0

yxh

U =∆

Sub-set 0 Sub-set 1 Sub-set 2

=−=

==

=

)()()(),0(

0),(

0),(

0)0,(

111

11

1

1

ysyqyfyU

yLU

lxU

xU)()(),(1 yYxXyxU =

BC:

SOV:

Then 0111 =+=∆ yyxx UUU

Sub-problem 1

8

=+

=−

=−=

=+

0"

0"

""

0""

2

2

2

YY

XX

Y

Y

X

X

XYYX

λλ

λ

The PDE becomes:

0),(

00)0,(

)cos()sin()(

1

21

21

nllxU

KxU

yKyKyY

n

=⇒=

=⇒=

+=

πλ

λλEigenfunction in y *

Boundary Conditions

Eigenfunctions

9

))(sinh().sin(),(

00),(

))(cosh())(sinh()(

11

411

1413

xLl

ny

l

nKyxU

KyLU

xLKxLKxX

n −=

=⇒=

−+−=

ππ

λλEigenfunction in x *

Boundary Conditions

Therefore

∑

∑

==

−=

∞+

=

+∞

=

nn

nn

Ll

ny

l

nKyAysyU

xLl

ny

l

nKyAyxU

11

11

1

11

11

)sinh().sin().()(),0(

))(sinh().sin().(),(

ππ

ππ

Eigensolution

Solution of the Sub-problem 1

Boundary Conditionand

10

∫

∫

∫∫

=

==

l

l

n

ll

mn

dyyl

nL

l

n

dyyl

nys

yA

dyxl

mx

l

ndyYY

0

21

0 11

00

)(sin)sinh(

)sin().()(

0)sin()sin(

ππ

π

ππ

andOrthogonality of the Y eigenfunctions

if n = m

Coefficients

We expand the function h(x,y) in Fourier series in x

with an and bn coefficients:

)2

sin()()2

cos()(),(1 10 1

xL

nybx

L

nyayxh

nn

nn ∑∑

+∞

=

+∞

=

+= ππ

=

=

=

∫

∫

∫

1

1

1

011

01

0

22

)2

cos(),(2

)(

),(1

)(

L

L

n

L

n

dxxL

nyxh

Lya

dxyxhL

ya

π

π

Particular Solution

11

Particular Solution An and Bn coefficient verify

= ∫

1

011

)2

sin(),(2

)(L

n dxxL

nyxh

Lyb

π

)2

sin()(

)2

cos()(),(

1 1

0 1

0

xL

nyB

xL

nyAyxU

nn

nn

∑

∑∞+

=

+∞

=

+

=

π

π

=−

=−

=

)()2

(

)()2

(

)(

2

1

"

2

1

"

0"0

ybBL

nB

yaAL

nA

yaA

nnn

nnn

π

π

Final Solution

+∑∑

+∞+∞

)2

sin()()2

cos()( nn xL

nyBx

L

nyA

ππ

Final Solution of the PDE in the domain 1

12

+

−+=

∑

∑

∑∑

∞+

=

∞+

=

==

1

'2

11

1

1 10 1

)sinh().sin().(

))(sinh().sin().()sin(),,(

nn

nn

nn

nn

xl

ny

l

nKyA

xLl

ny

l

nKyA

LL

tAtyxu

ππ

ππω

Interferograms of wallvibrations of the violin withsoundpost at the second airmode A1 at 460 Hz of (c)

Experimentation

13

Source: Vibration modes of the violin forced via the bridge and action of the soundpost (2)

mode A1 at 460 Hz of (c)the top plate and (d) theback plate. A loudspeakerwas used to excite the violin

• Finish MATLAB code

• Calculate the eigensolution for the

Next Steps

14

• Calculate the eigensolution for the second domain

• Implement the eigensolution in the code and modeling the node lines

1) Solution of Partial Differential Equations for Engineers, (Chapter 4)Eigenfunction expansion in linear problems, W. C. Reynolds, StanfordUniversity, Preliminary Edition – Winter 1999 version

2) Vibration modes of the violin forced via the bridge and action of thesoundpost, Henrik O. Saldner and Nils-Erik Molin, Lulea° University ofTechnology,Sweden,Erik V. Jansson,Royal Institute of Technology,

References

15

Technology,Sweden,Erik V. Jansson,Royal Institute of Technology,Sweden, accepted 29 March 1996

3) On the “Bridge Hill” of the Violin , J. Woodhouse, Cambridge UniversityEngineering Department, accepted 7June 2004

4) Body Vibration of the Violin – What Can a Maker Expect to Control?, J.Woodhouse, Cambridge University Engineering Department,May 2002

5) Physical modeling by directly solving wave PDE, Marco Palumbi,Lorenzo Seno, Centro Ricerche Via Lamarmora, Roma, Italy

Questions

Thank you

16

Thank you

Violin Concerto No1 g-moll op26(Final), Max Bruch Violin: Frederieke Saeijs

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