Modeling a vibrating string terminatedagainst a bridge with arbitrary geometry

Dmitri Kartofelev, Anatoli Stulov,Heidi-Maria Lehtonen and Vesa Välimäki

Institute of Cybernetics at Tallinn University of Technology,Centre for Nonlinear Studies (CENS),

Tallinn, Estonia&

Department of Signal Processing and Acoustics,Aalto University,Espoo, Finland

August 3, 2013

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 1 / 18

Motivation

In numerous musical instruments the collision of avibrating string with rigid spatial obstacles, such as fretsor a bridge is present.

Biwa Shamisen Sitar Jawari (bridge)

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 2 / 18

Motivation

Medieval and Renaissance bray harp and bray pins

Audio example of bray harp timbre (15 s)

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 3 / 18

Motivation

Capo bar (Capo d’astro) of the piano cast iron frame

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 4 / 18

String description

String u(x,t)

∂2u

∂t2= c2

∂2u

∂x2, c =

√T

µ(1)

u(0, t) = u(L, t) = 0 (2)

Solution to Eq. (1) is famous d’Alembert’s solution:

u(x, t) =1

2[ur(x− ct) + ul(x+ ct)] (3)

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 5 / 18

String description

String u(x,t)

∂2u

∂t2= c2

∂2u

∂x2, c =

√T

µ(1)

u(0, t) = u(L, t) = 0 (2)

Solution to Eq. (1) is famous d’Alembert’s solution:

u(x, t) =1

2[ur(x− ct) + ul(x+ ct)] (3)

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 5 / 18

Geometric termination condition (TC)

TC is an absolutely rigid unilateral constraint of thestring’s transverse deflection.Support profile geometry is described by an arbitraryfunction U(x).

StringU(x)

a) parabolic profile

StringU(x)

b) linear profile

StringU(x)

c) Sine-like profile

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 6 / 18

Geometric termination condition (TC)

TC is an absolutely rigid unilateral constraint of thestring’s transverse deflection.Support profile geometry is described by an arbitraryfunction U(x).

StringU(x)

a) parabolic profile

StringU(x)

b) linear profile

StringU(x)

c) Sine-like profile

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 6 / 18

Geometric termination condition (TC)

TC is an absolutely rigid unilateral constraint of thestring’s transverse deflection.Support profile geometry is described by an arbitraryfunction U(x).

StringU(x)

a) parabolic profile

StringU(x)

b) linear profile

StringU(x)

c) Sine-like profile

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 6 / 18

Bridge-string interaction model

Since the termination is rigid, it must hold

u(x∗, t) 6 U(x∗). (4)

In order to satisfy condition (4) for u(x∗, t) > U(x∗) areflected traveling wave is introduced

ur

(t− x

∗

c

)= U(x∗)− ul

(t+

x∗

c

), (5)

here the waves ul and ur correspond to any waves thathave reflected from the terminator earlier.

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 7 / 18

Bridge-string interaction model

Since the termination is rigid, it must hold

u(x∗, t) 6 U(x∗). (4)

In order to satisfy condition (4) for u(x∗, t) > U(x∗) areflected traveling wave is introduced

ur

(t− x

∗

c

)= U(x∗)− ul

(t+

x∗

c

), (5)

here the waves ul and ur correspond to any waves thathave reflected from the terminator earlier.

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 7 / 18

Bridge-string interaction model

Since the termination is rigid, it must hold

u(x∗, t) 6 U(x∗). (4)

In order to satisfy condition (4) for u(x∗, t) > U(x∗) areflected traveling wave is introduced

ur

(t− x

∗

c

)= U(x∗)− ul

(t+

x∗

c

), (5)

here the waves ul and ur correspond to any waves thathave reflected from the terminator earlier.

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 7 / 18

Single wave reflection from the terminator

ur(t− x∗/c) = U(x∗)− ul(t+ x∗/c)u(x∗, t) = U(x∗) = ur(t− x∗/c) + ul(t+ x∗/c) (6)

x

A

-A

∆x ∆x ∆x ∆x ∆x ∆x

x

A

-A

∆x ∆x ∆x ∆x ∆x ∆xx

A

-A

∆x ∆x ∆x ∆x ∆x ∆x

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 8 / 18

Single wave reflection from the terminator

ur(t− x∗/c) = U(x∗)− ul(t+ x∗/c)u(x∗, t) = U(x∗) = ur(t− x∗/c) + ul(t+ x∗/c) (6)

x

A

-A

∆x ∆x ∆x ∆x ∆x ∆xx

A

-A

∆x ∆x ∆x ∆x ∆x ∆x

x

A

-A

∆x ∆x ∆x ∆x ∆x ∆x

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 8 / 18

Single wave reflection from the terminator

ur(t− x∗/c) = U(x∗)− ul(t+ x∗/c)u(x∗, t) = U(x∗) = ur(t− x∗/c) + ul(t+ x∗/c) (6)

x

A

-A

∆x ∆x ∆x ∆x ∆x ∆xx

A

-A

∆x ∆x ∆x ∆x ∆x ∆xx

A

-A

∆x ∆x ∆x ∆x ∆x ∆x

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 8 / 18

Single wave reflection from the terminator

ur(t− x∗/c) = U(x∗)− ul(t+ x∗/c)u(x, t) = ur(t− x/c) + ul(t+ x/c) (7)

0.0 0.2 0.4 0.6 0.8 1.0Distance along the string x

1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.40

0.0 0.2 0.4 0.6 0.8 1.0Distance along the string x

1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.700.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.800.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 1.00

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 9 / 18

Single wave reflection from the terminator

ur(t− x∗/c) = U(x∗)− ul(t+ x∗/c)u(x, t) = ur(t− x/c) + ul(t+ x/c) (7)

0.0 0.2 0.4 0.6 0.8 1.0Distance along the string x

1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.400.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.70

0.0 0.2 0.4 0.6 0.8 1.0Distance along the string x

1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.800.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 1.00

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 9 / 18

Single wave reflection from the terminator

ur(t− x∗/c) = U(x∗)− ul(t+ x∗/c)u(x, t) = ur(t− x/c) + ul(t+ x/c) (7)

0.0 0.2 0.4 0.6 0.8 1.0Distance along the string x

1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.400.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.700.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.80

0.0 0.2 0.4 0.6 0.8 1.0Distance along the string x

1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 1.00

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 9 / 18

Single wave reflection from the terminator

ur(t− x∗/c) = U(x∗)− ul(t+ x∗/c)u(x, t) = ur(t− x/c) + ul(t+ x/c) (7)

0.0 0.2 0.4 0.6 0.8 1.0Distance along the string x

1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.400.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.700.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 0.800.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x1.0

0.5

0.0

0.5

1.0

Disp

lace

men

t u(x,t)

Time 1.00

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 9 / 18

Single wave reflection from the terminator

0.0 0.2 0.4 0.6 0.8 1.0Distance along the string [1]

1.0

0.5

0.0

0.5

1.0Am

plitu

de [1

]

Time 0.00

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 10 / 18

Model application: Biwa

String length L = 0.8 mString plucking point l = 3/4L = 0.6 m

Linear mass density of the string µ = 0.375 g/mString tension T = 38.4 N

Velocity of the traveling waves c = 320 m/sFundamental frequency f0 = 200 Hz

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 11 / 18

Bridge profiles studied

Profile shapes

Case 1: Linearbridge withsharp edge

Case 2: Linearbridge withcurved parabolicedge

Case 3: Bridgewith minordefect

0 5 10 xc 200.00.51.0

case 1

0 5 xb 15 200.00.51.0

U(x

) (m

m)

case 2

0 xa 5 xb 15 20Distance along the srtring x (mm)

0.00.51.0

case 3

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 12 / 18

Result: Time series u(l, t)

63036 case 1

63036

Disp

lace

men

t u(l,t) (

mm

)

case 2

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35Time t (s)

63036 case 3

Nonperiodic and almost periodic vibration regimes.

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 13 / 18

Case 1: Linear bridge with sharp edge

Spectrograms of the string vibration u(l, t).

0.0 0.1 0.2 0.3Time t (s)

0.00.51.01.52.02.53.0

Freq

uenc

y f

(kHz

) 0102030405060

Figure: Linear case, no TC

0.0 0.1 0.2 0.3Time t (s)

0.00.51.01.52.02.53.0 0

102030405060 R

elat

ive

leve

l (dB

)

Figure: Case 1. Transition betweenthe vibration regimes is shown bydashed line at tnp = 0.13 s.

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 14 / 18

Case 2: Linear bridge with curved edge

Spectrograms of the string vibration u(l, t).

0.0 0.1 0.2 0.3Time t (s)

0.00.51.01.52.02.53.0

Freq

uenc

y f

(kHz

) 0102030405060

Figure: Linear case, no TC

0.0 0.1 0.2 0.3Time t (s)

0.00.51.01.52.02.53.0 0

102030405060 R

elat

ive

leve

l (dB

)

Figure: Case 2. Transition betweenthe vibration regimes is shown bydashed line at tnp = 0.16 s.

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 15 / 18

Case 3: Bridge with minor defect

Spectrograms of the string vibration u(l, t).

0.0 0.1 0.2 0.3Time t (s)

0.00.51.01.52.02.53.0

Freq

uenc

y f

(kHz

) 0102030405060

Figure: Linear case, no TC

0.0 0.1 0.2 0.3Time t (s)

0.00.51.01.52.02.53.0 0

102030405060 R

elat

ive

leve

l (dB

)

Figure: Case 3. Transition betweenthe vibration regimes is shown bydashed line at tnp = 0.3 s.

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 16 / 18

Case 2: Animation

0 100 200 300 400 500 600 700 800Distance along the string x (mm)

10

5

0

5

10Di

spla

cem

ent u

(x,t) (

mm

)

Time 0.0 ms (nonperiodic regime)

Case 1Linear caseCase 1

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 17 / 18

Conclusions

A relatively simple method for modeling theTC-string interaction problem was presented.

Two distinct vibration regimes in the case of thelossless string: strongly nonlinear nonperiodic andalmost periodic regimes.

Duration of the nonperiodic vibration regimedepended on the bridge profile and on the pluckingcondition.

A minor imperfection of the bridge profile geometryleads to prolonged nonperiodic vibration regime.

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa Välimäki (CENS)SMAC SMC 2013 August 3, 2013 18 / 18