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Modeling a vibrating string terminated against a bridge with - - PowerPoint PPT Presentation

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Modeling a vibrating string terminated against a bridge with arbitrary geometry Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V alim aki Institute of Cybernetics at Tallinn University of Technology, Centre for Nonlinear

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SLIDE 1

Modeling a vibrating string terminated against a bridge with arbitrary geometry

Dmitri Kartofelev, Anatoli Stulov, Heidi-Maria Lehtonen and Vesa V¨ alim¨ aki

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 V¨ alim¨ aki (CENS) SMAC SMC 2013 August 3, 2013 1 / 18

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SLIDE 2

Motivation

In numerous musical instruments the collision of a vibrating string with rigid spatial obstacles, such as frets

  • r a bridge is present.

Biwa Shamisen Sitar Jawari (bridge)

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

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SLIDE 3

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 V¨ alim¨ aki (CENS) SMAC SMC 2013 August 3, 2013 3 / 18

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SLIDE 4

Motivation

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

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

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SLIDE 5

String description

∂2u ∂t2 = c2∂2u ∂x2, c =

  • T

µ (1) u(0, t) = u(L, t) = 0 (2)

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

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SLIDE 6

String description

∂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 V¨ alim¨ aki (CENS) SMAC SMC 2013 August 3, 2013 5 / 18

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SLIDE 7

Geometric termination condition (TC)

TC is an absolutely rigid unilateral constraint of the string’s transverse deflection. Support profile geometry is described by an arbitrary function U(x).

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

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SLIDE 8

Geometric termination condition (TC)

TC is an absolutely rigid unilateral constraint of the string’s transverse deflection. Support profile geometry is described by an arbitrary function U(x).

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

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SLIDE 9

Geometric termination condition (TC)

TC is an absolutely rigid unilateral constraint of the string’s transverse deflection. Support profile geometry is described by an arbitrary function U(x).

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

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SLIDE 10

Bridge-string interaction model

Since the termination is rigid, it must hold u(x∗, t) U(x∗). (4)

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

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SLIDE 11

Bridge-string interaction model

Since the termination is rigid, it must hold u(x∗, t) U(x∗). (4) In order to satisfy condition (4) for u(x∗, t) > U(x∗) a reflected traveling wave is introduced ur

  • t − x∗

c

  • = U(x∗) − ul
  • t + x∗

c

  • ,

(5)

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

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SLIDE 12

Bridge-string interaction model

Since the termination is rigid, it must hold u(x∗, t) U(x∗). (4) In order to satisfy condition (4) for u(x∗, t) > U(x∗) a reflected 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 that have reflected from the terminator earlier.

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

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SLIDE 13

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)

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

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SLIDE 14

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)

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

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SLIDE 15

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)

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

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SLIDE 16

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 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.40

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.70

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.80

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 1.00

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

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SLIDE 17

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 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.40

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.70

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.80

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 1.00

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

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SLIDE 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 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.40

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.70

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.80

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 1.00

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

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SLIDE 19

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 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.40

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.70

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 0.80

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string x

1.0 0.5 0.0 0.5 1.0

Displacement u(x,t) Time 1.00

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

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SLIDE 20

Single wave reflection from the terminator

0.0 0.2 0.4 0.6 0.8 1.0

Distance along the string [1]

1.0 0.5 0.0 0.5 1.0

Amplitude [1] Time 0.00

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

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Model application: Biwa

String length L = 0.8 m String plucking point l = 3/4L = 0.6 m Linear mass density of the string µ = 0.375 g/m String tension T = 38.4 N Velocity of the traveling waves c = 320 m/s Fundamental frequency f0 = 200 Hz

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

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Bridge profiles studied

Profile shapes Case 1: Linear bridge with sharp edge Case 2: Linear bridge with curved parabolic edge Case 3: Bridge with minor defect

5 10 xc 20 0.0 0.5 1.0 case 1 5 xb 15 20 0.0 0.5 1.0

U(x) (mm)

case 2 xa 5 xb 15 20

Distance along the srtring x (mm)

0.0 0.5 1.0 case 3

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

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Result: Time series u(l, t)

6 3 3 6 case 1 6 3 3 6

Displacement u(l,t) (mm)

case 2 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35

Time t (s)

6 3 3 6 case 3

Nonperiodic and almost periodic vibration regimes.

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

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SLIDE 24

Case 1: Linear bridge with sharp edge

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

0.0 0.1 0.2 0.3

Time t (s)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Frequency f (kHz)

10 20 30 40 50 60

Figure: Linear case, no TC

0.0 0.1 0.2 0.3

Time t (s)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 20 30 40 50 60

Relative level (dB)

Figure: Case 1. Transition between the vibration regimes is shown by dashed line at tnp = 0.13 s.

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

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Case 2: Linear bridge with curved edge

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

0.0 0.1 0.2 0.3

Time t (s)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Frequency f (kHz)

10 20 30 40 50 60

Figure: Linear case, no TC

0.0 0.1 0.2 0.3

Time t (s)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 20 30 40 50 60

Relative level (dB)

Figure: Case 2. Transition between the vibration regimes is shown by dashed line at tnp = 0.16 s.

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

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SLIDE 26

Case 3: Bridge with minor defect

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

0.0 0.1 0.2 0.3

Time t (s)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

Frequency f (kHz)

10 20 30 40 50 60

Figure: Linear case, no TC

0.0 0.1 0.2 0.3

Time t (s)

0.0 0.5 1.0 1.5 2.0 2.5 3.0 10 20 30 40 50 60

Relative level (dB)

Figure: Case 3. Transition between the vibration regimes is shown by dashed line at tnp = 0.3 s.

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

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SLIDE 27

Case 2: Animation

100 200 300 400 500 600 700 800

Distance along the string x (mm)

10 5 5 10

Displacement u(x,t) (mm) Time 0.0 ms (nonperiodic regime)

Case 1

Linear case Case 1

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

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Conclusions

A relatively simple method for modeling the TC-string interaction problem was presented. Two distinct vibration regimes in the case of the lossless string: strongly nonlinear nonperiodic and almost periodic regimes. Duration of the nonperiodic vibration regime depended on the bridge profile and on the plucking condition. A minor imperfection of the bridge profile geometry leads to prolonged nonperiodic vibration regime.

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