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Dynamic modal characterization of musical instruments using digital holography

Nazif Demoli and Ivan Demoli

Open Access Open Access

Abstract

This study shows that a dynamic modal characterization of musical instruments with membrane can be carried out using a low-cost device and that the obtained very informative results can be presented as a movie. The proposed device is based on a digital holography technique using the quasi-Fourier configuration and time-average principle. Its practical realization with a commercial digital camera and large plane mirrors allows relatively simple analyzing of big vibration surfaces. The experimental measurements given for a percussion instrument are supported by the mathematical formulation of the problem.

©2005 Optical Society of America

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Supplementary Material (2)

Media 1: AVI (3631 KB)     
Media 2: AVI (2048 KB)     

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Figures (4)
Fig. 1.
Fig. 1. Experimental setup. M and M’ denote mirrors, L lenses, VBS variable beam splitter, COL collimator, LS loudspeaker, VS vibration surface, Ob object, DC digital camera, and PC personal computer.

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Fig. 2.
Fig. 2. An example of the colored digital hologram (left) and its portion (right).

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Fig. 3.
Fig. 3. A movie composed from experimentally obtained data for modal characteristics of a drum. [Media 1]

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Fig. 4.
Fig. 4. A movie obtained numerically for a membrane analog to the drum. [Media 2]

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Tables (2)

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Table 1. Resonant frequencies of the circular membrane used in this work

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Table 2. Resonant frequencies measured experimentally

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Equations (13)

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U ( x 1 , y 1 , t ) = δ ( x 1 − X , y 1 − Y ) + s ( x 1 , y 1 , t ) ,
U ( x 2 , y 2 , t ) ∝ exp [ i ( π ⁄ λ d ) ( x 2 2 + y 2 2 ) ] F { U ( x 1 , y 1 , t ) exp⁡[i(π/λd) ( x 1 2 + y 1 2 ) ] } ,
U ( x 2 , y 2 , t ) ∝ exp { i ( π ⁄ λ d ) [ ( x 2 − X ) 2 + ( y 2 − Y ) 2 ] } + exp [ i ( π ⁄ λ d ) ( x 2 2 + y 2 2 ) ]
× F { s ( x 1 , y 1 ) exp [ i ( 4 π ⁄ λ ) h ( x 1 , y 1 ) sin 2 π ft + i ( π ⁄ λ d ) ( x 1 2 + y 1 2 ) ] } .
E 1 ( x 2 , y 2 ) ∝ exp [ − i ( 2 π ⁄ λ d ) ( x 2 X + y 2 Y ) ]
× F { s ( x 1 , y 1 ) J 0 [ ( 4 π ⁄ λ ) h ( x 1 , y 1 ) ] exp [ i ( π ⁄ λ d ) ( x 1 2 + y 1 2 ) ] }
∣ U ′ ( x 1 , y 1 ) ∣ ∝ ∣ s ( x 1 − X , y 1 − Y ) ∣ ∣ J 0 [ ( 4 π ⁄ λ ) h ( x 1 − X , y 1 − Y ) ] ∣ .
[ ( ∂ 2 ∂ r 2 + 1 r ∂ ∂ r + 1 r 2 ∂ 2 ∂ φ 2 ) − 1 v 2 ∂ 2 ∂ t 2 ] h ( r , φ , t ) = 0
h mn ( r , φ , t ) = A mn J m ( x mn r ⁄ a ) cos ( m φ − φ 0 ) cos ( ω mn t − δ ) ,
h ( r , φ , t ) = ∑ m = 0 ∞ ∑ m = 1 ∞ h mn ( r , φ , t ) .
d 2 T ( t ) dt 2 + γ dT ( t ) dt + ω mn 2 T ( t ) = F ( t ) ,
∣ T p ( ω ) ∣ = P 0 ⁄ [ ( ω mn 2 − ω 2 ) 2 + γ 2 ω 2 ] 1 ⁄ 2 ,
h ( r , φ , ω ) = P 0 ∑ m = 0 ∞ ∑ n = 1 ∞ { 1 ⁄ [ ( ω mn 2 − ω 2 ) 2 + γ 2 ω 2 ] 1 ⁄ 2 } J m ( x mn r ⁄ a ) cos m φ ,
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