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

1. Introduction

Dynamic modal characteristics (DMCs) of a musical instrument with membrane are defined by its physical properties such as: (i) the resonator structure, (ii) the membrane size, shape, and material, (iii) the membrane tension, mass density, and propagation velocity, and particularly (iv) the membrane inhomogeneities. For an ideal membrane system, these characteristics can be studied applying numerical modeling, but for a real musical instrument they must be measured experimentally. The DMCs reveal the sound quality of a particular instrument and may have considerable influence on the instrument design.

The holography based techniques such as the time-averaged holographic interferometry, classical [1] and digital [2], and the electronic speckle pattern interferometry [3], are recognized as particularly well suited for studying vibrating surfaces of musical instruments [4], since they are full-field, noncontact, and possess adequate sensitivity. Although highly utilized (for example: on the string instruments [5, 6], percussion instruments [7], or wind instruments [8]), these classical techniques are time and means consuming since they use either complicated procedures or expensive equipment. We propose the use of a digital holography technique by which we avoid wet processing and other complications, retaining at the same time the measuring simplicity and the processing possibilities inherent to handling with numerical data. Drawbecks of digital holography are related to the recording conditions of low numerical aperture which leads to large speckle noise and overlapping of zero- and first-order reconstruction terms. These conditions limit the size of investigated objects. An effective solution to overcome these difficulties named subtraction digital holography was reported [9] and demonstrated through several applications [10, 11]. We applied that technique in this work to achieve clean reconstructions of digital holograms.

The aim of this study was to present the DMCs of a membrane system as a movie showing changes of vibration patterns in a given frequency range. The movie should contain information about the measured resonant frequencies, the corresponding modal structure, the transient vibration structure, the phase shifts, and regularity of the vibration patterns. Any real membrane system would thus be possible to characterize uniquely. We use two approaches, theoretical, to numerically describe an ideal membrane system and, experimental, to measure the DMCs of a real instrument. For experimental measurements we constructed a low-cost device based on digital holography technique with compacted architecture and equipped with commercial elements.

Section 2 contains theoretical work including description of the quasi-Fourier digital holography technique as well as the calculation of the natural frequencies and DMCs for circular membranes. Numerical and experimental results for a percussion instrument are presented in section 3, while the conclusions are drawn in section 4.

2. Theory

2.1 Time-averaged quasi-Fourier digital holography

Digital holography is a holographic technique based on optical hologram recording by an array photodetector and numerical reconstruction. In quasi-Fourier setup, both a reference point source and an object are placed at the input plane P 1 [coordinates (x 1, y 1)], while the detector is placed at the hologram plane P 2 [coordinates (x 2, y 2)]. For vibration study, we describe input as

U(x1,y1,t)=δ(x1X,y1Y)+s(x1,y1,t),

where (X,Y) is the position of the point source. We assume that only out-of-plane harmonic vibrations of the object occur and that the illumination is normal to the object surface. Then, the object wave front is in the form: s(s 1,x 1,y1 ,t)=s(x 1,y 1) exp[i (4π/λ) h(x 1,y 1) sin2π ft], where s(x 1,y 1) denotes the static wave front, h(x1,y1) the vibration amplitude (the maximum deviation from the surface equilibrium), λ the wavelength, and f the vibration frequency. The diffracted field at the plane P 2 distanced d from the input plane P 1 can be calculated according to Fresnel approximation as

U(x2,y2,t)exp[i(πλd)(x22+y22)]F{U(x1,y1,t)exp[i(π/λd)(x12+y12)]},

where F denotes the Fourier transform operation and the constant terms are omitted. Thus,

U(x2,y2,t)exp{i(πλd)[(x2X)2+(y2Y)2]}+exp[i(πλd)(x22+y22)]
×F{s(x1,y1)exp[i(4πλ)h(x1,y1)sin2πft+i(πλd)(x12+y12)]}.

The exposure recorded by the detector can be described as an integral over the time τ: E(x 2,y 2)=0τ|U(x 2,y 2,t)|2 dt. From the condition τ≫f one of additive terms becomes

E1(x2,y2)exp[i(2πλd)(x2X+y2Y)]
×F{s(x1,y1)J0[(4πλ)h(x1,y1)]exp[i(πλd)(x12+y12)]}

where J0 is the zero-order Bessel function of the first kind. Modulus of the inverse Fourier transform of relation (4) yields the reconstruction

U(x1,y1)s(x1X,y1Y)J0[(4πλ)h(x1X,y1Y)].

Relation (5) shows that the object reconstruction is modulated by the fringes depending on the vibration amplitude h(x 1,y 1). The mathematical interpretation of fringes is that each fringe represents a line of constant displacement. The positions with zero amplitude define nodes and the positions of local maxima are antinodes.

2.2 Circular membrane: natural motion

The motion of a perfectly compliant circular membrane fixed at the radius a can be described in cylindrical coordinates (r,φ) by the wave Eq.

[(2r2+1rr+1r22φ2)1v22t2]h(r,φ,t)=0

with the boundary condition: h(a,φ,t)=0. The parameter v denotes the propagation velocity on the membrane, v=(Ft/σ)1/2, where Ft is the membrane tension (in N/m) and σ is the areal mass density of the membrane (in kg/m2). Differential Eq. (6) can be solved using variable separation method. Thus, for h(r,φ,t)=R(r)Φ(φ)T(t) and the separation constants -k 2 and m 2, the solution is

hmn(r,φ,t)=AmnJm(xmnra)cos(mφφ0)cos(ωmntδ),

where jm (x) is the m-th order Bessel function of the first kind with the n-th zero xmn and where ωmn is the membrane resonant frequency, ωmn=vkmn=vxmn/a (m=0, 1, 2, …, n=1, 2, 3, …). The coefficients Amn and the phase shifts φ 0 and δ are determined by the initial and boundary conditions. The complete solution for the motion of the circular membrane is

h(r,φ,t)=m=0m=1hmn(r,φ,t).

Calculated without a driving force, these solutions show the natural behavior of an idealized membrane system differing generally from the behavior of a real membrane with the same parameters.

2.3 Circular membrane: dynamic modal characteristics

The DMCs are obtained by determining the two-dimensional arrays of vibration amplitude values within a given range of frequencies ω: ω minωω max. In practice, the membrane is set into vibration with the frequency ω and its motion can be described by the time dependent inhomogeneous differential Eq.

d2T(t)dt2+γdT(t)dt+ωmn2T(t)=F(t),

where the force F(t) is harmonic, i.e. F(t)=P 0 cos(ωt+ϑ 0). In the case of applying a sound pressure, the constant P 0 is proportional to the intensity of a sound source. The force F(t) is compensated by the parameter γ that describes the energy loss and the resulting damping of the membrane. Generally, the membrane damping is small compared to ω and usually decreases as ω increases [4].

The steady-state solution for Eq. (9) is actually a particular solution which can be obtained for a function Tp (t)=A exp[i(ωt+ϑ 0)]. Inserting Tp (t) in Eq. (9) and calculating the constant A, it follows: Tp (t)={P 0/[(ωmn2-ω 2)+iγω]}exp[i(ωt+ϑ 0)]. The amplitude of this function,

Tp(ω)=P0[(ωmn2ω2)2+γ2ω2]12,

shows the dependence of the vibration amplitude on the frequency ω. This amplitude reaches its maxima at the frequencies ω=(ωmn2-γ 2/2)1/2, which is (due to γωmn for every m, n) approximately equal to the resonant frequencies ωmn .

Although the DMCs of a real membrane system ought to be determined experimentally due to the specific physical properties and particularly imperfections of the system parts, the insight into the modal properties of the analog ideal membrane system can be obtained from the solution given by Eq. (7). We replace the temporal term of Eq. (7) by the right-hand side of Eq. (10) and we take the values: Amn =1 and φ 0=0. Thus, the qualitative solution for the vibration amplitude is given by:

h(r,φ,ω)=P0m=0n=1{1[(ωmn2ω2)2+γ2ω2]12}Jm(xmnra)cosmφ,

and the DMCs of a membrane with radius a, velocity v, and damping γ can be presented as a movie where the frames change with ω from ωmin to ωmax .

3. Results

We determined the DMCs for a 6-inch drumhead with parameters: a=0.0762m and v=82.6 m/s, and in the frequency range from ωmin =2724 Hz (fmin =434 Hz) to ω max=11284 Hz (f max=1796 Hz). The resonant frequencies are given in Table 1. The damping of the membrane γ and the corresponding Q-factors were not investigated in this study. We synthesized two movies, one from the experimentally measured data and the other numerically. The duration of each movie is 27 seconds.

Tables Icon

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

3.1 Experimental

Figure 1 shows the scheme (left) and the photograph (right) of the experimental setup. It is composed of a laser, a variable beam splitter, a beam collimator, an array sensor, and various lenses and mirrors. We used a He-Ne laser (λ=632.8 nm) as a light source and a color digital camera (Olympus E-1, objective lens removed, sensor: 2560×1920 pixels of a size 6.8 µm×6.8 µm, primary RGB filter) as an array photo-detector. The beam emerging from the laser is split into an object and a reference beam. These beams are expanded and steered by mirrors, as depicted in Fig. 1, to form the interference pattern at the detector plane. The membrane is excited externally with the help of a loudspeaker whose frequency and intensity are controlled by a computer. To shorten the total length of the device, we placed two large plane mirrors between the object and the sensor. Thus, digital holograms are recorded at a distance of 6 m (3×2 m) from the object, using the exposure time of 1 second and the beam intensity ratio (reference vs. object) between 2.5 and 3.5.

The movie was synthesized by the following procedure: (i) the colored images (digital holograms) were transformed into the 8-bit gray scale images, (ii) these images were then Fourier-transformed, (iii) each obtained image was cut and we saved only one reconstruction, and (iv) the movie was composed from the sequence of these reconstructions. An illustration of a raw colored digital hologram with its magnified portion is given in Fig. 2. Figure 3 shows the first frame of the movie composed from the experimental data. The movie contains the relevant information. For example, the resonant frequencies estimated at the maxima described by Eq. (10) are shown in Table 2, where Δfmn denotes the difference between the calculated and measured values. Other data, such as the vibration structure, phase shifts, or pattern regularity can be easily extracted from the movie.

 figure: 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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Tables Icon

Table 2. Resonant frequencies measured experimentally

 figure: Fig. 2.

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

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3.1 Numerical

To synthesize numerical movie, we made a program in Mathcad and calculated the vibration amplitudes according to Eq. (11). We have chosen a graphic representation in which the lines of equal amplitude are constant in number. Thus, the transformation between the resonant modes is clearly visible and the relative amplitudes between frames (which depend on the unknown parameter γ) are irrelevant. Figure 4 shows the first frame of the numerical movie.

 figure: Fig. 3.

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

Download Full Size | PPT Slide | PDF

 figure: Fig. 4.

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

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4. Discussion and conclusions

We constructed a low-cost device that is: (i) based on the quasi-Fourier digital holography technique, (ii) equipped with a commercial color digital camera (objective removed), and (iii) compacted with large plane mirrors. We tested the device in a study of dynamic modal characterization of musical instruments with a membrane. We noticed minor degradation in the hologram recordings compared to using a professional monochromatic CCD sensor and long setup without the large mirrors (these mirrors introduce additional fringes in hologram recordings). These effects, however, do not disturb the measured DMCs. By showing the DMCs as a movie, it is demonstrated that such very informative measurements can be presented in a compact form.

We measured the DMCs for a percussion instrument. The obtained results are compared with the results of numerical analysis that was supported by the mathematical formulation of the problem. As expected, the DMCs measured experimentally differ from the ones obtained numerically. These differences, occurring in the resonant frequency values and the regularity of the vibration structures, are clearly pronounced using our approach.

Acknowledgments

This research has been supported by the Croatian Ministry of Science, Education and Sports under the project no. 0035005.

References and links

1. R. L. Powel and K. A. Stetson, “Interferometric vibration analysis by wavefront reconstruction,” J. Opt. Soc. Am. 55, 1593–1598 (1965). [CrossRef]  

2. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time-averaged digital holography,” Opt. Lett. 28, 1900–1902 (2003). [CrossRef]   [PubMed]  

3. J. N. Butters and J. A. Leendertz, “Holographic and video techniques applied to engineering measurement,” Meas. Control 4, 349–354 (1971).

4. N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments (Springer-Verlag, New York, 1999).

5. E. V. Jansson, “A study of acoustical and hologram interferometric measurements on the top plate vibrations of a guitar,” Acustica 25, 95–100 (1971).

6. C. M. Hutchins, “The acoustics of violin plates,” Scientific American 245, 170–186 (1981). [CrossRef]  

7. T. D. Rossing, I. Bork, H. Zhao, and D. Fystrom, “Acoustics of snare drums,” J. Acoust. Soc. Am. 92, 84–94 (1992). [CrossRef]  

8. O. J. Lokberg and O. K. Ledang, “Vibration of flutes studied by electronic speckle pattern interferometry,” Appl. Opt. 23, 3052–3058 (1984). [CrossRef]   [PubMed]  

9. N. Demoli, J. Meštrović, and I. Sović, “Subtraction digital holography,” Appl. Opt. 42, 798–804 (2003). [CrossRef]   [PubMed]  

10. N. Demoli, D. Vukicevic, and M. Torzynski, “Dynamic digital holographic interferometry with three wavelenths,” Opt. Express 11, 767–774 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-767. [CrossRef]   [PubMed]  

11. N. Demoli and D. Vukicevic, “Detection of hidden stationary deformations of vibrating surfaces by use of time-averaged digital holographic interferometry,” Opt. Lett. 29, 2423–2425 (2004). [CrossRef]   [PubMed]  

References

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  1. R. L. Powel and K. A. Stetson, “Interferometric vibration analysis by wavefront reconstruction,” J. Opt. Soc. Am. 55, 1593–1598 (1965).
    [Crossref]
  2. P. Picart, J. Leval, D. Mounier, and S. Gougeon, “Time-averaged digital holography,” Opt. Lett. 28, 1900–1902 (2003).
    [Crossref] [PubMed]
  3. J. N. Butters and J. A. Leendertz, “Holographic and video techniques applied to engineering measurement,” Meas. Control 4, 349–354 (1971).
  4. N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments (Springer-Verlag, New York, 1999).
  5. E. V. Jansson, “A study of acoustical and hologram interferometric measurements on the top plate vibrations of a guitar,” Acustica 25, 95–100 (1971).
  6. C. M. Hutchins, “The acoustics of violin plates,” Scientific American 245, 170–186 (1981).
    [Crossref]
  7. T. D. Rossing, I. Bork, H. Zhao, and D. Fystrom, “Acoustics of snare drums,” J. Acoust. Soc. Am. 92, 84–94 (1992).
    [Crossref]
  8. O. J. Lokberg and O. K. Ledang, “Vibration of flutes studied by electronic speckle pattern interferometry,” Appl. Opt. 23, 3052–3058 (1984).
    [Crossref] [PubMed]
  9. N. Demoli, J. Meštrović, and I. Sović, “Subtraction digital holography,” Appl. Opt. 42, 798–804 (2003).
    [Crossref] [PubMed]
  10. N. Demoli, D. Vukicevic, and M. Torzynski, “Dynamic digital holographic interferometry with three wavelenths,” Opt. Express 11, 767–774 (2003), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-11-7-767.
    [Crossref] [PubMed]
  11. N. Demoli and D. Vukicevic, “Detection of hidden stationary deformations of vibrating surfaces by use of time-averaged digital holographic interferometry,” Opt. Lett. 29, 2423–2425 (2004).
    [Crossref] [PubMed]

2004 (1)

2003 (3)

1992 (1)

T. D. Rossing, I. Bork, H. Zhao, and D. Fystrom, “Acoustics of snare drums,” J. Acoust. Soc. Am. 92, 84–94 (1992).
[Crossref]

1984 (1)

1981 (1)

C. M. Hutchins, “The acoustics of violin plates,” Scientific American 245, 170–186 (1981).
[Crossref]

1971 (2)

J. N. Butters and J. A. Leendertz, “Holographic and video techniques applied to engineering measurement,” Meas. Control 4, 349–354 (1971).

E. V. Jansson, “A study of acoustical and hologram interferometric measurements on the top plate vibrations of a guitar,” Acustica 25, 95–100 (1971).

1965 (1)

Bork, I.

T. D. Rossing, I. Bork, H. Zhao, and D. Fystrom, “Acoustics of snare drums,” J. Acoust. Soc. Am. 92, 84–94 (1992).
[Crossref]

Butters, J. N.

J. N. Butters and J. A. Leendertz, “Holographic and video techniques applied to engineering measurement,” Meas. Control 4, 349–354 (1971).

Demoli, N.

Fletcher, N. H.

N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments (Springer-Verlag, New York, 1999).

Fystrom, D.

T. D. Rossing, I. Bork, H. Zhao, and D. Fystrom, “Acoustics of snare drums,” J. Acoust. Soc. Am. 92, 84–94 (1992).
[Crossref]

Gougeon, S.

Hutchins, C. M.

C. M. Hutchins, “The acoustics of violin plates,” Scientific American 245, 170–186 (1981).
[Crossref]

Jansson, E. V.

E. V. Jansson, “A study of acoustical and hologram interferometric measurements on the top plate vibrations of a guitar,” Acustica 25, 95–100 (1971).

Ledang, O. K.

Leendertz, J. A.

J. N. Butters and J. A. Leendertz, “Holographic and video techniques applied to engineering measurement,” Meas. Control 4, 349–354 (1971).

Leval, J.

Lokberg, O. J.

Meštrovic, J.

Mounier, D.

Picart, P.

Powel, R. L.

Rossing, T. D.

T. D. Rossing, I. Bork, H. Zhao, and D. Fystrom, “Acoustics of snare drums,” J. Acoust. Soc. Am. 92, 84–94 (1992).
[Crossref]

N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments (Springer-Verlag, New York, 1999).

Sovic, I.

Stetson, K. A.

Torzynski, M.

Vukicevic, D.

Zhao, H.

T. D. Rossing, I. Bork, H. Zhao, and D. Fystrom, “Acoustics of snare drums,” J. Acoust. Soc. Am. 92, 84–94 (1992).
[Crossref]

Acustica (1)

E. V. Jansson, “A study of acoustical and hologram interferometric measurements on the top plate vibrations of a guitar,” Acustica 25, 95–100 (1971).

Appl. Opt. (2)

J. Acoust. Soc. Am. (1)

T. D. Rossing, I. Bork, H. Zhao, and D. Fystrom, “Acoustics of snare drums,” J. Acoust. Soc. Am. 92, 84–94 (1992).
[Crossref]

J. Opt. Soc. Am. (1)

Meas. Control (1)

J. N. Butters and J. A. Leendertz, “Holographic and video techniques applied to engineering measurement,” Meas. Control 4, 349–354 (1971).

Opt. Express (1)

Opt. Lett. (2)

Scientific American (1)

C. M. Hutchins, “The acoustics of violin plates,” Scientific American 245, 170–186 (1981).
[Crossref]

Other (1)

N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments (Springer-Verlag, New York, 1999).

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.
Fig. 2.
Fig. 2. An example of the colored digital hologram (left) and its portion (right).
Fig. 3.
Fig. 3. A movie composed from experimentally obtained data for modal characteristics of a drum. [Media 1]
Fig. 4.
Fig. 4. A movie obtained numerically for a membrane analog to the drum. [Media 2]

Tables (2)

Tables Icon

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

Tables Icon

Table 2. Resonant frequencies measured experimentally

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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