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Axisymmetric vibrations of a cylindrical resonator measured by holographic interferometry

Oscillatory displacements of a special sonic resonator, measured by holographic interferometry, are compared with theoretical results

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Abstract

A circular, cylindrical, ultrasonic resonator excited at one of its resonant frequencies is studied by holographic interferometry. Displacement distributions associated with the axisymmetric oscillations of the resonator are measured with the aid of time-average holograms, and are compared with a simple one-dimensional theory of rod vibrations, corrected for radial inertia. Analysis shows the overall error bounds on measured displacements to be ±9 percent of the maximum displacement at the resonator tip. Although the accuracy of measurements could be increased by refinements in experimental techniques, the work reported here represents substantial improvement in measuring the vibratory motion characteristics of ultrasonic devices over the point-by-point technique used heretofore.

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Abbreviations

A :

integration constant

a :

resonator radius, m

B :

integration constant

E :

Young’s modulus, N/m2

f n :

nth natural frequency, Hz

I :

light intensity of vibrating point on reconstructed image, W/m2

I stM :

light intensity of stationary point on reconstructed image, W/m2

k :

equivalent spring constant of piezoelectric ceramics, N/m

L :

half length of resonator, m

\(\hat n_i\) :

unit vector from a surface point toward source of illumination

\(\hat n_o\) :

unit vector from a surface point toward point of observation

N r :

radial component of sensitivity vector

N z :

axial component of sensitivity vector

p n :

nth eigenvalue of resonator model

\(\hat r\) :

unit vector in radial direction

t :

time

U(z) :

radial-displacement amplitude at lateral surface, m

\(\underset{\raise0.3em\hbox{$\smash{\scriptscriptstyle\thicksim}$}}{U}\) :

\(\left\{ {\begin{array}{*{20}c} {W(z)} \\ {U(z)} \\ \end{array} } \right\} = displacement{\text{ }}vector{\text{ }}of{\text{ }}a{\text{ }}surface{\text{ }}point,{\text{ }}m\)

W(z) :

longitudinal-displacement amplitude, m

u(z,t) :

radial displacement at lateral surface, m

w(z,t) :

longitudinal displacement, m

\(\hat z\) :

unit vector in axial direction

Δϕ:

phase change due to displacement of surface point, rad.

k 1 :

correction multiplier,k 1=1

Λ:

wavelength of coherent light, 632.8 nm

λ:

Lamé’s constant, N/m2

μ:

Lamé’s constant, N/m2

\(\omega _n\) :

nth circular frequency, 1/s

ρ:

density, kg/m3

\(\hat \theta\) :

unit vector in circumferential direction

\(\zeta _i\) :

angle enclosed by\(\hat n_i\) and\(\hat z\)

\(\xi _i\) :

angle enclosed by\(\hat n_i\) and\(\hat r\), at θ=0

\(\zeta _o\) :

angle enclosed by\(\hat n_0\) and\(\hat z\)

\(\xi _o\) :

angle enclosed by\(\hat n_0\) and\(\hat r\), at θ=0

v :

Poisson’s ratio

References

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Author information

Authors and Affiliations

  1. Department Engineer, Engineering Development Laboratory, E. I. du Pont de Nemours and Co., 19898, Wilmington, DE

    P. A. Tuschak

  2. Corning Glass Co., Corning, N.Y.

    R. A. Allaire

Authors
  1. P. A. Tuschak
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  2. R. A. Allaire
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Additional information

P. A. Tuschak and R. A. Allaire were Assistant Professor and Teaching Associate, respectively, Department of Engineering Mechanics, The Ohio State University, Columbus, OH 43210, when this paper was prepared.

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Tuschak, P.A., Allaire, R.A. Axisymmetric vibrations of a cylindrical resonator measured by holographic interferometry. Experimental Mechanics 15, 81–88 (1975). https://doi.org/10.1007/BF02320636

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  • DOI: https://doi.org/10.1007/BF02320636

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