# Acoustics and Vibration Animations

Daniel A. Russell, Graduate Program in Acoustics, The Pennsylvania State University

Based on a work at http://www.acs.psu.edu/drussell/demos.html.

The content of this page was originally posted on January 24, 2012. The HTML code was modified to be HTML5 compliant on March 16, 2013.

# Vibrational Modes of a Tuning Fork

The tuning fork vibrational modes shown below were extracted from a COMSOL Multiphysics computer model built by one of my former students (Eric Rogers) as part of the final project for the structural vibration component of PHYS-485, Acoustic Testing & Modeling, a course that I taught for several years while I was a member of the physics faculty at Kettering University.

## Clang Mode (2585 Hz)

This is the second most commonly heard vibrational mode. It results from striking the tines of the fork against a hard object. This mode is the second mode shape for a clamped-free bar, and it has a much higher frequency (roughly 6.26 times higher than the fundamental). The clang tone may sound louder than the fundamental because the human ear is much more sensitive to frequencies between 1000 Hz and 4000 Hz while the ear does not hear frequencies below 500 Hz very well.

This is also a symmetric mode, since the two tines are mirror images of each other.

## Asymmetric Modes (in-plane bending)

In addition to the familiar symmetric fundamental and clang modes, a tuning fork can also exhibit a family of in-plane bending modes, similar to a the vibrational modes of a clamped-free solid bar. Instead of each tine oscillating as a separate clamped-free bar, the entire fork vibrates as one object. The animations at right show the first three such clamped-free mode shapes for the tuning fork. The frequencies are (from left to right) 385 Hz, 2171 Hz, and 4772 Hz. Notice that the first of these modes has a frequency lower than that of the fundamental mode at 426 Hz.

The first in-plane bending mode (385 Hz) radiates sound as a dipole source.[2] The plot at left shows the measured directivity pattern (dots) representing the sound pressure level as a function of angle around the fork along with the theoretical model (curve) for a dipole source.[2]

## Out-of-Plane Bending Modes

In addition to the in-plane bending modes, a tuning fork will also exhibit several out-of-plane bending modes where the fork acts like a solid bar, vibrating perpendicularly to the plane of the tines. Frequencies (left to right): 457 Hz, 2861 Hz. The lowest of these modes is very close to the fundamental frequency.

The first out-of-plane bending mode (457 Hz) radiates sound as a dipole source. The plot at left shows the measured directivity pattern (dots) representing the sound pressure level as a function of angle around the fork along with the theoretical model (curve) for a dipole source.[2]

## Asymmetric Out-of-Plane Bending Modes

A fork, clamped at the stem, will also exhibit asymmetric out-of-plane modes where the two tines oscillate perpendicular to the plane of the fork, but in opposite directions to each other. The fork essentially twists back and forth - rather like the torsional twisting modes of a solid bar. The frequencies for these two modes of vibration are 537 Hz and 3102 Hz. This vibrational mode is discussed briefly by Backus in his text on musical acoustics.[4]

The first out-of-plane bending mode (537 Hz) radiates sound as a lateral quadrupole source. The plot at left shows the measured directivity pattern (dots) representing the sound pressure level as a function of angle around the fork along with the theoretical model (curve) for a lateral quadrupole.[2]

References

1. R. M. Sillitto, "Angular distribution of the acoustic radiation from a tuning fork," Am. J. Phys. 34: 639–644 (1966).
2. D.A. Russell, "On the sound field radiated by a tuning fork," Am. J. Phys., 68(12), 1139-45 (2000).
3. Animated GIFs to accompany "On the sound field radiated by a tuning fork".
4. John Backus, The Acoustical Foundations of Music (W.W. Norton, 1969), p. 67.