Nonlinear Shell Vibration: Cymbals

Maybe the most dramatic example of the effect of a nonlinearity in all of musical acoustics is the cymbal crash…here, a linear model doesn’t even begin to capture this effect. But the incorporation of a rather simple nonlinearity does…see the text for much more on this, and have a listen to the sound examples below.

 

A cymbal may be modelled as a curved plate, or shell, under free edge conditions, and under a variety of possible conditions at the center (close to clamped in the case of a high-hat, pivoting in the case of a ride cymbal, or perhaps free as for an orchestral crash cymbal. .

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The curvature of the shell has a rather large effect on the resulting sound---consider, for the moment, the case of linear vibration of a spherical cap, in which case a decomposition into modes makes sense. As the curvature increases, one family of modes exhibits an upward shift in frequency, as shown below. The result is a spectrum with a dense clustering of partials in the mid-frequency range, leading to the characteristic noisy timbre of the cymbal (compared with flatter percussion instruments such as gongs).

 

            Low curvature                                              Medium Curvature                                        High Curvature

 

The response of the shell model at high amplitudes is quite different from at low frequencies…the “modes” interact, leading to a dramatic increase in high-frequency energy…see the comparison below between spectrograms of synthetic sound output from a linear model, and a nonlinear model:

 

 

                               Low amplitude response                                           High amplitude response

 

Finite difference schemes can be employed over a grid in polar coordinates:

 

 

 

A time domain finite difference scheme can then be used to simulate the evolution of the state of the shell subject to, say, a localized force.

 

 

 

It’s interesting to compare sounds produced using a linear model with those produced by a nonlinear model:

 

LINEAR SOUND EXAMPLE

 

NONLINEAR SOUND EXAMPLE

 

As expected, the linear model produces a bell-like tone, lacking in character, especially in the attack portion. You can really hear the importance of the nonlinear effect when you listen to a full cymbal gesture…

 

CYMBAL SOUND EXAMPLE

 

ANOTHER CYMBAL SOUND EXAMPLE

 

You can also listen to the effect of increased curvature on the timbre…

 

FLAT CIRCULAR PLATE SOUND EXAMPLE

 

LIGHTLY CURVED SHELL SOUND EXAMPLE

 

STRONGLY CURVED SHELL SOUND EXAMPLE