I had the chance to attend a presentation by Cyril Touzé from ENSTA ParisTech, about non linear vibration of thin plates. Usually the term « Non linear » is a show stopper for most engineers, It just means that the mathematical description of a system can not or no longer be simplified using linear relations (things like : If I strike a structure with 2x more energy, I will make it vibrate 2x more, if I drive a structure with a given frequency content, I will obtain a result only in the driving frequency range…), therefore modelling the behaviour to predict sound output of a structure for example is more complex than in the « Linear world ».
For plates, as soon as the amplitude of vibration is in the order of magnitude of the thickness, non linear behaviour occurs.
For instance, even if you strike a gong with a soft mallet – injecting energy therefore « only » in low frequency, the resulting sound will be composed of much higher frequencies because the overall level of vibration will cause the gong to get in a « saturated/turbulent » state, and the more energy you will try to inject further, the higher the maximum frequency will be : energy flows from the blows of the mallet into the whole bandwidth of the structure, and « modal » behaviour is long gone.
Gong players sometimes « prime » their gongs to raise the vibratory energy inside and therefore put the gong closer to the turbulent / saturated state and ensure the characteristic sound.
Mr Touzé has presented a very interesting experiment, where a 50cm gong was driven by an electrodynamic shaker exactly at one of the gong natural frequency. Then the level of injected force was slowly raised, as radiated sound and vibration fo the gong were recorded.
with very low efforts injected, the modal behavior was prevalent with vibration and sound carrying only the excitation frequency : Purely linear state.
Then as the force increased, some other modes showed up, and some « cycling » occurred between them : the structure was exchanging energy from one mode to another, and in this still « periodic » state, simple relations between the appearing modes frequencies and driving frequency were verified.
Then suddenly, as the shaker was pushing even more energy into the structure, all the tonal content disappeared in wideband energy and the whole spectrum filled up as the gong was entering its « turbulent » state radiating the shimmering full and rich characteristic sound.
in « Idiophones à plaques et coques. Deuxième partie, non-linéarité forte : cymbales, tam-tams et plaque tonnerre » by Cyril Touzé, Enseignant-chercheur, Unité de Mécanique (UME), ENSTA-ParisTech, Palaiseau ; Olivier Thomas, Enseignant-chercheur, Laboratoire de Mécanique des Structures et Systèmes Couplés (LMSSC), CNAM, Paris
Sonagram of a Gong acceleration (frequency content vs time as level of force increases) : Three distincts states are observed : at low level of force (left side of the first dotted line) the response in only found in the driving frequency (556 Hz) and first harmonic.
Then a first bifurcation occurs and the gong enters in the quasi periodic state where it is still driven by the modal behavior : many partials are showing up, note that the sum of the frequencies of the two lower modes of higher amplitudes (around 250 and 300 Hz) are exactly equal to the driving frequency. The overall response (middle curve in blue) shows time fluctuation as modes are swapping energy between them.
Finally, second bifurcation for higher level of efforts and turbulent behavior prevails, with ultra rich frequency content, notably higher in frequency than the driving force.
What happens when you strike a structure ?
From a mallet or stick standpoint :
Depending on the « tool » selected to strike a structure, the frequency content or ability to produce high frequency will depend on the « softness » of the striking surface (assuming this one is way « softer » than the receiving cymbal or gong which is not always true) as shown in the table below.
The following table summarise the same phenomenon either seen from a time domain standpoint, or a geometrical standpoint, or a frequency content standpoint : The softer the mallet, the longer the duration of the shock and the lower the frequency content (works as inverse of shock duration).
In the space domain, the softer the mallet, the larger the contact area and the lower the highest wavelenght generated (the mallet or sticks blocks the struck surface and kills any generation of wavelength shorter than the contact area).
From a Struck structure standpoint :
If we take for instance thin cymbals and plate and hit them with sufficient energy, those will be almost instantaneously driven into their non linear turbulent state with the characteristic crash sound. Even if a soft mallet is used with a sufficiently intense blow, the whole spectrum will be heard.
As the structure is saturated, if you continue injecting energy (such as playing a roll with increasing intensity) the energy will spread to a higher frequency.
With thicker cymbals (« un-crashable ») like some rides, bells, etc…the amplitude of vibration even with heavy hits will stay an order of magnitude under the plate thickness and the vibration thus the sound radiated will be tonally characterised with a distinctive pitch, and a quasi linear (modal) behaviour.