# Vibration following an impact

This high-speed video shows a cymbal's response to impact. I'm interested in how the observed vibration relates to the modes and frequencies that an eigenvalue vibration analysis would predict. My interest is not specific to cymbals, the video just illustrates a point very well.

I suspect that since excitation frequency here is much higher than the fundamental frequency of the body, and displacements are very large, eigenvalue analysis is not appropriate. Is this reasoning correct?

My guess would be that the initial response to the impact is transient (wave propagation) and would eventually converge to the combination of modes predicted by the eigenvalue analysis. Is that correct? If so, what influences the speed of this convergence?

• Eigenvalue analysis is most certainly appropriate. For the simplest case of an instantaneous impact, The amplitude of each mode shape depends on the so-called "mode participation factor" for each mode, which depends on the mode shape and the point of impact. For example, if the impact occurs at a nodal point of a particular mode where its amplitude is zero, that mode will not be excited at all. Apr 6 '21 at 19:23
• Musician here: the modes you excite depend very strongly on where (radially) you strike the cymbal. In general, the energy decays (propagated into the air, primarily) long before the excited modes "collapse" into more stable modes. <-- which is basically a qualitative restatement of @alephzero 's comment about mode excitation. Apr 7 '21 at 12:23

Regarding your first question IMHO eigenvalue analysis is appropriate and valid as long as the material is within the elastic range. Also another misconception is that, the application of eigenvalue analysis depend on how close the excitation frequency to the fundamental frequency. (In actuality, IMHO you can regard the cymbals case as an impulse, not as a harmonic excitation).

Regarding the second part of the question:

My guess would be that the initial response to the impact is transient (wave propagation) and would eventually converge to the combination of modes predicted by the eigenvalue analysis. Is that correct? If so, what influences the speed of this convergence?

Eigenvalue analysis is not about the steady state response. In many aspects, it is mainly about the transient response. Somehow I got the feeling that you interpret, the eigenmodes (or mode shapes) like standing waves (which in my opinion they are not). IMHO, eigenvalue analysis is about expanding the concept of the natural frequency of a SDOF to a MDOF or continuous system.

In the SDOF system, the transient response is depended on the natural frequency, and the steady state response is depended on the excitation frequency. So, if you extend that to an MDOF, the natural frequencies are dominant on the transient response. Also, since the excitation of the cymbals is impulsive (not harmonic), their response also qualifies (IMHO) as transient.

• Creating a transient dynamic model using (damped) vibration modes is a standard procedure, but since we have no idea what background knowledge the OP has (e.g. Fourier transforms and convolution) writing an answer giving the math might not mean very much. Apr 7 '21 at 0:43
• @alephzero I knew I was forgetting something important, as I was writing the answer. There were many things I did not agree with the original post, that it was hard to write a properly worded answer. Although (you probably will be able to clarify this for me), in all fairness, if there is a harmonic excitation (not a general periodic) eventually you do have a steady state, even in MDOF systems. Isn't that right or am I oversimplifying by extending the SDOF to the MDOF? Apr 7 '21 at 3:17
• @alephzero I would appreciate the math, if someone could write it up or link to a reference. I am familiar with Fourier transforms and convolution.
– ik2
Apr 7 '21 at 15:35