Energy is an integral parameter of motion over the whole frequency range (modes for MDOF systems). In the case of certain spectral components attenuation, you have to use other criteria.

It depends... effectively, it depends on what you want to achieve / why you want to minimize the vibration.

As you already note, going with (kinetic / potential / total) energy gives you a scalar quantity that is well suited as cost function for minimization, so if you are just interested in some optimization cost function, energy is a valid approach.

As long as you are in the linear domain, you might even have a spectral look into energy (see Pavlo Krot ).

However, if the final issue / problem is with say, secondary noise radiation or vibration amplitudes at some point (potential mechanical contact) or leaving the linear vibration range or instationary signals or high crest factors or ... (you name it), you really need to have a more detailed look into your problem as minimizing some kind of energy for a specific simulation situation (as you rarely obtain all these vector quantities from real life measurements) is likely to not be sufficient.

Dear Sina, the response can only be YES as each vibration cycle achieves a total transfer between the structure's kinetic energy (at the zero amplitude of the cycle) and stress (potential) energy (at the maximum amplitude): minimizing one will minimize the other. This approach through the kinetic energy grounds in particular the "statistic energy analysis" (SEA) developed to model the high frequency vibration distribution in complex structural assemblies - from what you describes, if your plate presents already a significant modal density at the frequency you consider, you could use it to model your system. I disagree with the previous comments, it is not "it depends" but a loud YES. However, measuring it is not that easy because you have to weight the squared velocity with the proper "local mass" estimate, which strongly depends of the frequency for continuous structures. If your plate is homogeneous, you might just implement a regular mesh of miniature accelerometers or scan the plate with a laser vibrometer, and minimize the squared acceleration or velocity

Hello Bernard Garnier ,

I absolutely agree to your answer - as long as your implicit assumptions are valid, i.e. that a kind of "energetic" approach is adequate to the actual problem under consideration. For many problems, that is the case. However, as the original question is quite general, understanding the implicit assumptions and therefor the range / limits a specific approach is valid, is quite important.

Probably, I am a bit concerned about this personally: Having a physics background, I often see people from an engineering background to look for "just a method to do something" without paying consideration to the limits of such a method. I totally agree that this approach is sufficient, valid and important for many real world problems, however, not for all. That often is where the trouble begins...

dear Rolf, on your right grounds, you might have a look at the SEA methodology, its building from solid physics, and is applicable as soon as the response of several modes is involved in the frequency band of the investigation - which seems the case for Sina... Energy conservation is something engineers acknowledge too!

Do we minimize it? Or we share it between two systems?

- If the system is isolated, energy conservation implies that the only thing you can do is to share it in a different way allowing to minimize it where you need and increase it where it remains harmless - e.g. by fitting a tuned vibration absorber (TVA) that will concentrate kinetic energy on its resonance

- If you introduces decoupling between your various structural components, then you increase the kinetic energy on the source side and you decrease it downstream the isolators (impedance mismatch reduces the coupling between the components) - you remain an isolated system as a whole

- If you introduce damping, then you degrades dome of the kinetic energy into heat - the system is no more isolated, you export the heat

- if you increases the sound radiation efficiency, then you reduces the kinetic energy by radiating sound energy away - again you are no more an isolated system

- active control allows you to introduce energy from outside to counteract the original vibration - again you are no more an isolated system

(I don't see any other option...)

Modal damping ratio, based on the modal strain energy method (MSEM), is energy ratio ,including kinetic energy .Modal damping ratio can be used to evaluate the frequency band and degree of vibration attenuation quickly.and we can get the analytical formulas.

Yin WH, Sun FF. Analytical method of the modal damping ratio of the beam with distributed dissipative oscillators and application in broadband vibration mitigation. Mech. Syst. Signal Proc, 2022;166:108470.

Kinetic energy is a good criterion for vibration damping. However, the amplitude criterion (vibration transmissibility) is a better criterion for vibration isolation. See Vibration Control for Optomechanical Systems, World Scientific, 2021, ISBN 9789811237331, especially pages 101-103, 121-124, and 203-205.

Hello and dear all, I recommend to take a look to the publications of Michela Basili

in one of these publication she considered three objective function and one of these objective functions is the energy criterion .

Best regards,