transient vibration

[hellen8]

I have read with interest and, often, amusement the posts regarding impact forces. My question is an academic one. Immediately upon impact there can be a variable number of rather high frequency, low amplitude vibrations before the system settles to a steady state. I should like to model this mathematically using a second order nonlinear differential equation incorporating both the Duffing and van der Pol nonlinear functions. I am interested, perhaps obviously, in the immediate, transient response. I can get a nice, smooth transient but can I simulate the vibrations which may occur in the real world upon impact?

 

[MikeyP]

To simulate a "real world" respose you need a "real world" applied force (although I have never seen a mechanical Van der Pol oscillator that I could hit with a hammer!). The ideal impulse forcing function is a Dirac delta function where there is equal signal power at all forcing frequencies. In practice, this is not possible (it requires infinite energy!).

Assume for the moment that the impactor and impactee are both elastic and that all energy is transferred from the impactor to the impactee. For a stiff impactor, the energy is spread over a wide range of frequencies. If the impactor is less stiff, then the energy is concentrated over a narrower range of frequencies. The shape of this power spectrum is relatively flat in those ranges and then rolls off slowly. From years of doing impact modal testing with various hammers on various structures, I would suggest therefore that a reasonable first stab at a "real world" impact force would be a Dirac delta function (or a sampled representation of one) which has been passed through a 1st order low pass filter. Of course the actual spectrum depends on both the stiffness of the impactor AND the stiffness (ie the dynamic point impedance) of the impactee.

If you need something more sophisticated then I suggest you use a high end FE package which can do decent contact analysis. Although if you are studying such fundamental systems as the 1DOF Duffing and Van der Pol systems then, perhaps my approach described above is more appropriate.

M

--
Dr Michael F Platten

 

[hellen8]

Thank you very much. I shall take awhile to digest that. The less sophisticated is what I shall start trying to deal with.