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Relaxation of a solid geometry with initial displacement to fundamental normal mode
Posted Jan 7, 2020, 6:57 a.m. EST MEMS & Nanotechnology, Structural & Acoustics, Acoustics & Vibrations Version 5.4 2 Replies
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For a given solid structure (Cantilever beam) with size: length=8mm width=4mm and thickness=0.1mm, I want to investigate the transient behaviour for a given initial displacement. The initial displacement is everywhere zero excepted form a small area on the top surface where the displacement has a 2D gaussian distribution with radius of about 50µm. Right now, I am using a time dependent study, but I was wondering if there is a better method to investigate such Problems?
Is it perhaps possible to look at the exited normal modes with a frequency-based study? If yes: Can such a study show how the intensity of a particular normal modes will change over time? And will such a Problem at last always relax into a vibration of the fundamental mode?
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There is problem involving string vibration that will give you some insight. You will find this is many standard textbooks.
Consider a string between two fixed supports with a prescribed initial displacement that is not a normal mode. That initial displacement can be expressed as a sum of normal mode displacements (this turns out to be a Fourier series). We know how each of those normal modes will evolve in time, each with its own frequency. At any given time we can sum the displacements of all the normal modes, to find the actual displacement at that time.
You could in principle do the same thing using the eigenmodes of your system.
And as far as what will happen- at t>0 the solution will contain many modes (although possibly at very low amplitude) unless the initial displacement corresponds to the fundamental mode.
D.W. Greve DWGreve Consulting
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Thanks for the reply Dave, I looked up the string vibration problem. However, I am not sure if this really fits to my problem. The (analytic) solution for the string vibration contains only higher harmonics of a sin and cos function so if I understood it right this means that after a time period defined by the speed and the boundary conditions the shape of the initial shape should be recovered. For the (analytical) solution of a Cantilevered beam the functions are not just simple harmonics and the 3D case which I am considering is even more complex. In my FEM calculations it seems to me that at last a vibration of fundamental modes are domination. But the initial shape sems to be never recovered after t>0.