Energy Method
It is possible to calculate the frequency of different mode shape of system manually by the energy method. For a given mode shape of a multiple degree of freedom system you can find an "equivalent" mass, stiffness and applied force for a single degree of freedom system. For simple structures the basic mode shapes can be found by inspection, but it is not a conservative method. Rayleigh's principle states:
"The frequency ω of an arbitrary mode of vibration, calculated by the energy method, is always greater than - or equal to - the fundamental frequency ωn."
For an assumed mode shape, of a structural system with mass M; stiffness, EI (Young's modulus, E, multiplied by the second moment of area, I); and applied force, F(x):
then, as above:
Read more about this topic: Structural Dynamics, Modal Analysis
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