Interdisciplinary Applied Mathematics

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Linear Modes of Vibration

The basic idea in this method is to represent the physical variable, e.g., the deformed shape of the microdevice, as a summation of the linear normal mode shapes. This results in the transformation from the nodal coordinates to the time-dependent coefficients of the mode shapes, called modal coordinates (Ananthasuresh et al., 1996). This approach also eliminates the coupling between the inertia and stiffness matrices of the governing equations. Assuming that higher modes of vibration have negligible effect on the system’s response, a reduced-order model is obtained by using only the first few modes. Instead of the original system of N coupled equations, being the total number of degrees of freedom, only n equations need to be solved in the reduced-order model, where n is the number of modes considered. The number of modes considered determines both the accuracy and the computation time of the system. A general procedure for this method is given as (Ananthasuresh et al., 1996):

1.    Derive    basis    functions    from    an    initially    meshed    structure    by    solving

for the small-amplitude (linear) modes of a structure.

2.    Form a basis set that is orthonormal in the state space.

3. Consider the first few modes to represent the physical variable (s) of interest (e.g., structural deformation).

4. Represent the solution to the system as a linear combination of the modes with time-dependent coefficients.

The undamped dynamical behavior of a fully meshed structure is given by

Mx + Kx = F(x,t),    (17.17)

where M is the global inertial matrix, K is the global stiffness matrix, and F(x,t) is    the    nonlinear    external    force. Let    S be    the    modal matrix,    i.e.,    an

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