# Interdisciplinary Applied Mathematics

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tion states (large deflection case). In this method, an equivalent damping and stiffness matrix that captures the true dependency between structural velocities and fluid pressure is computed in the modal coordinates. The damping Cjiqi and the stiffness coefficients Kjiqi of such a matrix representation can be obtained from the following modal force balance equation:

Cjiqi + Kjiqi    ^j ^ N р(ф1 qi)dA:

where qi is the modal coordinate, ф1 is the ith eigenvector (mode), and NT is the vector of finite element shape functions. Here Cji and Kji state the dependency between structural wall velocities caused by mode i and the reacting fluid forces that act on mode j. The damping and the squeeze coefficients of each mode are the main diagonal terms. Off-diagonal terms represent the fluidic crosstalk among modes, which happens in case of asymmetric gap separation. The following steps are performed to obtain the coefficients of C and K:

1. The squeezed film model is excited by wall velocities that are equal to the values of the first eigenvector (mode).

2. A harmonic response analysis is performed to compute the pressure response in the entire frequency range.

3.    The    real    and    the    imaginary    parts    of    the    element pressure    are    inte

grated and the complex nodal force vector computed for each frequency.

4.    The    scalar    product    of all    eigenvectors and    the    nodal    force    vector    of

step 3 is computed. The resulting numbers are modal forces, which indicate how much of the pressure distribution acts on each mode.

5. The damping and the stiffness coefficients are extracted from the real and the imaginary parts of the modal forces.

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