Abstract
We demonstrate the importance of using the method of superposition for the precise calculation of frequencies and forms of axially symmetric elastic modes of test masses in gravitational antennas LIGO for detailed analysis of the effect of parametric oscillatory instability. The method allows one to construct analytical expressions for particular solutions to the medium equations of motion in cylindrical coordinates, which allow one to satisfy all zero boundary conditions. The obtained results allow considerably more reliable methods (as compared to meshing methods) to be used for predicting the number of combinations of optical and elastic modes that lead to the undesirable effect of the parametric oscillatory instability.
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