We present an application of the eigenvalue method with symmetry for solving polynomial systems arising in the vibration analysis of mechanical structures with symmetry properties. The search for solutions is conducted by the so called multiplication matrix method in which the symmetry of the system is taken into account by introducing a symmetry group and by working with the set of invariant polynomials under the action of this group. Using this method, we compute the periodic solutions of a simple dynamic system modeling a cyclic mechanical structure subjected to nonlinearities.
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