On a Method of Solution for the Coupled Hill Type Equations and Its Application to the Study of the Stability of Nonlinear Vibrations

摘要

In this paper, a method of the stability analysis for large amplitude steady state response of a nonlinear beam and flat plate under periodic excitation. The nonlinearity is attributed to the membrane tension which is developed when the beam and plate deflections are not small in comparison to their thickness. This problem is analyzed by the application of a Galerkin method, in which the effect of multi-mode participation is considered, and an unspecified function of the time resulting in nonlinear coupled ordinary differential equations of motion is solved by the harrnonic balance method and the Newton Raphson method. The stability question is investigated by studyingthe behavior of a small perturbation of the steady state response. The perturbation equations of the present method of solution reduce to the coupled Hill type equation. Assuming the solution of the form as a product of characteristic component and a Fourier series which represents the periodicity of motion and application of the harmonic balance method can transform the stability problem into the eigenvalue problem of a nonsymmetric matrix. After a proper transformation, the eigenvalues can be calculated on a digital computer by the QR double step method. The effectiveness and the accuracy of the proposed method are examined for a Mathieu equation whose stability has been worked out in detail and the application to stability analysis of the nonlinear vibrations of beams are presented.