Linear and nonlinear modal analysis of the axially moving continua based on the invariant manifold method

By studying the transverse dynamics of the axially moving beam, the linear and nonlinear complex modes of gyroscopic continua are investigated based on the invariant manifold method. The nonlinear partial differential equations are truncated into a set of ordinary differential equations by the Galerkin method. The invariant manifold method is then used to obtain the linear and nonlinear complex modes. The gyroscopic effect due to the axially moving speed is discussed by comparing the proportions of the trial functions in the complex mode during the modal motions. The 'traveling wave' phenomena are found for such a gyroscopic continuum. The power of the invariant manifold method in the application to gyroscopic systems is demonstrated and discussed by taking the axially moving continuum as an example.