Multidimensional Lindstedt–Poincaré method for nonlinear vibration of axially moving beams

The multidimensional Lindstedt–Poincaré (MDLP) method is extended to the nonlinear vibration analysis of axially moving systems. Galerkin method is used to discretize the governing equations. The forced response of an axially moving beam with internal resonance between the first two transverse modes is studied. The fundamental harmonic resonance is studied. The response curves exhibit the same internal resonance characteristics as that of non-transferring thin plates and beams because all these systems possess cubic nonlinearity and similar frequency distribution. The examples show that the results of the MDLP method agree reasonably well with that obtained by the incremental harmonic balance (IHB) method. However, the former is more straightforward and efficient for obtaining the solution.