Since the doubly clamped beam-type N/MEMS subjected to the van der Waals attraction is transformed into a mathematical model through Galerkin method, which is a nonlinear differential equation, the spreading residue harmonic balance method is introduced to solve the approximate solution of the nonlinear problem in this paper. The spreading residue harmonic balance method is developed on the basis of the residue harmonic balance method. The disadvantage of the approach is small parameter assumption is needed. In order to improve the accuracy of the approximate solution, the residual of the former order harmonic approximation is introduced. Besides, we compare the second-order spreading residue harmonic balance method solution with the numerical one by the Runge-Kutta method. This proves the availability and validity of spreading residue harmonic balance method.

The exact solutions of the nonlinear vibration systems are extremely complicated to be received, so it is crucial to analyze their approximate solutions. This paper employs the spreading residue harmonic balance method (SRHBM) to derive analytical approximate solutions for the fifth-order nonlinear problem, which corresponds to the strongly nonlinear vibration of an elastically restrained beam with a lumped mass. When the SRHBM is used, the residual terms are added to improve the accuracy of approximate solutions. Illustrative examples are provided along with verifying the accuracy of the present method and are compared with the HAM solutions, the EBM solutions, and exact solutions in tables. At the same time, the phase diagrams and time history curves are drawn by the mathematical software. Through analysis and discussion, the results obtained here demonstrate that the SRHBM is an effective and robust technique for nonlinear dynamical systems. In addition, the SRHBM can be widely applied to a variety of nonlinear dynamic systems.