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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.

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A two-degree-of-freedom coupled Duffing system with time delay is studied by a modified homotopy analysis method. First, a detailed calculation is given for the system. Second, single periodic and period-doubling solutions are obtained by solving the constructed nonlinear algebraic equation. Third, upon comparison of the periodic solution obtained by the multi-frequency homotopy analysis method with the numerical solutions obtained by the numerical method, it is found that the approximate solution agrees well with the numerical solution for the system. Finally, we discuss the convergence control parameters and convergence control function.