A Direction Interpolation Method Based on the Trigonometric Collocation Method and the Runge–Kutta Method

Abstract The direction interpolation method (DIM) based on the trigonometric collocation method and the Runge–Kutta (RK) method is developed in this work to track periodic responses in nonlinear systems. For traditional semi-numerical and semi-analytical methods such as the harmonic balance (HB) and incremental harmonic balance (IHB) methods, the Newton–Raphson iteration is essential. However, the iteration has a strict requirement about initial values, and the iteration may not converge when initial values are far away from exact solutions. In contrast, the DIM utilizes the RK method to provide the high-accuracy solution, thereby eliminating the need for the Newton–Raphson iteration and enhancing the robustness and efficiency of the response-tracking procedure. The method begins by employing the trigonometric collocation approach to obtain periodic solutions, which serve as initial conditions for the DIM. Then, a set of differential equations governing the evolution of Fourier coefficients are constructed, and these equations are solved using an order-four RK scheme. To demonstrate the effectiveness of the method, three nonlinear systems are investigated by the DIM, the trigonometric collocation method, and the continuation method: the van der Pol oscillator, the van der Pol–Mathieu oscillator with external excitation, and a slider-pendulum system under external harmonic force excitation. The first two examples mainly verify that the accuracy of the order-four DIM is o(h4) and the DIM takes slightly less time than the continuation method calculating same responses. The third example demonstrates that the DIM with most steps can obtain the full response while the continuation method with small steps can obtain full responses. Furthermore, solutions obtained via the DIM and the trigonometric collocation method show excellent agreement with results of the numerical integration (NI) by the RK method.