The Partial Derivative of 3D Hysteresis Spring Model for Harmonic Balance Method

Dry friction is widely used for reducing vibrations in turbine engines; however, the nonlinear forces at contact surfaces make solving the vibration problems of these structures challenging. The hysteresis spring contact model is commonly used for calculating contact forces. This paper presents a prediction-correction procedure for calculating contact forces based on displacement using a three-dimensional hysteresis spring contact model. The Harmonic Balance Method (HBM) is typically employed to analyze the dynamic behavior of structures with dry friction. During the application of HBM, the Newton-Raphson method is used to solve the dynamic equation. This procedure requires the partial derivative of the Fourier series of the friction force with respect to the Fourier series of the displacement. An analytical method for calculating this partial derivative is proposed. Compared to the finite difference method, the analytical method is more accurate, requiring less steps to converge, and faster, which can accelerate the solving procedure by 38% to 50%.