A harmonic-energy balance method is put forward to solve the strong nonlinear pendulum equation. The difference from the classical perturbation method is that the harmonic balance method does not account the differential equation and initial conditions separately, but it considers both simultaneously. Through the harmonic-balance method, two-order ordinary differential equations describing dynamic systems become a set of nonlinear algebraic equations with the variables of angular frequency and amplitude. Considering the balance of energy, the close equations with angular frequency and amplitude as the variables can be solved. The harmonic-energy balance method is a combination of harmonic-balance and energy balance. It overcomes the shortcomings of both methods and takes their advantages. A case study also shows that the harmonicenergy balance method is simpler with higher precision although it takes less harmonics.
李银山,李彤,韦炳威,李欣业.用谐波-能量平衡法求解单摆方程[J].动力学与控制学报,2016,14(3):197~204; Li Yinshan, Li Tong, Wei Bingwei, Li Xinye. Harmonic-energy balance method for solving pendulum equation[J]. Journal of Dynamics and Control,2016,14(3):197-204.