A Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Solutions of a Nonlinear Partial Differential Equation With a Linear Complex Boundary Condition

A spatial and temporal harmonic balance (STHB) method is demonstrated in this work by solving periodic solutions of a nonlinear string equation with a linear complex boundary condition, and stability analysis of the solutions is conducted by using the Hill’s method. In the STHB method, sine functions are used as basis functions in the space coordinate of the solutions, so that the spatial harmonic balance procedure can be implemented by the fast discrete sine transform. A trial function of a solution is formed by truncated sine functions and an additional function to satisfy the boundary conditions. In order to use sine functions as test functions, the method derives a relationship between the additional coordinate associated with the additional function and generalized coordinates associated with the sine functions. An analytical method to derive the Jacobian matrix of the harmonic balanced residual is also developed, and the matrix can be used in the Newton method to solve periodic solutions. The STHB procedures and analytical derivation of the Jacobian matrix make solutions of the nonlinear string equation with the linear spring boundary condition efficient and easy to be implemented by computer programs. The relationship between the Jacobian matrix and the system matrix of linearized ordinary differential equations (ODEs) that are associated with the governing partial differential equation is also developed, so that one can directly use the Hill’s method to analyze the stability of the periodic solutions without deriving the linearized ODEs. The frequency-response curve of the periodic solutions is obtained and their stability is examined.

The Spatial and Temporal Harmonic Balance Method for Obtaining Periodic Responses of a Nonlinear Partial Differential Equation With a Linear Complex Boundary Condition

The spatial and temporal harmonic balance (STHB) method is used to solve the periodic solution for a nonlinear partial differential equation (PDE) demonstrated by a nonlinear string equation with a linear complex boundary condition, and stablity analysis is conducted for the periodic solutions using Hill’s method. In order to avoid the integration procedure for discretizing the PDE to obtain the ordinary differential equations (ODEs), spatial and temporal harmonic balance procedures are conducted simultaneously, which can be efficiently achieved by the discrete sine transform and the fast Fourier transform. An additional coordinate associated with the generalized coordinates of the trial functions for the spatial discretization is introduced to make the solution satisfy all boundary conditions, and a relationship of the additional coordinate and the generalized coordinates is developed and used in the STHB method so that the test functions can be the same to the trial functions. Jacobian matrix of the harmonic balanced residual is obtained analytically, which can be used in Newton method for solving the periodic response. The STHB method and Jacobian matrix make the calculation of the periodic solution for the nonlinear string with a linear spring boundary condition efficient and easy to be implemented by computer programs. The relationship between the Jacobian matrix and the system matrix of the linearized ODEs are developed, so that one can directly obtain the Toeplitz form of the system matrix, and Hill’s method can be used to analyze the stability with the eigenvalues of the Toeplitz-form system matrix without the derivation of the ODEs. The frequency curve of the periodic solutions is obtained and their stability is indicated by the method in this work.