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.

Wet multi-disc friction components heat dissipation capability and optimal oil supply under continuous braking condition

Purpose – The purpose of this study is to investigate wet multi-disc brake temperature field and optimal oil supply under continuous braking condition. The oil supply of wet multi-disc brake has a direct impact on the drivability and fuel economy for tracked vehicles. Too small flow will result in the higher temperature and failure of brake while excessive one will lead to slow engagement increasing disengaged torque and the transmission efficiency could decline notably. The optimal oil supply and brake temperature field were obtained in this research. Design/methodology/approach – This article investigated on the heat dissipation capability and optimal oil supply of the brake by the means of CFX model. The working condition was continuous braking and the lubricating and cooling factors were included in the model. Findings – That the complex trends with increased oil flow is inconsistent with the traditional formula in which the effects of grooves were neglected. The fitting curve of optimal oil supply can predict various needed oil flow in various rotating speed and it provides a theoretical guidance for oil supply design. Originality/value – Traditional empirical formula of heat transfer coefficient and Reynolds equation solved by different methods could be difficult to deal with the complex boundary conditions of wet multi-disc brake. CFX model can solve the problem of complex boundary condition. The optimal oil supply curve can provide a theoretical guidance for oil supply design.

Numerical Study of Damage Model for Reinforced Concrete Slab under Complex Boundary Condition Subjected to Air Blast Load

This study applies the nonlinear finite element analysis software LS-DYNA to conduct a numerical simulation of damage mode of reinforced concrete slab under complex boundary condition subjected to air blast load. The factors affecting damage model of reinforced concrete slab, standoff distance and explosives location, were analyzed. The results show that: With the increase of the standoff distance, the failure mode of RC slab gradually changes from localized punching failure to shear failure at the support. With the location for explosive changing from the center of RC slab to the beam plate combination, the failure mode of RC slabs gradually changes from localized punching failure to bending-shear failure under close-in blast loading.

Flood Risk Map Analysis Based on Dam-Break Simulation

The flood risk map is useful to make a regional flood emergency plan and analyze the dam-break flood risk. A three-dimensional mathematical model is applied to simulate the floods evolution with consideration of the dam-break process and the complex boundary condition synthetically, and the finite volume method is used to get a discrete solution. The flood evolution process of Dongwushi reservoir in the instantaneous collapse conditions is taken as a case, and the flood risk map is obtained based on the results of calculation.

Boundary Condition Updating of Bridge Structure Based on Mode Parameters

According to the complex boundary condition of real bridge structure, appendix restraint parameters of structure boundary condition were detailed analyzed. The appendix horizontal and torsional restraint parameters were selected as research object. The least square constrained optimization inversion method was used to establish the question of reverse of bridge structure boundary appendix parameters based on tested and calculated mode parameters. The first order perturbation of eigen-value and eigen-vector based on matrix perturbation theory were substituted into the constrained optimization inversion equation, so the solving efficiency was improved greatly, and the iteration method was used to improve the precision. The corresponding program was made, and the example of a maglev railway girder shows that fairly well precise bridge structure boundary appendix parameters can be reserved with only former several modes parameters through four to seven times of iteration calculation.