Semi-numerical analysis of a two-stage series composite planetary transmission considering incremental harmonic balance and multi-scale perturbation methods

This study conducts an analytical investigation of the dynamic response characteristics of a two-stage series composite system (TsSCS) with a planetary transmission consisting of dual-power branches. An improved incremental harmonic balance (IHB) method, which solves the closed solution of incremental parameters passing through the singularity point of the analytical path, based on the arc length extension technique, is proposed. The results are compared with those of the numerical integration method to verify the feasibility and effectiveness of the improved method. Following that, the multi-scale perturbation (MsP) method is applied to the TsSCS proposed in this subject to analyze the parameter excitation and gap nonlinear equations and then to obtain the analytical frequency response functions including the fundamental, subharmonic, and superharmonic resonance responses. The frequency response equations of the primary resonance, subharmonic resonance, and superharmonic resonance are solved to generate the frequency response characteristic curves of the planetary gear system (PGS) in this method. A comparison between the results obtained by the MsP method and the numerical integration method proves that the former is ideal and credible in most regions. Considering the parameters of TsSCS excitation frequency and damping, the nonlinear response characteristics of steady-state motion are mutually converted. The effects of the time-varying parameters and the nonlinear deenthing caused by the gear teeth clearance on the amplitude–frequency characteristics of TsSCS components are studied in this special topic.

An Efficient Dimension-Adaptive Numerical Integration Method for Stochastic Dynamic Analysis of Structures with Uncertain Parameters

This paper presents a novel dimension-adaptive numerical integration method for dynamic analysis of structures with stochastic parameters subjected to deterministic excitations. First, an efficient dimension-adaptive algorithm is proposed to detect the importance of each random parameter involved in the structural model, based on which the quadrature nodes used for numerical integration can be collocated more reasonably. Then, the Gaussian quadrature formulas are used to evaluate the structural response statistics. To further improve the robustness and efficiency of the proposed method, the dimension-adaptive integration is only used to calculate the structural displacement response statistics. The velocity and acceleration response statistics are further evaluated using the finite difference formulas based on the concept of stochastic difference. Such a strategy is especially attractive when evaluating the response statistics of the derivative processes requires more quadrature nodes than that of the original process. Finally, two numerical examples encountered in civil engineering, including a shear frame with stochastic parameters subjected to a seismic ground motion and an Euler beam with unidimensional stochastic field of material properties (discretized via the Karhunen–Loève expansion) subjected to a moving load are studied to illustrate the performance of the proposed method. Via the numerical results, the accuracy and efficiency of the proposed method are verified.

Taylor Series Based Numerical Integration Method

This article deals with a high order integration method based on the Taylor series. The paper shows many positive properties of this method on a set of technical initial value problems. These problems can be transformed into the autonomous systems of ordinary differential equations for both linear and nonlinear problems. The MATLAB implementation of the method is compared with state-ofthe-art MATLAB solvers.

An Efficient Method for Calculating the Lightning Electromagnetic Field Over Perfectly Conducting Ground

In the implementation of the Cooray–Rubinstein formula, the calculation of a lightning electromagnetic field over perfectly conducting ground accounted for most of the computation time. Commonly, evaluating the ideal lightning electromagnetic field is based on the numerical integration method. In practice, only a sufficiently small discretization step is essential to get an accurate result, which leads to a relatively large number of calculations and results in a lengthy computation time. Besides, the programming is relatively complicated because the propagation of the lightning current along the channel must be considered. In order to increase the efficiency and simplify the programming, an improved method is proposed in this paper. In this method, the evaluation of the ideal lightning electromagnetic field is equated with a summation of analytical formulae and a simple integral operation, so it would be more efficient and easily programmed. The validation of the proposed method is demonstrated by some simulation examples.