A Universal Nonlinear Analyzer for Rigid Multibody Systems Based on the Efficient Galerkin Averaging-Incremental Harmonic Balance Method

2021 ◽  
Author(s):  
Ren Ju ◽  
Wei Fan ◽  
Weidong Zhu
Keyword(s):  
Harmonic Balance ◽  
Multibody Systems ◽  
Nonlinear Dynamic Analysis ◽  
Response Curves ◽  
Multibody Dynamic ◽  
Incremental Harmonic Balance ◽  
Modeling Theory ◽  
Rigid Multibody Systems ◽  
Rigid Multibody ◽  
Ihb Method

Abstract The bridge between the multibody dynamic modeling theory and nonlinear dynamic analysis theory is built for the first time in this work by introducing an efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method for steady-state nonlinear dynamic analysis of index-3 differential algebraic equations (DAEs) for general rigid multibody systems. The multibody dynamic modeling theory has made significant advances in generality and simplicity, and multibody systems are usually governed by DAEs. Since the fast Fourier transform and EGA are used, the EGA-IHB method has excellent robustness and computational efficiency. Since the Floquet theory cannot be directly used for stability analysis of periodic responses of DAEs, a new stability analysis procedure is developed, where perturbed, linearized DAEs are reduced to ordinary differential equations with use of independent generalized coordinates. A modified arc-length continuation method with a scaling strategy is used for calculating response curves and conducting parameter studies. Three examples are used to show the performance and capability of the current method. Periodic solutions of DAEs from the EGA-IHB method show excellent agreement with those from numerical integration methods. Amplitude-frequency and amplitude-parameter response curves are generated, and stability and period-doubling bifurcations are analyzed. The EGA-IHB method can be used as a universal solver and nonlinear analyzer for obtaining steady-state periodic responses of DAEs for general multibody systems.

An Efficient Galerkin Averaging-Incremental Harmonic Balance Method Based on the Fast Fourier Transform and Tensor Contraction

2020 ◽  
Author(s):  
R. Ju ◽  
W. Fan ◽  
W. D. Zhu
Keyword(s):  
Fourier Transform ◽  
Fast Fourier Transform ◽  
Harmonic Balance ◽  
High Efficiency ◽  
Jacobian Matrix ◽  
Sampling Rate ◽  
Harmonic Balance Method ◽  
Incremental Harmonic Balance ◽  
Tensor Contraction ◽  
Ihb Method

Abstract An efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method is developed based on the fast Fourier transform (FFT) and tensor contraction to increase efficiency and robustness of the IHB method when calculating periodic responses of complex nonlinear systems with non-polynomial nonlinearities. As a semi-analytical method, derivation of formulae and programming are significantly simplified in the EGA-IHB method. The residual vector and Jacobian matrix corresponding to nonlinear terms in the EGA-IHB method are expressed using truncated Fourier series. After calculating Fourier coefficient vectors using the FFT, tensor contraction is used to calculate the Jacobian matrix, which can significantly improve numerical efficiency. Since inaccurate results may be obtained from discrete Fourier transform-based methods when aliasing occurs, the minimal non-aliasing sampling rate is determined for the EGA-IHB method. Performances of the EGA-IHB method are analyzed using several benchmark examples; its accuracy, efficiency, convergence, and robustness are analyzed and compared with several widely used semi-analytical methods. The EGA-IHB method has high efficiency and good robustness for both polynomial and nonpolynomial nonlinearities, and it has considerable advantages over the other methods.

Rigid Multibody Systems: The Plastic Hinge Approach

2001 ◽  
pp. 205-237 ◽  
Author(s):  
J. A. C. Ambrósio ◽  
M. Seabra Pereira ◽  
J. F. A. Milho
Keyword(s):  
Multibody Systems ◽  
Plastic Hinge ◽  
Rigid Multibody Systems ◽  
Rigid Multibody

On Steady-State Solutions of a Wave Equation by Solving a Delay Differential Equation With an Incremental Harmonic Balance Method

2018 ◽  
Author(s):  
Xuefeng Wang ◽  
Mao Liu ◽  
Weidong Zhu
Keyword(s):  
Differential Equation ◽  
Wave Propagation ◽  
Wave Equation ◽  
Steady State ◽  
Delay Differential Equation ◽  
Harmonic Balance ◽  
Incremental Harmonic Balance ◽  
Delay Differential ◽  
Ihb Method ◽  
Steady State Solutions

For wave propagation in periodic media with strong nonlinearity, steady-state solutions can be obtained by solving a corresponding nonlinear delay differential equation (DDE). Based on the periodicity, the steady-state response of a repeated particle or segment in the media contains the complete information of solutions for the wave equation. Considering the motion of the selected particle or segment as a variable, motions of its adjacent particles or segments can be described by the same variable function with different phases, which are delayed variables. Thus, the governing equation for wave propagation can be converted to a nonlinear DDE with multiple delays. A modified incremental harmonic balance (IHB) method is presented here to solve the nonlinear DDE by introducing a delay matrix operator, where a direct approach is used to efficiently and automatically construct the Jacobian matrix for the nonlinear residual in the IHB method. This method is presented by solving an example of a one-dimensional monatomic chain under a nonlinear Hertzian contact law. Results are well matched with those in previous work, while calculation time and derivation effort are significantly reduced. Also there is no additional derivation required to solve new wave systems with different governing equations.

A New Incremental Harmonic Balance Method With Two Time Scales for Quasi-Periodic Motions of an Axially Moving Beam With Internal Resonance Under Single-Tone External Excitation

2019 ◽  
Author(s):  
Jianliang Huang ◽  
Weidong Zhu
Keyword(s):  
Fundamental Frequency ◽  
Time Scales ◽  
Harmonic Balance ◽  
Axially Moving Beam ◽  
Periodic Motions ◽  
Moving Beam ◽  
Frequency Components ◽  
Incremental Harmonic Balance ◽  
Axially Moving ◽  
Ihb Method

Abstract In this work, a new incremental harmonic balance (IHB) method with two time scales, where one is a fundamental frequency, and the other is an interval distance of two adjacent frequencies, is proposed for quasi-periodic motions of an axially moving beam with three-to-one internal resonance under singletone external excitation. It is found that the interval frequency of every two adjacent frequencies, located around the fundamental frequency or one of its integer multiples, is fixed due to nonlinear coupling among resonant vibration modes. Consequently, only two time scales are used in the IHB method to obtain all incommensurable frequencies of quasi-periodic motions of the axially moving beam. The present IHB method can accurately trace from periodic responses of the beam to its quasi-periodic motions. For periodic responses of the axially moving beam, the single fundamental frequency is used in the IHB method to obtain solutions. For quasi-periodic motions of the beam, the present IHB method with two time scales is used, along with an amplitude increment approach that includes a large number of harmonics, to determine all the frequency components. All the frequency components and their corresponding amplitudes, obtained from the present IHB method, are in excellent agreement with those from numerical integration using the fourth-order Runge-Kutta method.

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