Nonlinear Vibrations of Piecewise-Linear Systems by Incremental Harmonic Balance Method

1992 ◽  
Vol 59 (1) ◽  
pp. 153-160 ◽  
Author(s):  
S. L. Lau ◽  
W.-S. Zhang
Keyword(s):  
Harmonic Balance ◽  
Nonlinear Vibrations ◽  
Piecewise Linear ◽  
Practical Interest ◽  
Harmonic Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Incremental Harmonic Balance ◽  
Stiffness Characteristics ◽  
Ihb Method ◽  
Piecewise Linear Stiffness

The incremental harmonic balance (IHB) method is extended to analyze the periodic vibrations of systems with a general form of piecewise-linear stiffness characteristics. An explicit formulation has been worked out. This development is of significance as many structural and mechanical systems of practical interest possess a piecewise-linear stiffness. Typical examples show that the IHB method is very effective for analyzing this kind of systems under steady-state vibrations.

scholarly journals Calculation of nonlinear vibrations of piecewise-linear systems using the shooting method

2012 ◽  
Vol 34 (3) ◽  
pp. 157-167
Author(s):  
Nguyen Van Khang ◽  
Hoang Manh Cuong ◽  
Nguyen Thai Minh Tuan
Keyword(s):  
Harmonic Balance ◽  
Shooting Method ◽  
Piecewise Linear ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Periodic Attractors ◽  
Incremental Harmonic Balance ◽  
Excited Oscillator ◽  
Ihb Method

In this paper, an explicit formulation of the shooting scheme for computation of multiple periodic attractors of a harmonically excited oscillator which is asymmetric with both stiffness and viscous damping piecewise linearities is derived. The numerical simulation by the shooting method is compared with that by the incremental harmonic balance method (IHB method), which shows that the shooting method is in many respects distinctively advantageous over the incremental harmonic balance method.

A Modified Two-Timescale Incremental Harmonic Balance Method for Steady-State Quasi-Periodic Responses of Nonlinear Systems

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
R. Ju ◽  
W. Fan ◽  
W. D. Zhu ◽  
J. L. Huang
Keyword(s):  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Response Surfaces ◽  
Algebraic Equations ◽  
Incremental Harmonic Balance Method ◽  
Calculation Time ◽  
High Order Harmonic ◽  
Nonlinear Algebraic Equations ◽  
Incremental Harmonic Balance ◽  
Ihb Method

A modified two-timescale incremental harmonic balance (IHB) method is introduced to obtain quasi-periodic responses of nonlinear dynamic systems with combinations of two incommensurate base frequencies. Truncated Fourier coefficients of residual vectors of nonlinear algebraic equations are obtained by a frequency mapping-fast Fourier transform procedure, and complex two-dimensional (2D) integration is avoided. Jacobian matrices are approximated by Broyden's method and resulting nonlinear algebraic equations are solved. These two modifications lead to a significant reduction of calculation time. To automatically calculate amplitude–frequency response surfaces of quasi-periodic responses and avoid nonconvergent points at peaks, an incremental arc-length method for one timescale is extended for quasi-periodic responses with two timescales. Two examples, Duffing equation and van der Pol equation with quadratic and cubic nonlinear terms, both with two external excitations, are simulated. Results from the modified two-timescale IHB method are in excellent agreement with those from Runge–Kutta method. The total calculation time of the modified two-timescale IHB method can be more than two orders of magnitude less than that of the original quasi-periodic IHB method when complex nonlinearities exist and high-order harmonic terms are considered.

A Modified Incremental Harmonic Balance Method Combined with Tikhonov Regularization for Periodic Motion of Nonlinear System

2021 ◽  
pp. 1-16
Author(s):  
Ze-chang Zheng ◽  
Zhong-rong Lu ◽  
Chen Yanmao ◽  
Ji-Ke Liu ◽  
Guang Liu
Keyword(s):  
Nonlinear System ◽  
Tikhonov Regularization ◽  
Harmonic Balance ◽  
Duffing Oscillator ◽  
Harmonic Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Numerical Examples ◽  
Initial Values ◽  
Incremental Harmonic Balance ◽  
Ihb Method

Abstract In this paper, a modified incremental harmonic balance (IHB) method combined with Tikhonov regularization has been proposed to achieve the semi-analytical solution for the periodic nonlinear system. To the best of our knowledge, the convergence of the traditional IHB method is bound up with the iterative initial values of harmonic coefficients, especially near the bifurcation point. To this end, the Tikhonov regularization is introduced into the linear incremental equation to tackle the ill-posed situation in the iteration. To this end, the convergence performance of the traditional IHB method has been improved significantly. Moreover, convergence proof of the proposed method also has been given in this paper. Finally, a van der Pol–Duffing oscillator with external excitation and a cubic nonlinear airfoil system with the external store are adopted as numerical examples to illustrate the efficiency and the performance of the presented modified IHB method. The numerical examples show that the results achieved by the proposed method are in excellent agreement with the Runge–Kutta method, and the accuracy is not significantly reduced compared with the traditional IHB method. Especially, the modified IHB method also can converge to the exact solution from the initial values that the traditional IHB method cannot converge in both examples.

Comparison Between the Incremental Harmonic Balance Method and Alternating Frequency/Time-Domain Method

2020 ◽  
Vol 143 (2) ◽  
Author(s):  
R. Ju ◽  
W. Fan ◽  
W. D. Zhu
Keyword(s):  
Fourier Transform ◽  
Time Domain ◽  
Harmonic Balance ◽  
Sampling Rate ◽  
Harmonic Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Acceptable Accuracy ◽  
Incremental Harmonic Balance ◽  
First Time ◽  
Ihb Method

Abstract Two widely used semi-analytical methods: the incremental harmonic balance (IHB) method and alternating frequency/time-domain (AFT) method are compared, and some long-standing discussions on frameworks of these two methods are cleared up. The IHB and AFT methods are proved for the first time to be theoretically equivalent when spectrum aliasing does not occur in the AFT method. Based on this equivalence, the minimal nonaliasing sampling rate for the AFT and fast Fourier transform (FFT)-based IHB methods can be obtained for a system with polynomial nonlinearities. While spectrum aliasing is theoretically inevitable for nonpolynomial nonlinearities, a sufficiently large sampling rate can be usually used with acceptable accuracy and efficiency for many systems. Convergence and efficiency of the IHB method, AFT method, and several FFT-based IHB methods are compared. Accuracy and convergence can be affected when the sampling rate is insufficient. This comparison can provide some insights to avoid misuse of these methods and choose which methods to use in engineering applications.

The Incremental Harmonic Balance Method Applied to Coupled Van der Pol Oscillators

2010 ◽  
Author(s):  
Wei Zhang ◽  
Hailiang Hu ◽  
Youhua Qian
Keyword(s):  
Numerical Method ◽  
Harmonic Balance ◽  
Harmonic Balance Method ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Van Der Pol ◽  
Van Der Pol Oscillators ◽  
Incremental Harmonic Balance ◽  
Ihb Method ◽  
Good Agreement

The incremental harmonic balance (IHB) method is used to investigate coupled Van der Pol oscillators. An effective way for calculating the coefficient matrices and selecting the appropriate initial values is presented. The results of the IHB method are in good agreement with the results of the numerical method.

An Incremental Harmonic Balance Method With Two Time Scales for Quasi-Periodic Motion of Nonlinear Systems With Cubic Nonlinearity

2019 ◽  
Author(s):  
Jianliang Huang ◽  
Weidong Zhu ◽  
Shuhui Chen
Keyword(s):  
Numerical Integration ◽  
Time Scales ◽  
Periodic Motion ◽  
Harmonic Balance ◽  
Degrees Of Freedom ◽  
Harmonic Balance Method ◽  
Cubic Nonlinearity ◽  
Incremental Harmonic Balance Method ◽  
Incremental Harmonic Balance ◽  
Ihb Method

Abstract A new incremental harmonic balance (IHB) method with two time scales procedure is used to analyze quasi-periodic motion of multiple degrees of freedom systems with cubic nonlinearity. An amplitude increment algorithm is adapted to deal with cases where the two frequencies are unknown a priori, in order to automatically trace frequency response of quasi-periodic motion of the system and accurately calculate all frequency components and their corresponding amplitudes. Results of application of the present IHB method to quasi-periodic free vibration of the nonlinear system are shown and compared with previously published results with Lau method and those from numerical integration. While differences are noted between results predicted by the present IHB method and Lau method, excellent agreement is achieved between results from the present IHB method and numerical integration even in cases of strongly nonlinear vibration.

Bearing Looseness Fault Characteristics Analysis of Rotor System Based on Incremental Harmonic Balance Method

2013 ◽  
Vol 397-400 ◽  
pp. 517-523 ◽  
Author(s):  
Guo Liang Xiong ◽  
Qin Xiang ◽  
Qing Song Cao
Keyword(s):  
Harmonic Balance ◽  
Piecewise Linear ◽  
Harmonic Balance Method ◽  
Analytic Solutions ◽  
Rotor System ◽  
Balance Method ◽  
Incremental Harmonic Balance Method ◽  
Characteristics Analysis ◽  
Incremental Harmonic Balance ◽  
The Time Domain

Support or pedestal looseness fault exists in mechanical system typically.A new dynamical model of an eccentric motor rotor system with a piecewise-linear stiffness was proposed, and the incremental harmonic balance method was employed to obtain analytic solutions of the model. Then, the time domain waveform and amplitude spectrogram were achieved by applying the analytic solutions to program and iterate in the MATLAB software. Finally, the characteristics of looseness was analyzed. Results show that this research method is feasible, and this paper may provide a theoretical basis for the solution of complex bearing looseness model analytically and the diagnosis of its fault.

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