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

Tensor Contraction ◽

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.

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

Journal of Vibration and Acoustics ◽

10.1115/1.4047235 ◽

2020 ◽

Vol 142 (6) ◽

Cited By ~ 1

Author(s):

Ren Ju ◽

Wei Fan ◽

Weidong Zhu

Keyword(s):

Fourier Transform ◽

Fast Fourier Transform ◽

Harmonic Balance ◽

High Efficiency ◽

Jacobian Matrix ◽

Sampling Rate ◽

Harmonic Balance Method ◽

Tensor Contraction ◽

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 non-polynomial nonlinearities, and it has considerable advantages over the other methods.

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

Journal of Vibration and Acoustics ◽

10.1115/1.4048173 ◽

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 ◽

Acceptable Accuracy ◽

First Time ◽

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.

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

10.1115/1.0004469v ◽

2021 ◽

Author(s):

Ren Ju ◽

Wei Fan ◽

Weidong Zhu

Keyword(s):

Fourier Transform ◽

Fast Fourier Transform ◽

Harmonic Balance ◽

Harmonic Balance Method ◽

Balance Method ◽

Tensor Contraction

A Modified Incremental Harmonic Balance Method for 2-DOF Airfoil Aeroelastic Systems with Nonsmooth Structural Nonlinearities

Mathematical Problems in Engineering ◽

10.1155/2020/5767451 ◽

2020 ◽

Vol 2020 ◽

pp. 1-19

Author(s):

Ying-Ge Ni ◽

Wei Zhang ◽

Yi Lv ◽

Stylianos Georgantzinos

Keyword(s):

Fourier Transform ◽

Fast Fourier Transform ◽

Harmonic Balance ◽

Harmonic Balance Method ◽

Free Play ◽

Balance Method ◽

Aeroelastic System ◽

Structural Nonlinearity ◽

Incremental Harmonic Balance

A modified incremental harmonic balance method is presented to analyze the aeroelastic responses of a 2-DOF airfoil aeroelastic system with a nonsmooth structural nonlinearity. The current method, which combines the traditional incremental harmonic balance method and a fast Fourier transform, can be used to obtain the higher-order approximate solution for the aeroelastic responses of a 2-DOF airfoil aeroelastic system with a nonsmooth structural nonlinearity using significantly fewer linearized algebraic equations than the traditional method, and the dominant frequency components of the response can be obtained by a fast Fourier transform of the numerical solution. Thus, periodic solutions can be obtained, and the calculation process can be simplified. Furthermore, the nonsmooth nonlinearity was expanded into a Fourier series. The procedures of the modified incremental harmonic balance method were demonstrated using systems with hysteresis and free play nonlinearities. The modified incremental harmonic balance method was validated by comparing with the numerical solutions. The effect of the number of harmonics on the solution precision as well as the effect of the free-play and stiffness ratio on the response amplitude is discussed.

A modified incremental harmonic balance method based on the fast Fourier transform and Broyden’s method

Nonlinear Dynamics ◽

10.1007/s11071-015-2045-x ◽

2015 ◽

Vol 81 (1-2) ◽

pp. 981-989 ◽

Cited By ~ 25

Author(s):

X. F. Wang ◽

W. D. Zhu

Keyword(s):

Fourier Transform ◽

Fast Fourier Transform ◽

Harmonic Balance ◽

Harmonic Balance Method ◽

Balance Method ◽

Broyden’S Method ◽

Broyden's Method

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

Journal of Applied Mechanics ◽

10.1115/1.2899421 ◽

1992 ◽

Vol 59 (1) ◽

pp. 153-160 ◽

Cited By ~ 81

Author(s):

S. L. Lau ◽

W.-S. Zhang

Keyword(s):

Harmonic Balance ◽

Nonlinear Vibrations ◽

Piecewise Linear ◽

Practical Interest ◽

Harmonic Balance Method ◽

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.

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

Journal of Computational and Nonlinear Dynamics ◽

10.1115/1.4036118 ◽

2017 ◽

Vol 12 (5) ◽

Cited By ~ 4

Author(s):

R. Ju ◽

W. Fan ◽

W. D. Zhu ◽

J. L. Huang

Keyword(s):

Harmonic Balance ◽

Harmonic Balance Method ◽

Response Surfaces ◽

Algebraic Equations ◽

Calculation Time ◽

High Order Harmonic ◽

Nonlinear Algebraic Equations ◽

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

Journal of Applied Mechanics ◽

10.1115/1.4052573 ◽

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 ◽

Numerical Examples ◽

Initial Values ◽

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.

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

Vietnam Journal of Mechanics ◽

10.15625/0866-7136/34/3/2357 ◽

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 ◽

Periodic Attractors ◽

Excited Oscillator ◽

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.

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

Volume 13: Sound, Vibration and Design ◽

10.1115/imece2010-37351 ◽

2010 ◽

Author(s):

Wei Zhang ◽

Hailiang Hu ◽

Youhua Qian

Keyword(s):

Numerical Method ◽

Harmonic Balance ◽

Harmonic Balance Method ◽

Balance Method ◽

Van Der Pol ◽

Van Der Pol Oscillators ◽

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.