Elliptic harmonic balance method for two degree-of-freedom self-excited oscillators

Chen, Y.M.; Liu, J.K.

doi:10.1016/j.cnsns.2007.11.008

Cited by 17 publications

(5 citation statements)

References 22 publications

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“…so that A denotes the amplitude of the motion defined by Equation (8). In the remaining part of this Section, periodic solutions of (5) are obtained for the cubic nonlinearity.…”

Section: Applicationmentioning

confidence: 99%

Refined analytical approximations to limit cycles for non-linear multi-degree-of-freedom systems

Lewis

2019

International Journal of Non-Linear Mechanics

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This paper presents analytical higher order approximations to limit cycles of an autonomous multidegree-of-freedom system based on an integro-differential equation method for obtaining periodic solutions to nonlinear differential equations. The stability of the limit cycles obtained was then investigated using a method for carrying out Floquet analysis based on developments to extensions of the method for solving Hill's Determinant arising in analysing the Mathieu equation, which have previously been reported in the literature. The results of the Floquet analysis, together with the limit cycle predictions, have then been used to provide some estimates of points on the boundary of the domain of attraction of stable equilibrium points arising from a sub-critical Hopf bifurcation. This was achieved by producing a local approximation to the stable manifold of the unstable limit cycle that occurs. The integro-differential equation to be solved for the limit cycles involves no approximations. These only arise through the iterative approach adopted for its solution in which the first approximation is that which would be obtained from the harmonic balance method using only fundamental frequency terms. The higher order approximations are shown to give significantly improved predictions for the limit cycles for the cases considered. The Floquet analysis based approach to predicting the boundary of domains of attraction met with some success for conditions just following a sub-critical Hopf bifurcation. Although this study has focussed on cubic non-linearities, the method presented here could equally be used to refine limit cycle predictions for other non-linearity types.

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“…so that A denotes the amplitude of the motion defined by Equation (8). In the remaining part of this Section, periodic solutions of (5) are obtained for the cubic nonlinearity.…”

Section: Applicationmentioning

confidence: 99%

Refined analytical approximations to limit cycles for non-linear multi-degree-of-freedom systems

Lewis

2019

International Journal of Non-Linear Mechanics

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“…A nonlinear vibration isolation system with a negative stiffness mechanism was investigated using the averaging method [29]. However, when the elliptic harmonic balance method is applied to solve two or multiple DOFs systems, the number of equations obtained by harmonic balancing is not equal to that of unknowns [30]. For this issue, Chen and Liu analysed a 2-DOF self-excited oscillator with strongly cubic nonlinearity by an additional equation prior to harmonic balancing using Jacobi elliptic functions [30].…”

Section: Introductionmentioning

confidence: 99%

“…However, when the elliptic harmonic balance method is applied to solve two or multiple DOFs systems, the number of equations obtained by harmonic balancing is not equal to that of unknowns [30]. For this issue, Chen and Liu analysed a 2-DOF self-excited oscillator with strongly cubic nonlinearity by an additional equation prior to harmonic balancing using Jacobi elliptic functions [30]. Using an elliptic balance method, Elias-Zuniga and Beatty obtained the forced response of a 2-DOF undamped system with cubic nonlinearity [31].…”

Section: Introductionmentioning

confidence: 99%

The Elliptic Harmonic Balance Method for the Performance Analysis of a Two‐Stage Vibration Isolation System with Geometric Nonlinearity

Tang

2021

Shock and Vibration

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This study develops a modified elliptic harmonic balance method (EHBM) and uses it to solve the force and displacement transmissibility of a two-stage geometrically nonlinear vibration isolation system. Geometric damping and stiffness nonlinearities are incorporated in both the upper and lower stages of the isolator. After using the relative displacement of the nonlinear isolator, we can numerically obtain the steady-state response using the first-order harmonic balance method (HBM1). The steady-state harmonic components of the stiffness and damping force are modified using the Jacobi elliptic functions. The developed EHBM can reduce the truncation error in the HBM1. Compared with the HBM1, the EHBM can improve the accuracy of the resonance regimes of the amplitude-frequency curve and transmissibility. The EHBM is simple and straightforward. It can maintain the same form as the balancing equations of the HBM1 but performs better than it.

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“…A review of this method, with applications to single degree-of-freedom second order systems, has been carried out by Kovacic et al [24]. The method, in the form of elliptic harmonic balance, has been applied to aeroelastic systems possessing cubic nonlinearities [25,26]. There has also been interest in the application of the Groebner basis method to solving the coupled polynomial equations which may arise from the harmonic balance method [27,28].…”

Section: Introductionmentioning

confidence: 99%

Approximations to limit cycles for a nonlinear multi-degree-of-freedom system with a cubic nonlinearity through combining the harmonic balance method with perturbation techniques

Lewis

2020

International Journal of Non-Linear Mechanics

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This paper presents an approach to obtaining higher order approximations to limit cycles of an autonomous multi-degree-of-freedom system with a single cubic nonlinearity based on a first approximation involving first and third harmonics obtained with the harmonic balance method. This first approximation, which is similar to one which has previously been reported in the literature, is an analytical solution, except that the frequency has to be obtained numerically from a polynomial equation of degree 16. An improved solution is then obtained in a perturbation procedure based on the refinement of the harmonic balance solution. The stability of the limit cycles obtained is then investigated using Floquet analysis.The capability of this approach to refine the results obtained by the harmonic balance first approximation is demonstrated, by direct comparison with time domain simulation and frequency components obtained using the Discrete Fourier Transform. The particular case considered was based on an aeroelastic analysis of an all-moving control surface with a nonlinearity in the torsional degreeof-freedom of the root support, and parameters corresponding to air speed, together with linear stiffness and viscous damping of the root support were varied. It is also shown, for the cases considered, how the method can reveal further bifurcational behaviour of the system beyond the initial Hopf bifurcations which first lead to the onset of limit cycle oscillations.

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Elliptic harmonic balance method for two degree-of-freedom self-excited oscillators

Cited by 17 publications

References 22 publications

Refined analytical approximations to limit cycles for non-linear multi-degree-of-freedom systems

Refined analytical approximations to limit cycles for non-linear multi-degree-of-freedom systems

The Elliptic Harmonic Balance Method for the Performance Analysis of a Two‐Stage Vibration Isolation System with Geometric Nonlinearity

Approximations to limit cycles for a nonlinear multi-degree-of-freedom system with a cubic nonlinearity through combining the harmonic balance method with perturbation techniques

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