Computation of periodic orbits for piecewise linear oscillator by Harmonic Balance Methods

Pei, Lijun; Chong, Antonio S.E.; Pavlovskaia, Ekaterina; Wiercigroch, Marian

doi:10.1016/j.cnsns.2021.106220

Cited by 32 publications

(6 citation statements)

References 52 publications

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“…Miguel et al [168] proposed a new method for approximating both the Bouc-Wen and LuGre hysteresis loops analytically through a truncated Taylor series as a simple and effective smoothing procedure to enable the use of the HB method. Pei et al [169] applied the IHB and AFT-HB methods to the analysis of the dynamics of the piecewise linear oscillator and confirmed the robustness of the HB-like methods. Chen et al [170] presented a study on the non-smooth system of a floating network structure using the IHB method, especially the periodic solutions of bifurcation analysis.…”

Section: Non-smooth Systemsmentioning

confidence: 93%

Harmonic Balance Methods: A Review and Recent Developments

Zipu¹,

Dai²,

Wang³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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The harmonic balance (HB) method is one of the most commonly used methods for solving periodic solutions of both weakly and strongly nonlinear dynamical systems. However, it is confined to low-order approximations due to complex symbolic operations. Many variants have been developed to improve the HB method, among which the time domain HB-like methods are regarded as crucial improvements because of their fast computation and simple derivation. So far, there are two problems remaining to be addressed. i) A dozen of different versions of HB-like methods, in frequency domain or time domain or in hybrid, have been developed; unfortunately, misclassification pervades among them due to the unclear borderlines of different methods. ii) The time domain HB-like methods suffer from non-physical solutions, which have been shown to be caused by aliasing (mixture of the high-order into the low-order harmonics). Although a series of dealiasing techniques have been developed over the past two decades, the mechanism of aliasing and the final solution to dealiasing are still not well known to the academic community. This paper aims to provide a comprehensive review of the development of HB-like methods and enunciate their principal differences. In particular, the time domain methods are emphasized with the famous aliasing phenomenon clearly addressed. KEYWORDSHarmonic balance; frequency domain HB-like method; time domain HB-like method; dealiasing technique; HB algebraic equation NomenclatureThe commonly used abbreviations of the computing methods are shown below. Abbreviations AFT-HBAlternating frequency-time harmonic balance ANM Asymptotic numerical method ETDC Extended time domain collocation GOIA Global optimal iterative algorithm HB Harmonic balance

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Section: Non-smooth Systemsmentioning

confidence: 93%

Harmonic Balance Methods: A Review and Recent Developments

Zipu¹,

Dai²,

Wang³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

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“…According to the harmonic balance principle [27–34], we can obtain

- m Λ ω^{2} + k Λ + \frac{3}{4} β {normalΛ}^{3} \cos ωτ + K Λ ω^{α} \cos \frac{italicαπ}{2} = f \cos γ,

c Λ ω - \frac{3}{4} β {normalΛ}^{3} \sin ωτ + K Λ ω^{α} \sin \frac{italicαπ}{2} = f \sin γ .

…”

Section: Steady‐state Solutionmentioning

confidence: 99%

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

Yang

Jiao

2023

Math Methods in App Sciences

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The dynamic characteristics of the fractional Duffing system with nonlinear time delay term are studied according to the average method and the harmonic balance method. First, the steady‐state solutions of the fractional Duffing system with a nonlinear time delay term are obtained by applying the average method and the harmonic balance method. The analytical expressions of the amplitude–frequency characteristic curves under the average method and the harmonic balance method are also established. Second, the stability criterion of the system is obtained according to the indirect method of studying the motion stability problem, and the concept of the stability condition parameter is proposed in this process. Third, the transient solution of the fractional Duffing system with a nonlinear time delay term is obtained utilizing the average method, and the approximate analytical solution of the fractional Duffing system with nonlinear time delay term is obtained. Finally, the amplitude–frequency characteristic curves of the different fractional differential terms, coefficients of different fractional differential terms, magnitude of different time delay quantities, and feedback strength of different delay are analyzed by numerical simulation. Furthermore, the amplitude–frequency characteristic curves obtained by the average method and the harmonic balance method under different order of fractional differential term are compared and analyzed by numerical simulation. The time sequence and phase diagrams of the system under different order of fractional differential term and different time delay quantities are analyzed by numerical simulation. In addition, the correctness of the analytical analysis is verified by comparing the analytical results with the numerical simulation results under the average method and the harmonic balance method, respectively, by numerical simulation.

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“…Other solutions methods for nonlinear systems with piecewise functions include Harmonic Balance Method [27][28][29], coordinate transformations [30][31][32], and path-following continuation methods [1,2,33]. For the last category, authors generally resort to Numerical Continuation toolboxes, like MatCont [34], COCO [35] and AUTO-07 [36].…”

Section: Introductionmentioning

confidence: 99%

Approximating piecewise nonlinearities in dynamic systems with sigmoid functions: advantages and limitations

2023

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In the industry field, the increasingly stringent requirements of lightweight structures are exposing the ultimately nonlinear nature of mechanical systems. This is extremely true for systems with moving parts and loose fixtures which show piecewise stiffness behaviours. Nevertheless, the numerical solution of systems with ideal piecewise mathematical characteristics is associated with time-consuming procedures and a high computational burden. Smoothing functions can conveniently simplify the mathematical form of such systems, but little research has been carried out to evaluate their effect on the mechanical response of multi-degree-of-freedom systems. To investigate this problem, a slightly damped mechanical two-degree-of-freedom system with soft piecewise constraints is studied via numerical continuation and numerical integration procedures. Sigmoid functions are adopted to approximate the constraints, and the effect of such approximation is explored by comparing the results of the approximate system with the ones of the ideal piecewise counter-part. The numerical results show that the sigmoid functions can correctly catch the very complex dynamics of the proposed system when both the above-mentioned techniques are adopted. Moreover, a reduction in the computational burden, as well as an increase in numerical robustness, is observed in the approximate case.

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Computation of periodic orbits for piecewise linear oscillator by Harmonic Balance Methods

Cited by 32 publications

References 52 publications

Harmonic Balance Methods: A Review and Recent Developments

Harmonic Balance Methods: A Review and Recent Developments

Dynamics analysis for a class of fractional Duffing systems with nonlinear time delay terms

Approximating piecewise nonlinearities in dynamic systems with sigmoid functions: advantages and limitations

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