Periodic solution and bifurcation of a suspension vibration system by incremental harmonic balance and continuation method

Li, Yuanping; Chen, Siyu

doi:10.1007/s11071-015-2378-5

Cited by 9 publications

(5 citation statements)

References 30 publications

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“…( 9) may decrease the degree of nonlinearity of the residual term. Thus, it has been widely used, especially in mechanical systems [9,[61][62][63]. Lau et al [64] analysed the periodic vibrations of systems with a general form of piecewise-linear stiffness characteristics by using the IHB method.…”

Section: Incremental Harmonic Balance Methodsmentioning

confidence: 99%

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: Incremental Harmonic Balance Methodsmentioning

confidence: 99%

Harmonic Balance Methods: A Review and Recent Developments

Zipu¹,

Dai²,

Wang³

et al. 2023

Computer Modeling in Engineering &Amp; Sciences

View full text Add to dashboard Cite

show abstract

“…Lau et al [61] developed the methodology for the IHB method which is later on applied to different problems [62][63][64][65][66]. In general, IHB is the semi-analytical method that can be used for finding the solutions of differential equations appearing in mathematical models of various problems in mechanics.…”

Section: Instability Regions and Periodic Solution: Ihb Methodsmentioning

confidence: 99%

“…(15) into the system of Eq. (14) and applying the Galerkin procedure presented in [61][62][63][64][65][66][67][68], we obtain the system of linear algebraic equations in terms of unknown increments of amplitudes A i in the following form: …”

Section: Instability Regions and Periodic Solution: Ihb Methodsmentioning

confidence: 99%

Dynamic stability of a nonlinear multiple-nanobeam system

2018

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We use the incremental harmonic balance (IHB) method to analyse the dynamic stability problem of a nonlinear multiple-nanobeam system (MNBS) within the framework of Eringen's nonlocal elasticity theory. The nonlinear dynamic system under consideration includes MNBS embedded in a viscoelastic medium as clamped chain system, where every nanobeam in the system is subjected to time-dependent axial loads. By assuming the von Karman type of geometric nonlinearity, a system of m nonlinear partial differential equations of motion is derived based on the Euler-Bernoulli beam theory and D' Alembert's principle. All nanobeams in MNBS are considered with simply supported boundary conditions. Semi-analytical solutions for time response functions of the nonlinear MNBS are obtained by using the single-mode Galerkin discretization and IHB method, which are then validated by using the numerical integration method. Moreover, Floquet theory is employed to determine the sta- method, we obtain an incremental relationship with the frequency and amplitude of time-varying axial load, which defines stability boundaries. Numerical examples show the effects of different physical and material parameters such as the nonlocal parameter, stiffness of viscoelastic medium and number of nanobeams on Floquet multipliers, instability regions and nonlinear amplitude-frequency response curves of MNBS. The presented results can be useful as a first step in the study and design of complex micro/nanoelectromechanical systems.

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“…Such behaviour is quite different from the model in Yuanping and Siyu’s paper. 1 The difference is likely induced by the coupling effects in the springs along the X- and Y-directions.…”

Section: Nonlinear Characteristicsmentioning

confidence: 99%

“…The mathematical model of an air compressor can be simplified as a parametric pendulum system. 1 In this case, a suspension mass is connected to a vertical spring and two horizontal springs, and a pendulum is added as an unbalanced mass to simulate the internal eccentricity of a compressed air generator. The system is generally configured as a geometrical negative stiffness to reduce the dynamic stiffness.…”

Section: Introductionmentioning

confidence: 99%

Nonlinear dynamic response and global stability of an air compressor vibration system

Wang

Chen

et al. 2018

Journal of Low Frequency Noise, Vibration and Active Control

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This study proposes a strategy for the vibration isolation mounting of an air compressor to attenuate the vibration near the primary resonance region by using a system with dynamic negative stiffness. The vibration system is modelled as a parametric pendulum system. The nonlinear dynamic responses, including the global stability of the air compressor vibration system, are investigated analytically. The efficiency of the proposed vibration isolation strategy is numerically demonstrated over the original device. To analyse the bifurcation of the nonlinear response of the pendulum system, the phase portrait, bifurcation diagram and maximum Lyapunov exponent of the pendulum system are obtained numerically. Furthermore, the Floquet multiplier level is obtained by solving the perturbation equation numerically and can be used to determine the global stability of the air compressor vibration system.

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Periodic solution and bifurcation of a suspension vibration system by incremental harmonic balance and continuation method

Cited by 9 publications

References 30 publications

Harmonic Balance Methods: A Review and Recent Developments

Harmonic Balance Methods: A Review and Recent Developments

Dynamic stability of a nonlinear multiple-nanobeam system

Nonlinear dynamic response and global stability of an air compressor vibration system

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