Dynamical stability of the response of oscillators with discontinuous or steep first derivative of restoring characteristic

Wolf, Hinko; Terze, Zdravko; Sušić, Aleksandar

doi:10.1016/j.euromechsol.2004.08.001

Cited by 6 publications

(4 citation statements)

References 14 publications

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“…For periodic liner differential equations, stability condition is provided by Floquet theory [17]. The 8) and expanding the non-linear function in Taylor's series about the periodic solution while neglecting non-linear incremental terms, one gets a linear homogeneous differential equation.…”

Section: Analysis Methods Of Dynamic Responsesmentioning

confidence: 99%

Periodic and Chaotic Dynamic Responses of Face Gear Transmission System

Tang

Chen

2013

AMR

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The periodic and chaotic dynamic responses of face gear transmission system considering time-varying mesh stiffness and backlash nonlinearity are studied. Firstly, a nonlinear time-varying dynamic model of face gear pair is developed and the motion equations are presented, the real accurate mesh stiffness is obtained by applying Finite element approach. Then, the dynamic equations are solved using Runge-Kutta numerical integral method and bifurcation diagrams are presented and analyzed. The stability properties of steady state responses are illustrated with Floquet multipliers and Lyapunov exponents. The results show that a process of periodic-chaotic-periodic motion exists with the dimensionless pinion rotational frequency as control parameters. The analysis can be a reference to avoid the chaotic motion and unstable periodic motion through choosing suitable rotational frequency.

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Section: Analysis Methods Of Dynamic Responsesmentioning

confidence: 99%

Periodic and Chaotic Dynamic Responses of Face Gear Transmission System

Tang

Chen

2013

AMR

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“…A numerical method is applied to obtain the approximate transition matrix because it is di±cult to derive the transition matrix analytically. 28 For that purpose, the interval ½0; 2 is divided into M equal intervals with a width of Á ¼ 2=M. Inside each of the intervals, the time-varying matrix ½HðÞ can be replaced by its average value ½H j , j ¼ 1; 2; .…”

Section: Stability Analysismentioning

confidence: 99%

Resonance of a Quasi-Zero Stiffness Vibration System Under Base Excitation with Load Mismatch

Cheng

Wang

et al. 2018

Int. J. Str. Stab. Dyn.

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The primary resonance and 1/3 subharmonic resonance of a quasi-zero stiffness (QZS) vibration system under base excitation with load mismatch are studied in this research. The incremental harmonic balance (IHB) method is applied to obtain highly accurate solutions involving more dynamic behaviors. The effect of the offset displacement mainly caused by overloading on the primary resonance and displacement transmissibility is investigated. The results indicate that the system exhibits a softening characteristic under certain conditions. Although the isolation performance of the QZS system deteriorates, it still outperforms the equivalent linear system for excitation amplitudes that are not too large. The parametric analysis of the 1/3 subharmonic resonance shows that the response is unbounded, and interesting dynamic behaviors can be observed, such as the jump phenomenon. Moreover, the 1/3 subharmonic resonance can be avoided by applying a larger damping or reducing the excitation amplitude to a lower level.

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“…However, the author did not evaluate the extension of the error caused by the smoothing function, focusing the numerical observations on chaotic responses. Wolf et al [47,48] studied the dynamical stability of piecewise systems; in particular, they analysed the effect of smoothing functions on the stability of an SDOF mechanical system with free-play gaps when the Harmonic Balance Method is utilised [44]. They demonstrated that the smoothing functions introduce only limited inaccuracies in the computation of the system stability and that the effect of the number of harmonics is more important than the effect of the smoothing approximation.…”

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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Dynamical stability of the response of oscillators with discontinuous or steep first derivative of restoring characteristic

Cited by 6 publications

References 14 publications

Periodic and Chaotic Dynamic Responses of Face Gear Transmission System

Periodic and Chaotic Dynamic Responses of Face Gear Transmission System

Resonance of a Quasi-Zero Stiffness Vibration System Under Base Excitation with Load Mismatch

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

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