The influence of neglecting small harmonic terms on estimation of dynamical stability of the response of non-linear oscillators

Wolf, Hinko; Stegić, Milenko

doi:10.1007/s004660050511

Cited by 7 publications

(3 citation statements)

References 13 publications

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“…The effects of neglecting small harmonic terms on estimation of dynamical stability of the steady state solution determined in the frequency domain are considered in the paper [13]. For that purpose, a simple single-degree-of-freedom piecewise linear system excited by a harmonic excitation is analyzed.…”

Section: Literature Reviewmentioning

confidence: 99%

Study of forced vibrations transition processes of vibration protection devices with rolling-contact bearings

Bissembayev

Sultanova

2020

JMMCS

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Many seismic isolation and vibration protection devices use asan essential element the various types of rolling-contact bearings. The rolling-contact bearing is used for creation of moving base of body protected against vibration. The most dynamic disturbances acting in the constructions and structures have highly complex and irregular nature. This article considers the oscillation of a solid body on kinematic foundations, the main elements of which are rolling bearers bounded by the high order surfaces of rotation at horizontal displacement of the foundation. It is ascertained that the equations of motion are highly nonlinear differential equations. Stationary and transitional modes of the oscillatory process of the system have been investigated. It is determined that several stationary regimes of the oscillatory process exist. Equations of motion have been investigated also by quantitative methods. In this paper the cumulative curves in the phase plane are plotted, a qualitative analysis for singular points and study of them for stability is performed. In the Hayashi plane a cumulative curve of body protected against vibration forms a closed path which does not tend to the stability of singular point. This means that the vibration amplitude of body protected against vibration is not remain constant in steady-state, but changes periodically.

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Section: Literature Reviewmentioning

confidence: 99%

Study of forced vibrations transition processes of vibration protection devices with rolling-contact bearings

Bissembayev

Sultanova

2020

JMMCS

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“…The idea of seismic isolation of buildings in case of earthquakes with the help of rolling element linear guides is one of the simplest and most efficient ideas in the history of earthquake-proof construction [20].…”

Section: Numerical Studiesmentioning

confidence: 99%

Analysis of the Oscillating Motion of a Solid Body on Vibrating Bearers

et al. 2019

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This article considers the oscillation of a solid body on kinematic foundations, the main elements of which are rolling bearers bounded by high-order surfaces of rotation at horizontal displacement of the foundation. Equations of motion of the vibro-protected body have been obtained. It is ascertained that the obtained equations of motion are highly nonlinear differential equations. Stationary and transitional modes of the oscillatory process of the system have been investigated. It is determined that several stationary regimes of the oscillatory process exist. Equations of motion have been investigated also by quantitative methods. In this paper the cumulative curves in the phase plane are plotted, a qualitative analysis for singular points and a study of them for stability are performed. In the Hayashi plane a cumulative curve of a body protected against vibration forms a closed path which does not tend to the stability of a singular point. This means that the vibration amplitude of a body protected against vibration does not remain constant in a steady state, but changes periodically.

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“…References 2, 6, 8 and 9), the harmonic balance method (HBM; e.g. References 1,11,[13][14][15][16][17][18][21][22][23] and numerical integration methods (e.g. References 5 and 7 for brief descriptions of numerical methods to solve the non-linear differential equations of motion and to determine the monodromy matrix, and References 34 and 35 for more detailed descriptions of numerical solution of non-linear differential equations).…”

Section: Introductionmentioning

confidence: 99%

Stability of multi-degree-of-freedom Duffing oscillators

Ribeiro¹

2009

Proceedings of the Institution of Civil Engineers - Engineering

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Different methods of examining the stability of periodic solutions of non-linear multi-degree-of-freedom systems are compared in this paper. The methods are particularly suited for solutions computed with the harmonic balance method and the non-linear systems considered are represented by systems of coupled Duffing-type equations; that is, with cubic non-linear expressions. The latter systems appear frequently in models of thin-walled structures, particularly in straight beams and flat plates. In the first method reviewed in this paper the harmonic balance procedure is applied to define an eigenvalue problem, from which the characteristic exponents are determined. If the real part of any of these characteristic exponents is greater than zero, then the solution is unstable. The second method relies on the sign of the determinant of a Jacobian matrix; a matrix that is also often used to solve the equations of motion. The last method is based on a perturbation procedure and with it a closed-form expression for stable regions is achieved. The advantages and shortcomings of the different methods are discussed. INTRODUCTIONA non-linear dynamic system excited by a harmonic force can have periodic and non-periodic solutions. With regard to the periodic solutions, it is possible that more than one solution exists for a single excitation frequency and the initial conditions determine to which solution the system converges. An unstable steady-state solution does not occur in practice: if a solution is unstable, the system will evolve to another attractor. Hence, it is important to determine whether or not a solution is stable. The local stability can be determined by investigating the evolution of a perturbed solution that results from the addition of a small disturbance. A perturbation near an unstable equilibrium condition leads to a departure from this condition and the inverse occurs near a stable equilibrium condition. Several types and orders of instability can occur and this is a subject widely analysed 1-33 but still of considerable interest.

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The influence of neglecting small harmonic terms on estimation of dynamical stability of the response of non-linear oscillators

Cited by 7 publications

References 13 publications

Study of forced vibrations transition processes of vibration protection devices with rolling-contact bearings

Study of forced vibrations transition processes of vibration protection devices with rolling-contact bearings

Analysis of the Oscillating Motion of a Solid Body on Vibrating Bearers

Stability of multi-degree-of-freedom Duffing oscillators

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