Parametric resonance in systems with weak dissipation (original) (raw)
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Latin American Journal of Solids and Structures, 2018
In this paper, a nonlinear three-degrees-of-freedom dynamical system consisting of a variable-length pendulum mass attached by a massless spring to the forced slider is investigated. Numerical solution is preceded by application of Euler-Lagrange equation. Various techniques like time histories, phase planes, Poincaré maps and resonance plots are used to observe and identify the system responses. The results show that the variable-length spring pendulum suspended from the periodically forced slider can exhibit quasi-periodic, and in a resonance state, even chaotic motions. It was concluded that near the resonance the influence of coupling of bodies on the system dynamics can lead to unpredictable dynamical behavior.
Theory of parametric resonance: modern results
2003 IEEE International Workshop on Workload Characterization (IEEE Cat. No.03EX775), 2003
Linear dynatnical systems with many degrecs of freedom with periodic coefficients also depending on constmt parameters arc considcred. Stability of the trivial solution is studied with the use of the Floqiet theory. First arid second order derivativcs of the Floqqet matrix with rcspect to parameters are derivcd in terms of matriciants of the main and adjoint problems and derivatives of the system matrix. This allows to find &e derivatives of simple multipliers, responsible for svability of the system, with respect to parameters and predict llicir bchavior with a change of parameters. lliis shown how to use this information in gradipt procedures for stabilization or destabilization of the system. As a numerical example, the system described by Carsson-Cambi equation is considered. Then, strong and week interactions of multipliers on the complex plane are studied, and geometric interpretation o f thesc intcractions is giveo. As application or the dcvcloped theory the resonance domains for Hill's equation with damping are studied. It is shown that they represunt halves of cones in the three-parameter space. Then, parametric resonance of a pendulum with damping &d vibrating suspension point following arbitrary periodic law is considered, and the parametric resoniince domains are Sound. Another important application of damped Hill's equation is connected with the study:of stability of periodic motions in non-linear dynamical systems. I t is shown how to find stable and unstable regimes for harmonically excited Duffing's equation. Then, linear vihrational systems with periodic coefficients depending on three independent parametdrs: frequency and amplitude of pcriodic excitation, and damping parameter are considered with the assumption that the last two quantities arc small. Instability of $he trivial solution of thc system (parametric resonance);is studied. For arbitrary matrix of periodic excitation 4id positive definitc damping matrix general expressions for domains of the main (simple) and coinbinatibn resonances arc dcrived. Two important specific casesiof excitatioii matrix are studied: a symmetric matrix and a stationary matrix multiplied by a scalar perioclic function. I t is shown that in both cases the resonancc domains arc halves of cones in the thrce-dimensional space with the boundary surface coefficients depcnding only on the cigenfrequencies, eigenmodes and system matrices. The obtained relations allow to analyze influence of growing eigenfrequencies and resonance number on resonance domains. Two mechanical problems arc considered and solved Bolotin's problem of dynamic stability of a beam loadcd by periodic bending moments, and parametric resonance of a nonuniform column loaded by periodic longitudinal force. The lecture is a review of the recent results on parametric resonance obtained by the author with We consider a system of linear difrerential equations where G = G(1,p) is a real square matrix of dimension rn , which is smoothly depending on a vector of real parameters p = (pI,pz, ...p,,) and is a continuous periodic function of the time G(t,p) = c(t + T,p), T being a period. We denote linear independent solutions of system (I) as x l ( t ) , x 2 ( t ) , ..., x , ( t ) and Sorm out of them a fundamental matrix X(t) = [xl(t),X*(t) ,..., X,(t)].
The Stability Analysis of a Vibrating Auto-Parametric Dynamical System Near Resonance
Applied Sciences, 2022
This paper examines a new vibrating dynamical motion of a novel auto-parametric system with three degrees of freedom. It consists of a damped Duffing oscillator as a primary system attached to a damped spring pendulum as a secondary system. Lagrange’s equations are utilized to acquire the equations of motion according to the number of the system’s generalized coordinates. The perturbation technique of multiple scales is applied to provide the solutions to these equations up to a higher order of approximations, with the aim of obtaining more accurate novel results. The categorizations of resonance cases are presented, in which the case of primary external resonance is examined to demonstrate the conditions of solvability of the steady-state solutions and the equations of modulation. The time histories of the achieved solutions, the resonance curves in terms of the modified amplitudes and phases, and the regions of stability are outlined for various parameters of the considered system...
On the boundaries of the parametric resonance domain
Journal of Applied Mathematics and Mechanics, 2000
The problem of stability for a system of linear differential equations with coefficients which are periodic in time and depend on the parameters is considered. The singularities of the general position arising at the boundaries of the stability and instability (parametric resonance) domains in the case of two and three parameters are listed. A constructive approach is proposed which enables one, in the first approximation, to determine the stability domain in the neighbourhood of a point on the boundary (regular or singular) from the information at this point. This approach enables one to eliminate a tedious numerical analysis of the stability region in the neighbourhood of the boundary point and can be employed to construct the boundaries of parametric resonance domains. As an example, the problem of the stability of the oscillations of an articulated pipe conveying fluid with a pulsating velocity is considered. In the space of three parameters (the average fluid velocity and the amplitude and frequency of pulsations) a singularity of the boundary of the stability domain of the "dihedral angle" type is obtained and the tangential cone to the stability domain is calculated. 0 2001 Elsevier Science Ltd. All rights reserved.
Dynamics of an electromechanical system forced near the resonance
In this paper an electromechanical system with parametric excitation and also externally excited near the resonance is studied. The existence and stability of periodic orbits are rigorously obtained. Some results, in a special case, such as existence of periodic orbits and the relation 2:1 between the period of oscillation of the cart, which is a part of the electromechanical system, and the period of the current, are compatible with earlier numerical findings when there is only a constant external excitation.
Parametric Resonance in a Linear Oscillator
2003
The phenomenon of parametric resonance in a linear system arising from a periodic modulation of its parameter is investigated both analytically and with the help of a computer simulation based on the educational software package PHYSICS OF OSCILLATIONS (see in the web http://www.aip.org/pas). The simulation experiments aid greatly an understanding of basic principles and peculiarities of parametric excitation and complement the analytical study of the subject in a manner that is mutually reinforcing. The parametric excitation is studied on the example of the rotary oscillations of a mechanical torsion spring pendulum caused by periodic variations of its moment of inertia. Conditions and characteristics of parametric resonance and of parametric regeneration are discussed in detail. Ranges of frequencies within which parametric excitation is possible are determined. Stationary oscillations on the boundaries of these ranges are investigated.
Resonance in a Rigid Rotor with Elastic Support
ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik, 1993
Fur einen Rotor mit elastischer Halterung in axialer wie auch seitlicher Richtung wird ein Modell formuliert, das auf das Stabilitutsproblem fur kleine Vertikalschwingungen in senkrechter Position fihrt. Dies ist ein autopurametrisches Erregungsproblem, das in Gestalt zweier gekoppelter Mathieu-iihnlicher Gleichungen formuliert werden kann. Die Analyse der Speziulfille ohne und mit (hearer) Dampfung sowie der Full nichtlinearer Dampfung werden ausgefuhrt, wobei Mittelung und numerische Bijiurkationstechniken angewendet werden, die im nichtlinearen Fall auf Hysteresis und Phasensynchronisierung fiihren. A model is formulated for a rotor with elastic support in axial and lateral directions which leads to the problem of stability of small vertical oscillations in the upright position. This is an autoparametric excitation problem which can be formulated as two coupled Mathieu-like equations. The analysis of the cases without and with (linear) damping and the case of nonlinear damping is carried out using averaging and numerical bifurcation techniques leading in the nonlinear case to hysteresis and phase-locking.
TRANSIENT RESONANCE OSCILLATIONS OF A MECHANICAL SYSTEM WITH REGARD TO NON-LINEAR DAMPING EFFECTS
Proceedings of DETC’99 1999 ASME Design Engineering Technical Conferences September 12-15, 1999, Las Vegas, Nevada, USA
Transient, forced vibrations of a mechanical system are considered. Dissipative effects such as material damping, aerodynamic damping and damping at interfaces are taken into account. For the modeling of these effects the set of isolated weakly non-linear single degree of freedom oscillators with damping dependent non-linearities is used. The superposition of their responses approximates the resulting response of the system. The justification of this assumption is discussed. The Krylov-Bogoljubov asymptotic method is applied for the investigation of the transient resonance response. Numerical calculations are provided to demonstrate the validity of the Krylov-Bogoljubov first approximation.
Weakly nonlinear and symmetric periodic systems at resonance
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Effective-parametric resonance in a non-oscillating system
EPL (Europhysics Letters), 2012
We present a mechanism for the generation of oscillations and nonlinear parametric amplification in a non-oscillating system, which we term effective-parametric resonance. Sustained oscillations appear at a controlled amplitude and frequency, related directly to the external forcing parameters. We present an intuitive explanation for this phenomenon, based on an effective equation for a driven oscillation and discuss its relation to other approaches. More precisely, a highfrequency forcing can generate an effective oscillator, which may have a parametric resonance with the applied forcing. We point out the main ingredients for the development of effective-parametric resonance in non-oscillating systems and show its existence in a simple model. Theoretically, we calculate the appearance of this nonlinear oscillation by computing its stability curve, which is confirmed by numerical simulations and experimental studies on a vertically driven pendulum.