Averaging in vibration suppression by parametric stiffness excitation

Stability investigations of vibration suppression employing the concept of actuators with a variable stiffness are presented. Systems with an arbitrary number of degrees of freedom with linear spring- and damping-elements are considered that are subject to self-excitation as well as parametric stiffness excitation. General conditions for full vibration suppression and conditions of instability are derived analytically by applying a singular perturbation of first and second order. The analytical predictions are compared for exemplary systems by numerical time integration and show a great improvement of former results. These basic results obtained can be used for accurate design of a control strategy for actuators. The first author gratefully acknowledges the mobility grant of Vienna University of Technology for visiting the University of Utrecht during which preliminary results were obtained.     

Suppression of self-excited vibrations by a random parametric excitation

Previous theoretical and experimental studies have shown that some vibrating systems can be stabilized by zero-averaged periodic parametric excitations. It is shown in this paper that some zero-mean random parametric excitations can also be useful for this stabilization. Under some conditions, they can be even more efficient compared to the periodic ones. Two-mass mechanical system with self-excited vibrations is considered for this comparison. The so-called bounded noise is used as a model of the random parametric excitation. The mean-square stability diagrams are obtained numerically by considering an eigenvalue problem for large matrices.     

Enhanced damping of a cantilever beam by axial parametric excitation

A uniform cantilever beam under the effect of a time-periodic axial force is investigated. The beam structure is discretized by a finite-element approach. The linearised equations of motion describing the planar bending vibrations of the beam structure lead to a system with time-periodic stiffness coefficients. The stability of the system is investigated by a numerical method based on Floquet’s theorem and an analytical approach resulting from a first-order perturbation. It is demonstrated that the parametrically excited beam structure exhibits enhanced damping properties, when excited near a specific parametric combination resonance frequency. A certain level of the forcing amplitude has to be exceeded to achieve the damping effect. Upon exceeding this value, the additional artificial damping provided to the beam is significant and works best for suppression of vibrations of the first vibrational mode of the cantilever beam.     

Nonlinear oscillations with parametric excitation solved by homotopy analysis method

An analytical technique, namely the homotopy analysis method （HAM）, is used to solve problems of nonlinear oscillations with parametric excitation. Unlike perturbation methods, HAM is not dependent on any small physical parameters at all, and thus valid for both weakly and strongly nonlinear problems. In addition, HAM is different from all other analytic techniques in providing a simple way to adjust and control convergence region of the series solution by means of an auxiliary parameter h. In the present paper, a periodic analytic approximations for nonlinear oscillations with parametric excitation are obtained by using HAM, and the results are validated by numerical simulations.     

Multi-bifurcation cascaded bursting oscillations and mechanism in a novel 3D non-autonomous circuit system with parametric and external excitation

Nonlinear Dynamics - In this paper, multi-timescale dynamics and the formation mechanism of a 3D non-autonomous system with two slowly varying periodic excitations are systematically investigated....     

Asynchronous parametric suppression of resonance vibrations of hyghly nonlinear systems

Conclusions The investigations have shown that the auxiliary asynchronous parametric excitation ca be used to effectively suppress resonance vibrations in systems with highly nonlinear elast characteristics. Either local or wideband suppression of the resonance regime can be achieved, depending on the relation between the frequencies of the primary external and auxiliary parametric excitations (/=const or /const); also, the width and positions of the instability intervals can be controlled. This affords the possibility of using auxiliary parametric excitation not only to enhance the efficiency of nonlinear antivibration systems (by wideband suppression of resonance vibrations), but also for the design of fundamentally new resonance devices to monitor the frequency and amplitude of vibrations by exploiting the effect of local instability of the resonance regime.Polytechnic Institute, Riga. Translated from Prikladnaya Mekhanika, Vol. 27, No. 5, pp. 102–107, May, 1991.     

Stability of elastic systems under a stochastic parametric excitation

An efficient method to investigate the stability of elastic systems subjected to the parametric force in the form of a random stationary colored noise is suggested. The method is based on the simulation of stochastic processes, numerical solution of differential equations, describing the perturbed motion of the system, and the calculation of top Liapunov exponents. The method results in the estimation of the almost sure stability and the stability with respect to statistical moments of different orders. Since the closed system of equations for moments of desired quantities y _j (t) cannot be obtained, the statistical data processing is applied. The estimation of moments at the instant t _n is obtained by statistical average of derived from the solution of equations for the large number of realizations. This approach allows us to evaluate the influence of different characteristics of random stationary loads on top Liapunov exponents and on the stability of system. The important point is that results found for filtered processes, are principally different from those corresponding to stochastic processes in the form of Gaussian white noises.     

The dynamic stability of degenerate systems under parametric excitation

Summary The parametric resonance in a system having two modes of the same frequency is studied. The simultaneous occurence of the instabilities of the first and second kind is examined, by using a generalized perturbation procedure. The region of instability in the first approximation is obtained by using the Sturm's theorem for the roots of a polynomial equation.

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Übersicht Es wird die parametrische Resonanz eines Systems betrachtet, dessen zwei Eigenfrequenzen gleich sind. Das gleichzeitige Auftreten von Instabilitäten erster und zweiter Art wird durch eine verallgemeinerte Störungsrechnung untersucht. Der Instabilitätsbereich läßt sich dann in erster Näherung unter Benutzung des Sturmschen Satzes über die Wurzeln eines Polynoms berechnen.

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The author is indebted to Prof. Dr. E. Mettler, Universität Karlsruhe, for his useful suggestions. The author's grateful thanks are due to the German Academic Exchange Service (DAAD) for the financial support at the Institut für Mechanik, Universität Karlsruhe.     

Local bifurcation theory of nonlinear systems with parametric excitation

This paper summarizes the authors' research on local bifurcation theory of nonlinear systems with parametric excitation since 1986. The paper is divided into three parts. The first one is the local bifurcation problem of nonlinear systems with parametric excitation in cases of fundamental harmonic, subharmonic and superharmonic resonance. The second one is the experiment investigation of local bifurcation solutions in nonlinear systems with parametric excitation. The third one is the universal unfolding study of periodic bifurcation solutions in the nonlinear Hill system, where the influence of every physical parameter on the periodic bifurcation solution is discussed in detail and all the results may be applied to engineering.     

Flutter suppression of a plate-like wing via parametric excitation

The present work is motivated by the well known stabilizing effect of parametric excitation of some dynamical systems such as the inverted pendulum. The possibility of suppressing wing flutter via parametric excitation along the plane of highest rigidity in the neighborhood of combination resonance is explored. The nonlinear equations of motion in the presence of incompressible fluid flow are derived using Hamilton's principle and Theodorsen's theory for modeling aerodynamic forces. In the presence of air flow, the bending and torsion modes possess nearly the same frequency. Under parametric excitation and in the absence of air flow, each mode oscillates at its own natural frequency. In the neighborhood of combination resonance, the nonlinear response is determined using the multiple scales method at the critical flutter speed and at slightly higher airflow speed. The domains of attraction and bifurcation diagrams are obtained to reveal the conditions under which the parametric excitation can provide stabilizing effect. The basins of attraction for different values of excitation amplitude reveal the stabilizing effect that takes place above a critical excitation level. Below that level, the response experiences limit cycle oscillations, cascade of period doubling, and chaos. For flow speed slightly higher than the critical flutter speed, the response experiences a train of spikes, known as ‘firing,’ a term that is borrowed from neuroscience, followed by ‘refractory’ or recovery effect, up to an excitation level above which the wing is stabilized. The results of the multiple scales method are verified using numerical simulation of the original nonlinear differential equations.     

On die swell: Some theoretical results

Tanner's [1] theory for capillary die swell¹ is amended, taking account of a suggestion of Graessley et al. [2]. The calculations for a Maxwell fluid show no significant changes in predictions. The calculations are also carried out for a long annular die, the results being new. An alternative theory based on affine recovery from entry flow for short capillaries is developed. Surprisingly, for spherically symmetric sink flow, the predictions are barely distinguishable from the Tanner, or modified Tanner, theories. Comparison with experiment and further discussion will be given in a later paper.     

Threshold of parametric excitation of waves on a fluid surface

Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 4, pp. 61–67, July–August, 1988.     

The effect of parametric excitation on a self-excited three-mass system

This contribution deals with the quenching of self-excited vibrations by means of parametric excitation due to periodic variation of spring stiffness. A three-mass chain system is investigated in detail. It is shown that the self-excitation can be fully or partly suppressed in a particular frequency interval.     

Combination,principal parametric and internal resonances of an accelerating beam under two frequency parametric excitation

This study analyses the nonlinear transverse vibration of an axially moving beam subject to two frequency excitation. Focus has been made on simultaneous resonant cases i.e. principal parametric resonance of first mode and combination parametric resonance of additive type involving first two modes in presence of internal resonance. By adopting the direct method of multiple scales, the governing nonlinear integro-partial differential equation for transverse motion is reduced to a set of nonlinear first order ordinary partial differential equations which are solved either by means of continuation algorithm or via direct time integration. Specifically, the frequency response plots and amplitude curves, their stability and bifurcation are obtained using continuation algorithm. Numerical results reveal the rich and interesting nonlinear phenomena that have not been presented in the existent literature on the nonlinear dynamics of axially moving systems.     

Finite-element simulation of myocardial electrical excitation

Based on a single-domain model of myocardial conduction, isotropic and anisotropic finite element models of the myocardium are developed allowing excitation wave propagation to be studied. The Aliev-Panfilov phenomenological equations were used as the relations between the transmembrane current and the transmembrane potential. Interaction of an additional source of initial excitation with an excitation wave that passed and the spread of the excitation wave are studied using heart tomograms. A numerical solution is obtained using a splitting algorithm that allows the nonlinear boundary-value problem to be reduced to a sequence of simpler problems: ordinary differential equations and linear boundary-value problems in partial derivatives.     

An investigation of the parametric excitation of an eccentrically tensioned bar

Results of an investigation of the parametric excitation of an eccentrically tensioned bar are presented. Two distinct types of resonant behavior are observed. For one, the excitation-response frequency ratio is one and, for the other, the ratio is two. In germs of oscillation amplitude and size of the region of parameter space in which resonance occurs, the latter response exhibits the greater potential for producing damage. The transition from small to large amplitude oscillations occurs abruptly and coincides with the occurrence of a snap-through behavior.