Three-mode interactions in harmonically excited systems with quadratic nonlinearities

An investigation is presented of the response of a three-degree-of-freedom system with quadratic nonlinearities and the autoparametric resonances ₃2₂ and ₂2₁ to a harmonic excitation of the third mode, where the _m are the linear natural frequencies of the system. The method of multiple scales is used to determine six first-order nonlinear ordinary differential equations that govern the time variation of the amplitudes and phases of the interacting modes. The fixed points of these equations are obtained and their stability is determined. For certain parameter values, the fixed points are found to lose stability due to Hopf bifurcations and consequently the system exhibits amplitude-and phase-modulated motions. Regions where the amplitudes and phases display periodic, quasiperiodic, and chaotic time variations and hence regions where the overall system motion is periodically, quasiperiodically, and chaotically modulated are determined. Using various numerical simulations, we investigated nonperiodic solutions of the modulation equations using the amplitudeF of the excitation as a control parameter. As the excitation amplitudeF is increased, the fixed points of the modulation equations exhibit an instability due to a Hopf bifurcation, leading to limit-cycle solutions of the modulation equations. AsF is increased further, the limit cycle undergoes a period-doubling bifurcation followed by a secondary Hopf bifurcation, resulting in either a two-period quasiperiodic or a phase-locked solution. AsF is increased further, there is a torus breakdown and the solution of the modulation equations becomes chaotic, resulting in a chaotically modulated motion of the system.     

On methods for continuous systems with quadratic and cubic nonlinearities

Methods for determining the response of continuous systems with quadratic and cubic nonlinearities are discussed. We show by means of a simple example that perturbation and computational methods based on first discretizing the systems may lead to erroncous results whereas perturbation methods that attack directly the nonlinear partial-differential equations and boundary conditions avoid the pitfalls associated with the analysis of the discretized systems. We describe a perturbation technique that applies either the method of multiple scales or the method of averaging to the Lagrangian of the system rather than the partial-differential equations and boundary conditions.     

Dynamic analysis of two-degree-of-freedom airfoil with freeplay and cubic nonlinearities in supersonic flow

The nonlinear aeroelastic response of a two-degree-of-freedom airfoil with freeplay and cubic nonlinearities in supersonic flows is investigated. The second-order piston theory is used to analyze a double-wedge airfoil. Then, the fold bifurcation and the amplitude jump phenomenon are detected by the averaging method and the multi-variable Floquet theory. The analytical results are further verified by numerical simulations. Finally, the influence of the freeplay parameters on the aeroelastic response is analyzed in detail.     

多自由度参激系统稳定性的数值分析

提出多自由度周期参激系统稳定性的数值直接法。通过将扰动方程表示成状态方程形式,再根据Flo-quet理论将扰动解表示成指数特征分量与周期分量之积,并将其周期分量与系统周期系数展成Fourier级数,导出一系列代数方程,建立矩阵特征值问题,从而由数值求解特征值可直接确定参激系统的稳定性。该方法可用于一般周期参激阻尼系统,特征值矩阵不含逆子阵。应用于斜拉索在支座周期运动激励下的参激振动不稳定性分析,数值结果表明该方法的有效性。     

Bifurcations and chaos in parametrically excited single-degree-of-freedom systems

The behavior of single-degree-of-freedom systems possessing quadratic and cubic nonlinearities subject to parametric excitation is investigated. Both fundamental and principal parametric resonances are considered. A global bifurcation diagram in the excitation amplitude and excitation frequency domain is presented showing different possible stable steady-state solutions (attractors). Fractal basin maps for fundamental and principal parametric resonances when three attractors coexist are presented in color. An enlargement of one region of the map for principal parametric resonance reveals a Cantor-like set of fractal boundaries. For some cases, both periodic and chaotic attractors coexist.     

Global bifurcations in externally excited two-degree-of-freedom nonlinear systems

In this study we examine the global dynamics associated with a generic two-degree-of-freedom (2-DOF), coupled nonlinear system that is externally excited. The method of averaging is used to obtain the second order approximation of the response of the system in the presence of one-one internal resonance and subharmonic external resonance. This system can describe a variety of physical phenomena such as the motion of an initially deflected shallow arch, pitching vibrations in a nonlinear vibration absorber, nonlinear response of suspended cables etc. Using a perturbation method developed by Kovai and Wiggins (1992), we show the existence of Silnikov type homoclinic orbits which may lead to chaotic behavior in this system. Here two different cases are examined and conditions are obtained for the existence of Silnikov type chaos.An earlier version of this paper was presented in the workshop on Applications of Pattern Formation at the Fields Institute of Mathematical Sciences, Waterloo, Canada, March 1993.     

Stochastic stability of parametrically excited random systems

Multidegree-of-freedom dynamic systems subjected to parametric excitation are analyzed for stochastic stability. The variation of excitation intensity with time is described by the sum of a harmonic function and a stationary random process. The stability boundaries are determined by the stochastic averaging method. The effect of random parametric excitation on the stability of trivial solutions of systems of differential equations for the moments of phase variables is studied. It is assumed that the frequency of harmonic component falls within the region of combination resonances. Stability conditions for the first and second moments are obtained. It turns out that additional parametric excitation may have a stabilizing or destabilizing effect, depending on the values of certain parameters of random excitation. As an example, the stability of a beam in plane bending is analyzed.Published in Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 135–144, October 2004.     

Considerations on self- and parametrically excited vibrational systems

Summary In order to enhance our ability to predict and detect the behaviour characteristics of a parametrically excited vibratory system, one has to grasp the fundamental features concerning the mode interactions between self-excitation and parametric excitation. In this regard, two types of nonlinear models are proposed, and the correlation and the interaction between both excitations are discussed. These models differ from those obtained earlier. Taking into account the effect of exciting terms y ⁿ cos (2t) and y ^n–1y × cos (2t) (n=1, 3) where y represents a deflection and is the frequency, the features of these models are considered in comparison with those in previous works. Resonance phenomena of orders 1 and 1/2 and the behaviour in the neighborhood of resonances are mainly investigated by the averaging method.

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Einige Bemerkungen zu selbst- und parametererregten Schwingungssystemen

Übersicht Zur Vorhersage und Charakterisierung des Verhaltens parametererregter Schwingungssysteme ist es hilfreich, die Wechselwirkungen zwischen Selbsterregung und Parametererregung zu untersuchen. Zu diesem Zweck werden am Beispiel zweier nichtlinearer Modelle diese Wechselwirkungen studiert. Dabei werden Erregerterme der Form y ⁿ cos (2t) und y ^n–1y cos (2t) (n=1, 3) betrachtet, wobei die Erregerfrequenz ist. Die typischen Eigenschaften der untersuchten nichtlinearen Systeme werden mit Resultaten früherer Arbeiten verglichen. Die Resonanzen der Ordnungen 1 und 1/2 und das Verhalten der Systeme in der Nachbarschaft dieser Resonanzen werden untersucht, wobei die Mittelwertmethode Anwendung findet.

     

On stability of parametrically excited linear stochastic systems

The dynamic stability of a coupled two-degrees-of-freedom system subjected to parametric excitation by a harmonic action superimposed by an ergodic stochastic process is investigated. For the stability analysis, the method of moment functions is used. Explicit expressions for the stability of the second moments are obtained when the frequency of the harmonic excitation lies in the vicinity of the combination sum of the natural frequencies. Good agreement between the analytical and numerical results is obtained. As an application, the example of the flexural-torsional instability of a thin elastic beam under dynamic loading is considered     

Hopf bifurcation of an oscillator with quadratic and cubic nonlinearities and with delayed velocity feedback

This paper studies the local dynamics of an SDOF system with quadratic and cubic stiffness terms, and with linear delayed velocity feedback. The analysis indicates that for a sufficiently large velocity feedback gain, the equilibrium of the system may undergo a number of stability switches with an increase of time delay, and then becomes unstable forever. At each critical value of time delay for which the system changes its stability, a generic Hopf bifurcation occurs and a periodic motion emerges in a one-sided neighbourhood of the critical time delay. The method of Fredholm alternative is applied to determine the bifurcating periodic motions and their stability. It stresses on the effect of the system parameters on the stable regions and the amplitudes of the bifurcating periodic solutions. The project supported by the National Natural Science Foundation of China (19972025)     

A new numerical technique for the analysis of parametrically excited nonlinear systems

A new computational scheme using Chebyshev polynomials is proposed for the numerical solution of parametrically excited nonlinear systems. The state vector and the periodic coefficients are expanded in Chebyshev polynomials and an integral equation suitable for a Picard-type iteration is formulated. A Chebyshev collocation is applied to the integral with the nonlinearities reducing the problem to the solution of a set of linear algebraic equations in each iteration. The method is equally applicable for nonlinear systems which are represented in state-space form or by a set of second-order differential equations. The proposed technique is found to duplicate the periodic, multi-periodic and chaotic solutions of a parametrically excited system obtained previously using the conventional numerical integration schemes with comparable CPU times. The technique does not require the inversion of the mass matrix in the case of multi degree-of-freedom systems. The present method is also shown to offer significant computational conveniences over the conventional numerical integration routines when used in a scheme for the direct determination of periodic solutions. Of course, the technique is also applicable to non-parametrically excited nonlinear systems as well.     

Global transient dynamics of nonlinear parametrically excited systems

In this paper we examine the response of a typical nonlinear system that is subjected to parametric excitation. Particular attention is paid to how basins of attraction evolve such that the global transient stability of the system may be assessed. We show that at a forcing level that is considerably smaller than that at which the steady-state attractor loses its stability, there may exist a rapid erosion and stratification of the basin, signifying a global loss of engineering integrity of the system.We also show, for a system near its equilibrium state, that the boundaries in parameter space can become fractal. The significance of such an analysis is not only that it corresponds to a failure locus for a system subjected to a sudden pulse of excitation, but since the phase-space basin is often eroded throughout its central region, the determination of basin boundaries in control space can often reflect the characteristics of the phase-space basin structure, and hence on the macroscopic level they provide information regarding the global transient stability of the system.     

Arbitrary amplitude periodic solutions for parametrically excited systems with time delay

Periodic solutions for parametrically excited system under state feedback control with a time delay are investigated. Using the asymptotic perturbation method, two slow-flow equations for the amplitude and phase of the parametric resonance response are derived. Their fixed points correspond to limit cycles (phase-locked periodic solutions) for the starting system. In the system without control, periodic solutions (if any) exist only for fixed values of amplitude and phase and depend on the system parameters and excitation amplitude. In many cases, the amplitudes of periodic solutions do not correspond to the technical requirements. On the contrary, it is demonstrated that, if the vibration control terms are added, stable periodic solutions with arbitrarily chosen amplitude and phase can be accomplished. Therefore, an effective vibration control is possible if appropriate time delay and feedback gains are chosen.     

Controlling erosion of safe basin in nonlinear parametrically excited systems

This paper considers the dynamical behavior of a Duffing-Mathieu type system with a cubic single-well potential during the principal parametric resonance. Both the cases of constant and time-dependent excitation amplitude are used to observe the variation of the extent and the rate of the erosion in safe basins. It is evident that the appearance of fractal basin boundaries heralds the onset of the losing of structural integrity. The minimum value of control parameter to prevent the basin from erosion is given along with the excitation amplitude varying. The results show the time-dependence of excitation amplitude can be used to control the extent and the rate of the erosion and delay the first occurrence of heteroclinic tangency. The project supported by the National Natural Science Foundation of China and PSF of China.     

Vibration control for parametrically excited Liénard systems

We apply a new vibration control method for time delay non-linear oscillators to the principal resonance of a parametrically excited Liénard system under state feedback control with a time delay. Using the asymptotic perturbation method, we obtain two slow flow equations on the amplitude and phase. Their fixed points correspond to limit cycles for the Liénard system. Vibration control and high-amplitude response suppression can be performed with appropriate time delay and feedback gains. Using energy considerations, we investigate existence and characteristics of limit cycles of the slow flow equations. A limit cycle corresponds to a two-period quasi-periodic modulated motion for the starting system and in order to reduce the amplitude peak of the parametric resonance and to exclude the existence of two-period quasi-periodic motion, we find the appropriate choices for the feedback gains and the time delay.     

Elliptic balance solution of two-degree-of-freedom, undamped, forced systems with cubic nonlinearity

The solution of a system of two coupled, nonhomogeneous undamped, ordinary differential equations with cubic nonlinearity and sinusoidal driving force is obtained by the use of Jacobian elliptic functions and the elliptic balance method. To assess the accuracy of our proposed solution, we consider an example that arises in the study of the finite amplitude, nonlinear vibration of a simple shear suspension system. It is shown that the analytical results exhibit good agreement with the numerical integration solutions even for moderate values of the system parameters.     

On primary resonances of weakly nonlinear delay systems with cubic nonlinearities

We implement the method of multiple scales to investigate primary resonances of a weakly nonlinear second-order delay system with cubic nonlinearities. In contrast to previous studies where the implementation is confined to the assumption of linear delay terms with small coefficients (Hu et al. in Nonlinear Dyn. 15:311, 1998; Ji and Leung in Nonlinear Dyn. 253:985, 2002), in this effort, we propose a modified approach which alleviates that assumption and permits treating a problem with arbitrarily large gains. The modified approach lumps the delay state into unknown linear damping and stiffness terms that are functions of the gain and delay. These unknown functions are determined by enforcing the linear part of the steady-state solution acquired via the method of multiple scales to match that obtained directly by solving the forced linear problem. We examine the validity of the modified procedure by comparing its results to solutions obtained via a harmonic balance approach. Several examples are discussed demonstrating the ability of the proposed methodology to predict the amplitude, softening-hardening characteristics, and stability of the resulting steady-state responses. Analytical results also reveal that the system can exhibit responses with different nonlinear characteristics near its multiple delay frequencies.     

A parametrically excited pendulum with irrational nonlinearity

In this paper, we propose a parametrically excited pendulum with irrational nonlinearity which comprises a simple pendulum linked by a linear spring under base excitation. This parametric vibration system exhibits bistable state and discontinuous characteristics due to the geometry configuration. For small oscillations, this system can be described by Mathieu equation coupled with SD (Smooth and Discontinuous) oscillator whose dynamic response is examined analytically by using the averaging method in both smooth and discontinuous case. Numerical simulations are carried out to demonstrate the complicated dynamic behavior of multiple periodic motions and different types of chaotic motions.     

Non-smooth stability analysis of the parametrically excited impact oscillator

The aim of this paper is to give a Lyapunov stability analysis of a parametrically excited impact oscillator, i.e. a vertically driven pendulum which can collide with a support. The impact oscillator with parametric excitation is described by Hill's equation with a unilateral constraint. The unilaterally constrained Hill's equation is an archetype of a parametrically excited non-smooth dynamical system with state jumps. The exact stability criteria of the unilaterally constrained Hill's equation are rigorously derived using Lyapunov techniques and are expressed in the properties of the fundamental solutions of the unconstrained Hill's equation. Furthermore, an asymptotic approximation method for the critical restitution coefficient is presented based on Hill's infinite determinant and this approximation can be made arbitrarily accurate. A comparison of numerical and theoretical results is presented for the unilaterally constrained Mathieu equation.