Three-to-One Internal Resonances in Parametrically Excited Hinged-Clamped Beams

The nonlinear planar response of a hinged-clamped beam to a principal parametric resonance of either its first or second mode or a combination parametric resonance of the additive type of its first two modes is investigated. The analysis accounts for mid-plane stretching, a static axial load, a restraining spring at one end, and modal damping. The natural frequency of the second mode is approximately three times the natural frequency of the first mode for a range of static axial loads, resulting in a three-to-one internal resonance. The method of multiple scales is used to attack directly the governing nonlinear integral-partial-differential equation and associated boundary conditions and derive three sets of four first-order nonlinear ordinary-differential equations describing the modulation of the amplitudes and phases of the first two modes in the cases of (a) principal parametric resonance of either the first or the second mode, and (b) a combination parametric resonance of the additive type of these modes. Periodic motions and periodically and chaotically modulated motions of the beam are determined by investigating the equilibrium and dynamic solutions of the modulation equations. For the case of principal parametric resonance of the first mode or combination parametric resonance of the additive type, trivial and two-mode solutions are possible, whereas for the case of parametric resonance of the second mode, trivial, single, and two-mode solutions are possible. The trivial and two-mode equilibrium solutions of the modulation equations may undergo either a supercritical or a subcritical Hopf bifurcation, depending on the magnitude of the axial load. For some excitation parameters, we found complex responses including period-doubling bifurcations and blue-sky catastrophes.     

一类含平方、立方非线性项1:2内共振两自由度系统的全局分岔

研究了一类含平方、立方非线性项的两自由度系统的全局分岔。首先应用多尺度法求解其平均方程 ,然后通过一系列变换得到一个近似可积的两自由度系统。应用能量 相位准则 ,确定了在哈密顿共振时Silnikov轨道存在的条件。通过数值计算验证了此条件。     

Resonance Capture in a Three Degree-of-Freedom Mechanical System

We study the phenomena of resonance capture in a three degree-of-freedom dynamical system modeling the dynamics of an unbalanced rotor, subject to a small constant torque, supported by orthogonal, linearly elastic supports, which is constrained to move in the plane. In the physical system the resonance exists between translational motions of the frame and the angular velocity of the unbalanced rotor. These equations, valid in the neighborhood of the resonance, possess a small parameter which is related to the imbalance. In the limit 0, the unperturbed system possesses a homoclinic orbit which separates bounded periodic motion corresponding to resonant solutions from unbounded motion which corresponds to solutions passing through the resonance. Using a generalized Melnikov integral, we characterize the splitting distance between the invariant manifolds which govern capture and escape from resonance for 0. It is shown that as certain slowly varying parameters evolve, the separation distance alternates sign, indicating that both capture into, and escape from resonance occur. We find that although a measurable set of initial conditions enter into a sustained resonance, as the system further evolves the orientation of the manifolds reverses and many of these captured solutions will subsequently escape.     

Homoclinic Saddle-Node Bifurcations in Singularly Perturbed Systems

In this paper we study the creation of homoclinic orbits by saddle-node bifurcations. Inspired on similar phenomena appearing in the analysis of so-called localized structures in modulation or amplitude equations, we consider a family of nearly integrable, singularly perturbed three dimensional vector fields with two bifurcation parameters a and b. The O() perturbation destroys a manifold consisting of a family of integrable homoclinic orbits: it breaks open into two manifolds, W ^s() and W ^u(), the stable and unstable manifolds of a slow manifold . Homoclinic orbits to correspond to intersections W ^s()W ^u(); W ^s()W ^u()= for a<a*, a pair of 1-pulse homoclinic orbits emerges as first intersection of W ^s() and W ^u() as a>a*. The bifurcation at a=a* is followed by a sequence of nearby, O( ²(log)²) close, homoclinic saddle-node bifurcations at which pairs of N-pulse homoclinic orbits are created (these orbits make N circuits through the fast field). The second parameter b distinguishes between two significantly different cases: in the cooperating (respectively counteracting) case the averaged effect of the fast field is in the same (respectively opposite) direction as the slow flow on . The structure of W ^s()W ^u() becomes highly complicated in the counteracting case: we show the existence of many new types of sometimes exponentially close homoclinic saddle-node bifurcations. The analysis in this paper is mainly of a geometrical nature.     

Bifurcations of double homoclinic flip orbits with resonant eigenvalues

Concerns double homoclinic loops with orbit flips and two resonant eigen- values in a four-dimensional system.We use the solution of a normal form system to construct a singular map in some neighborhood of the equilibrium,and the solution of a linear variational system to construct a regular map in some neighborhood of the double homoclinic loops,then compose them to get the important Poincarémap.A simple cal- culation gives explicitly an expression of the associated successor function.By a delicate analysis of the bifurcation equation,we obtain the condition that the original double homoclinic loops are kept,and prove the existence and the existence regions of the large 1-homoclinic orbit bifurcation surface,2-fold large 1-periodic orbit bifurcation surface, large 2-homoclinic orbit bifurcation surface and their approximate expressions.We also locate the large periodic orbits and large homoclinic orbits and their number.     

Resonance and Stochastic Layer in a Parametrically Excited Pendulum

The (2M:1)-librational and (M:1)-rotational resonances are discovered in the stochastic layer of a parametrically excited pendulum. The analytical conditions for the onset of a resonance in the stochastic layer are derived. Numerical predictions of the appearance of resonance in thestochastic layer are also completed. Illustrations of the stochasticlayer in the parametrically excited pendulums are given through thePoincaré mapping sections. This methodology can be used for resonantlayers in nonlinear Hamiltonian systems. However, the analyticalapproaches need to be improved for the better predictions of theresonant characteristics in the stochastic layer.     

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.     

Nonlinear Nonplanar Dynamics of Parametrically Excited Cantilever Beams

The nonlinear nonplanar response of cantilever inextensional metallic beams to a principal parametric excitation of two of its flexural modes, one in each plane, is investigated. The lowest torsional frequencies of the beams considered are much larger than the frequencies of the excited modes so that the torsional inertia can be neglected. Using this condition as well as the inextensionality condition, we develop a Lagrangian whose variation leads to two integro-partial-differential equations governing the motions of the beams. The method of time-averaged Lagrangian is used to derive four first-order nonlinear ordinary-differential equations governing the modulation of the amplitudes and phases of the two interacting modes. These modulation equations exhibit symmetry properties. A pseudo arclength scheme is used to trace the branches of the equilibrium solutions and an investigation of the eigenvalues of the Jacobian matrix is used to assess their stability. The equilibrium solutions experience pitchfork, saddle-node, Hopf, and codimension-2 bifurcations. A detailed bifurcation analysis of the dynamic solutions of the modulation equations is presented. Five branches of dynamic (periodic and chaotic) solutions were found. Two of these branches emerge from two Hopf bifurcations and the other three are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold, and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises.     

An Investigation of Internal Resonance in Aeroelastic Systems

Although the study of internal resonance in mechanical systems has been given significant consideration, minimal attention has been given to internal resonance for systems which consider the presence of aerodynamic forces. Herein, the investigators examine the possible existence of internal resonances, and the related nonlinear pathologies that such responses may have, for an aeroelastic system which possesses nonlinear aerodynamic loads. Evidence of internal resonance is presented for specific classes of aeroelastic systems, and such adverse response indicates nonlinearities may lead to aeroelastic instabilities that are not predicted by traditional (linear) approaches.     

Detection of Homoclinic Bifurcations in Resonance Zones of Forced Oscillators

We study homoclinic behaviour in resonance zones of nonconservative,forced oscillators represented as one-parameter families of periodicperturbations of planar Hamiltonian systems. We use a Melnikov-typetechnique and obtain a simple condition under which separatrixsplittings with exponentially small upper bounds may occur. Todemonstrate our theory, we give two examples for Duffing-typeoscillators and compare the theoretical results with numericalsimulations. In these examples, homoclinic behaviour that was notreported in early work is detected.     

Global Bifurcations and Chaotic Dynamics in Nonlinear Nonplanar Oscillations of a Parametrically Excited Cantilever Beam

This paper presents the analysis of the global bifurcations and chaotic dynamics for the nonlinear nonplanar oscillations of a cantilever beam subjected to a harmonic axial excitation and transverse excitations at the free end. The governing nonlinear equations of nonplanar motion with parametric and external excitations are obtained. The Galerkin procedure is applied to the partial differential governing equation to obtain a two-degree-of-freedom nonlinear system with parametric and forcing excitations. The resonant case considered here is 2:1 internal resonance, principal parametric resonance-1/2 subharmonic resonance for the in-plane mode and fundamental parametric resonance–primary resonance for the out-of-plane mode. The parametrically and externally excited system is transformed to the averaged equations by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is applied to find the explicit formulas of normal forms associated with a double zero and a pair of pure imaginary eigenvalues. Based on the normal form obtained above, a global perturbation method is utilized to analyze the global bifurcations and chaotic dynamics in the nonlinear nonplanar oscillations of the cantilever beam. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Silnikov type single-pulse homoclinic orbit in the averaged equation for the nonlinear nonplanar oscillations of the cantilever beam. These results show that the chaotic motions can occur in the nonlinear nonplanar oscillations of the cantilever beam. Numerical simulations verify the analytical predictions.     

一类含平方、立方非线性质项1：2内共振两自由度系统的全局分岔

研究了一类含平方、立方非线性项的两自由度系统全局分岔。首先应用多尺度法求解其平均方程，然后通过一系列变换得到一个近似可积的两自由度系统。应用能量-相位准则，确定了在哈密顿共振时Silnikov轨道存在的条件。通过数值计算验证了此条件。     

Self-Excited Oscillators with Asymmetric Nonlinearities and One-to-Two Internal Resonance

An analysis is presented on the dynamics of asymmetric self-excited oscillators with one-to-two internal resonance. The essential behavior of these oscillators is described by a two degree of freedom system, with equations of motion involving quadratic nonlinearities. In addition, the oscillators are under the action of constant external loads. When the nonlinearities are weak, the application of an appropriate perturbation approach leads to a set of slow-flow equations, governing the amplitudes and phases of approximate motions of the system. These equations are shown to possess two different solution types, generically, corresponding to static or periodic steady-state responses of the class of oscillators examined. After complementing the analytical part of the work with a method of determining the stability properties of these responses, numerical results are presented for an example mechanical system. Firstly, a series of characteristic response diagrams is obtained, illustrating the effect of the technical parameters on the steady-state response. Then results determined by the application of direct numerical integration techniques are presented. These results demonstrate the existence of other types of self-excited responses, including periodically-modulated, chaotic, and unbounded motions.     

Chaos in Perturbed Planar Non-Hamiltonian Integrable Systems with Slowly-Varying Angle Parameters

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一类慢变参数振子系统的同宿分叉及其安全盆侵蚀

分析一个具有慢变参数的非线性系统,利用Ｍｅｌｎｋｏｖ方法,分析了系统在参数发生变化时的同宿分叉,同时利用分叉结果,数值讨论了当系统参数发生变化时安全盆的侵蚀及分叉,混沌的联系。     

Regular and Chaotic Vibrations of a Parametrically and Self-Excited System Under Internal Resonance Condition

Vibrations of a parametrically and self-excited system with two degrees of freedom have been analysed in this paper. The system is constituted by two parametrically coupled oscillators characterised by self-excitation and nonlinear Duffing’s type nonlinearities. Synchronisation phenomenon has been determined near the principal resonances in the neighbourhood of the first p₁ and the second p₂ natural frequencies, and near the combination resonance (p₁+p₂)/2. Vibrations have been investigated for parameters which satisfy the internal resonance condition p₂/p₁=3. The existence and break down of the synchronisation phenomenon have been revealed analytically by the multiple time scale method, whilst transition of the system to chaotic motion has been carried out numerically.     

Predicting Homoclinic Bifurcations in Planar Autonomous Systems

An analytical method to predict the homoclinic bifurcation in a planar autonomous self-excited weakly nonlinear oscillator is presented. The method is mainly based on the collision between the periodic orbit undergoing the homoclinic bifurcation and the saddle fixed point. To illustrate the analytical predictive criteria, two typical examples are investigated. The results obtained in this work are then compared to Melnikov's technique and to a previous criterion based on the vanishing of the frequency. Numerical simulations are also provided.     

Global Dynamics of a Parametrically and Externally Excited Thin Plate

Both the local and global bifurcations of a parametrically andexternally excited simply supported rectangular thin plate are analyzed.The formulas of the thin plate are derived from the vonKármán equation and Galerkin's method. The method ofmultiple scales is used to find the averaged equations. The numericalsimulation of local bifurcation is given. The theory of normal form,based on the averaged equations, is used to obtain the explicitexpressions of normal form associated with a double zero and a pair ofpurely imaginary eigenvalues from the Maple program. On the basis of thenormal form, global bifurcation analysis of a parametrically andexternally excited rectangular thin plate is given by the globalperturbation method developed by Kovacic and Wiggins. The chaotic motionof the thin plate is found by numerical simulation.     

Bifurcation Control of a Parametrically Excited Duffing System

A linear time-delayed feedback control is used to delay the occurrenceof pitchfork bifurcations and to eliminate saddle-node bifurcations,which may arise in the nonlinear response of a parametrically excitedDuffing system under the principal parametric resonance. The feedbackgains and the time delay are chosen by analyzing the modulationequations of the amplitude and the phase. It is shown that by using anappropriate feedback control, the stable region of the trivial solutionscan be broadened, a discontinuous bifurcation can be transformed into acontinuous one, and the jump phenomenon in the resonance response can beremoved.     

Global bifurcations in externally excited autoparametric systems

We consider an autoparametric system consisting of an oscillator coupled with an externally excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in primary resonance. The method of second-order averaging is used to obtain a set of autonomous equations of the second-order approximations to the externally excited system with autoparametric resonance. The Šhilnikov-type homoclinic orbits and chaotic dynamics of the averaged equations are studied in detail. The global bifurcation analysis indicates that there exist the heteroclinic bifurcations and the Šhilnikov-type homoclinic orbits in the averaged equations. The results obtained above mean the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the externally excited system with autoparametric resonance. Furthermore, a detailed bifurcation analysis of the dynamic (periodic and chaotic) solutions of the averaged equations is presented. Nine branches of dynamic solutions are found. Two of these branches emerge from two Hopf bifurcations and the other seven are isolated. The limit cycles undergo symmetry-breaking, cyclic-fold and period-doubling bifurcations, whereas the chaotic attractors undergo attractor-merging and boundary crises. Simultaneous occurrence of the limit cycle and chaotic attractors, homoclinic orbits, intermittency chaos and homoclinic explosions are also observed.