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.     

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。     

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.     

Global bifurcations in the motion of parametrically excited thin plates

In this paper we investigate global bifurcations in the motion of parametrically excited, damped thin plates. Using new mathematical results by Kovai and Wiggins in finding homoclinic and heteroclinic orbits to fixed points that are created in a resonance resulting from perturbation, we are able to obtain explicit conditions under which Silnikov homoclinic orbits occur. Furthermore, we confirm our theoretical predictions by numerical simulations.     

Sine-Gordon方程的截断系统的同宿轨道

研究Ｓｉｎｅ Ｇｏｒｄｏｎ方程的广义渐近惯性流形上的常微分方程组，证实了在一定参数条件下存在Ｗｉｇｇｉｎｓ［１］意义下的同宿轨道．计算表明，与Ｂｉｓｈｏｐ［２］用数值计算得到的Ｓｉｎｅ Ｇｏｒｄｏｎ方程产生混沌的参数值尚有差别，考虑到同宿出现参数值往往低于混沌出现参数值，故结果在定性上正确，而且改进了文［１］中的结果．     

Two mode sub-harmonic and harmonic vibrations of a non-linear beam forced by a two mode harmonic load

The non-linear transverse vibrations of a uniform beam with ends restrained to remain a fixed distance apart and forced by a two mode function which is harmonic in time, are studied by a corresponding two mode approach. The existence of sub-harmonic response of order 1/3 and harmonic response in the sub-harmonic resonance region of the forcing frequency is proved. Approximate solutions are found by Urabe's numerical method applied to Galerkin's procedure and by an analytical harmonic balance-perturbation method. Error bounds of the Galerkin approximations are given.     

Dynamic near three-to-one internal resonance in composite stiffened panels

Nonlinear dynamic of composite stiffened panels to parametric and three-to-one internal resonances is investigated. The ordinary differential equation of two mode shapes is established by using Galerkin method and the condition of three-to-one internal resonance between the first mode (1,3) and the second mode (3,1) is examined near the principal resonance 2:1 of the first mode. Then, the nonlinear behavior of the two buckling mode shapes is analyzed using a perturbation analysis. We show the existence of jump phenomena for the two modes indicating a complex dynamic of the structure near the three-to-one internal resonance for the HM Graphite/epoxy materials.     

Homoclinic orbits in a shallow arch subjected to periodic excitation

Homoclinic orbits in a shallow arch subjected to periodic excitation are investigated in the presence of 1:1 internal resonance and external resonance. The method of multiple scales is used to obtain a set of near-integrable systems. The geometric singular perturbation method and Melnikov method are employed to show the existence of the one-bump and multi-bump homoclinic orbits that connect equilibria in a resonance band of the slow manifold. These orbits arise from singular homoclinic orbits and are composed of alternating slow and fast pieces. The result obtained imply the existence of the amplitude-modulated chaos for the Smale horseshoe sense in the class of shallow arch systems.     

Orbits homoclinic to resonances in a harmonically excited and undamped circular plate

Orbits homoclinic to resonances in mode interactions of an imperfect circular plate with 1:1 internal resonance are investigated. The case of primary resonance is considered. The damping force is not included in the analysis. The energy-phase criterion is used to give a fairly complete picture of the complex dynamics associated with orbits homoclinic to the resonances. A saddle-node bifurcation of homoclinic orbits occurs. The existence of homoclinic orbits in the unperturbed system may lead to chaos in the sense of Smale horseshoes under perturbation.     

多储液腔航天器刚液耦合动力学与复合控制

采用复合控制方法对充液航天器的姿态和轨道机动进行高精度控制.通过傅里叶-贝塞尔级数展开法,将低重力环境下液体的弯曲自由表面的动态边界条件转化为简单的微分方程,其中耦合液体晃动方程的状态向量由相对势函数的模态坐标和波高的模态坐标组成.通过广义准坐标下的拉格朗日方程得到航天器刚体部分运动和液体燃料晃动的耦合动力学方程,提出了自适应快速终端滑模策略和输入整形技术相结合的复合控制器,并分别用于控制携带有一个燃料腔和四个燃料腔航天器的轨道机动和姿态机动.通过数值模拟来验证控制器的效率和精度.结果表明,对于多储液腔航天器,如果在设计航天器的姿态和轨道控制器时没有充分考虑燃料晃动效应,那么在受控航天器系统中将会出现刚-液-控耦合问题并导致航天器姿态不稳定.而本研究中的复合自适应终端滑模控制器可以实现航天器机动的高精度控制并有效抑制液体燃料晃动.     

Dynamics of a Cubic Nonlinear Vibration Absorber

We study the dynamics of a nonlinear active vibration absorber. We consider a plant model possessing curvature and inertia nonlinearities and introduce a second-order absorber that is coupled with the plant through user-defined cubic nonlinearities. When the plant is excited at primary resonance and the absorber frequency is approximately equal to the plant natural frequency, we show the existence of a saturation phenomenon. As the forcing amplitude is increased beyond a certain threshold, the response amplitude of the directly excited mode (plant) remains constant, while the response amplitude of the indirectly excited mode (absorber) increases. We obtain an approximate solution to the governing equations using the method of multiple scales and show that the system possesses two possible saturation values. Using numerical techniques, we perform stability analyses and demonstrate that the system exhibits complicated dynamics, such as Hopf bifurcations, intermittency, and chaotic responses.     

Stability of Equilibrium Solutions of Autonomous and Periodic Hamiltonian Systems with <Emphasis Type="Italic">n</Emphasis>-Degrees of Freedom in the Case of Single Resonance

This paper concerns with the study of the stability of an equilibrium solution of an analytic Hamiltonian system in a neighborhood of the equilibrium point with n-degrees of freedom, in the autonomous and periodic case under the presence of a single resonance. Our Main Theorem generalizes several results existing in the literature and we also give a geometrical interpretation of the hypotheses involved there. In particular, our Main Theorem provides necessary and sufficient conditions for the stability of the equilibrium solutions under the existence of a single resonance, depending on the coefficients of the Hamiltonian function.     

Global bifurcations and chaotic motions of a flexible multi-beam structure

Global bifurcations and multi-pulse chaotic motions of flexible multi-beam structures derived from an L-shaped beam resting on a vibrating base are investigated considering one to two internal resonance and principal resonance. Base on the exact modal functions and the orthogonality conditions of global modes, the PDEs of the structure including both nonlinear coupling and nonlinear inertia are discretized into a set of coupled autoparametric ODEs by using Galerkin’s technique. The method of multiple scales is applied to yield a set of autonomous equations of the first order approximations to the response of the dynamical system. A generalized Melnikov method is used to study global dynamics for the “resonance case”. The present analysis indicates multi-pulse chaotic motions result from the existence of Šilnikov’s type of homoclinic orbits and the critical parameter surface under which the system may exhibit chaos in the sense of Smale horseshoes are obtained. The global results are finally interpreted in terms of the physical motion of such flexible multi-beam structure and the dynamical mechanism on chaotic pattern conversion between the localized mode and the coupled mode are revealed.     

Multi-pulse orbits and chaotic dynamics in a nonlinear forced dynamics of suspended cables

The global bifurcations in mode of a nonlinear forced dynamics of suspended cables are investigated with the case of the 1:1 internal resonance. After determining the equations of motion in a suitable form, the energy phase method proposed by Haller and Wiggins is employed to show the existence of the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the two cases of Hamiltonian and dissipative perturbation. Furthermore, some complex chaos behaviors are revealed for this class of systems.     

Multi-physics dynamics of a mechanical oscillator coupled to an electro-magnetic circuit

The dynamics of a non-linear electro-magneto-mechanical coupled system is addressed. The non-linear behavior arises from the involved coupling quadratic non-linearities and it is explored by relying on both analytical and numerical tools. When the linear frequency of the circuit is larger than that of the mechanical oscillator, the dynamics exhibits slow and fast time scales. Therefore the mechanical oscillator forced (actuated) via harmonic voltage excitation of the electric circuit is analyzed; when the forcing frequency is close to that of the mechanical oscillator, the long term damped dynamics evolves in a purely slow timescale with no interaction with the fast time scale. We show this by assuming the existence of a slow invariant manifold (SIM), computing it analytically, and verifying its existence via numerical experiments on both full- and reduced-order systems. In specific regions of the space of forcing parameters, the SIM is a complicated geometric object as it undergoes folding giving rise to hysteresis mechanisms which create a pronounced non-linear resonance phenomenon. Eventually, the roles played by the electro-magnetic and mechanical components in the resulting complex response, encompassing bifurcations as well as possible transitions from regular to chaotic motion, are highlighted by means of Poincaré sections.     

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.     

Design of CNF-based nonlinear integral sliding surface for matched uncertain linear systems with multiple state-delays

This paper presents an approach of achieving high performance and robustness for matched uncertain multi-input multi-output linear systems with external disturbances and multiple state-delays, which are often encountered in practice and are frequently the sources of instability. This scheme is based on composite nonlinear feedback and integral sliding mode control methods. The selection of nonlinear function and the existence of sliding mode are two important issues, which have been addressed. The control law is designed to guarantee the existence of the sliding mode around the nonlinear surface, and the damping ratio of the closed-loop system is increased as the output approaches the set-point. Simulation results are presented to show the effectiveness of the proposed method as a promising way for controlling similar nonlinear systems.     

No-chatter variable structure control for fractional nonlinear complex systems

This paper concerns the problem of robust control of uncertain fractional-order nonlinear complex systems. After establishing a simple linear sliding surface, the sliding mode theory is used to derive a novel robust fractional control law for ensuring the existence of the sliding motion in finite time. We use a nonsmooth positive definitive function to prove the stability of the controlled system based on the fractional version of the Lyapunov stability theorem. In order to avoid the chattering, which is inherent in conventional sliding mode controllers, we transfer the sign function of the control input into the first derivative of the control signal. The proposed sliding mode approach is also applied for control of a class of nonlinear fractional-order systems via a single control input. Simulation results indicate that the proposed fractional variable structure controller works well for stabilization of hyperchaotic and chaotic complex fractional-order nonlinear systems. Moreover, it is revealed that the control inputs are free of chattering and practical.     

HOMOTOPY PERTURBATION SOLUTION AND PERIODICITY ANALYSIS OF NONLINEAR VIBRATION OF THIN RECTANGULAR FUNCTIONALLY GRADED PLATES

In this paper nonlinear analysis of a thin rectangular functionally graded piate is formulated in terms of von-Karman＇s dynamic equations. Functionaily Graded Material （FGM） properties vary through the constant thickness of the plate at ambient temperature. By expansion of the solution as a series of mode functions, we reduce the governing equations of motion to a Duffing＇s equation. The homotopy perturbation solution of generated Duffing＇s equation is also obtained and compared with numerical solutions. The sufficient conditions for the existence of periodic oscillatory behavior of the plate are established by using Green＇s function and Schauder＇s fixed point theorem.     

Dynamics of Nonlinear Oscillators under Simultaneous Internal and External Resonances

An analysis is presented for a class of two degree of freedom weakly nonlinear oscillators, with symmetric restoring force. Conditions of one-to-three internal resonance and subharmonic external resonance of the lower vibration mode are assumed to be satisfied simultaneously. As a consequence, the second vibration mode may also be under the action of external primary resonance. Initially, a set of slow-flow equations is derived, governing the amplitudes and phases of approximate long time response of these oscillators, by applying an asymptotic analytical method. Determination of several possible types of steady-state motions is then reduced to solution of sets of algebraic equations. For all these solution types, appropriate stability analysis is also performed. In the second part of the study, this analysis is applied to an example mechanical system. First, a systematic search is performed, revealing effects of system parameters on the existence and stability properties of periodic motions. Frequency-response diagrams are presented and attention is focused on understanding the evolution and interaction of the various solution branches as the external forcing and nonlinearity parameters are varied. Finally, numerical integration of the equations of motion demonstrates that the system exhibits quasiperiodic or chaotic response for some parameter combinations.