Global bifurcations and homoclinic trees in motion of a thin rectangular plate on a nonlinear elastic foundation

The global bifurcations and chaotic dynamics of a thin rectangular plate on a nonlinear elastic foundation subjected to a harmonic excitation are investigated. On the basis of the amplitude and phase modulation equations derived by the method of multiple scales, a near integrable two-degree-of-freedom Hamiltonian system is obtained by a transformation. The energy-phase method proposed by Haller and Wiggins is employed to analyze the global bifurcations for the thin rectangular plate. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case, which implies that chaotic motions may occur for this class of systems. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. To illustrate the theoretical predictions, we present visualizations of these complicated structures and numerical evidence of chaotic motions.     

Multi-pulse chaotic motions of high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate

This paper investigates the multi-pulse global bifurcations and chaotic dynamics of the high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate by using an extended Melnikov method in the resonant case. Using the von Karman type equations, Reddy’s third-order shear deformation plate theory and Hamilton’s principle, the equations of motion are derived for the laminated composite piezoelectric rectangular plate with combined parametric excitations and transverse excitation. Applying the method of multiple scales and Galerkin’s approach to the partial differential governing equation, the four-dimensional averaged equation is obtained for the case of 1:2 internal resonance and primary parametric resonance. From the averaged equations obtained, the theory of normal form is used to derive the explicit expressions of normal form with a double zero and a pair of pure imaginary eigenvalues. Based on the explicit expressions of normal form, the extended Melnikov method is used for the first time to investigate the Shilnikov type multi-pulse homoclinic bifurcations and chaotic dynamics of the laminated composite piezoelectric rectangular plate. The necessary conditions of the existence for the Shilnikov type multi-pulse chaotic dynamics of the laminated composite piezoelectric rectangular plate are analytically obtained. Numerical simulations also illustrate that the Shilnikov type multi-pulse chaotic motions can also occur in the laminated composite piezoelectric rectangular plate. Overall, both theoretical and numerical studies demonstrate that the chaos in the Smale horseshoe sense exists for the laminated composite piezoelectric rectangular plate.     

MULTI-PULSE ORBITS AND CHAOTIC DYNAMICS OF A COMPOSITE LAMINATED RECTANGULAR PLATE

This paper presents an analysis on the nonlinear dynamics and multi-pulse chaotic motions of a simply-supported symmetric cross-ply composite laminated rectangular thin plate with the parametric and forcing excitations. Firstly, based on the Reddy’s third-order shear deformation plate theory and the model of the von Karman type geometric nonlinearity, the nonlinear governing partial difirential equations of motion for the composite laminated rectangular thin plate are derived by using the Hamilton’s principle. Then, using the second-order Galerkin discretization, the partial differential governing equations of motion are transformed to nonlinear ordinary differential equations. The case of the primary parametric resonance and 1:1 internal resonance is considered. Four-dimensional averaged equation is obtained by using the method of multiple scales. From the averaged equation obtained here, the theory of normal form is used to give the explicit expressions of normal form. Based on normal form, the energy phase method is utilized to analyze the global bifurcations and multi-pulse chaotic dynamics of the composite laminated rectangular thin plate. The theoretic results obtained above illustrate the existence of the chaos for the Smale horseshoe sense in a parametrical and forcing excited composite laminated thin plate. The chaotic motions of the composite laminated rectangular thin plate are also found by using numerical simulation, which also indicate that there exist different shapes of the multi-pulse chaotic motions for the composite laminated rectangular thin plate.     

Multi-pulse chaotic dynamics of six-dimensional non-autonomous nonlinear system for a composite laminated piezoelectric rectangular plate

Global bifurcations and multi-pulse chaotic dynamics are studied for a four-edge simply supported composite laminated piezoelectric rectangular plate under combined in-plane, transverse, and dynamic electrical excitations. Based on the von Karman type equations for the geometric nonlinearity and Reddy’s third-order shear deformation theory, the governing equations of motion for a composite laminated piezoelectric rectangular plate are derived. The Galerkin method is employed to discretize the partial differential equations of motion to a three-degree-of-freedom nonlinear system. The six-dimensional non-autonomous nonlinear system is simplified to a three-order standard form by using the method of normal form. The extended Melnikov method is improved to investigate the six-dimensional non-autonomous nonlinear dynamical system in mixed coordinate. The global bifurcations and multi-pulse chaotic dynamics of the composite laminated piezoelectric rectangular plate are studied by using the improved extended Melnikov method. The multi-pulse chaotic motions of the system are found by using numerical simulation, which further verifies the result of theoretical analysis.     

Global bifurcations of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation

The global bifurcations in mode interaction of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation are investigated with the case of the 1:1 internal resonance, the average equations representing the evolution of the amplitudes and phases of the interacting normal modes exhibiting complex dynamics. A global perturbation method, i.e., the higher-dimensional Melnikov method and its extensions proposed by Kova?i? and Wiggins, is utilized to analyze the global bifurcations for the rectangular metallic plate. A sufficient condition for the existence of a Silnikov-type homoclinic orbit is obtained, which implies that chaotic motions may occur for this class of rectangular metallic plates. Finally, numerical results are presented to confirm these analytical predictions.     

Global bifurcations and multi-pulse chaotic dynamics of rectangular thin plate with one-to-one internal resonance

Global bifurcations and multi-pulse chaotic dynamics for a simply supported rectangular thin plate are studied by the extended Melnikov method.The rectangular thin plate is subject to transversal and in-plane excitation.A two-degree-of-freedom nonlinear nonautonomous system governing equations of motion for the rectangular thin plate is derived by the von Karman type equation and the Galerkin approach.A one-toone internal resonance is considered.An averaged equation is obtained with a multi-scale method.After transforming the averaged equation into a standard form,the extended Melnikov method is used to show the existence of multi-pulse chaotic dynamics,which can be used to explain the mechanism of modal interactions of thin plates.A method for calculating the Melnikov function is given without an explicit analytical expression of homoclinic orbits.Furthermore,restrictions on the damping,excitation,and detuning parameters are obtained,under which the multi-pulse chaotic dynamics is expected.The results of numerical simulations are also given to indicate the existence of small amplitude multi-pulse chaotic responses for the rectangular thin plate.     

Multi-pulse chaotic motions of functionally graded truncated conical shell under complex loads

The global bifurcations and multi-pulse orbits of an aero-thermo-elastic functionally graded material (FGM) truncated conical shell under complex loads are investigated with the case of 1:2 internal resonance and primary parametric resonance. The method of multiple scales is utilized to obtain the averaged equations. Based on the averaged equations obtained, the normal form theory is employed to find the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method developed by Haller and Wiggins is used to analyze the multi-pulse homoclinic bifurcations and chaotic dynamics of the FGM truncated conical shell. The analytical results obtained here indicate that there exist the multi-pulse Shilnikov-type homoclinic orbits for the resonant case which may result in chaos in the system. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. The diagrams show a gradual breakup of the homoclinic tree in the system as the dissipation factor is increased. Numerical simulations are presented to illustrate that for the FGM truncated conical shell, the multi-pulse Shilnikov-type chaotic motions can occur. The influence of the structural-damping, the aerodynamic-damping, and the in-plane and transverse excitations on the system dynamic behaviors is also discussed by numerical simulations. The results obtained here mean the existence of chaos in the sense of the Smale horseshoes for the FGM truncated conical shell.     

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.     

Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance

Nonlinear flexural vibrations of a rectangular plate with uniform stretching are studied for the case when it is harmonically excited with forces acting normal to the midplane of the plate. The physical phenomena of interest here arise when the plate has two distinct linear modes of vibration with nearly the same natural frequency. It is shown that, depending on the spatial distribution of the external forces, the plate can undergo harmonic motions either in one of the two individual modes or in a mixed-mode. Stable single-mode and mixed-mode solutions can also coexist over a wide range in the amplitudes and frequency of excitation. For low damping levels, the presence of Hopf bifurcations in the mixed-mode response leads to complicated amplitude-modulated dynamics including period doubling bifurcations, chaos, coexistence of multiple chaotic motions, and crisis, whereby the chaotic attractors suddenly disappear and the plate resumes small amplitude harmonic motions in a single-mode. Numerical results are presented specifically for 1 : 1 resonance in the (1, 2) and (3, 1) plate modes.     

Multi-pulse homoclinic orbits and chaotic dynamics for an axially moving viscoelastic beam

The multi-pulse homoclinic orbits and chaotic dynamics for an axially moving viscoelastic beam are investigated in the case of 1:2 internal resonance. On the basis of the modulation equations derived by the method of multiple scales, the theory of normal form is utilized to find the explicit formulas of normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method is employed to analyze the global bifurcations for the axially moving viscoelastic beam. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case, leading to chaos in the system. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. To illustrate the theoretical predictions, we present visualizations of these complicated structures.     

Global bifurcations and multi-pulse orbits of a parametric excited system with autoparametric resonance

We consider an autoparametric system which consists of an oscillator coupled with a parametrically excited subsystem. The oscillator and the subsystem are in one-to-one internal resonance. The excited subsystem is in principal parametric resonance. The system contains the most general type of quadratic and cubic non-linearities. The method of second-order averaging is used to yield a set of autonomous equations of the second-order approximations to the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse 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 Shilnikov-type multi-pulse homoclinic orbits in the averaged equations. The results obtained above mean the existence of amplitude-modulated chaos in the Smale horseshoe sense in the parametric excited system with autoparametric resonance. The Shilnikov-type multi-pulse chaotic motions of the parametric excited system with autoparametric resonance are also found by using numerical simulation.     

Nonlinear vibrations of a composite laminated cantilever rectangular plate with one-to-one internal resonance

The nonlinear vibrations of a composite laminated cantilever rectangular plate subjected to the in-plane and transversal excitations are investigated in this paper. Based on the Reddy??s third-order plate theory and the von Karman type equations for the geometric nonlinearity, the nonlinear partial differential governing equations of motion for the composite laminated cantilever rectangular plate are established by using the Hamilton??s principle. The Galerkin approach is used to transform the nonlinear partial differential governing equations of motion into a two degree-of-freedom nonlinear system under combined parametric and forcing excitations. The case of foundational parametric resonance and 1:1 internal resonance is taken into account. The method of multiple scales is utilized to obtain the four-dimensional averaged equation. The numerical method is used to find the periodic and chaotic motions of the composite laminated cantilever rectangular plate. It is found that the chaotic responses are sensitive to the changing of the forcing excitations and the damping coefficient. The influence of the forcing excitation and the damping coefficient on the bifurcations and chaotic behaviors of the composite laminated cantilever rectangular plate is investigated numerically. The frequency-response curves of the first-order and the second-order modes show that there exists the soft-spring type characteristic for the first-order and the second-order modes.     

Resonant chaotic motions of a buckled rectangular thin plate with parametrically and externally excitations

Resonant chaotic motions of a simply supported rectangular thin plate with parametrically and externally excitations are analyzed using exponential dichotomies and an averaging procedure for the first time. The formulas of the rectangular thin plate are derived by a von Karman type equation and the Galerkin’s approach. The critical condition to predict the onset of chaotic motions for the full system is obtained by developing a Melnikov function containing terms from the non-hyperbolic mode. We prove that the non-hyperbolic mode of the thin plate does not affect the critical condition for the occurrence of chaotic motions in the resonant case. Simulations also show that the chaotic motions of the hyperbolic subsystem are shadowed by the chaotic motions for the full system of the rectangular thin plate.     

Complicated nonlinear responses of a simply supported FGM rectangular plate under combined parametric and external excitations

In this paper, we use the asymptotic perturbation method based on the Fourier expansion and the temporal rescaling to investigate the nonlinear oscillations and chaotic dynamics of a simply supported rectangular plate made of functionally graded materials (FGMs) subjected to a through-thickness temperature field together with parametric and external excitations. Material properties are assumed to be temperature-dependent. Based on the Reddy’s third-order plate theory, the governing equations of motion for the plate are derived using the Hamilton’s principle. The Galerkin procedure is employed to obtain a two-degree-of-freedom nonlinear system including the quadratic and cubic nonlinear terms. The resonant case considered here is 1:2 internal resonance, principal parametric resonance-1/2 subharmonic resonance. Based on the averaged equation in polar coordinate form, the stability of steady state solutions is analyzed. The phase portrait, waveform and Poincaré map are used to analyze the periodic and chaotic motions of the FGM rectangular plate. It is found that the FGM rectangular plate exhibits the chaotic motions under certain circumstances. It is seen that the nonlinear dynamic responses of the FGM rectangular plate are more sensitive to transverse excitation. The excitation force can be used as a controlling factor which can change the response of the FGM rectangular plate from periodic motion to the chaotic motion.     

Global bifurcations and chaos of a composite laminated cylindrical shell in supersonic air flow

Nonlinear Dynamics - The global bifurcations and chaotic motions in mode interaction of a composite laminated cylindrical shell subjected to complex loads are investigated in the case of...     

Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory

In this paper, an analysis on the nonlinear dynamics and chaos of a simply supported orthotropic functionally graded material (FGM) rectangular plate in thermal environment and subjected to parametric and external excitations is presented. Heat conduction and temperature-dependent material properties are both taken into account. The material properties are graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. Based on the Reddy’s third-order share deformation plate theory, the governing equations of motion for the orthotropic FGM rectangular plate are derived by using the Hamilton’s principle. The Galerkin procedure is applied to the partial differential governing equations of motion to obtain a three-degree-of-freedom nonlinear system. The resonant case considered here is 1:2:4 internal resonance, principal parametric resonance-subharmonic resonance of order 1/2. Based on the averaged equation obtained by the method of multiple scales, the phase portrait, waveform and Poincare map are used to analyze the periodic and chaotic motions of the orthotropic FGM rectangular plate. It is found that the motions of the orthotropic FGM plate are chaotic under certain conditions.     

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.     

Analysis of global dynamics in a parametrically excited thin plate

The global bifurcations and chaos of a simply supported rectangular thin plate with parametric excitation are analyzed. The formulas of the thin plate are derived by von Karman type equation and Galerkin's approach. The method of multiple scales is used to obtain the averaged equations. Based on the averaged equations, the theory of the normal form is used to give the explicit expressions of the normal form associated with a double zero and a pair of pure imaginary eigenvalues by Maple program. On the basis of the normal form, a global bifurcation analysis of the parametrically excited rectangular thin plate is given by the global perturbation method developed by Kovacic and Wiggins. The chaotic motion of thin plate is also found by numerical simulation. The project supported by the National Natural Science Foundation of China (10072004) and by the Natural Science Foundation of Beijing (3992004)     

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.     

高维非线性系统的全局分岔和混沌动力学研究

综述了Melnikov方法的发展历史, 从1963年苏联学者Melnikov提出该方法开始, 一直到目前广义Melnikov方法的提出和发展. Melnikov方法的发展历程可以概括为3 个阶段, 分别综述了每一个阶段Melnikov方法的扩展和应用, 论述了国内外在该方向的研究现状和所获得的主要结果, 指出了各种Melnikov方法之间的联系、存在的问题和不足. 为了对比两种研究高维非线性系统多脉冲混沌动力学的理论, 本文综述了另外一种全局摄动理论, 即能量相位法, 总结了该方法十几年来的发展历史以及国内外的理论研究成果和工程应用实例, 阐述了能量相位法发展的根源以及与Melnikov方法的内在联系, 比较了能量相位法和广义Melnikov方法两种理论研究对象的差别, 以及各自所存在的不足和问题. 简要论述了能量相位法和广义Melnikov方法的理论体系, 并利用广义Melnikov方法分析了四边简支矩形薄板的多脉冲混沌动力学, 数值模拟进一步验证了理论研究的结果. 最后, 详细综述了两种理论的缺点和不足, 说明今后全局摄动理论的发展方向.