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.     

Subharmonic Melnikov method of six-dimensional nonlinear systems and application to a laminated composite piezoelectric rectangular plate

In this paper, the bifurcations of subharmonic orbits are investigated for six-dimensional non-autonomous nonlinear systems using the improved subharmonic Melnikov method. The unperturbed system is composed of three independent planar Hamiltonian systems such that the unperturbed system has a family of periodic orbits. The key problem at hand is the determination of the sufficient conditions on some of the periodic orbits for the unperturbed system to generate the subharmonic orbits after the periodic perturbations. Using the periodic transformations and the Poincaré map, an improved subharmonic Melnikov method is presented. Two theorems are obtained and can be used to analyze the subharmonic dynamic responses of six-dimensional non-autonomous nonlinear systems. The subharmonic Melnikov method is directly utilized to investigate the subharmonic orbits of the six-dimensional non-autonomous nonlinear system for a laminated composite piezoelectric rectangular plate. Using the subharmonic Melnikov method, the bifurcation function of the subharmonic orbit is obtained. Numerical simulations are used to verify the analytical predictions. The results of the numerical simulation also indicate the existence of the subharmonic orbits for the laminated composite piezoelectric rectangular plate.     

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.     

复合材料层合板1:1参数共振的分岔研究

针对复合材料对称铺设各向异性矩形层合板的物理模型,在同时考虑了材料、阻尼和几何等非线性因素后,建立了二自由度非线性参数振动系统动力学控制方程,并应用多尺度法求得基本参数共振下的近似解析解,利用数值模拟分析了系统的分岔和混沌运动.指出了伽辽金截断对系统动力学分析的影响,以及系统进入混沌的途径.     

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.     

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.     

压电复合材料层合梁的分岔、混沌动力学与控制

研究了简支压电复合材料层合梁在轴向、横向载荷共同作用下的非线性动力学、分岔和混沌动力学响应. 基于vonKarman理论和Reddy高阶剪切变形理论,推导出了压电复合层合梁的动力学方程. 利用Galerkin法离散偏微分方程,得到两个自由度非线性控制方程,并且利用多尺度法得到了平均方程. 基于平均方程,研究了压电层合梁系统的动态分岔,分析了系统各种参数对倍周期分岔的影响及变化规律. 结果表明,压电复合材料层合梁周期运动的稳定性和混沌运动对外激励的变化非常敏感,通过控制压电激励,可以控制压电复合材料层合梁的振动,保持系统的稳定性,即控制系统产生倍周期分岔解,从而阻止系统通过倍周期分岔进入混沌运动,并给出了控制分岔图.      

Analysis on nonlinear dynamics of a deploying composite laminated cantilever plate

This paper presents the analysis on the nonlinear dynamics of a deploying orthotropic composite laminated cantilever rectangular plate subjected to the aerodynamic pressures and the in-plane harmonic excitation. The third-order nonlinear piston theory is employed to model the transverse air pressures. Based on Reddy’s third-order shear deformation plate theory and Hamilton’s principle, the nonlinear governing equations of motion are derived for the deploying composite laminated cantilever rectangular plate. The Galerkin method is utilized to discretize the partial differential governing equations to a two-degree-of-freedom nonlinear system. The two-degree-of-freedom nonlinear system is numerically studied to analyze the stability and nonlinear vibrations of the deploying composite laminated cantilever rectangular plate with the change of the realistic parameters. The influences of different parameters on the stability of the deploying composite laminated cantilever rectangular plate are analyzed. The numerical results show that the deploying velocity and damping coefficient have great effects on the amplitudes of the nonlinear vibrations, which may lead to the jumping phenomenon of the amplitudes for first-order and second-order modes. The increase of the damping coefficient can suppress the increase of the amplitudes of the nonlinear vibration.     

Multi-pulse jumping orbits and homoclinic trees in motion of a simply supported rectangular metallic plate

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 modulation equations representing the evolution of the amplitudes and phases of the interacting normal modes exhibit complex dynamics. The energy-phase method proposed by Haller and Wiggins is employed to analyze the global bifurcations for the rectangular metallic plate. The results obtained here indicate that there exist the Silnikov-type multi-pulse orbits homoclinic to certain invariant sets for the resonant case in both Hamiltonian and dissipative perturbations, which imply 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.     

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 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 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.     

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.     

亚音速气流中复合材料悬臂板的非线性振动响应研究

随着材料科学的发展,越来越多的新型材料应用到了工程实践中.在气流激励的作用下,对于以航空航天工程为背景、采用复合材料的板壳结构的非线性动力学问题仍是动力学领域的研究热点.本文研究了复合材料悬臂板在亚音速气流条件下的非线性振动和响应.根据理想不可压缩流体的流动条件和 Kutta--Joukowski升力定理,基于升力面理论,利用涡格法计算了三维有限长平板机翼上的亚音速气动升力.将亚音速气动力施加到复合材料悬臂板上,利用Hamilton原理,考虑Reddy三阶剪切变形理论并引入冯$\cdot$卡门非线性应变位移关系,建立了有限长平板的非线性动力学微分方程.利用有限元方法考察了不同几何参数下层合板悬臂板的固有特性,通过比较不同材料和几何参数的线性系统的固有频率,得到不同比例的内共振关系.利用Galerkin方法将偏微分方程截断为两自由度非线性常微分方程,在这里考虑了1:2的内部共振关系并利用多尺度法进行了摄动分析.对应多个选取参数,得到了频率响应曲线.结果展示了硬化弹簧型行为和跳跃现象.      

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.     

Three-dimensional nonlinear vibrations of composite beams — III. Chordwise excitations

Three nonlinear integro-differential equations of motion derived in Part I are used to investigate the forced nonlinear vibration of a symmetrically laminated graphite-epoxy composite beam. The analysis focuses on the case of primary resonance of the first in-plane flexural (chordwise) mode when its frequency is approximately twice the frequency of the first out-of-plane flexural-torsional (flapwise-torsional) mode. A combination of the fundamental-matrix method and the method of multiple scales is used to derive four first-order ordinary-differential equations describing the modulation of the amplitudes and phases of the interacting modes with damping, nonlinearity, and resonances. The eigenvalues of the Jacobian matrix of the modulation equations are used to determine the stability of their constant solutions, and Floquet theory is used to determine the stability and bifurcations of their limit-cycle solutions. Hopf bifurcations, symmetry-breaking bifurcations, period-multiplying sequences, and chaotic motions of the modulation equations are studied. The results show that the motion can be nonplanar although the input force is planar. Nonplanar responses may be periodic, periodically modulated, or chaotically modulated motions.     

Nonlinear active control of damaged piezoelectric smart laminated plates and damage detection

Considering mass and stiffness of piezoelectric layers and damage effects of composite layers, nonlinear dynamic equations of damaged piezoelectric smart laminated plates are derived. The derivation is based on the Hamilton＇s principle, the higher- order shear deformation plate theory, von Karman type geometrically nonlinear straindisplacement relations, and the strain energy equivalence theory. A negative velocity feedback control algorithm coupling the direct and converse piezoelectric effects is used to realize the active control and damage detection with a closed control loop. Simply supported rectangular laminated plates with immovable edges are used in numerical computation. Influence of the piezoelectric layers＇ location on the vibration control is in- vestigated. In addition, effects of the degree and location of damage on the sensor output voltage are discussed. A method for damage detection is introduced.     

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.     

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.