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.     

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.     

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.     

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.      

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.     

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.     

Theoretical and experimental studies on nonlinear oscillations of symmetric cross-ply composite laminated plates

The nonlinear oscillations and resonant responses of the symmetric cross-ply composite laminated plates are investigated theoretically and experimentally. The governing equations of motion for the composite laminated plate are derived by using the von Karman type equation, Reddy’s third-order shear deformation plate theory, and Galerkin method with the geometric nonlinearity. The four-dimensional averaged equation is obtained by using the method of multiple scales. The frequency-response functions are analyzed under the consideration of strongly coupled of two modes. The influences of the resonance case on the softening and hardening type of nonlinearity are analyzed with different parameters for the composite laminated plates. The numerical results indicate that there exist the hardening and softening types of the composite laminated plate in the specific resonant case. The variation of the response amplitudes is studied for the composite laminated plate under combined the transverse and in-plane excitations. A sweep frequency experiment is performed to obtain the hardening and softening nonlinearities of a composite laminated plate. The experimental results coincide with the numerical results qualitatively. The influences of the excitation amplitudes on the softening and hardening types of nonlinearity are also analyzed for the composite laminated plate. The amplitude spectrums of the test plate also demonstrate that the change of the nonlinear dynamic responses may be caused by the subharmonic resonance.     

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.     

VIBRATION SUPPRESSION OF CANTILEVER LAMINATED COMPOSITE PLATE WITH NONLINEAR GIANT MAGNETOSTRICTIVE MATERIAL LAYERS

In this paper, a nonlinear and coupled constitutive model for giant magnetostrictive materials(GMM) is employed to predict the active vibration suppression process of cantilever laminated composite plate with GMM layers. The nonlinear and coupled constitutive model has great advantages in demonstrating the inherent and complicated nonlinearities of GMM in response to applied magnetic field under variable bias conditions(pre-stress and bias magnetic field).The Hamilton principle is used to derive the nonlinear and coupled governing differential equation for a cantilever laminated composite plate with GMM layers. The derived equation is handled by the finite element method(FEM) in space domain, and solved with Newmark method and an iteration process in time domain. The numerical simulation results indicate that the proposed active control system by embedding GMM layers in cantilever laminated composite plate can efficiently suppress vibrations under variable bias conditions. The effects of embedded placement of GMM layers and control gain on vibration suppression are discussed respectively in detail.     

Nonlinear dynamic response of a functionally graded plate with a through-width surface crack

A nonlinear vibration analysis of a simply supported functionally graded rectangular plate with a through-width surface crack is presented in this paper. The plate is subjected to a transverse excitation force. Material properties are graded in the thickness direction according to exponential distributions. The cracked plate is treated as an assembly of two sub-plates connected by a rotational spring at the cracked section whose stiffness is calculated through stress intensity factor. Based on Reddy’s third-order shear deformation plate theory, the nonlinear governing equations of motion for the FGM plate are derived by using the Hamilton’s principle. The deflection of each sub-plate is assumed to be a combination of the first two mode shape functions with unknown constants to be determined from boundary and compatibility conditions. The Galerkin’s method is then utilized to convert the governing equations to a two-degree-of-freedom nonlinear system including quadratic and cubic nonlinear terms under the external excitation, which is numerically solved to obtain the nonlinear responses of cracked FGM rectangular plates. The influences of material property gradient, crack depth, crack location and plate thickness ratio on the vibration frequencies and transient response of the surface-racked FGM plate are discussed in detail through a parametric study.     

Nonlinear harmonic response characteristics and experimental investigation of cantilever hard-coating plate

The nonlinear harmonic response of a cantilever hard-coating plate which is made of a layer of anisotropic hard-coating material and isotropic metal substrate is investigated based on the theory of high-order shear deformation of plate. Firstly, based on the theories of von Karman and Reddy’s three-order shear deformation, the nonlinear dynamic equations of hard-coating plate are built by Hamilton variation principle. Secondly, to obtain nonlinear governing equation of hard-coating plate under transverse load, these equations are discretized in Galerkin method. The system averaged equations with 1:3 internal resonances are obtained by the method of multiple scales, and the multi-periodic responses behavior of cantilever hard-coating plate under transverse loading could be presented. Finally, the vibration response experiment of hard-coating plate is conducted, and the multi-periodic responses are also present for the hard-coating plate with three-to-one internal resonance. Besides, through the vibration response experiment of uncoated titanium alloy plate, the damping characteristic of hard coating is further analyzed.     

Nonlinear dynamic instability analysis of laminated composite thin plates subjected to periodic in-plane loads

In this paper, the dynamic instability of thin laminated composite plates subjected to harmonic in-plane loading is studied based on nonlinear analysis. The equations of motion of the plate are developed using von Karman-type of plate equation including geometric nonlinearity. The nonlinear large deflection plate equations of motion are solved by using Galerkin’s technique that leads to a system of nonlinear Mathieu-Hill equations. Dynamically unstable regions, and both stable- and unstable-solution amplitudes of the steady-state vibrations are obtained by applying the Bolotin’s method. The nonlinear dynamic stability characteristics of both antisymmetric and symmetric cross-ply laminates with different lamination schemes are examined. A detailed parametric study is conducted to examine and compare the effects of the orthotropy, magnitude of both tensile and compressive longitudinal loads, aspect ratios of the plate including length-to-width and length-to-thickness ratios, and in-plane transverse wave number on the parametric resonance particularly the steady-state vibrations amplitude. The present results show good agreement with that available in the literature.     

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.      

Time-varying nonlinear dynamics of a deploying piezoelectric laminated composite plate under aerodynamic force

Using Reddy’s high-order shear theory for laminated plates and Hamilton’s principle, a nonlinear partial differential equation for the dynamics of a deploying cantilevered piezoelectric laminated composite plate, under the combined action of aerodynamic load and piezoelectric excitation, is introduced. Two-degree of freedom (DOF) nonlinear dynamic models for the time-varying coefficients describing the transverse vibration of the deploying laminate under the combined actions of a first-order aerodynamic force and piezoelectric excitation were obtained by selecting a suitable time-dependent modal function satisfying the displacement boundary conditions and applying second-order discretization using the Galerkin method. Using a numerical method, the time history curves of the deploying laminate were obtained, and its nonlinear dynamic characteristics, including extension speed and different piezoelectric excitations, were studied. The results suggest that the piezoelectric excitation has a clear effect on the change of the nonlinear dynamic characteristics of such piezoelectric laminated composite plates. The nonlinear vibration of the deploying cantilevered laminate can be effectively suppressed by choosing a suitable voltage and polarity.     

Nonlinear vibration of a rotating laminated composite circular cylindrical shell: traveling wave vibration

In this paper, the large-amplitude (geometrically nonlinear) vibrations of rotating, laminated composite circular cylindrical shells subjected to radial harmonic excitation in the neighborhood of the lowest resonances are investigated. Nonlinearities due to large-amplitude shell motion are considered using the Donnell’s nonlinear shallow-shell theory, with account taken of the effect of viscous structure damping. The dynamic Young’s modulus which varies with vibrational frequency of the laminated composite shell is considered. An improved nonlinear model, which needs not to introduce the Airy stress function, is employed to study the nonlinear forced vibrations of the present shells. The system is discretized by Galerkin’s method while a model involving two degrees of freedom, allowing for the traveling wave response of the shell, is adopted. The method of harmonic balance is applied to study the forced vibration responses of the two-degrees-of-freedom system. The stability of analytical steady-state solutions is analyzed. Results obtained with analytical method are compared with numerical simulation. The agreement between them bespeaks the validity of the method developed in this paper. The effects of rotating speed and some other parameters on the nonlinear dynamic response of the system are also investigated.     

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.     

LARGE AMPLITUDE FREE VIBRATIONS OF LAMINATED COMPOSITE PLATES

This paper deals with large amplitude free flexural vibrations of laminated composite plates using a 9-node Heterosis degenerated isoparametric quadrilateral element, including the effects of transverse shear and rotary inertia. The nonlinear dynamic equations of the plates are formulated in von Karman's sense. Amplitude-frequency relationships are obtained through dynamic response history using the Newmark numerical integration scheme. Detailed numerical results based on various parameters are presented for orthotropic laminated plates with different boundary conditions. The rectangular anti-symmetric cross-ply plates show the softening type of nonlinearity for initial small amplitudes. The displacement amplitudes decrease and nonlinear frequencies increase with the increment of time. Supported by the NNSFC (No. 19672033), the National Key Project on Basic Research and Applied Research (PD9521904) and the Doctoral Training Foundation of Education Commission of China(No. 98000304).     

A nonlinear composite beam theory

Presented here is a general theory for the three-dimensional nonlinear dynamics of elastic anisotropic initially straight beams undergoing moderate displacements and rotations. The theory fully accounts for geometric nonlinearities (large rotations and displacements) by using local stress and strain measures and an exact coordinate transformation, which result in nonlinear curvature and strain-displacement expressions that contain the von Karman strains as a special case. Extensionality is included in the formulation, and transverse shear deformations are accounted for by using a third-order theory. Six third-order nonlinear partial-differential equations are derived for describing one extension, two bending, one torsion, and two shearing vibrations of composite beams. They show that laminated beams display linear elastic and nonlinear geometric couplings among all motions. The theory contains, as special cases, the Euler-Bernoulli theory, Timoshenko's beam theory, the third-order shear theory, and the von Karman type nonlinear theory.     

Stress Distribution in a Rigidly Clamped Composite Plate with Locally Curved Structures under Forced Vibration

A problem on the forced vibrations of a rectangular composite plate with locally curved structures is formulated using the exact three-dimensional equations of continuum mechanics and continuum theory. A technique for numerical solution of the problem is developed based on the semianalytic finite-element method. Numerical results are given for the stress distribution in the plate under forced vibrations. The results obtained are analyzed to study the effect of the curvature in the structure of the plate on the distribution of stress amplitudes. It is shown that the curvatures change significantly the stress pattern under either static or dynamic loading     

Natural vibrations of a cantilever thin elastic orthotropic cylindrical shell

The natural vibrations of a cantilever thin elastic orthotropic circular cylindrical shell are studied. Dispersion equations for the determination of possible natural frequencies of cantilever closed shells and open shells with Navier hinged boundary conditions at the longitudinal edges are derived from the classical dynamic theory of orthotropic cylindrical shells. It is proved that there are asymptotic relationships between these problems and the problems for a cantilever orthotropic strip plate and for a cantilever rectangular plate and the eigenvalue problem for a semi-infinite closed orthotropic cylindrical shell with free end and for the same but open shell with Navier hinged boundary conditions at the longitudinal edges. A procedure to identify types of vibrations is presented. Orthotropic cylindrical shells with different radii and lengths are used as an example to find approximate values of the dimensionless natural frequency and damping factor for vibration modes __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 5, pp. 68–91, May 2008.