Nonlinear damping and forced vibration analysis of laminated composite beams

The purpose of the present work it to study the damping and forced vibrations of three-layered, symmetric laminated composite beams. In the analytical formulation, both normal and shear deformations are considered in the core by using the higher-order zig-zag theories. The harmonic balance method is coupled with a one mode Galerkin procedure for a simply supported beam. The geometrically nonlinear coupling leads to a nonlinear frequency amplitude equation governed by several complex coefficients. In the first part of the paper, linear and nonlinear damping parameters of laminated composite beams are obtained. In the second part, nonlinear forced vibration analysis is carried out for small and large vibration amplitudes. The frequency response curves are presented and discussed for various geometric and material properties.     

Vibration analyses of laminated composite beams using refined higher-order shear deformation theory

The objective of the paper is to analyze the free vibration of laminated composite beams using a refined higher-order shear deformation theory. The influences of parabolic transverse shear strain, transverse normal strain and Poisson effect are included in the present formulation. The governing differential equations of motion for coupled vibrations of laminated beams are derived using the Hamilton’s principle. In the case of simply supported composite beams, the closed-form solutions for the natural frequency of free harmonic vibration are obtained. The correctness and accuracy of the present theory are validated by comparing the present results with those previously published in the literature and ANSYS solutions.     

Non‐linear finite element modal approach for the large amplitude free vibration of symmetric and unsymmetric composite plates

A theoretical analysis is presented for the large amplitude vibration of symmetric and unsymmetric composite plates using the non‐linear finite element modal reduction method. The problem is first reduced to a set of Duffing‐type modal equations using the finite element modal reduction method. The main advantage of the proposed approach is that no updating of the non‐linear stiffness matrices is needed. Without loss of generality, accurate frequency ratios for the fundamental mode and the higher modes of a composite plate at various values of maximum deflection are then determined by using the Runge–Kutta numerical integration scheme. The procedure for obtaining proper initial conditions for the periodic plate motions is very time consuming. Thus, an alternative scheme (the harmonic balance method) is adopted and assessed, as it was employed to formulate the large amplitude free vibration of beams in a previous study, and the results agreed well with the elliptic solution. The numerical results that are obtained with the harmonic balance method agree reasonably well with those obtained with the Runge–Kutta method. The contribution of each linear mode to the maximum deflection of a plate can also be obtained. The frequency ratios for isotropic and composite plates at various maximum deflections are presented, and convergence of frequencies with the number of finite elements, number of linear modes, and number of harmonic terms is also studied. Copyright © 2005 John Wiley & Sons, Ltd.     

模态数量对薄壁短圆柱壳振动响应分析的影响

采用振型叠加法研究薄壁短圆柱壳受谐波激励的振动响应分析时模态截断数量的影响,且考虑简支-简支、固支-固支和固支-自由三种约束条件。首先基于Love壳体理论建立薄壁短圆柱壳的动力学模型。然后,给出采用振型叠加法进行薄壁短圆柱壳受径向谐波激励时的振动响应求解方法。在三种不同约束条件下,计算采用不同模态截断数量时的稳态响应的振动幅值,对比其一致性,并与实测结果进行比较。结果表明,在这三种约束条件下计算受谐波激励薄壁短圆柱壳的振动响应,需要截取前8阶模态函数用于表征位移模式;模态数据超过8阶对响应计算的精度没有明显改善,实测振动响应结果与解析结果基本吻合。     

叠层板在两项简谐激励作用下的组合共振分析

研究了在四边简支的边界条件下,正交各向异性矩形叠层板在两项横向简谐激励作用下的非线性组合共振及其稳定性问题。在给出了正交各向异性叠层板的振动微分方程的基础上,利用伽辽金法导出了相应的无量纲化达芬型非线性强迫振动方程。应用多尺度法对组合共振问题进行求解,得到了系统在稳态运动下的幅频响应方程。基于李雅普诺夫稳定性理论,得到了解的稳定性判定条件。通过数值算例,对几种复合材料薄板的共振特性进行了分析,分别给出了不同条件下系统运动的响应图、幅频图和动相平面图,讨论了不同参数对系统非线性动力学行为的影响。     

Exact three-dimensional free vibration analysis of thick homogeneous plates coated by a functionally graded layer

Exact closed-form solutions are carried out for both in-plane and out-of-plane free vibration of thick homogeneous simply supported rectangular plates coated by a functionally graded (FG) layer, based on three-dimensional elasticity theory. The elasticity modulus and mass density of the FG coating are assumed to vary exponentially through the thickness of the coating layer, whereas Poisson’s ratio is remaining constant. The equations of motion are solved using two proposed displacement fields for the in-plane and out-of-plane vibration modes. By inserting the displacement fields in the 3-D elasto-dynamic equations, some independent ordinary equations are obtained and solved analytically. Natural frequencies are extracted by satisfying boundary conditions of interface and surfaces of the structure. The solution procedure is validated by comparing the obtained results with corresponding results of a 3-D finite element analysis. Finally, the influence of the FG coating layer on the natural frequencies of the structure is investigated and discussed. Clearly, the present closed-form solutions can exactly predict both in-plane and out-of-plane vibration modes of thick FG coated plates.     

Semi-analytical solution for free transverse vibrations of Euler–Bernoulli nanobeams with manifold concentrated masses

In the present study, transverse vibrations of nanobeams with manifold concentrated masses, resting on Winkler elastic foundations, are investigated. The model is based on the theory of nonlocal elasticity in the presence of concentrated masses applied to Euler–Bernoulli beams. A closed-form expression for the transverse vibration modes of Euler–Bernoulli beams is presented. The proposed expressions are provided explicitly as the function of two integrated constants, which are determined by the standard boundary conditions. The utilization of the boundary conditions leads to definite terms of natural frequency equations. The natural frequencies and vibration modes of the concerned nanobeams with different numbers of concentrated masses in different positions under some typical boundary conditions (simply supported, cantilevered, and clamped–clamped) have been analyzed by means of the proposed closed–form expressions in order to show their efficiency. It is worth mentioning that the effect of various nonlocal length parameters and Winkler modulus on natural frequencies and vibration modes are also discussed. Finally, the results are compared with those corresponding to a classical local model.     

Collapse of clamped and simply supported composite sandwich beams in three-point bending

Composite sandwich beams, comprising glass–vinylester face sheets and a PVC foam core, have been manufactured and tested quasi-statically. Clamped and simply supported beams were tested in three-point bending in order to investigate the initial collapse modes, the mechanisms that govern the post-yield deformation and parameters that set the ultimate strength of these beams. Initial collapse is by three competing mechanisms: face microbuckling, core shear and indentation. Simple formulae for the initial collapse loads of clamped and simply supported beams along with analytical expressions for the finite deflection behaviour of clamped beams are presented. The simply supported beams display a softening post-yield response, while the clamped beams exhibit hardening behaviour due to membrane stretching of the face sheets. Good agreement is found between the measured, analytical and finite element predictions of the load versus deflection response of the simply supported and clamped beams. Collapse mechanism maps with contours of initial collapse load and energy absorption are plotted. These maps are used to determine the minimum mass designs of sandwich beams comprising woven glass face sheets and a PVC foam core.     

Dynamic Stability Optimization of Laminated Composite Plates under Combined Boundary Loading

Dynamic stability and design optimization of laminated simply supported plates under planar conservative boundary loads are investigated in current study. Examples can be found in internal connecting elements of spacecraft and aerospace structures subjected to edge axial and shear loads. Designation of such elements is function of layup configuration, plate aspect ratio, loading combinations, and layup thickness. An optimum design aims maximum stability load satisfying a predefined stable vibration frequency. The interaction between compound loading and layup angle parameter affects the order of merging vibration modes and may stabilize the dynamic response. Laminated plates are assumed to be angle-plies symmetric to mid-plane surface. Dynamic equilibrium PDE has been solved using kernel integral transformation for modal frequency values and eigenvalue-based orthogonal functions for critical stability loads. The dictating dynamic stability mode is shown to be controlled by geometric stiffness distributions of composite plates. Solution of presented design optimization problem has been done using analytical approach combined with interior penalty multiplier algorithm. The results are verified by FEA approach and stability zones of original and optimized plates are stated as final data. Presented method can help designers to stabilize the dynamic response of composite plates by selecting an optimized layup orientation and thickness for prescribed design circumstances.     

Experimental and computed natural frequencies of square pultruded GRP plates: effects of anisotropy, hole size ratio and edge support conditions

Experiments have been carried out to determine the free vibration frequencies and mode shapes of 3.2 mm thick, pultruded GRP, square plates with six combinations of clamped (C), simply supported (S) and free (F) edge supports. Comparison of experimental and theoretical/numerical frequencies confirms that thin homogeneous orthotropic/anisotropic plate theory provides a reasonable model for predicting the free vibration response of pultruded GRP plates. Additional vibration experiments were carried out on plates with central circular cutouts. The hole size ratios were varied from about 0.1 to 0.4 for three combinations of clamped (C) and simply supported (S) edge conditions. Finite-element (FE) frequency and mode shape predictions based on orthotropic plate theory were again shown to be in reasonable agreement with the experimental frequencies and modes.     

基于自重影响的连续轴扭/弯耦合振动的研究

考虑轴的自重的影响,研究轴的质量中心分布线与旋转中心线不重合时的扭/弯耦合振动情况。研究中,基于轴的基本理论建立了轴的扭/弯耦合振动的方程组,并根据振型叠加法以及模态的正交性进行了耦合方程组的求解。对简支轴仅考虑自重情况下轴的扭/弯耦合振动进行了自由振动和强迫振动的研究,对与理论研究条件相同的实际简支轴进行了耦合振动响应的实测研究。研究结果表明:第一、三、五…阶弯曲振动是跟第二、四、六…阶扭转振动相耦合的,而第二、四、六…阶弯曲振动则是跟第一、三、五…阶扭转振动相耦合。     

Thermomechanical bending analysis of functionally graded sandwich plates with both functionally graded face sheets and functionally graded cores

The bending response of functionally graded material (FGM) sandwich plates subjected to thermomechanical loads is investigated using a four-variable refined plate theory. A new type of FGM sandwich plate, namely, both FGM face sheets and an FGM hard core, is considered. Containing only four unknown functions, the governing equations are deduced based on the principle of virtual work and then these equations are solved via the Navier approach. Analytical solutions are obtained to predict the deflections and stresses of simply supported FGM sandwich plates. Benchmark comparisons of the solutions obtained for a degradation model (functionally graded face sheets and homogeneous cores) with ones computed by several other theories are conducted to verify the accuracy and efficiency of the present approach. The influences of volume fraction distribution, geometrical parameters, and thermal load on dimensionless deflections and normal and shear stresses of the FGM sandwich plates are studied.     

Nonlinear vibrations of angle-ply laminated circular cylindrical shells: Skewed modes

Skewed modes in geometrically nonlinear forced vibrations of angle-ply laminated circular cylindrical shells are investigated in the present study by using the Amabili–Reddy higher-order shear deformation theory. An harmonic force excitation is applied in radial direction and simply supported boundary conditions are assumed. The equations of motion are obtained by using an energy approach based on Lagrange equations that retains dissipation. Numerical results are obtained by using the pseudo-arclength continuation method and bifurcation analysis.     

功能梯度矩形板的强非线性共振分析

针对陶瓷-金属功能梯度矩形板,在给出非均匀材料应力-应变关系及非线性几何方程的基础上,应用虚功原理导出了横向简谐激励力作用下功能梯度板的非线性振动偏微分方程。对于四边简支约束功能梯度矩形板,通过位移函数的设定,利用伽辽金积分法推得了关于时间自变量的达芬型强非线性振动方程。针对强非线性系统的主共振问题,应用改进的多尺度法进行解析求解,得到了稳态运动下的幅频响应方程。通过数值算例,给出了功能梯度矩形板共振下的幅频曲线图和相图,讨论了激励幅值及频率等参数对系统非线性振动特性的影响,并对改进多尺度法和经典多尺度法的结果进行了比较。     

简支板振动无线主动控制及驱动器优化布置研究

推导光电层合简支板结构动力学方程及模态控制方程,以规格化后模态控制力指数作为遗传算法的适应度函数,基于二进制编码的遗传算法对用于简支板振动控制的单对、双对光致伸缩驱动器布局进行优化,计算机仿真结果表明优化后的驱动器布局方案可有效提高板结构振动控制的有效性。在此基础上进一步对板结构多模态振动控制进行探讨,提出适用于板结构多模态振动控制的驱动器布局优化方法及振动控制方案,仿真算例表明该方法可有效地对简支板前二阶模态进行振动无线控制。     

随从力作用下功能梯度矩形板的非线性振动

本文研究了切向均布随从力作用下简支FGM矩形板的非线性振动问题。按照材料组份体积分数的简单幂率分布规律,FGM板的材料常数仅沿厚度连续变化。由大挠度的von Karman理论建立了以应力函数和挠度函数表示的运动偏微分方程组,再由Galerkin法转化成非线性常微分方程。对随从力作用下的四边简支陶瓷/金属矩形板,讨论了随从力、梯度指标和边长比对板的动力特性的影响,得到了各种条件下板中心振幅与非线性基频的关系。     

正交异性蜂窝夹层板的动力学分析

采用Reddy 高阶剪切理论对正交各向异性蜂窝夹层板进行了分层研究, 推导出蜂窝夹层板的动力学基本方程, 并且对正交各向异性单向蜂窝夹层板的自由振动进行了深入的研究, 给出了对边简支时的频率方程及振型函数, 同时分析了夹芯厚度及厚跨比对其频率的影响, 并与低阶剪切理论作了比较, 结果证明高阶理论较好。      

Buckling and vibrations of two-directional FGM Mindlin circular plates under hydrostatic peripheral loading

An analysis is presented for the effect of hydrostatic peripheral loading on the free axisymmetric vibrations of functionally graded moderately thick circular plates on the basis of first-order shear deformation theory. The mechanical properties of the plate material are supposed to vary according to a power-law in both the radial and transverse directions. The numerical solution of the governing differential equation derived by using Hamilton's energy principle for such simply supported and clamped boundary conditions has been obtained employing the harmonic differential quadrature method choosing zeros of Chebyshev–Gauss–Lobatto as the grid points. The effect of different parameters has been analyzed on the frequency parameter for the first three modes of vibration. The critical buckling loads for both the plates have been computed by putting the frequencies to zero. Three-dimensional mode shapes for particular plate have been plotted. Obtained results have been compared.     

多加筋圆柱壳体振动特性的导纳法研究

弹性多加筋圆柱壳体由于其结构的特点，广泛应用于工程实际，因此对其进行振动研究具有重要的工程价值和应用意义。文章利用薄板和梁的导纳公式推导了多加筋圆柱壳体在任意位置简谐力激励下的弯曲振动响应，并对特定参数的多加筋圆柱壳体的振动能量分布进行了数值仿真，结果表明振动能量在加筋处呈现急剧衰减。还对不加筋圆柱壳以及加不同数目筋的圆柱壳的振动响应进行了对比，得到了加筋和加不同叛目的筋对结构振动能量分布和传播的影响。     

Convergence to higher symmetric modes in impulsively loaded rigid-plastic beams

The question of convergence onto higher modes in impulsively loaded rigid, perfectly plastic beams is considered. Two specific examples are studied; a fixed end and a simply supported beam in which the velocity field is symmetric and in which there are three hinges in the interior of the span. The equations of motion are written in terms of shape parameters, and the trajectories of the solutions of these equations show geometrically how the shape of the velocity field changes with time. Difficulty is experienced, however, in determining the trajectories which pass through points defining higher modes.