Free vibration analysis of sandwich beams using improved dynamic stiffness method

In this paper, free vibration of three-layered symmetric sandwich beam is investigated using dynamic stiffness and finite element methods. To determine the governing equations of motion by the present theory, the core density has been taken into consideration. The governing partial differential equations of motion for one element contained three layers are derived using Hamilton’s principle. This formulation leads to two partial differential equations which are coupled in axial and bending deformations. For the harmonic motion, these equations are combined to form one ordinary differential equation. Closed form analytical solution for this equation is determined. By applying the boundary conditions, the element dynamic stiffness matrix is developed. They are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by the use of numerical techniques and the known Wittrick–Williams algorithm. After validation of the present model, the effect of various parameters such as density, thickness and shear modulus of the core for various boundary conditions on the first natural frequency is studied.     

High-order free vibration analysis of sandwich beams with a flexible core using dynamic stiffness method

In this paper, high-order free vibration of three-layered symmetric sandwich beam is investigated using dynamic stiffness method. The governing partial differential equations of motion for one element are derived using Hamilton’s principle. This formulation leads to seven partial differential equations which are coupled in axial and bending deformations. For the harmonic motion, these equations are divided into two ordinary differential equations by considering the symmetrical sandwich beam. Closed form analytical solutions of these equations are determined. By applying the boundary conditions, the element dynamic stiffness matrix is developed. The element dynamic stiffness matrices are assembled and the boundary conditions of the beam are applied, so that the dynamic stiffness matrix of the beam is derived. Natural frequencies and mode shapes are computed by use of numerical techniques and the known Wittrick–Williams algorithm. Finally, some numerical examples are discussed using dynamic stiffness method.     

Wave propagation in anisotropic poroelastic beam with axial–flexural coupling

The exact solution of the governing partial differential equations describing the motion of an anisotropic porous beam with axial-flexural coupling is presented. The motion of the beam is described by the classical Euler–Bernoulli theory. The pore-fluid pressure is governed by the generalized Darcy’s law with relaxation and retardation time parameters to account for the inertia and viscosity of the fluid. Solutions are sought in the frequency domain where the governing equations are converted into a polynomial eigenvalue structure and solved exactly. The wavenumbers and group speeds of propagating waves in the beam are studied in detail. It is found that the presence of fluid-filled porous micro-structure introduces three additional propagating modes, other than the axial and bending modes predicted by the classical beam theory. The effect of diffusion boundary conditions on the transverse motion of a porous beam is investigated in detail. It is also found that the material parameters have considerable influence on the magnitude of the transverse velocity, the group speed of propagation and the behavior of the pressure resultants.     

高速旋转飞行器弯-扭-轴向变形耦合与拍振

以高速旋转大长径比飞行器为对象，考虑发射过程中弹体在各种激励作用下的弯曲，扭转和轴向耦合运动与变形，运用Euler-Bernoulli旋转梁理论，给出弹体空间运动的一般描述，通过Hamilton原理建立顺的三维耦合动力学方程，并利用参数摄动方法和假设振型方法将非线性偏微分方程化为一组线性常分方程，数值模拟并分析了转速，长径比，弯－扭－轴向度变形耦合等因素对飞行器瞬态响应的影响，揭示了发射过程中由弹体耦合变形诱导的拍振现象。     

Coupled bending–torsional dynamic stiffness matrix for axially loaded beam elements

Analytical expressions for the coupled bending–torsional dynamic stiffness matrix elements of an axially loaded uniform beam element are derived in an exact sense by solving the governing differential equations of motion of the beam. The influence of axial force on the coupled bending–torsional frequencies of a cantilever and hinged–hinged beam of thin-walled section is demonstrated by numerical results. Application of the developed theory includes coupled bending–torsional frequency and mode calculations of helicopter, turbine and propeller blades, plane and space frames, and grillages consisting of axially loaded beam elements with non-coincident mass centre and shear centre.     

Stochastic modal interaction in linear and nonlinear aeroelastic structures

The linear and autoparametric modal interactions in a three defree-of-freedom structure under wide band random excitation are examined. For a structure with constant parameters the linear response is obtained in a closed form. When the structure stiffness matrix involves random fluctuations, the governing equations of motion, in terms of the normal coordinates, are found to be coupled through parametric terms. The structural response is mainly governed by the condition of mean square stability. The boundary of stable-unstable responses is obtained as a function of the internal detuning parameter. The results of the linear system with constant parameters are used as a reference to measure the deviation of the system response when the nonlinear inertia coupling is included. In the neighbourhood of combination internal resonance the system random response is determined by using the Fokker Planck equation approach together with the Gaussian closure scheme. This approach results in 27 coupled first order differential equations in the first and second response moments. These equations are solved numerically. The response is found to deviate significantly from the linear solution when the system internal detuning is close to the exact internal resonance. The autoparametric interaction is found to depend significantly on the system damping ratios and a nonlinear coupling parameter. In the vicinity of combination internal resonance, the second normal mode mean square exhibits an increase associated with a corresponding decrease in the first and third normal modes.     

On finite element modelling of surface tension Variational formulation and applications – Part I: Quasistatic problems

This paper describes variational formulation and finite element discretization of surface tension. The finite element formulation is cast in the Lagrangian framework, which describes explicitly the interface evolution. In this context surface tension formulation emerges naturally through the weak form of the Laplace–Young equation.The constitutive equations describing the behaviour of Newtonian fluids are approximated over a finite time step, leaving the governing equations for the free surface flow function of geometry change rather than velocities. These nonlinear equations are then solved by using Newton-Raphson procedure.Numerical examples have been executed and verified against the solution of the ordinary differential equation resulting from a parameterization of the Laplace-Young equation for equilibrium shapes of drops and liquid bridges under the influence of gravity and for various contact angle boundary conditions.     

Nonlinear deterministic and random response of shallow shells

Deterministic and random vibrations are considered for the case of shear deformable shallow shells composed of multiple perfectly bonded layers. The nontrivial generalization of the flat plate vibrations is expressed by the fact of “small amplitude” vibrations existing about the curved equilibrium position together with the snap-through and snap-buckling type large amplitude vibrations about the flat position. The geometrically nonlinear vibrations are treated by applying Berger’s approximation to the generalized von Karman-type plate equations considering hard hinged supports of the straight boundary segments of skew or even more generally shaped polygonal shells. Shear deformation is considered by means of Mindlin’s kinematic hypothesis and a distributed lateral force loading is applied. Application of a multi-mode expansion in the Galerkin procedure to the governing differential equation, where the eigenfunctions of the corresponding linear plate problem are used as space variables, renders a coupled set of ordinary time differential equations for the generalized coordinates with cubic and quadratic non-linearities. For reasons of convergence, a light viscous modal damping is added. The nonlinear steady-state response of shallow shells subjected to a time-harmonic lateral excitation is investigated and the phenomenon of primary resonance is studied by means of the “perturbation method of multiple scales”. The use of a nondimensional formulation and introduction of the eigen-time of the basic mode of the associated linearized problem provides a unifying result with respect to the planform of the shell. Within the scope of random vibrations, it is assumed that the effective forces can be modelled by uncorrelated, zero-mean wide-band noise processes. Considering the set of modal equations to be finite, the Fokker-Planck-Kolmogorov (FPK) equation for the transition probability density of the generalized coordinates and velocities is derived. Its stationary solution gives the probability of eventual snapping after a long time has elapsed. However, the probability of first occurrence follows from the (approximate) integration of the nonstationary FPK equation. The probability of first dynamic snap-through is derived for a single mode approximation with the influence of higher modes taken into account. Using the two-mode expansion, the probability distribution of the asymmetric snap-buckling is also evaluated.     

计及非齐次边界条件的面内变速运动黏弹性板的稳态响应

研究非齐次边界条件和1∶3内共振下面内平动黏弹性板的横向非线性1∶2主参数振动的稳态响应。考虑黏弹性对边界条件的影响,建立了面内平动板的偏微分运动方程和相应的非齐次边界条件。采用直接多尺度法建立了次谐波参数共振时的可解性条件,并根据Routh-Hurvitz判据判别了系统幅频响应的稳定性。讨论了速度扰动幅值和黏弹性系数对幅频响应的影响,对比了齐次和非齐次边界条件下稳态响应的差异。最后,引入微分求积法验证直接多尺度法的近似解析结果。     

非线性粘弹性桩耦合运动中的混沌分析

研究轴向周期载荷作用下非线性粘弹性桩纵横向耦合运动中的混沌运动。桩体材料满足Leaderman非线性粘弹性本构关系和近似的非线性几何关系，考虑桩体发生纵横向运动的耦合，得到的方程为耦合的非线性偏微分一积分方程；利用Galerkin方法将方程简化并进行数值计算，揭示非线性粘弹性桩的混沌运动和分岔等动力学行为。     

双轴受压反对称角铺设复合材料层合板在固支边界下的后屈曲和模态跃迁

为有效分析双轴受压反对称角铺设复合材料层压板在固支边界下的后屈曲性能, 由渐近修正几何非线性理论推导其双耦合四阶偏微分方程(即应变协调方程和稳定性控制方程), 通过双Fourier级数将耦合非线性控制偏微分方程转换为系列非线性常微分方程, 从而获得相对简单的求解方法。使用广义Galerkin方法求解与角交铺设复合层合板相关的边界值问题, 研究了模态跃迁前后不同复杂程度的后屈曲模式。对四层固支边界复合层合板的数值模拟结果表明: 该解析法与有限元方法在主后屈曲区域的线性屈曲荷载计算结果吻合良好; 有限元方法在解靠近二次分岔点时失去收敛性, 而解析方法可深入后屈曲区域, 准确捕捉模态跃迁现象; 对于反对称角铺设层合板, 可仅用纯对称模态来定性预测主后屈曲分支、二次分岔荷载及远程跃迁路径。      

Large deflection analysis of beams with variable stiffness

Summary. In this paper, the Analog Equation Method (AEM), a BEM-based method, is employed to the nonlinear analysis of a Bernoulli-Euler beam with variable stiffness undergoing large deflections, under general boundary conditions which maybe nonlinear. As the cross-sectional properties of the beam vary along its axis, the coefficients of the differential equations governing the equilibrium of the beam are variable. The formulation is in terms of the displacements. The governing equations are derived in both deformed and undeformed configuration and the deviations of the two approaches are studied. Using the concept of the analog equation, the two coupled nonlinear differential equations with variable coefficients are replaced by two uncoupled linear ones pertaining to the axial and transverse deformation of a substitute beam with unit axial and bending stiffness, respectively, under fictitious load distributions. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. Several beams are analyzed under various boundary conditions and loadings to illustrate the merits of the method as well as its applicability, efficiency and accuracy.     

波浪场中水中悬浮隧道动力响应的研究

针对水中悬浮隧道在波浪力作用下动力响应的问题,通过Hamilton原理推导得到了悬浮隧道管段和锚索的运动控制方程,同时考虑了锚索横向和轴向变形之间的耦合作用,建立了悬浮隧道的动力响应模型,在时间域内采用逐步积分法迭代求解其运动控制方程。波浪力采用Airy线性波理论和Morison方程计算。计算结果表明:当锚索长细比较大时,锚索的自振模态会被激发,其横向和轴向变形之间的耦合作用不可忽略。随着入射波高或悬浮隧道重浮比的增加,悬浮隧道的横荡位移以及横摇角增大,但结构的垂荡位移以及锚索中的应力受波浪的影响较小。     

内外共振联合作用下索－梁组合结构非线性振动分析

研究内共振和外共振联合作用下的索－梁组合结构非线性振动问题。利用Ｈａｍｉｌｔｏｎ原理推导索－梁组合结构非线性动力学方程,同时考虑索的垂度以及由梁和索之间模态耦合引起的非线性影响。利用Ｇａｌｅｒｋｉｎ方法将索－梁组合结构非线性运动偏微分方程离散为一组常微分方程。最后对数学模型进行数值计算,得到了不同内外共振联合作用下梁和索的模态时程曲线。研究表明,梁的稳态运动呈现周期性振荡,而索在不同的内外共振联合作用下,分别呈现出混沌或周期性振荡,并且索和梁之间持续的模态交替现象只能在特定的内外共振下出现。     

Flexural-torsional Vibrations of Beams by BEM

In this paper a boundary element method is developed for the general flexural-torsional vibrations of Euler–Bernoulli beams of arbitrarily shaped constant cross-section. The beam is subjected to arbitrarily transverse and/or torsional distributed or concentrated loading, while its edges are restrained by the most general linear boundary conditions. The resulting initial boundary value problem, described by three coupled partial differential equations, is solved using the analog equation method, a BEM based method. Besides the effectiveness and accuracy of the developed method, a significant advantage is that the displacements as well as the stress resultants are computed at any cross-section of the beam using the respective integral representations as mathematical formulae. The general character of the proposed method is verified from the fact that all basic equations are formulated with respect to an arbitrary coordinate system, which is not restricted to the principal one. Both free and forced vibrations are examined. Several beams are analysed to illustrate the method and demonstrate its efficiency and wherever possible its accuracy. The range of applicability of the thin-tube theory is also investigated through examples with great practical interest.     

Free vibration analysis of composite beams via refined theories

This paper presents a free-vibration analysis of simply supported, cross-ply beams via several higher-order as well as classical theories. The three-dimensional displacement field is approximated along the beam cross-section in a compact form as a generic N-order polynomial expansion. Several higher-order displacements-based theories accounting for non-classical effects can be, therefore, formulated straightforwardly. Classical beam models, such as Euler–Bernoulli’s and Timoshenko’s, are obtained as particular cases. The governing differential equations and the boundary conditions are derived by variationally imposing the equilibrium via the principle of virtual displacements. Thanks to the compact form of the displacement field approximation, governing equations are written in terms of a fundamental nucleo that does not depend upon the approximation order. A Navier-type, closed form solution is adopted in order to derive the governing algebraic equations. Besides the fundamental natural frequency, natural frequencies associated to higher modes (such as torsional, axial, shear and mixed ones) are investigated. A half waves number equal to one is considered. The effect of the length-to-thickness ratio, lamination, aspect ratio and material properties on: (1) the accuracy of the proposed theories and (2) the natural frequencies and modes is presented and discussed. For the latter case, the modes change in order of appearance (modes swapping) and in shape (modes mutation) is investigated. Results are assessed towards three-dimensional FEM solutions. Numerical results show that, upon the choice of the appropriate approximation order, very accurate results can be obtained for all the considered modes.     

Nonlinear free vibration analysis of generic coupled induced strain actuated piezo-laminated beams

Piezo-laminated thin beams have been analyzed with induced strain actuation using Kirchhoff’s hypothesis and von Kármán strain displacement relations. Extremizing the Lagrangian of the system derives the governing nonlinear partial differential equations for the beam. Eliminating the in-plane displacement, an integro-partial differential equation of motion is obtained in terms of the transverse displacement. A deflection function that satisfies the simply supported boundary conditions is assumed to get the system equation as a nonlinear second order ordinary differential equation in time, which is of Duffing’s type. The solution of the problem is obtained through exact integration. Results are presented for frequency and amplitude for surface bonded PZT-5A layer in composite beams with various stacking sequences.     

The analog equation method for large deflection analysis of membranes. A boundary-only solution

 In this paper the analog equation method (AEM) is applied to nonlinear analysis of elastic membranes with arbitrary shape. In this case the transverse deflections influence the inplane stress resultants and the three partial differential equations governing the response of the membrane are coupled and nonlinear. The present formulation, being in terms of the three displacements components, permits the application of geometrical inplane boundary conditions. The membrane is prestressed either by prescribed boundary displacements or by tractions. Using the concept of the analog equation the three coupled nonlinear equations are replaced by three uncoupled Poisson's equations with fictitious sources under the same boundary conditions. Subsequently, the fictitious sources are established using a procedure based on BEM and the displacement components as well as the stress resultants are evaluated from their integral representations at any point of the membrane. Several membranes are analyzed which illustrate the method and demonstrate its efficiency and accuracy. Moreover, useful conclusions are drawn for the nonlinear response of the membranes. The method has all the advantages of the pure BEM, since the discretization and integration is limited only to the boundary. Received 21 November 2000     

转动柔性梁横向振动的一种近似解法

通过随体坐标系的建立分析做定轴转动的刚柔耦合系统的变形运动,在考虑柔性梁轴向、横向变形和截面弯转的情况下,采用Green应变理论分析系统的几何非线性.然后用微元法从应力-应变的角度得出了系统的动力学方程.在考虑梁的几何非线性的同时,通过忽略其轴向变形,得出一个描述转动梁横向振动的强非线性方程.最后采用一种改进的L-P法求得了方程的一阶近似解,通过与能量法所得结果的比较表明,所得近似解有较好的精度.