Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam with time-dependent velocity

Non-linearly parametric resonances of an axially moving viscoelastic sandwich beam are investigated in this paper. The beam is moving with a time-dependent velocity, namely a harmonically varied velocity about the mean velocity. The partial differential equation is discretized into nonlinear ordinary differential equations via the method of Galerkin truncation and then the steady-state response is obtained using the method of multiple scales, an approximate analytical method. The tuning equations are obtained by eliminating secular terms and the amplitude of the vibration is derived from the tuning equations expressed in polar form, and two bifurcation points are obtained as well. Additionally, the stability conditions of trivial and nontrivial solutions are analyzed using the Routh–Hurwitz criterion. Eventually, the effects of various parameters such as the thickness of core layer, mean velocity, initial tension, and the amplitude of axially moving velocity on amplitude–frequency response curves and unstable regions are investigated.     

Vibration of axially moving hyperelastic beam with finite deformation

In this paper, we study the vibration of an axially moving hyperelastic beam under simply supported condition. The kinematic of the axially moving beam have been described by Eulerian-Lagrangian formulation. In continuum mechanics frame, the finite deformation formula and a higher order shear deformation beam theory are applied to describe the deformation of the axially moving hyperelastic beam. In these formulas the material parameter, shear deformation and the geometric non-linearity have been taken into account. Through the Hamilton principle, the governing equations of nonlinear vibration are obtained, where the transverse vibration is coupled with the longitudinal vibration. When the velocity is a constant, the critical speed and natural frequencies are determined by solving the corresponding linear equations. Meantime, effects of the geometrical and material parameters on the critical speed and natural frequencies have been investigated. Comparisons among the critical velocities of the hyperelastic and Euler linear beam are also made. The results show that the critical velocity of hyperelastic beam is larger than that of linear Euler–Bernoulli beam. For the natural frequencies, we have the same conclusions. Lastly, by the multiple scales method, the leading order analytical solutions of the equilibrium state of axially moving hyperelastic beam in the supercritical regime are obtained. Furthermore the amplitudes of analytical solutions of the hyperelastic beam have been compared with that of linear Euler–Bernoulli beam. The effects of the material and geometrical parameters on the asymptotic solutions and the amplitude has been analyzed.     

变速度轴向运动粘弹性梁的动态稳定性

研究速度变化的轴向运动粘弹性梁在亚谐波共振及组合共振范围内的参数振动．通过平均法,在运动参数激励频率为2倍固有频率或为两阶固有频率之和附近时得到了自治的常微分方程组．在参数激励频率和激励振幅平面上,可以找到由于共振而产生的失稳区域,并应用数值方法验证了理论推导结果的正确性．分析了粘弹性阻尼,速度和预紧张力对失稳区域的影响．粘弹性阻尼使得共振失稳区域减小,而速度和预紧张力使共振失稳区域在频率-振幅平面上发生漂移．     

Vibrations and stability of axially traveling laminated beams

In this paper, vibrations and stability of an axially traveling laminated composite beam are investigated analytically via the method of multiple scales. Based on classical laminated beam theory, the governing equations of motion for a time-variant axial speed are obtained using Newton’s second law of motion and constitutive relations. The method of multiple scales, an approximate analytical method, is applied directly to the gyroscopic governing equations of motion and complex eigenfunctions and natural frequencies of the system are obtained. The stability boundaries of the system near resonance are determined via the Routh-Hurwitz criterion. Finally, a parametric study is conducted which considers the effects of laminate type and configuration as well as the mean speed and amplitude of speed fluctuations on the vibration response, natural frequencies and stability boundaries of the system.     

末端质量作用下变长度轴向运动梁的振动特性

对带集中质量,变长度(或速度)轴向运动梁的振动特性采用两种精确方法求解.首先,对变长度轴向运动Euler(欧拉)梁横向自由振动方程进行化简,通过复模态分析得到本征方程,并在有集中质量的边界条件下得到频率方程,用数值方法求解固有频率和模态函数.然后,采用有限元方法建立运动梁自由振动的方程,求解矩阵方程得到复特征值和复特征向量,结合形函数得到复模态位移.最后,将两种方法的计算结果进行了分析和对比.数值算例的结果表明:不同的轴向运动速度和集中质量对变长度轴向运动梁的振动特性有显著影响,两种计算方法的结果接近且均有效.     

Vibration and stability analysis of a simply-supported Rayleigh beam with spinning and axial motions

The vibration and stability of a simply supported beam are analyzed when the beam has an axially moving motion as well as a spinning motion. When a beam has spinning and axial motions, rotary inertia plays an important role on the lateral vibration. Compared to previous studies, the present study adopts the Rayleigh beam theory and derives more exact kinetic energy and equations of motion. The rotary inertia terms derived by the present study are compared to those of the previous studies. We investigate the natural frequencies between the present and previous studies. In addition, the critical speed and stability boundary for the spinning and moving speeds are also analyzed. It can be observed from the computed natural frequencies and dynamic responses that the present equations of motion are more reliable than the previous equations because the present equations fully consider the rotary inertia terms.     

粘弹性运动带动力响应分析

基于Kelvin粘弹性材料本构模型及带运动方程,建立了运动带非线性动力学分析模型.基于该模型和Lie群分析方法推导了匀速运动及简谐运动带线性问题的解析解;基于该非线性模型的数值仿真讨论了运动带材料参数、带稳态运动速度、扰动速度对系统动态响应的影响.结果表明:1)当带匀速运动时,无论系统是线性还是非线性,运动带横向振动"频率"都随着带运动稳态速度增加而减小.2)随着材料粘性增加,系统耗散能力逐渐增强,动态响应逐渐减小.3)当带运动速度简谐波动时,系统动态响应随扰动速度增大而增大.扰动频率对带横向振动影响较大.     

Dynamic response of clamped axially moving beams: Integral transform solution

The generalized integral transform technique (GITT) is employed to obtain a hybrid analytical-numerical solution for dynamic response of clamped axially moving beams. The use of the GITT approach in the analysis of the transverse vibration equation leads to a coupled system of second order differential equations in the dimensionless temporal variable. The resulting transformed ODE system is then solved numerically with automatic global accuracy control by using the subroutine DIVPAG from IMSL Library. Excellent convergence behavior is shown by comparing the vibration displacement of different points along the beam length. Numerical results are presented for different values of axial translation velocity and flexural stiffness. A set of reference results for the transverse vibration displacement of axially moving beam is provided for future co-validation purposes.     

Non-linear parametric vibration and stability analysis for two dynamic models of axially moving Timoshenko beams

Non-linear parametric vibration and stability of an axially moving Timoshenko beam are considered for two dynamic models; the first one, with considering only the transverse displacement and the second one, with considering both longitudinal and transverse displacements. The set of non-linear partial-differential equations of both models are derived using an energy approach. The method of multiple scales is applied directly to both models, and using the equation order one, the mode shape equations and natural frequencies are obtained. Then, for the equation order epsilon, the solvability conditions are considered for the resonance case and the stability boundaries are formulated analytically via Routh–Hurwitz criterion. Eventually, some numerical examples are provided to show the differences in the behavior of the above-mentioned non-linear models.     

Bifurcation and chaos of an axially accelerating viscoelastic beam

This paper investigates bifurcation and chaos of an axially accelerating viscoelastic beam. The Kelvin–Voigt model is adopted to constitute the material of the beam. Lagrangian strain is used to account for the beam's geometric nonlinearity. The nonlinear partial–differential equation governing transverse motion of the beam is derived from the Newton second law. The Galerkin method is applied to truncate the governing equation into a set of ordinary differential equations. By use of the Poincaré map, the dynamical behavior is identified based on the numerical solutions of the ordinary differential equations. The bifurcation diagrams are presented in the case that the mean axial speed, the amplitude of speed fluctuation and the dynamic viscoelasticity is respectively varied while other parameters are fixed. The Lyapunov exponent is calculated to identify chaos. From numerical simulations, it is indicated that the periodic, quasi-periodic and chaotic motions occur in the transverse vibrations of the axially accelerating viscoelastic beam.     

Vibration and stability of an axially moving Rayleigh beam

In this paper, the vibration and stability of an axially moving beam is investigated. The finite element method with variable-domain elements is used to derive the equations of motion of an axially moving beam based on Rayleigh beam theory. Two kinds of axial motions including constant-speed extension deployment and back-and-forth periodical motion are considered. The vibration and stability of beams with these motions are investigated. For vibration analysis, direct time numerical integration, based on a Runge–Kutta algorithm, is used. For stability analysis of a beam with constant-speed axial extension deployment, eigenvalues of equations of motion are obtained to determine its stability, while Floquet theory is employed to investigate the stability of the beam with back-and-forth periodical axial motion. The effects of oscillation amplitude and frequency of periodical axial movement on the stability of the beam are discussed from the stability chart. Time histories are established to confirm the results from Floquet theory.     

轴向运动三参数黏弹性梁弱受迫振动的渐近分析

研究了轴向运动三参数黏弹性梁的弱受迫振动．建立了轴向运动三参数黏弹性梁受迫振动的控制方程．使用多尺度法渐近分析了运动梁的稳态响应,导出了解稳定性边界方程、稳态振幅的表达式以及稳态响应非零解的存在条件．依据Routh-Hurwitz定律决定了非线性稳态响应非零解的稳定性．     

微曲输流管道振动固有频率分析与仿真北大核心CSCD

首次建立了基于Timoshenko梁理论的微曲输流管道横向振动的动力学模型,并分析了流体流动影响下微曲管道横向自由振动的固有特征.采用广义Hamilton原理,导出了考虑流体影响的微曲管道横向振动的控制方程,通过Galerkin截断对控制方程离散化,再由广义本征值问题得到管道横向振动的固有频率,并研究了液体流速和弯曲幅度对管道横向固有振动特征的影响.发展了基于等效刚度和等效阻尼方法的考虑流体影响的微曲管道振动分析的有限元仿真计算方法,并通过有限元软件实现数值仿真,验证了Galerkin截断的分析结果以及所建立的Timoshenko微曲管道动力学模型的有效性.研究表明,流体的流速以及管道的弯曲幅度对管道横向振动固有频率均有显著影响.     

Post-buckling and nonlinear free vibration analysis of geometrically imperfect functionally graded beams resting on nonlinear elastic foundation

In this paper, post-buckling and nonlinear vibration analysis of geometrically imperfect beams made of functionally graded materials (FGMs) resting on nonlinear elastic foundation subjected to axial force are studied. The material properties of FGMs are assumed to be graded in the thickness direction according to a simple power law distribution in terms of the volume fractions of the constituents. The assumptions of a small strain and moderate deformation are used. Based on Euler–Bernoulli beam theory and von-Karman geometric nonlinearity, the integral partial differential equation of motion is derived. Then this partial differential equation (PDE) problem, which has quadratic and cubic nonlinearities, is simplified into an ordinary differential equation (ODE) problem by using the Galerkin method. Finally, the governing equation is solved analytically using the variational iteration method (VIM). Some new results for the nonlinear natural frequencies and buckling load of the imperfect functionally graded (FG) beams such as the effects of vibration amplitude, elastic coefficients of foundation, axial force, end supports and material inhomogeneity are presented for future references. Results show that the imperfection has a significant effect on the post-buckling and vibration response of FG beams.     

Transverse nonlinear dynamics of axially accelerating viscoelastic beams based on 4-term Galerkin truncation

This paper investigates bifurcation and chaos in transverse motion of axially accelerating viscoelastic beams. The Kelvin model is used to describe the viscoelastic property of the beam material, and the Lagrangian strain is used to account for geometric nonlinearity due to small but finite stretching of the beam. The transverse motion is governed by a nonlinear partial-differential equation. The Galerkin method is applied to truncate the partial-differential equation into a set of ordinary differential equations. When the Galerkin truncation is based on the eigenfunctions of a linear non-translating beam subjected to the same boundary constraints, a computation technique is proposed by regrouping nonlinear terms. The scheme can be easily implemented in practical computations. When the transport speed is assumed to be a constant mean speed with small harmonic variations, the Poincaré map is numerically calculated based on 4-term Galerkin truncation to identify dynamical behaviors. The bifurcation diagrams are present for varying one of the following parameter: the axial speed fluctuation amplitude, the mean axial speed and the beam viscosity coefficient, while other parameters are unchanged.     

轴向变速运动粘弹性弦线横向振动的复模态Galerkin方法

在考虑初始张力和轴向速度简谐涨落的情况下,利用含预应力三维变形体的运动方程,建立了轴向变速运动弦线横向振动的非线性控制方程,材料的粘弹性行为由Kelvin模型描述．利用匀速运动线性弦线的模态函数构造了变速运动非线性弦线复模态Galerkin方法的基底函数,并借助构造出来的基底函数研究了复模态Galerkin方法在轴向变速运动粘弹性弦线非线性振动分析中的应用．数值结果表明,复模态Galerkin方法相比实模态Galerkin方法对变系数陀螺系统有较高的收敛速度．     

轴向变速运动弦线的非线性振动的稳态响应及其稳定性

研究具有几何非线性的轴向运动弦线的稳态横向振动及其稳定性．轴向运动速度为常平均速度与小简谐涨落的叠加．应用Hamilton原理导出了描述弦线横向振动的非线性偏微分方程．直接应用于多尺度方法求解该方程．建立了避免出现长期项的可解性条件．得到了近倍频共振时非平凡稳态响应及其存在条件．给出数值例子说明了平均轴向速度、轴向速度涨落的幅值和频率的影响．应用Liapunov线性化稳定性理论,导出倍频参数共振时平凡解和非平凡解的不稳定条件．给出数值算例说明相关参数对不稳定条件的影响．     

Analytical analysis of free vibration of non-uniform and non-homogenous beams: Asymptotic perturbation approach

This paper applies the asymptotic perturbation approach (APA) to obtain a simple analytical expression for the free vibration analysis of non-uniform and non-homogenous beams with different boundary conditions. A linear governing equation of non-uniform and non-homogeneous beams is obtained based on the Euler–Bernoulli beam theory. The perturbative theory is employed to derive an asymptotic solution of the natural frequency of the beam. Finally, numerical solutions based on the analytical method are illustrated, where the effect of a variable width ratio on the natural frequency is analyzed. To verify the accuracy of the present method, two examples, piezoelectric laminated trapezoidal beam and axially functionally graded tapered beam, are presented. The results are compared with those results obtained from the finite element method (FEM) simulation and the published literature, respectively, and a good agreement is observed for lower-order beam frequencies.     

Large-amplitude free vibrations of functionally graded beams by means of a finite element formulation

The large-amplitude free vibration analysis of functionally graded beams is investigated by means of a finite element formulation. The Von-Karman type nonlinear strain–displacement relationships are employed where the ends of the beam are constrained to move axially. The effects of the transverse shear deformation and rotary inertia are included based upon the Timoshenko beam theory. The material properties are assumed to be graded in the thickness direction according to the power-law distribution. A statically exact beam element which devoid the shear locking effect with displacement fields based on the first order shear deformation theory is used to study the geometric nonlinear effects on the vibrational characteristics of functionally graded beams. The finite element method is employed to discretize the nonlinear governing equations, which are then solved by the direct numerical integration technique in order to obtain the nonlinear vibration frequencies of functionally graded beams with different boundary conditions. The influences of power-law exponent, vibration amplitude, beam geometrical parameters and end supports on the free vibration frequencies are studied. The present numerical results compare very well with the results available from the literature where possible. Some new results for the nonlinear natural frequencies are presented in both tabular and graphical forms which can be used for future references.     

Transverse free vibration and stability of axially moving nanoplates based on nonlocal elasticity theory

Transverse dynamical behaviors of axially moving nanoplates which could be used to model the graphene nanosheets or other plate-like nanostructures with axial motion are examined based on the nonlocal elasticity theory. The Hamilton's principle is employed to derive the multivariable coupling partial differential equations governing the transverse motion of the axially moving nanoplates. Subsequently, the equations are transformed into a set of ordinary differential equations by the method of separation of variables. The effects of dimensionless small-scale parameter, axial speed and boundary conditions on the natural frequencies in sub-critical region are discussed by the method of complex mode. Then the Galerkin method is employed to analyze the effects of small-scale parameter on divergent instability and coupled-mode flutter in super-critical region. It is shown that the existence of small-scale parameter contributes to strengthen the stability in the super-critical region, but the stability of the sub-critical region is weakened. The regions of divergent instability and coupled-mode flutter decrease even disappear with an increase in the small-scale parameter. The natural frequencies in sub-critical region show different tendencies with different boundary effects, while the natural frequencies in super-critical region keep constants with the increase of axial speed.