Enhanced damping of a cantilever beam by axial parametric excitation

A uniform cantilever beam under the effect of a time-periodic axial force is investigated. The beam structure is discretized by a finite-element approach. The linearised equations of motion describing the planar bending vibrations of the beam structure lead to a system with time-periodic stiffness coefficients. The stability of the system is investigated by a numerical method based on Floquet’s theorem and an analytical approach resulting from a first-order perturbation. It is demonstrated that the parametrically excited beam structure exhibits enhanced damping properties, when excited near a specific parametric combination resonance frequency. A certain level of the forcing amplitude has to be exceeded to achieve the damping effect. Upon exceeding this value, the additional artificial damping provided to the beam is significant and works best for suppression of vibrations of the first vibrational mode of the cantilever beam.     

梁主共振情形下索梁结构1∶1内共振分析

研究梁产生主共振情形下索梁组合结构的1∶1内共振问题。基于斜拉桥中的索梁组合结构模型,忽略索梁纵向惯性力的影响,考虑弯曲刚度、几何非线性及垂度等因素,利用索梁连接处的变形协调条件,采用Hamilton变分原理建立了索梁结构面内耦合非线性偏微分方程,运用Galerkin离散和多尺度法研究了梁主共振情形下索梁的1∶1相互作用问题,获得了内共振时的平均方程和分叉响应曲线方程。以某斜拉桥中索梁结构参数为例,研究了内共振时索梁结构之间的相互影响及时程曲线。结果表明,索容易出现共振情形,并呈现出较强的非线性特点;梁振动对索振动影响显著,索振动对梁振动影响较小;索梁内共振时能量相互交换,索梁振幅呈现此消彼长的现象。     

Non-linear vibrations of suspension bridges with external excitation

Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equations. Free and forced vibrations with damping are investigated in detail. Amplitude and phase modulation equations are obtained. The dependence of non-linear frequency on amplitude is described. Steady-state solutions are analyzed. Frequency-response equation is derived and the jump phenomenon in the frequency-response curves resulting from non-linearity is considered. Effects of initial amplitude and phase values, amplitude of excitation, and damping coefficient on modal amplitudes, are determined.     

粘弹性轴向运动梁的非线性动力学行为

本文研究了带有小脉动的轴向运动粘弹性梁的分岔及混沌现象。建立了系统的动力学模型。通过二阶Galerkin截断，把描述系统运动的偏微分方程离散化。利用数值方法分别分析了几种运动脉动频率时，梁随轴向运动脉动幅值，平均速度及粘弹性系数等几个参数变化时的运动分岔行为。利用Lyapunov指数识别系统的动力学行为，区分准周期振动和混沌运动。     

Transient response analysis of a steel beam with vacuum packed particles

In this paper the method of semi-active damping of vibrations is presented. Free vibrations of a cantilever steel beam encapsulated in a sleeve, filled with the granular material are investigated. Various values of the partial vacuum generated in the granular structure allow to control the global dissipative properties of the discussed system. The loose grains encapsulated in the hermetic, polyvinyl chloride (PVC) envelope transform into a rigid, viscoplastic body as the jamming mechanism occurs when the underpressure is generated. Such phenomenon enables original strategies for semi-active damping. A detailed discussion related to the experimental results concerning the amplitude of vibration, damping, stiffness, and frequency of the continuous granular beam system is provided. The simplified Finite Element Model succeeded in describing the dynamic response of the structure.     

Single-mode response and control of a hinged–hinged flexible beam

The problem of suppressing the vibrations of a hinged–hinged flexible beam that is subjected to primary and principal parametric excitations is tackled. Different control laws are proposed, and saturation phenomenon is investigated to suppress the vibrations of the system. The dynamics of the beam are modeled with a second-order nonlinear ordinary-differential equation. The method of multiple scales is used to derive two-first ordinary differential equations that govern the time variation of the amplitude and phase of the response. These equations are used to determine the steady-state responses and their stability. The results of perturbation solution have been verified through numerical simulations, where different effects of the system parameters on the steady-state amplitude and on saturation phenomena at resonance have been reported.     

DYNAMIC STABILITY OF AXIALLY MOVING VISCOELASTIC BEAMS WITH PULSATING SPEED

IntroductionThe class of systems with axially moving materials involves power transmission chains,band saw blades and paper sheets during processing. Vibration of such systems is generallyundesirable. The traveling tensioned Euler-Bernoulli beam is the pr…     

Wavelet approach to vibratory analysis of surface due to a load moving in the layer

The paper analyses theoretically the surface vibration induced by a point load moving uniformly along a infinitely long beam embedded in a two-dimensional viscoelastic layer. The beam is placed parallel to the traction-free surface and the layer under the beam is assumed to be a half space. The response due to a harmonically varying load is investigated for different load frequencies. The influence of the layer damping and moving load speed on the level of vibrations at the surface is analysed and analytical closed form solutions in the integral form for the displacement amplitude and the amplitude spectra are derived. Approximate displacement values depending on Young’s modulus and mass density of layers are obtained. The mathematical model is described by the Euler–Bernoulli beam equation, Navier’s elastodynamic equation of motion for the elastic medium and appropriate boundary and continuity conditions. A special approximation method based on the wavelet theory is used for calculation of the displacements at the surface.     

Damping of Forced Vibrations in Systems of Connected Rigid Bodies

The problem of damping forced vibrations in a system of bodies linked by one-degree-of-freedom elastic joints is considered. The vibrations are damped by introducing additional bodies into the system. The controlled motion of these bodies compensates the effect of the external perturbation. The motion of a system of three bodies is studied in detail. To determine the sensitivity of the perturbation compensation method to the damping parameters, a numerical simulation is carried out     

Non-linear dynamics of a slender beam carrying a lumped mass with principal parametric and internal resonances

The non-linear behaviour of a slender beam carrying a lumped mass subjected to principal parametric base excitation is investigated. The dimension of the beam–mass system and the position of the attached mass are so adjusted that the system exhibits 3 : 1 internal resonance. Multi-mode discretization of the governing equation which retains the cubic non-linearities of geometrical and inertial type is carried out using Galerkin’s method. The method of multiple scales is used to reduce the second-order temporal differential equation to a set of first-order differential equations which is then solved numerically to obtain the steady-state response and the stability of the system. The linear first-order perturbation results show new zones of instability due to the presence of internal resonance. For low amplitude of excitation and damping Hopf bifurcations are observed in the trivial steady-state response. The multi-branched non-trivial response curves show turning point, pitch-fork and Hopf bifurcations. Cascade of period and torus doubling, crises as well as the Shilnikov mechanism for chaos are observed. This is the first natural physical system exhibiting a countable infinity of horseshoes in a neighbourhood of the homoclinic orbit.     

T型槽端面密封气膜热弹流润滑动态稳定性

对T型槽端面密封气膜热弹流润滑动态稳定进行了分析. 考虑端面热变形和弹性变形以及辅助密封的阻尼特性,数值分析了不同振动频率下密封气膜动态压力分布和温度分布规律,并利用小扰动方法分析了外界扰动频率对气膜刚度、阻尼和振幅的影响规律. 结果表明:高压和高速条件下,密封端面的弹性变形和热变形产生发散间隙,导致密封气膜厚度显著降低;外界扰动产生附加压力和温度分布,刚度随扰动频率的增加而迅速增加,阻尼随扰动频率的增加而迅速下降;一定扰动频率范围内,轴向振幅与扰动频率成对数线性关系增加,辅助密封阻尼使得密封气膜的振幅显著上升.      

Nonlinear-forced vibrations of piezoelectrically actuated viscoelastic cantilevers

This paper aims to study the nonlinear-forced vibrations of a viscoelastic cantilever with a piecewise piezoelectric actuator layer on its top surface using the method of Multiple Scales. The governing equation of motion is a second-order nonlinear ordinary differential equation with quadratic and cubic nonlinearities which appear in stiffness, inertia, and damping terms. The nonlinear terms are due to the piezoelectricity, viscoelasticity, and geometry of the system. Forced vibrations of the system are investigated in the cases of primary resonance and non-resonance hard excitation including subharmonic and superharmonic resonances. Analytical expressions for frequency responses are derived, and the effects of different parameters including damping coefficient, thickness to width ratio of the beam, length and position of the piezoelectric layer, density of the beam, and the piezoelectric coefficient on the frequency-response curves are discussed for each case. It is shown that in all these cases, the response of the system follows a softening behavior due to the existence of the piezoelectric layer. The piezoelectric layer provides an effective tool for active control of vibration. In addition, the effect of the viscoelasticity of the beam on passive control of amplitude of vibration is illustrated.     

NONLINEAR VIBRATIONS OF AXIALLY ACCELERATING VISCOELASTIC TIMOSHENKO BEAMS

研究了轴向加速黏弹性Timoshenko梁的非线性参数振动。参数激励是由径向变化张力和轴向速度波动引起的。引入了取决于轴向加速度的径向变化张力,同时还考虑了有限支撑刚度对张力的影响。应用广义哈密尔顿原理建立了Timoshenko梁耦合平面运动的控制方程和相关的边界条件。黏弹性本构关系采用Kelvin模型并引入物质时间导数。耦合方程简化为具有随时间和空间变化系数的积分-偏微分型非线性方程。采用直接多尺度法分析了Timoshenko梁的组合参数共振。根据可解性条件得到了Timoshenko梁的稳态响应,并应用Routh-Hurvitz判据确定了稳态响应的稳定性。最后通过一系列数值例子描述了黏弹性系数、平均轴向速度、剪切变形系数、转动惯量系数、速度脉动幅值、有限支撑刚度参数以及非线性系数对稳态响应的影响。     

非齐次边界条件下轴向运动梁的非线性振动

轴向运动系统的横向非线性振动一直是国内外研究的热点课题之一.目前相关研究大都是针对齐次边界条件的.但是在工程实际中,非齐次边界条件更为常见,而针对非齐次边界条件的研究相对较少.为深入研究非齐次边界条件对轴向运动系统横向非线性振动的影响,本文以轴向变速运动黏弹性Euler梁为例,引入由黏弹性引起的非齐次边界条件,同时还引入由轴向加速度引起的径向变化张力,建立梁横向振动的积分-偏微分型运动方程,并导出了相应的非齐次边界条件.采用直接多尺度法分析了梁的次谐波参数共振.由可解性条件得到了梁的稳态响应,并根据Routh-Hurvitz判据确定了系统稳态响应的稳定性.通过数值例子讨论了黏弹性系数,轴向运动速度,轴向速度脉动幅值和非线性系数对幅频响应的影响,并详细对比分析了非齐次边界条件和齐次边界条件对幅频响应的影响.结果表明:随着黏弹性系数的增大,非齐次边界条件下的零解失稳区域和稳态响应幅值比齐次边界条件下的失稳区域和幅值大,非齐次边界条件对高阶次谐波参数共振的影响更加显著.最后,引入微分求积法来验证直接多尺度法的近似解结果.      

Two-dimensional rheological element in modelling of axially moving viscoelastic web

To model the axially moving viscoelastic web material a two-dimensional rheological element is used in this paper. This model is formed by elastic region and viscoelastic region. Using two-dimensional rheological model and the plate theory the differential equation of motion in the form of the eighth-order linear partial differential equation that governs the transverse vibrations of the system is derived. The Galerkin method is applied to simplify the governing equation into two-order truncated system defined by the set of ordinary differential equations. Numerical investigations of dynamic stability of the paper web were carried out. The effects of the transport speed and the internal damping on the dynamic behaviour of the axially moving web are presented in this paper.     

Nonlinear dynamics of a viscoelastic beam traveling with pulsating speed,variable axial tension under two-frequency parametric excitations and internal resonance

This paper investigates nonlinear combined parametric transverse vibrations of a traveling viscoelastic beam. The combined parametric excitations originate from the time dependency of axial velocity as well as axial tension. Two parametric excitations are enforced into the system amid the internal resonance. Two-frequency parametric resonance is assumed to be comprised of combination parametric resonance of first two modes due to the time dependency of axial velocity, and the principal parametric resonance of first mode due to the variable tension in the axial direction in the presence of internal resonance for viscoelastic beam is considered for the first time. The higher-order integro-partial differential equation of motion is solved through direct method of multiple scales. Continuation algorithm is employed to explore the stability and various bifurcations of the nonlinear dynamic system. Focus has been made to study the effect of variations of fluctuating tension component, fluctuating velocity component independently and when combined, internal and parametric frequency detuning parameters and damping on the system response. Frequency response equilibrium curves are complex and unique in shapes which are embodied with various bifurcations. Such steady-state behavior is not seen in the existent literature. With variation in fluctuating velocity component, the number of steady-state nontrivial equilibrium curves increases to three and with variation in fluctuating axial tension, they become four. In this process, significant changes in stability, number and position of various bifurcations like supercritical and subcritical pitchfork, Hopf and saddle node are observed. Unlike the previous study, the shape, stability and bifurcations of equilibrium curves under the combined effect of axial velocity and tension closely match with the case of fluctuating axial tension component. The effect of variation in internal and parametric frequency detuning parameter is more realized for second mode compared to first mode. A comparison of the present work with a previous one where axial tension is variable reveals many qualitative and quantitative similarities and dissimilarities. But when compared with earlier work where axial velocity is constant, significant dissimilarities are surfaced. The system displays a wide ranging dynamic behavior including stable periodic, quasiperiodic and unstable chaotic behavior. The numerical computation depicts various nonlinear characteristics and oscillatory behaviors which are not found so far in the existent literature.     

Nonlinear dynamics of axially moving beam with coupled longitudinal–transversal vibrations

In this study, the nonlinear vibrations of an axially moving beam are investigated by considering the coupling of the longitudinal and transversal motion. The Galerkin method is used to truncate the governing partial differential equations into a set of coupled nonlinear ordinary differential equations. By detuning the axially velocity, the exact parameters with which the system may turn to internal resonance are detected. The method of multiple scales is applied to the governing equations to study the nonlinear dynamics of the steady-state response caused by the internal–external resonance. The saturation and jump phenomena of such system have been reported by investigating the nonlinear amplitude–response curves with respect to external excitation, internal, and external detuning parameters. The longitudinal external excitation may trigger only longitudinal response when excitation amplitude is weak. However, beyond the critical excitation amplitude, the response energy will be transferred from the longitudinal motion to the transversal motion even the excitation is employed on the longitudinal direction. Such energy transfer due to saturation has the potential to be used in the vibration suppression.     

Bifurcation and chaos of an axially moving viscoelastic string

In this paper, bifurcation and chaos of an axially moving viscoelastic string are investigated. The 1-term and the 2-term Galerkin truncations are respectively employed to simplify the partial-differential equation that governs the transverse motions of the string into a set of ordinary differential equations. The bifurcation diagrams are presented in the case that the transport speed, the amplitude of the periodic perturbation, or the dynamic viscosity is respectively varied while other parameters are fixed. The dynamical behaviors are numerically identified based on the Poincare maps. Numerical simulations indicate that periodic, quasi-periodic and chaotic motions occur in the transverse vibrations of the axially moving viscoelastic string.     

Exploiting internal resonance for vibration suppression and energy harvesting from structures using an inner mounted oscillator

The flexural vibration of a symmetrically laminated composite cantilever beam carrying a sliding mass under harmonic base excitations is investigated. An internally mounted oscillator constrained to move along the beam is employed in order to fulfill a multi-task that consists of both attenuating the beam vibrations in a resonance status and harvesting this residual energy as a complementary subtask. The set of nonlinear partial differential equations of motion derived by Hamilton’s principle are reduced and semi-analytically solved by the successive application of Galerkin’s and the multiple-scales perturbation methods. It is shown that by properly tuning the natural frequencies of the system, internal resonance condition can be achieved. Stability of fixed points and bifurcation of steady-state solutions are studied for internal and external resonances status. It results that transfer of energy or modal saturation phenomenon occurs between vibrational modes of the beam and the sliding mass motion through fulfilling an internal resonance condition. This study also reveals that absorbers can be successfully implemented inside structures without affecting their functionality and encumbering additional space but can also be designed to convert transverse vibrations into internal longitudinal oscillations exploitable in a straightforward manner to produce electrical energy.     

Stability and bifurcations of an axially moving beam with an intermediate spring support

The forced non-linear vibrations of an axially moving beam fitted with an intra-span spring-support are investigated numerically in this paper. The equation of motion is obtained via Hamilton??s principle and constitutive relations. This equation is then discretized via the Galerkin method using the eigenfunctions of a hinged-hinged beam as appropriate basis functions. The resultant non-linear ordinary differential equations are then solved via either the pseudo-arclength continuation technique or direct time integration. The sub-critical response is examined when the excitation frequency is set near the first natural frequency for both the systems with and without internal resonances. Bifurcation diagrams of Poincaré maps obtained from direct time integration are presented as either the forcing amplitude or the axial speed is varied; as we shall see, a sequence of higher-order bifurcations ensues, involving periodic, quasi-periodic, periodic-doubling, and chaotic motions.