Stability of axially accelerating viscoelastic beams: asymptotic perturbation analysis and differential quadrature validation

An asymptotic perturbation method is proposed to investigate stability of an axially accelerating viscoelastic beam. The material time derivative is used in the viscoelastic constitutive relation. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The stability condition can be determined via the asymptotic perturbation method. The differential quadrature scheme is developed to solve numerically the equation of axially accelerating viscoelastic beams with simple supports. The stability boundaries are numerically located in the summation parametric resonance and the principal parametric resonance. Numerical examples show the effects of the beam viscoelasticity and the mean axial speed. The numerical calculations validate the analytical results in the principal parametric resonance.     

Stability of axially accelerating viscoelastic beams: multi-scale analysis with numerical confirmations

Stability is investigated for an axially accelerating viscoelastic beam. The material time derivative is used in the viscoelastic constitutive relation, not simply the partial time derivative. The method of multiple scales is applied directly to the governing equation without discretization. When the axial speed is characterized as a simple harmonic variation about the constant mean speed, the instability conditions are presented for axially accelerating viscoelastic beams constrained by simple supports with rotational springs in parametric resonance. The finite difference schemes are developed to solve numerically the equation of axially accelerating viscoelastic beams with fixed supports for the instability regions in the principal parametric resonance. The numerical calculations confirm the analytical results. Numerical examples show the effects of the constraint stiffness, the mean axial speed, and the viscoelasticity.     

时变张力作用下轴向运动梁的分岔与混沌

轴向运动结构的横向参激振动一直是非线性动力学领域的研究热点之一. 目前研究较多的是轴向速度摄动的动力学模型,参数激励由速度的简谐波动产生. 但在工程应用中,存在轴向张力波动的运动结构较为广泛,而针对轴向张力摄动的模型研究较少. 本文研究了时变张力作用下轴向变速运动黏弹性梁的分岔与混沌. 考虑随着时间周期性变化的轴向张力,计入线性黏性阻尼,采用Kelvin模型的黏弹性本构关系,给出了梁横向非线性 振动的积分--偏微分控制方程. 首先应用四阶Galerkin截断方法将控制方程离散化,然后采用四阶Runge-Kutta方法计算系统的数值解,进而确定其动力学行为. 基于梁中点的横向位移和速度的数值结果,仿真了梁沿平均轴速、张力摄动幅值、张力摄动频率以及黏弹性系数变化的倍周期分岔与混 沌运动,并且通过计算系统的最大李雅普诺夫指数来识别其混沌行为. 结果表明:较小的平均轴速有助于梁的周期运动,梁在临界速度附近容易发生倍周期分岔与混沌行为. 随着张力摄动幅值的增大,梁的振动幅值的混沌区间不断增大. 较小的黏弹性系数和张力摄动频率更容易使梁发生混沌运动. 最后,给出时程图、频谱图、相图以及Poincaré 映射图来确定梁的混沌运动.      

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.     

Analysis of frequency-domain vibration response of thin plate attached with viscoelastic free layer damping

To predict the vibration response of viscoelastic composite structure, two key issues need to be conducted, one is introducing the constitutive model of viscoelastic material into the analysis model and the other is describing the real damping behavior of viscoelastic composite structure. The emphasis of this study is to obtain the effects of frequency dependence on the vibration response of viscoelastic composite structure and the method of introducing two kinds of damping (viscoelastic material damping and remaining equivalent viscous damping). Vibration response analysis in frequency domain was investigated for viscoelastically damped plate. A cantilever plate attached with the ZN_1 viscoelastic free layer damping (FLD) was chosen to demonstrate the developed method. Frequency-domain response of the composite plate were solved and the obtained results were compared with the experimental values for the purpose of assessing the rationality of the proposed method. In addition, in order to obtain the effects of viscoelastic material parameters on vibration response of viscoelastic composite structure, a detailed parametric analysis was performed. This study shows that the frequency dependent characteristic of viscoelastic material has significant influence on the vibration response in the resonant region and acceptable results can be achieved in the non-resonant region if frequency dependent parameters are substituted by average values of the viscoelastic parameters reasonably in the analysis process.     

DYNAMIC STABILITY OF A BEAM-MODEL VISCOELASTIC PIPE FOR CONVEYING PULSATIVE FLUID

The dynamic stability in transverse vibration of a viscoelastic pipe for conveying pulsative fluid is investigated for the simply-supported case.The material property of the beam- model pipe is described by the Kelvin-type viscoelastic constitutive relation.The axial fluid speed is characterized as simple harmonic variation about a constant mean speed.The method of mul- tiple scales is applied directly to the governing partial differential equation without discretization when the viscoelastic damping and the periodical excitation are considered small.The stability conditions are presented in the case of subharmonic and combination resonance.Numerical results show the effect of viscosity and mass ratio on instability regions.     

轴向变速运动黏弹性梁稳定性渐近分析和数值验证

研究了轴向变速运动黏弹性梁参数振动的稳定性.对黏弹性本构关系采用物质时间导数,轴向速度用关于恒定平均速度的简单谐波变化来描述.发展浙近摄动法确定稳定性条件.应用微分求积法数值求解简支边界条件下的轴向变速运动黏弹性梁方程,并进而确定次谐波参数共振的稳定性边界.数值结果显示了梁的黏性阻尼和轴向平均速度的影响并验证了次谐波共振的解析结果.     

连续梁在周期荷载下的动力稳定性简化分析方法

对于一般任意支撑的连续梁结构动力稳定性问题，已有的计算方法求解过程都很复杂，给工程设计带来极大的不便．本文提出了一个简化的分析方法，利用现有的商业软件，只需求得连续梁的自然频率及静力屈曲（失稳）荷载，就可容易得到结构的动力失稳区域，当考虑结构阻尼对不稳定区域的影响时，可将阻尼矩阵表达为Rayleigh阻尼的形式．研究结果表明：采用本文计算方法与已有的理论计算方法得到的连续梁主参数共振的不稳定边界非常吻合，而本文计算方法更为简单，计算结果可靠，计算精度高，可满足工程设计的需要．     

Dynamic stability of axially accelerating Timoshenko beam: Averaging method

This study investigates dynamic stability in transverse parametric vibrations of an axially accelerating tensioned beam of Timoshenko model on simple supports. The axial speed is assumed as a harmonic fluctuation about the constant mean speed. The Galerkin method is applied to discretize the governing equation into a finite set of ordinary differential equations. The method of averaging is applied to analyze the instability phenomena caused by subharmonic and combination resonance. Numerical examples demonstrate the effects of the mean axial speed, bending stiffness, rotary inertia and shear modulus on the instability boundaries.     

Stability in parametric resonance of an axially moving beam constituted by fractional order material

The stability of an axially moving beam constituted by fractional order material under parametric resonances is investigated. The governing equation is derived from Newton??s second law and the fractional derivative Kelvin constitutive relationship. The time-dependent axial speed is assumed to vary harmonically about a constant mean velocity. The resulting principal parametric resonances and summation resonances are investigated by the multi-scale method. It is found that instabilities occur when the frequency of axial speed fluctuations is close to two times the natural frequency of the beam or when the frequency is close to the sum of any two natural frequencies. Moreover, Numerical results show that the larger fractional order and the viscoelastic coefficient lead to the larger instability threshold of speed fluctuation for a given detuning parameter. The regular axially moving beam displays a higher stability than the beam constituted by fractional order material.     

黏弹性阻尼作用下轴向运动Timoshenko梁振动特性的研究

黏弹性阻尼一直是轴向运动系统的研究热点之一.以往研究轴向运动系统大都没有考虑黏弹性阻尼的影响.但在工程实际中, 存在黏弹性阻尼的轴向运动体系更为普遍.本文研究了黏弹性阻尼作用下轴向运动Timoshenko梁的振动特性.首先, 采用广义Hamilton原理给出了轴向运动黏弹性Timoshenko梁的动力学方程组和相应的简支边界条件.其次, 应用直接多尺度法得到了轴速和相关参数的对应关系, 给出了前两阶固有频率和衰减系数在黏弹性作用下的近似解析解.最后, 采用微分求积法分析了在有无黏弹性作用下前两阶固有频率和衰减系数随轴速的变化; 给出了前两阶固有频率和衰减系数在黏弹性作用下的近似数值解, 验证了近似解析解的有效性.结果表明: 随着轴速的增大, 梁的固有频率逐渐减小.梁的固有频率和衰减系数随着黏弹性系数的增大而逐渐减小, 其中衰减系数与黏弹性系数成正比关系, 黏弹性系数对第一阶衰减系数和固有频率的影响很小, 对第二阶衰减系数和固有频率的影响较大.      

粘弹性夹芯层合旋转圆板的气动弹性动力稳定性分析

基于哈密尔顿原理,考虑作用于圆板上的空气动力载荷以及粘弹性复模量本构模型,本文建立了含有粘弹性夹芯层的旋转圆板的空气动力学模型,并给出了相应的稳定性的控制方程及边界条件。运用Galerkin方法,文中数值求解并给出了夹芯层合旋转圆板的动力学基本特征,以及结构几何参数、材料参数对于其前、后行波振动频率、阻尼和动力稳定性的影响。结果表明：通过优化合理选取粘弹性夹芯层合旋转圆板的几何参数和材料参数,可提高旋转圆板的临界转速和颤振速度,从而增强其稳定性。     

Parametric instability of cylindrical thin shell with periodic rotating speeds

Parametric instability of a cylindrical thin shell with periodically time-varying rotating speeds is studied in the paper. Energy formulation based upon Love's thin shell theory and the assumed mode method is utilized to obtain the governing equations of a rotating cylindrical shell under simply supported condition. Considering the time-varying rotating speeds, the second order differential equations of the system have time-periodic gyroscopic and stiffness coefficients. The multiple scales method is utilized to obtain the boundaries of both primary and combination instabilities analytically. The primary instability occurs when the excitation frequency is near twice of the natural frequency. The excitation frequency close to the sum of two natural frequencies might lead to the occurrence of combination instability. Numerical simulations are conducted to verify the analytical results. It is shown that the primary instability regions for each mode always appear in the periodically rotating cylindrical shell. Their widths increase continually with excitation amplitude of the time-periodic rotating speed. For certain modes, the combination instability region might not exist. The conditions for its existence are obtained analytically and verified by numerical simulations. Increasing the constant rotating speed would greatly enhance the instability regions. Moreover, it might also cause the appearance of combination instability region.     

Nonlinear dynamics of a viscoelastic sandwich beam with parametric excitations and internal resonance

Nonlinear dynamical behaviors of an axially accelerating viscoelastic sandwich beam subjected to three-to-one internal resonance and parametric excitations resulting from simultaneous velocity and tension fluctuations are investigated. The direct method of multiple scales is adopted to obtain a set of first-order ordinary differential equations and associated boundary conditions. The frequency and amplitude response curves along with their stability and bifurcation are numerically studied. A great number of dynamic behaviors are presented in the form of phase portraits, time traces, Poincaré sections, and FFT power spectra. Due to modal interaction, various periodic, quasiperiodic, and chaotic behaviors are displayed, depending on the initial conditions. The largest Lyapunov exponent is carried out to determine the midly chaotic response by the convergent form of exponents. Numerical results show various oscillatory behaviors indicating the influence of internal resonance and coupled effects of fluctuating axial velocity and tension.     

Nonlinear transverse vibration of axially accelerating strings with exact internal resonances and longitudinally varying tensions

This work explores the steady-state periodic transverse responses with their stabilities of axially accelerating viscoelastic strings. Longitudinally varying tension due to the axial acceleration is recognized in the modeling, while the tension was approximatively assumed to be longitudinally uniform in previous investigations. Exact internal resonances are highlighted in the analysis, while the resonances have been neglected in all available works. A governing equation of transverse nonlinear vibration is derived from the generalized Hamilton principle and the Kelvin viscoelastic model on the assumption that the string deformation is not infinitesimal, but still small. The axial speed is supposed to be a small simple harmonic fluctuation about the constant mean axial speed. The method of multiple scales is applied to solve the governing equation in the parametric resonances when the axial speed fluctuation frequency approaches the first three natural frequencies of the linear generating system based on 1–3 term truncations. The amplitude, the existence conditions, and the stability are determined, and the effects of the viscosity, the mean axial speed, the axial speed fluctuation amplitude, and the axial support rigidity on the amplitude and the existence are examined via the numerical examples. It is found that the 1-term, the 2-term, and the 3-term truncations yield the qualitatively same and the quantitatively close results in the case that there exist the exact internal resonances among the first three frequencies.     

Effect of rotary inertia on stability of axially accelerating viscoelastic Rayleigh beams

The dynamic stability of axially moving viscoelastic Rayleigh beams is presented. The governing equation and simple support boundary condition are derived with the extended Hamilton’s principle. The viscoelastic material of the beams is described as the Kelvin constitutive relationship involving the total time derivative. The axial tension is considered to vary longitudinally. The natural frequencies and solvability condition are obtained in the multi-scale process. It is of interest to investigate the summation parametric resonance and principal parametric resonance by using the Routh-Hurwitz criterion to obtain the stability condition. Numerical examples show the effects of viscosity coefficients, mean speed, beam stiffness, and rotary inertia factor on the summation parametric resonance and principle parametric resonance. The differential quadrature method (DQM) is used to validate the value of the stability boundary in the principle parametric resonance for the first two modes.     

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。     

Flutter analysis of a viscoelastic tapered wing under bending–torsion loading

The dynamic stability of a tapered viscoelastic wing subjected to unsteady aerodynamic forces is investigated. The wing is considered as a cantilever tapered Euler–Bernoulli beam. The beam is made of a linear viscoelastic material where Kelvin–Voigt model is assumed to represent the viscoelastic behavior of the material. The governing equations of motion are derived through the extended Hamilton’s principle. The resulting partial differential equations are solved via Galerkin’s method along with the classical flutter investigation approach. The developed model is validated against the well-known Goland wing and HALE wing and good agreement is obtained. Different solution methods, namely; the k method, the p-k method, and the flutter determinant method are compared for the case of elastic wing. However, when the viscoelastic damping is introduced, the k and p-k methods become less effective. The flutter determinant method is modified and employed to carry out non-dimensional parametric study on the Goland wing. The study includes the effects of parameters such as the taper ratio, the density ratio, the viscoelastic damping of wing structure and many other parameters on the flutter speed and flutter frequency. The study reveals that a tapered wing would be more dynamically stable than a uniform wing. It is also observed that the viscoelastic damping provides wider stability region for the wing. The investigation shows that the density ratio, bending-to-torsion frequency ratio, and the radius of gyration have significant effects on the dynamic stability of the wing. Based on the obtained results, a wing with an elastic center and inertial center that are located closer to the mid-chord would be more dynamically stable.     

An experimental study of the parametric excitation of a tensioned sheet with a cracklike opening

The problem of the parametric excitation of a thins tensioned sheet with a cracklike opening is discussed Data obtained from an experimental investigation are presented and they indicated that both principal and secondary regions of instability are developed. Plts of the stability boundaries are presented in terms of excitation frequency, mean tensile load and alternating load. The principal region is observed to be significantly larger than the secondary region and the amplitudes of the oscillations associated with the principal region are also much larger than those of the secondary region. Oscillation amplitudes of the order of twelve times the thickness are reported and amplitude vs. excitation-frequency data are shown to exhibit an overhang behavior in the direciton of increasing frequency. This indicates the presence of a nonlinear stiff effect which is attributed to middle-surface stretching due to bending. Although damping and membrane effects were found to prevent the development of unbounded oscillations, it is noted that the large deflections associated with the principal region of instability could be expected to have a deterious effect on both crack nucleation and crack propagation.     

Nonlinear Dynamics of a Flexible Spinning Disc Coupled to a Precompressed Spring

This paper analyses the nonlinear transverse vibrations of a rotating, clamped-free, flexible disc coupled to a precompressed spring. This is representative of a large class of loadings in rotating disc systems such as air jet and electromagnetic excitation commonly used in experiments. Such a loading induces a simultaneous critical speed resonance and parametric instability. The disc is modelled as a Von Kármán plate, and the equations of motion are discretised by a Galerkin projection onto a pair of 1:1 internally resonant modes. The large amplitude wave motions and their stabilities are studied using the averaging method and via numerical continuation techniques. The analysis is carried out in a co-rotating as well as a ground-fixed frame. Numerical simulations are used to verify the above analyses. The response predicted by these analyses is substantially different from that arising from a critical speed resonance or of a parametric instability alone. As many as five equilibrium solutions can coexist at supercritical speed. Two distinct regimes of large amplitude response appear to exist depending on the relationship between the strength of the parametric excitation and the damping. The existence of these regimes underscores the subtle competition between critical speed resonance and parametric instability that is likely to be observed in experiments near critical speed in such systems.Contributed by Prof. A.K. Bajaj.