Stability of a deploying/extruding beam in dense fluid

The equations of motion of a flexible slender cantilevered beam with uniform circular cross-section, extending axially in a horizontal plane at a known rate while immersed in an incompressible fluid are derived. An “axial added mass coefficient” is implemented in these equations in order to better approximate the mass of fluid which stays attached to the oscillating beam while moving in the axial direction. Realistic initial conditions are given to the system and numerical solutions are obtained. The dynamical behaviour of the system is observed for cases of constant extension rate and for a trapezoidal deployment rate profile.In the case of low constant extension rates, the system displays a phase of oscillation with increasing amplitude and decreasing frequency until the motion is strongly damped and later becomes statically unstable. For faster deployment rates, the beam has a short flutter phase at the beginning of the deployment, followed by a brief phase of damped oscillation until it exhibits static divergence. For fast enough deployment rates, the system is unstable from the beginning and never stabilizes. The effect the axial added mass coefficient has on the system is studied and it is found that it plays two roles in the stability of the system. The trapezoidal deployment rate profile is studied because it is deemed more representative of real-life applications. For long deployment times, the system behaves in a very similar manner to one with low constant extension rate, except that it does not become statically unstable. For shorter deployment times, the maximum amplitude of the tip displacement is usually attained after the beam has stopped extruding.     

Theoretical and experimental study on the transverse vibration properties of an axially moving nested cantilever beam

An axially moving nested cantilever beam is a type of time-varying nonlinear system that can be regarded as a cantilever stepped beam. The transverse vibration equation for the axially moving nested cantilever beam with a tip mass is derived by D’Alembert?s principle, and the modified Galerkin?s method is used to solve the partial differential equation. The theoretical model is modified by adjusting the theoretical beam length with the measured results of its first-order vibration frequencies under various beam lengths. It is determined that the length correction value of the second segment of the nested beam increases as the structural length increases, but the corresponding increase in the amplitude becomes smaller. The first-order decay coefficients are identified by the logarithmic decrement method, and the decay coefficient of the beam decreases with an increase in the cantilever length. The calculated responses of the modified model agree well with the experimental results, which verifies the correctness of the proposed calculation model and indicates the effectiveness of the methods of length correction and damping determination. Further studies on non-damping free vibration properties of the axially moving nested cantilever beam during extension and retraction are investigated in the present paper. Furthermore, the extension movement of the beam leads the vibration displacement to increase gradually, and the instantaneous vibration frequency and the vibration speed decrease constantly. Moreover, as the total mechanical energy becomes smaller, the extension movement of the nested beam remains stable. The characteristics for the retraction movement of the beam are the reverse.     

Nonlinear dynamics of higher-dimensional system for an axially accelerating viscoelastic beam with in-plane and out-of-plane vibrations

In this paper, the bifurcations and chaotic motions of higher-dimensional nonlinear systems are investigated for the nonplanar nonlinear vibrations of an axially accelerating moving viscoelastic beam. The Kelvin viscoelastic model is chosen to describe the viscoelastic property of the beam material. Firstly, the nonlinear governing equations of nonplanar motion for an axially accelerating moving viscoelastic beam are established by using the generalized Hamilton’s principle for the first time. Then, based on the Galerkin’s discretization, the governing equations of nonplanar motion are simplified to a six-degree-of-freedom nonlinear system and a three-degree-of-freedom nonlinear system with parametric excitation, respectively. At last, numerical simulations, including the Poincare map, phase portrait and Lyapunov exponents are used to analyze the complex nonlinear dynamic behaviors of the axially accelerating moving viscoelastic beam. The bifurcation diagrams for the in-plane and out-of-plane displacements via the mean axial velocity, the amplitude of velocity fluctuation and the frequency of velocity fluctuation are respectively presented when other parameters are fixed. The Lyapunov exponents are calculated to identify the existence of the chaotic motions. From the numerical results, it is indicated that the periodic, quasi-periodic and chaotic motions occur for the nonplanar nonlinear vibrations of the axially accelerating moving viscoelastic beam. Observing the in-plane nonlinear vibrations of the axially accelerating moving viscoelastic beam from the numerical results, it is found that the nonlinear responses of the six-degree-of-freedom nonlinear system are much different from that of the three-degree-of-freedom nonlinear system when all parameters are same.     

Nonlinear dynamic behaviors of a deploying-and-retreating wing with varying velocity

In this paper, the nonlinear dynamical behaviors of deploying-and-retreating wings in supersonic airflow are investigated. A cantilever laminated composite beam, which is axially moving at a known rate, is implemented to model the deploying-and-retreating wing. Associated with Reddy's third-order theory and von Karman type equations of large deformation, the nonlinear governing equations of motion of the deploying-and-retreating wing are derived based on the Hamilton's principle. The nonlinear partial differential equations of motion are transformed into a set of the ordinary differential equations using Galerkin's method. The nonlinear dynamical behaviors of the deployable-and-retreating wing are investigated in the cases of three different axially moving rates during deploying process and retreating process using the numerical simulations.     

Analysis of the coupled thermoelastic vibration for axially moving beam

The coupled thermoelstic vibration characteristics of the axially moving beam are investigated. The differential equation of motion of the axially moving beam under the thermoelastic coupling is established based to the equilibrium equation and the thermal conduction equation involving deformation term. The eigenequation is deduced and the dimensionless complex frequencies of the axially moving beam with different boundary conditions under the coupled thermoelastic case are calculated by the differential quadrature method. The curves of the real parts and imaginary parts of the first three-order dimensionless complex frequencies versus the dimensionless axially moving speed are obtained. The effects of the dimensionless coupled thermoelastic factor, the ratio of length to height, the dimensionless moving speed on the stability of the beam are analyzed.     

The effects of rotation,tip mass,and hub radius on the stability of beck's column

The stability of a cantilever beam subjected to a follower force at its free end and rotating at a uniform angular velocity is investigated. The beam is assumed to be offset from the axis of rotation, carries a tip mass at its free end, and undergoes deflection in a direction perpendicular to the plane of rotation. The equations of motion are formulated within the Euler-Bernoulli and Timoshenko beam theories for the case of a Kelvin model viscoelastic beam. The associated adjoint boundary value problems are derived and appropriate adjoint variational principles are introduced. These variational principles are used for the purpose of determining approximately the values of the critical flutter load of the system as it depends upon its damping parameters, tip mass and its rotary inertia, hub radius, and speed of rotation. The variation of the critical flutter load with these parameters is revealed in a series of several graphs. The numerical results show that the critical load can be reduced significantly due to (a) the transverse and rotary inertia of the tip mass and (b) increasing values of the internal damping parameter associated with the transverse shear deformation of the rotating beam.     

A general solution procedure for the forced vibrations of a system with cubic nonlinearities: Three-to-one internal resonances with external excitation

A general vibrational model of a continuous system with arbitrary linear and cubic operators is considered. Approximate analytical solutions are found using the method of multiple scales. The primary resonances of the external excitation and three-to-one internal resonances between two arbitrary natural frequencies are treated. The amplitude and phase modulation equations are derived. The steady-state solutions and their stability are discussed. The solution algorithm is applied to two specific problems: (1) axially moving Euler-Bernoulli beam, and (2) axially moving viscoelastic beam.     

旋转内接悬臂梁的刚柔耦合动力学特性分析

对固结于旋转刚环上内接柔性梁的刚柔耦合动力学特性进行了研究. 在精确描述柔性梁非线性变形基础上, 利用Hamilton变分原理和假设模态法, 在计入柔性梁由于横向变形而引起的轴向变形二阶耦合量的条件下, 推导出一次近似耦合模型. 忽略柔性梁纵向变形的影响,给出一次近似简化模型,引入无量纲变量, 对简化模型做无量纲化处理. 首先分析在非惯性系下内接悬臂梁的动力学响应, 并与外接悬臂梁进行比较; 其次研究内接悬臂梁的稳定性;最后分析内接悬臂梁失稳临界转速的收敛性. 研究发现, 与外接悬臂梁存在动力刚化效应不同,内接悬臂梁存在着动力柔化效应; 给出了内接悬臂梁无条件稳定的临界径长比以及失稳的临界转速的计算方法; 若第一阶固有频率随转速增大而减小,则该内接悬臂梁处于有条件稳定; 随着模态截断数的增加,内接悬臂梁失稳的临界转速减小且有收敛值. 关键词： 内接悬臂梁 一次近似简化模型 动力柔化 临界转速     

A conserved quantity and the stability of axially moving nonlinear beams

Free nonlinear transverse vibration is investigated for an axially moving beam modeled by an integro-partial-differential equation. Based on the equation, a conserved quantity is defined and confirmed for axially moving beams with pinned or clamped ends. The conserved quantity is applied to demonstrate the Lyapunov stability of the straight equilibrium configuration in transverse nonlinear of beam with a low axial speed.     

Dynamic stability in parametric resonance of axially accelerating viscoelastic Timoshenko beams

This paper investigates dynamic stability of an axially accelerating viscoelastic beam undergoing parametric resonance. The effects of shear deformation and rotary inertia are taken into account by the Timoshenko thick beam theory. The beam material obeys the Kelvin model in which the material time derivative is used. The axial speed is characterized as a simple harmonic variation about the constant mean speed. The governing partial-differential equations are derived from Newton's second law, Euler's angular momentum principle, and the constitutive relation. The method of multiple scales is applied to the equations to establish the solvability conditions in summation and principal parametric resonances. The sufficient and necessary condition of the stability is derived from the Routh-Hurvitz criterion. Some numerical examples are presented to demonstrate the effects of related parameters on the stability boundaries.     

Longitudinal,transverse, and torsional vibrations and stabilities of axially moving single-walled carbon nanotubes

Thanks to the brilliant mechanical properties of single-walled carbon nanotubes (SWCNTs), they are suggested as high speed nanoscale vehicles. To date, various aspects of vibrations of SWCNTs have been addressed; however, vibrations and instabilities of moving SWCNTs have not been thoroughly assessed. Herein, vibrational properties of an axially moving SWCNT with simply supported ends are studied using nonlocal Rayleigh beam theory. Employing assumed mode and Galerkin methods, the discrete governing equations pertinent to longitudinal, transverse, and torsional motions of the moving SWCNT are obtained. The resulting eigenvalue equations are then numerically solved. The speeds corresponding to the initiation of the instability within the moving nanostructure are calculated. The roles of the speed of the moving SWCNT, small-scale parameter, and aspect ratio on the characteristics of longitudinal, transverse, and torsional vibrations of axially moving SWCNTs are scrutinized. The obtained results show that the appearance of the small-scale parameter would result in the occurrence of both divergence and flutter instabilities at lower levels of the speed.     

Stabilization of an axially moving web via regulation of axial velocity

In this paper, a novel control algorithm for suppression of the transverse vibration of an axially moving web system is presented. The principle of the proposed control algorithm is the regulation of the axial transport velocity of an axially moving beam so as to track a profile according to which the vibration energy decays most quickly. The optimal control problem that generates the proposed profile of the axial transport velocity is solved by the conjugate gradient method. The Galerkin method is applied in order to reduce the partial differential equation describing the dynamics of the axially moving beam into a set of ordinary differential equations (ODEs). For control design purposes, these ODEs are rewritten into state-space equations. The vibration energy of the axially moving beam is represented by the quadratic form of the state variables. In the optimal control problem, the cost function modified from the vibration energy function is subjected to the constraints on the state variables, and the axial transport velocity is considered as a control input. Numerical simulations are performed to confirm the effectiveness of the proposed control algorithm.     

Stability and bifurcation of an axially moving beam tuned to three-to-one internal resonances

This study analyzed the nonlinear vibration of an axially moving beam subject to periodic lateral force excitations. Attention is paid to the fundamental and subharmonic resonances, since the excitation frequency is close to the first two natural frequencies of the system. The incremental harmonic balance (IHB) method was used to evaluate the nonlinear dynamic behaviour of the axially moving beam. The stability and bifurcations of the periodic solutions for given parameters were determined by the multivariable Floquet theory using Hsu’s method. The solutions obtained from the IHB method agreed very well with those obtained from numerical integration. Furthermore, numerical examples are given to illustrate the effects of the three-to-one internal resonance on the response of the system.     

轴向加速运动黏弹性梁受迫振动中的混沌动力学

研究了黏弹性轴向运动梁在外部激励和参数激励共同作用下横向振动的混沌非线性动力学行为. 引入有限支撑刚度, 并考虑黏弹性本构关系取物质导数, 同时计入由梁轴向加速度引起的沿径向变化的轴力, 建立轴向运动黏弹性梁横向非线性振动的偏微分-积分模型. 通过Galerkin截断方法研究了外部激励的频率和因速度简谐脉动引起的参数激励的频率在不可通约关系时轴向运动连续体的非线性动力学行为, 并对不同截断阶数的数值预测进行了对比. 基于对控制方程的Galerkin截断, 得到离散化的常微分方程组, 使用四阶Runge-Kutta方法求解. 基于此数值解, 运用非线性动力学时间序列分析方法, 通过Poincaré 映射, 观察到轴向运动梁随扰动速度幅值的倍周期分岔现象, 并比较了有无外部激励对倍周期分岔的影响. 分别在低速以及近临界高速运动状态下, 从相平面图、Poincaré 映射以及频谱分析的角度识别了系统中存在的准周期运动形态. 关键词： 轴向运动梁 非线性 混沌 分岔     

Instability of vibrations of a mass that moves uniformly along a beam on a periodically inhomogeneous foundation

The stability of vibrations of a mass that moves uniformly along an Euler-Bernoulli beam on a periodically inhomogeneous continuous foundation is studied. The inhomogeneity of the foundation is caused by a slight periodical variation of the foundation stiffness. The moving mass and the beam are assumed to be always in contact. With the help of a perturbation analysis it is shown analytically that vibrations of the system may become unstable. The physical phenomenon that lies behind this instability is parametric resonance that occurs because of the periodic (in time) variation of the foundation stiffness under the moving mass. The first instability zone is found in the system parameters within the first approximation of the perturbation theory. The location of the zone is strongly dependent on the spatial period of the inhomogeneity and on the weight of the moving mass. The larger this period is and/or the smaller the mass, the higher the velocity is at which the instability occurs.     

Parametric instability of a cantilever beam subjected to two electromagnetic excitations: Experiments and analytical validation

An electromagnetic device, acting like a spring with alternating stiffness, has been designed to parametrically excite the cantilever beam. However, only one parametric excitation (induced by one electromagnetic device) was considered in current research, and the effects of the design parameters of the device upon the instability were studied inadequately. Actually, multiple parametric excitations with various phases and amplitudes would bring significant impacts to the system instability. The electromagnetic device with various design parameters could cause the unstable regions to change evidently. Thus, the parametric instability of a cantilever beam subjected to two electromagnetic excitations is studied experimentally and analytically in the paper. The governing equations for the beam system are established utilizing the assumed mode method, and then verified through a DC current test. Based upon these, the instability experiments for the cantilever beam with one or two electromagnetic excitations are conducted in detail. Two design parameters of the device (magnet spacing and device location) are investigated, respectively, for their effects upon the instability regions. When two electromagnetic devices operate together to bring two parametric stiffness excitations with various phases and amplitudes to the cantilever beam, the variations of both simple and combination instability regions with coil current are observed and discussed. The above experimental results are all found to agree well with the analytical ones.     

On utilizing delayed feedback for active-multimode vibration control of cantilever beams

We present a single-input single-output multimode delayed-feedback control methodology to mitigate the free vibrations of a flexible cantilever beam. For the purpose of controller design and stability analysis, we consider a reduced-order model consisting of the first n vibration modes. The temporal variation of these modes is represented by a set of nonlinearly coupled ordinary-differential equations that capture the evolving dynamics of the beam. Considering a linearized version of these equations, we derive a set of analytical conditions that are solved numerically to assess the stability of the closed-loop system. To verify these conditions, we characterize the stability boundaries using the first two vibration modes and compare them to damping contours obtained by long-time integration of the full nonlinear equations of motion. Simulations show excellent agreement between both approaches. We analyze the effect of the size and location of the piezoelectric patch and the location of the sensor on the stability of the response. We show that the stability boundaries are highly dependent on these parameters. Finally, we implement the controller on a cantilever beam for different controller gain-delay combinations and assess the performance using time histories of the beam response. Numerical simulations clearly demonstrate the controller ability to mitigate vibrations emanating from multiple modes simultaneously.     

On the dynamics of interaction between a moving mass and an infinite one-dimensional elastic structure at the stability limit

The paper herein deals with the study of the dynamic behaviour generated by the instability of the vibration of a loaded mass, uniformly moving along an Euler-Bernoulli beam on a viscoelastic foundation, induced by the anomalous Doppler waves excited in the beam. This issue is relevant for the case of modern trains travelling along a track with soft soil when the trains speed exceeds the phase velocity of the waves induced in the track. The model corresponds to a railway vehicle reduced to a loaded wheel running along a (half) track. The beam takes account of the bending stiffness of the rail and the mass of the track, including the mass of the rail, semi-sleepers and half of the ballast layer, where the viscoelastic foundation represents the subgrade. The model includes the wheel/rail Hertzian contact and it allows the simulation of the possibility of contact loss. The nonlinear equations of motion are integrated using a numerical approach based on the Green’s function method. When the vibration becomes unstable, the system evolution is a limit cycle characterised by a succession of shocks, due to the action of two opposite factors: the anomalous Doppler waves that pump energy at the interface between the moving mass and the beam, thus forcing the mass to take off, and the static load that push the mass downwards. The frequency of the shocks increases at higher velocity and the magnitude of the impact force decreases; the most dangerous velocity is the critical one, which represents the stability limit of the linear approximation of the motion equations. The transient behaviour that precedes the limit cycle appearance is being analysed. The Hertzian contact influences the time history of the limit cycle and the magnitude of the impact force and, therefore, it is essential to be included in the model. To the authors’ knowledge, this problem has never been dealt with.     

THE DYNAMIC ANALYSIS OF A BEAM-MASS SYSTEM DUE TO THE OCCURRENCE OF TWO-COMPONENT PARAMETRIC RESONANCE

The objective of this paper is an analytical and numerical study of the dynamics of a beam--mass system. Special attention is given to the phenomena arising due to the motion of the attached mass and modal interactions produced by the existence of multi-component, specifically two-component, parametric resonance under primary resonance. The two-component parametric resonance occurs when the sums or the differences among internal frequencies are the same, or close, as the dimensionless speed parameter of the moving mass. The effects of two-component parametric resonance post on dynamic condition are investigated. Resonance generated by more than two-component modes are neglected due to its remote probability of occurrence in nature.The mechanics of the problem is Newtonian. Based on the assumption that when the moving mass is set in motion the mass is assumed to be rolling on the beam, the mechanics, including the effects due to friction and convective accelerations, of the interface between the moving mass and the beam are determined.Based on the Bernoulli-Euler beam theory, the coupled non-linear equations of motion of an inextensible beam with an attached moving mass are derived. By employing Galerkin procedure, the partial differential equations which describe the motion of a beam-mass system are reduced to an initial-value problem with finite dimensions. The method of multiple time scales is applied to consider the solutions and the occurrence of internal resonance of the resulting multi-degree-of-freedom beam--mass system with time dependent coefficients.     

Dynamic contact analysis of a tensioned beam with a moving mass–spring system

The dynamic contact problem of a tensioned beam with clamped-pinned ends is analyzed when the beam contacts a moving mass–spring system. The contact and contact loss conditions are expressed in terms of constraint equations after considering the dynamic contact between the beam and the moving mass. Using these constraints and equations of motion for the beam and moving mass, dynamic contact equations are derived and then discretized using the finite element method, which is based on the Lagrange multiplier method. The time responses for the contact forces are computed from these discretized equations. The contact force variations and contact loss are investigated for the variations of the moving mass velocity, the beam tension, the moving mass, and the stiffness of the moving mass–spring system. In addition, the possibility of contact loss and safe contact conditions between the moving mass and the tensioned beam are also studied.