部分浸入水中的柔性梁非线性自由振动

研究了浸入水中的柔性梁非线性自由振动,假设其底端具有线弹性扭转弹簧支撑,顶端附有不计体积的集中质量块.推导了梁的运动控制方程和边界条件,由于考虑了大挠度,法向运动和轴向运动是非线性耦合的,使用Morison方程给出了流体力的表达式,利用有限差分法和Runge-Kutta法数值分析了梁在真空中和在水中的自由振动,讨论了参数对振动模态、固有频率等的影响.     

Chaotic vibrations of a beam with non-linear boundary conditions

Forced vibrations of an elastic beam with non-linear boundary conditions are shown to exhibit chaotic behavior of the strange attractor type for a sinusoidal input force. The beam is clamped at one end, and the other end is pinned for the tip displacement less than some fixed value and is free for displacements greater than this value. The stiffness of the beam has the properties of a bi-linear spring. The results may be typical of a class of mechanical oscillators with play or amplitude constraining stops. Subharmonic oscillations are found to be characteristic of these types of motions. For certain values of forcing frequency and amplitude the periodic motion becomes unstable and nonperiodic bounded vibrations result. These chaotic motions have a narrow band spectrum of frequency components near the subharmonic frequencies. Digital simulation of a single mode mathematical model of the beam using a Runge-Kutta algorithm is shown to give results qualitatively similar to experimental observations.     

Transient vibrations of a fractional Zener viscoelastic cantilever beam with a tip mass

Meccanica - The presented work concerns the kinematically excited transient vibrations of a cantilever beam with a mass element fixed to its free end. The Euler–Bernoulli beam theory and the...     

Free vibrations of circular cylindrical shells with a small added concentrated mass

The effect of a small added mass on the frequency and shape of free vibrations of a thin shell is studied using shallow shell theory. The proposed mathematical model assumes that mass asymmetry even in a linear formulation leads to coupled radial flexural vibrations. The interaction of shape-generating waves is studied using modal equations obtained by the Bubnov–Galerkin method. Splitting of the flexural frequency spectrum is found, which is caused not only by the added mass but also by the wave-formation parameters of the shell. The ranges of the relative lengths and shell thicknesses are determined in which the interaction of flexural and radial vibrations can be neglected.     

Consideration of the masses of helical springs in forced vibrations of damped combined systems

The common point about many systems modeled as Bernoulli-Euler beams with attachments is that the own masses of the helical springs are neglected. Some researchers accounted for the masses of the springs during free vibrations of those systems. Further to these studies, the present study deals with the investigation of the effect of not taking into account the masses of the helical springs in damped combined systems during their forced vibrations. It is shown that the spring mass effect may be important in regions near to the resonance frequencies. Further, this effect influences the phase angles more than the amplitudes.     

Eigenfrequencies of the three dimensional euler-bernoulli beam system with dissipative joints

In this paper, we will compute the transfer matrices to find the eigenfrequencies for the vibrations of the general non-collinear Euler-Bernoulli or Timoshenko beam structure with dissipative joints. We will allow the structure to be three dimensional, and thus we must consider all types of vibrations simultaneously, including longitudinal and torsional vibrations. The general structure considered will consist of any number of beams joined end to end to form a chain. Many different kinds of dampers are allowed, even within the same structure. We also will allow different materials within the structure as well as different beam widths. We then will show that asymptotic estimates can be used to find the eigenfrequencies approximately.     

Finite amplitude oscillations of a simple rubber support system

Exact solution of the nonlinear problem of undamped, finite amplitude, free vertical oscillations of a mass supported by a rubber spring made of a neo-Hookean material is presented for both suspension and compression supports. The motion in the special case of free fall of the mass from rest at the unstretched state is characterized in terms of elliptic integrals, and it is shown that the periodic time may be expressed universally in terms of the tabulated Heuman lambda-function. The finite amplitude, free vibrational frequency and the dynamic deflection of a neo-Hookean oscillator are compared with those for a linear spring oscillator having the same constant stiffness; and both upper and lower bounds on the ratio of these frequencies are presented. Numerical values for several cases are illustrated, and the physical results are described graphically. General solutions for the free vibrations with arbitrary initial data are obtained in terms of certain generalized lambda and beta-functions, and some transformation identities relating these functions are derived.     

Characteristics and comparative analysis of piezoelectric-electromagnetic energy harvesters from vortex-induced oscillations

Nonlinear Dynamics - The vortex-induced vibrations of a circular cylinder attached as a tip mass at the end of a cantilever beam are investigated for hybrid energy harvesting using two different...     

Dynamic and quasi-static bending of saturated poroelastic Timoshenko cantilever beam

Based on the three-dimensional Gurtin-type variational principle of the incompressible saturated porous media, a one-dimensional mathematical model for dynamics of the saturated poroelastic Timoshenko cantilever beam is established with two assumptions, i.e., the deformation satisfies the classical single phase Timoshenko beam and the movement of the pore fluid is only in the axial direction of the saturated poroelastic beam. Under some special cases, this mathematical model can be degenerated into the Euler-Bernoulli model, the Rayleigh model, and the shear model of the saturated poroelastic beam, respectively. The dynamic and quasi-static behaviors of a saturated poroelastic Timoshenko cantilever beam with an impermeable fixed end and a permeable free end subjected to a step load at its free end are analyzed by the Laplace transform. The variations of the deflections at the beam free end against time are shown in figures. The influences of the interaction coefficient between the pore fluid and the solid skeleton as well as the slenderness ratio of the beam on the dynamic/quasi-static performances of the beam are examined. It is shown that the quasi-static deflections of the saturated poroelastic beam possess a creep behavior similar to that of viscoelastic beams. In dynamic responses, with the increase of the slenderness ratio, the vibration periods and amplitudes of the deflections at the free end increase, and the time needed for deflections approaching to their stationary values also increases. Moreover, with the increase of the interaction coefficient, the vibrations of the beam deflections decay more strongly, and, eventually, the deflections of the saturated poroelastic beam converge to the static deflections of the classic single phase Timoshenko beam.     

The influences of both offset and mass moment of inertia of a tip mass on the dynamics of a centrifugally stiffened visco-elastic beam

The present study is concerned with the out-of-plane vibrations of a rotating, internally damped (Kelvin-Voigt model) Bernoulli-Euler beam carrying a tip mass. The centroid of the tip mass, possessing also a mass moment of inertia is offset from the free end of the beam and is located along its extended axis. This system can be thought of as an extremely simplified model of a helicopter rotor blade or a blade of an auto-cooling fan. The differential eigenvalue problem is solved by using Frobenius method of solution in power series. The characteristic equation is then solved numerically. The simulation results are tabulated for a variety of the nondimensional rotational speeds, tip mass, tip mass offset, mass moment of inertia and internal damping parameters. These are compared with the results of a conventional finite element modeling as well, and excellent agreement is obtained. Some numerical results are given in graphical form. The numerical results obtained, indicate clearly that the tip mass offset and mass moment of inertia are important parameters on the eigencharacteristics of rotating beams so that they have to be included in the modeling process.     

Impulsive loading of ideal fibre-reinforced rigid-plastic beams—III Cantilever beam

The theory outlined in Part I is applied to the problem of a cantilever beam struck transversely at any point by a mass which subsequently adheres to the beam. In the subsequent motion, slope and velocity discontinuities propagate outwards from the point of impact. Solutions for the velocity and deflection of the various segments of the beam are obtained for the case of linear strain-hardening, and simpler approximate solutions are derived for the case of low impact velocity and/or slight strain-hardening. The discontinuity propagating towards the free end of the beam always comes to rest before it reaches this end, but for sufficiently high values of impact mass and velocity, and a strain-hardening parameter, one or more reflections of the discontinuity may occur at the fixed end of the beam and at the point of impact.     

钝圆柱头质量对简支梁的弹性次碰撞研究

该文采用赫兹碰撞理论、三维动力有限元方法和自行研制的次碰撞实验装置,研究了钝圆柱头质量一次坠落碰撞简支钢梁过程的次碰撞现象、次碰撞过程的持续时间和次碰撞发生的条件．实验测试和数值模拟均清楚地观察到了复杂的次碰撞现象．研究结果表明：（1）当梁初始处于静止状态时,将实验测试和数值模拟结果与理论分析结果对比后发现,首个次碰撞过程由局部接触变形主导,并可以采用 Hertz弹性碰撞理论来描述;（2）实验和数值模拟结果展现了与首个次碰撞过程特征迥异的后续次碰撞过程,后续次碰撞过程的持续时间出现大范围的随机变化;（3）进一步分析发现,后续次碰撞过程由局部接触变形和梁的整体变形运动共同主导,梁的整体变形运动使得碰撞力响应变得异常复杂;（4）通过数值计算结果发现,当出现次碰撞现象时,碰撞位移响应中的一阶模态幅值占比会突然降低,相位角发生明显的变化;（5）数值计算结果表明,次碰撞发生的条件与质量比、碰撞初速度、碰撞动量、碰撞位置、测试梁长度和厚度等有关;（6）次碰撞发生的条件若用质量比来度量,则发现存在一个质量比阀值．上述研究表明,局部接触变形和梁的整体变形运动相互作用,产生了复杂的耦合效应,使得次碰撞现象呈现了丰富的力学行为．因此,进一步通过理论、实验和数值仿真研究复杂的次碰撞现象,对于深入理解柔性结构的碰撞行为很有必要．     

Lock-in of the vortex-induced vibrations of a long tensioned beam in shear flow

The occurrence of lock-in, defined as the local synchronization between the vortex shedding frequency and the cross-flow structural vibration frequency, is investigated in the case of a tensioned beam of length to diameter ratio 200, free to move in both the in-line and cross-flow directions, and immersed in a linear shear current. Direct numerical simulation is employed at three Reynolds numbers, from 110 to 1100, so as to include the transition to turbulence in the wake. The Reynolds number influences the response amplitudes, but in all cases we observed similar fluid–structure interaction mechanisms, resulting in high-wavenumber vortex-induced vibrations consisting of a mixture of standing and traveling wave patterns.Lock-in occurs in the high oncoming velocity region, over at least 30% of the cylinder length. In the case of multi-frequency response, at any given spanwise location lock-in is principally established at one of the excited vibration frequencies, usually the locally predominant one. The spanwise patterns of the force and added mass coefficients exhibit different behaviors within the lock-in versus the non-lock-in region. The spanwise zones where the flow provides energy to excite the structural vibrations are located mainly within the lock-in region, while the flow damps the structural vibrations in the non-lock-in region.     

On modelling and linear vibrations of arbitrarily sagged inclined cables in a quiescent viscous fluid

A theory of free linear vibrations of arbitrarily sagged inclined cables in a viscous fluid is presented in the framework of the heavy fluid loading concept. The static equilibrium shape of the cable is found by using the model of inextensible catenary and the validity ranges of this approximation are assessed. The dynamics of the viscous fluid is described by the linearised Navier–Stokes equations and their solution is pursued analytically by formulating the fluid field variables via potential functions. The vibration problem of a submerged cable is solved by Galerkin's method and the modal added mass and modal viscous damping coefficients are calculated. As a prerequisite for this analysis, the free vibrations of a cable in vacuum are addressed and a very good agreement with known results is observed. The physical interpretation of the dependence of modal added mass and modal damping coefficients on the ‘design variables’ for a fluid-loaded cable is given and the possible extensions of the suggested theory to capture weakly nonlinear effects are highlighted.     

Chaotic vibrations of a post-buckled L-shaped beam with an axial constraint

Analytical results are presented on chaotic vibrations of a post-buckled L-shaped beam with an axial constraint. The L-shaped beam is composed of two beams which are a horizontal beam and a vertical beam. The two beams are firmly connected with a right angle at each end. The beams joint with the right angle is attached to a linear spring. The other ends are firmly clamped for displacement. The L-shaped beam is compressed horizontally via the spring at the beams joint. The L-shaped beam deforms to a post-buckled configuration. Boundary conditions are required with geometrical continuity of displacements and dynamical equilibrium with axial force, bending moment, and share force, respectively. In the analysis, the mode shape function proposed by the senior author is introduced. The coefficients of the mode shape function are fixed to satisfy boundary conditions of displacements and linearized equilibrium conditions of force and moment. Assuming responses of the beam with the sum of the mode shape function, then applying the modified Galerkin procedure to the governing equations, a set of nonlinear ordinary differential equations is obtained in a multiple-degree-of-freedom system. Nonlinear responses of the beam are calculated under periodic lateral acceleration. Nonlinear frequency response curves are computed with the harmonic balance method in a wide range of excitation frequency. Chaotic vibrations are obtained with the numerical integration in a specific frequency region. The chaotic responses are investigated with the Fourier spectra, the Poincaré projections, the maximum Lyapunov exponents and the Lyapunov dimension. Applying the procedure of the proper orthogonal decomposition to the chaotic responses, contribution of vibration modes to the chaotic responses is confirmed. The following results have been found: The chaotic responses are generated with the ultra-subharmonic resonant response of the two-third order corresponding to the lowest mode of vibration. The Lyapunov dimension shows that three modes of vibration contribute to the chaotic vibrations predominantly. The results of proper orthogonal decomposition confirm that the three modes contribute to the chaos, which are the first, second, and third modes of vibration. Moreover, the results of the proper orthogonal decomposition are evaluated with velocity which is equivalent to kinetic energy. Higher modes of vibration show larger contribution to the chaotic responses, even though the first mode of vibration has the largest contribution ratio.     

Numerical analysis of free vibrations of a beam with oscillators

The problem of free vibrations of a beam with free ends of variable cross section and mass, from which point masses (oscillators) are suspended by bars, is considered. It is shown that parametric resonances can occur in this oscillating system. Numerical examples showing the efficiency of the calculation method proposed are given. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 135–144, July–August, 2006.     

A Motion Amplifier Using an Axially Driven Buckling Beam: II. Modeling and Analysis

In this paper, the nonlinear behavior of a motion amplifier used to obtain large rotations from small linear displacements produced by a piezoelectric stack is studied. The motion amplifier uses elastic (buckling) and dynamic instabilities of an axially driven buckling beam. Since the amplifier is driving a large rotary inertia at the pinned end and the operational frequency is low compared to the resonant frequencies of the beam, the mass of the buckling beam and the dynamics of the PZT stack are neglected and the system is modeled as a single-degree-of-freedom, nonlinear system. The beam simply behaves as a nonlinear rotational spring having a prescribed displacement on the input end and a moment produced by the inertial mass acting on the output end. The moment applied to the mass is then a function of the beam end displacement and the mass rotation. The system can, thus, be modeled simply as a base-excited, spring–mass oscillator. Results of the response for an ideal beam using this reduced-order model agree with the experimental data to a high degree. Inclusion of loading and geometric imperfections show that the response is not particularly sensitive to these imperfections. Parameter studies for the ideal buckling beam amplifier were conducted to provide guidance for improving the design of the motion amplifier and finding the optimal operating conditions for different applications. An erratum to this article can be found at     

Multi-frequency vortex-induced vibrations of a long tensioned beam in linear and exponential shear flows

The multi-frequency vortex-induced vibrations of a cylindrical tensioned beam of aspect ratio 200, free to move in the in-line and cross-flow directions within first a linearly and then an exponentially sheared current are investigated by means of direct numerical simulation, at a Reynolds number equal to 330. The shape of the inflow profile impacts the spectral content of the mixed standing-traveling wave structural responses: narrowband vibrations are excited within the lock-in area, which is limited to a single region lying in the high flow velocity zone, for the linear shear case; in contrast, the lock-in condition occurs at several spanwise locations in the exponential shear case, resulting in broadband responses, containing a wide range of excited frequencies and spatial wavenumbers. The broadband in-line and cross-flow vibrations occurring for the exponential shear current have a phase difference that lies within a specific range along the entire span; this differs from the phase drift noted for narrowband responses in linear shear flow. Lower vibration amplitudes, time-averaged and fluctuating in-line force coefficients are observed for the exponential shear current. The cross-flow force coefficient has comparable magnitude for both inflow profiles along the span, except in zones where the broadband vibrations are under the lock-in condition but not the narrowband ones. As in the narrowband case, the fluid forces associated with the broadband responses are dominated by high frequencies related to high-wavenumber vibration components. Considerable variability of the effective added mass coefficients along the span is noted in both cases.     

Predicting buckling loads from vibration data

There are analytical methods for predicting the buckling loads of columns with the boundaries ideally fixed, i.e., simply supported or built-in, or partially fixed. Vibration-test results may furnish a practical method of measuring the fixity. In this investigation a beam, that may or may not be loaded as a column, is assumed to have a torsional spring at each end such that a zero torsional stiffness corresponds to a simply supported end and an infinite torsional stiffness corresponds to a built-in end. From a Rayleigh-Ritz analysis, the buckling load and the fundamental frequency of the beam are each computed as a function of the torsional stiffness. This procedure leads to a one-to-one nondimensional relationship between the buckling load and the natural frequency. From these calculations, it is seen that regardless of the degree of clamping of one end relative to the other end, all that is needed to predict the buckling load within a 15-percent range is a knowledge of the theoretical buckling load of the simply supported column; the theoretical fundamental frequency of the simply supported beam; and the experimental fundamental frequency. Experimental results are presented to support the theory.     

EIGENFREQUENCIES OF THE THREE DIMENSIONAL EULER－BERNOULLI BEAM SYSTEM WITH DISSIPATIVE JOJNTS

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