Theoretical development and closed-form solution of nonlinear vibrations of a directly excited nanotube-reinforced composite cantilevered beam

The nonlinear equations of motion of planar bending vibration of an inextensible viscoelastic carbon nanotube (CNT)-reinforced cantilevered beam are derived. The viscoelastic model in this analysis is taken to be the Kelvin–Voigt model. The Hamilton principle is employed to derive the nonlinear equations of motion of the cantilever beam vibrations. The nonlinear part of the equations of motion consists of cubic nonlinearity in inertia, damping, and stiffness terms. In order to study the response of the system, the method of multiple scales is applied to the nonlinear equations of motion. The solution of the equations of motion is derived for the case of primary resonance, considering that the beam is vibrating due to a direct excitation. Using the properties of a CNT-reinforced composite beam prototype, the results for the vibrations of the system are theoretically and experimentally obtained and compared.     

Experimental and theoretical analysis of the transverse vibrations of a beam having bilinear support

Experimental and theoretical investigations have been conducted to study the transverse vibrations of a beam having nonlinear constraint. One end of the beam is fixed while the other is supported on a bilinear spring and carries a concentrated mass. Free-vibration curves, obtained for different values of the spring constants and the end mass, indicate that free periodic vibrations with frequencies which can lie within any one of an infinite number of ranges may occur. Forced harmonic response may exhibit the multiplicity of jump phenomena within the frequency ranges of free vibrations.     

Nonlinear motion of a beam

The investigation reported herein analyzes the vibration of a uniform beam with hinged ends which are restrained. The beam is subjected to a linearly-varying distributed load which has a maximum intensity w ₀ at the center and is released from rest when the load is suddenly removed. The motion is found to be inherently nonlinear, even for small vibrations, and there is dynamic mode-coupling. The mode frequencies are functionally related to initial conditions, particularly the amplitudes of all modes.     

Nonlinear vibration of a rotating cantilever beam in a surrounding magnetic field

The nonlinear vibrations of a rotating cantilever beam made of magnetoelastic materials surrounded by a uniform magnetic field are investigated. The kinetic energy, potential energy and work done by the electromagnetic force are obtained. A nonlinear dynamic model, based on the Hamilton principle, which includes the stretching vibration and bending vibration is presented. The Galerkin method is adopted to discretize the dynamic equations. The proposed method is validated by comparison with the literature. The nonlinear behaviors of the responses are studied. Then simulations for different kinds of magnetic field are conducted. The effects of magnetic field parameters, including the amplitude, plane angle, spatial angle and time-varying frequency, on the dynamic behaviors of the stretching motion and bending motion are investigated in detail. The results illustrate that the interaction effects between the rotating cantilever beam and the magnetic field will increase the vibration amplitude and fluctuation of the beam. In particular, we found that: collinear magnetic fields with equal amplitude lead to the same dynamic responses; the amplitude of magnetic field intensity increases the dynamic responses remarkably; the response amplitude changes nonlinearly with the plane angle and spatial angle of the magnetic field; and the increase of time-varying frequency enhances dynamic responses of the rotating cantilever beam.     

On the dynamic behaviour of a beam with an accelerating mass

Summary The dynamic behaviour of an Euler beam traversed by a moving concentrated mass, is analyzed for the general case of a mass moving with a varying speed. The equation of motion in a matrix form is formulated using the Lagrangian approach and the assumed mode method. The dimensionless form of the equation enables the numerical results to be applicable for a wide range of system parameters. The possibility of the mass separating from the beam is analyzed by examining the contact forces between the mass and the beam during the motion.     

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.     

Stability analysis for nonplanar free vibrations of a cantilever beam by using nonlinear normal modes

Stability analysis of nonplanar free vibrations of a cantilever beam is made by using the nonlinear normal mode concept. Assuming nonplanar motion of the beam, we introduce a nonlinear two-degree-of-freedom model by using Galerkin’s method based on the first mode in each direction. The system turns out to have two normal modes. Using Synge’s stability concept, we examine the stability of each mode. In order to check the validity of the stability criterion obtained analytically, we plot a Poincaré map of the motions neighboring on each mode obtained numerically. It is found that the maps agree with the stability criterion obtained analytically.     

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.     

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.     

Experimental Method to Evaluate Effective Dynamic Properties of a Meta-Structure for Flexural Vibrations

In this study, a laboratory method to evaluate the effective dynamic properties for flexural vibrations of a meta-structure is presented. The flexural vibration of a beam with periodically spaced resonators was investigated using wave propagation analysis. After analyzing vibration interactions between the resonators and the homogeneous beam, the effective dynamic properties of the meta-structure were evaluated using the transfer function method. The comparison of the measured and predicted results allowed for an understanding of the proposed vibration control mechanism. The effective bending stiffness was not affected by the attached resonators. The effective mass exhibited significant frequency-dependent variation near the natural frequency of the resonators. The effective mass becomes complex when the stiffness of the resonators is viscoelastic. The reflected wave from the metamaterial was completely blocked when the real part of the effective mass became negative. The effective mass decreased as the loss factor of the attached resonators increased. The proposed evaluation method can be used to analyze the effects of resonators on structural vibrations.     

On the limitations of linear beams for the problems of moving mass-beam interaction using a meshfree method

This paper deals with the capabilities of linear and nonlinear beam theories in predicting the dynamic response of an elastically supported thin beam traversed by a moving mass. To this end, the discrete equations of motion are developed based on Lagrange’s equations via reproducing kernel particle method (RKPM). For a particular case of a simply supported beam, Galerkin method is also employed to verify the results obtained by RKPM, and a reasonably good agreement is achieved. Variations of the maximum dynamic deflection and bending moment associated with the linear and nonlinear beam theories are investigated in terms of moving mass weight and velocity for various beam boundary conditions. It is demonstrated that for majority of the moving mass velocities, the differences between the results of linear and nonlinear analyses become remarkable as the moving mass weight increases, particularly for high levels of moving mass velocity. Except for the cantilever beam, the nonlinear beam theory predicts higher possibility of moving mass separation from the base beam compared to the linear one. Furthermore, the accuracy levels of the linear beam theory are determined for thin beams under large deflections and small rotations as a function of moving mass weight and velocity in various boundary conditions.     

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.     

Thermo-mechanical nonlinear vibration analysis of a spring-mass-beam system

Thermo-mechanical vibrations of a simply supported spring-mass-beam system are investigated analytically in this paper. Taking into account the thermal effects, the nonlinear equations of motion and internal/external boundary conditions are derived through Hamilton’s principle and constitutive relations. Under quasi-static assumptions, the equations governing the longitudinal motion are transformed into functions of transverse displacements, which results in three integro-partial differential equations with coupling terms. These are solved using the direct multiple-scale method, leading to closed-form solutions for the mode functions, nonlinear natural frequencies and frequency–response curves of the system. The influence of system parameters on the linear and nonlinear natural frequencies, mode functions, and frequency–response curves is studied through numerical parametric analysis. It is shown that the vibration characteristics depend on the mid-plane stretching, intra-span spring, point mass, and temperature change.     

The Dynamics of a Beam on an Elastic Base under a Moving Force and Moment (Timoshenko Model)

The problem on the motion of concentrated forces over a beam on an elastic base is solved on the basis of the Timoshenko beam theory. Motion modes are constructed and three critical values of the velocity are found. The solution obtained is compared with that for the Kirchhoff beam     

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...     

Transverse vibrations of a single-span beam with the motion of a bending moment

Conclusions Analysis of the dynamic action of a moving bending moment on a single-span beam-type system showed that, with v=(0.2–0.8v ₁ ⁰ , taking account of the inertial forces of the load does not enter into the margin of strength of the construction, and these forces must be taken into consideration in dynamic calculations. The greatest deflections of the beam, when the mass of the load M=0.5ml, exceed the static deformations, taking account of the inertial forces of the load, by 2.5 times. The value of the velocity here is v=0.6v ₁ ⁰ .The maximal coefficient of the dynamics, calculated without taking account of the weight of the load, is equal to 1.95 and occurs with v=0.8v ₁ ⁰ . We note that, with the motion of a vertical force along the beam, the maximal value of the dynamic coefficient is equal to 1.77 and is observed with v=0.6v ₁ ⁰ [3].If v<0.6v ₁ ⁰ , where the mass of the load is not introduced into the calculations, and v<0.4v ₁ ⁰ , where account is taken of the inertial forces of the load, then the maximal deformations of the beam take place during the process of forced vibrations at the moment that the load is located in the construction. With large values of v, the greatest deflections are observed after passage of the load, during the period of free vibrations of the system.In distinction from the solution of the problem of the vibrations of a beam under the action of a moving force (load), where a sufficient degree of exactness of the computations assures taking account of the first form of the vibrations of the construction, with an analysis of dynamic deformations of a beam, brought about by the action of a moving bending moment, the higher forms of the vibrations of the system must be taken into consideration.Leningrad Institute of Railroad Engineers. Translated from Prikladnaya Mekhanika, Vol. 14, No. 1, pp. 111–115, January, 1978.     

Analysis of nonlinear dynamic response and dynamic buckling for laminated shallow spherical thick shells with damage

Based on the nonlinear theory of shallow spherical thick shells and the damage mechanics, a set of nonlinear equations of motion for the laminated shallow spherical thick shells with damage subjected to a normal concentrated load on the top are established. According to Hertz law, the contact force acted upon the shells is determined due to the impact of a mass, and it is related to the mass and initial velocity of the striking object, the geometrical and physical character of the shell. By using the finite difference method and the time increment procedure, the nonlinear equations are resolved. In the numerical examples, the effects of the damage, the initial velocity, and mass of the striking object, the shells’ geometrical parameters on the dynamic responses and dynamic buckling of the laminated shallow spherical thick shells are discussed. Research of Y. Fu, Z. Gao and F. Zhu was supported by National Natural Science Foundation of China (No. 10572049).     

Resonant Interaction of a Beam and an Elastic Foundation During the Motion of a Periodic System of Concentrated Loads

The vibrations of a beam on an elastic foundation under the action of a periodic system of moving concentrated forces are analyzed against different spatial periods and different velocity ranges separated by three critical values. It is established that for velocities ranging from zero to the minimum critical value, resonance does not occur and the beam deflection weakly depends on the force velocity. For velocities exceeding the minimum critical value, a dense spectrum of resonant frequencies is observed__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 5, pp. 116–123, May 2005.     

Amplitude-frequency characteristics of non-linear vibrations of clamped elliptic plates carrying a concentrated mass

The influence of a concentrated mass on the amplitude-frequency characteristics of large amplitude-transverse vibrations of an isotropic clamped elliptic plate has been investigated by the application of the von Karman equations.     

On aspects of damping for a vertical beam with a tuned mass damper at the top

In this paper, the wind-induced, horizontal vibrations of a vertical Euler–Bernoulli beam will be considered. At the top of the beam, a tuned mass damper (TMD) has been installed. The horizontal vibrations can be described by an initial-boundary value problem. Perturbation methods will be applied to construct approximations of the solutions of the initial-boundary value problem, and it will be shown that the TMD uniformly damps the oscillation modes of the beam. In the analysis, it will be assumed that damping, wind-force, and gravity effects are small but not negligible.