Dynamic responses of axially moving viscoelastic beam under a randomly disordered periodic excitation

We investigate dynamic responses of axially moving viscoelastic beam subject to a randomly disordered periodic excitation. The method of multiple scales is used to derive the analytical expression of first-order uniform expansion of the solution. Based on the largest Lyapunov exponent, the almost sure stability of the trivial steady-state solution is examined. Meanwhile, we obtain the first-order and the second-order steady-state moments for the non-trivial steady-state solutions. Specially, we discuss the first mode theoretically and numerically. Results show that under the same conditions of the parameters, as the intensity of the random excitation increases, non-trivial steady-state solution fluctuation will become strenuous, which will result in the non-trivial steady-state solution lose stability and the trivial steady-state solution can be a possible. In the case of parametric principal resonance, the stochastic jump is observed for the first mode, which indicates that the stationary joint probability density concentrates at the non-trivial solution branch when the random excitation is small, but with the increase of intensity of the random excitation, the probability of the trivial steady-state solution will become larger. This phenomenon of stochastic jump can be defined as a stochastic bifurcation.     

Dynamic torsion from a linear lagrangian

Following the example of electromagnetism, a torsion potential is introduced. By considering variations with respect to the metric tensor and the torsion potential, a theory of gravitation with propagating torsion is obtained. Sources are introduced and the equation of motion is obtained.     

Vibrations of a beam with an absorber consisting of a viscoelastic beam and a spring-viscous damper

The effectiveness of the dynamic vibration absorber which consists of a double-cantilever viscoelastic beam and a spring-viscous damper is studied. The absorber is attached to the centre of the main beam. The ends of the main beam are built in and excited sinusoidally by the base motion. In the numerical example, the displacement transmissibility, i.e., the ratio of the displacement at the centre of the main beam to that of the base, is investigated. Variations of the resonant peaks are shown when the absorber parameters are changed. Values of the optimum tuning design parameters are presented, and it is verified that two of the main beam resonances are optimized simultaneously.     

Dynamic orientational phase transition in a two-layer magnetically coupled structure

The magnetization dynamics was studied for a system of two magnetic layers separated by a nonmagnetic spacer providing their antiferromagnetic coupling. A new effect of orientational phase transition occurring upon a change in the amplitude (frequency) of a microwave field was observed near the edge of the orientational hysteresis loop.     

Acoustic radiation from a viscoelastic beam impacted by a steel sphere

The acoustic radiation from a viscoelastic beam impacted by a steel sphere has been studied both theoretically and experimentally. Transverse vibrations of free-free viscoelastic beams have been analyzed by employing the modal analysis technique and an approximate method, with the Hertz theory used to evaluate impact forces. The wave equation was solved to determine the acoustic pressure radiated from impacted beams of circular and elliptical cross-sections. The theoretical predictions are compared with the experimental results for the radiation from PMMA beams of circular and rectangular cross-sections. It is shown that for beams of circular cross-sections the theoretical and experimental results are in good agreement and that for beams of rectangular cross-sections the radiation is well predicted by modeling them as beams with elliptical cross-sections.     

Dynamic magnetic susceptibility of a two-layer film in a strong magnetic field

A theoretical study is reported of the effect of interlayer exchange coupling on the resonance properties of a two-layer magnetic film with “easy-axis” and “easy-plane” anisotropic layers in a strong tilted magnetic field. The dependence of the resonance fields on the tilting angle of the external magnetic field to the film has been obtained, the tensor of integrated high-frequency film susceptibility has been found, and its dependence on the strength and orientation of the external field, as well as on layer thickness, has been analyzed. The results obtained agree with the available experimental data.     

Dispersion curves for a viscoelastic Timoshenko beam with fractional derivatives

The Kramers–Kronig dispersion relation, often used as a viscoelastic constitutive law for polymeric materials, is based on purely mathematical properties of linearity, convergence of improper integrals, and causality; thus, it may also be valid as a viscoelastic constitutive law for general structural materials. Accordingly, the motion equation of a Timoshenko beam composed of conventional elastic structural materials is extended to one composed of viscoelastic materials. From the derived governing equation, a dispersive equation is derived for a viscoelastic Timoshenko beam. By plotting phase velocity curves and group velocity curves for a beam of solid circular cross-section composed of a viscoelastic material (polyvinyl chloride foam), the influence of the fractional order of viscoelasticity is examined. As a result, it is found that, in the high frequency range, only the first mode of a Timoshenko beam converged to the propagation velocity of the Rayleigh wave, which takes account of the fractional order of viscoelasticity. In addition, the phase velocity and the group velocity were found to increase as the fractional order approaches 0, and to decrease as the fractional order approaches 1. Furthermore, the rate of velocity change becomes greater as the fractional order approaches 0, and becomes smaller as the fractional order approaches 1.     

Measurement of viscoelastic properties from the vibration of a compliantly supported beam

A laboratory method is presented by which the viscoelastic properties of compliant materials are measured over a wide frequency range. The test setup utilizes a flexible beam clamped at one end and excited by a shaker at the free end. A small specimen of a compliant material is positioned to support the beam near its midpoint. The deformation from gravity is minimized since the specimen is not loaded by an attached mass. Forced vibration responses measured at two locations along the beam are used to derive a transfer function from which the dynamic properties of compliant materials are directly determined by use of a theoretical procedure investigating the effects of specimen stiffness on the propagation of flexural waves. Sensitivity of the measured properties to experimental uncertainties is investigated. Young's moduli and associated loss factors are determined for compliant materials ranging from low-density closed-cell foams to high-density damping materials. The method is validated by comparing the measured viscoelastic properties to those from an alternative dynamic test method.     

An investigation of electron-photon fluxes after two-layer barriers irradiated by a beam of fast monoenergetic electrons

The dependences of the electron and photon fluxes on the thickness of two-layer barriers irradiated with fast electrons were compared. The results were examined as a source of information about the composition of the absorber, which led to a new method of measuring the thickness of two-layer objects and of determining definitely the layer in which a change of thickness occurs.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 131–135, February, 1978.     

An experimental-theoretical correlation study of non-linear bending and torsion deformations of a cantilever beam

An experimental study of the large deformation of a cantilevered beam under a gravity tip load has been made. The beam root is rotated so that the tip load is oriented at various angles with respect to the beam principal axes. Static twist and bending deflections of the tip and bending natural frequencies have been measured as a function of tip load magnitude and orientation. The experimental data are compared with the results of a recently developed non-linear structural theory. Agreement is reasonably good when bending deflections are small compared to the beam span, but systematic differences occur for larger deflections.     

A finite element for the study of coupled bending-prevented torsion of a straight beam

Prevented torsion has been the subject of numerous works, yet curiously in classical finite element programs there appears to be an absence of elements enabling one to take into account the stiffness in warping or the fact that shear centre and surface centre of a section do not always coincide. The object of this article is to present an element which has been experimentally tested and which could enable one to fill the present gap and thus improve accuracy in calculations of structures formed of straight cylindrical beams of any section.     

Polarizing beam splitter of two-layer dielectric rectangular transmission gratings in Littrow mounting

We theoretically investigated the design of polarizing beam splitters (PBS) of two-layer dielectric rectangular transmission gratings in Littrow mounting. The PBS grating is aimed to separate TE and TM polarized waves into two different transmitted orders, i.e. the 0th and −1st orders. The polarization-dependent diffraction process is analyzed by using a simplified modal method. The mode reflection and transmission at the interface of the two layers are illustrated. Design equations are given based on the average mode indices of the even and odd modes inside the grating region, which are verified by using rigorous coupled wave analysis. Moreover, with simulated annealing algorithm, two kinds of two-layer PBS gratings are optimized for operation over the C-band (1520-1570 nm). Compared with the single-layer transmission PBS gratings, such two-layer scheme exhibits a great flexibility of design. The PBSs of two-layer dielectric transmission gratings should be useful for practical applications.     

高斯光束在双层云中传输的蒙特卡罗模拟

基于辐射传输理论, 利用蒙特卡罗方法模拟了无限窄(冲击函数)准直光束入射到典型水云以及冰水双层云时的后向散射特性, 进而将得到的冲击响应与高斯光束卷积, 得到高斯光束在云层中传输的多次散射特性. 文中给出了两种波束入射时水云以及冰水双层云的反射函数随径向r和天顶角α的变化关系, 并给出了光强在云层内部的二维分布图. 计算结果表明, 高斯光束入射时, 云层反射函数的特点与无限窄准直光束入射时有较大区别. 因此在利用激光雷达进行云层探测时需要考虑激光的散斑, 文中的方法可以为此提供理论依据.     

Dynamic strain of a longitudinally vibrating viscoelastic rod with an end mass

The dynamic strain of a viscoelastic rod excited into longitudinal simple harmonic vibration, a constant amplitude displacement being maintained by a shaker at one end and a mass terminating the other end, is theoretically investigated in this paper. The general equation of the strain distribution in the rod is derived by solving the one-dimensional wave equation. It is shown that the strain in the rod is the largest at the excited end, at the antiresonant frequencies. Simplified approximate equations are derived for the low loss case to calculate the largest strain in the rod with knowledge of the loss factor.     

Dynamic stability of a rotating beam with a constrained damping layer

The dynamic behavior and dynamic instability of the rotating sandwich beam with a constrained damping layer subjected to axial periodic loads are studied by the finite element method. The influences of rotating speed, thickness ratio, setting angle and hub radius ratio on the resonant frequencies and modal system loss factors are presented. The regions of instability for simple and combination resonant frequencies are determined from the Mathieu equation that is obtained from the parametric excitation of the rotating sandwich beam. The regions of dynamic instability for various parameters are presented.     

Vibration of an excitation system supported flexibly on a viscoelastic sandwich beam at its mid-point

A vibration analysis of an excitation system supported flexibly on a three layer sandwich beam is presented in this paper. The flexibly supported excitation system, which is essentially the primary system, consists of a mass, a spring and a dash-pot. The beam is analyzed separately as a continuous system in a classical way and then its dynamic stiffness at the junction point is combined with that of the primary system to obtain the resultant dynamic stiffness, which in turn is used to develop the expressions for the response of the primary system and the transmissibility provided by the whole system. Both response and transmissibility are evaluated for different geometrical and physical parameters of the sandwich beam. The solution to this problem is also obtained by approximating the sandwich beam by a lumped mass supported on a spring and dash-pot. The results in the two cases are compared. Results obtained from an experimental test-rig substantiate the theoretical results.     

Dynamic analysis of a beam with an intermediate elastic support

A survey of the literature shows that the title problem has not been studied to a great extent. In the present paper an approximate solution is obtained in the case of a beam with ends elastically restrained against rotation and an intermediate elastic support. When dealing with the forced vibrations situation a sinusoidally varying excitation is assumed.     

Dynamic coefficient of a two-layered thick beam with imperfect bonding

This paper is concerned with the impulse response of a two-layered prestressed beam with flexible bonding resting on an elastic foundation. Two dissimilar layers of the beam are assumed to bend according to the Timoshenko beam theory. Numerical results are given for simply supported beams subjected to a uniformly distributed step load. The dynamic coefficient (= dynamical maximum moment/static moment in the layer) is calculated as a function of the shear bond stiffness (ranging from zero to infinity) for several values of the axial prestress and the foundation stiffness parameters. The results are also compared with those from the Euler beam theory.     

Dynamic behaviour of a beam subjected to a force of time-dependent frequency

It is a well-known fact that, for an ideal system in which damping is ignored, the deflection of a beam becomes infinite for the case where the constant frequency of a steady-state external force is equal to the critical frequency of the beam. In this paper, the external force is assumed to be proportional to $\text{sin}\text{(}\text{at}{}^2\text{2}\text{+}\text{bt}{}^3\text{3}\text{)}$ with respect to time, where a and b are constants, and the effect of time-dependent frequency on the dynamic behaviour of the beam is investigated. Integrations involved in the theoretical results are carried out by Simpson's rule. From the results of the theoretical analysis, it is evident that the maximum values of the dynamic deflection are much affected by the time-dependence of the frequency of the external force when it passes through the critical frequency, increasing or decreasing.