Dynamic analysis of an axially moving finite-length beam with intermediate spring supports

This paper presents the analysis for the transverse vibration of an axially moving finite-length beam inside which two points are supported by rotating rollers. In this study, the rollers are modeled as uniaxial springs in the transverse direction. Hamilton?s principle is applied to derive the equations of motion and boundary conditions of the system. The equations of motion include translational and rotational motions as well as flexible motion. These equations are discretized using Galerkin?s method, and then the dynamic characteristics of a flexible beam with spring supports are studied by solving an eigenvalue problem. The veering phenomenon of natural frequency loci and mode exchanges are investigated for different positions of the springs and various values of the spring stiffness. In addition, the mode localization is also analyzed using the peak amplitude ratio. It is found in this study that the first mode is localized in one of the beam spans if an appropriate value of the spring constant is selected. Furthermore, it is shown that mode localization can be used to reduce the vibration transferred from one span to the other span while a beam moves axially.     

Dynamic analysis of an elastic plate on a thin,elastic foundation

An improved theory for the motion of an elastic foundation is developed. Included is the interaction of two or more plates resting on a common foundation. A simplified model is deduced which appears useful for some applications.     

Acoustic scattering of a high-order Bessel beam by an elastic sphere

The exact analytical solution for the scattering of a generalized (or “hollow”) acoustic Bessel beam in water by an elastic sphere centered on the beam is presented. The far-field acoustic scattering field is expressed as a partial wave series involving the scattering angle relative to the beam axis and the half-conical angle of the wave vector components of the generalized Bessel beam. The sphere is assumed to have isotropic elastic material properties so that the nth partial wave amplitude for plane wave scattering is proportional to a known partial-wave coefficient. The transverse acoustic scattering field is investigated versus the dimensionless parameter ka(k is the wave vector, a radius of the sphere) as well as the polar angle θ for a specific dimensionless frequency and half-cone angle β. For higher-order generalized beams, the acoustic scattering vanishes in the backward (θ = π) and forward (θ = 0) directions along the beam axis. Moreover it is possible to suppress the excitation of certain resonances of an elastic sphere by appropriate selection of the generalized Bessel beam parameters.     

Transient response of an elastic beam with discrete impedances from decelerating boundaries

Analyzed is transient response of an elastic beam from decelerating boundaries. It is shown that passive attenuation of peak transient deceleration is possible by coupling the beam to a tuned one-degree-of-freedom (1-dof) oscillator. For a fixed oscillator mass, optimum spring stiffness yields the lowest deceleration response. Furthermore, over a finite range of spring stiffness, the second coupled system mode disappears yielding an additional sharp reduction in transient deceleration response. The disappearance of the second mode is independent of boundary conditions from the weakest simple supports to the strongest clamped ends.     

Transverse vibrations of beams with variable moment of inertia and an intermediate support

The above problem is tackled using the Ritz method and approximating the deflection function by means of polynomial co-ordinate functions which satisfy the essential boundary conditions. The presence of an axial force and concentrated masses is considered. Fundamental frequency coefficients are determined in the case of a rather complex structure taking into account different types of boundary conditions.     

Interactions of an elastic solid with a viscous fluid: Eigenmode analysis

In this paper we study the interaction of a viscous fluid with an elastic solid. Of particular interest are the eigenmodes of the coupled system. Starting from the Navier-Stokes equations for the fluid and the linear elasticity equations for the solid, we derive the linear equations governing the motion of the system. It is shown how a variational formulation of the problem may be obtained by re-scaling the displacement unknowns. The finite-element technique is then used to discretize the equations. The resulting quadratic eigenvalue problem is solved by means of an inverse iteration procedure.     

Computation and analysis of the Bessel beam scattering by a submerged elastic sphere

Taking an elastic sphere for example,the acoustic scattering of a submerged object illuminated by a Bessel beam is studied.The partial wave series representation for an elastic sphere has been extended to the case of Bessel beam scattering.Referring to the scattering of a plane wave,the peak to peak intervals in backscattering form function caused by the interference of the specular wave and the Franz wave have been analyzed in geometry.The influence of the characteristic parameterβof a Bessel beam on the peak to peak intervals has been indicated,and a predictive formula of the peak to peak intervals has been established for the first time.Meanwhile the elastic scattering of each partial wave has been separated based on the Resonance Scattering Theory.The influence ofβon the pure elastic resonance has been studied further.The results show that selecting specificβcan reduce the contribution of a certain partial wave.Therefore the resonance at the corresponding frequency and the nearby region in the backscattering is remarkably suppressed.The work of this paper could be helpful to the applications of Bessel beams on the acoustic detection of submerged objects.     

Proposal for an x-ray free electron laser oscillator with intermediate energy electron beam

Harmonic lasing of low-gain free electron laser oscillators has been experimentally demonstrated in the terahertz and infrared regions. Recently, the low-gain oscillator has been reconsidered as a promising candidate for hard x-ray free electron lasers, through the use of high reflectivity, high-resolution x-ray crystals. In this Letter, it is proposed to utilize a crystal-based cavity resonant at a higher harmonic of the undulator radiation, together with phase shifting, to enable harmonic lasing of the x-ray free electron laser oscillator, and hence allow the generation of hard x-ray radiation at a reduced electron beam energy. Results show that fully coherent free electron laser radiation with megawatt peak power, in the spectral region of 10-25 keV, can be generated with a 3.5 GeV electron beam.     

Dynamic performance of the beam position monitor support at the SSRF

Electron beam stability is very important for third‐generation light sources, especially for the Shanghai Synchrotron Radiation Facility whose ground vibrations are much larger than those for other light sources. Beam position monitors (BPMs), used to monitor the position of the electron beam, require a greater stability than other mechanical structures. This paper concentrates on an investigation of the dynamic performance of the BPM support prototype. Modal and response analyses have been carried out by finite‐element (FE) calculations and vibration measurements. Inconsistent results between calculation and measurement have motivated a change in the soft connections between the support and the ground from a ground bolt in the initial design to full grout. As a result the mechanical stability of the BPM support is greatly improved, showing an increase in the first eigenfrequency from 20.2 Hz to 50.2 Hz and a decrease in the ratio of the root‐mean‐square displacement (4–50 Hz) between the ground and the top of the support from 4.36 to 1.23 in the lateral direction. An example is given to show how FE analysis can guide the mechanical design and dynamic measurements (i.e. it is not just used as a verification method). Similar ideas can be applied to improve the stability of other mechanical structures.     

Phase shift analysis of heavy-ion elastic scattering measured at intermediate energies

An elastic scattering phase shift analysis based on McIntyre parametrization is presented. It has been applied to¹²C+¹²C system at 360- and 1,016 MeV incident energies as well as to¹⁶O scattering on¹²C,⁴⁰Ca,⁹⁰Zr and²⁰⁸Pb targets at 1,503 MeV. The success of this model allows the interplay between diffraction and refraction to be pinned down: a presence of a nuclear rainbow is clearly evidenced in several cases.     

Bifurcation analysis of an impact oscillator with a one-sided elastic constraint near grazing

In this paper a linear oscillator undergoing impact with a secondary elastic support is studied experimentally and semi-analytically for near-grazing conditions. The experimentally observed bifurcations are explained with help from simulations based on mapping solutions between locally smooth subspaces. Smooth as well as nonsmooth bifurcations are observed, and the resulting atypical bifurcations are explained, often as an interplay between them. In order to understand the observed bifurcation scenarios, a global analysis is required, due to the influence of stable and unstable orbits which are born in distant bifurcations but become important at near-grazing conditions. The good degree of correspondence between experiment and theory fully justifies the modelling approach.     

Vibration of a cantilever beam on a composite support with base excitation

An analysis of natural frequency and dynamic response of a cantilever beam subjected to base motion is presented. The analysis is extended to a case in which the cantilever is mounted on a composite support. Typical numerical results for the natural frequency and the deflection of the beam are given for both cases to illustrate the attenuation of motion by the composite mount.     

Non-linear vibrations of a simple-simple beam with a non-ideal support in between

A simply supported Euler-Bernoulli beam with an intermediate support is considered. Non-linear terms due to immovable end conditions leading to stretching of the beam are included in the equations of motion. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the intermediate support is assumed to allow small deflections. An approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique. Ideal and non-ideal frequencies as well as frequency-response curves are contrasted.