Random vibration of a point-driven two-span beam on an elastic foundation

In this study, we investigate the random vibrations of a point-driven two-span beam of length 2L on a linear elastic foundation. The normal mode approach is utilized. The natural frequencies are falling in two groups: the first group corresponds to the beam of length L that is simply supported at its ends; the second group is associated with the beam of length L that is simply supported at one end and clamped at the other end. The mean-square velocity of the beam is written in terms of auto- and cross-correlations within groups, as well as in terms of cross-correlations between the modes of two groups. In addition, the two-span beam experiences the clustering of frequencies with increased modal density. The effect of cross-correlations between the modal responses corresponding to closely spaced natural frequencies turns out to be extremely significant.     

On the three-dimensional vibrations of elastic prisms with skew cross-section

Three-dimensional (3-D) free vibration of an elastic prism with skew cross-section is investigated using an elasticity-based variational Ritz procedure. Specifically, the associated energy functional minimized in the Ritz procedure is formulated using a simple coordinate mapping to transform the solid skew elastic prism into a unit cube computational domain. The displacements of the prism in each direction are approximately expressed in the form of variable separation. As an enhancement to conventional use of algebraic polynomials trial series in related solid body vibration studies in the associated literature, the assumed skew prism displacement, u, v and w in the computational ξ–η–ζ skew coordinate directions, respectively, are approximated by a set of generalized coefficients multiplied by a finite triplicate Chebyshev polynomial series and boundary functions in ξ–η–ζ to ensure the satisfaction of the geometric boundary conditions of the prism. Upon invoking the stationary condition of the Lagrangian energy functional for the skew elastic prism with respect to the assumed generalized coefficients, the usual characteristic frequency equations of natural vibrations of the skew elastic prism are derived. Upper bound convergence of the first eight non-dimensional frequencies accurate to four significant figures is achieved by using up to 10–15 terms of the assumed skew prism displacement functions. First known 3-D vibration characteristics of skew elastic prisms are examined showing the effects of varying prism length ratios (ranging from skew solids to skew slender beams), as well as, varying cross-sectional side ratios and skewness, which collectively can serve as benchmark studies against which vibration modes predicted by classical Euler and shear deformable skew beam theories as well as alternative methodologies used in elastic prism vibrations of mechanical and structural components.     

On a vibrating plate with a concentrated mass

We consider the vibrations of an elastic plate that contains a small region whose size depends on a small parameter ε. The density is of order O(ε^−m) in the small region, the concentrated mass, and it is of order O(1) outside; m is a positive parameter. Depending on the value of m (m<2, m=2 and m>2) we describe the asymptotic behaviour, as ϵ→0, of the eigenvalues and eigenfunctions of the corresponding spectral problem. For m>2 the vibrations associated with the low frequencies affect asymptotically only a neighbourhood of the concentrated mass; we also consider the asymptotic behaviour of the eigenfunctions associated with the high frequencies.     

Transverse Vibrations of Laminated Beams in Three-Dimensional Formulation

A new asymptotic method is proposed to describe the free and forced transverse vibrations of elastic laminated beams of arbitrary cross section using the three-dimensional elastic equations without additional hypotheses and constraints. For beams with layers of equal Poisson's ratio, the zero-order natural frequencies are equal to the natural frequencies predicted by classical beam theory based on the Bernoulli hypothesis. The method makes it possible to calculate the frequencies of free vibrations and amplitudes of forced vibrations with prescribed accuracy for the first natural modes __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 6, pp. 56–71, June 2005.     

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation

Parametric vibrations and stability of an axially accelerating string guided by a non-linear elastic foundation are studied analytically. The axial speed, as the source of parametric vibrations, is assumed to involve a mean speed, along with small harmonic variations. The method of multiple scales is applied to the governing non-linear equation of motion and then the natural frequencies and mode shape equations of the system are derived using the equation of order one, and satisfying the compatibility conditions. Using the equation of order epsilon, the solvability conditions are obtained for three distinct cases of axial acceleration frequency. For all cases, the stability areas of system are constructed analytically. Finally, some numerical simulations are presented to highlight the effects of system parameters on vibration, natural frequencies, frequency-response curves, stability, and bifurcation points of the system.     

ENERGY TRAPPING OF THICKNESS-SHEAR AND THICKNESS-TWIST MODES IN A PARTIALLY ELECTRODED AT-CUT QUARTZ RESONATOR

The thickness-shear and thickness-twist vibrations of a finite and partially electroded AT-cut quartz resonator are investigated. The equations of anisotropic elasticity are used with the omission of the small elastic constant c 56 . An analytical solution is obtained using Fourier series from which the free vibration resonant frequencies, mode shapes, and energy trapping are calculated and examined.     

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.     

考虑刚体运动与弹性运动耦合影响的旋转叶片振动有限元分析

本文研究了旋转叶片的纵向振动和双向横振动,考虑了刚体运动和弹性振动的耦合关系,利用有限元法推导出离散系统动力学方程,从而引出陀螺特征值问题。本文就某一特例了计算了在不同转速时叶片振动的自然频率,讨论了转速对振动频率的影响。     

Anomalous dependence of the vibration frequencies of a rod in an elastic medium on the rod length

We use numerical-analytic methods to study the influence of the length of a thin inhomogeneous rod on its natural frequencies and the shapes of its plane transverse vibrations. We found that the existence of an external elastic medium described by the Winkler model can lead to an anomalous effect, i.e., to an increase in the natural frequencies of the vibration lower modes as the rod length increases continuously. We discovered rather subtle properties of this phenomenon in the case of variations in the length, the mode number, and the fixation method. We separately studied vibrations for the standard boundary conditions: fixation, hinged fixation, tangential fixation, and free end. We calculated several simple examples illustrating the anomalous dependence of the frequency of the rod natural vibrations in a strongly inhomogeneous elastic medium with different boundary conditions.     

粘弹性地基上弹性梁的自由振动分析

本文将文克尔弹性地基梁模型中的弹簧用粘弹性元件来替代,建立了三元件文克尔粘弹性地基止粘弹性梁的静力和动力本构方程,求出了粘弹性地基上弹性梁的自由振动的级数解。并且对不同的振动情况进行讨论,最后给出了算例及结论。     

Dissipative dynamics of geometrically nonlinear Bernoulli-Euler beams

We consider nonlinear dissipative vibrations of the Bernoulli-Euler beam. We find that, under the action of a transverse alternating load, the vibrations may become chaotic. We study a scenario in which harmonic vibrations become chaotic, namely, the Feigenbaum scenario, and find the Feigenbaumconstant. In the present paper, we paymuch attention to the reliability of the results obtained. To this end, we use two methods, the finite difference O(h ²) method and the finite element method in the Bubnov-Galerkin form, and verify the convergence of these methods.     

Coupled bending and torsional vibration of nonsymmetrical axially loaded thin-walled Bernoulli–Euler beams

The dynamic transfer matrix is formulated for a straight uniform and axially loaded thin-walled Bernoulli–Euler beam element whose elastic and inertia axes are not coincident by directly solving the governing differential equations of motion of the beam element. Bernoulli–Euler beam theory is used, and the cross section of the beam does not have any symmetrical axes. The bending vibrations in two perpendicular directions are coupled with torsional vibration and the effect of warping stiffness is included. The dynamic transfer matrix method is used for calculation of exact natural frequencies and mode shapes of the nonsymmetrical thin-walled beams. Numerical results are given for a specific example of thin-walled beam under a variety of end conditions, and exact numerical solutions are tabulated for natural frequencies and solutions calculated by the other method are also tabulated for comparison. The effects of axial force and warping stiffness are also discussed.     

Das Verhalten eines elastischen Trägers mit nichtlinearen Randbedingungen bei Durchgang durch die Resonanz

Übersicht Es wird der Resonanzdurchgang der Schwingungen eines elastischen Trägers untersucht. Der Träger besitzt zwei elastische Auflager mit nichtlinearer Kennlinie und steht unter der Einwirkung einer äußeren periodischen Kraft mit konstanter Amplitude und veränderlicher Frequenz. Die Ergebnisse lassen zugleich auch das Verhalten dieses Systems im stationären Zustand erkennen.

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Summary The vibrations of an elastic beam passing trough the resonance frequency are investigated. The beam ist subjected to periodic forces with constant amplitude and varying frequency. The two bearings at the ends of the beam are assumed to be elastic with a nonlinear displacement characteristic. Some of the results indicate the properties of the system in the case of stationary vibrations.

     

ROBUST CONTROL OF VORTEX-INDUCED VIBRATION OF A RIGID CYLINDER SUPPORTED BY AN ELASTIC BEAM USING μ -SYNTHESIS

For lightweight and flexible structures, it is important to suppress the vibrations induced by interactions between fluid and structures. This paper presents the robust control of the vortex-induced vibration of a rigid circular cylinder supported by an elastic cantilever beam in which the fluid force is considered as an external excitation on the structure. For the problems considered here, the excitation frequency is assumed to be equal to the natural frequency of the structure or the “lock-in” frequency. The natural frequencies of this analytical model are calculated by using the modal analysis method and then modal coordinates are introduced to obtain the state equations of the structural system. A pair of piezoelectric devices fixed under the base plate, on which the elastic beam is clamped, were used as actuators. A robust controller satisfying the nominal performance and robust performance is designed using μ -synthesis theory based on the structured singular value. Simulation and experiment were carried out with the designed controller and the effectiveness of the robust control strategy was verified by both experimental and simulation results.     

“Non-linear normal modes” and the generalized Ritz method in the problems of vibrations of non-linear elastic continuous systems

In the study of natural vibrations of non-linear elastic systems it is shown that the mode shape of the vibration can vary with the amplitude as well as the frequency, and that the amplitude frequency relation is strongly affected by constraints imposed on the mode shape in an approximate solution. A method is developed which assumes the approximate solution in the form of a truncated series in which, instead of the set of coefficients, the set of functions of spatial variables is unknown and then determined by a procedure that can be regarded as a generalization of the Ritz method. The problem of variations of the normal mode shapes and of the associated natural frequencies with the amplitude is illustrated by two examples of beams with non-linear boundary conditions, and the amplitude-frequency relation is compared to that corresponding to the a priori assumed linear normal mode solution. Further possible consequences of the mode shape amplitude variations in forced, resonant motion of nonconservative systems are also indicated.     

On the three-dimensional vibrations of a hollow elastic torus of annular cross-section

This paper offers a three-dimensional elasticity-based variational Ritz procedure to examine the natural vibrations of an elastic hollow torus with annular cross-section. The associated energy functional minimized in the Ritz procedure is formulated using toroidal coordinates (r,q, j)({r,\theta , \varphi}) comprised of the usual polar coordinates (r, θ) originating at each circular cross-sectional center and a circumferential coordinate j{\varphi} around the torus originating at the torus center. As an enhancement to conventional use of algebraic–trigonometric polynomials trial series in related solid body vibration studies in the associated literature, the assumed torus displacement, u, v and w in the r, θ and j{\varphi } toroidal directions, respectively, are approximated in this work as a triplicate product of Chebyshev polynomials in r and the periodic trigonometric functions in the θ and j{\varphi} directions along with a set of generalized coefficients. Upon invoking the stationary condition of the Lagrangian energy functional for the elastic torus with respected to these generalized coefficients, the usual characteristic frequency equations of natural vibrations of the elastic torus are derived. Upper bound convergence of the first seven non-dimensional frequency parameters accurate to at least five significant figures is achieved by using only ten terms of the trial torus displacement functions. Non-dimensional frequencies of elastic hollow tori are examined showing the effects of varying torus radius ratio and cross-sectional radius ratio.     

Reducing cross-flow vibrations of underflow gates: Experiments and numerical studies

An experimental study is combined with numerical modelling to investigate new ways to reduce cross-flow vibrations of hydraulic gates with underflow. A rectangular gate section placed in a flume was given freedom to vibrate in the vertical direction. Horizontal slots in the gate bottom enabled leakage flow through the gate to enter the area directly under the gate which is known to play a key role in most excitation mechanisms.For submerged discharge conditions with small gate openings the vertical dynamic support force was measured in the reduced velocity range 1.5<V_r<10.5 for a gate with and without ventilation slots. The leakage flow significantly reduced vibrations. This attenuation was most profound in the high stiffness region at 2<V_r<3.5.Two-dimensional numerical simulations were performed with the Finite Element Method to assess local velocities and pressures for both gate types. A moving mesh covering both solid and fluid domain allowed free gate movement and two-way fluid–structure interactions. Modelling assumptions and observed numerical effects are discussed and quantified. The simulated added mass in still water is shown to be close to experimental values. The spring stiffness and mass factor were varied to achieve similar response frequencies at the same dry natural frequencies as in the experiment. Although it was not possible to reproduce the vibrations dominated by impinging leading edge vortices (ILEV) at relatively low V_r, the simulations at high V_r showed strong vibrations with movement-induced excitation (MIE). For the latter case, the simulated response reduction of the ventilated gate agrees with the experimental results. The numerical modelling results suggest that the leakage flow diminishes pressure fluctuations close to the trailing edge associated with entrainment from the wake into the recirculation zone directly under the gate that most likely cause the growing oscillations of the ordinary rectangular gate.     

The dimensional reduction modelling approach for 3D beams: Differential equations and finite-element solutions based on Hellinger–Reissner principle

This paper illustrates an application of the so-called dimensional reduction modelling approach to obtain a mixed, 3D, linear, elastic beam-model.We start from the 3D linear elastic problem, formulated through the Hellinger–Reissner functional, then we introduce a cross-section piecewise-polynomial approximation, and finally we integrate within the cross section, obtaining a beam model that satisfies the cross-section equilibrium and could be applied to inhomogeneous bodies with also a non trivial geometries (such as L-shape cross section). Moreover the beam model can predict the local effects of both boundary displacement constraints and non homogeneous or concentrated boundary load distributions, usually not accurately captured by most of the popular beam models.We modify the beam-model formulation in order to satisfy the axial compatibility (and without violating equilibrium within the cross section), then we introduce axis piecewise-polynomial approximation, and finally we integrate along the beam axis, obtaining a beam finite element. Also the beam finite elements have the capability to describe local effects of constraints and loads. Moreover, the proposed beam finite element describes the stress distribution inside the cross section with high accuracy.In addition to the simplicity of the derivation procedure and the very satisfying numerical performances, both the beam model and the beam finite element can be refined arbitrarily, allowing to adapt the model accuracy to specific needs of practitioners.     

Optimum design of band-gap beam structures

The design of band-gap structures receives increasing attention for many applications in mitigation of undesirable vibration and noise emission levels. A band-gap structure usually consists of a periodic distribution of elastic materials or segments, where the propagation of waves is impeded or significantly suppressed for a range of external excitation frequencies. Maximization of the band-gap is therefore an obvious objective for optimum design. This problem is sometimes formulated by optimizing a parameterized design model which assumes multiple periodicity in the design. However, it is shown in the present paper that such an a priori assumption is not necessary since, in general, just the maximization of the gap between two consecutive natural frequencies leads to significant design periodicity.The aim of this paper is to maximize frequency gaps by shape optimization of transversely vibrating Bernoulli–Euler beams subjected to free, standing wave vibration or forced, time-harmonic wave propagation, and to study the associated creation of periodicity of the optimized beam designs. The beams are assumed to have variable cross-sectional area, given total volume and length, and to be made of a single, linearly elastic material without damping. Numerical results are presented for different combinations of classical boundary conditions, prescribed orders of the upper and lower natural frequencies of maximized natural frequency gaps, and a given minimum constraint value for the beam cross-sectional area.To study the band-gap for travelling waves, a repeated inner segment of the optimized beams is analyzed using Floquet theory and the waveguide finite element (WFE) method. Finally, the frequency response is computed for the optimized beams when these are subjected to an external time-harmonic loading with different excitation frequencies, in order to investigate the attenuation levels in prescribed frequency band-gaps. The results demonstrate that there is almost perfect correlation between the band-gap size/location of the emerging band structure and the size/location of the corresponding natural frequency gap in the finite structure.     

Effect of magnetic field and non-homogeneity on the radial vibrations in hollow rotating elastic cylinder

In this paper, the natural frequencies of the radial vibrations of a hollow cylinder with different boundary conditions under influences of magnetic field, rotation and non-homogeneity have been studied. The solution of the problem is obtained by using technique of variables separation. In the present paper three different boundary conditions are considered, namely the free, fixed and mixed boundary conditions. The displacement and stresses components have been obtained in analytical form involving Bessel function of first and second kind and of order n. The determination is concerned with the eigenvalues of the natural frequencies of the radial vibrations for different boundary conditions in the case of harmonic vibrations. Numerical results are given and illustrated graphically for each case considered. Comparisons are made with the results in the absence of magnetic field, rotation and non-homogeneity. The results indicate that the effect of magnetic field, rotation and non-homogeneity are very pronounced.