Vibrations of Piezoceramic Rods

Presently, most of the research on vibrations of monolithic piezoelectric rods at weak electric fields is restricted to longitudinal oscillations of such structural members where free and forced vibrations have been dealt with and in the case of resonance conditions not only linear but also nonlinear effects within the constitutive relations have been incorporated. On the other hand, bending and torsional vibrations of piezoceramic one‐parametric rods have not been examined yet. The present contribution develops a linear vibration theory of rods with a focus to bending vibrations taking into account rotatory inertia and shear deformation. The governing boundary value problem for beams with both longitudinal and as well transversal polarization is derived, in particular free vibrations are analyzed. Also nonlinear extensions not only of physical nature but also geometrical ones are addressed. A possible technical application is given. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Investigation of Damping of Vibrations in a System With Two‐Disc Inseparable Clutch

This paper presents the results of tests on free and forced harmonic torsional vibrations in a system with a two‐disc inseparable clutch, with structural friction taken into account. Nonlinear histeresis loop describing the frictional‐elastic properties of the system was introduced into the model. The mathematical model of the vibrating system containing two disks inseparable clutch was built. During free vibrations of the system, its damping characteristics were tested by a digital simulation method. The vibration damping decrement as a function of amplitude torsional displacement was determined. When vibrations were harmonically forced, the amplitude ‐ frequency characteristics of the system were determined numerically. The system was used as a nonlinear torsional vibration damper in a linear system with a harmonic force. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Damping of Vibrations in a SystemWith Two Beams by Structural Friction

This paper presents the results of tests on free and forced harmonic vibrations in a system with two beams with structural friction taken into account. The beams are clamped together with uniform unitary pressure. The hysteresis loop describing the frictional-elastic properties of the system has a form of a parallelogram. The autor created a mathematical model of the vibrating system with two beams. During free vibrations of the system, its damping characteristics were tested by a digital simulation method. The vibration damping decrement as a function of amplitude displacement was determined. When vibrations were harmonically forced, the amplitude - frequency characteristics of the system were determined numerically. The system was used as a nonlinear vibration damper in a linear system with a harmonic force. The equations of motion of the nonlinear two-degree of freedom system were solved by means of a digital simulation method. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Nonlinear dynamics and contact interactions of the structures composed of beam-beam and beam-closed cylindrical shell members

Nonlinear beam-beam and beam-cylindrical shell contact interactions, where a beam is subjected to harmonic uniform load, are studied. First, the nonlinear dynamics governed by four nonlinear PDEs including a switch function controlling the contact pressure between the mentioned structural members are presented. Relations between dimensional and dimensionless quantities are derived, and the original problem of infinite dimension has been reduced to that of oscillator chains via the FDM (Finite Difference Method). Time histories, FFT (Fast Fourier Transform), phase portraits, Poincaré maps, and Morlet wavelets are applied to discover novel nonlinear chaotic and synchronization phenomena of the interacting structural members. Numerous bifurcations, full-phase synchronization of the beam-shell vibrations, the evolution of energy of the vibrating members, damped vibrations of the analyzed conservative system of the beam and the shell surface deformations for various time instants, as well as the buckling of the shell induced by impacts are illustrated and discussed, among others.In addition, we have detected that in all studied cases, in spite of analyzing a large set of nonlinear ODEs approximating the behavior of interacting structural members, the scenario of transition from regular to chaotic dynamics follows the Ruelle-Takens-Newhouse scenario.     

Evolution of the resonance vibrations of an oscillator with dry friction

We study the problem of free and forced vibrations of a mechanical system modeling the functioning of seismic isolation systems of “kinematic foundation” type in the case when the sliding friction damper is not centrally located. We determine the damping characteristics in the case of free vibrations and the critical values of the damping parameter for the case of forced vibrations. For the case of resonance vibrations with one degree of freedom we establish the relation of the increase in the amplitude of the vibrations to time at subcritical values of the friction. Translated fromDinamicheskie Sistemy, Vol. 11, 1992.     

文克尔地基上阶梯式矩形薄板的振动

用奇异函数建立文克尔地基上阶梯式矩形薄板自由振动和强迫振动的微分方程并求得其通解,用Ｗ算子给出振型函数的表达式及常见支承条件下板的频率方程,用广义函数的富里叶展开,讨论板在不同载荷作用下的强迫振动响应     

One-way and two-way couplings of CFD and structural models and application to the wake-body interaction

The current study focuses on the wake-body interaction of a circular cylinder, whose transverse free vibration is modeled by a mass-spring-damper system coupled to a computational fluid dynamics (CFD) model for the flow and wake. We first simulate the free vibration of the elastically-mounted cylinder and the wake, and analyze the transverse load it exerts on the cylinder and its phase with the vibration. We vary the damping by three orders of magnitude and examine the difference in the wake-body interaction for slightly-damped and highly-damped systems. We then use the spectral properties of the free vibration and use them to construct two different types of forced vibrations: one consists only of the fundamental component of the free vibration, and the other accounts for all spectral properties of it. We compare the wake load for each type to that corresponding to the free vibration. The forced vibrations correspond to a one-way coupling and the information is communicated from the CFD model to the structural model, whereas the free vibration corresponds to a two-way coupling of the models. By comparing the spectral properties of the wake load, including the phase relation of its components with the vibration, which we obtained for the free vibration and for the equivalent forced vibration, we identify the effects of the wake feedback. The findings show that a forced vibration does not reproduce exactly the wake load at small and intermediate levels of structural damping. As the damping increases, the vibration changes from being in-phase with the wake load to being 90° out-of-phase with it, corresponding to two different wake states, and the forced vibration gives wake load that is very close to the one occurring in the case of full wake-body interaction.     

Frequency-dependent vibration analysis of symmetric cross-ply laminated plate of Levy-type by spectral element and finite strip procedures

This research describes spectral finite element formulation for vibration analysis of rectangular symmetric cross-ply laminated composite plates of Levy-type based on classical lamination plate theory (CLPT). Formulation based on SFEM includes partial differential equations of motion, spectral displacement field, dynamic shape functions, and spectral element stiffness matrix (SESM). In this paper, vibration analysis of composite plate is investigated in two sections: free vibrations and forced vibrations. In free vibrations, natural frequencies are calculated for different Young’s moduli ratios and boundary conditions. In forced vibrations, plate vibrations are investigated under high-frequency concentrated impulsive loads. The resulting responses due to spectral element formulation are compared with those of (time-domain) finite element and analytical formulations, whenever available. The results demonstrate the superiority of SFEM with respect to FEM, in reducing computational burden, simultaneously increasing numerical accuracy, specifically for excitations of high-frequency content. By reducing the time duration of impulsive loads, and consequently increasing the modal contribution of higher modes in the transient response of plate, the accuracy of FEM responses decreases substantially accompanied with a high volume of computations, while the accuracy of the SFEM response results is very high and simultaneously, with a low computational burden. Practically, SFEM follows very closely exact analytical solutions.     

The Effect of Structural Curvings on the Stress Distribution in a Rigidly Fixed Composite Plate under Forced Vibration

Based on the exact three-dimensional equations of continuum mechanics and the Akbarov-Guz' continuum theory, the problem on forced vibrations of a rectangular plate made of a composite material with a periodically curved structure is formulated. The plate is rigidly fixed along the Ox ₁ axis. Using the semi-analytic method of finite elements, a numerical procedure is elaborated for investigating this problem. The numerical results on the effect of structural curvings on the stress distribution in the plate under forced vibrations are analyzed. It is shown that the disturbances of the stress ₂₂ in a hinge-supported plate are greater than in a rigidly fixed one. Also, it is found that the structural curvings considerably affect the stress distribution in plates both under static and dynamic loading.     

Numerical-experimental investigation of resonance characteristics of a sounding board

The paper presents results of numerical and experimental investigations into the vibrations of thin-walled structures, considering such their features as the complexity of geometry, the laminated structure of walls, the anisotropy of materials, the presence of stiffeners, and the initial stresses. The object of the study is the sounding board of an acoustic guitar, the main structural material of which is a three-layer birch veneer. Based on the finite-element method, a corresponding calculation model is created, and the steady-state regimes of forced vibrations of the sounding board are investigated. A good correspondence between calculation results and experimental data is found to exist. __________ Translated from Mekhanika Kompozitnykh Materialov, Vol. 43, No. 3, pp. 399–410, May–June, 2007.     

Stationarity and stochastic differential equations

A linear differential equation of order N with stochastic process coefficients and excitation are studied. The objective of this paper is to demonstrate that, by using an expansion method, when the coefficients and excitation are strict sense stationary processes, the response is also a strict sense stationary process. Such problems occur frequently in the engineering sciences and are very important. Applications include parametric random vibrations, turbulent environment rotorcraft dynamics, and dynamics of axially loaded structural members, among others. An example application is provided.     

Triple scale analysis of periodic solutions and resonance of some asymmetric non linear vibrating systems

We consider small solutions of a vibrating mechanical system with smooth non-linearities for which we provide an approximate solution by using a triple scale analysis; a rigorous proof of convergence of the triple scale method is included; for the forced response, a stability result is needed in order to prove convergence in a neighbourhood of a primary resonance. The amplitude of the response with respect to the frequency forcing is described and it is related to the frequency of a free periodic vibration.     

Identification of the Running-State of Railway Wheelsets

High-speed trains require a reliable monitoring of the current running-state to detect exceptional running-conditions which can occur due to track irregularities, wear in the wheel-rail contact or chassis damages. A sophisticated monitoring not only classifies the running-state in terms of safe operation but also derives the causes to enable condition-based maintenance. The running-state can be characterized by the dynamic interaction in the contact patch inducing forces and torques and leading to forced and self-excited structural vibrations of the wheelset. It is very difficult to measure structural vibrations directly and therefore it is convenient to observe them by accelerations in longitudinal, transversal and vertical direction at the left and right axle-boxes of each wheelset. The transient behavior of structural vibrations can be well analyzed by timefrequency transforms. A decomposition into the principal eigenmodes is achieved by model-based preprocessing. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Utilizing CFD-Simulations to Assess Flow-Induced Vibrations of Slender Beams in Cross-Flow

Flow-induced vibrations of a slender U-beam in cross-flow are assessed, using the proprietary CFD-solver ANSYS-CFX. The U-beam has two degrees of freedom: heaving motion and rotation. The flow- and motion-governing equations are solved together, using an iterative coupling method. We find, that using this iterative coupling does not ensure a valid long-time behaviour of the solution. The results of free vibration simulations would allow investigation of the excitation mechanisms. But for determining possibility of large amplitude vibrations, methods relying on forced motion are deemed more promising. (© 2013 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Forced vibrations of an oscillator in the presence of dry friction

We study the vibrations of an oscillator with two degrees of freedom in the presence of dry friction. We compare the nature of the damping of the free oscillations in a straight line with the general case. For the forced vibrations we determine the way in which the critical values of friction at which there exist periodic motions depend on the parameters of external action in resonance mode.Translated fromDinamicheskie Sistemy, No. 6, 1987, pp. 49–54.     

Effect of memory dependent derivative on forced transverse vibrations in transversely isotropic thermoelastic cantilever nano-Beam with two temperature

The present research deals with the study of forced vibrations in transversely isotropic thermoelastic (TIT) nanoscale beam with two temperature (2T). Memory dependent derivative theory of thermoelasticity for clamped-free/cantilever nano-beam has been considered. The mathematical model is prepared for the nanoscale beam in a closed form with the application of Euler Bernoulli beam theory. Laplace transform method is employed to solve the problem. Forced vibrations due to exponential decaying time varying load acting vertically downward along the thickness direction of the nano-beam, Uniform load, Time harmonic load have been considered. Dynamic analysis for these forced vibrations and Static analysis has been carried out in this research. The dimensionless expressions for lateral deflection, thermal moment, temperature change, and axial stress are solved for these three forced vibrations. Response ratio has also been calculated. The analytical results have been numerically analysed using programming in MATLAB. The effect of kernel function has been depicted graphically on the lateral deflection, thermal moment, temperature change, axial stress and response ratio for all the three types of forced vibrations. Some particular cases have also been discussed.     

弹性扁壳由移动质量引起的强迫振动

本文用变分法讨论在弹性扁壳上有移动的质量所引起的壳体的强迫振动.文中讨论了关于强迫振动、共振条件及临界速度等一系列问题.     

Über nichtlineare Schwingungen sphärisch schwingender Gasblasen in Flüssigkeiten unter Berücksichtigung der Kompressibilität des Fluides

Zusammenfassung Es werden nichtlineare erzwungene Schwingungen einer Gasblase in einer kompressiblen Flüssigkeit untersucht. Die von Herring hergeleitete Differentialgleichung wird für die erzwungene Schwingung ergänzt und numerisch gelöst. Die Ergebnisse werden mit dem RPNNP-Modell, welches die Flüssigkeit als inkompressibel voraussetzt, verglichen. Es werden für die erzwungene Schwingung Resonanzkurven berechnet. Die Ergebnisse zeigen, daß die Kompressibilität mit berücksichtigt werden muß.

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Summary The nonlinear forced vibrations of a gas-bubble in a compressible liquid are investigated. The differential equation given by Herring for the case of the forced vibrations is completed and solved numerically. The results are compared with the known incompressible RPNNP-model. The resonance curves under forced vibrations are calculated. The results have shown that the fluid compressibility must be taken into consideration.

     

A Modal Analysis of Resonance during the Whole-Body Vibration

The mechanism of resonance of a damping system of multi‐degrees of freedom such as the human body and the dependence of resonance on system parameters, particularly on the damping level, are studied in terms of detailed mathematical solutions of both the whole‐body vibrations and the eigen modes for a simple model. It is revealed that resonance would only occur near the eigen frequencies of neutral modes for which the complex eigen frequencies of the corresponding damping modes for the given damping level of the system have not moved far from the starting point (damping‐free case) along the corresponding tracks in the plane of complex eigen frequency yet. The major resonance would occur near the eigen frequency of the neutral mode where the modulus of the characteristic function of the system has the strongest, i.e., the deepest and sharpest, local minimum. For the present model, this neutral mode is the lowest neutral mode. It is found that the resonance and eigen frequencies increase with the stiffness of muscles and decrease with the body mass, with the portion of wobbling mass in the upper body, and with the portion of upper body mass in the whole body. Both the modal analysis and the analysis of the whole‐body vibration show that the phase differences among different parts of the system are still small at the unique or the lowest resonance frequency and increase dramatically only when the frequency of the vibrating source goes beyond the resonance frequency. Thus, some effects of body vibrations, e.g., internal loads, may reach their maximum not at the resonance frequency, but at a frequency somewhat higher than the resonance frequency. This may account for the fact that the frequency ranges for abdominal pain and for lumbosacral pain caused by body vibrations are not exactly the same as the frequency range for major body resonance but shifted to somewhat higher frequency ranges. It is therefore suggested that the frequency used for strength training in terms of vibrating devices should be above 20 Hz in order to avoid not only the major resonance but also the maximal internal loads.     

On optimal design of vibrating structures

Optimal design problems of linearly elastic vibrating structural members have been formulated in two ways. One is to minimize the total mass holding the frequency fixed; the other is to maximize the fundamental frequency holding the total mass fixed. Generally, these two formulations are equivalent and lead to the same solution. It is shown in this work that the equivalence is lost when the design variable (the specific stiffness) appears linearly in Rayleigh's quotient and when there is no nonstructural mass. The maximum-frequency formulation then is a normal Lagrange problem, whereas the minimum-mass problem is abnormal. The lack of recognition of this can lead to incorrect conclusions, particularly concerning existence of solutions. It is shown that existence depends directly on the boundary conditions and, when a sandwich beam has a free end, a solution to the maximum-frequency problem does not exist.