Transversal vibrations of the axially moving sandwich belts

Inspired by literature on free transversal vibrations of one axially moving belt, we derive and solve analytically coupled partial differential equations of the transversal vibrations of an axially moving sandwich double-belt system. A numerical experiment and visualization are carried out.     

On the application of the constant deflection-contour method in nonlinear vibrations of elastic plates

Summary  This paper concerns the application of the constant deflection-contour method to problems involving nonlinear vibrations. Two specific problems are considered: a clamped circular plate and an annular plate with free inner boundary. For the linear case, the results obtained offer excellent agreement with previous studies, indicating significant potential for the utilization of this method in different nonlinear cases. The analysis may be applied to other types of geometrical structures. Notwithstanding the fact that only a first-term approximation has been made for the deflection function, in conjunction with the Galerkin procedure, excellent agreement has been found. Additional analytical calculations could be made to improve accuracy, indicating that the method could prove particularly useful when employed with a symbolic manipulation package. Received 13 June 2001; accepted for publication 6 November 2001     

Transversal forced vibrations of an axially moving sandwich belt system

Based on the author’s previously published results for transversal free vibrations of axially moving sandwich belts described by coupled partial differential equations, which are derived and analytically solved, this paper contains new analytical results, for forced vibrations of the same system excited by transversal external excitation. The transversal forced vibrations of the axially moving sandwich belts are described by the coupled partial nonhomogeneous differential equations. The partial differential equations are analytically solved. Bernoulli’s method of particular integrals and Lagrange’s method of the variations of the constants are used.     

Instability of vibrations of a moving two-mass oscillator on a flexibly supported Timoshenko beam

Summary  Transversal vibrations of a uniformly moving two-mass oscillator on a Timoshenko beam of infinite length supported by a viscoelastic foundation are studied. By using integral transforms, the characteristic equation for the oscillator's vibrations is obtained. It is shown that the equation may have a root with a positive real part. The existence of such a root leads to the exponential increase of the amplitude of the oscillator vibrations, i.e. to instability. The reasons for the instability to occur are discussed. By employing the method of D-decomposition, the instability domains are found in the space of the system parameters. Received 30 October 2000; accepted for publication 28 March 2001     

Purely electrical damping of vibrations in arbitrary PEM plates: a mixed non-conforming FEM-Runge-Kutta time evolution analysis

Summary A new numerical code, based upon a mixed FEM-Runge-Kutta method, is used for the analysis and design of plane 2-D smart structures. The code is applied to the study of arbitrarily shaped PEM plates, based on a weak formulation of their governing equations, [17]. The optimal parameters needed to synthesize appropriate electric networks are computed, and the overall performances of such plates are investigated. Two examples are studied: firstly, a simple case is used to test the main features of the code; secondly, a more complex PEM plate is designed and analyzed by means of the proposed numerical approach. The ideas developed in this paper were discussed with Prof. Vincenzo Ciampi, Stefano Vidoli, PhD, Corrado Maurini, MS, and Maurizio Porfiri, MS. Their friendly help and constructive criticism were really useful.     

The analysis of arbitrarily-damped linear mechanical systems

Summary The analysis of arbitrarily-damped linear mechanical systems is the subject of this paper, the main issue being the analysis of nondecouplable systems. It is well known that decouplable systems occur when the damping matrix happens to be a linear combination of the mass and stiffness matrices. Systems with this type of damping are said to have proportional damping, which nevertheless seldom occurs in practice, but many a damped system is analyzed under the assumption that it is proportionally damped. In fact, this property allows the analyst to study these systems using the same approach as that applicable to their undamped counterparts. In this paper, we show that proportional damping need not be assumed in order to analyze the system at hand with the same approach as used to analyze undamped systems. Moreover, we propose an algorithm to determine the natural frequencies, the damped frequencies and the damping ratios of an n-degree-of-freedom damped system, that does not require the casting of the system into first-order form. In this way, the characteristic equation is derived naturally as a 2n-degree polynomial, computing its roots being then straightforward. Furthermore, we propose a semigraphical method to ease this calculation, which should be attractive to practicing engineers. Received 5 April 1999; accepted for publication 7 May 1999     

The effect of friction reduction in the presence of in-plane vibrations

A reduction of friction by vibrations has been observed in various experiments. This effect can be applied to actively control frictional forces by modulating vibrations. Moreover, common methods of controlling friction rely on lubricants and suitable material combinations. The superimposition of vibrations can further reduce the friction force. This study presents a theoretical approach based on the Dahl friction model that describes the friction reduction observed in the presence of the tangential vibrations at an arbitrary angle. Analysis results indicated that the tangential compliance should be considered in modeling the effect of vibrations in reducing friction. At any vibration angle, the tangential compliance of the contacts reduces the friction reduction effect. The vibrations parallel to the macroscopic velocity are most effective for friction reduction.     

Vibration of a continuous beam with multiple elastic supports excited by a moving two-axle system with separation

This paper studies the dynamics of a two-axle system travelling along a continuous elastic beam resting on elastic supports modelled as linear springs. During its travel along the vibrating beam, the vibrating two-axle system may separate from the beam for certain time intervals when there is no interaction between the two-axle system and the beam. As a result, the dynamics of the whole system can be very different from the case when separation is ignored. The study is focused on conditions that facilitate the onset of separation between the moving and supporting structures and subsequent possible reattachment. Numerical results show that the occurrence of separation may be influenced to a large extent by the presence of intermediate elastic supports.     

Free vibration analysis of a plate on foundation with completely free boundary by finite integral transform method

The theoretical solutions of eigenfrequencies and vibration modes of a rectangular thin plate on an elastic foundation with completely free boundary are derived by using a double finite cosine integral transform method. In the analysis procedure, the elastic foundation is regarded as a Winkler elastic foundation model. Because the basic dynamic elasticity equations of the thin plate on elastic foundation are only used, it is not needed to select the deformation function arbitrarily. Therefore, the solution developed in the present paper is more reasonable and more accurate. To prove the correctness of the solutions, numerical results obtained using the present solutions are compared with those in the literatures.     

Effect of small-amplitude vibrations on viscoelastic flows in a plane slider bearing

The qualitative changes of dynamic lift and friction forces caused by small-amplitude harmonic vibrations superimposed on flows in a plane slider bearing are considered for simple viscous and viscoelastic lubricating fluids. Low- and high-frequency disturbances are analysed in greater detail and the most beneficial situations discussed.     

A new class of flexure-free torsional vibrations of annular rods

It is shown that the system of partial differential equations governing small amplitude vibrations of an elastic ring has solutions describing motions in which the axial curve C of the ring remains rigid, but executes a rocking motion while the cross-sections undergo torsional rotations about C that vary periodically both in time and in distance along C. This type of flexure-free torsional vibration can occur both in rings that are stress-free in a circular equilibrium configuration and in rings formed by bringing together and sealing, with or without the addition of twist, the ends of rods that are stress-free when straight.

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Sommario Si mostra che le equazioni che governano le oscillazioni di piccola ampiezza di travi elastiche anulari hanno soluzioni che corrispondono a moti nei quali l'asse C della trave è sottoposto a un'oscillazione rigida e le sezioni trasversali subiscono una rotazione torsionale che varia periodicamente nel tempo e con l'ascissa lungo C. Questo tipo di vibrazione torsionale, non accompagnata da flessione dell'asse, si presenta sia quando la configurazione anulare è priva di sforzi sia quando tale configurazione può pensarsi ottenuta inflettendo una trave che in uno stato privo di sforzi è ad asse rettilineo, imprimendo ad essa una eventuale deformazione torsionale, e collegando fra loro le sue estremità.

     

On the flexural vibrations of sandwich plates

A simplified approach to sandwich plate free vibrations, based on the superposition method, can provide approximations to the lowest modes for various boundary conditions, showing a characteristic influence of the bending-shear stiffness ratio.

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Sommario Un'analisi semplificata delle vibrazioni naturali di pannelli sandwich in diverse condizioni di vincolo, con il metodo di sovrapposizione degli effetti, consente l'individuazione approssimata dei modi più bassi, mostrando un comportamento vibratorio caratteristico al variare del rapporto tra le rigidezze di flessione e taglio.

     

Nonlinear damped vibrations of simply-supported rectangular sandwich plates

The nonlinear, forced, damped vibrations of simply-supported rectangular sandwich plates with a viscoelastic core are studied. The general, nonlinear dynamic equations of asymmetrical sandwich plates are derived using the virtual work principle. Damping is taken into account by modelling the viscoelastic core as a Voigt-Kelvin solid. The harmonic balance method is employed for solving the equations of motion. The influence of the thickness of the layers and material properties on the nonlinear response of the plates is studied.     

Application of conservation laws to Dirichlet problem for elliptic quasilinear systems

In the present work we show the possibility of using of conservation laws to solve the Dirichlet problem for elliptic quasilinear systems. As a result the integral representation of solution is obtained. For the system of filtration of aerated fluid in porous medium and for system of elastic–plastic torsion of prismatic rods corresponding conservation laws are calculated in explicit form and the Dirichlet problems are solved.     

Time domain simulation of vortex-induced vibrations in stationary and oscillating flows

This paper focuses on the further development of a previously published semi-empirical method for time domain simulation of vortex-induced vibrations (VIV). A new hydrodynamic damping formulation is given, and the necessary coefficients are found from experimental data. It is shown that the new model predicts the observed hydrodynamic damping in still water and for cross-flow oscillations in stationary incoming flow with high accuracy. Next, the excitation force model, which is one component of the total hydrodynamic force model, is optimized by simulating the VIV response of an elastic cylinder in a series of experiments with stationary flow. The optimization is performed by repeating the simulations until the best possible agreement with the experiments is found. The optimized model is then applied to simulate the cross-flow VIV of an elastic cylinder in oscillating flow, without introducing any changes to the hydrodynamic force modeling. By comparison with experiment, it is shown that the model predicts the frequency content, mode and amplitude of vibration with a high level of realism, and the amplitude modulations occurring at high Keulegan–Carpenter numbers are well captured. The model is also utilized to investigate the effect of increasing the maximum reduced velocity and the mass ratio of the elastic cylinder in oscillating flow. Simulations show that complex response patterns with multiple modes and frequencies appear when the maximum reduced velocity is increased. If, however, the mass ratio is increased by a factor of 5, a single mode dominates. This illustrates that, in oscillating flows, the mass ratio is important in determining the mode participation at high maximum reduced velocities.     

Chaotic vibrations of beams on nonlinear elastic foundations subjected to reciprocating loads

Chaotic vibration of beams resting on a foundation with nonlinear stiffness is investigated in this paper. Cosine–cosine function is employed in modeling of the reciprocating load. The equation of motion is derived and solved to obtain corresponding Poincaré section in phase–space. Lyapunov exponent as a criterion for chaos indication is obtained. Dynamic behavior of the beam is examined in resonance condition. Homoclinic orbits are captured and their corresponding Melnikov's functions are established. A parametric study is then carried out and effects of linear and nonlinear parameters on the chaotic behavior of the system are studied.     

Vibration analysis of cracked plates using the extended finite element method

In the present paper, the extended finite element method (X-FEM) is adopted to analyze vibrations of cracked plates. Mindlin’s plate theory taking into account the effects of shear deformation and rotatory inertia is included in the development of the model. First, conventional FEM without any discontinuity is carried out, then the enrichment proposed by Moës et al. (Int J Numer Methods Eng 46, 131–150, 1999) of nodal elements containing cracks is added to the FEM formulation. Numerical implementation of enriched elements by discontinuous functions is performed, and thus dynamic equations (stiffness and mass matrices) are established. A FORTRAN computer code based on the X-FEM formulation is hence developed. Rectangular and square plates containing through-edge and central cracks with different boundary conditions are considered. The subspace iteration method is used to solve the eigenvalue problem. Natural frequencies as well as the corresponding eigenfunctions are consequently calculated as a function of the crack length. The obtained results show that the X-FEM is an efficient method in the dynamic analysis of plates containing discontinuities.     

Vibration control for parametrically excited Liénard systems

We apply a new vibration control method for time delay non-linear oscillators to the principal resonance of a parametrically excited Liénard system under state feedback control with a time delay. Using the asymptotic perturbation method, we obtain two slow flow equations on the amplitude and phase. Their fixed points correspond to limit cycles for the Liénard system. Vibration control and high-amplitude response suppression can be performed with appropriate time delay and feedback gains. Using energy considerations, we investigate existence and characteristics of limit cycles of the slow flow equations. A limit cycle corresponds to a two-period quasi-periodic modulated motion for the starting system and in order to reduce the amplitude peak of the parametric resonance and to exclude the existence of two-period quasi-periodic motion, we find the appropriate choices for the feedback gains and the time delay.     

Studying the free vibrations of multilayer plates with a complex planform

An efficient method is developed to solve the free-vibration problems for arbitrarily shaped orthotropic multilayer plates in a refined formulation. The method is based on the R-function and Ritz methods. Sequences of coordinate functions satisfying kinematic boundary conditions are constructed in an analytic form. The method is used to solve the vibration problem for multilayer square and arbitrarily shaped plates. The results obtained for square plates are analyzed. A comparison of these results with those available in the literature demonstrates the efficiency of the method __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 119–125, January 2006.     

The bending stability and vibrations of cantilever rectangular plates

This paper discusses the problems of the bending, stability and vibrations of cantilever rectangular plates by means of the variational method. In the text a good many calculating examples are illustrated.