Double plate system with a discontinuity in the elastic bonding layer

The double plate system with a discontinuity in the elastic bonding layer of Winker type is studied in this paper. When the discontinuity is small, it can be taken as an interface crack between the bi-materials or two bodies (plates or beams). By comparison between the number of multifrequencies of analytical solutions of the double plate system free transversal vibrations for the case when the system is with and without discontinuity in elastic layer we obtain a theory for experimental vibration method for identification of the presence of an interface crack in the double plate system. The analytical analysis of free transversal vibrations of an elastically connected double plate systems with discontinuity in the elastic layer of Winkler type is presented. The analytical solutions of the coupled partial differential equations for dynamical free and forced vibration processes are obtained by using method of Bernoulli’s particular integral and Lagrange’s method of variation constants. It is shown that one mode vibration corresponds an infinite or finite multi-frequency regime for free and forced vibrations induced by initial conditions and one-frequency or corresponding number of multi-frequency regime depending on external excitations. It is shown for every shape of vibrations. The analytical solutions show that the discontinuity affects the appearance of multi-frequency regime of time function corresponding to one eigen amplitude function of one mode, and also that time functions of different vibration basic modes are coupled. From final expression we can separate the new generalized eigen amplitude functions with corresponding time eigen functions of one frequency and multi-frequency regime of vibrations. The English text was polished by Keren Wang.     

Energy transfer in the hybrid system dynamics (energy transfer in the axially moving double belt system)

First, as an introduction, using the author’s published references, a short survey of an analytical study of the energy transfer between two coupled subsystems, as well as between a linear and nonlinear oscillators of a hybrid system, in the free and forced vibrations of a different type of inter connections between subsystems is presented. Second, as author’s new research result, an analytical study of the energy transfer between two coupled like-string belts interconnected by light pure elastic layer in the axially moving sandwich double belt system, in the free vibrations is presented. On the basis of the obtained analytical expressions for the kinetic and potential energy of the belts and potential energy of the of light pure elastic distributed layer numerous conclusions are derived. In the pure linear elastic double belt system no transfer energy between different eigen modes of transversal vibrations of the axially moving double belt system, but in every from of the set of the infinite numbers eigen modes, there are transfer energy between belts. Each of the eigen modes of the free transversal vibrations are like two-frequency. The change of the potential energy of the booth belts is four frequency, and interaction part of the potential energy is one frequency in the each eigen mode. Changes of the kinetic energy of the both belts of the sandwich double axially moving bet system is two frequency like oscillatory regimes with two time multiplicities of the eineg frequencies of the corresponding eigen amplitude mode.     

Transversal vibrations of double-plate systems

This paper presents an analytical and numerical analysis of free and forced transversal vibrations of an elastically connected double-plate system. Analytical solutions of a system of coupled partial differential equations, which describe corresponding dynamical free and forced processes, are obtained using Bernoulli’s particular integral and Lagrange’s method of variation constants. It is shown that one-mode vibrations correspond to two-frequency regime for free vibrations induced by initial conditions and to three-frequency regime for forced vibrations induced by one-frequency external excitation and corresponding initial conditions. The analytical solutions show that the elastic connection between plates leads to the appearance of two-frequency regime of time function, which corresponds to one eigenamplitude function of one mode, and also that the time functions of different vibration modes are uncoupled, for each shape of vibrations. It has been proven that for both elastically connected plates, for every pair of m and n, two possibilities for appearance of the resonance dynamical states, as well as for appearance of the dynamical absorption, are present. Using the MathCad program, the corresponding visualizations of the characteristic forms of the plate middle surfaces through time are presented.The English text was polished by Keren Wang.     

Energy analysis in a nonlinear hybrid system containing linear and nonlinear subsystems coupled by hereditary element

Energy transfer between subsystems coupled by standard light hereditary element in hybrid system is very important for different engineering applications, especially for dynamical absorption. An analytical study of the energy transfer between coupled linear and nonlinear oscillators in the free vibrations of a viscoelastically connected double-oscillator system as a new hybrid nonlinear system with two and half degrees of freedom is pointed out. The analytical study shows that the viscoelastic–hereditary connection between oscillators causes the appearance of like two-frequency regimes of subsystem's vibrations and that the energy transfer between subsystems appears. The Lyapunov exponents corresponding to each of two eigenmodes of the hybrid system, as well as to the subsystems are obtained and expressed by using energy of the corresponding eigentime components. The Lyapunov exponents are measures of the vibration processes stability in the hybrid system and in component subsystem vibrations. In Honor of Giuseppe Rega and Fabrizio Vestroni on the Occasion of their 60th Birthday.     

The effect of laminations on the vibrational modes of circular annular plates

The purpose of this article is to present an experimental study of the effect of laminations on the vibrations of circular annular plates. To obtain a basis for comparison with experimental data, the natural frequencies and mode shapes of a series of solid circular annular plates were calculated using the finite element method. An extensive range of experiments were performed on both a series of solid models and a series of laminated models under a range of normal clamping pressures. Based on the analytical and experimental results, it was found that the vibrational behavior of the laminated plates was dominated by that of the individual plate of which they were composed and that the effects of the laminations on vibrations were mode type dependent. The effects on the transverse vibrational modes were dependent on both the normal clamping pressure and the number of plates. The amplitude of the frequency response function for these modes reduced quickly, and the resonant frequency of such modes shifted higher as the clamping pressure or the number of plates increased. For the in-plane vibrational modes, the amplitude of the frequency response function reduced slightly as the number of plates increased; the resonant frequency of such modes could be considered to be a constant and independent of both the clamping pressure and the number of plates.     

Geometrically non-linear transverse vibrations of C-S-S-S and C-S-C-S rectangular plates

This paper is concerned with a new improved formulation of the theoretical model previously developed by Benamar et al. based on Hamilton's principle and spectral analysis, for the geometrically non-linear vibrations of thin structures. The problem is reduced to a non-linear algebraic system, the solution of which leads to determination of the amplitude-dependent fundamental non-linear mode shapes, the frequency parameters, and the non-linear stress distributions. The cases of C-S-C-S and C-S-S-S rectangular plates are examined, and the results obtained are in a good qualitative and quantitative agreement with the previous available works, based on various methods. In order to obtain explicit analytical solutions for the first non-linear mode shapes of C-S-C-S RP² and C-S-S-S RP, which are expected to be very useful in engineering applications and in further analytical developments, the improved version of the semi-analytical model developed by El Kadiri et al. For beams and fully clamped rectangular plates, has been slightly modified, and adapted to the above cases, leading to explicit expressions for the higher basic function contributions, which are shown to be in a good agreement with the iterative solutions, for maximum non-dimensional vibration amplitude values up to 0.75 and 0.6 for the first non-linear mode shapes of C-S-C-S RP and C-S-S-S RP, respectively.     

Nonlinear vibration of circular sandwich plate under the uniformed load

ＩｎｔｒｏｄｕｃｔｉｏｎＳｏｆａｒ,ｏｎｌｙａｆｅｗｐｅｏｐｌｅｈａｖｅｓｔｕｄｉｅｄｔｈｅｌａｒｇｅｄｅｆｌｅｃｔｉｏｎｐｒｏｂｌｅｍｓｏｆｓａｎｄｗｉｃｈｐｌａｔｅｓａｎｄｓｈｅｌｌｓｂｅｃａｕｓｅｏｆｔｈｅｄｉｆｆｉｃｕｌｔｙｏｆｎｏｎｌｉｎｅａｒｍａｔｈｅｍａｔｉｃｓ.ＬｉｕＲｅｎｈｕａｉｈａｓｄｏｎｅｍｕｃｈｔｏｆｉｎｄａｓｅｒｉｅｓｏｆｒｅｓｕｌｔｓｗｉｔｈｔｈｅｖａｌｕｅｏｆａｐｐｌｉｃａｔｉｏｎｉｎｅｎｇｉｎｅｅｒｉｎｇｐｒａｃｔｉｃｅ[1～5].Ａｕｔｈｏｒ[6,7]ｈａｄｔｈｅｉｎｉｔｉ…     

Non-linear vibrations of free-edge thin spherical shells: modal interaction rules and 1:1:2 internal resonance

This paper is devoted to the derivation and the analysis of vibrations of shallow spherical shell subjected to large amplitude transverse displacement. The analog for thin shallow shells of von Kármán’s theory for large deflection of plates is used. The validity range of the approximations is assessed by comparing the analytical modal analysis with a numerical solution. The specific case of a free edge is considered. The governing partial differential equations are expanded onto the natural modes of vibration of the shell. The problem is replaced by an infinite set of coupled second-order differential equations with quadratic and cubic non-linear terms. Analytical expressions of the non-linear coefficients are derived and a number of them are found to vanish, as a consequence of the symmetry of revolution of the structure. Then, for all the possible internal resonances, a number of rules are deduced, thus predicting the activation of the energy exchanges between the involved modes. Finally, a specific mode coupling due to a 1:1:2 internal resonance between two companion modes and an axisymmetric mode is studied.     

Designing a Linear Structure with a Local Nonlinear Attachment for Enhanced Energy Pumping

We present a design procedure for enhancing nonlinear energy pumping from a mode of a linear-damped substructure to a weakly coupled, essentially nonlinear oscillator. By this we denote the one way, irreversible passive transfer of vibrational energy from the mode to the nonlinear attachment. The design relies in the asymptotic expansion for large energies of a nonlinear normal mode of the underlying conservative system that provides an analytic estimate of the level of the amplitude reached by the nonlinear attachment in the energy pumping regime. The analytical findings are validated by direct numerical simulations.     

Load’s temporal characteristics for annulling forced vibrations of linear elastic plates

We consider trapezoidal load-time pulses with linearly increasing and affinely decreasing durations equal to integer multiples of the time period of the first bending mode of vibration of a linearly elastic structure. For arbitrary spatial distributions of loads applied to monolithic and laminated orthotropic plates, it is shown through numerical solutions that plates’ vibrations become miniscule after the load is removed. This phenomenon is independent of the dwell time (i.e., the time duration between the rising and the falling portions) during which the load is kept constant. The primary reason for this response is that for such time-dependent loads, nearly all of plate’s strain energy is concentrated in deformations corresponding to the fundamental bending mode of vibration. Thus plate’s deformations can be studied by taking the mode shape of the 1st bending mode as the basis function and reducing the problem to that of solving a single second-order ordinary differential equation. We have verified this postulate by comparing strain energies computed from the 3-dimensional deformations of different plate geometries and boundary conditions with those determined by using the single degree of freedom (DoF) model. Thus for trapezoidal time-dependent loads applied on plates, the 1 DoF model provides reasonably accurate results and saves considerable computational effort.     

Large free vibration of thin plates: Hierarchic finite Element Method and asymptotic linearization

The aim of this work is to study the free dynamic response of thin plates characterized by geometrical nonlinearities. To achieve this task, the equation of motion of the plate is first carried out through modeling by hierarchical finite element method whose interpolating shape functions are sinusoidal. Then, the study of the nonlinear vibrations was carried out by the development of asymptotic linearization and equivalent linearization methods in modal space. The nonlinear angular frequencies are successively deduced by exciting the corresponding vibrating mode of the structure. The confrontation of these results to those obtained by the iterative method in the physical space and to those found in the literature, showed a very good agreement between the various methods. From the elementary nonlinear frequencies we showed that there exists an equivalent linear dynamical system characterized by only one equivalent linear stiffness matrix. Numerical experiments were carried out on beams and thin plates of various dimensions ratios and boundary conditions. These numerical test simulations, whether in time space or frequency space, have showed that the nonlinear elastic energy is restored by the equivalent linear dynamical system. Nevertheless, we have to say that the dynamic effects of modes above the excited one are neglected.     

Non-linear free periodic vibrations of variable stiffness composite laminated plates

Steady-state free vibrations, with large amplitude displacements, of variable stiffness composite laminated plates (VSCL) are analysed. The intentions of this research are: (1)?to find out how the natural frequencies and (mode) shapes evolve with the displacement amplitude in this new type of laminated composite material; (2)?to describe modal interactions in VSCL due to energy interchanges under the coupling induced by non-linearity; (3)?to compare the VSCL with traditional, constant stiffness, laminated plates. The VSCL of interest here have curvilinear fibres and the numerical analysis carried out is based on a recently developed p-version finite element with hierarchic basis functions. The element follows first-order shear deformation theory and considers Von Kármán??s non-linear terms. The time domain equations of motion are first reduced using the linear modes of vibration and then transformed to the frequency domain via the harmonic balance method. These frequency domain equations are solved by an arc-length continuation method.     

One-to-two global-local interaction in a cable-stayed beam observed through analytical, finite element and experimental models

The mechanism of cable end angle-variation induced oscillations in the non-linear interactions between beams and cables in stayed-systems is first explained by a proposed analytical model. It is then verified by both experimental and finite element models. The non-linear interaction maximizes its effects for cable oscillations when inherent quadratic coupling between local and global modes produces energy transfer from low to high frequency vibrations by means of a one-to-two global-local autoparametric resonance. The response of the analytical model is fully described using a continuation method applied directly to the reduced two degree of freedom discrete model showing that, for a selected one-to-two global-local resonant system, primary harmonic excitation of the global mode produces large oscillations of the local mode at twice the excitation frequency. Detailed comparisons between the responses of the analytical model, experimental results and finite element simulations show excellent agreement both in the qualitative behaviour and in the calculated/measured response amplitudes.     

On the modal equations of large amplitude flexural vibration of beams,plates, rings and shells

In a simple and straight forward manner the modal equations applicable for the large amplitude flexural vibrations of plates and shells are obtained by the Lagrange's method. These equations can easily be specialised to obtain the corresponding equations applicable for beams and rings. The basic nature of the modal equations for beams and plates on the one hand and rings and shells on the other hand are shown to exhibit hard and soft spring characteristics, respectively.     

Two mode sub-harmonic and harmonic vibrations of a non-linear beam forced by a two mode harmonic load

The non-linear transverse vibrations of a uniform beam with ends restrained to remain a fixed distance apart and forced by a two mode function which is harmonic in time, are studied by a corresponding two mode approach. The existence of sub-harmonic response of order 1/3 and harmonic response in the sub-harmonic resonance region of the forcing frequency is proved. Approximate solutions are found by Urabe's numerical method applied to Galerkin's procedure and by an analytical harmonic balance-perturbation method. Error bounds of the Galerkin approximations are given.     

Experimental and theoretical study on large amplitude vibrations of clamped rubber plates

In this paper, the large amplitude forced vibrations of thin rectangular plates made of different types of rubbers are investigated both experimentally and theoretically. The excitation is provided by a concentrated transversal harmonic load. Clamped boundary conditions at the edges are considered, while rotary inertia, geometric imperfections and shear deformation are neglected since they are negligible for the studied cases. The von Kármán nonlinear strain-displacement relationships are used in the theoretical study; the viscoelastic behaviour of the material is modelled using the Kelvin-Voigt model, which introduces nonlinear damping. An equivalent viscous damping model has also been created for comparison. In-plane pre-loads applied during the assembly of the plate to the frame are taken into account. In the experimental study, two rubber plates with different material and thicknesses have been considered; a silicone plate and a neoprene plate. The plates have been fixed to a heavy rectangular metal frame with an initial stretching. The large amplitude vibrations of the plates in the spectral neighbourhood of the first resonance have been measured at various harmonic force levels. A laser Doppler vibrometer has been used to measure the plate response. Maximum vibration amplitude larger than three times the thickness of the plate has been achieved, corresponding to a hardening type nonlinear response. Experimental frequency-response curves have been very satisfactorily compared to numerical results. Results show that the identified retardation time increases when the excitation level is increased, similar to the equivalent viscous damping but to a lesser extent due to its nonlinear nature. The nonlinearity introduced by the Kelvin-Voigt viscoelasticity model is found to be not sufficient to capture the dissipation present in the rubber plates during large amplitude vibrations.     

Nonlinear normal modes of a self-excited system driven by parametric and external excitations

Vibrations of nonlinear coupled parametrically and self-excited oscillators driven by an external harmonic force are presented in the paper. It is shown that if the force excites the system inside the principal parametric resonance then for a small excitation amplitude a resonance curve includes an internal loop. To find the analytical solutions, the problem is reduced to one degree of freedom oscillators by applications of Nonlinear Normal Modes (NNMs). The NNMs are formulated on the basis of free vibrations of a nonlinear conservative system as functions of amplitude. The analytical results are validated by numerical simulations and an essential difference between linear and nonlinear modes is pointed out.     

Interaction of free and forced nonlinear normal modes in two-DOF dissipative systems under resonance conditions

The resonance dynamics of a dissipative spring-mass and of a dissipative spring-pendulum system is studied. Internal resonance case is considered for the first system; both external resonances and simultaneous external and internal resonance are studied for the second one. Analysis of the systems resonance behavior is made on the base of the concept of nonlinear normal vibration modes (NNMs) by Kauderer and Rosenberg, which is generalized for dissipative systems. The multiple time scales method under resonance conditions is applied. The resulting equations are reduced to a system with respect to the system energy, arctangent of the amplitudes ratio and the difference of phases of required solution in the resonance vicinity. Equilibrium positions of the reduced system correspond to nonlinear normal modes; in energy dissipation case they are quasi-equilibriums. Analysis of the equilibrium states of the reduced system permits to investigate stability of nonlinear normal modes in the resonance vicinity and to describe transfer from unstable vibration mode to stable one. New vibration regimes, which are called transient nonlinear normal modes (TNNMs) are obtained. These regimes take place only for some particular levels of the system energy. In the vicinity of values of time, corresponding to these energy levels, the TTNM attract other system motions. Then, when the energy decreases, the transient modes vanish, and the system motions tend to another nonlinear normal mode, which is stable in the resonance vicinity. The reliability of the obtained analytical results is confirmed by numerical and numerical-analytical simulations.     

Rotation of the apparent vibration plane of a swinging spring at the 1:1:2 resonance

Nonlinear spatial vibrations of a mass point on a weightless elastic suspension (pendulum on a spring) are considered. The frequency of vertical vibrations is assumed to be equal to the doubled swinging frequency (the 1:1:2 resonance). In this case, as numerical calculations and experiments show, the vertical vibrations are unstable, which leads to the vertical vibration energy transfer to the pendulum swinging energy. The vertical vibrations of the mass point decay and, after a certain time period, the pendulum starts swinging in a certain vertical plane. This swinging is also unstable, which results in the reverse energy transfer into the vertical vibration mode. The vertical vibrations are again repeated. But after the second transfer of the vertical vibration energy to the pendulum swinging energy, the apparent plane of vibrations rotates by a certain angle. These effects are described analytically; namely, the energy transfer period, the time variations in the amplitudes of both modes, and the variations in the angle of the apparent vibration plane are determined. An asymptotic solution is also constructed for the mass point trajectory in the orbit elements. In projection on the horizonal plane, the mass point moves in a nearly elliptic trajectory. The ellipse semiaxes slowly vary with time, so that their product remains constant, and the major semiaxis slowly rotates at a constant sectorial velocity. The obtained analytic time dependence of the ellipse semiaxes and the precession angle agree well with the results of numerical calculations.