Modal interactions in the vibrations of a heated annular plate

We investigate the non-linear forced vibrations of a thermally loaded annular plate with clamped–clamped immovable boundary conditions in the presence of a three-to-one internal resonance between the first and second axisymmetric modes. We consider the in-plane thermal load to be axisymmetric and excite the plate externally by a harmonic force near primary resonance of the second mode. We then use the non-linear von Kármán plate equations to model the behavior of the system and apply the method of multiple scales to investigate its responses. We found that the response can be periodic oscillations consisting of both modes, with a large component from the first mode. Moreover, the periodic solutions may undergo Hopf bifurcations, which lead to aperiodic oscillations of the plate.     

Transition to chaotic vibrations for harmonically forced perfect and imperfect circular plates

The transition from periodic to chaotic vibrations in free-edge, perfect and imperfect circular plates, is numerically studied. A pointwise harmonic forcing with constant frequency and increasing amplitude is applied to observe the bifurcation scenario. The von Kármán equations for thin plates, including geometric non-linearity, are used to model the large-amplitude vibrations. A Galerkin approach based on the eigenmodes of the perfect plate allows discretizing the model. The resulting ordinary-differential equations are numerically integrated. Bifurcation diagrams of Poincaré maps, Lyapunov exponents and Fourier spectra analysis reveal the transitions and the energy exchange between modes. The transition to chaotic vibration is studied in the frequency range of the first eigenfrequencies. The complete bifurcation diagram and the critical forces needed to attain the chaotic regime are especially addressed. For perfect plates, it is found that a direct transition from periodic to chaotic vibrations is at hand. For imperfect plates displaying specific internal resonance relationships, the energy is first exchanged between resonant modes before the chaotic regime. Finally, the nature of the chaotic regime, where a high-dimensional chaos is numerically found, is questioned within the framework of wave turbulence. These numerical findings confirm a number of experimental observations made on shells, where the generic route to chaos displays a quasiperiodic regime before the chaotic state, where the modes, sharing internal resonance relationship with the excitation frequency, appear in the response.     

Non-Linear vibrations and chaos in harmonically excited rectangular plates with one-to-one internal resonance

Nonlinear flexural vibrations of a rectangular plate with uniform stretching are studied for the case when it is harmonically excited with forces acting normal to the midplane of the plate. The physical phenomena of interest here arise when the plate has two distinct linear modes of vibration with nearly the same natural frequency. It is shown that, depending on the spatial distribution of the external forces, the plate can undergo harmonic motions either in one of the two individual modes or in a mixed-mode. Stable single-mode and mixed-mode solutions can also coexist over a wide range in the amplitudes and frequency of excitation. For low damping levels, the presence of Hopf bifurcations in the mixed-mode response leads to complicated amplitude-modulated dynamics including period doubling bifurcations, chaos, coexistence of multiple chaotic motions, and crisis, whereby the chaotic attractors suddenly disappear and the plate resumes small amplitude harmonic motions in a single-mode. Numerical results are presented specifically for 1 : 1 resonance in the (1, 2) and (3, 1) plate modes.     

Two modes nonresonant interaction for rectangular plate with geometrical nonlinearity

The vibrations of thin rectangular plate with geometrical nonlinearity are analyzed. The models of plate vibrations with different numbers of degrees-of-freedom are derived. It is deduced that two degrees-of-freedoms are enough to describe low-frequency nonlinear dynamics of plates. Nonlinear normal modes are used to analyze the system dynamics. If vibrations amplitudes are increased, single-mode plate vibrations are transformed into two mode ones. In this case, internal resonance conditions are not observed. Such transformation of vibration is described using Kauderer?CRosenberg nonlinear normal modes.     

Vibrations of Circular Sandwich Plates under Resonance Loads

The paper studies axisymmetric resonance vibrations of an elastic circular sandwich plate under local periodic surface loads of rectangular, sinusoidal, and parabolic forms. The hypotheses of broken normal are used to describe the kinematics of the plate, which is asymmetric in thickness. The core is assumed to be light. The initial–boundary-value problems are solved analytically. The solutions are analyzed     

Finite amplitude vibrations of an orthotropic circular plate with an isotropic core

The free and forced non-linear vibrations of a fixed orthotropic circular plate, with a concentric core of isotropic material, are studied. Existence of harmonic vibrations is assumed and thus the time variable is eliminated by a Ritz-Kantorovich method. Hence, the governing non-linear partial equations for the axisymmetric vibration of the composite circular plate are reduced to a set of ordinary differential equations which form a non-linear eigen-value problem. Solutions are obtained by utilizing the related initial-value problems in conjunction with Newton's integration method. The results reveal the effects of finite amplitude and anisotropy of materials upon the dynamic responses. Further, the method developed in this paper, which is used to solve the title problem, is one of some generality. It can be applied to many differential eigenvalue problems with piecewise continuous functions.     

Nonlinear Dynamics of a Flexible Spinning Disc Coupled to a Precompressed Spring

This paper analyses the nonlinear transverse vibrations of a rotating, clamped-free, flexible disc coupled to a precompressed spring. This is representative of a large class of loadings in rotating disc systems such as air jet and electromagnetic excitation commonly used in experiments. Such a loading induces a simultaneous critical speed resonance and parametric instability. The disc is modelled as a Von Kármán plate, and the equations of motion are discretised by a Galerkin projection onto a pair of 1:1 internally resonant modes. The large amplitude wave motions and their stabilities are studied using the averaging method and via numerical continuation techniques. The analysis is carried out in a co-rotating as well as a ground-fixed frame. Numerical simulations are used to verify the above analyses. The response predicted by these analyses is substantially different from that arising from a critical speed resonance or of a parametric instability alone. As many as five equilibrium solutions can coexist at supercritical speed. Two distinct regimes of large amplitude response appear to exist depending on the relationship between the strength of the parametric excitation and the damping. The existence of these regimes underscores the subtle competition between critical speed resonance and parametric instability that is likely to be observed in experiments near critical speed in such systems.Contributed by Prof. A.K. Bajaj.     

Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry

Non-linear vibrations of free-edge shallow spherical shells are investigated, in order to predict the trend of non-linearity (hardening/softening behaviour) for each mode of the shell, as a function of its geometry. The analog for thin shallow shells of von Kármán's theory for large deflection of plates is used. The main difficulty in predicting the trend of non-linearity relies in the truncation used for the analysis of the partial differential equations (PDEs) of motion. Here, non-linear normal modes through real normal form theory are used. This formalism allows deriving the analytical expression of the coefficient governing the trend of non-linearity. The variation of this coefficient with respect to the geometry of the shell (radius of curvature R, thickness h and outer diameter 2a) is then numerically computed, for axisymmetric as well as asymmetric modes. Plates (obtained as R→∞) are known to display a hardening behaviour, whereas shells generally behave in a softening way. The transition between these two types of non-linearity is clearly studied, and the specific role of 2:1 internal resonances in this process is clarified.     

Non-linear vibrations of free-edge thin spherical shells: Experiments on a 1:1:2 internal resonance

This study is devoted to the experimental validation of a theoretical model of large amplitude vibrations of thin spherical shells described in a previous study by the same authors. A modal analysis of the structure is first detailed. Then, a specific mode coupling due to a 1:1:2 internal resonance between an axisymmetric mode and two companion asymmetric modes is especially addressed. The structure is forced with a simple-harmonic signal of frequency close to the natural frequency of the axisymmetric mode. The experimental setup, which allows precise measurements of the vibration amplitudes of the three involved modes, is presented. Experimental frequency response curves showing the amplitude of the modes as functions of the driving frequency are compared to the theoretical ones. A good qualitative agreement is obtained with the predictions given by in the model. Some quantitative discrepancies are observed and discussed, and improvements of the model are proposed.     

Flexural Vibrations and Heating of a Circular Bimorph Piezoplate under Electric Excitation Applied to Nonuniformly Electroded Surfaces

The flexural vibrations and dissipative heating of a circular bimorph piezoceramic plate are studied. The plate is excited by a harmonic electric field applied to nonuniformly electroded surfaces. The viscoelastic behavior of piezoceramics is described in terms of temperature-dependent complex moduli. The nonlinear coupled problem of thermoviscoelasticity is solved by step-by-step integration in time, using the discrete-orthogonalization method to solve the mechanics equations and the finite-differences method to solve the heat-conduction equations. A numerical analysis is conducted for TsTStBS-2 piezoceramics to study the influence of the nonuniform electroding on the resonant frequency, amplitude, and modes of flexural vibrations and the amplitude- and temperature-frequency characteristics of the plate __________ Translated from Prikladnaya Mekhanika, Vol. 41, No. 9, pp. 94–100, September 2005.     

Resonant many-mode periodic and chaotic self-sustained aeroelastic vibrations of cantilever plates with geometrical non-linearities in incompressible flow

The plates interacting with inviscid, incompressible, potential gas flow are analyzed. Many modes interaction is considered to describe self-sustained vibrations of plates. The singular integral equation is solved to obtain gas pressures acting on the plate. The Von Karman equations with respect to three displacements are used to describe the plate geometrical non-linear vibrations. The Galerkin method is applied to each partial differential equation to obtain the finite-degree-of-freedom model of the plate vibrations. Self-sustained vibrations, which take place due to the Hopf bifurcation, are investigated. These vibrations undergo the Naimark?CSacker bifurcation and the periodic motions are transformed into the almost periodic ones. If the stream velocity is increased, almost periodic motions are transformed into chaotic ones. As a result of the internal resonance, the saturation of the vibration mode is observed. The non-linear dynamics of low- and high-aspect-ratio plates is analyzed.     

Transverse vibrations and vibrational heating of a rectangular bimorphous plate of dissipative piezoelectric material

The problem of induced resonance vibrations and dissipation heating in a rectangular bimorphous plate made of a dissipative piezoelectric material under a harmonic potential difference is tackled. The edges of the plate are considered to be hinged and ideally thermally insulated. The dissipation properties of the material are taken into account on the basis of the concept of complex characteristics, which are assumed to be temperature-independent. An exact solution is found for the problem. The critical value of the load parameter is determined when the maximum temperature reaches the Curie point. A finite-element method has been developed for investigating the dynamic behavior and temperature of vibrational heating that bimorphous plates made of a viscoelastic material undergo under a harmonic load. The results obtained for the electromechanical vibrations of plates by finite-element calculations and by an analytical solution are compared. Translated from Prikladnaya Mekhanika, Vol. 35, No. 9, pp. 85–93, September, 1999.     

Study of the Resonant Vibrations and Dissipative Heating of Thin-Walled Elements Made of a Physically Nonlinear Material

An approximate formulation is given to a dynamic coupled thermomechanical problem for physically nonlinear inelastic thin-walled structural elements within the framework of a geometrically linear theory and the Kirchhoff–Love hypotheses. A simplified model is used to describe the vibrations and dissipative heating of inhomogeneous physically nonlinear bodies under harmonic loading. Nonstationary vibroheating problem is solved. The dissipative function obtained from the solution for steady-state vibrations is used to simulate internal heat sources. For the partial case of forced vibrations of a beam, the amplitude–frequency characteristics of the field quantities are studied within a wide frequency range. The temperature characteristics for the first and second resonance modes are compared.     

Asymptotic analysis of nonlinear dynamics of simply supported cylindrical shells

Donnell equations are used to simulate free nonlinear oscillations of cylindrical shells with imperfections. The expansion, which consists of two conjugate modes and axisymmetric one, is used to analyze shell oscillations. Amplitudes of the axisymmetric motions are assumed significantly smaller, than the conjugate modes amplitudes. Nonlinear normal vibrations mode, which is determined by shell imperfections, is analyzed. The stability and bifurcations of this mode are studied by the multiple scales method. It is discovered that stable quasiperiodic motions appear at the bifurcations points. The forced oscillations of circular cylindrical shells in the case of two internal resonances and the principle resonance are analyzed too. The multiple scales method is used to obtain the system of six modulation equations. The method for stability analysis of standing waves is suggested. The continuation algorithm is used to analyze fixed points of the system of the modulation equations.     

Axisymmetric free vibrations of infinite micropolar thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Axisymmetric free vibrations of infinite thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Resonant effects of local loads on circular sandwich plates on an elastic foundation

The axisymmetric forced vibrations of a circular sandwich plate on an elastic foundation are studied. The plate is subjected to axisymmetric surface and mechanical loads with frequency equal to one of the natural frequencies of the plate. The foundation reaction is described by the Winkler model. To describe the kinematics of an asymmetric sandwich, the hypothesis of broken normal is used. The core is assumed to be light. The analytical solution of the problem is obtained and numerical results are analyzed     

Harmonic vibrations and waves in a cylindrical helically anisotropic shell

A Kirchhoff-Love type applied theory is used to study the specific characteristics of harmonic waves and vibrations of a helically anisotropic shell. Special attention is paid to axisymmetric and bending vibrations. In both cases, the dispersion equations are constructed and a qualitative and numerical analysis of their roots and the corresponding elementary solutions is performed. It is shown that the skew anisotropy in the axisymmetric case generates a relation between the longitudinal and torsional vibrations which is mathematically described by the amplitude coefficients of homogeneous waves. In the case of a shell with rigidly fixed end surfaces, the dependence of the first two natural frequencies on the shell length and the helical line slope α, i.e., the geometric parameter of helical anisotropy, is studied. A boundary value problem in which longitudinal vibrations are generated on one of the end surfaces and the other end is free of forces and moments is considered to analyze the degree of transformation of longitudinal vibrations into longitudinally torsional vibrations. In the case of bending vibrations, two problems for a half-infinite shell are studied as well. In the first problem, the waves are excited kinematically by generating harmonic vibrations of the shell end surface in the plane of the axial cross-section, and it is shown that the axis generally moves in some closed trajectories far from the end surface. In the second problem, the reflection of a homogeneous wave incident on the shell end is examined. It is shown that the “boundary resonance” phenomenon can arise in some cases.     

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.     

Evolution of the planar vibrations of a rectangular piezoceramic plate as its aspect ratio is changed

The evolution of the planar vibrations of a rectangular piezoceramic plate as its aspect ratio is changed starting with 1 is studied. Experimental data are obtained using an integrated technique based on Meson’s circuit, Onoe’s circuit, and a piezotransformer transducer. As the aspect ratio increases (square plate becomes rectangular), the intensity of electromagnetic vibrations rapidly increases at the first longitudinal resonance and gradually decreases in the first radial mode. When the aspect ratio is changed so that the length of one of the plate sides remains constant, the resonant frequencies of all vibration modes change too __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 7, pp. 98–106, July 2007.