Single-frequency vibrations and vibrational heating of a piezoelectric circular sandwich plate under monoharmonic electromechanical loading

The forced monoharmonic bending vibrations and dissipative heating of a piezoelectric circular sandwich plate under monoharmonic mechanical and electrical loading are studied. The core layer is passive and viscoelastic. The face layers (actuators) are piezoelectric and oppositely polarized over the thickness. The plate is subjected to harmonic pressure and electrical potential. The viscoelastic behavior of the materials is described by complex moduli dependent on the temperature of heating. The coupled nonlinear problem is solved numerically. A numerical analysis demonstrates that the natural frequency, amplitude of vibrations, mechanical stresses, and temperature of dissipative heating can be controlled by changing the area and thickness of the actuator. It is shown that the temperature dependence of the complex moduli do not affect the electric potential applied to the actuator to compensate for the mechanical stress __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 1, pp. 79–89, January 2008.     

Monoharmonic vibrations and vibrational heating of an electromechanically loaded circular plate with piezoelectric actuators subject to shear strain

The paper addresses the forced flexural vibrations and dissipative heating of a circular viscoelastic plate with piezoactive actuators under axisymmetric loading. A refined formulation of this coupled problem is considered. The viscoelastic behavior of materials is described using the concept of complex moduli dependent on the temperature of dissipative heating. The electromechanical behavior of the plate is modeled based on the Timoshenko hypotheses for the mechanical variables and analogous hypotheses for the electric-field variables in the piezoactive layers of the actuator. The temperature is assumed constant throughout the thickness. The nonlinear problem is solved by a time stepping method using, at each step, the discrete-orthogonalization and finite-difference methods to solve the elastic and heat-conduction equations, respectively. A numerical study is made of the effect of the shear strain, the temperature dependence of the material properties, fixation conditions, and geometrical parameters of the plate on the vibrational characteristics and the electric potential applied to the actuator electrodes to balance the mechanical load Translated from Prikladnaya Mekhanika, Vol. 44, No. 9, pp. 104–114, September 2008.     

Resonant flexural vibrations and dissipative heating of a piezoceramic ring plate with nonuniformly electroded surfaces

The forced flexural vibrations and dissipative heating of a bimorph ring plate are studied. The plate is made of viscoelastic piezoceramics and is polarized across the thickness. The outer surfaces of the plate are nonuniformly electroded, and harmonic electric excitation is applied to the electrodes. The viscoelastic behavior of the material is described using the concept of temperature-dependent complex moduli. The coupled nonlinear problem of thermoviscoelasticity is solved by time iteration using, at each iteration, the discrete-orthogonalization method to integrate the mechanics equations and the explicit finite-difference method to solve the heat-conduction equation with a nonlinear heat source. Numerical calculations demonstrate that by changing the size of the ring electrode we can influence the natural frequency, stress and displacement distributions, dissipative-heating temperature, and amplitude-and temperature-frequency characteristics. With certain boundary conditions, there is an optimal electrode configuration that produces deflections of maximum amplitudes when an electric excitation is applied __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 3, pp. 102–109, March 2006.     

Resonant vibration and heating of ring plates with piezoactuators under electromechanical loading and shear deformation

The bending vibration and dissipative heating of a viscoelastic isotropic ring plate with piezoceramic actuators under electromechanical loading and shear deformation are studied by solving a coupled problem. The temperature dependence of the complex characteristics of the passive and piezoactive materials is taken into account. The nonlinear problem of thermoviscoelasticity is solved by time stepping with discrete orthogonalization used at each iteration to integrate the equations of elasticity and using an explicit finite-difference scheme to solve the heat-conduction equation with a nonlinear heat source. The effect of shear deformation, fixation conditions for the plate, the geometry of the piezoactuators, and the dissipative-heating temperature on the active damping of the forced vibration of a circular plate subject to uniform transverse monoharmonic compression is studied Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 124–132, February 2009.     

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.     

Damping the vibrations of a rectangular plate with piezoelectric actuators

The paper addresses the active damping of nonstationary vibrations of a hinged rectangular plate with distributed piezoelectric actuators. The problem is solved by two methods: (i) the classical method of balancing the fundamental vibration modes by applying the appropriate potential difference to the actuator and (ii) the dynamic-programming method that reduces the problem to an algebraic Riccati equation. The results produced by both approaches are presented and compared __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 78–84, February 2008.     

Basic equations for thermoviscoelastic plates with distributed actuators under monoharmonic loading

The basic equations for thin-walled thermoviscoelastic plates with distributed piezoelectric actuators under monoharmonic mechanical and electric loads are derived. The thermomechanical behavior of materials is described using the concept of complex characteristics. Variational methods of solving nonlinear problems of active damping of the bending vibrations of plates are considered. The effect of dissipative heating on the damping of the axisymmetric bending vibrations of a circular plate is examined as an example Translated from Prikladnaya Mekhanika, Vol. 45, No. 2, pp. 107–123, February 2009.     

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.     

Planar electroelastic vibrations of piezoceramic rectangular plate and half-disk

An attempt is made to systematize experimental data for a rectangular piezoceramic plate and to compare them with those on planar vibrations of a thin piezoceramic half-disk. Experimental data on planar vibrations of a half-disk are discussed for the first time. Neighboring vibration modes of a rectangular plate with solid electrodes demonstrate strong superposition and coupling __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 5, pp. 89–96, May 2007.     

Nonlinear vibrations of a viscoelastic plate with concentrated masses

The problem of vibrations of a viscoelastic plate with concentrated masses is studied in a geometrically nonlinear formulation. In the equation of motion of the plate, the action of the concentrated masses is taken into account using Dirac δ-functions. The problem is reduced to solving a system of Volterra type ordinary nonlinear integrodifferential equations using the Bubnov-Galerkin method. The resulting system with a singular Koltunov-Rzhanitsyn kernel is solved using a numerical method based on quadrature formulas. The effect of the viscoelastic properties of the plate material and the location and amount of concentrated masses on the vibration amplitude and frequency characteristics is studied. A comparison is made of numerical calculation results obtained using various theories. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 6, pp. 158–169, November–December, 2007.     

Analysis of free non-linear vibrations of a viscoelastic plate under the conditions of different internal resonances

Non-linear free damped vibrations of a rectangular plate described by three non-linear differential equations are considered when the plate is being under the conditions of the internal resonance one-to-one, and the internal additive or difference combinational resonances. Viscous properties of the system are described by the Riemann-Liouville fractional derivative of the order smaller than unit. The functions of the in-plane and out-of-plane displacements are determined in terms of eigenfunctions of linear vibrations with the further utilization of the method of multiple scales, in so doing the amplitude functions are expanded into power series in terms of the small parameter and depend on different time scales, but the fractional derivative is represented as a fractional power of the differentiation operator. It is assumed that the order of the damping coefficient depends on the character of the vibratory process and takes on the magnitude of the amplitudes’ order. The time-dependence of the amplitudes in the form of incomplete integrals of the first kind is obtained. Using the constructed solutions, the influence of viscosity on the energy exchange mechanism is analyzed which is intrinsic to free vibrations of different structures being under the conditions of the internal resonance. It is shown that each mode is characterized by its damping coefficient which is connected with the natural frequency of this mode by the exponential relationship with a negative fractional exponent. It is shown that viscosity may have a twofold effect on the system: a destabilizing influence producing unsteady energy exchange, and a stabilizing influence resulting in damping of the energy exchange mechanism.     

Dissipative Heating of a Viscoelastic Cylinder under a Load Steadily Moving over Its Surface

The Fourier method is used to find the analytical solutions to two-dimensional quasistatic problems of stationary polyharmonic vibrations and dissipative heating of a linearly viscoelastic cylinder. The influence of the cylinder thickness and the width of the loading area on the thermomechanical state of the cylinder is studied based on numerical data__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 23–32, February 2005.     

Radial Vibrations and Heating of a Ring Piezoplate under Electric Excitation Applied to Nonuniformly Electroded Surfaces

Radial vibrations and dissipative heating of a polarized piezoceramic ring plate are studied. The plate is excited by a harmonic electric field applied to nonuniformly electroded surfaces of the plate. The viscoelastic behavior of piezoceramics is described in terms of complex quantities. An analytical solution is found in the case of quasistatic harmonic loading. The dynamic nonlinear problem of coupled thermoviscoelasticity is solved with regard for the temperature dependence of the properties of piezoceramics by step-by-step integration in time, using the numerical methods of discrete orthogonalization and finite differences. A numerical analysis is conducted for TsTStBS-2 piezoceramics to study the influence of partial electroding on the stress–strain distribution, natural frequency, and amplitude–frequency and temperature–frequency characteristics     

Vibrations and Dissipative Heating of a Cylindrical Panel under a Periodic Moving Load

An analytic solution is obtained to describe the vibrations and dissipative heating of a simply supported infinite cylindrical panel under periodic normal loads moving over its surface with a constant velocity. Special attention is focused on resonant vibrations, which result in the most intensive dissipative heating. It is additionally assumed that the material of the panel is viscoelastic, its properties are independent of temperature, and Poisson’s ratio is real. The influence of thickness, radius of curvature, load velocity, and viscoelastic properties on the thermal state of the panel is analysed against the thermal state of the plate__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 4, pp. 100–109, April 2005.     

Modeling the thermomechanical behavior of physically nonlinear materials under monoharmonic loading

We analyze approximate approaches to the modeling of the thermomechanical behavior of physically nonlinear materials under harmonic loading. The approaches are based on various harmonic-linearization schemes and the concept of complex moduli. Mechanical and mathematical features of various schemes are considered. Some modifications of the model are proposed to account for various aspects of material behavior under harmonic loading. The problems of vibration and dissipative heating of physically nonlinear bodies are formulated. The main thermomechanical characteristics are analyzed for some classes of problems.The study was partially sponsored by the State Fund for Basic Research of the Ministry of Education and Science of Ukraine (Grant No. 01.07/00050).Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 3–34, September 2004.     

Experimental study of large amplitude vibrations of a thin plate in contact with sloshing liquids

The linear and geometrically nonlinear (large amplitude) dynamical response of a thin plate in contact with water on one or both sides has been experimentally studied, considering different filling levels. The free liquid surface is free to slosh and the water is delimited by practically rigid walls, except for the thin plate. An experimental method for monitoring and measuring the free surface waves of the fluid has been also used in order to analyze the behavior of the liquid free surface during the nonlinear vibration of the plate forced by harmonic excitation. The plate deflection due to hydrostatic pressure plays a significant role in changing the plate nonlinearity, but tests with liquid on both sides eliminating this effect have been also presented. For excitation in the frequency neighborhood of the fundamental mode of the plate, the oscillation of the free surface of the liquid is characterized by a very large 1/2-subharmonic component.     

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.     

Effect of static loading on the nonlinear vibrations of a three-time redundant portal frame: Analytical and numerical studies

An analytical study of the nonlinear vibrations in a three-time redundant portal frame is presented herewith, considering the effect of the axial forces caused by the static loading upon the first anti-symmetrical mode (sway) and the first symmetrical mode natural frequencies. It is seen that the axial forces may play an important role in tuning the sway mode and the first symmetrical mode into a 1:2 internal resonance. Harmonic support excitations resonant with the first symmetrical mode are then introduced and the amplitudes of nonlinear steady states are computed based upon a multiple scales solution. Comparisons with numerical analyses using a finite-element program developed by the authors show good qualitative agreement.     

Asymmetric vibrations and stability of a rotating annular plate loaded by a torque

The paper investigates transverse vibration of a thin annular plate clamped at its inner edge to a rigid shaft, while its outer edge is clamped to a rigid cylinder. The shaft and the outer edge of the plate are loaded by torques of the same intensity, but of opposite directions. The whole structure rotates at a constant angular speed. The solution has been determined using Galerkin’s method. The obtained results illustrate the impact of the torque, angular speed and inner and outer radia ratio to transverse asymmetric vibration frequency of the plate. Stability of the plate has been examined and critical values of angular speed and torque leading to the loss of stability of the plate have been determined. Some mode shapes have been drawn and the influence of torque and angular speed on nodal lines has been shown.