Influence of boundary conditions and temperature of dissipative heating on active damping of forced axisymmetric resonant bending vibrations of circular viscoelastic plates by piezoelectric sensors and actuators

The problem of forced monoharmonic, axisymmetric, bending vibrations and dissipative heating of circular viscoelastic plates with piezoelectric sensors and actuators is considered. We describe the viscoelastic behavior of a passive (without the piezoeffect) and a piezoactive material according to the concept of complex modules depending on temperature. The nonlinear coupled problem of electrothermoviscoelasticity is solved by numerical methods. The influence of boundary conditions and temperature of dissipative heating on the active damping of forced resonant vibrations of circular viscoelastic plates using piezoelectric sensors and actuators is investigated.     

Numerical modeling of the active damping of forced thermomechanical resonance vibrations of viscoelastic shells of revolution with the help of piezoelectric inclusions

We consider the problem of active damping of forced resonance vibrations of viscoelastic shells of revolution with the help of piezoelectric sensors and actuators. Here, the interaction of electromechanical and thermal fields is taken into account. For modeling of vibrations, we use the Kirchhoff–Love hypotheses as well as hypotheses adequate to them and describing the distribution of temperature and electric field quantities. The shell temperature increases as a result of dissipative heating. For the active damping of vibrations, piezoelectric sensors and actuators are used. It is supposed that the electromechanical characteristics of materials depend on the temperature. The solution of this complex nonlinear problem has been obtained by the iterative method and finite element method. We have investigated the influence of temperature of dissipative heating on the efficiency of active damping of vibrations of a viscoelastic cylindrical panel with rigid restraint of its edges.     

Active damping of forced resonance vibrations of an isotropic shallow viscoelastic cylindrical panel under the action of an unknown mechanical load

Using a new approach, we consider the problem of active damping of the forced resonance vibrations of a viscoelastic isotropic cylindrical panel with simply supported edge faces. The mechanical load is assumed to be unknown; it can be found from the experimental data of a sensor. The problem is solved by the Bubnov–Galerkin method. A relation was obtained for the potential difference that has to be supplied to an actuator for the damping of resonance vibrations of the panel. We also study the influence of sizes of sensors and actuators, the dissipative properties of materials, and temperature on the efficiency of active damping of the forced resonance vibrations of a cylindrical panel.     

Influence of physical nonlinearity on resonance vibrations and dissipative heating of inelastic rectangular plates

On the basis of a concept of complex characteristics, we present a statement of nonlinear conjugate problems of forced resonance vibrations and dissipative heating of inelastic plates of physically nonlinear materials, for which the real and imaginary parts of mechanical complex characteristics depend on the amplitude of deformations. Numerical-analytical methods for the solution of the indicated nonlinear problems are presented. An analysis of the influence of physical nonlinearity on the amplitude–frequency and temperature–frequency characteristics and on the coefficient of damping of vibrations of rectangular plates is performed.     

Parametric vibrations of cylindrical shells with a viscoelastic core

The method of calculating the axisymmetric and nonaxisymmetric parametric vibrations of a cylindrical shell bonded to an elastic core [2] is extended to the case of hollow and solid viscoelastic cores by substituting for the material moduli in the equations of motion of the core integral operators with kernels in the form of an exponential and a sum of exponentials. Expressions are given for the reaction of the viscoelastic core, together with the equation of the boundaries of the spectrum of principal regions of dynamic instability. The effect of relaxation time and the long-term modulus of elasticity of the core on the shape and location of the regions of dynamic instability is analyzed.     

Investigation of frequencies of natural vibrations of a transversally isotropic cylindrical panel with a circular hole

We consider the problem of natural vibrations of a hingedly supported transversally isotropic cylindrical panel with a circular hole. The deformation of the shell is described by modified equations of the Timoshenko theory of shells. The numerical solution of the problem is constructed by the indirect method of boundary integral equations based on the sequential representation of Green functions.     

Effect of ultrasonic vibrations on the flow of a viscoelastic liquid

The rheological characteristics of a viscoelastic liquid flow, on which finite-amplitude ultrasonic shear vibrations have been superimposed, are investigated. It is shown that at periodic deformation amplitudes of 5 µ or more there is a thixotropic reduction of the viscosity and elasticity of the system owing to the destruction of certain types of structural bonds. The experimental apparatus is described.Moscow Institute of Chemical Machine Building. Translated from Mekhanika Polimerov, No. 6, pp. 1097–1099, November–December, 1971.     

Determination of the physicomechanical characteristics of viscoelastic materials from the velocity of propagation and damping of longitudinal and transverse vibrations in a thin rod

The author examines the propagation of longitudinal and transverse waves in a thin viscoelastic rod. Relations are obtained for determining the components of the complex moduli and and Poisson's ratio from the propagation velocities and decrements of the longitudinal and transverse vibrations, together with the static and relaxational moduli of elasticity, the relaxation time, and the equivalent viscosity for materials whose law of deformation corresponds to a standard linear solid.Mekhanika Polimerov, Vol. 2, No. 3, pp. 453–460, 1966     

An asymptotic calculation of the vibrations of a viscoelastic cylinder with a slowly varying internal boundary

The vibrational problem is considered for a long hollow viscoelastic cylinder with a variable internal boundary, this being enclosed in an elastic shell. An asymptotic method based on averaging is used to find the resonant vibrations in response to a small uniformly distributed internal pressure that varies periodically with time. The calculation is carried through to quadrature working formulas.Moscow Institute of Electric Machine Construction. Translated from Mekhanika Polimerov, No. 5, pp. 935–939, September–October, 1973.     

Analysis of the static response of cross-ply simply supported plates and shells based on a higher-order theory

The boundary-discontinuous double Fourier series-based solution methodology is used to solve the problem of higher-order shear deformation of cross-ply plates and doubly curved panels, which are characterized by a system of five highly coupled linear partial differential equations with mixed-type simply supported boundary conditions prescribed at all four their edges. The present solution is related to a number of unsolved boundary-value problems and can serve as a tool in particular for early design stages and for benchmark comparisons and verifications of numerical results. The analytical results obtained are compared with finite-element calculations, and a good agreement is found to exist between them.     

Speed-optimized control of heating of a thermoelastic plate with the help of internal heat sources

Speed-optimized control of the heating of a plate with restrictions on the thermal stresses is constructed as a junction of the distribution of internal heat sources by the method of the inverse problem of heat conduction.Translated from Matematicheskie Methody i Fiziko-mekhanicheskie Polya, No. 26, pp. 55–59, 1987.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution

In this paper, we consider a nonlinear viscoelastic wave equation with nonlinear boundary damping and source terms. Under some appropriate assumptions on the relaxation function $g$ and with certain initial data, the global existence of solutions and a general decay for the energy have been established.