Dynamic stability of a viscoelastic plate with concentrated masses

The paper addresses the geometrically nonlinear problem of dynamic stability of a viscoelastic plate with concentrated masses. The Bubnov-Galerkin method based on polynomial approximation is used to reduce the problem to a system of nonlinear Volterra-type integro-differential equations with singular relaxation kernels. This system is solved by numerical method based on quadrature formulas. The critical loads are found and their dependence on the arrangement and number of concentrated masses is studied for a wide range of mechanical and geometrical parameters of the plate. The choice of a relaxation kernel for dynamic problems for viscoelastic thin-walled plate-like structures is justified. Results produced by different theories are compared __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 109–118, February 2008.     

Nonlinear flutter of viscoelastic rectangular plates and cylindrical panels of a composite with a concentrated masses

The problem of flutter of viscoelastic rectangular plates and cylindrical panels with concentrated masses is studied in a geometrically nonlinear formulation. In the equation of motion of the plate and panel, the effect of concentrated masses is accounted for using the δ-Dirac function. The problem is reduced to a system of nonlinear ordinary integrodifferential equations by using the Bubnov-Galerkin method. The resulting system with a weakly singular Koltunov-Rzhanitsyn kernel is solved by employing a numerical method based on quadrature formulas. The behavior of viscoelastic rectangular plates and cylindrical panels is studied and the critical flow velocities are determined for real composite materials over wide ranges of physicomechanical and geometrical parameters.     

Dynamic Stability of Viscoelastic Plates under Increasing Compressing Loads

The dynamic stability problem of viscoelastic orthotropic and isotropic plates is considered in a geometrically nonlinear formulation using the generalized Timoshenko theory. The problem is solved by the Bubnov-Galerkin procedure combined with a numerical method based on quadrature formulas. The effect of viscoelastic and inhomogeneous properties of the material on the dynamic stability of a plate is discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 2, pp. 165–175, March–April, 2006.     

Nonlinear vibrations of viscoelastic cylindrical shells taking into account shear deformation and rotatory inertia

The vibration problem of a viscoelastic cylindrical shell is studied in a geometrically nonlinear formulation using the refined Timoshenko theory. The problem is solved by the Bubnov–Galerkin procedure combined with a numerical method based on quadrature formulas. The choice of relaxation kernels is substantiated for solving dynamic problems of viscoelastic systems. The numerical convergence of the Bubnov–Galerkin procedure is examined. The effect of viscoelastic properties of the material on the response of the cylindrical shell is discussed. The results obtained by various theories are compared.     

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.     

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.     

Nonlinear vibrations and stability of elastic and viscoelastic systems under random stationary loads

The paper deals with numerical analysis of nonlinear vibrations of viscoelastic systems under a stochastic action in the form of a Gaussian stationary process with rational spectral density. The analysis is based on numerical simulation of the original stationary process, numerical solution of the differential equations describing the motion of the system, and computation of the maximum Lyapunov exponent if the stability of this motion is studied. An example of a plate subjected to a random stationary load applied in its plane is used to consider specific issues concerning the application of the proposed method and the peculiarities of the behavior of geometrically nonlinear elastic and viscoelastic stochastic systems. Special attention is paid to the interaction of a deterministic periodic action and a stochastic action from the viewpoint of stability of the system motion. It is shown that in some cases imposing a “colored” noise may stabilize an unstable system subjected to a periodic load.     

Stability of a viscoelastic plate in fluid flow

The stability of an infinite viscoelastic plate on an elastic foundation in a viscous incompressible flow is studied. The Navier-Stokes system is linearized for an exponential velocity profile. The problem is reduced by a Fourier-Laplace transform to a system of ordinary differential equations, whose solution is found in the form of convergent series. The roots of the dispersion relation that characterize the stability of the system are found numerically. The effect of the viscosities of the fluid and the plate on the stability of the waves propagating upstream and downstream is studied. The results are compared with available data on the stability of a viscoelastic plate in an ideal fluid flow. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 66–74, July–August, 2006.     

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.     

Nonlinear vibration analysis of viscoelastic plates based on a refined Timoshenko theory

Nonlinear vibrations of viscoelastic orthotropic and isotropic shells are mathematically modeled using a geometrically nonlinear Timoshenko theory. Nonlinear problems are solved by using the Bubnov-Galerkin method and a numerical method based on quadrature formulas. Results obtained from different theories are compared and analyzed. For each problem, the Bubnov-Galerkin method is tested for convergence. The influence of the viscoelasticity and inhomogeneity of materials on the vibrations of plates is demonstrated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 5, pp. 120–131, May 2006.     

Resonant vibrations of a clamped viscoelastic rectangular plate

The paper examines the effect of dissipative heating on the performance of a sensor in a viscoelastic rectangular plate undergoing resonant vibrations. The thermoviscoelastic behavior of materials is described using the concept of complex characteristics. The coupling of the electromechanical and thermal fields is taken into account. The nonlinear problem is solved by the Bubnov–Galerkin method. The effect of the mechanical boundary conditions and dissipative-heating temperature on the performance of the sensors is analyzed     

Computing flow-induced stresses of injection molding based on the Phan–Thien–Tanner model

A numerical approach is introduced to solve the viscoelastic flow problem of filling and post-filling in injection molding. The governing equations are in terms of compressible, non-isothermal fluid, and the constitutive equation is based on the Phan–Thien–Tanner model. By introducing some hypotheses according to the characteristics of injection molding, a quasi-Poisson type equation about pressure is derived with part integration. Besides, an analytical form of flow-induced stress is also generalized by using the undermined coefficient method. The conventional Galerkin approach is employed to solve the derived pressure equation, and the ‘upwind’ difference scheme is used to discrete the energy equation. Coupling is achieved between velocity and stress by super relax iteration method. The flow in the test mold is investigated by comparing the numerical results and photoelastic photos for polystyrene, showing flow-induced stresses are closely related to melt temperatures. The filling of a two-cavity box is also studied to investigate the viscoelastic effects on real injection molding.     

Parameter determination and response analysis of viscoelastic material

The parameter determination of viscoelastic material is a multi-variable, multi-aim nonlinear optimization problem, which made the optimization process very complicated. In this paper a hybrid optimal algorithm was proposed to determine the viscoelastic parameters in the constitutive relation according to the experimentally obtained mechanical properties. This algorithm merges the Broydon–Fletcher–Goldfarb–Shanno search into a genetic algorithm framework as a basic operator in order to enhance the local search capability. The proposed hybrid algorithm not only can reduce the iterative times greatly but can abolish the limitation of initial parameter values. Nonlinear material characteristic curve-fitting was carried out using the proposed algorithm and other existing approaches. And the comparison results show this algorithm is accurate and effective. The numerical simulation and experimental study of viscoelastic cantilever beam also indicates that the finite element formulation and the calculative viscoelastic model parameters are reliable. The proposed optimization method can be extended to further complex parameter estimation researches.     

Flexural vibrations and vibrational heating of a ring plate with thin piezoceramic pads under single-frequency electromechanical loading

The paper deals with the coupled problem of flexural vibrations and dissipative heating of a viscoelastic ring plate with piezoceramic actuators under monoharmonic electromechanical loading. The temperature dependence of the complex characteristics of passive and piezoactive materials is taken into account. The coupled nonlinear problem of thermoviscoelasticity is solved by an iterative method. At each iteration, orthogonal discretization is used to integrate the equations of elasticity and an explicit finite-difference scheme is used to solve the heat-conduction equation with a nonlinear heat source. The effect of the dissipative heating temperature, boundary conditions, and the thickness and area of the actuator on the active damping of the forced vibrations of the plate under uniform transverse harmonic pressure is examined __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 99–108, February 2008.     

Stress-strain state of an anisotropic plate with an elliptic hole and thin rigid inclusions

The stress-strain state of an anisotropic plate containing an elliptic hole and thin, absolutely rigid, curvilinear inclusions is studied. General integral representations of the solution of the problem are constructed that satisfy automatically the boundary conditions on the elliptic-hole contour and at infinity. The unknown density functions appearing in the potential representations of the solution are determined from the boundary conditions at the rigid inclusion contours. The problem is reduced to a system of singular integral equations which is solved by a numerical method. The effects of the material anisotropy, the degree of ellipticity of the elliptic hole, and the geometry of the rigid inclusions on the stress concentration in the plate are studied. The numerical results obtained are compared with existing analytical solutions. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 4, pp. 173–180, July–August, 2007.     

Nonlinear vibration analysis of isotropic cantilever plate with viscoelastic laminate

The nonlinear vibration of an isotropic cantilever plate with viscoelastic laminate is investigated in this article. Based on the Von Karman’s nonlinear geometry and using the methods of multiple scales and finite difference, the dimensionless nonlinear equations of motion are analyzed and solved. The solvability condition of nonlinear equations is obtained by eliminating secular terms and, finally, nonlinear natural frequencies and mode-shapes are obtained. Knowing that the linear vibration of this type of plate does not have exact solution, Ritz method is employed to obtain semi-analytical nonlinear mode-shapes of transverse vibration of this plate. Airy stress function and Galerkin method are employed to reduce nonlinear PDEs into an ODE of duffing type. Stability of plate and chaotic behavior are investigated by Runge–Kutta method. Poincare section diagrams are in good agreement with results of Lyapunov criteria.     

变温场中具损伤粘弹性矩形板的非线性动力响应分析

基于热粘弹性理论、Von Karman板理论和连续损伤力学,导出了二维状态下各向同性材料的变温粘弹性本构方程,建立了含损伤效应的各向同性粘弹性矩形板在变温场中的非线性运动控制方程,且应用有限差分法对问题进行求解.算例中,讨论了损伤演化及温度场等因素对粘弹性矩形板非线性动力学行为的影响,得出一些有意义的结论.     

Natural vibration of a rectangular plate of composite material with periodically bent structures

The natural vibration of a thin rectangular plate made of a composite material with periodically bent structures is studied. The problem is examined by the finite-element method using the variation principle. The effect of the bending parameters on the natural frequency of the plate is analyzed using the numerical results obtained. Institute of Mathematics and Mechanics, Academy of Sciences of Azerbaijan, Baku, Azerbaijan. Translated from Prikladnaya Mekhanika, Vol. 35, No. 10, pp. 68–73, October, 1999.