Nonlinear damped vibrations of simply-supported rectangular sandwich plates

The nonlinear, forced, damped vibrations of simply-supported rectangular sandwich plates with a viscoelastic core are studied. The general, nonlinear dynamic equations of asymmetrical sandwich plates are derived using the virtual work principle. Damping is taken into account by modelling the viscoelastic core as a Voigt-Kelvin solid. The harmonic balance method is employed for solving the equations of motion. The influence of the thickness of the layers and material properties on the nonlinear response of the plates is studied.     

Nonlinear vibrations of FGM rectangular plates in thermal environments

Geometrically nonlinear vibrations of FGM rectangular plates in thermal environments are investigated via multi-modal energy approach. Both nonlinear first-order shear deformation theory and von Karman theory are used to model simply supported FGM plates with movable edges. Using Lagrange equations of motion, the energy functional is reduced to a system of infinite nonlinear ordinary differential equations with quadratic and cubic nonlinearities. A pseudo-arclength continuation and collocation scheme is used and it is revealed that, in order to obtain the accurate natural frequency in thermal environments, an analysis based on the full nonlinear model is unavoidable since the plate loses its original flat configuration due to thermal loads. The effect of temperature variations as well as volume fraction exponent is discussed and it is illustrated that thermally deformed FGM plates have stronger hardening behaviour; on the other hand, the effect of volume fraction exponent is not significant, but modal interactions may rise in thermally deformed FGM plates that could not be seen in their undeformed isotropic counterparts. Moreover, a bifurcation analysis is carried out using Gear’s backward differentiation formula (BDF); bifurcation diagrams of Poincaré maps and maximum Lyapunov exponents are obtained in order to detect and classify bifurcations and complex nonlinear dynamics.     

Nonlinear vibrations of rectangular plates with linearly varying thickness

Simplified nonlinear governing differential equations proposed by Berger for static cases and extended by Nash and Modeer for dynamic cases are used to analyse the title problem. Steady-state harmonic oscillations are assumed and the time variable is eliminated by a Kantorovich averaging method. The enclosure or comparison theorem of Collatz is then applied to the reduced equations to obtain the upper and lower bounds for the fundamental nonlinear frequency of simply-supported rectangular plates with linearly varying thickness. The fundamental eigenvalues are given for several taper and aspect ratios.Nomenclature a, b dimensions of plates - A _i series coefficients - D Eh ³/12(1– ²) flexural rigidity - D ₀ Eh ₀ ³ /12(1– ²) - E Young's modulus - h thickness, h ₀(1+x) - h ₀ thickness parameter - N _x , N _y stress resultants in the X and Y directions - N (N _x +N _y )/(1+) - P ₁, P ₂, ... parameters - Q ₁, Q ₂, ... parameters - R[X, (A/h ₀)²] bounding function - t time - u, v in-plane displacements - lateral deflections of plate - X=x/a dimensionless co-ordinate - x, y rectangular co-ordinates - y _n (X) series related to - thickness taper ratio - parameter in the neighbourhood of - error-function associated with differential equation - eigenvalue relating to frequency - Poisson's ra-tio - plate material specific weight - (X) function related to plate deflection - (X) admissible functions - circular frequency     

Active damping of the resonant vibrations of a flexible viscoelastic rectangular plate

The active damping of the resonant vibrations of a hinged flexible viscoelastic rectangular plate with distributed piezoelectric sensors and actuators is considered. It is shown that it is possible to considerably decrease the amplitude of resonant vibrations by choosing the appropriate feedback factor. The collective effect of geometrical nonlinearity and dissipative properties of the material on the effectiveness of active damping of the resonance vibrations of rectangular plates with sensors and actuators is analyzed     

Resonant vibrations of a hinged viscoelastic rectangular plate

The paper examines the effect of dissipative heating on the performance of a sensor in a hinged thermoviscoelastic rectangular plate undergoing resonant flexural 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 variational and Bubnov–Galerkin methods. The effect of the dissipative-heating temperature and the dimensions of the sensor on its performance during resonant vibrations of the plate is analyzed.     

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     

The bending stability and vibrations of cantilever rectangular plates

This paper discusses the problems of the bending, stability and vibrations of cantilever rectangular plates by means of the variational method. In the text a good many calculating examples are illustrated.     

Nonlinear buckled states of rectangular plates

A constructive method is developed to establish the existence of buckled states of a thin, flat elastic plate that is rectangular in shape, simply supported along its edges, and subjected to a constant compressive thrust applied normal to its two short edges. Under the assumption that the stress function and the deformation of the plate are described by the nonlinear von Kármán equations, the approach used yields information regarding not only the number of buckled states near an eigenvalue of the linearized problem, but also the continuous dependence of such states on the load parameter and the possible selection of that buckled state “preferred” by the plate. In particular, the methods used provide a rigorous approach to studying the existence of buckled states near the first eigenvalue of the linearized problem (that is, near the “buckling load”) even when the first eigenvalue is not simple.     

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.     

The vibrations of rectangular plates rib-reinforced in one direction

The exact solution of equilibrium equations is found on the assumption that ribs are symmetrical about the middle plane of a plate and work in bending from this plane. The necessary and sufficient conditions of the applicability of the theory of structurally orthotropic shells to the solution of the problem are formulated. This study has been carried out in accordance with Project 614 of the Ukrainian Scientific and Technological Center. S. P. Timoshenko Institute of Mechanics, National Academy of Sciences of Ukraine, Kiev. Translated from Prikladnaya Mekhanika, Vol. 36, No. 3, pp. 90–94, March, 2000.     

Natural vibrations of rectangular plates with a through crack

Summary The paper presents a finite element model of the rectangular plate with a through crack. The crack occurring in the plate is nonpropagating and open. It was assumed that the crack changes only the stiffness of the plate, whereas the mass is unchanged. The method of the formation of the stiffness matrix of a finite element for the rectangular, cracked plate is presented. The effects of the crack location and its length on the changes of the eigenfrequencies of the simply supported and cantilever plate are studied. The results of numerical computations are compared with results of theoretical and experimental data presented in the literature.

---

Eigenschwingungen von rechteckigen Platten mit Transverslriß

Übersicht Der Beitrag präsentiert ein Finite-Element-Modell einer rechteckigen Platte mit Transverslriß. Der Riß pflanzt sich nicht fort und bleibt offen. Es wird angenommen, daß der Riß nur die Steifigkeit der Platte verändert, wogegen die Masse konstant bleibt. Eine Methode zur Aufstellung der Steifigkeitsmatrix für ein finites Element der rechteckigen Platte wird vorgestellt. Der Einfluß der Lage des Risses und der Rißlänge auf die Eigenschwingungen der einfach gestützten und freitragenden Platte wird untersucht. Abschließend erfolgt ein Vergleich der numerischen Berechnungen mit den Ergebnissen von theoretischen und experimentellen Literaturdaten.

     

Nonlinear vibration of viscoelastic laminated composite plates

The dynamic behavior of laminated composite plates undergoing moderately large deflection is investigated by considering the viscoelastic properties of the material. Based on von Karman's nonlinear deformation theory and Boltzmann's superposition principle, nonlinear and hereditary type governing equations are derived through Hamilton's principle. Finite element analysis and the method of multiple scales are applied to examine the effect of large amplitude on the dissipative nature as well as on the natural frequency of viscoelastic laminated plates. Numerical experiments are performed for the nonlinear elastic case and linear viscoelastic case to check the validity of the procedure presented in this paper. Limitations of the method are discussed also. It is shown that the geometric nonlinearity does not affect the dissipative characteristics in the cases that have nonlinearity of perturbed order.