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.     

Nonlinear vibrations and damping of fractional viscoelastic rectangular plates

Nonlinear Dynamics - A new model-free robust control scheme for payload swing angle attenuation of two-dimensional crane systems with varying rope length is introduced in this work. The proposed...     

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.     

Numerical investigation of nonlinear vibrations of viscoelastic plates and cylindrical panels in a gas flow

Nonlinear vibrations of viscoelastic elements of aviation structures are studied. A method and an algorithm for the numerical solution of integrodifferential equations are proposed. The critical velocity of the flow past viscoelastic plates is determined. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 48, No. 2, pp. 156–162, March–April, 2007.     

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.     

Some aspects of flutter of cylindrical panels

Summary Pursuing a previous study, the investigation of some aspects of the flutter of weakly curved panels is taken a stage further. A new system of visual presentation of the results evolved for this makes the analog computer study easier and more rewarding. Further, an improvement in the calculating circuit permits continuous variation both of the flutter parameter and of the compression load during calculation.

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Sommario Proseguendo una ricerca già iniziata si approfondiscono alcuni aspetti del flutter dei pannelli a debole curvatura. Per questo è stato messo a punto un nuovo sistema di presentazione visiva dei risultati che ha reso più agevole e profiquo lo studio condotto con il calcolatore analogico; inoltre si è migliorato il circuito di calcolo ottenendo la possibilità di variare in modo continuo, durante il calcolo, oltre che il parametro di flutter, anche il carico di compressione.

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Natural vibrations of a cylindrical shell reinforced with rectangular plates

This paper outlines a technique of determining the natural frequencies and modes of an elastic structure consisting of a cylindrical shell and noncrossing rectangular plates inserted into it. The influence of the position, number, and thickness of the plates on the natural frequencies and modes is analyzed __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 3, pp. 89–96, March 2006.     

Stress state of saturated viscoelastic masses with cylindrical workings

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 25, No. 2, pp. 76–82, February, 1989.     

Postbuckling analysis of imperfect non-linear viscoelastic cylindrical panels

The postbuckling behavior of imperfect cylindrical panels made of a non-linear viscoelastic material is investigated. The material is modeled according to the Schapery representation of non-linear viscoelasticity. Solutions are developed to calculate the growth of the initial imperfection in time by using the Donnell equilibrium equations for geometrically non-linear cylindrical panels. The equations are derived symbolically using Mathematica in the form of a system of first-order non-linear differential equations. A numerical example is presented and discussed.     

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 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     

Vibration analysis of laminated composite cylindrical panels via a meshfree approach

In this paper, the free vibration of laminated composite cylindrical panels is solved by the meshfree (meshless) approach. The reproducing kernel particle approximation is employed to model the two-dimensional displacement functions. The effects of the particle distribution and the size of the domain of influence, on the convergence behavior, are studied. This study examines in detail the effects of different boundary conditions on the frequency characteristics of the cylindrical panels. The effects of the curvature of the cylindrical panels as well as the lamination scheme, on the frequencies of the panels, are also investigated. The present results are verified by comparison against results available in the open literature, with very good agreement being attained.     

Buckling delamination of a rectangular plate containing a rectangular crack and made from elastic and viscoelastic composite materials

This paper studies a three-dimensional buckling delamination problem for a rectangular plate made from elastic and viscoelastic composite material. It is assumed that the plate contains a rectangular band-crack (Case 1) and a rectangular edge-crack (Case 2) and that the edge-surfaces of these cracks have an initial infinitesimal imperfection. The evolution of this initial imperfection with an external compressive loading, acting along the crack (for an elastic composite) or with duration of time (for a viscoelastic composite under fixed external loading) is investigated within the framework of three-dimensional geometrically non-linear field equations of the theory of the viscoelasticity for anisotropic bodies. To determine the values of the critical force or critical time as well as the buckling delamination mode, the initial imperfection criterion is used. The corresponding boundary-value problems are solved by employing boundary form perturbation techniques, Laplace transform and FEM (Finite Element Method). The influence of the materials and/or the geometrical parameters of the plate on the critical values are discussed. In particular, it is established that for the considered change range of the problem parameters, the buckling form depends only on the initial infinitesimal imperfection mode of the crack edges.