Generalized non-linear free vibrations of prestressed plates and shells

This paper investigates the non-linear free vibration of prestressed plates and shells in a general form. The analysis includes the effects of in-plane inertia. The analysis is based on the non-linear equations of motion and uses a perturbation procedure. No assumption is made a priori for the form of the time or space mode. The boundary conditions are treated in a general manner including boundary conditions where non-linear stress resultants are specified. The method is illustrated by three examples.     

Calculation of vibrations of thin plates by partitioning into elastically connected undeformable units

A mathematical model of a thin plate in the form of a system of elastically connected undeformable rectangular units is proposed and substantiated. With necessary additions, the basic statements of the method can be extended to the case of a thick plate. Novosibirsk State Engineering University, Novosibirsk 630092. Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 39, No. 5, pp. 159–168, September–October, 1998.     

On the non-linear dynamic stability problem for thin elastic plates

In this paper the non-linear response of thin elastic plates under parametric excitation is investigated. A new analytical method is proposed. It gives the possibility to obtain all the characteristic features of the phenomenon considered, which are known from experiments—the existing of beats, their dependence on the excitation parameter, the influence of the initial conditions, the typical character of the vibrations in the different regions. Analog computer studies are carried out, and they show clearly the influence of different parameters on the output of the problem considered.     

On the energy method in the non-linear theory of thin plates

Summary The equilibrium configuration of a thin plate under normal pressure is studied. A variational non-linear treatment of the problem is considered, taking into account the bending stresses and allowing large deflections.Existence, uniqueness, and regularity of solutions are obtained.

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Sommario Si studia il problema non lineare della configurazione di equilibrio di una piastra sottile caricata normalmente nel suo piano medio, tenendo conto delle tensioni di natura flessionale e in un regime di spostamenti moderatamente grandi.Con una tecnica variazionale si studiano l'esistenza, l'unicità, e la regolarità delle soluzioni del problema.

     

An analytical study of the non-linear vibrations of cylindrical shells

The primary resonance response of simply supported circular cylindrical shells is investigated using the perturbation method. Donnell's non-linear shallow-shell theory is used to derive the governing partial differential equations of motion. The Galerkin technique is then employed to transform the equations of motion into a set of temporal ordinary differential equations. Considering only the primary resonance case, the method of multiple scales is used to study the periodic solutions and their stability. The necessary and sufficient conditions for appearance of the so-called companion mode are also discussed. To this end, a range of the possible multi-mode solution is obtained for response and excitation amplitudes and also excitation frequency as a function of damping, geometry and material properties of the shell. Parametric studies are performed to illustrate the effect of different values of thickness, length and material composition on the possibility of the companion mode participation in primary resonance response.     

Photoelastic and strain-gage investigation of penetration of thin plates

An experimental investigation of the central normal impact and penetration of plates of Columbia Resin No. 39, simply supported at the corners, was conducted using both photoelastic and strain-gage techniques. Both conically-pointed and blunt-nosed steel projectiles of 1/4-in. diam were employed at initial velocities ranging from 191 to 272 fps. Sequences of fringe photographs were obtained by repeating shots on different plates, using various predetermined time intervals between impact and photograph. A dynamic value of Young's modulus has been calculated from the velocity of propagation of tension waves, and a strain-optic coefficient of 398 microinches per fringe for the material was determined from a quasi-static test. Both fringe order and strain histories for various impact conditions have been ascertained and are compared.Paper was presented at 1961 SESA Spring Meeting held in Philadelphia, Pa., on May 10–12.     

The effect of uncertain material properties on free vibrations of thin periodic plates

Thin periodic plates with uncertain properties in a periodicity cell are investigated. To describe dynamics of these plates the non-asymptotic tolerance modelling method, cf. Wo?niak and Wierzbicki (Averaging techniques in thermomechanics of composite solids. Wydawnictwo Politechniki Cz?stochowskiej, Cz?stochowa, 15), Wo?niak et al. (eds.) (Thermomechanics of microheterogeneous solids and structures. Tolerance averaging approach. Wydawnictwo Politechniki ?ódzkiej, ?ód?, 16), for those plates is applied. The governing equations of tolerance models based on this method take into account the effect of period lengths on the overall behaviour of the plate. Hence, the additional effects of the periodicity can be analysed, as higher order vibrations. Moreover, properties of the plate in the periodicity cell are determined uncertainly. To analyse an influence of random variables of the properties with a fixed probability distribution on vibrations of the plate the Monte Carlo analysis is applied.     

On the equations of non-linear vibrations

The purpose of this short paper is to show that equations of non-linear vibrations whose linear parts involve two coefficient matrices can be simplified so that the linear portions become uncoupled. Equivalence transformations are utilized in the simplification, which can substantially streamline any subsequent analysis.     

New exact solutions for free vibrations of rectangular thin plates by symplectic dual method

The separation of variables is employed to solve Hamiltonian dual form of eigenvalue problem for transverse free vibrations of thin plates, and formulation of the natural mode in closed form is performed. The closed-form natural mode satisfies the governing equation of the eigenvalue problem of thin plate exactly and is applicable for any types of boundary conditions. With all combinations of simplysupported （S） and clamped （C） boundary conditions applied to the natural mode, the mode shapes are obtained uniquely and two eigenvalue equations are derived with respect to two spatial coordinates, with the aid of which the normal modes and frequencies are solved exactly. It was believed that the exact eigensolutions for cases SSCC, SCCC and CCCC were unable to be obtained, however, they are successfully found in this paper. Comparisons between the present results and the FEM results validate the present exact solutions, which can thus be taken as the benchmark for verifying different approximate approaches.     

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.