Vibrations of annular plates including the effects of rotatory inertia and transverse shear deformation

An investigation of the natural vibrations of isotropic annular plates of uniform thickness has been made by considering the effects of rotatory inertia and shear deformation. The frequency determinantal equations are derived in explicit form for nine sets of common boundary conditions. Numerical results for the frequency parameters of annular plates having various thickness ratios and inner to outer radii ratios have been obtained. The results are compared with those given by the classical plate theory wherever possible. Among the effects of transverse shear deformation and rotatory inertia, the effect of shear deformation has been found to be more prominent.     

Shear and rotatory inertia effects on large amplitude vibration of skew plates

The large amplitude free flexural vibration of elastic, isotropic skew plates is investigated, the effects of transverse shear and rotatory inertia being included. By use of Galerkin's method and the extended Berger approximation, solutions are obtained on the basis of an assumed vibration mode. The non-linear period vs. amplitude behavior is of the hardening type and the non-linear period is found to increase when the effects of transverse shear and rotatory inertia are considered in the analysis. The influence of these effects on aspect ratios and skew angles of thin and moderately thick skew plates is investigated both at small and large amplitudes.     

Vibrations of initially imperfect circular plates including the shear and rotatory inertia effects

An analysis of the free flexural vibrations of elastic circular plates with initial imperfections is presented. The analysis includes the effects of transverse shear and rotatory inertia. The vibration amplitudes are assumed to be large, and two non-linear differential equations are obtained for free vibration of the plate and solved numerically. The period of the plate has been calculated as a function of the initial amplitude for four typical supporting conditions.     

Non-linear axisymmetric vibration of orthotropic thin circular plates on elastic foundations

This study deals with the large amplitude axisymmetric free vibrations of cylindrically orthotropic thin circular plates resting on elastic foundations. Geometric non-linearity due to moderately large deflections has been included. Movable and immovable simply supported plates and immovable clamped plates resting on Winkler, Pasternak and non-linear Winkler foundations have been considered. The von Kármán type governing equations have been employed. Harmonic vibrations are assumed and the time t is eliminated by the Kantorovich averaging method. An orthogonal point collocation method is used for spatial discretization. Numerical results are presented for the linear natural frequency of the first axisymmetric mode and for the ratio of the non-linear period to the linear period of natural vibration. The effects of foundation parameters, the orthotropic parameter and the edge conditions on the non-linear vibration behaviour have been investigated.     

Large amplitude vibration of an initially stressed moderately thick plate

Non-linear equations of motion for a transversely isotropic moderately thick plate in a general state of non-uniform initial stress where the effects of transverse shear and rotary inertia are included are derived. The large amplitude flexural vibration of a simply supported rectangular moderately thick plate subjected to initial stress is investigated. The initial stress is taken to be a combination of a pure bending stress plus an extensional stress in the plane of the plate. These equations are used to solve the vibrations problem by the Galerkin method. The effects of various parameters on the non-linear vibration frequencies are studied.     

Analytical approach for free vibration analysis of two-layer Timoshenko beams with interlayer slip

In this paper, an analytical procedure for free vibrations of shear-deformable two-layer beams with interlayer slip is developed. The effect of transverse shear flexibility of two layers is taken into account in a general way by assuming that each layer behaves as a Timoshenko beam element. Therefore, the layers have independent shear strains that depend indeed on their own shear modulus. This is the main improvement of the proposed model compared to existing models where the transverse shear flexibility is ignored or taken into account in a simplified way in which the shear strains of both layers are assumed to be equal whatever the shear modulus of the layers. In the proposed model, the two layers are connected continuously and the partial interaction is considered by assuming a continuous relationship between the interface shear flow and the corresponding slip. Based on these key assumptions, the governing differential equation of the problem is derived using Hamilton's principle and is analytically solved. The solutions for the eigenfrequencies and eigenmodes of four single span two-layer beams with classical Euler boundary conditions, i.e. pinned-pinned, clamped-clamped, clamped-pinned and clamped-free, are presented. Next, some numerical applications dealing with these four beams are carried out in order to compare the eigenfrequencies obtained with the proposed model against two existing models which consider different kinematic assumptions. Finally, a parametric study is conducted with the aim to investigate the influence of varying material and geometric parameters on the eigenfrequencies, such as shear stiffness of the connectors, span-to-depth ratios, flexural-to-shear moduli ratios and layer shear moduli ratios.     

Free vibration analysis of plates by using a four-node finite element formulated with assumed natural transverse shear strain

A study on the free vibration analysis of plates is described in this paper. In order to investigate vibrational characteristics of plates, a four-node plate element is developed by using the assumed natural strains on the basis of Reissner-Mindlin (RM) assumptions which allows us to consider the shear deformation and rotatory inertia effect. All terms related to the plate finite element formulation are consistently defined in the natural domain. Assumed natural strains are derived to alleviate the locking phenomena inherited in the RM plate elements. In particular, the explicit expression of assumed natural transverse shear strain is described in this paper. The natural constitutive equation is used in conjunction with the natural strain terms. Several numerical examples are carried out and their results are then compared with the existing reference solutions.     

A second order beam theory

A second order beam theory which takes into account shear curvature, transverse direct stresses and rotatory inertia is presented. The governing differential equation is similar in form to the Timoshenko beam equation but contains two coefficients, one of which depends on cross-sectional warping just as does Cowper's expression while the second, although similar in form, also includes terms dependent on the transverse direct stresses. Comparison is made with exact and other approximate theories for particular cases.     

THE EFFECTS OF LARGE VIBRATION AMPLITUDES ON THE MODE SHAPES AND NATURAL FREQUENCIES OF THIN ELASTIC SHELLS. PART II: A NEW APPROACH FOR FREE TRANSVERSE CONSTRAINED VIBRATION OF CYLINDRICAL SHELLS

The non-linear dynamic behaviour of infinitely long circular cylindrical shells in the case of plane strains is examined and results are compared with previous studies. A theoretical model based on Hamilton's principle and spectral analysis previously developed for non-linear vibration of thin straight structures (beams and plates) is extended here to shell-type structures, reducing the large-amplitude free vibration problem to the solution of a set of non-linear algebraic equations. In the present work, the transverse displacement is assumed to be harmonic and is expanded in the form of a finite series of functions corresponding to the constrained vibrations, which exclude the axisymmetric displacements. The non-linear strain energy is expressed by taking into account the non-linear terms due to the considerable stretching of the shell middle surface induced by large deflections. It has been shown that the model presented here gives new results for infinitely long circular cylindrical shells and can lead to a good approximation for determining the fundamental longitudinal mode shape and the associated higher circumferencial mode shapes (n>3) of simply supported circular cylindrical shells of finite length. The non-linear results at small vibration amplitudes are compared with linear experimental and theoretical results obtained by several authors for simply supported shells. Numerical results (non-linear frequencies, vibration amplitudes and basic function contributions) of infinite shells associated to the first four mode shapes of free vibrations, are obtained, using a multi-mode approach and are summarized in tables. Good agreement is found with results from previous studies for both small and large amplitudes of vibration. The non-linear mode shapes are plotted and discussed for different thickness to radius ratios. The distributions of the bending stresses associated with the mode shapes are given and compared with those obtained via the linear theory.     

The influence of rotatory inertia,tip mass,and damping on the stability of a cantilever beam on an elastic foundation

The stability of a uniform viscoelastic cantilever resting on an elastic foundation, carrying a tip mass, and subjected to a follower force at its free end is investigated. The effects of the rotatory inertia of the beam, the transverse and rotatory inertias of the tip mass, and the foundation modulus, which characterizes a Winkler type of elastic foundation, are included in the partial differential equation of motion and boundary conditions, and the influence of these quantities on the value of the critical flutter load parameter Q_f is sought. The exact forms of the fundamental frequency equations are derived for the cases of a viscoelastic and a purely elastic beam, and these equations are solved numerically for Q_f These numerical results reveal that Q_f depends strongly upon the foundation modulus for the cantilever carrying a tip mass or possessing rather small internal damping. In the absence of damping and a tip mass, the value of Q_f, computed upon the inclusion of the rotatory inertia of the beam in the formulation of the equation of motion, is decreased slightly and continues to decrease in essentially a linear manner as the value of the foundation modulus parameter κ is decreased. Moreover, when the effect of very small internal damping is included, the value of Q_f computed when the rotatory inertia of the beam is neglected increases slowly in an essentially linear fashion as x increases, whereas, when the effect of rotatory inertia is retained, the value of Q_f decreases as κ is increased. Additional numerical results are reported graphically.     

VIBRATION ANALYSIS OF A DAMPED ARCH USING AN ITERATIVE LAMINATE MODEL

A new model is presented for the dynamic analysis of a laminated circular ring segment. The differential equations which govern the free vibrations of a circular ring segment and the associated boundary conditions are derived by Hamilton's principle having consideration for the bending and shear deformation of all layers. The author uses a new iterative process to successively refine the stress/strain field in the sandwich arch. The model includes the effects of transverse shear and rotatory inertia. The iterative model is used to predict the modal frequencies and damping of simply-supported sandwich circular arch. The solutions for a three-layer circular arch are compared with a three-layer approximate model.     

Vibration and damping analysis of multilayered rectangular plates with constrained viscoelastic layers

Governing equations of motion for vibrations of a general multilayered plate consisting of an arbitrary number of alternate stiff and soft layers of orthotropic materials are derived by using variational principles. Extension, bending and in-plane shear deformations in stiff layers and only transverse shear deformations in soft layers are considered as in conventional sandwich structural analysis. In addition to transverse inertia, longitudinal translatory and rotary inertias are included, as such analysis gives higher order modes of vibration and leads to accurate results for relatively thick plates. Vibration and damping analysis of rectangular simply supported plates consisting of alternate elastic and viscoelastic layers is carried out by taking a series solution and applying the correspondence principle of linear viscoelasticity. The damping effectiveness, in term of the system loss factor, for all families of modes for three-, five- and seven-layered plates is evaluated and its variations with geometrical and material property parameters are investigated.     

Vibration analysis of laminated cross-ply oval cylindrical shells

Here, free vibrations and transient dynamic response analyses of laminated cross-ply oval cylindrical shells are carried out. The formulation is based on higher order theory that accounts for the transverse shear and the transverse normal deformations, and includes zig-zag variation in the in-plane displacements across the thickness of the multi-layered shells. The contributions of inertia effect due to in-plane and rotary motions, and the higher order function arising from the assumed displacement models are included. The governing equations obtained using Lagrangian equations of motion are solved through finite element approach. A detailed parametric study is conducted to bring out the influence of different shell geometry, ovality parameter, lay-up and loading environment on the vibration characteristics related to different modes of vibrations of oval shell.     

Refined theory for flexural motions of isotropic elastic plates

A technical theory for the flexural motions of isotropic elastic plates has been developed, taking into account the influence of transverse normal strain and transverse normal stress, together with rotatory inertia and transverse shear. The theory is tested by studying the classical wave propagation problem and results indicate the influence of the transverse normal strain on the wave speed at large values of $\text{h}\text{λ}$. In addition, a constant magnitude for the shear coefficient $\text{κ}{}^2\text{=}\text{5}\text{6}$ is obtained, which is in contrast to an undetermined coefficient form in previous flexural motion formulations but consistent with the value obtained in the Reissner static technical theory of plate bending.     

Some problems of analysis and optimization of plates and shells

The classical optimization problems of plates and shells to satisfy a priori given geometry and dynamical characteristics are considered. Orthotropic plates and shells with variable thickness and low transverse stiffness are analyzed. First, some useful theorems and their proofs are given. Then the finite approximation of the problem related to optimization of free vibrations of shells with transverse deformation and rotary inertia is discussed. The varational iteration (MVI) and Bubnov-Galerkin (MB) methods are applied, and their convergence and suitability for application to plates and shells analysis are discussed and numerically evaluated.     

Forced axisymmetric motion of circular,viscoelastic plates

Williams' method for forced motion of elastic systems is applied to circular, viscoelastic plates where the effects of rotatory inertia, transverse shear and time-dependent boundary conditions are included. The viscoelastic material is assumed to have a constant Poisson's ratio. A particular problem is solved for a symmetrically loaded, completely free plate. The material used is vulcanized rubber where the viscoelastic behavior in shear is used in specifying the material parameters of a three-element solid.     

Non-linear axisymmetric transient analysis of an orthotropic thin circular plate with an elastically restrained edge

Results are presented for the geometrically non-linear axisymmetric transient elastic stress and deflection responses of a cylindrically orthotropic thin circular plate with an elastically restrained edge, including both rotational and in-plane displacements. In the analysis the dynamic analogue of the von Kárman governing differential equations in terms of the normal displacement w and the stress function ψ are employed. The displacement w and stress function ψ are expanded in finite power series. The orthogonal point collocation method in the space domain and the Newmark-β scheme in the time domain are used. Four types of uniformly distributed transient loadings have been considered: step function, sinusoidal and N-shaped pulses, and exponentially decaying loads. The influence of the orthotropic parameter β and the elastic rotational and in-plane edge restraint parameters (K_b, K_i) on the large amplitude response has been investigated. The effect of a prescribed in-plane displacement on the non-linear transient response has also been studied.     

The dynamic transient analysis of axisymmetric circular plates by the finite element method

The linear elastic, dynamic transient, analysis of some circular plate bending problems is considered by using axisymmetric, parabolic isoparametric, elements with an explicit time marching scheme. The effects of rotatory inertia and transverse shear deformation are included. A special mass lumping scheme and the use of a reduced integration technique allow the treatment of thin as well as thick plates. Several numerical examples are presented and compared with results from other sources.     

ASYMMETRIC NON-LINEAR FORCED VIBRATIONS OF FREE-EDGE CIRCULAR PLATES. PART 1: THEORY

In this article, a detailed study of the forced asymmetric non-linear vibrations of circular plates with a free edge is presented. The dynamic analogue of the von Kàrmàn equations is used to establish the governing equations. The plate displacement at a given point is expanded on the linear natural modes. The forcing is harmonic, with a frequency close to the natural frequency ω_kn of one asymmetric mode of the plate. Thus, the vibration is governed by the two degenerated modes corresponding to ω_kn, which are one-to-one internally resonant. An approximate analytical solution, using the method of multiple scales, is presented. Attention is focused on the case where one configuration which is not directly excited by the load gets energy through non-linear coupling with the other configuration. Slight imperfections of the plate are taken into account. Experimental validations are presented in the second part of this paper.     

Accurate free vibration analysis of thick laminated circular plates with attached rigid core

This paper deals with the free vibration behavior of laminated transversely isotropic circular plates with axisymmetric rigid core attached at the center. The governing equations of motion are obtained based on Mindlin's first-order shear deformation plate theory. Two possible categories of vibration modes related to up-down translation of the core and wobbly rotation of the core about a diameter are studied. Accurate natural frequencies hitherto not reported in the literature are presented for a wide range of thickness-to-radius ratio, inner-to-outer radius ratio, mass and moment of inertia ratios of the core and various boundary conditions at the outer edge of the plate. Numerical results are compared with those of a three-dimensional finite element method (3-D FEM) to demonstrate the high accuracy and reliability of the current analysis.