Natural vibration of rectangular plates with an internal line hinge using the first order shear deformation plate theory

This paper presents exact solutions for vibration of rectangular plates with an internal line hinge. The rectangular plate is simply supported on two parallel edges and the remaining two edges may take any combination of support conditions. The line hinge is perpendicular to the two simply supported parallel edges. The Lévy type solution method and the state-space technique are employed in connection with the first order shear deformation plate theory (FSDT) to study natural vibration of rectangular plates with an internal line hinge. In particular, exact vibration frequencies are obtained for rectangular plates of different aspect ratios and edge support conditions. The influence of the internal line hinge on the vibration behavior of rectangular plates is studied.     

Large amplitude flexural vibration of eccentrically stiffened plates

The large amplitude free flexural vibrations of thin, orthotropic, eccentrically and lightly stiffened elastic rectangular plates are investigated. Clamped boundary conditions with movable in-plane edge conditions are assumed. A simple modal form of one-term transverse displacement is used and in-plane displacements are made to satisfy the in-plane equilibrium equations. By using Lagrange's equation, the modal equations for the nonlinear free vibration of stiffened plates are obtained for the cases when the stiffeners are assumed to be smeared out over the entire surface of the plate, and when the stiffeners are located at finite intervals. Numerical results are obtained for various possibilities of stiffening and for different aspect ratios of the plate. By particularizing the problem to different known cases, the results obtained here are compared with available analytical and experimental results, and the agreement is good.     

IMPROVEMENT OF THE SEMI-ANALYTICAL METHOD, FOR DETERMINING THE GEOMETRICALLY NON-LINEAR RESPONSE OF THIN STRAIGHT STRUCTURES: PART II—FIRST AND SECOND NON-LINEAR MODE SHAPES OF FULLY CLAMPED RECTANGULAR PLATES

In a previous series of papers, a semi-analytical model based on Hamilton's principle and spectral analysis has been developed for geometrically non-linear free vibrations occurring at large displacement amplitudes of clamped-clamped beams and fully clamped rectangular homogeneous and composite plates. In Part I of this series of papers, concerned with geometrically non-linear free and forced vibrations of various beams, a practical simple “multi-mode theory”, based on the linearization of the non-linear algebraic equations, written in the modal basis, in the neighbourhood of each resonance has been developed. Simple explicit formulae, ready and easy to use for analytical or engineering purposes have been derived, which allows direct calculation of the basic function contributions to the first three non-linear mode shapes of the beams considered. Also, various possible truncations of the series expansion defining the first non-linear mode shape have been considered and compared with the complete solution, which showed that an increasing number of basic functions has to be used, corresponding to increasingly sized intervals of vibration amplitudes; starting from use of only one function, i.e., the first linear mode shape, corresponding to very small amplitudes, for which the linear theory is still valid, and ending by the complete series, involving six functions, corresponding to maximum vibration amplitudes at the beam middle point up to once the beam thickness. For higher amplitudes, a complementary second formulation has been developed, leading to reproduction of the known results via the solution of reduced linear systems of five equations and five unknowns. The purpose of this paper is to extend and adapt the approach described above to the geometrically non-linear free vibration of fully clamped rectangular plates in order to allow direct and easy calculation of the first, second and higher non-linear fully clamped rectangular plate mode shapes, with their associated non-linear frequencies and non-linear bending stress patterns. Also, numerical results corresponding to the first and second non-linear modes shapes of fully clamped rectangular plates with an aspect ratio α=0·6 are presented. Data concerning the higher non-linear modes, the aspect ratio effect, and the forced vibration case will be presented later.     

Nonlinear dynamics of harmonically excited circular cylindrical shells containing fluid flow

In the present study, the geometrically nonlinear vibrations of circular cylindrical shells, subjected to internal fluid flow and to a radial harmonic excitation in the spectral neighbourhood of one of the lowest frequency modes, are investigated for different flow velocities. The shell is modelled by Donnell's nonlinear shell theory, retaining in-plane inertia and geometric imperfections; the fluid is modelled as a potential flow with the addition of unsteady viscous terms obtained by using the time-averaged Navier-Stokes equations. A harmonic concentrated force is applied at mid-length of the shell, acting in the radial direction. The shell is considered to be immersed in an external confined quiescent liquid and to contain a fluid flow, in order to reproduce conditions in previous water-tunnel experiments. For the same reason, complex boundary conditions are applied at the shell ends simulating conditions intermediate between clamped and simply supported ends. Numerical results obtained by using pseudo-arclength continuation methods and bifurcation analysis show the nonlinear response at different flow velocities for (i) a fixed excitation amplitude and variable excitation frequency, and (ii) fixed excitation frequency by varying the excitation amplitude. Bifurcation diagrams of Poincaré maps obtained from direct time integration are presented, as well as the maximum Lyapunov exponent, in order to classify the system dynamics. In particular, periodic, quasi-periodic, sub-harmonic and chaotic responses have been detected. The full spectrum of the Lyapunov exponents and the Lyapunov dimension have been calculated for the chaotic response; they reveal the occurrence of large-dimension hyperchaos.     

On the dynamics and optimization of a non-smooth bistable oscillator – Application to energy harvesting

Bistable nonlinear oscillators can transform slow sinusoidal excitations into higher frequency periodic or quasi-periodic oscillations. This behaviour can be exploited to efficiently convert mechanical oscillations into electrical power, but being nonlinear, their dynamical behaviour is relatively complicated. In order to better understand the dynamics of bistable oscillators, an approximate bilinear analytical model, which is valid for narrow potential barriers, is developed. This model is expanded to the case of wider potential with experimental verification. Indeed, the model is verified by numerical simulations and a suitable Poincaré section that the analytical model captures most of bifurcations for large amplitude vibrations and can be used to optimize the harvested power of such devices. The method of Shaw and Holmes [1] is enhanced by exploiting symmetry to obtain closed form expressions of the Poincaré section and mapping.     

Experiments and simulations for large-amplitude vibrations of rectangular plates carrying concentrated masses

Large-amplitude (geometrically nonlinear) forced vibrations of a stainless-steel thin rectangular plate carrying different concentrated masses are experimentally studied. The experimental boundary conditions are close to those of a clamped plate. The plate is vertically and horizontally tested in order to investigate the gravity effect. Harmonic excitation is applied by using electrodynamic exciter and the plate vibration is measured by using a laser Doppler vibrometer with displacement decoder. The harmonic excitation is controlled in closed-loop in order to keep constant the desired force and is increased (or decreased) by very small discrete steps. Numerical simulations on reduced-order models, obtained by using Von Kármán nonlinear plate theory and global discretization, are also carried out and compared to experiments in order to better understand the system. Results show that concentrated masses have no effect on the trend of nonlinearity of the vertical plate, while they play a role in case of horizontal plate due to the static flexural deflection caused by gravity, which reduces the hardening-type nonlinearity. Initial geometric imperfection (deviation from flat surface in vertical position) of the plate is measured and taken into account; it plays a significant role.     

An approximate solution for the bending vibrations of a combination of rectangular thin plates

The appropriate method often used for calculating the bending vibration of a single rectangular plate is extended to calculate the bending vibrations of a global system of combinations of rectangular plates with elastically supported and damped non-coupled edges. Two examples, a series of T-combinations and an L-combination of rectangular thin isotropic plates, are considered and the input and transfer mobilities due to point excitation derived. Numerical results are presented for the case of combinations of concrete plates and the effects of varying the material damping of the plates and edge damping are investigated.The eigenfrequencies of an L-combination of plates with one plate of very high bending stiffness are calculated and results compare well with the eigenfrequencies of a single plate calculated by means of the classical Ritz-Rayleigh method.     

Geometrically nonlinear dynamic response of stiffened plates with moving boundary conditions

An approach is presented to investigate the nonlinear vibration of stiffened plates. A stiffened plate is divided into one plate and some stiffeners, with the plate considered to be geometrically nonlinear, and the stiffeners taken as Euler beams. Lagrange equation and modal superposition method are used to derive the dynamic equilibrium equations of the stiffened plate according to energy of the system. Besides, the effect caused by boundary movement is transformed into equivalent excitations. The first approximation solution of the non-resonance is obtained by means of the method of multiple scales. The primary parametric resonance and primary resonance of the stiffened plate are studied by using the same method. The accuracy of the method is validated by comparing the results with those of finite element analysis via ANSYS. Numerical examples for different stiffened plates are presented to discuss the steady response of the non-resonance and the amplitude-frequency relationship of the primary parametric resonance and primary resonance. In addition, the analysis on how the damping coefficients and the transverse excitations influence amplitude-frequency curves is also carried out. Some nonlinear vibration characteristics of stiffened plates are obtained, which are useful for engineering design.     

Large amplitude vibration of an initially stressed cross ply laminated plates

In this paper, nonlinear equations of large amplitude vibration for a laminated plate in a general state of nonuniform initial stress are derived. The equations include the effects of transverse shear and rotary inertia. Using these derived governing equations, the large amplitude vibration behaviour of an initially stressed cross-ply laminated plate is studied. The initial stress is taken to be a combination of pure bending stress plus an extensional stress in the plane of the plate. The Galerkin method is used to reduce the governing nonlinear partial differential equations to ordinary nonlinear differential equations and the Runge-Kutta method is used to obtain the nonlinear to linear frequencies. The frequency responses of nonlinear vibration are sensitive of the vibration amplitude, aspect ratio, thickness ratio, modulus ratio, stack sequence, layer number and state of initial stresses. The effects of various parameters on the large amplitude free vibrations are presented.     

Periodic geometrically nonlinear free vibrations of circular plates

The geometrically nonlinear free vibrations of thin isotropic circular plates are investigated using a multi-degree-of-freedom model, which is based on thin plate theory and on Von Kármán's nonlinear strain-displacement relations. The middle plane in-plane displacements are included in the formulation and the common axisymmetry restriction is not imposed. The equations of motion are derived by the principle of the virtual work and an approximated model is achieved by assuming that the in-plane and transverse displacement fields are given by weighted series of spatial functions. These spatial functions are based on hierarchical sets of polynomials, which have been successfully used in p-version finite elements for beams and rectangular plates, and on trigonometric functions. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. Convergence with the number of shape functions and of harmonics is analysed. The numerical results obtained are presented and compared with available published results; it is shown that the hierarchical sets of functions provide good results with a small number of degrees of freedom. Internal resonances are found and the ensuing multimodal oscillations are described.     

Suppression of the primary resonance vibrations of a forced nonlinear system using a dynamic vibration absorber

In a single degree-of-freedom weakly nonlinear oscillator subjected to periodic external excitation, a small-amplitude excitation may produce a relatively large-amplitude response under primary resonance conditions. Jump and hysteresis phenomena that result from saddle-node bifurcations may occur in the steady-state response of the forced nonlinear oscillator. A simple mass-spring-damper vibration absorber is thus employed to suppress the nonlinear vibrations of the forced nonlinear oscillator for the primary resonance conditions. The values of the spring stiffness and mass of the vibration absorber are significantly lower than their counterpart of the forced nonlinear oscillator. Vibrational energy of the forced nonlinear oscillator is transferred to the attached light mass through linked spring and damper. As a result, the nonlinear vibrations of the forced oscillator are greatly reduced and the vibrations of the absorber are significant. The method of multiple scales is used to obtain the averaged equations that determine the amplitude and phases of the first-order approximate solutions to primary resonance vibrations of the forced nonlinear oscillator. Illustrative examples are given to show the effectiveness of the dynamic vibration absorber for suppressing primary resonance vibrations. The effects of the linked spring and damper and the attached mass on the reduction of nonlinear vibrations are studied with the help of frequency response curves, the attenuation ratio of response amplitude and the desensitisation ratio of the critical amplitude of excitation.     

Thermally induced transitions to chaos in plate vibrations

The geometrically nonlinear vibrations of linear elastic and isotropic plates under the combined effect of thermal fields and mechanical excitations are analysed. With this purpose, a model based on a p-version, hierarchical, first-order shear deformation finite element is employed. The equations of motion are solved in the time domain by Newmark's implicit time integration method. The temperature and the amplitude of the mechanical excitation are varied, and transitions from periodic to non-periodic motions are found.     

Structural-acoustic coupling effect on the nonlinear natural frequency of a rectangular box with one flexible plate

The nonlinear natural frequency of a rectangular box, which consists of one flexible plate and five rigid plates, is studied in this paper. The flexible plate is assumed to vibrate like a simple piston. The behavior of the structural-acoustic coupling between the flexible plate and the air cavity is analyzed by using the proposed finite element modal method. The system finite element equation is reduced and expressed in terms of the modal coordinates with small degrees of freedom by using the proposed reduction method. The system nonlinear stiffness matrix representing the large amplitude vibration can be transformed to be a constant modal matrix. The natural frequencies are determined by using the harmonic balance method to solve the eigenvalue equations of the structural-acoustic system. The effect of the cavity depth on the natural frequencies and convergence studies are discussed in detail.     

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.     

Transverse vibrations of thin,elastic plates with concentrated masses and internal elastic supports

The fundamental frequency of vibration of a plate carrying concentrated masses and with internal elastic supports is determined. The case of an orthotropic, rectangular plate elastically restrained against rotation along the four edges is tackled first by using simple polynomial approximations and the Galerkin method. Then, vibrations of clamped and simply supported isotropic plates of regular polygonal shape are studied by using the conformal mapping technique coupled with the variational method. Finally the case of a circular plate elastically restrained against translation and rotation is considered.     

Geometrically nonlinear vibrations of rectangular plates carrying a concentrated mass

Nonlinear forced vibrations of rectangular plates carrying a central concentrated mass are studied. The plate is assumed to have immovable edges and rotational springs; numerical results are presented for clamped plates. The Von Kármán nonlinear plate theory is used, but in-plane inertia in both the plate and the mass is retrained. The problem is discretized into a multi-degree-of-freedom (dof) system by using an energy approach and Lagrange equations taking damping into account. A pseudo-arclength continuation method is used in order to obtain numerical solutions. Results are presented as both (i) frequency-amplitude curves and (ii) time domain responses. The effect of gravity and the effect of the consequent initial plate deflection are also 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.     

Large amplitude axisymmetric vibrations of geometrically imperfect circular plates

This paper deals with the effects of geometric imperfections on the large amplitude vibrations of circular plates. It is found that geometric imperfections of the order of a fraction of the plate thickness may significantly raise the linear vibration frequencies. Furthermore, such imperfections may even change the inherent non-linear hard-spring character of the circular plates and cause them to exhibit soft-spring behavior. The effects of various boundary conditions are examined.     

Nonlinear free vibrations of beams in space due to internal resonance

The geometrically nonlinear free vibrations of beams with rectangular cross section are investigated using a p-version finite element method. The beams may vibrate in space, hence they may experience longitudinal, torsional and non-planar bending deformations. The model is based on Timoshenko’s theory for bending and assumes that, under torsion, the cross section rotates as a rigid body and is free to warp in the longitudinal direction, as in Saint-Venant’s theory. The geometrical nonlinearity is taken into account by considering Green’s nonlinear strain tensor. Isotropic and elastic beams are investigated and generalised Hooke’s law is used. The equation of motion is derived by the principle of virtual work. Mostly clamped–clamped beams are investigated, although other boundary conditions are considered for validation purposes. Employing the harmonic balance method, the differential equations of motion are converted into a nonlinear algebraic form and then solved by a continuation method. One constant term, odd and even harmonics are assumed in the Fourier series and convergence with the number of harmonics is analysed. The variation of the amplitude of vibration with the frequency of vibration is determined and presented in the form of backbone curves. Coupling between modes is investigated, internal resonances are found and the ensuing multimodal oscillations are described. Some of the couplings discovered lead from planar oscillations to oscillations in the three dimensional space.     

Transverse vibrations of a non-uniform rectangular plate on an elastic foundation

Free transverse vibrations of an isotropic rectangular plate of variable thickness resting on an elastic foundation has been studied on the basis of classical plate theory. The fourth-order differential equation governing the motion is solved by using the quintic spline interpolation technique. Characteristic equations for plates of exponentially varying thickness have been obtained for three combinations of boundary conditions at the edges. Frequencies, mode shapes and moments have been computed for different values of the taper constant and the foundation moduli for the first three modes of vibration.