Natural Frequency Coefficients Research Articles

VIBRATIONS AND ELASTIC STABILITY OF THIN CIRCULAR PLATES WITH VARIABLE PROFILE

Circular plates of thicknesses varying to the functional relation h o[1 + γ( r/ a) n ], where nis a positive integer, are studied in the present paper. Uniform, elastic boundary restraints are considered and the first two natural frequency coefficients corresponding to axisymmetric modes and the buckling, in plane pressure are determined using simple polynomial co-ordinate functions which identically satisfy the boundary conditions and the classical Rayleigh-Ritz method. An independent solution is also obtained by means of the differential quadrature technique.

Free vibrations of beams of bilinearly varying thickness

Free vibrations of Bernoulli beams of bilinearly varying thickness are studied using: (a) the optimized Rayleigh-Ritz method, (b) the differential quadrature technique and (c) the finite element approach. It is shown that the fundamental frequencies of vibration of the structural elements as determined by the methodologies (a) and (c) are in excellent agreement while rather considerable discrepancies are found when using the differential quadrature technique. The values of the second and third natural frequency coefficients determined via the finite element method are also presented.

Plate Characteristic Functions and Natural Frequencies of Vibration of Plates by Iterative Reduction of Partial Differential Equation

Natural frequency coefficients of rectangular plates and the corresponding plate characteristic functions are obtained by reduction of plate partial differential equation to an ordinary differential equation and solving it exactly. The reduction is carried out by assuming a deflection shape in one direction consistent with the boundary conditions and applying Galerkin’s averaging technique to eliminate the variable. The reduction method, commonly known as Kantorovich method, is applied sequentially on either directions of the plate and iterated until convergence is achieved for the natural frequency coefficients. The resulting plate characteristic functions are very good approximations to the normal modes of the plate. The results are tabulated for plates with combination of clamped, simply-supported, and free edge conditions.

Natural Frequencies and Mode Shapes of Beams Carrying a Two Degree-of-Freedom Spring-Mass System

An exact solution for the natural frequencies and mode shapes for a beam elastically constrained at its ends and to which a rigid mass is elastically mounted is obtained. The attached mass can both translate and rotate. The general solution is obtained using the Laplace transform with respect to the spatial variable and yields the exact solutions to several previously published simpler configurations that were obtained using approximate methods. Numerous numerical results are presented for the natural frequency coefficients that extend previously reported results and that show the transition between various limiting cases. In addition, values are presented for the lowest two natural frequency coefficients for a beam that is clamped at both ends and is carrying a two dof spring-mass system. Representative mode shapes at selected values of the system’s parameters are also given.

Vibrations of a timoshenko beam of non-uniform cross-section elastically restrained at one end and carrying a finite mass at the other

An exact solution of the title problem is obtained in the case where one of the beam ends is elastically restrained against rotation and translation. Natural frequency coefficients are determined for a significant range of the parameters that come into play.

Vibrations of a timoshenko beam clamped at one end and carrying a finite mass at the other

The present paper deals with an exact solution of the title problem. Modal shapes and natural frequency coefficients are determined for a significant range of the mechanical and geometric parameters that come into play. When the parameter I/AL 2 (where I is cross-sectional moment of inertia, A is cross-sectional area, and L beam length) approaches zero, the beam dynamic characteristics agree with values already available in the open literature.

Free, in-plane vibrations of circular rings with radial supports taking into account shear and rotatory inertia effects

Natural frequency coefficients and modal shapes corresponding to some of the lower modes of vibration of the structural systems described in the title are determined using a finite element algorithmic procedure.

On the relative accuracy and relative difficulties of vibrations and buckling problems of structural elements

Some analytical and numerical experiments on the determination of natural frequency coefficients and buckling loads of some structural elements are presented. It is concluded that, when dealing with approximate analytical procedures such as Galerkin's or the Ritz method or with numerical methodologies such as the universally used finite element method (FEM) the accuracy achieved is, in general, considerably better when determining frequency coefficients when the same amount of labour or computer time is invested in both phenomena. It is shown in the case of the FEM that the precision of the frequency coefficients is of approximately twice the accuracy when determining buckling loads. No claim of originality is made by the authors, but it is hoped that designers of mechanical systems will find the considerations made in this study useful. It is also felt that the subject matter is of a fundamental nature, and that there is considerable room for basic studies on this question, which is also extremely important from a practical engineering viewpoint.

Free and forced vibrations of a simply supported rectangular plate partially embedded in an elastic foundation

The present note deals with an approximate solution of the title problem in the case where a portion of the plate is subject to a p o cos ωt-type excitation. The lower natural frequency coefficients and dimensionless values of displacements and stress resultants are determined for several combinations of the mechanical parameters coming into play.

A note on free flexural vibrations of a non-uniform elliptical ring in its plane

An approximate solution for the title problem is obtained by using the Rayleigh-Ritz method. Two different types of co-ordinate functions are used: sinusoids, and polynomial expressions which contain an undetermined exponential parameter which allows for minimization of the natural frequency coefficients. Continuous and discontinuous variation of the ring thickness are considered. The analytical predictions are compared with values obtained by means of the finite element method, and good engineering agreement is shown to exist. No claim of originality is made, but it is hoped that present results will be of interest to designers.

Analytical and experimental investigation on vibrating circular plates with stepped thickness over a concentric circular region

Transverse vibrations of the structural elements described in the title are studied for the case where the edges are elastically restrained against translation and rotation. Since finding an exact solution is a difficult task, it was considered convenient to approximate the response of the plate in the case of free, axisymmetric vibrations by means of a summation of simple polynomial coordinate functions that satisfy the governing boundary conditions. The Ritz method is used in order to generate the frequency equation. The natural frequency coefficients are optimized by minimizing each eigenvalue with respect to an undetermined exponential parameter included in each coordinate function.

Axisymmetric vibration of bimodulus thick circular and annular plates

The fundamental natural frequencies of axisymmetric circular and annular plates subjected to a combination of a pure bending stress and extensional stress in the plane of the plate are investigated. The thick plate ring element model which includes the effect of transverse shear deformation is created for axisymmetric free vibration problems. The obtained results of non-dimensional natural frequency coefficient compared with the closed form solutions for ordinary plates are shown to be very good. The effects of various parameters on the natural frequency and neutral surface locations are studied. The bimodulus properties are shown to have significant influence on the natural frequency.

Definition And Dynamic Portrait Of Large SpaceStructures For Control System Synthesis

A new definition of large space structures (LSS) is given, yielding a mathematical model of minimal order for three-axis attitude control system synthesis. Then, the dynamic portrait is introduced, allowing the structure to be designed with minimal excitation of certain vibration modes by the control variables. The theory is developed by considering space structures having a branched configuration near the centre of which are located attitude sensors and actuators collocated with an orthogonal control axis set to be orientated. It is well known that the complete set of space structures comprises two subsets, one in which rigid body dynamics may be assumed and the other, referred to as the Large Space Structures (LSS), for which one or more flexure modes, typically with very low natural frequencies, must be taken into account. This paper provides a much needed quantitative boundary between the two subsets, given by the definition that a structure is a LSS if the inequalities, k > 2.47 and co < 2k , are satisfied for any i, where co and k are, respectively, the natural frequency and excitability coefficient of the i* flexure mode. The approach is based on a comparison of the flexure mode motion with the rigid-body mode motion in the phase double-plane (i* modal phase-plane superposed on the rigid-body phase-plane) of a structural model in the modal state representation to which is applied a step control variable. Hence, the model derived is suitable for designing control systems employing discontinuous on-off thrusters as well as continuous actuators. The Lagrangian and Modal dynamic models of the structure are then used together to derive the dynamic portrait as a set of Transactions on the Built Environment vol 19, © 1996 WIT Press, www.witpress.com, ISSN 1743-3509 256 Structures in Space graphs, [w ](%,) and [k.](X), where A, is a selected physical parameter of the spacecraft. The structure may be designed, where practicable, to correspond with minima in these graphs to simplify the control problem. The new method is demonstrated by examples.

Elastic Plane-Strain Vibrations of a Hollow Cylinder Bonded to a Thin Shell

Frequency equations for elastic plane-strain vibrations of an infinitely long hollow elastic cylinder bonded on the outer curved surface to a thin elastic casing are discussed and compared with similar equations appropriate for hollow cylinders with (i) traction-free surfaces and (ii) one traction-free and one clamped surface. The axisymmetric made uncouples into a shear mode and a radial-extensional mode. The effects of shell thickness, mass density, and elastic properties on the natural circular frequency, coefficients associated with the shear and extensional modes are discussed, and graphs of the natural circular frequency coefficients and of a displacement ratio are included.

Elastic Axial Shear Vibrations of a Hollow Cylinder Bonded to a Thin Shell

Frequency equations and displacement ratios for elastic axial shear vibrations of an infinitely long hollow elastic cylinder bonded on its outer curved surface to a thin elastic casing are derived and compared with similar equations appropriate for hollow cylinders with (i) traction free surfaces and (ii) one traction-free and one clamped surface. Various plots of the natural circular frequency coefficients and displacement ratios have been made in order to display their dependence on shell thickness, core thickness, mass density, and elastic properties.