Eigenfrequencies of generally restrained beams (original) (raw)
Related papers
Vibration frequencies for a beam with a rotational restraint in an adjustable position
Applied Acoustics, 1999
The problem of free vibrations of uniform beams having a combination of clamped, pinned and free ends supports, has been extensively studied by several investigators [1±4]. Transverse vibration of uniform beams with elastically restrained ends have also been extensively investigated [5±10]. In contrast to the body of information available, there is only a very limited amount of information for uniform beam elastically restrained at some point . The present work deals with the exact and the approximate determination of frequency coecients of a uniform beam, with one end spring-hinged and a rotational restraint in a variable position. The approximate eigenvalues are obtained by means of the Rayleigh±Schmidt method, and the results obtained were compared with the exact values. It is shown that a one term approximation with various undetermined exponents yields fundamental frequency values which are in good agreement with exact values. #
A note on vibrations of generally restrained beams
Journal of Sound and Vibration, 1989
Exact frequency and normal mode shape expressions are derived in this note for generally restrained Bernoulli-Euler beams with unsymmetrical translations and rotations at either end. The eigenfrequencies and mode shape parameters are presented for a wide range of restraint parameters. Several degenerate cases have been studied, and some of these have been compared with those available in the published literature. It is believed that the results presented in this paper will be of use in design of beams, shafts and piping under dynamic considerations.
A VARIATIONAL APPROACH TO THE VIBRATION OF TAPERED BEAMS WITH ELASTICALLY RESTRAINED ENDS
Journal of Sound and Vibration, 1996
The determination of natural frequencies in transverse vibrations of tapered beams with elastically restrained ends is a problem that has been extensively studied by several investigators. To review the literature is not attempted here and only a few references will be mentioned. Mabie and Rogers [1-4] studied several cases of tapered beams with different end conditions. Laura and co-workers [5-8] treated various cases of non-uniform beams with different conditions of end restraints. Grossi and Bhat [9-10] analyzed the case of linearly tapered beams with ends elastically restrained against rotation. The problem of determination of frequencies for beams with both ends elastically restrained against rotation and translation has been studied by Kameswara Rao and Mirza , but for the case of uniform beams. The present work is concerned with the use of the Rayleigh-Schmidt method in the determination of frequencies corresponding to the first two modes of free vibration of a linearly tapered beam with both ends elastically restrained against rotation and translation. It is shown that adopting, for the assumed mode shapes, functions with several adjustable exponents leads to straightforward and simple algorithm which, in the case of the fundamental frequency coefficient, yields very accurate results. An interesting feature of the present variational approach is that it increases the accuracy of numerical results through a refinement of the shape functions by optimizing the adjustable exponents rather than by increasing the terms of approximation. In this paper, several examples are solved and the results obtained are compared with previously published results to demonstrate the accuracy and flexibility of the algorithm developed. New results are also determined for tapered beams with generally restrained ends.
Computation of eigenfrecuencies for elastic beams, a comparative approach
2003
In this manuscript we present an extensive study on mathematical and numerical modeling of ßexural vibrations of elastic beams. We consider classical one dimensional models. Three fundamental effects are considered; Bending, Rotary Inertia and Shear. Based on the Wave Propagation Method (WPM), we propose an asymptotic method corrected with a root Þnding technique to compute eigenfrequencies to any desired accuracy. This method is applied successfully to equations involving bending and either rotary inertia and shear.
Vibrations of elastically restrained frames
Journal of Sound and Vibration, 2005
This paper deals with the determination of eigenfrequencies of a frame which consists of a beam supported by a column and is submitted to intermediate elastic constraints. The ends of the frame are elastically restrained against rotation and translation. The individual members of the frame are assumed to be governed by the transverse and axial vibration theory of an Euler-Bernoulli beam. The boundary and eigenvalue problem which governs the dynamical behavior of the frame structure is derived using the techniques of calculus of variations. Exact values of eigenfrequencies are determined by the application of the separation of variables method. Also, results are obtained by the use of the finite element method. The natural frequencies and mode shapes are presented for a wide range of values of the restraint parameters. Several particular cases are presented and some of these have been compared with those available in the literature.
The International Journal of Acoustics and Vibration
This note deals with the free transverse vibration of a beam with two arbitrarily located internal hinges, four intermediate elastic restraints, and ends elastically restrained against rotation and translation. The method of separation of variables is used for the determination of the exact frequencies and mode shapes. New results are presented for different boundary conditions and restraint conditions in the internal hinges. The mathematical model is also used to study the influence on the frequencies and mode shapes of varying intermediate supports that are located at the nodal points of higher modes. A detailed numerical study on the effects of the locations of intermediate translational restraints and their stiffness on the natural frequencies and mode shapes is performed for different boundary conditions. The effect of the presence of the internal hinges is also analysed. Graphs and tables of the non-dimensional frequencies and the corresponding mode shapes are given in order to illustrate the behaviour of frequency parameters and the presence of mode shape switching.
MATEC Web of Conferences, 2012
The purpose of the present paper is the development of a physically discrete model for free transverse constrained vibrations of beams. The discrete model consists on an N-degree of freedom system made of masses placed at the end of solid bars connected with spiral springs. The calculations made involve two tensors, namely the mass tensor [m i j ] and the linear rigidity tensor [k i j ]. The results obtained by the physically discrete model show a good agreement and a quick convergence to the equivalent continuous beam for various end conditions for both the natural frequencies and the corresponding mode shapes. The model proposed in the present paper, which has been validated here using classical cases, may be easily applied to the flexural vibration of beams with various types of discontinuities, and to beams carrying concentrated masses.
Transverse Vibration Analysis of Uniform Beams under Various Ends Restraints
APCBEE Procedia, 2014
The beam analysis, based on the assumptions of the Bernoulli-Euler theory, in free vibration has been largely investigated. Many researches focused on the transverse vibrations study, under the application of different boundary constraints where different theories were applied. The considered stiffness and mass matrices are those obtained by assembling the elementary ones resulting from the FEM use The Jacobi method allowed the solution of the eigenvalue problem. These well known concepts were applied to the study of beams with constant geometrical and mechanical characteristics having one to two overhangs with variable lengths. Murphy studied, by an algebraic solving approach, a simply supported beam with two overhangs of arbitrary length, which allows an experimental determination of the E elastic modulus. The advantage of our paper offers a possibility of extending this approach to many interesting problems formed by beams vibrating transversally with various ends restraints.
Symmetric Properties of Eigenvalues and Eigenfunctions of Uniform Beams
Symmetry, 2020
In this paper, the models of Euler–Bernoulli beams on the Winkler foundations are considered. The novelty of the research is in consideration of the models with an arbitrary variable coefficient of foundation. Qualitative results that influence the symmetry of the coefficient of foundation on the spectral properties of the corresponding problems are obtained, for which specific variable coefficients of foundation are tested using numerical calculations. Three types of fixing at the ends are studied: clamped-clamped, hinged-hinged and free-free. The conditions of the stiffness and types of beam fixing have been found for the set of eigenvalues of boundary value problems on a full segment and can be represented as two groups of the eigenvalues of certain problems on a half segment. Such qualitative spectral properties of a mechanical system can contribute to the creation of various algorithms for nondestructive testing, which are widely used in technical acoustics.