Nonlinear free and forced vibration of Euler-Bernoulli beams resting on intermediate flexible supports (original) (raw)
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Geometrically nonlinear free and forced vibrations of Euler-Bernoulli multi-span beams
MATEC Web of Conferences
The objective of this paper is to establish the formulation of the problem of nonlinear transverse forced vibrations of uniform multi-span beams, with several intermediate simple supports and general end conditions, including use of translational and rotational springs at the ends. The beam bending vibration equation is first written at each span and then the continuity requirements at each simple support are stated, in addition to the beam end conditions. This leads to a homogeneous linear system whose determinant must vanish in order to allow nontrivial solutions to be obtained. The formulation is based on the application of Hamilton's principle and spectral analysis to the problem of nonlinear forced vibrations occurring at large displacement amplitudes, leading to the solution of a nonlinear algebraic system using numerical or analytical methods. The nonlinear algebraic system has been solved here in the case of a four span beam in the free regime using an approximate method developed previously (second formulation) leading to the amplitude dependent fundamental nonlinear mode of the multi-span beam and to the corresponding backbone curves. Considering the nonlinear regime, under a uniformly distributed excitation harmonic force, the calculation of the corresponding generalised forces has led to the conclusion that the nonlinear response involves predominately the fourth mode. Consequently, an analysis has been performed in the neighbourhood of this mode, based on the single mode approach, to obtain the multi-span beam nonlinear frequency response functions for various excitation levels.
Journal of Vibration Engineering & Technologies, 2019
Purpose The geometrically non-linear free and forced vibrations of a multi-span beam resting on an arbitrary number of supports and subjected to a harmonic excitation force is investigated. Methods The theoretical model developed here is based on the Euler-Bernoulli beam theory and the von Kármán geometrical non-linearity assumptions. Assuming a harmonic response, the non-linear beam transverse displacement function is expanded as a series of the linear modes, determined by solving the linear problem. The discretised expressions for the beam total strain and kinetic energies are then derived, and by applying Hamilton's principle, the problem is reduced to a non-linear algebraic system solved using an approximate method (the so-called second formulation). The basic function contribution coefficients to the structure deflection function and the corresponding backbone curves giving the non-linear amplitude-frequency dependence are determined. Considering the non-linear forced response, an approximate multimode approach has been used in the neighbourhood of the predominant mode, to obtain numerical results, for a wide range of vibration amplitudes. Results The effects on the non-linear forced dynamic response of the support number and locations, the excitation frequency and the level of the applied harmonic force (a centered point force or a uniformly distributed force) have been investigated and illustrated by various examples.
Non-linear vibrations of a simple–simple beam with a non-ideal support in between
Journal of Sound and Vibration, 2003
A simply supported Euler-Bernoulli beam with an intermediate support is considered. Non-linear terms due to immovable end conditions leading to stretching of the beam are included in the equations of motion. The concept of non-ideal boundary conditions is applied to the beam problem. In accordance, the intermediate support is assumed to allow small deflections. An approximate analytical solution of the problem is found using the method of multiple scales, a perturbation technique. Ideal and non-ideal frequencies as well as frequency-response curves are contrasted. r
Nonlinear free vibrations of multispan beams on elastic supports
Computers & Structures, 1989
Abstract@This paper presents a numerical solution for geometrically nonlinear free vibrations of multispan beams on elastic supports. The horizontally and rotary inertia forces have been neglected and the beams are considered as distributed mass systems. The variational approach, dynamic finite element method and the iterative procedure are used for determining the frequencies and nonlinear mode of vibrations. Several examples show the usefulness of the method and moreover the influence of supports flexibility on the frequency-amplitude relation is examined.
A numerical method for solving free vibration of Euler-Bernoulli beam
A method of using He's variational iteration method to solve free vibration problems of Euler-Bernoulli beam under various supporting conditions is presented in this paper. By employing this technique, the beam's natural frequencies and mode shapes can be solved and a rapid convergent sequence is obtained during the solution. The obtained results are the same as the results obtained by the Adomian decomposition method. It is verified that the present method is accurate and it provides a simple and efficient approach in solving the vibration problems of uniform Euler-Bernoulli beams. A robust and efficient algorithm is also programmed using Matlab based on the present method, which can be easily used to solve Euler Bernoulli beam problem. Keywords—Euler-Bernoulli beam, He's variational iteration method, free vibration, natural frequency, mode shape. I. INTRODUCTION The vibration problems of uniform Euler-Bernoulli beams have been solved by different approaches. Smith et a...
The governing equation of motion for free vibration of a uniform Euler-Bernoulli beam is given as
2017
EVERAL techniques have been used to carry out the vibration analysis of beams with a view to determining their vibration characteristics. Lai, et al. [1] analysed the free vibration of uniform Euler-Bernoulli beam with different elastically supported conditions using Adomian decomposition method (ADM). Li [2] had earlier studied the vibration characteristics of a beam having general boundary conditions. The displacement of the beam was sought in form of a linear combination of a Fourier series and an auxiliary polynomial function. Kim and Kim [3] also applied Fourier series to determine the natural frequencies of beams having generally restrained boundary conditions. Later, Liu and Gurram [4] adopted the He’s variational iteration method to estimate the vibration frequencies of a uniform Euler-Bernoulli beam for various supporting end conditions. Natural frequencies for the first six modes of vibration were presented in their work. Malik and Dang [5] employed the differential transf...
Effect of Free Vibration Analysis on Euler-Bernoulli Beam with Different Boundary Conditions
2020
This paper presents an analysis of the effect of free vibrations of a free-free beam, fixed-fixed beam and simply supported beam using the series solution. It was found that the mode shape for each of the modes has effects on the displacement or deflection of such beam so that the deflection increases as the increase of the mode. Also, a Simply-Supported beam has a lower displacement compared to the free-free beam and fixed-fixed beam which almost have the same displacement. At mode one, it is seen that a Simply Supported beam has a higher amplitude, followed by a free-free beam and then a fixed-fixed beam.
Journal of physics, 2019
Geometrically nonlinear forced vibrations of fully clamped multi-span beams resting on multiple simple supports and carrying multiple masses may be encountered in many mechanical and civil engineering applications. The theoretical model developed here is based on the Euler-Bernoulli beam theory and the Von Karman geometrical non-linearity assumptions. Harmonic motion is assumed and the nonlinear beam transverse displacement function is expanded as a series of the linear modes, determined by solving first the linear problem. The discretised expressions for the beam total strain and kinetic energies are then derived, and by applying Hamilton's principle, the problem is reduced to a nonlinear algebraic system solved using an approximate method (the so-called second formulation). The basic function contribution coefficients to the structure non-linear response function and the corresponding backbone curves giving the non-linear amplitude-frequency dependence is determined. Numerical results are given in the neighbourhood of the predominant nonlinear mode shape, based on the single mode approach, for a wide range of vibration amplitudes, showing the effect of the added masses and their locations, as well as the applied uniformly distributed harmonic force on its non-linear dynamic response.
Vibration of initially stressed beam with discretely spaced multiple elastic supports
2004
Vibration behavior of an initially stressed beam on discretely spaced multiple elastic supports has been studied and a theoretical formulation of the system is derived using the variational principle. Unlike beams on an elastic foundation, discretely spaced supports can distort the beam mode shapes when the supports have rather large stiffness, i.e. usually expected beam modes cannot be obtained, but rather irregular mode shapes are observed. Conversely, irregular modes can be recovered by changing initial stress. Since support location is closely associated with the dynamic characteristics, this work also discusses eigenvalue sensitivity with respect to the support position and some numerical examples are investigated to illustrate the above findings.
International journal of engineering trends and technology, 2023
Purpose The geometrically non-linear free and forced vibrations of a multi-span beam resting on an arbitrary number of supports and subjected to a harmonic excitation force is investigated. Methods The theoretical model developed here is based on the Euler-Bernoulli beam theory and the von Kármán geometrical non-linearity assumptions. Assuming a harmonic response, the non-linear beam transverse displacement function is expanded as a series of the linear modes, determined by solving the linear problem. The discretised expressions for the beam total strain and kinetic energies are then derived, and by applying Hamilton's principle, the problem is reduced to a non-linear algebraic system solved using an approximate method (the so-called second formulation). The basic function contribution coefficients to the structure deflection function and the corresponding backbone curves giving the non-linear amplitude-frequency dependence are determined. Considering the non-linear forced response, an approximate multimode approach has been used in the neighbourhood of the predominant mode, to obtain numerical results, for a wide range of vibration amplitudes. Results The effects on the non-linear forced dynamic response of the support number and locations, the excitation frequency and the level of the applied harmonic force (a centered point force or a uniformly distributed force) have been investigated and illustrated by various examples.