Free vibration of spring-mounted beams
01 Jul 2019-Vol. 1264, Iss: 1, pp 012026
TL;DR: In this paper, a semi-analytical method based on Green's function of Euler-Bernoulli beam together with a Rayleigh-Ritz method was developed to determine the natural frequencies and mode shapes of a spring-mounted beam.
Abstract: In the present work, free vibration characteristics of a spring-mounted beam is studied. The objective is to determine the natural frequencies and mode shapes of this beam. Towards this end, a novel semi-analytical method is developed. The methodology is based on Green's function of Euler-Bernoulli beam together with a Rayleigh-Ritz method. A minimal set of basis functions for the Rayleigh-Ritz method is found, by generating the beam mode shapes for extreme values (zero and infinity) of the connecting spring stiffnesses. Thus, for a beam with n intermediate spring connections, 2 n beam modes are generated as basis functions. The Green's function approach is used to extract the beam mode shapes for the extreme values of the connecting spring stiffnesses. The results from the proposed formulation are compared with those from a finite element (FE) simulation. For most cases, the results obtained by the two methods are in excellent agreement. Guidelines for further improvement of the accuracy of the present results have been proposed. Detailed parametric studies on the effect of spring stiffness value and their location have been performed.
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TL;DR: In this paper, the Euler-Bernoulli and Timoshenko beams were analyzed under three boundary conditions, that is, mixing between being simply upheld and fixed while utilizing the differential quadrature method (DQM).
Abstract: This research studies the vibration analysis of Euler–Bernoulli and Timoshenko beams utilizing the differential quadrature method (DQM) which has wide applications in the field of basic vibration of different components, for example, pillars, plates, round and hollow shells, and tanks. The free vibration of uniform and nonuniform beams laying on elastic Pasternak foundation will be studied under three sets of boundary conditions, that is, mixing between being simply upheld and fixed while utilizing the DQM. The natural frequencies and deflection values were produced through the examination of both beam types. Results show great concurrence with solutions from previous research studies. The impact of the nonuniform cross-section area on the vibration was contemplated. A comparison between the results from both beams is obtained. The focus of this work is on studying the deflection difference between both beam theories at different beam dimensions as well as showing the shape of rotation of the cross section while applying a nodal point load equation to simulate the moving load. The results were discussed and a general contemplation about the theories was developed.
6 citations
01 Apr 2021
TL;DR: In this paper, the dynamics of an Euler-Bernoulli beam with multiple spring/mass attachments was investigated, and a novel analytical approach for determining the natural frequencies and mode shapes of the beam was proposed.
Abstract: The dynamics of an Euler–Bernoulli beam, with multiple spring/mass attachments, is investigated in this paper. A novel analytical approach for determining the natural frequencies and mode shapes of...
References
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TL;DR: In this article, a method for analyzing the free vibration of combined linear undamped dynamical systems attached at discrete points is presented, which uses separation of variables to exhibit the harmonic motion of the system and to derive a generalized differential equation for the normal modes.
Abstract: A method for analyzing the free vibration of combined linear undamped dynamical systems attached at discrete points is shown. The method uses separation of variables to exhibit the harmonic motion of the system and to derive a generalized differential equation for the normal modes. Green’s functions for the vibrating component systems are used to solve the generalized differential equation and derive the characteristic equation for the natural frequencies of the system. The characteristic equation can then be solved for the exact natural frequencies and exact normal modes. The method is demonstrated for two types of dynamical systems involving beams and oscillators. For two particular systems, approximate natural frequencies determined through a Galerkin’s method and the finite element method are compared to the exact natural frequencies. The generalized orthogonality relation for each system is derived.
98 citations
TL;DR: In this paper, the derivation of the frequency equation of a special combined dynamic system consisting of a clamped-free Bernoulli-Euler beam with a tip mass where a spring-mass system is attached to it is essentially carried out by means of the Lagrange multipliers method.
94 citations
TL;DR: In this article, the eigenvalue equation was derived analytically by using the expansion theorem and then numerically calculated eigenvalues and eigenvectors were calculated numerically.
92 citations
TL;DR: In this article, the free vibration of a combined dynamical system is governed by the solution of a generalized eigenvalue problem of orderN×N, whose stiffness and mass matrices consist of diagonal matrices modified by a total of spiral one matrices, which correspond to the number of attachment points.
30 citations