In this paper, the free vibration analysis of a double-beam system is investigated. This structure is formed by two beams with elastic restraints at one end and free at the other end. These beams are connected by a mass-spring device. First, the related eigenvalue problem is established in the frequency domain, employing the well-known Fourier transform. Then, by utilizing eight boundary and compatibility conditions and solving two differential equations, the eigenvalues of the system are found and tabulated for different amounts of the parameters. Furthermore, some mode shapes of the mechanical system under study are plotted for various values of the springs' stiffness. In order to verify the results, some special cases are analyzed, and their outcomes are compared with the available ones.

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This article focuses on the free vibration analysis of a non-uniform cantilever beam with an attached mass-spring system at the free end. One end of the beam is elastically restrained against rotation and translation. The height of the beam is assumed constant but the width of the beam exponentially varies. The governing differential equations of the beam, which is a partial differential equation with variable coefficients, and that of the mass-spring system, which is an ordinary differential equation, are found. The exact solution of the problem is then obtained using the pertinent boundary conditions. The eigenvalues and eigenfunctions of the problem which are frequencies and mode shapes of the system are derived for various properties of the system such as stiffness of springs and attached mass. Some limiting cases with available results in the literature are analyzed. The comparison of the proposed results and the reference results shows the accuracy of the solution.

This technical note addresses the free vibration problem of an elastically restrained Euler–Bernoulli beam with rotational spring-lumped rotary inertia system at its mid-span hinge. The governing differential equations and the boundary conditions of the beam are presented. Special attention is directed toward the conditions of the intermediate spring-mass system which plays a key role in the solution. Sample frequency parameters of the beam system are solved and tabulated. Mode shapes of the beam are also plotted for some spring stiffnesses.

This paper is concerned with the free vibration of a gabled frame with rotational springs at intersecting joints. Each member of the frame may have different mechanical or geometrical properties. The problem at hand is to solve the gabled frame vibration problem for the natural frequencies by applying the differential transform method (DTM). Special attention is directed towards the derivation of the governing differential equations along with boundary and compatibility conditions of the problem. Some plots of mode shapes are also presented. The problem was also solved by using the finite element method to provide an independent check on the solutions.

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