In the present paper the Differential Quadrature Method, DQM, and the domain decomposition are used to carry out the free transverse vibration analysis of non-uniform multi-span rotating Timoshenko beams with perfect and not perfect boundary conditions. The cross section could vary in a continuous or discontinuous fashion along the beam length. The material of the beam could be different in each beam span. The influence of elastically clamped boundary conditions at hub end are studied and discussed. The effect of an arbitrary hub radius is considered. The governing differential equations of motion for rotating Timoshenko beams come from the derivation of Hamilton's principle. The first six natural frequencies of vibration are obtained for many particular situations and for some of them the mode shapes are also available. The examples of applications of the method indicated its effectiveness. The results for particular cases are in excellent agreement with published results and results obtained by means of the finite element method.

This paper provides an analytical solution for free transverse vibrations of axially functionally graded beams with step changes in geometry and in material properties. The differential quadrature method using domain decomposition technique is used. Based on Timoshenko beam theory, the equations of motion are derived using Hamilton's principle. Material properties are assumed to vary along the beam in a continuous or an abrupt fashion. The combinations of classical boundary conditions (Free, Simply Supported and Clamped) are considered to determine the natural frequencies of many numerical examples. The results for different step locations with different axially functionally graded materials are presented. The phenomenon of dynamic stiffness of beams can be observed in various situations. As there are no available previous results of axially functionally graded beams with step changes, only the results for beams with no abrupt discontinuities are compared with published results. The developed differential quadrature solution has proved its simplicity and robustness to solve the problem presented in the title.

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The present study aims to provide some new information for the design of micro systems. It deals with free vibrations of Bernoulli–Euler micro beams with nonrigid supports. The study is based on the formulation of the modified couple stress theory. This theory is a nonclassical continuum theory that allows one to capture the small-scale size effects in the vibrational behavior of micro structures. More realistic boundary conditions are represented with elastic edge conditions. The effect of Poisson’s ratio on the micro beam characteristics is also analyzed. The present results revealed that the characterization of real boundary conditions is much more important for micro beams than for macro beams, and this is an assessment that cannot be ignored.

Vibration analysis of rotating beams is a topic of constant interest in mechanical engineering.The differential quadrature method (DQM) is used to obtain the natural frequencies of free transverse vibration of rotating beams. As it is known the DQM offers an accurate and useful method for solution of differential equations. And it is an effective technique for solving this kind of problems as it is shown comparing the obtained results with those available in the open literature and with those obtained by an independent solution using the finite element method. The beam model is based on the Timoshenko beam theory.