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In this paper, free vibration characteristics of a uniform Rayleigh beam are studied using the differential transform method. The procedure entails transforming the partial differential equation governing the motion of the beam under consideration and the associated boundary conditions. The transformation yields a set of difference equations. Some simple algebraic operations are performed on the resulting difference equations to determine any ith natural frequency and the closed-form series function for any ith mode shape. Finally, one problem is presented to illustrate the implementation of the present method and analyse the effect of mass per length on the natural frequencies of the beam.

The paper presents the transverse vibration of an open edge crack graded Rayleigh beam with axial motion. The vibration equation was obtained relying on Hamilton's precept and resolved by the approach of Galerkin's. The power-law is adopted to represent the gradient of properties of the material composing the beam along the direction of the thickness. Cracks were modeled as rotational massless spring. The effects of axial velocity, material property change index, location, and depth of cracking on the vibration characteristics are observed. Also, the corresponding mode shapes are found for cracked moving graded Rayleigh beam with simply supported, and fixed-fixed end supports. The results clear up that the natural frequencies drop due to the rise in the axial velocity, the crack depth, and the material property index

Functionally gradient materials and small-scale pipes have a great important in industry because of its wide applications in many engineering fields such as, fluid transport in fluidic devices. The aim of this work is to study the dynamic stability of double FGM micro pipes conveying fluid depending on a modified couple stress theory. The two micro pipes are connected together continuously through elastic spring. The vibration equations with boundary conditions are acquired based on Hamilton’s principle and subsequently, solved by Galerkin’s method. The results of this research were compared with results reported in the literature. A reasonable agreement was found. Also, the influences of a gradient index of the material, a parameter of a length scale, the outer diameter of micro-pipe on the critical flow velocity and resonant frequencies are discussed. The results displayed that the critical velocities and natural frequencies are increased hastily with an increase in a gradient index n

This paper presents a procedure for assessing the vibration analysis of type SCARA robots. The motion and running conditions of such robots are very complicated that leads to produce vibration and shock which are generated by arm profile in running conditions .In this study the vibration analysis gives the feasibility of the preview control was examined to improve the performance of the SCARA robots system. As it is important for containment the robot arm trajectories generated by the model to show satisfactory safe performance under vibration occurrence phenomena so that they completely avoid errors, the results obtained from such vibration analysis assessment procedure are considered to be valuable and reliable process not only with respect to vibration risk assessment but also for predicting kinematic analysis by investigating the robot arm motion using the kinematic and vibration methods. Forced vibrations is studied analytically help the designer to predict the behavior and design the robot hardware or control system. Theoretical results show reduction in both vibration amplitude and time history response.