Effect of Mass per Unit Length on freely vibrating Simply Supported Rayleigh Beam

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

Dynamic characteristics of cracked simply supported bidirectional functionally graded Rayleigh beam

This paper investigates the dynamic behavior of cracked Rayleigh beams constructed from bidirectional functionally graded (BDFG) materials under simple boundary conditions. A torsional massless spring is employed to model the beam's open crack type. The vibration equations are obtained using Hamilton's principle. The graded beam material properties are varied throughout the thickness based on the power-law distribution and in the longitudinal direction using the exponential material distribution. To solve the dynamic equations, Galerkin's approach is employed. The paper evaluates the impacts of the axial index, gradient property index, beam modulus ratio, and crack parameters on the natural frequencies of the FG beam. The results indicate that the dimensionless natural frequencies of intact graded beams decrease with increased gradient index k. In contrast, they increase with a rise in the modulus ratio. Additionally, the results demonstrate that an increase in the crack depth ratio decreases dimensionless natural frequencies