# NATURAL VIBRATION ANALYSIS OF A TAPERED ROTATING PRESTRESSED RAYLEIGH BEAM USING DIFFERENTIAL TRANSFORM METHOD

## Main Article Content

## Abstract

This study aims at carrying out the vibration analysis of a tapered rotating cantilever initially stressed Rayleigh beam. The governing partial differential equations of motion of a rotating tapered Rayleigh beam are reduced to ordinary differential equations using separation of variables. A semi-analytical approach known as Differential Transform Method (DTM) is then applied to the non-dimensional form of the governing equations of motion and a set of recursive algebraic equations are determined. Performing some direct algebraic operations on these derived equations and using some computer codes developed and implemented in MAPLE 18, the dimensionless natural frequencies and the associated mode shapes of the beam are obtained. In the illustrated example presented, the modulus of elasticity is assumed to vary along the beam's length while the mass density is kept constant. The effects of rotational speed, inverse of slenderness ratio and taper ratio on the dimensionless natural frequencies are also investigated. The results obtained are validated using six relevant existing studies and are found to compare favourably well with those in the open literature. It is found among others that (i) increasing the rotational speed and the inverse of slenderness ratio of the tapered rotating Rayleigh beam lead to an increase and a decrease respectively in its natural frequencies; and (ii) increasing the tapered ratio of the beam yields, generally, a decrease in its natural frequencies. Finally, it is shown that the convergence of the DTM for the governing ordinary differential equation holds.

**Keywords****:** Rotating tapered beam, natural vibration, Rayleigh beam theory and differential transform method.

Corresponding author. Tel.: +2348035785943 Email address: j.agbadeyan@yahoo.com

This study aims at carrying out the vibration analysis of a tapered rotating cantilever initially stressed Rayleigh beam. The governing partial differential equations of motion of a rotating tapered Rayleigh beam are reduced to ordinary differential equations using separation of variables. A semi-analytical approach known as Differential Transform Method (DTM) is then applied to the non-dimensional form of the governing equations of motion and a set of recursive algebraic equations are determined. Performing some direct algebraic operations on these derived equations and using some computer codes developed and implemented in MAPLE 18, the dimensionless natural frequencies and the associated mode shapes of the beam are obtained. In the illustrated example presented, the modulus of elasticity is assumed to vary along the beam's length while the mass density is kept constant. The effects of rotational speed, inverse of slenderness ratio and taper ratio on the dimensionless natural frequencies are also investigated. The results obtained are validated using six relevant existing studies and are found to compare favourably well with those in the open literature. It is found among others that (i) increasing the rotational speed and the inverse of slenderness ratio of the tapered rotating Rayleigh beam lead to an increase and a decrease respectively in its natural frequencies; and (ii) increasing the tapered ratio of the beam yields, generally, a decrease in its natural frequencies. Finally, it is shown that the convergence of the DTM for the governing ordinary differential equation holds.

## Article Details

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