**1. Introduction**

290 Mechanical Engineering

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Rotating beams – like structures are widely used in many engineering fields and are of great interest as they can be used to model blades of wind turbines, helicopter rotors, robotic manipulators, turbo-machinery and aircraft propellers. The governing differential equations of motion in free vibration of a non-uniform rotating Timoshenko beam, with general elastic restraints at the ends are solved using the differential quadrature method, (Bellman & Roth, 1986; Felix et al., 2008, 2009). The equations of motion are derived to include the effects of shear deformation, rotary inertia, hub radius, ends elastically restrained and non-uniform variation of the cross-sectional area of the beam. The presence of a centrifugal force due to the rotational motion is considered as Banerjee has developed, using Hamilton's principle to capture the centrifugal stiffening arising in fast rotating structures, (Banerjee, 2001). With the proposed model, a great number of different situations are admitted to be solved. Particular cases with classical restraints can be deduced for limiting values of the rigidities. Also step changes in cross-section are considered (Naguleswaran, 2004).

The natural vibration frequencies and mode shapes of rotating beams have been a topic of interest and have received considerable attention. A large number of researchers have studied the dynamic behavior of rotating uniform or tapered Euler-Bernoulli beams. (Yang el al., 2004; Özdemir & Kaya, 2006; Lin & Hsiao, 2001). Banerjee derived the dynamic stiffness matrix of a rotating Bernoulli-Euler beam using the Frobenius method of solution in power series and he includes the presence of an axial force at the outboard end of the beam in addition to the existence of the usual centrifugal (Banerjee, 2000).

Not so many studies have tackled the problem of rotating beams taking into account rotary inertia, shear deformation and their combined effects, hub radius and ends elastically restrained, (Bambill et al., 2010). In applications where the rotary inertia and the shear deformation effects are not significant, an analysis based on the Euler–Bernoulli beam theory can be used. However, Timoshenko theory allows describing the vibration of short beams, sandwich composite beams or high modes of a slender beam, (Rossi et al., 1991; Seon et al., 1999). (Banerjee et al., 2006) investigated the free bending vibration of rotating tapered Timoshenko beams by the dynamic stiffness method. (Ozgumus & Kaya, 2010) used the Differential Transform Method for free vibration analysis of a rotating, tapered Timoshenko beam.

Free Vibration Analysis of Centrifugally

*L*

 

Fig. 1. Rotating beam model

Fig. 2. Rotating beam segment *k* of length *Lk*

Finally, the tensile force can be written as

with

 

The total axial force at the cross section located at *R x k k* is

Stiffened Non Uniform Timoshenko Beams 293

2 2 () ( ) () 1 1 () ()

(3)

Rk xk dxk

 

Segment d

Ld

*k k k k k k kk k k kk k kkk k x x x N x R x A x dx F R A x dx A x x dx F*

*Fk*1 is the outboard force at the end of the segment *k* , due to the adjacent segments *k*+1 to *d*.

L

Rk xk dxk

Lk

<sup>2</sup> () () () () () *<sup>k</sup>* <sup>1</sup> *N x RV L L RV x x F k k kk k k k kk k k k*

dFk Fk+1

(4)

 

*k L L k k*

*k k k*

R1 L1 Lk

Segment 1 Segment k

The finite element method was used by (Hodges & Rutkowski, 1981). (Vinod et al., 2007) presented a study about spectral finite element formulation for a rotating beam subjected to small duration impact. (Gunda & Ganguli, 2008) developed a new beam finite element whose basis functions were obtained by the exact solution of the governing static homogenous differential equation of a stiff string, which resulted from an approximation in the rotating beam equation. (Singh et al., 2007) used the Genetic Programming to create an approximate model of rotating beams. (Gunda et al., 2007) introduced a low degree of freedom model for dynamic analysis of rotating tapered beams based on a numerically efficient superelement, developed using a combination of polynomials and Fourier series as shape functions. (Kumar & Ganguli, 2009) looked for rotating beams whose eigenpair, frequency and mode-shape, is the same as that of uniform non rotating beams for a particular mode. An interesting paper (Ganesh & Ganguli, 2011) presented physics based basis function for vibration analysis of high speed rotating beams using the finite element method. The basis function gave rise to shape functions which depend on position of the element in the beam, material, geometric properties and rotational speed of the beam.

The present study tries to provide not only solutions for practical engineering situations but they also may be useful as benchmark for comparing other numerical models. The proposed differential quadrature method, offers a useful and accurate procedure for the solution of linear and non linear partial differential equations. It was used by Bellman in the 1970's. He used this method to calculate the natural frequencies of transverse vibration of a rotating cantilever beam. (Bellman & Casti, 1971). Other authors have used the differential quadrature method and recognized it as an effective technique for solving this kind of problems, (Bert & Malik, 1996; Shu & Chen, 1999; Choi et al., 2000; Liu & Wu, 2001; Shu, 2000).

Numerical results are obtained for the natural frequencies of transverse vibration and the mode shapes of rotating beams considering the elastic restraints, with non uniform variation of the cross-sectional area. Some of those cases have also been solved using the finite element method, and the sets of results are in excellent agreement.
