**6. Conclusion**

310 Mechanical Engineering

The next Figures, 5 and 6, present the variation of the fundament frequency parameter Ω<sup>1</sup> with the variation of the non-dimensional rotational speed *η* and the spring constant *Kψ*1 .

Fig. 5. The fundamental frequency coefficient Ω1 of a one-span elastically restrained rotating Timoshenko beam versus the spring constant parameter of the rotational spring *Kψ*1, for

Fig. 6. The fundamental frequency coefficient Ω1 of a two-span elastically restrained rotating Timoshenko beam versus the spring constant parameter of the rotational spring *Kψ*1, for

different rotational speed parameters *η*. *Kw*1= 10; *Kwd*=1; *Kψd*=10

different rotational speed parameters *η*. *Kw*1= 10; *Kwd*=0; *Kψd*=0

The differential quadrature method proves to be very efficient to obtain frequencies and mode shapes of natural vibration, for the rotating Timoshenko beam model.

The versatility of the proposed beam model (variable cross section, step change in cross section, elastic restraints at both ends) allows to solve a large number of individual cases.

Something interesting to point out is that because the method directly solves two ordinary differential equations, additional restrictions are not generated. This does not happen in other methodologies, such as the dynamic stiffness method (Banerjee, 2000, 2001).

As a matter of fact, the differential quadrature method has the same advantage as the finite element method and it needs less computer memory requirements than the FEM.

In particular the present results show that the frequency coefficients vary more significantly when the translational spring stiffness changes at the end of the beam farthest from the axis of rotation *Kψd*.
