**3. Differential Quadrature Method, DQM**

In order to obtain the DQM analog equations from the governing equations of the rotating beam, the beam segment domain is discretized in a grid of *i* points, using the Chebyshev – Gauss - Lobato expression, (Shu, 2000). (See Fig. A.1 in Appendix A)

Free Vibration Analysis of Centrifugally

shapes of the rotating beam.

The term <sup>2</sup> () () *kk kk* 

**5. Numerical results** 

particular situation there are:

( 0.886364

beam:

 

**4. Finite element method, MEF** 

Stiffened Non Uniform Timoshenko Beams 299

1 1 1 1 1 11 1 1 11

*N a A W a lK W*

 ;

(0) (0) (0) 0

1 (1) 1 11 1 1 1 1

 

(1) (1) (1) 0; 2(1 ) 2(1 )

 

*d d nj dj d dn d Wd dn*

*N a A W a lK W*

*<sup>n</sup> <sup>d</sup> n dn nj dj d j*

The DQM linear equation system is used to determine the natural frequencies and mode

The number of terms taken in the summations had been studied for many situations and the

An independent set of results for the natural frequencies, was also obtained by a finite element code. (Bambill et al., 2010). The finite element model employed in the analysis has 3000 beam elements of two nodes in the longitudinal direction (Rossi, 2007). See Table 2.

The beam model also takes into account the shear deformation (Timoshenko beam's theory)

formulation. Probably for this reason some small differences between both sets of numerical

In the following examples some calculations were performed over elliptical cross sections.

 *)*. Without loss of generality, one may choose to keep constant width *ek=e* and vary the height ( ) *<sup>k</sup> h x* in each segment of the beam. The area and the second moment of area

*k*

*k*

The following formula is proposed to a quadratic variation of the height in each segment of

01 2 ( ) *k kk k hx c c x c x*

*Ix x* of equation (13.b) was not included in the finite element

*k*

*k*

3

3 ( ) ( ) <sup>64</sup> *k*

, and for this

*eh x I x* 

,

*eh x A x* 

> ( ) ( ) (0) *k*

*h x b x h* 

*k*

2

This number of elements was proved to be enough with a convergence analysis.

results (DQM and FEM) begin to appear when the rotational speed *η* increases.

( ) ( ) (0) *k*

*k h x*

*<sup>h</sup>* ;

and the increase in bending stiffness induced by the centrifugal force.

of the cross section of the beam will be ( ) ( ) <sup>4</sup>

*k*

*a x*

*j*

 

> 

(0) <sup>0</sup> *n*

(1)

 

(1) <sup>0</sup>

1

*j j W*

*j j*

(34a,b)

(35a,b)

(1)

*<sup>b</sup> K A l*

(1)

*<sup>b</sup> K A l*

1

1

*j*

system has acceptable convergence by *n*= 21 terms. (See Table 1)

*n*

*j*

2(1 ) 2(1 ) *n*

Equations (18, 19, 20) assumed the form:

$$N\_k(\mathbf{x}\_i) = \eta^2 \frac{l\_k^2}{s\_1^2} \left( R\_k \,\upsilon\_k(\mathbf{1}) + \phi\_k(\mathbf{1}) - R\_k \,\upsilon\_k(\mathbf{x}\_i) - \phi\_k(\mathbf{x}\_i) \right) + N\_{k+1} \tag{27}$$

$$Q\_k(\mathbf{x}\_i) = \left(N\_k(\mathbf{x}\_i) + \frac{\kappa}{2(1+\nu)} a\_k(\mathbf{x}\_i)\right) \sum\_{j=1}^n A\_{ij}^{(1)} \, \mathcal{W}\_{kj} - \frac{\kappa}{2(1+\nu)} a\_k(\mathbf{x}\_i) \Psi\_{ki} \tag{28}$$

$$M\_k(\mathbf{x}\_i) = b\_k(\mathbf{x}\_i) \sum\_{j=1}^n A\_{ij}^{(1)} \Psi\_{k\bar{j}} \tag{29}$$

The equations of motion (21) and (22) become:

 2 2 1 (1) 2 1 2 2 2 1 1 (2) 1 (1) 2 2 2 1 1 2 1 2 2 ( ) ( ) 2(1 ) ( ) ( ) ( ) 2(1 ) 2(1 ) ( ) ( ) 2(1 ) *<sup>n</sup> k i ki k i ij k j k j n n k i k i ij k j ki k ij j k k j j k k i ki k i k i k <sup>s</sup> da x ax R x A W l dx s s <sup>s</sup> N x ax A W ax A l l l <sup>s</sup> da x axW l dx* (30) 2 2 2 (1) 1 (2) 1 2 1 1 2 2 2 2 1 (1) 2 1 2 1 ( ) ( ) 2(1 ) ( ) () () ( ) 2(1 ) *n n kk i ij kj k i ij k j j j k <sup>n</sup> k i k k i k i ki ij k j k i ki k j <sup>s</sup> ssa x A W b x A l <sup>s</sup> db x ssa x b x A bx l dx* (31)

where the (1) *Ai <sup>j</sup>* and (2) *Ai <sup>j</sup>* are the weighting coefficients of linear algebraic equations. (See Appendix A.1 for more details).

Finally, the conditions (23) and (24) are replaced by:

$$\left| l\_k \,\,\mathcal{W}\_{k\,\,n} - l\_{k+1} \,\,\mathcal{W}\_{\{k+1\}} \,\, 1 = 0 \,\,\right| \,\, \Psi\_{k\,\,n} - \Psi\_{\{k+1\}} \,\, 1 = 0 \,\,\, \,\tag{32a,b}$$

$$a\_k \left( \left( N\_k \left( 1 \right) + \frac{\kappa}{2(1+\nu)} a\_k(1) \right) \sum\_{j=1}^n A\_{nj}^{(1)} W\_{kj} - \frac{\kappa}{2(1+\nu)} a\_k(1) \Psi\_{k,n} \right)$$

$$-a\_{k+1} \left( \left( N\_{k+1}(0) + \frac{\kappa}{2(1+\nu)} a\_{k+1}(0) \right) \sum\_{j=1}^n A\_{1j}^{(1)} W\_{(k+1)j} - \frac{\kappa}{2(1+\nu)} a\_{k+1}(0) \Psi\_{k,1} \right) = 0; \qquad \text{(33a,b)}$$

$$\frac{a\_k}{l\_k} b\_k(1) \sum\_{j=1}^n A\_{nj}^{(1)} \Psi\_{kj} - \frac{a\_{k+1}}{l\_{k+1}} b\_{k+1}(0) \sum\_{j=1}^n A\_{1j}^{(1)} \Psi\_{(k+1)j} = 0$$

and the boundary conditions (25) and (26) replaced by:

( ) (1) (1) ( ) ( ) *<sup>k</sup> k i kk k kk i k i k <sup>l</sup> N x Rv Rv x x N*

*Qx Nx ax A W ax*

() ()

*Mx bx A*

1 1 (2) 1 (1)

*<sup>n</sup> k i*

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

2

1 1

(1)

*n k k k j k j k k j*

1 1 1

*N a AW a*

 

1

*n k k k n j k j k kn j*

*N a AW a*

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

*n n <sup>k</sup> <sup>k</sup>*

*k k j j bA b A*

 

*l*

*n n kk i ij kj k i ij k j j j k*

*s s <sup>s</sup> N x ax A W ax A*

*k j*

*k k j j k*

1 () () () ( ) 2(1 ) 2(1 ) *n ki ki k i ij k j k i ki j*

> *n ki ki k ij j j*

1

1

*k i k i ij k j ki k ij j*

2 2 2 2 1 (1) 2

*<sup>s</sup> db x ssa x b x A bx l dx*

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

where the (1) *Ai <sup>j</sup>* and (2) *Ai <sup>j</sup>* are the weighting coefficients of linear algebraic equations. (See

(1) 1 1 1 1 ( 1) 1 1 1

 

(1) 1 (1)

*k nj kj k j k j*

(1) (0) 0

(0) (0) (0) 0; 2(1 ) 2(1 )

 

> 

2 1

(1)

(1)

1 1

 

1

1 ( 1) 1 0 *k kn k k lW l W* ; ( 1) 1 0 *kn k* ; (32a,b)

 

1 1 ( 1)

 

(33a,b)

*<sup>n</sup> k i k k i k i ki ij k j k i ki k j*

 

*n n*

(28)

 

> 

> >

(29)

(30)

(27)

(31)

Equations (18, 19, 20) assumed the form:

The equations of motion (21) and (22) become:

2

Appendix A.1 for more details).

2

*l dx*

*k*

2 2

*ki k i ij k j*

*<sup>s</sup> da x ax R x A W*

2 1 (1)

( ) ( ) 2(1 )

1 2

*<sup>s</sup> da x axW*

1 2

( ) ( ) 2(1 )

1 2

( ) ( ) 2(1 )

Finally, the conditions (23) and (24) are replaced by:

 and the boundary conditions (25) and (26) replaced by:

*l l*

*k i*

2

2

2 2 2

*l dx*

2 2 2

*l l l*

*ki k i k i*

2 2 (1) 1 (2)

*<sup>s</sup> ssa x A W b x A*

1

*s*

$$\begin{aligned} \left(N\_{1}(0) + \frac{\kappa}{2(1+\nu)}a\_{1}(0)\right) \sum\_{j=1}^{n} A\_{1j}^{(1)} \, \mathcal{W}\_{1j} - \frac{\kappa}{2(1+\nu)}a\_{1}(0) \Psi\_{11} - l\_{1}K\_{\mathbb{V}V1} \, \mathcal{W}\_{11} &= 0 \end{aligned} \tag{34a.b}$$

$$\begin{aligned} \, \mathcal{K}\_{\mathbb{W}1} \, \Psi\_{11} - \frac{b\_{1}(0)}{l\_{1}} \sum\_{j=1}^{n} A\_{1j}^{(1)} \, \Psi\_{1j} &= 0 \end{aligned} \tag{34a.b}$$

$$\begin{aligned} \left(N\_{d}\, \{1\} + \frac{\kappa}{2(1+\nu)}a\_{d}(1)\right) \sum\_{j=1}^{n} A\_{nj}^{(1)} \, \mathcal{W}\_{dj} - \frac{\kappa}{2(1+\nu)}a\_{d}(1) \, \Psi\_{d\eta} - l\_{d}K\_{\mathbb{V}\mathbb{W}d} \, \mathcal{W}\_{d\eta} &= 0; \\ \Psi\_{\mathbb{W}n} \, \Psi\_{d\eta} - \frac{b\_{d}(1)}{l\_{d}} \sum\_{j=1}^{n} A\_{nj}^{(1)} \, \Psi\_{dj} &= 0 \end{aligned} \tag{35a.b}$$

The DQM linear equation system is used to determine the natural frequencies and mode shapes of the rotating beam.

*d j*

1

The number of terms taken in the summations had been studied for many situations and the system has acceptable convergence by *n*= 21 terms. (See Table 1)
