**5. Numerical results**

In the following examples some calculations were performed over elliptical cross sections. ( 0.886364 *)*. 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

of the cross section of the beam will be ( ) ( ) <sup>4</sup> *k k eh x A x* , 3 ( ) ( ) <sup>64</sup> *k k eh x I x* , and for this

particular situation there are:

$$a\_k(\infty) = \frac{h\_k(\infty)}{h\_k(0)} \; ; \; b\_k(\infty) = \left(\frac{h\_k(\infty)}{h\_k(0)}\right)^3$$

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

$$h\_k(\mathbf{x}) = c\_{0k} + c\_{1k} \left| \mathbf{x} + c\_{2k} \right| \mathbf{x}^2$$

Free Vibration Analysis of Centrifugally

excellent.

 0.30 ; 0.886364 

Stiffened Non Uniform Timoshenko Beams 301

(Barnejee, 2006) when Banerjee´s parameter is n=1. As it can be observed the agreement is

0

10 3.38628165 11.7689336 26.5951854 46.6658427 71.0448001 100 3.37398143 11.7248502 26.4438604 46.1408176 69.5136708 1000 3.37385398 11.7243988 26.4423706 46.1357196 69.4986357 2000 3.37385302 11.7243954 26.4423593 46.1356810 69.4985219 3000 3.37385284 11.7243946 26.4423572 46.1356739 69.4985008

 10 10 11.6074237 25.8805102 44.0407905 66.3753084 92.6859627 100 11.6098042 25.7094320 43.5638284 65.4674874 90.8491237 1000 11.6098077 25.7074626 43.5585908 65.4579769 90.8301746 2000 11.6098078 25.7074476 43.5585511 65.4579049 90.8300310 3000 11.6098078 25.7074448 43.5585437 65.4578915 90.8300044

*A EI L* 0 0 /

for a one-span beam, *l L* <sup>1</sup> / 1 ;

using MEF.

*<sup>d</sup>* . The

Table 2. Convergence analysis of the frequency coefficients <sup>2</sup> *i i*

<sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup>

<sup>0</sup>DQM 3.82377 18.3171 47.2638 90.4468 147.992 (Barnejee,2006) 3.82379 18.3173 47.2648 90.4505 148.002 <sup>2</sup>DQM 4.43680 18.9365 47.8706 91.0589 148.609 (Barnejee,2006) 4.43680 18.9366 47.8717 91.0625 148.619 <sup>4</sup>DQM 5.87874 20.6850 49.6446 92.8693 150.444 (Barnejee,2006) 5.87877 20.6851 49.6456 92.8730 150.454 <sup>6</sup>DQM 7.65512 23.3091 52.4622 95.8054 153.450 (Barnejee,2006) 7.65514 23.3093 52.4632 95.8090 153.460 <sup>8</sup>DQM 9.55392 26.5435 56.1584 99.7601 157.555 (Barnejee,2006) 9.55396 26.5437 56.1595 99.7638 157.564

<sup>10</sup>DQM 11.5015 30.1825 60.5628 104.608 162.668 (Barnejee,2006) 11.5015 30.1827 60.5639 104.612 162.677

*A EI L*

All the calculations performed for the following Tables and Graphics used <sup>1</sup> *R* 0 ; and

The DQM results are determined using *n* = 21 in each segment of the beam, and the MEF

The beam considered in Table 4 has one segment and is elastically restrained at its outer end. The parameter of rotation speed *η* is taken equal to 10. The Table presents the frequency coefficients for the first five mode shapes which correspond to different sets of

elastically boundary conditions given by the spring constant parameters *KW d* and *K*

*<sup>d</sup>* .

<sup>1</sup> ; *KW d* 0 ; 0 *K*

1 1 (0)/ (0) *i i*

Table 3. Frequency coefficients <sup>2</sup>

(elliptical cross section).

other details of the beam are specified in the legend of the table.

<sup>1</sup>*s* 1000 ; / 1 /2 *B A h h* ; *KW* <sup>1</sup> ; *K*

results were obtained with 3000 elements.

Number of elements <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup>

And the slope is the derivative of this function

$$h\_k'(\mathbf{x}) = \frac{d h\_k(\mathbf{x})}{d \mathbf{x}} = c\_{1k} + \mathbf{2}c\_{2k} \ge 0$$

where 0*<sup>k</sup> c* , 1*<sup>k</sup> c* and <sup>2</sup>*<sup>k</sup> c* are constants, which are defined by the heights and slopes at both ends of each segment *k*. The heights and slopes at each end are identified with the subscript *A* for *x*=0: *Ak h* ; *Ak h* and with the subscript *B* for *x*=1: *Bk h* ; *Bk h* .

If the segment of the beam shows a linear variation of height , <sup>2</sup> 0 *<sup>k</sup> c* and

$$h\_{A\,k} = \mathfrak{c}\_{0\,k} \,\,\, \dot{\mathfrak{a}}\,\, h\_{B\,k} = \mathfrak{c}\_{0\,k} + \mathfrak{c}\_{1\,k} \,\, \dot{\mathfrak{a}}\,\, h'\_{A\,k} = \mathfrak{h}'\_{\partial k} = \mathfrak{c}\_{1\,k}$$

As it can be seen in Table 1, the frequency coefficients calculated by the Differential Quadrature Method, DQM, using a summation with *n* 19 (*i*= 1, 2, 3, …, *n*) points, show none significant improvement.


Table 1. Convergence analysis of the DQM, for a two-span rotating Timoshenko beam elastically restrained al both ends, with a quadratic variation of height.

The frequency coefficients in Table 1, correspond to a beam of two segments, rotating at speed 10 , whose characteristics are: elliptical cross section; 0.3 ; 0.886364 ; *R*1=0; *l Ll L* 1 2 / / 1/2 ; 1*s* 300 ; 1 1 / 1/2 *B A h h* ; 1 0 *Bh* ; 2 1 / 1/2 *A B h h* ; 2 2 / 1/2 *B A h h* ; 2 0 *Ah* ; 1 *KW* 10 ; 1 *K* 5 ; *KW d* 0.1 ; 1 *K<sup>d</sup>* .

In Table 2 the values obtained for the natural frequency coefficients using the finite element method are presented for <sup>2</sup> *A EI L* 0 0 / 0 and 10 . The number of elements is increased from 10 to 3000.

The model of the rotating beam of Table 2 has the following characteristics: one segment; rectangular cross section; 0.3 ; 10(1 ) /(12 11 ) =0.849673; *R*1=0; 1*s* 300 ; / 1/4 *B A h h* ; *h*'B=0 ; *KW* <sup>1</sup> ; *K* <sup>1</sup> ; *KW d* 0 ; 0 *K<sup>d</sup>* .

In the first examples it is assumed a perfect clamped condition at the axis of rotation, given by: *KW* <sup>1</sup> and *K*<sup>1</sup> . (Tables 3, 4 and 5).

Table 3 presents the effect of the rotational speed parameter *η* on the natural frequency coefficients of a rotating cantilever beam of one segment, ( *KW* <sup>1</sup> ; *K* <sup>1</sup> ; *KW d* 0 ; 0 *K<sup>d</sup>* ). The results correspond to a linear variation of height and a comparison is made with

( ) ( ) <sup>2</sup> *<sup>k</sup> k k k dh x hx c c x dx*

where 0*<sup>k</sup> c* , 1*<sup>k</sup> c* and <sup>2</sup>*<sup>k</sup> c* are constants, which are defined by the heights and slopes at both ends of each segment *k*. The heights and slopes at each end are identified with the subscript

*Ak k* <sup>0</sup> *h c* ; *Bk k k* 0 1 *hcc* ; *Ak Bk k*<sup>1</sup> *h hc*

As it can be seen in Table 1, the frequency coefficients calculated by the Differential Quadrature Method, DQM, using a summation with *n* 19 (*i*= 1, 2, 3, …, *n*) points, show

n <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup> 5 15.6861 29.2939 49.1602 63.9792 112.610 7 15.1981 28.9907 46.9070 64.9219 88.8670 9 14.9057 29.5079 47.4960 64.7054 87.4079 11 14.8340 29.6332 47.6579 64.7247 87.6724 13 14.8281 29.6467 47.6811 64.7310 87.7047 15 14.8291 29.6464 47.6820 64.7319 87.7079 17 14.8295 29.6460 47.6816 64.7320 87.7080 19 14.8296 29.6459 47.6815 64.7320 87.7080 21 14.8296 29.6459 47.6815 64.7320 87.7080

Table 1. Convergence analysis of the DQM, for a two-span rotating Timoshenko beam

10 , whose characteristics are: elliptical cross section; 0.3

coefficients of a rotating cantilever beam of one segment, ( *KW* <sup>1</sup> ; *K*

<sup>1</sup> . (Tables 3, 4 and 5).

The frequency coefficients in Table 1, correspond to a beam of two segments, rotating at

*R*1=0; *l Ll L* 1 2 / / 1/2 ; 1*s* 300 ; 1 1 / 1/2 *B A h h* ; 1 0 *Bh* ; 2 1 / 1/2 *A B h h* ;

In Table 2 the values obtained for the natural frequency coefficients using the finite element

The model of the rotating beam of Table 2 has the following characteristics: one segment;

In the first examples it is assumed a perfect clamped condition at the axis of rotation, given

Table 3 presents the effect of the rotational speed parameter *η* on the natural frequency

*<sup>d</sup>* ). The results correspond to a linear variation of height and a comparison is made with

 10(1 ) /(12 11 ) 

<sup>1</sup> ; *KW d* 0 ; 0 *K*

 *A EI L* 0 0 / 0 and

; *KW d* 0.1 ; 1 *K*

*<sup>d</sup>* .

 

*<sup>d</sup>* . 10 . The number of elements is

=0.849673; *R*1=0; 1*s* 300 ;

<sup>1</sup> ; *KW d* 0 ;

 ; 0.886364 

;

elastically restrained al both ends, with a quadratic variation of height.

;

2 2 / 1/2 *B A h h* ; 2 0 *Ah* ; 1 *KW* 10 ; 1 *K* 5

method are presented for <sup>2</sup>

1 2

And the slope is the derivative of this function

none significant improvement.

speed

increased from 10 to 3000.

by: *KW* <sup>1</sup> and *K*

0 *K*

rectangular cross section; 0.3

/ 1/4 *B A h h* ; *h*'B=0 ; *KW* <sup>1</sup> ; *K*

*A* for *x*=0: *Ak h* ; *Ak h* and with the subscript *B* for *x*=1: *Bk h* ; *Bk h* .

If the segment of the beam shows a linear variation of height , <sup>2</sup> 0 *<sup>k</sup> c* and


(Barnejee, 2006) when Banerjee´s parameter is n=1. As it can be observed the agreement is excellent.

Table 2. Convergence analysis of the frequency coefficients <sup>2</sup> *i i A EI L* 0 0 / using MEF.


Table 3. Frequency coefficients <sup>2</sup> 1 1 (0)/ (0) *i i A EI L* for a one-span beam, *l L* <sup>1</sup> / 1 ; <sup>1</sup>*s* 1000 ; / 1 /2 *B A h h* ; *KW* <sup>1</sup> ; *K* <sup>1</sup> ; *KW d* 0 ; 0 *K<sup>d</sup>* .

All the calculations performed for the following Tables and Graphics used <sup>1</sup> *R* 0 ; and 0.30 ; 0.886364 (elliptical cross section).

The DQM results are determined using *n* = 21 in each segment of the beam, and the MEF results were obtained with 3000 elements.

The beam considered in Table 4 has one segment and is elastically restrained at its outer end. The parameter of rotation speed *η* is taken equal to 10. The Table presents the frequency coefficients for the first five mode shapes which correspond to different sets of elastically boundary conditions given by the spring constant parameters *KW d* and *K <sup>d</sup>* . The other details of the beam are specified in the legend of the table.

Free Vibration Analysis of Centrifugally

*K*

0

1

10

Stiffened Non Uniform Timoshenko Beams 303

*<sup>d</sup> KWd* Method <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup>

<sup>0</sup>DQM 11.8651 24.5717 40.8347 59.8775 81.1573

0.1 DQM 15.2667 30.3903 49.8409 67.9228 90.5546

<sup>1</sup>DQM 15.6938 31.4585 52.2093 72.0062 98.7038

<sup>10</sup>DQM 15.7412 31.5756 52.4458 72.4214 99.4803

DQM 15.7466 31.5887 52.4718 72.4669 99.5627

<sup>0</sup>DQM 11.9142 25.1342 42.8878 62.4877 85.6040

0.1 DQM 16.2121 31.6526 50.5459 67.9979 90.8436

<sup>1</sup>DQM 16.9952 33.8476 55.0090 75.3283 102.166

<sup>10</sup>DQM 17.0842 34.0961 55.4704 76.2542 103.723

DQM 17.0942 34.1238 55.5205 76.3541 103.882

<sup>0</sup>DQM 11.9157 25.1505 42.9498 62.5733 85.7690

0.1 DQM 16.2528 31.7152 50.5831 68.0018 90.8571

<sup>1</sup>DQM 17.0528 33.9729 55.1728 75.5622 102.430

<sup>10</sup>DQM 17.1437 34.2281 55.6450 76.5200 104.034

DQM 17.1539 34.2566 55.6962 76.6231 104.197

<sup>0</sup>DQM 11.9158 25.1524 42.9569 62.5831 85.7880

0.1 DQM 16.2575 31.7225 50.5875 68.0023 90.8587

<sup>1</sup>DQM 17.0595 33.9876 55.1921 75.5901 102.461

<sup>10</sup>DQM 17.1506 34.2436 55.6656 76.5517 104.071

DQM 17.1609 34.2722 55.7169 76.6551 104.234

1 1 (0)/ (0) *i i*

restrained rotating Timoshenko beam, with elliptical cross section and quadratic height

*A EI L*

variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* ,

Table 5. First natural frequencies <sup>2</sup>

2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , *KW* <sup>1</sup> *K*

FEM 11.8796 24.5914 40.8559 59.9110 81.1826

FEM 15.2858 30.4064 49.8638 67.9498 90.5743

FEM 15.7140 31.4754 52.2362 72.0330 98.7265

FEM 15.7616 31.5927 52.4732 72.4482 99.5038

FEM 15.7669 31.6059 52.4993 72.4937 99.5862

FEM 11.9288 25.1532 42.9079 62.5196 85.6258

FEM 16.2314 31.6672 50.5682 68.0245 90.8635

FEM 17.0160 33.8634 55.0372 75.3520 102.190

FEM 17.1052 34.1120 55.4993 76.2779 103.748

FEM 17.1152 34.1398 55.5496 76.3778 103.907

FEM 11.9302 25.1695 42.9699 62.6051 85.7907

FEM 16.2721 31.7297 50.6053 68.0283 90.8770

FEM 17.0737 33.9886 55.2011 75.5857 102.453

FEM 17.1648 34.2440 55.6741 76.5435 104.059

FEM 17.1750 34.2725 55.7255 76.6466 104.222

FEM 11.9304 25.1713 42.9770 62.6149 85.8097

FEM 16.2768 31.7370 50.6097 68.0288 90.8786

FEM 17.0804 34.0033 55.2205 75.6136 102.485

FEM 17.1717 34.2595 55.6947 76.5752 104.096

FEM 17.1820 34.2881 55.7461 76.6786 104.259

 <sup>1</sup> , 10 .

for a two-span elastically


The beam model considered in Table 5 has two segments of equal length and similar conditions and parameters as Table 4.

Table 4. First natural frequencies <sup>2</sup> 1 1 (0)/ (0) *i i A EI L* for a one-span rotating Timoshenko beam, with elliptical cross section and quadratic height variation along the axis. 0.3 ; 1*s* 300 ; / 1 /2 *B A h h* ; 0 *Bh* ; *KW* <sup>1</sup> ; *K* <sup>1</sup> ; 10 .

The beam model considered in Table 5 has two segments of equal length and similar

*<sup>d</sup> KWd* Method Ω<sup>1</sup> Ω<sup>2</sup> Ω<sup>3</sup> Ω<sup>4</sup> Ω<sup>5</sup>

<sup>0</sup>DQM 11.2148 27.6174 50.0089 77.5866 108.472

0.1 DQM 15.4254 32.5178 52.8516 79.0733 109.357

<sup>1</sup>DQM 18.0157 40.3494 65.6538 92.2848 119.836

<sup>10</sup>DQM 18.3978 41.6361 69.0216 99.4111 131.859

DQM 18.4417 41.7750 69.3474 100.033 132.894

<sup>0</sup>DQM 11.3941 29.3678 53.0174 81.0192 112.024

0.1 DQM 15.6233 32.8965 55.0307 82.2247 112.825

<sup>1</sup>DQM 19.2962 41.3980 65.9339 92.3365 120.822

<sup>10</sup>DQM 19.9179 43.3987 70.5662 100.622 132.723

DQM 19.9899 43.6199 71.0558 101.509 134.136

<sup>0</sup>DQM 11.4913 30.3954 55.1815 84.0260 115.628

0.1 DQM 15.7621 33.1503 56.5688 84.8630 116.210

<sup>1</sup>DQM 20.4765 42.5961 66.2635 92.3899 121.730

<sup>10</sup>DQM 21.3539 45.5548 72.8197 102.560 134.141

DQM 21.4553 45.8807 73.5609 103.912 136.261

<sup>0</sup>DQM 11.5091 30.5860 55.6105 84.6706 116.454

0.1 DQM 15.7905 33.2010 56.8768 85.4233 116.975

<sup>1</sup>DQM 20.7557 42.9193 66.3549 92.4035 121.943

<sup>10</sup>DQM 21.6961 46.1510 73.5285 103.223 134.643

DQM 21.8045 46.5039 74.3452 104.735 137.026

1 1 (0)/ (0) *i i*

Timoshenko beam, with elliptical cross section and quadratic height variation along the

*A EI L*

Table 4. First natural frequencies <sup>2</sup>

; 1*s* 300 ; / 1 /2 *B A h h* ; 0 *Bh* ; *KW* <sup>1</sup> ; *K*

FEM 11.2375 27.6743 50.0711 77.6432 108.523

FEM 15.4438 32.5548 52.9087 79.1298 109.408

FEM 18.0465 40.3841 65.6882 92.3208 119.877

FEM 18.4315 41.6757 69.0611 99.4484 131.893

FEM 18.4757 41.8151 69.3878 100.071 132.929

FEM 11.4148 29.4104 53.0660 81.0662 112.068

FEM 15.6400 32.9308 55.0763 82.2710 112.868

FEM 19.3219 41.4295 65.9674 92.3723 120.859

FEM 19.9463 43.4345 70.6034 100.658 132.756

FEM 20.0187 43.6562 71.0937 101.546 134.170

FEM 11.5115 30.4328 55.2229 84.0663 115.665

FEM 15.7780 33.1835 56.6092 84.9031 116.247

FEM 20.4994 42.6248 66.2963 92.4255 121.765

FEM 21.3795 45.5875 72.8543 102.594 134.174

FEM 21.4813 45.9139 73.5962 103.947 136.294

FEM 11.5291 30.6228 55.6510 84.7101 116.491

FEM 15.8064 33.2340 56.9165 85.4625 117.012

FEM 20.7782 42.9475 66.3875 92.4392 121.978

FEM 21.7214 46.1832 73.5626 103.257 134.675

FEM 21.8302 46.5368 74.3801 104.769 137.059

for a one-span rotating

 <sup>1</sup> ; 10 .

conditions and parameters as Table 4.

*K*

0

1

10

axis. 0.3 


Table 5. First natural frequencies <sup>2</sup> 1 1 (0)/ (0) *i i A EI L* for a two-span elastically restrained rotating Timoshenko beam, with elliptical cross section and quadratic height variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* , 2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , *KW* <sup>1</sup> *K* <sup>1</sup> , 10 .

Free Vibration Analysis of Centrifugally

*K*

0

1

10

Stiffened Non Uniform Timoshenko Beams 305

*<sup>d</sup> KWd* Method <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup>

<sup>0</sup>DQM 11.3734 23.4059 39.2570 56.9363 78.4787

0.1 DQM 14.4551 28.9705 47.6260 65.1400 87.8678

<sup>1</sup>DQM 14.8320 29.9641 49.6507 69.1005 95.0494

<sup>10</sup>DQM 14.8738 30.0733 49.8536 69.5057 95.7100

DQM 14.8785 30.0856 49.8761 69.5501 95.7801

<sup>0</sup>DQM 11.4126 23.8883 41.1048 59.4030 82.7965

0.1 DQM 15.3087 30.2367 48.3683 65.2804 88.0547

<sup>1</sup>DQM 15.9925 32.2677 52.2109 72.5363 98.2033

<sup>10</sup>DQM 16.0702 32.4974 52.6080 73.4315 99.5067

DQM 16.0790 32.5231 52.6513 73.5282 99.6404

<sup>0</sup>DQM 11.4138 23.9023 41.1598 59.4837 82.9526

0.1 DQM 15.3450 30.2988 48.4074 65.2877 88.0636

<sup>1</sup>DQM 16.0434 32.3866 52.3585 72.7730 98.4379

<sup>10</sup>DQM 16.1228 32.6224 52.7651 73.6979 99.7796

DQM 16.1317 32.6487 52.8094 73.7975 99.9164

<sup>0</sup>DQM 11.4139 23.9039 41.1661 59.4929 82.9706

0.1 DQM 15.3493 30.3061 48.4120 65.2886 88.0646

<sup>1</sup>DQM 16.0493 32.4005 52.3759 72.8012 98.4661

<sup>10</sup>DQM 16.1289 32.6371 52.7836 73.7295 99.8122

DQM 16.1378 32.6635 52.8280 73.8295 99.9493

1 1 (0)/ (0) *i i*

restrained rotating Timoshenko beam, with elliptical cross section and quadratic height

*A EI L*

variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* ,

Table 7. First natural frequencies <sup>2</sup>

2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , <sup>1</sup> *KW* 10 , 1 *K* 10

FEM 11.3904 23.4273 39.2820 56.9721 78.5041

FEM 14.4773 28.9890 47.6542 65.1664 87.8894

FEM 14.8553 29.9837 49.6827 69.1264 95.0757

FEM 14.8972 30.0931 49.8861 69.5316 95.7371

FEM 14.9019 30.1054 49.9086 69.5761 95.8073

FEM 11.4297 23.9092 41.1294 59.4361 82.8192

FEM 15.3312 30.2539 48.3957 65.3061 88.0766

FEM 16.0166 32.2865 52.2436 72.5592 98.2306

FEM 16.0945 32.5165 52.6413 73.4545 99.5353

FEM 16.1032 32.5422 52.6847 73.5513 99.6691

FEM 11.4309 23.9232 41.1844 59.5168 82.9753

FEM 15.3676 30.3159 48.4348 65.3134 88.0854

FEM 16.0676 32.4055 52.3914 72.7958 98.4653

FEM 16.1471 32.6415 52.7985 73.7208 99.8083

FEM 16.1560 32.6679 52.8428 73.8204 99.9453

FEM 11.4310 23.9248 41.1908 59.5260 82.9933

FEM 15.3718 30.3232 48.4394 65.3142 88.0865

FEM 16.0735 32.4194 52.4088 72.8240 98.4934

FEM 16.1532 32.6562 52.8170 73.7524 99.8409

FEM 16.1622 32.6827 52.8615 73.8524 99.9782

 , 10 .

for a two-span elastically


Next Tables, 6 to 10, correspond to beams of two segments, elastically restrained at both ends and any particular details are expressed in each legend.

Table 6. First natural frequencies <sup>2</sup> 1 1 (0)/ (0) *i i A EI L* for a two-span elastically restrained rotating Timoshenko beam, with elliptical cross section and quadratic height variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* , 2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , *KW* <sup>1</sup> , 1 *K* 0.1 , 10 .

Next Tables, 6 to 10, correspond to beams of two segments, elastically restrained at both

*<sup>d</sup> KWd* Method <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup>

<sup>0</sup>DQM 9.98841 21.2706 37.3110 54.9224 77.7336

0.1 DQM 12.4181 26.7466 45.4389 63.7717 87.4947

<sup>1</sup>DQM 12.7051 27.6834 47.3154 67.8701 94.9448

<sup>10</sup>DQM 12.7370 27.7869 47.5065 68.2901 95.6398

DQM 12.7406 27.7985 47.5276 68.3362 95.7137

<sup>0</sup>DQM 10.0086 21.6505 39.0007 57.4853 82.1930

0.1 DQM 13.1047 28.0775 46.2626 63.9615 87.6768

<sup>1</sup>DQM 13.6220 29.9765 49.8847 71.5084 98.2767

<sup>10</sup>DQM 13.6812 30.1918 50.2664 72.4302 99.6646

DQM 13.6879 30.2159 50.3082 72.5299 99.8074

<sup>0</sup>DQM 10.0092 21.6615 39.0505 57.5689 82.3537

0.1 DQM 13.1336 28.1416 46.3059 63.9714 87.6854

<sup>1</sup>DQM 13.6618 30.0922 50.0321 71.7561 98.5256

<sup>10</sup>DQM 13.7222 30.3131 50.4233 72.7081 99.9556

DQM 13.7290 30.3379 50.4661 72.8107 100.102

<sup>0</sup>DQM 10.0093 21.6628 39.0562 57.5786 82.3722

0.1 DQM 13.1369 28.1491 46.3110 63.9726 87.6864

<sup>1</sup>DQM 13.6664 30.1057 50.0495 71.7856 98.5555

<sup>10</sup>DQM 13.7270 30.3273 50.4418 72.7411 99.9903

DQM 13.7338 30.3521 50.4847 72.8441 100.137

1 1 (0)/ (0) *i i*

restrained rotating Timoshenko beam, with elliptical cross section and quadratic height

*A EI L*

variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* ,

Table 6. First natural frequencies <sup>2</sup>

2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , *KW* <sup>1</sup> , 1 *K* 0.1

FEM 10.0246 21.3074 37.3506 54.9699 77.7658

FEM 12.4641 26.7810 45.4827 63.8054 87.5227

FEM 12.7526 27.7193 47.3634 67.9035 94.9788

FEM 12.7847 27.8230 47.5548 68.3236 95.6747

FEM 12.7883 27.8347 47.5760 68.3697 95.7487

FEM 10.0451 21.6876 39.0412 57.5291 82.2227

FEM 13.1521 28.1101 46.3049 63.9941 87.7052

FEM 13.6718 30.0120 49.9325 71.5384 98.3117

FEM 13.7312 30.2278 50.3147 72.4604 99.7012

FEM 13.7379 30.2519 50.3566 72.5601 99.8441

FEM 10.0457 21.6987 39.0910 57.6127 82.3835

FEM 13.1811 28.1741 46.3481 64.0039 87.7138

FEM 13.7116 30.1278 50.0799 71.7860 98.5607

FEM 13.7723 30.3491 50.4716 72.7381 99.9923

FEM 13.7792 30.3739 50.5145 72.8408 100.139

FEM 10.0458 21.6999 39.0968 57.6224 82.4020

FEM 13.1844 28.1817 46.3532 64.0051 87.7149

FEM 13.7163 30.1414 50.0973 71.8155 98.5906

FEM 13.7771 30.3634 50.4901 72.7711 100.027

FEM 13.7840 30.3882 50.5331 72.8741 100.174

 , 10 .

for a two-span elastically

ends and any particular details are expressed in each legend.

*K*

0

1

10


Table 7. First natural frequencies <sup>2</sup> 1 1 (0)/ (0) *i i A EI L* for a two-span elastically restrained rotating Timoshenko beam, with elliptical cross section and quadratic height variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* , 2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , <sup>1</sup> *KW* 10 , 1 *K* 10 , 10 .

Free Vibration Analysis of Centrifugally

*K*

0

1

10

Stiffened Non Uniform Timoshenko Beams 307

*<sup>d</sup> KWd* Method <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup>

<sup>0</sup>DQM 10.3650 21.7083 37.5771 54.9536 77.3897

0.1 DQM 12.9366 27.1514 45.6367 63.6520 86.9652

<sup>1</sup>DQM 13.2417 28.0899 47.4960 67.6756 94.0576

<sup>10</sup>DQM 13.2755 28.1934 47.6848 68.0876 94.7110

DQM 13.2794 28.2050 47.7057 68.1329 94.7804

<sup>0</sup>DQM 10.3892 22.1044 39.2705 57.4666 81.7463

0.1 DQM 13.6565 28.4599 46.4416 63.8427 87.1268

<sup>1</sup>DQM 14.2060 30.3646 50.0205 71.2606 97.2421

<sup>10</sup>DQM 14.2687 30.5803 50.3961 72.1634 98.5382

DQM 14.2757 30.6044 50.4372 72.2610 98.6715

<sup>0</sup>DQM 10.3900 22.1160 39.3204 57.5486 81.9025

0.1 DQM 13.6869 28.5231 46.4839 63.8527 87.1344

<sup>1</sup>DQM 14.2479 30.4798 50.1652 71.5043 97.4786

<sup>10</sup>DQM 14.3119 30.7011 50.5501 72.4364 98.8132

DQM 14.3191 30.7259 50.5922 72.5368 98.9498

<sup>0</sup>DQM 10.3901 22.1173 39.3262 57.5580 81.9205

0.1 DQM 13.6904 28.5305 46.4889 63.8539 87.1353

<sup>1</sup>DQM 14.2527 30.4933 50.1823 71.5333 97.5069

<sup>10</sup>DQM 14.3169 30.7153 50.5682 72.4688 98.8460

DQM 14.3241 30.7401 50.6105 72.5696 98.9829

1 1 (0)/ (0) *i i*

restrained rotating Timoshenko beam, with elliptical cross section and quadratic height

*A EI L*

variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* ,

Table 9. First natural frequencies <sup>2</sup>

2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , <sup>1</sup> *KW* 10 , 1 *K* 1

FEM 10.3937 21.7398 37.6121 54.9961 77.4183

FEM 12.9736 27.1805 45.6754 63.6816 86.9900

FEM 13.2800 28.1204 47.5385 67.7048 94.0875

FEM 13.3141 28.2242 47.7276 68.1169 94.7416

FEM 13.3179 28.2358 47.7485 68.1621 94.8111

FEM 10.4182 22.1360 39.3061 57.5054 81.7724

FEM 13.6946 28.4875 46.4789 63.8714 87.1519

FEM 14.2462 30.3947 50.0627 71.2866 97.2727

FEM 14.3091 30.6108 50.4388 72.1896 98.5700

FEM 14.3161 30.6350 50.4800 72.2872 98.7035

FEM 10.4190 22.1475 39.3560 57.5874 81.9286

FEM 13.7250 28.5506 46.5212 63.8813 87.1595

FEM 14.2881 30.5099 50.2075 71.5301 97.5092

FEM 14.3524 30.7317 50.5928 72.4624 98.8452

FEM 14.3596 30.7565 50.6350 72.5629 98.9819

FEM 10.4190 22.1488 39.3618 57.5968 81.9466

FEM 13.7286 28.5580 46.5261 63.8825 87.1604

FEM 14.2930 30.5235 50.2245 71.5591 97.5375

FEM 14.3574 30.7458 50.6110 72.4948 98.8780

FEM 14.3646 30.7707 50.6532 72.5957 99.0150

 , 10 .

for a two-span elastically


Table 8. First natural frequencies <sup>2</sup> 1 1 (0)/ (0) *i i A EI L* for a two-span elastically restrained rotating Timoshenko beam, with elliptical cross section and quadratic height variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* , 2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , <sup>1</sup> *KW* 10 , 1 *K* 5 , 10 .

*<sup>d</sup> KWd* Method <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup>

<sup>0</sup>DQM 11.0954 22.8658 38.6771 56.1987 78.0667

0.1 DQM 14.0189 28.3688 46.9190 64.5736 87.5306

<sup>1</sup>DQM 14.3726 29.3408 48.8766 68.5581 94.6820

<sup>10</sup>DQM 14.4118 29.4478 49.0737 68.9659 95.3401

DQM 14.4162 29.4598 49.0954 69.0107 95.4100

<sup>0</sup>DQM 11.1299 23.3178 40.4680 58.6771 82.4016

0.1 DQM 14.8296 29.6459 47.6815 64.7320 87.7080

<sup>1</sup>DQM 15.4691 31.6287 51.4150 72.0511 97.8483

<sup>10</sup>DQM 15.5418 31.8531 51.8028 72.9497 99.1495

DQM 15.5499 31.8782 51.8452 73.0468 99.2832

<sup>0</sup>DQM 11.1309 23.3309 40.5211 58.7582 82.5578

0.1 DQM 14.8640 29.7082 47.7217 64.7403 87.7163

<sup>1</sup>DQM 15.5170 31.7461 51.5610 72.2905 98.0837

<sup>10</sup>DQM 15.5912 31.9764 51.9582 73.2187 99.4233

DQM 15.5996 32.0021 52.0016 73.3186 99.5601

<sup>0</sup>DQM 11.1310 23.3324 40.5272 58.7675 82.5758

0.1 DQM 14.8680 29.7155 47.7264 64.7412 87.7173

<sup>1</sup>DQM 15.5225 31.7598 51.5783 72.3191 98.1119

<sup>10</sup>DQM 15.5970 31.9908 51.9765 73.2506 99.4560

DQM 15.6053 32.0166 52.0200 73.3510 99.5931

1 1 (0)/ (0) *i i*

restrained rotating Timoshenko beam, with elliptical cross section and quadratic height

*A EI L*

variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* ,

Table 8. First natural frequencies <sup>2</sup>

2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , <sup>1</sup> *KW* 10 , 1 *K* 5

FEM 11.1149 22.8898 38.7052 56.2367 78.0931

FEM 14.0444 28.3902 46.9506 64.6011 87.5532

FEM 14.3992 29.3634 48.9120 68.5850 94.7094

FEM 14.4386 29.4706 49.1095 68.9930 95.3684

FEM 14.4430 29.4826 49.1314 69.0377 95.4383

FEM 11.1495 23.3414 40.4960 58.7122 82.4254

FEM 14.8556 29.6659 47.7122 64.7587 87.7309

FEM 15.4967 31.6508 51.4509 72.0751 97.8766

FEM 15.5696 31.8754 51.8392 72.9737 99.1791

FEM 15.5778 31.9005 51.8817 73.0708 99.3129

FEM 11.1505 23.3545 40.5491 58.7932 82.5816

FEM 14.8900 29.7282 47.7523 64.7669 87.7393

FEM 15.5447 31.7682 51.5970 72.3143 98.1121

FEM 15.6191 31.9987 51.9947 73.2426 99.4531

FEM 15.6275 32.0245 52.0381 73.3426 99.5900

FEM 11.1506 23.3560 40.5552 58.8025 82.5996

FEM 14.8940 29.7355 47.7570 64.7679 87.7402

FEM 15.5503 31.7819 51.6143 72.3428 98.1403

FEM 15.6249 32.0132 52.0131 73.2745 99.4857

FEM 15.6333 32.0391 52.0566 73.3749 99.6230

 , 10 .

for a two-span elastically

*K*

0

1

10


Table 9. First natural frequencies <sup>2</sup> 1 1 (0)/ (0) *i i A EI L* for a two-span elastically restrained rotating Timoshenko beam, with elliptical cross section and quadratic height variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* , 2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , <sup>1</sup> *KW* 10 , 1 *K* 1 , 10 .

Free Vibration Analysis of Centrifugally





<sup>2</sup> 0 *Ah* ; 10 

a) 1 *KW* 10 , 1 *K* 5

while b) corresponds to 1 *KW* 10 , 1 *K* 5

0.01

0.02

0.03

0



0

0 0.2 0.4 0.6 0.8 1

Second modal shape

0 0.2 0.4 0.6 0.8 1

Third modal shape

0 0.2 0.4 0.6 0.8 1

Fourth modal shape

0 0.2 0.4 0.6 0.8 1

, *KW d* 0 , 0 *K*

kinds of boundary conditions: a) corresponds to 1 *KW* 10 , 1 *K* 5

Fig. 4. Natural frequencies mode shapes for a two-span elastically restrained rotating Timoshenko beams, with elliptical cross section and quadratic height variation along the axis. *l Ll L* 1 2 / / 1/2 ; 1 1 / 1/2 *B A h h* ; 1 0 *Bh* ; 2 1 / 1/2 *A B h h* ; 2 2 / 1/2 *B A h h* ;

Figure 4 shows the first four natural frequency mode shapes for beams, with two different

, *KW d* 0.1 , 1 *K*

First modal shape

Stiffened Non Uniform Timoshenko Beams 309

0

0.01




*<sup>d</sup>* b) <sup>1</sup> *KW* 10 , 1 *K* 5

0

0.01

0.02

0.01

0.02

0.03

0

0.02

0.03

0.04

0 0.2 0.4 0.6 0.8 1

Second modal shape

0 0.2 0.4 0.6 0.8 1

Third modal shape

0 0.2 0.4 0.6 0.8 1

Fourth modal shape

0 0.2 0.4 0.6 0.8 1

, *KW d* 0.1 , 1 *K*

, *KW d* 0 , 0 *K*

*<sup>d</sup>*

> *<sup>d</sup>* ,

*<sup>d</sup>* .

First modal shape


Table 10. First natural frequencies <sup>2</sup> 1 1 (0)/ (0) *i i A EI L* for a two-span elastically restrained rotating Timoshenko beam, with elliptical cross section and quadratic height variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* , 2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , <sup>1</sup> *KW* 10 , 1 *K* 0.1 , 10 .

*<sup>d</sup> KWd* Method <sup>1</sup> <sup>2</sup> <sup>3</sup> <sup>4</sup> <sup>5</sup>

<sup>0</sup>DQM 9.94295 21.1789 37.1287 54.4936 77.1422

0.1 DQM 12.3497 26.6244 45.1340 63.3202 86.7544

<sup>1</sup>DQM 12.6335 27.5522 46.9619 67.3571 93.8220

<sup>10</sup>DQM 12.6651 27.6547 47.1481 67.7705 94.4732

DQM 12.6686 27.6662 47.1687 67.8159 94.5425

<sup>0</sup>DQM 9.96265 21.5532 38.7860 57.0231 81.5044

0.1 DQM 13.0297 27.9504 45.9575 63.5235 86.9102

<sup>1</sup>DQM 13.5409 29.8282 49.4887 70.9743 97.0122

<sup>10</sup>DQM 13.5994 30.0410 49.8608 71.8782 98.3059

DQM 13.6060 30.0648 49.9016 71.9759 98.4391

<sup>0</sup>DQM 9.96324 21.5641 38.8347 57.1056 81.6606

0.1 DQM 13.0583 28.0142 46.0008 63.5342 86.9175

<sup>1</sup>DQM 13.5802 29.9428 49.6333 71.2194 97.2490

<sup>10</sup>DQM 13.6399 30.1611 50.0148 72.1525 98.5813

DQM 13.6467 30.1856 50.0566 72.2531 98.7177

<sup>0</sup>DQM 9.96331 21.5653 38.8403 57.1151 81.6785

0.1 DQM 13.0616 28.0216 46.0058 63.5354 86.9184

<sup>1</sup>DQM 13.5848 29.9563 49.6504 71.2486 97.2773

<sup>10</sup>DQM 13.6447 30.1752 50.0329 72.1851 98.6141

DQM 13.6514 30.1997 50.0748 72.2860 98.7509

*A EI L*

1 1 (0)/ (0) *i i*

variation along the axis. *l L* <sup>1</sup> / 1/2 *l L* <sup>2</sup> / 1/2 , 1 1 / 1/2 *B A h h* , 1 0 *Bh* ,

restrained rotating Timoshenko beam, with elliptical cross section and quadratic height

Table 10. First natural frequencies <sup>2</sup>

2 1 / 1/2 *A B h h* , 2 2 / 1/2 *B A h h* , 2 0 *Ah* , <sup>1</sup> *KW* 10 , 1 *K* 0.1

FEM 9.97877 21.2145 37.1668 54.5377 77.1717

FEM 12.3950 26.6574 45.1757 63.3506 86.7801

FEM 12.6802 27.5867 47.0072 67.3872 93.8529

FEM 12.7119 27.6894 47.1937 67.8007 94.5049

FEM 12.7155 27.7009 47.2144 67.8461 94.5743

FEM 9.99874 21.5890 38.8248 57.0634 81.5315

FEM 13.0763 27.9817 45.9976 63.5530 86.9362

FEM 13.5897 29.8623 49.5335 71.0012 97.0437

FEM 13.6484 30.0754 49.9060 71.9052 98.3387

FEM 13.6550 30.0993 49.9469 72.0030 98.4720

FEM 9.99933 21.5999 38.8736 57.1459 81.6876

FEM 13.1049 28.0454 46.0408 63.5635 86.9436

FEM 13.6291 29.9769 49.6781 71.2462 97.2806

FEM 13.6890 30.1956 50.0600 72.1794 98.6142

FEM 13.6958 30.2201 50.1018 72.2800 98.7508

FEM 9.99940 21.6011 38.8792 57.1553 81.7056

FEM 13.1082 28.0529 46.0459 63.5648 86.9444

FEM 13.6337 29.9904 49.6952 71.2753 97.3090

FEM 13.6938 30.2097 50.0781 72.2120 98.6471

FEM 13.7006 30.2342 50.1201 72.3129 98.7840

 , 10 .

for a two-span elastically

*K*

0

1

10

Fig. 4. Natural frequencies mode shapes for a two-span elastically restrained rotating Timoshenko beams, with elliptical cross section and quadratic height variation along the axis. *l Ll L* 1 2 / / 1/2 ; 1 1 / 1/2 *B A h h* ; 1 0 *Bh* ; 2 1 / 1/2 *A B h h* ; 2 2 / 1/2 *B A h h* ; <sup>2</sup> 0 *Ah* ; 10 

Figure 4 shows the first four natural frequency mode shapes for beams, with two different kinds of boundary conditions: a) corresponds to 1 *KW* 10 , 1 *K* 5 , *KW d* 0 , 0 *K <sup>d</sup>* , while b) corresponds to 1 *KW* 10 , 1 *K* 5 , *KW d* 0.1 , 1 *K<sup>d</sup>* .

Free Vibration Analysis of Centrifugally

**6. Conclusion** 

of rotation *Kψd*.

**7. Appendix A** 

Fig. A1. Grid of *n* points

Malik, 1996).

matrix of order *n*.

Stiffened Non Uniform Timoshenko Beams 311

The differential quadrature method proves to be very efficient to obtain frequencies and

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

As a matter of fact, the differential quadrature method has the same advantage as the finite

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

As Shu presents in his book (Shu, 2000), the differential quadrature method, DQM, is a

In order to obtain the DQM analog equations to the governing equations of the rotating beam and its boundary conditions, the beam domain is discretized in a grid of points using

1.0 *<sup>n</sup> <sup>x</sup>*<sup>1</sup> <sup>0</sup> *<sup>x</sup> <sup>i</sup>* 0.2 *<sup>x</sup>* 0.4 0.6 0.8

The weighting coefficients (1) *Ai <sup>j</sup>* and (2) *Ai <sup>j</sup>* , which appeared in the linear algebraic equations of quadrature (28-35), were determined using the explicit expressions cited by (Bert &

The coefficients (1) *Ai <sup>j</sup>* correspond to first order derivatives and can be arranged in a square

1 cos ( 1) /( 1)

where *n* is the number of discrete points or nodes and *<sup>i</sup> x* is the coordinate of node *i*.

*<sup>x</sup>*

node *i*

The matrix elements (1) *Ai <sup>j</sup>* with *i* <sup>≠</sup> *<sup>j</sup>* , are determined by:

*i n*

; 1,2,..., *i n*

mode shapes of natural vibration, for the rotating Timoshenko beam model.

other methodologies, such as the dynamic stiffness method (Banerjee, 2000, 2001).

element method and it needs less computer memory requirements than the FEM.

numerical technique for solving differential equations.

the Chebyshev – Gauss - Lobato expression, (Shu & Chen, 1999):

2 *<sup>i</sup>*

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

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*=0; *Kψd*=0
