The analysis presented in this chapter considers vibrations of a beam that could represent a blade of simple geometry. The beam is attached to a disc of radius 0rand the disc rotates with the angular velocity  (fig.1) the cross section of the beam is linearly varying and shear centre of each cross section does not coincides with the centre of gravity, consequently the torsional and bending oscillations are coupled. Since the cross section of the beam varies S, I and J are the functions of x.

The governing differential eqns. for coupled bending and torsional vibrations taken from Tomor and Dhole (22) are E

2

2 2 2 2 2 2 2

VIxxxx

    

   



G2

2 2 2 3 3

tIVtSxCxxJxxxxxxx

  

   

 

  

 (1)

Also in a steady flow of speed U, the blade will have some deformation due to aerodynamic force. Then the above differential eqns. become

E022222222   

LxMVtSxVIxxxx

G0222233 

NtIVtSxCxxJxxxxxxx (2)

WherexC is the warping rigidity and 22

xM

, L and N are given by

2 2

 

=

 

 



hxlxlrm02

222022

1

LCCUL2

2'





xCcUN022'

2



The coefficient LCand NC are the lift and moment coefficient about the leading edge which are expressed as and

cC4 1

8

Also,

 

The eqns. (2) when the effect of tensile force F is taken into consideration reduce to E

02

2 2 2 2 2 2 2 2

2      

    

VIxxxx

and (3) G

02

2 2 2 3

3   

   

 

   

tINVtSxCxxJxxxxxxx where

3

01 

xIIx





xJJx10

5

01 

xCCx

3

10 

xIIx





xx10

The equations are now put in terms of dimensionless variable Lx = and using the following substitutions

40 02



, c4

20 02

GJr

82 '

0 2

3 

cK

0

'0

II 

xK



4

104

 

012

'

20

5lSFK The eqn. (3) become





0.1.

1

1312212116161

221 22522222002 3202002223324432

  



 

and



01 11511

224132232' 223344451232

   

 



(4)

The solution of eqns. (4) is of the form tieAftV, tieBt, (5) Where A and B are constants which are not independent and fand satisfy all boundary conditions of the beam which are as follows.

0)1()1()1()1(3

3 3 3 2

2   

     

(6)

For an approximate determination of the fundamental frequency, fis chosen as the shape function for the fundamental mode of uncoupled bending vibrations and as the shape function for the fundamental mode of uncoupled torsional vibrations of a uniform cantilever beam. These shape functions satisfy the boundary condition (6) and are

And 2sin (7) Where 87510.1 Substitutions of eqns. (5) in (4) give

 

011 131221 2116161

2212112225 22002223202 002223324432

   





x

 

And (8)

 

0 1

15111

3421241232'

3342451222412

 



 





 

ddddCddddrAfUiKKfx



 The Stieltjes integrals may now be formed as follows:

 

011 131221 2116161

2212112225 22002223202 0022233244321

0

   



  



1

151113421241232'33424512224121

0

 



 



Or 010292872654321BUiaUaaAUiaaaaaaa (10) And

()(216151413121128aaaaaAUiaa

Where

fda4

4310211

fda2

32102216

fddfdlrlrlra221032020022231312212116

ra



1

0

30024

fdKa

1 02 2 55



1 0



1 0



1



1

0



1



1 0



1 0

213



1 04

451141



1 03

3411515



1 0



1 0



1 0

The homogeneous eqns. (10) admit vanishing solution A, B only if the determinant of their coefficients vanishes. This determinant being complex, both real and imaginary parts must vanish separately on setting the determinant to zero.

216151413121128

10292872654321 

 

equals 0. Or

0254223241UAAUAAA

And

028726UAAA

Where 286161aaaA 1110981761873aaaaaaaaA )(54321151413124aaaaaaaaaA )(54321175aaaaaaA 1181081861676aaaaaaaaA

)(54321181514131277aaaaaaaaaaaA

1191778aaaaA The second eqn. of (11) gives

6

2872

UAA

(12)

Substituting this expression into first eqn. of (11), we obtain:

024RQUPU (13)

Where

)(81638AAAAAP 2657638628712AAAAAAAAAAAQ 264271672AAAAAAAR

From eqn. (13), we obtain the value of the critical speed as given below:

PRQQU2

422

(14)

The right hand side of eqn. (14) is positive. Corresponding to two solutions of 2Ufrom Eqn. (14), there are two values of 2 from eqn. (12). Usually the smaller 2Uis associated with the higher value of 2.

The effect of tensile force on the fundamental frequency of the coupled torsional vibrations of a straight rotating slender beam of linearly varying

blade is taken as a channel-section of height 0band breadth 02band thickness1t.

Blezeno, C.B. And Gramemel, R. ‘Engineering Dynamics’ Blackie and Sons, London,

1954

Boyce, W.E. Effect of hub-radius on the vibrations of uniform bar. Jr. Appl. Mech. 1956 Carngie, W. Vibrations of pretwisted cantilever blade taking an additional effect due to torsion. Jr. of Applied Mechanics, May, 1962 Chun, K.R. Free vibrations of a beam w ith one end spring-binged and the other end free. Jr. of Applied Mechanics, Trans.. ASME. Vol. 94. E. Dec..

1972

Fund, Y.C. An Introduction to the theory of aero-elasticity. John Willey and Sons, New York 1955 Grant, D.A. Vibration frequencies for a uniform beam with one end classically supported and carrying a mass at the other end. Jr. of Applied Mechanics Trans. ASME. Dec.. 1975 Hau, La Effect of small hub-radius change on the bending frequencies of a rotating beam. Jr. Applied Mechanics

1960.

Jacobson, L.S. Natural periods of uniform cantilever beam. Trans. Amer. Soc. Civil Engg. 1939 Jacobson, L.S. Engineering vibrations. Mc Graw Hill and Ayre, R.S. Lee, T.W. Vibration frequencies for a uniform beam with one end spring-hinged and carrying a mass at the other end. Jr. of Applied Mechanics Trans. ASME. Vol. 94. E. sept..

1973. 813-816

Lentin, M. Numerical solution of the beam equation with non- uniform foundation classification. Louise, H. Jones The transverse vibration ofa rotating beam with tip mass, the method of internal equation. Quaterly Jr. of Applied Maths. Vol. 4. Oct..

1975