These analysis presented in this chapter considers vibrations of a slender beam that could represent a turbine blade of simple geometry. The oscillations of a beam whose elastic axis and the line of centre of gravity do not coincide are always coupled. The beam is attached to a disc of radius 0rand disc rotates with angular velocity . The beam in allowed to oscillate in a plane making an angle with the plane of rotation.

Timoshenko beam theory gives the following differential eqns for the coupled bending and torsional vibrations of slender beam

EI 02

2 4 4

 

And GJ

hmxCx

    

2

222

2 4

4'12

2

Where11C is the warping rigidity? When  =0 these above equns. Reduce to two independent ones f or the purely torsional and purely bending oscillations. If the centrifugal force effect is to be considered then above governing eqns become

EI02

2 2 2 2 2 4

4   



h And (1)

GJ02 222

2 4

411

2     





Where 22

xM

the load due to is centrifugal force and is given by

2 2

 

=

   

202

22202sin2

When we consider the beam under steady aerodynamic force eqns (1) becomes

EI02

2 2 2 2 2 4 4



  



h and (2) GJ

02

222

2 4

411

2     



Where L and N are given by

2

2

N= c2 The coefficient LC and NC are lift and moment coefficient about the leading edge which are expressed as



  

dCCLL04

311

and

cC4

1

 

The eqns (2) when the effect of shear force is taken into consideration, reduce to EI

04 4 1 2 2 2 2 2 2 2 4

4    



h And (3) GJ

02

222

2 4

411

2     



The equations (3) becomes

0 sin221

445321 222220222002442

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





and (4)

0224122221244222tUKthUUKKthtIT

Let the motion be harmonic represent able as follows tieAfth, (5) tieBt, Where  is real and A, B are complex constants. The function f () and  satisfy the boundary conditions of the beam which are as follows at=0 and (6)

03 3 2 3 2 2



  

     

at=0

For an approximate determination of the fundamental frequency f is 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 vibration in still air of cantilever beam of uniform cross section. Let the shape functions which satisfy the boundary conditions (6) be chosen as

63

2432f (7)

82

143

Substituting eqns (5) in (4), we obtain

0

sin221

21212 45122202222002442

   

And (8)



0342124121244222 412

  



 

For the solution of eqns (8), we multiply eqns (8) by f and  respectively and integrating the result with respect to  from 0 to 1, we obtain the following eqns Fung (5) 087262154542321UaiaUaBaUaiaaaaA (9) 01413122112109762aUaiaUaaBUaiaB Where

fda4

41021

ra

  

201023sin

dfa2104 dfKa21015 dfKa1017 dfKKa10218 dfKKa10419



da2

210210

dIa210211 dKKa2104112 dKKKKa210421313



dTa4

41014

dfKa210515 The determinant of the co-efficient of A and B is complex, both the real and imaginary parts must vanish. On setting the determinant equal to zero and separating the real and imaginary parts, we obtain the following 51542321aiaiaaaa 87262UaiaUa 962Uaia or

022122214EUBBAUA (10) 023212UCCC

66141511410151aaaaaaaaA

12152aaA

144113211041aaaaaaaaB 981351242aaaaaaB 14103211aaaaaE 131451aaaC 79145133211052aaaaaaaaaaC 1253aaC Second eqn of (10) gives us

1

2322CUCC (11)

Substituting this value of 2 in eqn (10) we get

(12)

PRQQU2

422

where

31232312CCACCCAP 2221322213112CAACCBCCCCBQ 211211221CCBCECAR

dfa104dfa106

eqn (12), there are two values of 2 from eqn (11). Usually the smaller 2Uis associated with the higher2 in the eqn. (11). Numerical example: A numerical example for the coupled vibration of a rotating slender beam involving shear and aerodynamics forces just described is presented here. Critical speeds and frequencies are computed from eqn. (11) & (12). The physical constants of the blade and other constants are given below Biezeno and Grammel [1] S = 0.14889 in2 A = 0.24885 in 1t = 0.19712 in m = 0.00011 lbs 6110.00189x10C  = 314 sec-1 2lbs/in m 0.095235I 4in 0.001846I 36lbs/in 11.53x10GG 36lbs/in x1067161.4G in 0.11311360x in 0.024

 4in 0.00193J

321K

26lbs/in 29.20x10E l = 4.41 in and 22U as 521102.7632627xU and 522x1054707.2U Substituting these values in eqn (10) we get the fundamental frequency as 5211x101.56117022 5229x101.56184728

The result thus obtained shows that the respective speed of the steady flow are the larger of two values of speeds 2Ucalculated from Eqn (12) i.e. the critical speed corresponding to torsional coupled vibration of the rotating slender beam under aerodynamics couplings. The larger of two values of 2Ugiven by equation (12) will provide the smaller value of 2, which will be the upper bound for the frequency of the fundamental mode of vibration. The smaller of two values of 2Uwill provide the larger value of 2which will be an upper bound for the next higher mode of vibration?

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

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. company, New York, 1958 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