PSI - Issue 2_B

Burago N.G. et al. / Procedia Structural Integrity 2 (2016) 1109–1116 Burago N.G. / Structural Integrity Procedia 00 (2016) 000–000

1112

4



 

 

(     n z

3

i t 

)sin

e

z

n







3

i t 

(

)sin

e

s z s z 

n

 





3

(

)sin

i t 

e

z   

z

n







3

zz

n

3

n

n

3

rr

n

n

1

1

n

n





1

n



 



 

3    The coefficients of the Fourier series are new unknown functions of the radial variable r , multiplier i t e  defines vibrations,  is a vibration frequency. If substitute the expressions for displacements and stresses in the original system and equate terms of like powers z up to z 3 then one can obtain the resolution system of ordinary differential equations for the auxiliary variables at different n.   / 2 2 / / 2 n n n n n n n r s r p u          / 2 2 / 2 / 2 n n n n n ns r r T v         3 0 ( )cos i t  r   n n n e z    z n    2 p p z p z   4 2 4 1 ( )sin i t  rz n n n n e n     2 n e T T z T z   4 2 4 0 ( )cos i t z   n n n n  

/

/ U r w      

u

/ / p nT r p r  /

2 w 

   

2

n

n

n

n

n

n

n

n

n

/

/ V r     /

v

/ w p 

/

u  

n

n

n

n

n

n

/ / p nT r p r   /

2 w 

3   

/ w p 

/

3 u  

2

2

2

3

2

n

n

n

n

n

2

2

3

n

n

n

/ u U r      / 2 w

s

/ n n T v nw r     n

/ 2 n n u U r w       / 2 n n

2

n

n

n

n

/

/ U r

4

u

w

 

 





/

2 

/ n r

3    ( n

) / s r p

u

  

 

3

3

3

4

n

n

n

n

3

3

3

4

3

n

n

n

n

n

/

/ V r     /

v

/

2 

/ ns r

3  2 / n

4 r T

v

3 

 



 

3

3

3

n

n

n

3

4

3

n

n

n

n

/ u U r  /

4

s

w

 





(      

2 h h h h 3 ) /

3

2

/

h

 

/ T nw r  

3

v





3

3

3

4

n

n

n

n

3

3

n

n

n

n

2

2

3

n

n

n

3 4 n n p h h h h p p h h        3 2 ( ) / ( ) / n n n 4 2

4

3 4 n n T h h h h T T h h        3 2 ( ) / ( ) / n n n 4 2

4

3 n n V nu v    . 3 n

n n V nu v    , 3

3 n n U nv u    , 3 3 n

n n n U nv u    , n

where

3. The boundary conditions for torsional vibrations For calculations of stress state for disk due to torsional vibrations of blades the boundary conditions for auxiliary variables (Fourier coefficients) on radial boundaries r a  and r b  are

0 n   ,

3 0 n   ,

0 n   ,

3 0 n   ,

0 n p  ,

0

2 n p 

r a  :

0 n   ,

3 0 n   , n

bn    , 3 0 n   ,

n bn p p  ,

2

/ p h   bn

p

r b  :

2

n

where bn  and bn p are predefined values of Fourier coefficients for disk boundary stresses in root sections of blades undergoing to torsion. For calculation of values bn  and bn p consider every blade as a plate of rectangular section of width d , intensity of torsion  and use the approximate solution of known task about torsion of plates of rectangular cross section:   2 2 0 ( ) 1 b Q       , 0 0.8 / Q d h    ,   

0 ( ) b p T     ,

,

0 T d  

  

where

/ (2 ) 1 d b    .