PSI - Issue 2_B

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

1111

3

Burago N.G. / Structural Integrity Procedia 00 (2016) 000–000

2

1

r u



r 

rr   













rz  





rr





2

r r

z

r

t











2

2           z r z r   

1

u



r 











2

r r

t







2

1

z u



z  







   



zz    rz

rz

2

r r

z

r

t











Stresses and strains are subjected to the Hooke’s law:

( 2 ) zz            zz            rr rr ( 2 ) zz rr

( 2 ) rr             zz

2           r r 2 z z

2 rz rz   

Relations between strains and displacements are:

1     r 

r u u 

u



1 1 2 r  

u u r r

r       u

  

 





r

r  



 

 

rr

r

r



u



1 2             r u u z r z

1 1 2 r              z u u z 





 



z

z

rz

zz

z



where  ,  are Lame elastic moduli,  is density of disk material. Further the dimensionless stresses are divided by 2    while the dimensionless spatial variables and displacements are divided by internal radius of a disk a . The boundary conditions on free boundaries at ( ) z h r   are: 0 rz rr h      0 z r h        0 zz rz h     

Internal boundary of disk is unloaded:

0 r    ,

0 rr   ,

0

rz  

r a  :

On the external disk boundary the periodic (along the angular coordinate  and time t ) applied loads

_ rz b  are

_ r b   and

produced by torsional vibrations of the blades:

r b  : rz rz b    Because of the periodicity of unknown functions along the angular coordinate  and time t the displacements are represented in the form of Fourier series: 0 rr   , _ r b      , r _

 

 

 

3

2 w w z w z   (

4

( v z v z 

)cos

)sin

i t 

i t 

u e  

n

u e 

n





3

( u z u z 

)sin

i t 

u e 

n



3

2

4

n

n

z

n

n

n

3

r

n

n

0

1

n

n

1





n



Appropriate representations of the stresses are: