PSI - Issue 28

K. Mysov et al. / Procedia Structural Integrity 28 (2020) 352–357 Author name / Structural Integrity Procedia 00 (2019) 000–000

356

5

1 2 tg x tg y tg h h h       , 1 , 2 2

(2.16)

It leads to the singular integral equation   1 

1  0

  , y yW x y dy

   y yS x y q dy F x С      , ,

0 

(2.17)

 

0

1

q

  0 , W x y is discontinuous Weber-Sonin kernel     2 1 1 q R F x G hx  

where

  1 r  

(2.18)

, R d

 

2

2

arctg hx

a

  

  

N 

  0 1 cos k

  , q k    

  0 k P

(2.19)

, , S x y q

2

0, R R P

cos

R

k 

  hx

  hy

2

2

2

arctg hx arctg hy

 

 

1

1

0

k

2

      2 3 2 , 2 1 X R arctgay hy 

  y

(2.20)

unknown constant 1 С is found from the crack closure condition     1 2 0 0 1 y y dy hy    

(2.21)

3. Numerical results The most important, from the point of view of mechanical applications is finding the cone’s eigen frequencies. With this aim the method proposed in Grinchenko and Meleshko (1981) was applied. The next parameters were used 7 N  , 10 2 80 10 / / G g cm s   , 3 7.86 / g cm   , 10 a cm  , 2 b a  , / 3    , 5 / 3.19 10 / c G cm s     , 3 0.1 V cm  , 0.786 m V g    ,   2 2 arcsin / 2 39.3 J m g cm     , 10 2 2 800 10 / M g cm s    , / 2 R a l   , / 2    , here V and m are the overlays of volume and mass respectively. There were found several first eigen frequencies of the cone. The values of   and w were given for different cone opening angles (Table 1, Table 2) and different linear length (Fig. 2). The boundary resonance effect was stated for cone with crack (Fig. 2. b ). This fact can be useful during the solving of the corresponding inverse problem.

Table 1. First eigen frequencies for different cone opening angles for cone without a crack.  2 , 1,..., 4 i i l i c      36  1.437 3.227 4.006 5.175 5.437 45  1.437 3.227 3.455 4.889 4.975 60  1.437 2.938 3.227 3.961 4.584 Table 2. First eigen frequencies for different cone opening angles for cone with a crack.  2 , 1,..., 4 i i l i c      36  0.937 1.814 2.361 3.224 4.006 45  0.851 1.814 2.327 3.224 3.455 60  0.759 1.814 2.323 2.938 3.224

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