PSI - Issue 28

K. Mysov et al. / Procedia Structural Integrity 28 (2020) 352–357

353

2

Author name / Structural Integrity Procedia 00 (2019) 000–000

1. Main text

Nomenclature a

radius at which cone’s bottom face is located radius at which cone’s upper face is located

b

R

radius at which crack is located cone’s opening angle crack’s opening angle torque applied to overlay steady state frequency cone’s shear modulus overlay’s known inertia moment





M

J



G

cone’s density



is the unknown rotation angle



Fig. 1. Geometry of the problem.

The twice truncated elastic cone is considered in the spherical coordinate system

, , a r b              

with a spherical crack , r R              (Fig. 1). The problem is stated for the case of steady state oscillations, thus for all mechanical characteristics representations     i , , , e , , t f r t f r        takes place, factor i e t  is omitted in all the next formulas. Bottom spherical face , , r a              is in adhesion with an absolutely rigid overlay through which the torsion dynamic moment impacts the cone:   r a w lF     (1.1) here     , , w r u r     , l b a   ,   F  is an arbitrary continuous function. The latter should be found from the movement equation of the overlay ,



3 0 2 sin ( , ) r a    2

2 a d M J

0

(1.2)

   

    

The upper spherical face of the cone

, , r b              is fixed 0 r b w  

(1.3)

The cone’s surface

, , a r b            is free from stress 0      

(1.4)