Calculation of the axisimmetric thermopower bending problem of rotating in the thermal field of the polar-orthotropic disc with variable thickness by Volterra integral equation of the second kind

V. V. Korolevich ; D. G. Medvedev

Journal of the Belarusian State University. Mathematics and Informatics, Tome 2 (2017), pp. 44-51.

Résumé

With the help of linear Volterra integral equation of the second kind the problem of the axisymmetric bending polar orthotropic annular disk of variable thickness, rotating around the normal axis with constant angular velocity $\omega_{0}$ in a inhomogeneous thermal field is solved in the general form. Under the influence of centrifugal forces and thermal field the disk will experience stretch on its plane. The effect of the axisymmetric flow of incandescent gas or steam, directed normal by to the median plane of the disk, as well as the boundary moments and shear forces cause axisymmetric bending. Thus, the disc will experience the stretch and flexural bending at the same time. It is assumed that the temperature field in the disk is known and it is axisymmetric. The elastic constants – Young's modulus and shear modulus – are linearly temperature dependent, and Poisson's coefficient are considered to be constant. Calculation of bending of the anisotropic thin disk is carried out according to the classical theory of bending of thin plates, based on Kirchhoff hypothesis. The problem of the axisymmetric bending polar-orthotropic annular disc of variable thickness is reduced to the integration of the ordinary differential equation of second order with variable coefficients for a rotation angle of a normal cell to the median plane of the disk. The resulting differential equation is reduced to a linear integral Volterra equation of the second kind. The general solution of the integral equation is written down the resolvent is used for this purpose. The conditions under which the integral equation has a unique continuous solution are given. Calculation formulas are given for the radial and tangential bending moments, transverse radial forces and radial deflection function through resolution function. Formula for the components of the radial, tangential and shear stresses, taking into account the simultaneous stretching and bending of anisotropic annular disc of variable thickness under the influence of applied loads is written.

Détail

Keywords: polar-orthotropic disc; inhomogeneous thermal field; temperature; radial, tangential and shear forces; the radial and tangential bending moments; deflection function; the rotation angle of the normal; differential and integral equations; resolvent; radial, tangential and shear stress components.

Bibliographie

[1] V. V. Vorobei, E. V. Morozov, O. V. Tatarnikov, “Raschet termonapryazhennykh konstruktsii iz kompozitsionnykh materialov”, 1992

[2] S. M. Durgaryan, “Temperaturnyi raschet ortotropnoi sloistoi plastinki pri uprugikh postoyannykh i koeffitsiente temperaturnogo rasshireniya, zavisyaschikh ot temperatury”, Izv. Akad. nauk Arm. SSR. Fiz.-mat. nauki, 13, # 2 (1960), 73–88

[3] V. V. Korolevich, D. G. Medvedev, “Integralnye uravneniya Volterry vtorogo roda v zadachakh izgiba vraschayuschikhsya polyarnoortotropnykh diskov peremennoi tolschiny”, Vestn. BGU. Fizika. Matematika. Informatika, # 3 (2012), 108–116

[4] V. V. Korolevich, D. G. Medvedev, “Reshenie osesimmetrichnoi ploskoi zadachi termouprugosti dlya vraschayuschegosya v teplovom pole polyarno-ortotropnogo diska peremennoi tolschiny metodom integralnogo uravneniya Volterry vtorogo roda”, Zhurn. Belorus. gos. un-ta. Matematika. Informatika, # 1 (2017), 47–52

[5] M. L. Krasnov, A. I. Kiselev, G. I. Makarenko, “Integralnye uravneniya: zadachi i primery s podrobnymi resheniyami”, 2007

[6] A. F. Verlan, V. S. Sizikov, “Integralnye uravneniya: metody, algoritmy, programmy”, Kiev, 1986

[7] S. V. Boyarshinov, “Osnovy stroitelnoi mekhaniki mashin”, 1973