The stresses in an eccentrically mounted rotating elliptical plate

Abstract

1. Introduction. In some mechanical applications a knowledge is required of the elastic behavior of a plane elliptical plate which is mounted on and rotating with a rigid shaft. The axis of the shaft is perpendicular to the plate and pierces it at an arbitrary point. The angular velocity of rotation is constant. If the cross-sectional area of the shaft is taken into account the geometry of the elastic area--an ellipse containing an arbitrarily placed circular hole -- is complicated to a degree which makes the determination of analytical expressions for the stress and displacement fields extremely difficult. This paper presents an analytic solution which approximates the actual physical conditions to the extent that the plate is assumed to rotate about a point instead of a shaft of finite cross-sectional area. In this form the problem is one of finding a solution to the First Boundary Value Problem (prescribed boundary stresses) for an elliptical area. A form of the solution of this Boundary Value Problem, which utilizes a knowledge of the forces on the boundary rather than the stresses, is availablO. For reasons of convenience the boundary conditions are expressed in this paper in terms of the stresses, not the forces, and accordingly the analysis of MuskhelishvilP is not directly applicable. In the process of finding the stress and displacement fields of immediate interest, the basic analysis of the First Boundary Value Problem for an elliptical area, in a form suitable for the stress boundary condition, is presented. The solution of this problem is effected by conformally mapping the elliptical area into a circular annulus and subsequently applying the complex variable technique developed by Muskhelishvili whose notation is used throughout. This analysis contains as special cases i) the rotation of a circular disc about an axis through its center, a very well known problem ~, ii) the rotation of a circular disc about an eccentric axis; this problem has been treated by Mindlin ~, Sen 4, and Ya. K. II'yn 5, iii) the rotation of an elliptical disc about an axis through its center; this case has been treated by Stevenson s. 2. Statement and Solution of Problem. Consider a thin elliptic plate with semi-major axis a and semi-minor axis b rotating with angular velocity oJ about an axis normal to the plate and passing through the point z 0 -= x 0 + i Yo as shown in Fig. 1. Because of the eccentricity of the axis of rotation, the inertia forces produce a reaction force P : X -i- i Y at the axis of rotation, where P-~ X-k i Y: q~ a b co ~ z0, (1) where q is the density of the plate. The contribution to the complex stress combinations in the plate due to this concentrated reaction force is taken in the form given by Muskhelishvili* and which for this problem gives