NONLINEAR DYNAMIC ANALYSIS OF VIBRO-MILLING IN MILLS WITH SPATIAL MOTION .1. FORMULATION OF NONLINEAR EQUATIONS OF MOTION

A rigid body system having a mass M representing a mechanical model of a vibrating mill is considered. This model has geometric and mechanical symmetry with respect to the coordinate axis Cz. This system follows a general motion and is supported by an even number of linear springs. The spatial position of the mill is defined by six independent variables. The motion of the vibrating mill takes place under the action of two eccentric mass m0 mounted on a bar along the axis of symmetry Cz of the rigid body system at equal distances from the center of mass for the system, C. The eccentric masses go through a circular motion at a constant angular speed in a stationary motion. Assuming that the system of acting and internal forces on the rigid body is known, the scalar differential equations governing the motion of the rigid body are obtained. These governing equations are derived by utilizing the general principle of 'impulse-moment' from classical mechanics, namely the 'theorem of impulse' or the 'theorem of kinetic moment'. The resulting system of equations is a nonlinear set of six simultaneous differential equations and are presented in this paper. The modeling of the vat/mill structure as a rigid body and as presented in this paper has never been done before.