Abstract
In the paper we present a mathematical model through which are determined the balance conditions, needed for the stability analysis of the oscillating movement of the main spindle at CNC lathe. We take into account Hamilton's variation principle, the axiom of impulse derivative and the axiom of kinetic moment derivative. We present the general movement equations that generate the oscillations based on the calculus hypotheses, performing the introduction of the external solicitations. Establishment of the balance configuration is done by imposing the conditions that the system of forces that act upon the ensemble spindle – bearings - tool causes a deformation of the spindle, without producing spindle vibration. We obtain the new differential equations of the movement, in which the forces and moments are determined from the static case, based on which we can determine the integration constants in the characteristic points of the main spindle.
Highlights
Turning plays a special role among cutting technological processes due to its high dynamic instability characteristic of the process
The peak values of the dynamic forces that appear under these circumstances produce an increased wear of lathe tools and the deformation of piece turned surface
For a spindle model corresponding to figure 1, the input of external forces F1,1, F2,1, F3,1 and couples M2,1, M3,1 represents the action of the bearing (i) upon the main spindle
Summary
Turning plays a special role among cutting technological processes due to its high dynamic instability characteristic of the process. 2. THE MATHEMATICAL MODEL FOR ESTABLISHING THE MOVEMENT EQUATIONS OF THE MAIN SPINDLE. We take into account the following: On the external surface of the spindle there are no forces or distributed superficial couples. The mathematical model takes into account the known principles (Hamilton’s variational principle) Based on this we compute the inertial terms and the resultant force and moment of the tensions in section [1]. For a spindle model corresponding to figure 1, the input of external forces F1,1, F2,1, F3,1 and couples M2,1, M3,1 represents the action of the bearing (i) upon the main spindle (figure 2). Taking into account the inertial forces and moments introduced by the rotation movement of the spindle, and projecting the movement with respect to a fixed reference system, the mathematical model that establishes the movement equations of the spindle is given by:.
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