Mathematical Analysis of Effect of Centrifugal Force on Turning Error

In turning, especially at high-speed turning, due to the unbalance of high-speed rotating parts, centrifugal force is inevitably generated. Analyzing and understanding the effect of centrifugal force on machining error is a problem that people often pay attention to. Comprehensive conclusions, there are generally two different views. First, it is believed that the centrifugal force will cause dimensional errors in the outer circle of the workpiece, ie, radius error: Another view is that it will affect the shape error of the outer circle. This article analyzes this problem from a mathematical point of view and proposes a completely different view from the above conclusions. 1 The mathematical expression of centrifugal force and its analysis Set the gravity of the workpiece to be W, the speed of the lathe spindle to n, and the distance from the unbalanced mass m to the center of rotation to r, then the centrifugal force FQ is: FQ=mrw2= W r( 2pn ) 2 g 60 (1) Set the process system stiffness to KXT. The spindle axis offset Ar under the action of centrifugal force is: Ar=Fo/KXT (2) Since the direction of FQ is constantly changing, we should pass Establish a coordinate system to study the law of change. As shown in the figure, establish the absolute coordinate system YOZ, O point is the ideal axis of the main shaft, and assume that the main shaft has no turning error. Because of the centrifugal force, the actual rotation axis O1 (yo1, zo1) of the spindle rotates in the YOZ plane with the angular velocity w of the spindle. We establish the dynamic coordinate system VO1W with the O1 instantaneous axis, and the instantaneous axis change caused by the centrifugal force. The mathematical expression is: { yo1 = Arcoswt = Arcosf zo1 = Arsinwt = Arsinf (3) where f is the instantaneous rotation angle of the centrifugal force. The coordinate transformation relation between the absolute coordinate system and the dynamic coordinate system is: (y ) = ( yo1 ) + (cosf -sinf )( V ) z zo1 sinf cosf w (4) (v ) = ( y -yo1 ) ( Cosf sinf ) wz -zo1 -sinf cosf (5) When turning, because the workpiece rotates with the spindle, the cross-sectional geometry of the workpiece is formed by the relative trajectory of the tool in the dynamic coordinate system. Let the coordinate position of the tool in the absolute coordinate system be: {y=r z=0 (6) where r is the workpiece machining radius. Substituting equations (6) and (5) into equation (5): {v=(r-Arcosf)cosf-Arsin2f=rcosf-Ar w=(r-Arcosf)(-sin f)-Arsinfcosf=rsinf ( 7) Available from equation (7): (v+Ar)2+w2=r2 (8)

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From equation (8), it can be clearly seen that the cross-section of the workpiece to be machined is still a circle with a radius of r, without any shape error and dimensional error, except that the center of the circle is offset by the Ar distance from the positioning reference O, and thus the machining is performed. Coaxiality errors occur between the surface and the positioning base. 2 Conclusion During turning, the centrifugal force generated by the unbalanced workpiece will not cause the dimensional error and shape error of the machined surface. It will only cause the coaxiality error between the machined surface and the positioning surface. The error of the coaxiality is the deformation caused by the centrifugal force. Ar. When measuring the size and shape of the machined surface, the measurement reference should be selected correctly. To measure the diameter of the dimension, to measure the radius and the shape of the outer circle, it is necessary to determine the measurement reference according to the inclusion principle. Centrifugal force can cause the vibration of the process system, affect the quality of the machined surface and the coaxiality error, and should take technical measures to limit the unbalanced quality of the machined workpiece. The above conclusion also applies to the case where the single dial rotates the outer circle (grinding the outer circle) when the workpiece rotates.

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