In high-speed turning operations, the imbalance of rotating components inevitably generates centrifugal force. Understanding and analyzing how this centrifugal force affects machining accuracy is a critical concern in the field of precision manufacturing. There are generally two prevailing viewpoints on this issue. One perspective suggests that centrifugal force leads to dimensional errors on the outer diameter of the workpiece, specifically radial errors. Another view claims that it causes shape deviations on the surface. However, this article presents a new mathematical analysis that challenges these conventional conclusions. To begin with, we consider the mathematical expression of centrifugal force. Let the weight of the workpiece be denoted as W, the spindle speed as n (in revolutions per minute), and the distance from the unbalanced mass m to the center of rotation as r. The centrifugal force FQ can be expressed as: $$ F_Q = mr\omega^2 = \frac{W r (2\pi n)^2}{g \cdot 60} $$ Assuming the stiffness of the machining system is KXT, the displacement of the spindle axis due to centrifugal force, denoted as Δr, is given by: $$ \Delta r = \frac{F_Q}{K_{XT}} $$ Since the direction of the centrifugal force changes continuously, it is necessary to establish a coordinate system to study its dynamic behavior. We define an absolute coordinate system YOZ, where point O represents the ideal axis of the spindle. Due to the influence of centrifugal force, the actual spindle axis O1 moves in the YOZ plane, rotating at an angular velocity ω. A dynamic coordinate system VO1W is established, representing the instantaneous position of the spindle axis. The mathematical expressions for the coordinates of O1 are: $$ y_{o1} = Ar \cos(\omega t) = Ar \cos f $$ $$ z_{o1} = Ar \sin(\omega t) = Ar \sin f $$ Where f represents the instantaneous rotation angle caused by the centrifugal force. The transformation between the absolute and dynamic coordinate systems is given by: $$ \begin{bmatrix} y \\ z \end{bmatrix} = \begin{bmatrix} y_{o1} \\ z_{o1} \end{bmatrix} + \begin{bmatrix} \cos f & -\sin f \\ \sin f & \cos f \end{bmatrix} \begin{bmatrix} v \\ w \end{bmatrix} $$ When the workpiece rotates along with the spindle, the geometry of the machined surface is determined by the relative motion of the cutting tool within the dynamic coordinate system. Assuming the tool’s position in the absolute coordinate system is: $$ \begin{cases} y = r \\ z = 0 \end{cases} $$ Substituting into the coordinate transformation equations, we derive the following expressions for the tool’s position in the dynamic system: $$ \begin{cases} v = (r - Ar \cos f)\cos f - Ar \sin^2 f = r \cos f - Ar \\ w = -(r - Ar \cos f)\sin f - Ar \sin f \cos f = r \sin f \end{cases} $$ From these equations, we obtain: $$ (v + Ar)^2 + w^2 = r^2 $$ This equation clearly shows that the cross-section of the workpiece remains a perfect circle with radius r, but its center is displaced by Ar from the original reference point. As a result, a coaxiality error occurs between the machined surface and the positioning base. ### Conclusion During turning, the centrifugal force generated by an unbalanced workpiece does not cause dimensional or shape errors on the machined surface. It only introduces a coaxiality error between the machined surface and the reference surface. This error is directly related to the deformation caused by the centrifugal force, Δr. When measuring the dimensions and shape of the machined part, it is crucial to select the correct measurement reference. For diameter measurements, the radius and outer surface profile must be evaluated based on the inclusion principle. Additionally, centrifugal force may induce vibrations in the machining system, affecting both surface quality and coaxiality. To mitigate these effects, technical measures should be implemented to minimize the unbalanced mass of the workpiece. These findings are also applicable when turning or grinding the outer surface of a rotating workpiece.

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