In high-speed turning, the imbalance of rotating components inevitably generates centrifugal force. Understanding the impact of this force on machining accuracy is a critical concern in the field of manufacturing. While there are generally two prevailing viewpoints regarding the effect of centrifugal force, this paper presents a new mathematical analysis that challenges these traditional perspectives.
The first perspective suggests that centrifugal force causes dimensional errors, specifically radial deviations, on the outer surface of the workpiece. The second view claims it leads to shape errors, such as ovality or out-of-roundness. However, through a detailed mathematical formulation, this study offers an alternative conclusion.
Let’s define the gravitational force acting on the workpiece as $ W $, the spindle speed as $ n $ (in revolutions per minute), and the distance from the unbalanced mass $ m $ to the axis of rotation as $ r $. The centrifugal force $ F_Q $ can be expressed as:
$$
F_Q = m r \omega^2 = \frac{W r (2\pi n)^2}{g \cdot 60}
$$
Assuming the stiffness of the machining system is $ K_{XT} $, the displacement of the spindle axis due to the centrifugal force is:
$$
\Delta r = \frac{F_Q}{K_{XT}}
$$
Since the direction of the centrifugal force changes continuously with the rotation of the spindle, a coordinate system must be established to analyze its dynamic behavior. A fixed coordinate system $ YOZ $ is set up, where point $ O $ represents the ideal centerline of the spindle. Due to the unbalance, the actual spindle axis $ O_1 $ rotates within the $ YOZ $ plane at angular velocity $ \omega $.
To model the motion, a dynamic coordinate system $ VO_1W $ is introduced, where the coordinates of the spindle axis are given by:
$$
\begin{cases}
y_{o1} = Ar \cos(\omega t) = Ar \cos f \\
z_{o1} = Ar \sin(\omega t) = Ar \sin f
\end{cases}
$$
where $ f = \omega t $ is the instantaneous rotation angle of the centrifugal force.
The transformation between the absolute and dynamic coordinate systems is defined as:
$$
\begin{bmatrix}
y \\
z
\end{bmatrix}
=
\begin{bmatrix}
y_{o1} \\
z_{o1}
\end{bmatrix}
+
\begin{bmatrix}
\cos f & -\sin f \\
\sin f & \cos f
\end{bmatrix}
\begin{bmatrix}
v \\
w
\end{bmatrix}
$$
During the turning process, the tool's position in the absolute coordinate system is assumed to be:
$$
\begin{cases}
y = r \\
z = 0
\end{cases}
$$
Substituting into the transformation equations, we find that the cross-section of the workpiece remains circular with radius $ r $, but its center is shifted by $ Ar $. This shift results in a coaxiality error between the machined surface and the reference axis.
From the derived equations, it becomes clear that centrifugal force does not introduce dimensional or shape errors in the workpiece. Instead, it only causes a shift in the axis of rotation, leading to coaxiality issues. Therefore, when measuring the dimensions and geometry of the machined part, it is essential to choose the correct measurement reference based on the inclusion principle.
Moreover, centrifugal force may also induce vibrations in the machining system, which can affect surface finish and overall accuracy. To mitigate these effects, measures should be taken to minimize the unbalanced mass of the workpiece.
This conclusion is applicable not only to turning operations but also to other processes involving rotational motion, such as grinding of external surfaces.
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