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Abstract: This thesis focuses on a general lathe spindle，quantitatively analyzes the machining precision with spindle swinging in geometry way and derives a mathematical expression of machining errors at different swing frequencies． By applying MATLAB to simulate the workpieces' cross-section at different swing frequencies，taper is analyzed and variation law is summarized．53444
1. Introduction
In the machine tool design and precision machining，spindle system in machine is the key to guarantee the machining accuracy． As the dominant vibration system among the multiple degree of freedom vibration systems，the error motion of a machine tool spindle includes the radial run-out，axial shifting and angular swinging． In the most of available literatures，they analyze the spindle angle and machining error with the assumption that the spindle swings and rotates at a same frequency． However，in the actual production，the swinging frequency of the main spindle is generally inconsistent with the rotation frequency，and few of paper has ever discussed it and been published． Therefore， this article focuses on discussing the processing error of mathematics and geometry in the main shaft plane with the swinging angles，then calculates and analyzes the work pieces's taper and its influencing factors at a series of swing frequencies．
2 Mathematical analysis and simulation atfixed swing frequencies
2.1 Mathematical expressions of spindle angular swinging
Establishing the static coordinate system XYZO of the main spindle at first，and OZ is the ideal axis of the spindle． Then，establishing the dynamic coordinate
system X' Y' Z' O， and shaft OZ' is the spindle instantaneous axis． The built spindle static /dynamic coordinate system is shown in Figure 1.
Figure 1 The spindle static /dynamic coordinate systems
The spindle rotates with the angular velocity ω，meanwhile，its axis line angularly swings at the same frequency in XOZ plane．
The rule of its swinging is: θ = θ0cosα
θ0 is the swinging amplitude of the spindle axis; α is the spindle's rotation angle，α=ωt.
2. 2 Trajectory analysis at fixed swing frequencies
Draw the above equation in polar coordinates with MATLAB．The spindle rotates one circle，and the cross section contour diagram is shown in Figure 2．Analysis shows that if the swinging frequency is equal to the rotational frequency，the profile of the workpiece is approximately a circle,although its center deviates from the ideal center after the rotation of the workpiece.Its roundness error is very small and negligible．
Figure 2 The trajectory of tool rotation when the swinging
frequency = rotational frequency
3 Calculation and analysis of taper
When it comes to the cylindrical workpieces，its taper is equal to ( larger diameter D minus the path d) over axial length． From the previous analysis we know that，when the spindle swinging frequency is low，the diameter of the workpiece is determined by the least square circle of cross-sectional profile ( the envelope curve) ; when spindle swing frequency is high，the diameter of the workpiece is determined by the ideal circle of cross-sectional profile．Let's assume that the length of workpiece machining is 100mm，the diameter is 30 mm，and the lathe spindle amplitude of angular swing is 0. 01．
3. 1 Taper analysis at low swing frequencies
Assuming n = 0. 1， the workpiece cross section diameter changes at different processing length as shown in Figure 9．
Figure 9 The cross-section ideal diameter under
the different processing length
Figure 9 shows that when the swinging frequency is constant，the cross-sectional diameter of workpiece decreases linearly with the growth of the length of