Mathematical processing of CNC lathes

The mathematical processing of part drawings mainly involves calculating the dimensions of the machining trajectory of the part, that is, calculating the coordinates of the base point and node of the machining contour of the part, or the coordinates of the base point and node of the tool center contour, in order to compile the machining program.
Calculation of base point coordinates
General CNC machine tools only have linear and circular interpolation functions. For a planar profile composed of straight lines and arcs, the main task of numerical calculation during programming is to find the coordinates of each base point.
1. The meaning of the base point
The intersection or tangent point of different geometric isolines that make up the contour of a part is called the base point. The base point can directly serve as the starting or ending point of its motion trajectory.
2. Directly calculated content
According to the requirements of filling out the machining program form, the content of direct calculation of the base point includes: the coordinates of the starting and ending points of each motion trajectory in the selected coordinate system, and the center coordinate value of the arc motion trajectory.
The method of directly calculating the base point is relatively simple, and can generally be manually completed based on the known conditions given in the part drawing. Using knowledge of algebra, trigonometry, geometry, or analytical geometry to directly calculate numerical values based on the dimensions given on the part drawing. When calculating, it is important to leave enough digits after the decimal point to ensure sufficient accuracy.
Calculation of node coordinates
For some plane profiles that are composed of non-circular equation curves Y=F (X), such as involutes, Archimedes spirals, etc., they can only be approximated by lines and arcs that can be processed. At this point, the task of numerical calculation is to calculate the coordinates of nodes.
1. Definition of nodes
When using CNC machine tools that do not have the function of non circular curve interpolation to process parts with non circular curve contours, in the programming work of the machining program, multiple straight line segments or arcs are commonly used to approximate non circular curves, which is called fitting processing. The intersection or tangent point of a fitted line segment is called a node.
2. Calculation of node coordinates
The calculation difficulty and workload of node coordinates are relatively high, so they are often completed by computers, and can also be manually calculated if necessary. The commonly used methods include line approximation (equal spacing method, equal step method, and equal error method) and arc approximation.