Curve And Surface Constructing Based On The Linear Differential System And Five-axis Machining Tool Orientation Optimization

With the development of multi-axis numerical control technology, the modelling of products is more and more complicated. The geometrical modelling of products is blending in with multiple disciplines gradually such as mechanics, and streamline modelling is more widely used. Streamline modelling product possesses not only beautiful appearance but also good aerodynamic and mechanical properties. NC machining is based on accurate expression of curve surface and the tangent vectors, normal vectors and some other information. Therefore, how to describe the surface of streamline modelling product is the premise of processing such products.In addition to processing factory and design institutions, it is difficult to obtain the design drawings and accurate size data of the products in design and research process. Therefore, reverse engineering is used to reconstruct the product appearance generally. By acquiring point cloud data on the surface of product, the data fitting and refactoring can be complete. In curve and surface reconstruction of scattered data, curve and surface construction that based on the vector and scatter can be converted into the solution of linear ordinary differential equation if the vector of each point has been given,. As a consequence, this paper puts forward a new express by differential equation. In this method, we select appropriate initial values in scattered points, use the integrktion solving method to process these points along the given direction of the vector field, then the integral curve is flowline. After that, we fit out the curve in the vertical direction of the flow using parameterized method and least squares shape the surface skeleton, and then construct the surface. Curve and surface generated by this way own rich geometry information such as tangent vector and curvature, which enabling the combination of the design and disciplines such as mechanics by using the related knowledge of the streamline. This new method conforms to the development trend of the streamline modeling for curve and surface.The integral curve, in curve and surface constructed by ordinary differential equation, can be used as the tool path for machining directly. In the five-axis machining, due to the two degrees of freedom increased on the cutter location, tool can travel to any position of the workpiece with arbitrary attitude theoretically. In high speed machining, it is easier to exceed the drive constraint and cause vibration or damage to tool and even damage to machine tool when there are drastic changes of the tool orientation in tool path,. Therefore, it is important to improve the machining process by optimizing the tool orientation which is of most significance for improving the stability of machining process as well as the quality and accuracy of workpiece. Concerning this issue, this paper presents a tool position and orientation optimization algorithm with multi-constraints for parametric curve. The constraint conditions considered by this algorithm covers the process characteristics, cutting properties and machine drive properties, including the velocity and tangential acceleration limits of cutter tip, the angular velocity and acceleration limits of tool axis, the velocity and acceleration limits of machine’s axes. The analytical relation between the constraint conditions and the cutter location information can be deduced by the kinematics theory of machine tool. Then, we can establish the nonlinear programming tool optimization model using the discrete tool path with a suitable parameter interval. The optimization goal of this model is to minimizing the weighted sum of the squares of the adjacent tool axis rotation speed.This paper takes blade as an example to carry on the simulation test of differential equation curve and surface fitting. Then it takes a streamline of curve and surface fitting as the tool path to carry on the simulation test of nonlinear tool optimization. The simulation results are analyzed after the simulation test. The result shows that differential equation curve and surface fitting method in this paper is simple, and can structure curve and surface which meet the requirements of the given data. And tool position optimization algorithm is able to meet all the given constraint conditions. The machining accuracy and machine tool dynamic characteristics has great improvement. To sum up, these two methods provide a good solution for model creation and tool position optimization in five-axis NC machining, which have a broad application prospect.