Abstract: In this work, a five step continuous method for the solution of third order ordinary differential equations was developed in block form using collocation and interpolation techniques of the shifted Legendre polynomial basis function. The method was found to be zero-stable, consistent and convergent. The application of the method in solving third order initial value problem of ordinary differential equations revealed that the method compared favorably with existing methods.
Abstract: Helical milling operations are used to generate or
enlarge boreholes by means of a milling tool. The bore diameter can be
adjusted through the diameter of the helical path. The kinematics of
helical milling on a three axis machine tool is analysed firstly. The
relationships between processing parameters, cutting tool geometry
characters with machined hole feature are formulated. The feed motion
of the cutting tool has been decomposed to plane circular feed and
axial linear motion. In this paper, the time varying cutting forces acted
on the side cutting edges and end cutting edges of the flat end cylinder
miller is analysed using a discrete method separately. These two
components then are combined to produce the cutting force model
considering the complicated interaction between the cutters and
workpiece. The time varying cutting force model describes the
instantaneous cutting force during processing. This model could be
used to predict cutting force, calculate statics deflection of cutter and
workpiece, and also could be the foundation of dynamics model and
predicting chatter limitation of the helical milling operations.