Date of Award

Spring 1997

Document Type


Degree Name

Doctor of Philosophy in Mechanical Engineering - (Ph.D.)


Industrial and Manufacturing Engineering

First Advisor

M. C. Leu

Second Advisor

Denis L. Blackmore

Third Advisor

Reggie J. Caudill

Fourth Advisor

Rajesh N. Dave

Fifth Advisor

Zhiming Ji


This research falls in two important areas in solid modeling and manufacturing automation: (1) swept-volume modeling; (2) computer-based NC (Numerically controled) machining simulation and verification. The swept volume is defined as the volume swept by an object undergoing an arbitrary motion. Modeling of 3D swept volumes includes the boundary computation and representation of a swept volume generated by a general object undergoing general motion in three dimensional space. The Sweep Differential Equation (SDE) and Sweep Envelope Differential Equation (SEDE) methods are two of the important swept volume modeling methods employed in this dissertation. They exploit differential equations to obtain the boundary points of a swept volume generated by a moving object. The application of SDE/SEDE methods is addressed to computer-based NC simulation and verification. Comparison of the SDE/SEDE approach with other swept volume modeling methods is conducted too. It has been shown that the SDE and SEDE methods have great benefits in calculating and representing general swept volumes and the research has substantially advanced existing manufacturing technologies.

The main contributions of the research are:

(1) The SDE method has been extended to three dimensional space to represent cutter swept volumes generated by moving five-axis NC milling tools. A SDE sweep generator, which can represent and analyze three-dimensional swept volumes generated by flat-end and ball-end tools for a typical five-axis NC milling machine graphically, has been developed. In the SDE sweep generator, a machine control data based interpolation method is uniquely used to describe the interpolation motion equation of a five-axis NC milling tool.

(2) The SEDE method is derived for a more efficient swept volume calculation. A SEDE-based algorithm for the numerical boundary computation of swept volume is described and combined with some novel smooth approximation formulas in order to calculate the swept volume generated by a general 7-parameter APT (Automatic Programming Tool) tool for a large class of sweeps that includes the motions encountered in five-axis NC milling processes. The SEDE approach for the most part reduces the computation to the determination of SEDE trajectories at the initial grazing points (the main part of the boundary of a swept volume) of the tool, and therefore appears to reduce computational cost as well as providing a natural connectivity for most points on the swept volume boundary.

(3) An SEDE-based program has been integrated with Deneb Robotics's Virtual NC commercial software. The SEDE module is used to replace Virtual NC's convex hull sweep algorithm for a more accurate geometrical tool swept volume representation. By using the Boolean subtractor and verifier in Virtual NC, material removal of five-axis NC milling process is simulated and analyzed in an interactive machining environment. Furthermore, the SDE/SEDE approach has been integrated with a five-axis NC milling CAD/CAM system at NJIT to perform part design, tool path generation, Cutter Location (CL) and NC code simulation and verification, and actual machining on a FADAL VMC-20 five-axis NC milling machine. Several examples including machining of a turbine impeller are given to illustrate the effectiveness of this integration approach.