Civil Engineering Reference
In-Depth Information
FIGURE 1.5
Irregular geometric objects and their dimensions.
variations in the work material and random vibrations [57]. The intricate interplay of these multifarious
dynamic effects lead to distinctive error structures in surface generation.
If we conceive the phase space of the manufacturing process, the dynamics of surface generation can
be visualized as a straight line. As the manufacturing process removes (or adds) material, the line evolves.
Deterministic effects are convergent and have a propensity to fold the line. Random factors are divergent
and are inclined to stretch the same line. From this dynamical synergy, a unique error structure emerges.
We conjecture that this error structure is well characterized by fractal parameters. This conjecture is
reinforced by shifting from the abstract to the physical, and probing certain specific aspects of machining
processes that have the potential to generate fractal profiles/surfaces.
A Laplacian Fracture Model
The primary surface generation mechanism in machining is fracture, caused by nucleation and propa-
gation of cracks. Visualize the workpiece material (prior to any machining) as a matrix of interlinked
cells (in the
plane), with the corners of each cell represented by nodes. As the cutting tool traverses
the work material, it displaces the cells in its vicinity, subjecting them to plastic deformation, and
ultimately leading to crack formation and chip separation [12]. Therefore each node is a prospective
separation point, experiencing a displacement
xy
).
The force equilibrium of any given node is governed by Laplace's equation, (1.1), with suitable
boundary conditions [57, 62]:
u
(
x
,
y
2
2
u
u
--------
--------
0
(1.1)
x
2
y
2
. In addition to the displacement caused by the cutting
tool, the links between successive nodes also have a fracture probability, depending on local factors, e.g.,
workpiece hardness. The final surface is a result of these combined effects, and will be fractal [36, 42].
Solving this equation gives the displacement,
u
Energy Cascade in Fracture
Retaining the definition of fracture as a crack propagation process, consider the energy associated with
the formation of a new surface. In metal cutting, the energy stored in the tool is expended for plastic
