ComputerAided_Design_Engineering_amp_Manufactur.pdf

The distance constraint from pt � (x, y, z) to plane is
Given a distance d, the positive or negative of d will determine the direction of variation along with
(A_d, B_d, C_d) or (�A_d, �B_d,�C_d).
Systems of Equations
Once constraints are translated into a set of equations, the next step is to solve for the variables part of
the equations. For a simplified example, to solve the system equation of point-to-plane (as described in
the previous section), if datum plane d is coincident with plane p, pt_1, pt_2, pt_3, pt_4, and d_2 are
given, then the value of point pt (x, y, z) can be solved directly. When multiple dimensions are applied
and with fewer fixed variables, the solution scheme becomes complicated. Two methods are available for
solving a system of equations. One is a direct solution using substitution algorithm and the other is
numerical iteration. The direct solution has the advantage that, in general, it is faster and more precise.
The direct solution has the main disadvantage that it may not be used to solve all systems of equations
because it may not be possible to define a variable explicitly in an expression for its subsequent substitution
in other expressions (Serrano, 1984). The Newton-Raphson method that can solve simultaneous
nonlinear equations does not have substitution problems. The disadvantage of the Newton-Raphson
method is that it requires the user to input the initial guess close to the actual solution in order to converge
to a solution, and it may not converge to the desired solution if multiple solutions are possible (Lin, 1981).
Direct Substitution Method
In order to determine if a direct solution is possible, it is necessary to determine if all the variables may
be defined explicitly by an expression. Given a set of system equations,
to solve this set of equations we need the same number of equations and unknowns, and explicit definition
for each unknown in terms of other variables; therefore, no slings should appear in the resulting digraph
(directed graph). A sling would be formed if an unknown is defined in terms of itself. The system must
be acyclic because the process of determining the order of execution is analogous to that of scheduling
a network of activities. An activity originating at a node cannot start before all activities terminating at
that node are completed. Using network path scheduling, it is possible to solve systems with more than
one root, i.e., systems with subsets of expressions that are not related among themselves. An adjacency
matrix as shown in Figure 9.14 is constructed to describe the digraph of the above system equations.
FIGURE 9.14 Adjacent matrix.
© 2001 by CRC Press LLC
A_px � B_py � C_pz � D�d_2 y
b
x
a
z
y � y( z)
b � b( x,z)
x � x( y,z)
a � a( x,y,z)
y b x a z
0 0 0 0 1
0 0 1 0 1
1 0 0 0 1
1 0 1 0 1
0 0 0 0 0