4569846498

120 Mutibody Systems Approach to Vehicle Dynamics
G(x)
x
Solution
(x 0 , G 0 )
Trial
solution
G/x
x
G
(x n , G n )
Fig. 3.37
Application of Newton–Raphson iteration
where
[x] is a set of unknown variables
[G] is a set of implicit functions dependent on [x]
Consider the solution of a single non-linear equation G(x) 0, where G is
an implicit function dependent on x. If we plot a graph of G(x) against x the
solution is the value of x where the curve intersects the x-axis as shown in
Figure 3.37.
The solution may be obtained using Newton–Raphson iteration based on the
assumption that very close to the solution the curve may be approximated to
a straight line where it crosses the x-axis. This is illustrated in Figure 3.37
where the line is determined by a trial solution, for say the nth iteration,
located at (x n , G n ) and the slope or derivative of the curve (∂G/∂x) at this
point.
If we take the point (x 0 , G 0 ) to be that at which the straight line crosses the
x-axis then the gradient ∂G/∂x is given by
∂G
∂x
G
x
0
0
G
x
n
n
G
x
(3.72)
Rearranging this we can write
(G/x) x G (3.73)
We can now extend this to demonstrate how the method can be used to iterate
and close in on the solution. This is shown graphically in Figure 3.38.
Extending the single equation G(x) to a set of simultaneous non-linear
equations we can write
⎡∂G
⎤
⎣
⎢ ∂x
⎦
⎥ [ x] [ G]
(3.74)