Engineering Optimization: Theory and Practice, Fourth Edition

5.1 Introduction 251

Figure 5.2 Contact stress between two spheres.

SOLUTION Forν 1 =ν 2 = 0. 3 , Eq. (E 1 ) educes tor

f (λ)=

0. 75

1 +λ^2

+ 0. 65 λtan−^1

1

λ

− 0. 65 (E 4 )

wheref=τzx/pmaxand λ=z/a. Since Eq.(E 4 ) s a nonlinear function of the distance,i
λ, the application of the necessary condition for the maximum off,df/dλ=0, gives
rise to a nonlinear equation from which a closed-form solution forλ∗cannot easily be
obtained. In such cases, numerical methods of optimization can be conveniently used
to find the value ofλ∗.

The basic philosophy of most of the numerical methods of optimization is to
produce a sequence of improved approximations to the optimum according to the
following scheme:

1.Start with an initial trial pointX 1.

2 .Find a suitable directionSi ( i= 1 to start with) that points in the general

direction of the optimum.

3.Find an appropriate step lengthλ∗i for movement along the directionSi.

4 .Obtain the new approximationXi+ 1 as

Xi+ 1 =Xi+λ∗iSi (5.1)

5 .Test whether Xi+ 1 is optimum. If Xi+ 1 is optimum, stop the procedure.

Otherwise, set a newi=i+1 and repeat step (2) onward.