Engineering Optimization: Theory and Practice, Fourth Edition

528 Geometric Programming

Figure 8.2 Cantilever beam of rectangular cross section.

SOLUTION The width and depth of the beam are considered as design variables.

The objective function (weight) is given by

f (X)=ρlx 1 x 2 (E 1 )

whereρis the weight density andlis the length of the beam. The maximum stress

induced at the fixed end is given by

σ=

Mc

I

=P l

x 2

2

1

1

12 x^1 x

3

2

=

6 Pl

x 1 x 22

(E 2 )

and the constraint becomes

6 P l

σy

x 1 −^1 x− 22 ≤ 1 (E 3 )

Example 8.10 Design of a Cone Clutch [8.23] Find the minimum volume design

of the cone clutch shown in Fig.1.18 such that it can transmit a specified minimum

torque.

SOLUTION By selecting the outer and inner radii of the cone,R 1 andR 2 , as design

variables, the objective function can be expressed as

f (R 1 , R 2 )=^13 π h(R^21 +R 1 R 2 +R 22 ) (E 1 )

where the axial thickness,h, is given by

h=

R 1 −R 2

tanα

(E 2 )

Equations(E 1 ) nd (Ea 2 ) ieldy

f (R 1 , R 2 )=k 1 (R 13 −R^32 ) (E 3 )

where

k 1 =

π

3 tanα

(E 4 )