Engineering Optimization: Theory and Practice, Fourth Edition

8.12 Applications of Geometric Programming 527

where

C 31 =a 2 Sm−^1 ax (E 9 )

If the constraint (E 8 ) s also included, the problem will have a degree of difficultyi
two. By takinga 2 = 1. 36 × 108 ,b 2 = − 1. 5 2,c 2 = 1. 004 , Smax= 001 μin.,Fmax=
0 .01,andPmax= 2. 0 in addition to the previous data, we obtain the following result:

f∗= 1 $.11 per piece, V∗= 11 ft 3 /min, F∗= 0. 0 046 in./rev

Example 8.8 Design of a Hydraulic Cylinder [8.11] The minimum volume design
of a hydraulic cylinder (subject to internal pressure) is considered by taking the pis-
ton diameter (d), force (f ), hydraulic pressure (p), stress (s), and the cylinder wall
thickness (t)as design variables. The following constraints are considered:

Minimum force required isF, that is,

f=p

π d^2

4

≥F (E 1 )

Hoopstress induced should be less thanS, that is,

s=

pd

2 t

≤S (E 2 )

Side constraints:

d+ 2 t≤D (E 3 )

p≤P (E 4 )

t≥T (E 5 )

whereDis the maximum outside diameter permissible,Pthe maximum pressure

of the hydraulic system andT the minimum cylinder wall thickness required.

Equations (E 1 ) o (Et 5 ) an be stated in normalized form asc

4

π

Fp−^1 d−^2 ≤ 1

1

2 S

− (^1) pdt− (^1) ≤ 1
D−^1 d+ 2 D−^1 t≤ 1
P−^1 p≤ 1
Tt−^1 ≤ 1
The volume of the cylinder per unit length (objective) to be minimized is given by
π t (d+t).
Example 8.9 Design of a Cantilever Beam Formulate the problem of determining
the cross-sectional dimensions of the cantilever beam shown in Fig. 8.2 for minimum
weight. The maximum permissible bending stress isσy.