Engineering Optimization: Theory and Practice, Fourth Edition

530 Geometric Programming

Table 8.3 Results for Example 8.10

Iteration Starting Ordinary geometric programming Solution

number design problem of OGP

1 x 1 =R 0 = 40 Minimizex 11 x 20 x 30 x 1 = 162. 5

x 2 =R 1 = 3 subject to x 2 = 5. 0

x 3 =R 2 = 3 0. 507 x− 10.^597 x^32 x− 31.^21 ≤ 1 x 3 = 2. 5

1. 667 (x 2 −^1 +x 3 −^1 )≤ 1

2 x 1 =R 0 = 162. 5 Minimizex 11 x 20 x 30 x 1 = 82. 2

x 2 =R 1 = 5. 0 subject to x 2 = 4. 53

x 3 =R 2 = 2. 5 0. 744 x− 10.^912 x 23 x− 30.^2635 ≤ 1 x 3 = 2. 265

3. 05 (x 2 −^0.^43 x 3 −^0.^571 +x 2 −^1.^43 x 30.^429 )≤ 1

2 x 2 −^1 x 3 ≤ 1

3 x 1 =R 0 = 82. 2 Minimizex 11 x 20 x 30 x 1 = 68. 916

x 2 =R 1 = 4. 53 subject to x 2 = 4. 2874

x 3 =R 2 = 2. 265 0. 687 x− 10.^876 x 23 x− 30.^372 ≤ 1 x 3 = 2. 1437

1. 924 x^01 x 2 −^0.^429 x− 30.^571 +

1. 924 x^01 x− 21.^492 x^03.^429 ≤ 1

2 x 2 −^1 x 3 ≤ 1

wherenis the number of active turns,Qthe number of inactive turns, andρthe weight

density of the spring. If the deflection of the spring isδ, we have

δ=

8 PC^3 n

Gd

or n=

Gdδ

8 PC^3

(E 3 )

whereGis the shear modulus,P the axial load on the spring, andCthe spring index

(C=D/d). Substitution of Eq. (E 3 ) nto (Ei 2 ) ivesg

f (X)=

π^2 ρGδ

32 P

d^6

D^2

+

π^2 ρQ

4

d^2 D (E 4 )

If the maximum shear stress in the spring (τ) is limited toτmax, the stress constraint

can be expressed as

τ=

8 KPC

π d^2

≤τmax or

8 KPC

π d^2 τmax

≤ 1 (E 5 )

whereKdenotes the stress concentration factor given by

K≈

2

C^0 5.^2

(E 6 )

The use of Eq. (E 6 ) n (Ei 5 ) esults inr

16 P

πτmax

D^3 /^4

d^11 /^4

≤ 1 (E 7 )