472 Nonlinear Programming III: Constrained Optimization Techniques
J= 2
{
x 1 x 2
√
2[
x 22
12+
(
x 1 +x 3
2) 2 ]}
σ(X)=6 P L
x 4 x^23δ(X)=4 PL^3
Ex 33 x 4Pc(X)=4. 013
√
EG(x 32 x 46 / 63 )
L^2(
1 −
x 3
2 L√
E
4 G
)
Data: P=6000 lb, L=14 in., E= 30 × 106 psi, G= 12 × 106 psi, τmax=
13 ,600 psi,σmax= 03 ,000 psi, andδmax= 0. 2 5 in.
Starting and optimum solutions:Xstart=
h
l
t
b
=
0. 4
6. 0
9. 0
0. 5
in., fstart= 5 $. 3904 , X∗=
h
l
t
b
∗=
0. 2444
6. 2177
8. 2915
0. 2444
in.,f∗= 2 $. 38107.22.4 Speed Reducer (Gear Train) Design
The design of the speed reducer, shown in Fig. 7.24, is considered with the face width
(b), module of teeth (m), number of teeth on pinion (z), length of shaft 1 between bear-
ings (l 1 ), length of shaft 2 between bearings (l 2 ), diameter of shaft 1 (d 1 ), and diameter
of shaft 2 (d 2 ) as design variablesx 1 , x 2 ,... , x 7 , respectively. The constraints include
limitations on the bending stress of gear teeth, surface stress, transverse deflections of
shafts 1 and 2 due to transmitted force, and stresses in shafts 1 and 2 [7.40, 7.41].Figure 7.24 Speed reducer (gear pair) [7.40].