8.12 Applications of Geometric Programming 529The axial force applied (F )and the torque developed (T) are given by [8.37]
F=
∫
p dAsinα=∫R 1
R 2p2 πr dr
sinαsinα=πp(R^21 −R 22 ) (E 5 )T=
∫
rfp dA=∫R 1
R 2rfp2 π r
sinαdr=2 πfp
3 sinα(R 13 −R^32 ) (E 6 )
wherep is the pressure,f the coefficient of friction, andA the area of contact.
Substitution ofpfrom Eq. (E 5 ) nto (Ei 6 ) eads tol
T=
k 2 (R^21 +R 1 R 2 +R^22 )
R 1 +R 2(E 7 )
where
k 2 =2 Ff
3 sinα(E 8 )
Sincek 1 is a constant, the objective function can be taken asf=R^31 −R^32. The min-
imum torque to be transmitted is assumed to be 5k 2. In addition, the outer radiusR 1
is assumed to be equal to at least twice the inner radiusR 2. Thus the optimization
problem becomes
Minimizef (R 1 , R 2 )=R^31 −R 23
subject to
R 12 +R 2 R 2 +R^22
R 1 +R 2≥ 5 (E 9 )
R 1
R 2
≥ 2
This problem has been solved using complementary geometric programming [8.23]
and the solution was found iteratively as shown in Table 8.3. Thus the final solution is
taken asR∗ 1 = 4. 2 874,R 2 ∗= 2. 1 437, andf∗= 86 .916.
Example 8.11 Design of a Helical Spring Formulate the problem of minimum
weight design of a helical spring under axial load as a geometric programming prob-
lem. Consider constraints on the shear stress, natural frequency, and buckling of the
spring.
SOLUTION By selecting the mean diameter of the coil and the diameter of the wire
as the design variables, the design vector is given by
X={
x 1
x 2}
=
{
D
d}
(E 1 )
The objective function (weight) of the helical spring can be expressed as
f (X)=π d^2
4(πD)ρ(n+Q) (E 2 )