8.12 Applications of Geometric Programming 531To avoid fatigue failure, the natural frequency of the spring (fn) is to be restricted to
be greater than (fn)min. The natural frequency of the spring is given by
fn=2 d
πD^2 n(
Gg
32 ρ) 1 / 2
(E 8 )
where g is the acceleration due to gravity. Using g= 9 .81 m/s^2 , G= 8. 56 ×
1010 N/m^2 , and (fn)min= 3, Eq. (E 1 8 ) ecomesb
13 (fn)minδG
288 , 800 Pd^3
D≤ 1 (E 9 )
Similarly, in order to avoid buckling, the free length of the spring is to be limited
as
L≤
11. 5 (D/ 2 )^2
P /K^1
(E 10 )
Using the relations
K^1 =
Gd^4
8 D^3 n(E 11 )
L=nd( 1 +Z) (E 12 )andZ= 0 .4, Eq. (E 10 ) an be expressed asc
0. 0527
(
Gδ^2
P)
d^5
D^5≤ 1 (E 13 )
It can be seen that the problem given by the objective function of Eq. (E 4 ) andcon-
straints of Eqs. (E 7 ) (E, 9 ) and (E, 13 ) s a geometric programming problem.i
Example 8.12 Design of a Lightly Loaded Bearing [8.29] A lightly loaded bearing
is to be designed to minimize a linear combination of frictional moment and angle of
twist of the shaft while carrying a load of 1000 lb. The angular velocity of the shaft is
to be greater than 100 rad/s.
SOLUTION
Formulation as a Zero-Degree-of-Difficulty Problem
The frictional moment of the bearing (M) and the angle of twist of the shaft (φ) are
given by
M=
8 π
√
1 −n^2μ
cR^2 L (E 1 )
φ=Sel
GR(E 2 )
whereμis the viscosity of the lubricant,nthe eccentricity ratio (=e/c),ethe eccentric-
ity of the journal relative to the bearing,cthe radial clearance,the angular velocity