Engineering Optimization: Theory and Practice, Fourth Edition

532 Geometric Programming

of the shaft,Rthe radius of the journal,Lthe half-length of the bearing,Sethe shear

stress,lthe length between the driving point and the rotating mass, andGthe shear

modulus. The load on each bearing (W) is given by

W=

2 μRL^2 n

c^2 ( 1 −n^2 )^2

[π^2 ( 1 −n^2 ) + 16 n^2 ]^1 /^2 (E 3 )

For the data W=1000 lb, c/R= 0 .0015, n= 0 .9, l=10 in., Se= 03 ,000 psi,

μ= 10 −^6 lb-s/in^2 , andG= 12 × 106 psi, the objective function and the constraint

reduce to

f (R, L)=aM+bφ= 0. 038 R^2 L + 0. 025 R−^1 (E 4 )

R−^1 L^3 = 11. 6 (E 5 )

≥ 100 (E 6 )

whereaandbare constants assumed to bea=b=1. Using the solution of Eq. (E 5 )

gives

= 11. 6 RL−^3 (E 7 )

theoptimization problem can be stated as

Minimizef (R, L)= 0. 45 R^3 L−^2 + 0. 025 R−^1 (E 8 )

subjectto

8. 62 R−^1 L^3 ≤ 1 (E 9 )

The solution of this zero-degree-of-difficulty problem can be determined asR∗=

0. 2 12 in.,L∗= 0. 2 91 in., andf∗= 0. 1 7.

Formulation as a One-Degree-of-Difficulty Problem

By considering the objective function as a linear combination of the frictional moment

(M), the angle of twist of the shaft (φ), and the temperature rise of the oil (T ),

we have

f=aM+bφ+cT (E 10 )

wherea,b, andcare constants. The temperature rise of the oil in the bearing is given

by

T= 0. 045

μR^2

c^2 n

√

( 1 −n^2 )

(E 11 )

By assuming that 1 in.-lb of frictional moment in bearing is equal to 0.0025 rad of angle

of twist, which, in turn, is equivalent to 1◦F rise in temperature, the constantsa,b,

andccan be determined. By using Eq. (E 7 ) the optimization problem can be stated,

as

Minimizef (R, L)= 0. 44 R^3 L−^2 + 01 R−^1 + 0. 592 RL−^3 (E 12 )