Engineering Optimization: Theory and Practice, Fourth Edition

8.12 Applications of Geometric Programming 535

The optimum values ofxican be found from Eqs. (8.62) and (8.63):

1 =

0. 188 y∗d∗

19. 8

1 = 1. 75 y∗h∗−^1 d∗−^1

1

2 =^009 y

∗−^2

1

2 =y

∗−^2 h∗^2

These equations give the solution:y∗= 24 .426,h∗= 0 in., and 3 d∗= .475 in. 2

Example 8.14 Design of a Four-bar Mechanism [8.24] Find the link lengths of the
four-bar linkage shown in Fig. 8.4 for minimum structural error.

SOLUTION Leta,b,c, andddenote the link lengths,θ the input angle, andφ
the output angle of the mechanism. The loop closure equation of the linkage can be
expressed as

2 adcosθ− 2 cdcosφ+(a^2 −b^2 +c^2 +d^2 )

− 2 accos(θ−φ)= 0 (E 1 )

Infunction-generating linkages, the value ofφgenerated by the mechanism is made
equal to the desired value,φd, only at some values ofθ.These are known asprecision
points. In general, for arbitrary values of the link lengths, the actual output angle (φi)
generated for a particular input angle (θi) involves some error (εi) compared to the
desired value (φdi), so that

φi=φdi+εi (E 2 )

whereεiis called thestructural error atθi. By substituting Eq. (E 2 ) nto (Ei 1 ) nda
assuming that sinεi≈εiand cosεi≈ for small values of 1 εi, we obtain

εi=

K+ 2 adcosθi − 2 cdcosθdi− 2 accosθicos (φdi−θi)

− 2 acsin(φdi−θi) − 2 cdsinφdi

(E 3 )

where

K=a^2 −b^2 +c^2 +d^2 (E 4 )

Figure 8.4 Four-bar linkage.