Engineering Optimization: Theory and Practice, Fourth Edition

Problems 373

q

Figure 6.22 Two-bar truss.

(b)Find the steepest descent direction,S 1 , offat the trial vectorX 1 =

{ 0

0

}

.

(c)Derive the one-dimensional minimization problem,f (λ), atX 1 along the direction

S 1.

(d)Find the optimal step lengthλ∗using the calculus method and find the new design

vectorX 2.

6.7 Three carts, interconnected by springs, are subjected to the loadsP 1 , P 2 , andP 3 as shown
in Fig. 6.23. The displacements of the carts can be found by minimizing the potential
energy of the system(f ):

f (X)=^12 XT[K]X−XTP

where

[K]=





k 1 +k 4 +k 5 −k 4 −k 5

−k 4 k 2 +k 4 +k 6 −k 6

−k 5 −k 6 k 3 +k 5 +k 6 +k 7 +k 8





P=







P 1

P 2

P 3







and X=







x 1

x 2

x 3







Derive the functionf (x 1 , x 2 , x 3 )for the following data:k 1 =5000 N/m ,k 2 =1500 N/m,

k 3 =2000 N/m,k 4 =1000 N/m,k 5 =2500 N/m,k 6 =500 N/m,k 7 =3000 N/m,k 8 =

3500 N/m,P 1 =1000 N,P 2 =2000 N, andP 3 =3000 N. Complete one iteration of

Newton’s method and find the equilibrium configuration of the carts. UseX 1 = {0 0 0}T.

6.8 Plot the contours of the following function over the region (− 5 ≤x 1 ≤ 5 ,− 3 ≤x 2 ≤6)
and identify the optimum point:

f (x 1 , x 2 )=(x 1 + 2 x 2 − 7 )^2 +( 2 x 1 +x 2 − 5 )^2