Engineering Optimization: Theory and Practice, Fourth Edition

370 Nonlinear Programming II: Unconstrained Optimization Techniques

Problems

6.1 A bar is subjected to an axial load,P 0 , as shown in Fig. 6.17. By using a one-finite-element

model, the axial displacement,u(x), can be expressed as [6.1]

u(x)= {N 1 (x) N 2 (x)}

{

u 1

u 2

}

whereNi(x)are called the shape functions:

N 1 (x)= 1 −

x

l

, N 2 (x)=

x

l

andu 1 andu 2 are the end displacements of the bar. The deflection of the bar at point

Qcan be found by minimizing the potential energy of the bar(f ), which can be

expressed as

f=

1

2

∫l

0

EA

(

∂u

∂x

) 2

dx−P 0 u 2

whereEis Young’s modulus andAis the cross-sectional area of the bar. Formulate the

optimization problem in terms of the variablesu 1 andu 2 for the caseP 0 l/EA=1.

6.2 The natural frequencies of the tapered cantilever beam(ω)shown in Fig. 6.18, based on

the Rayleigh-Ritz method, can be found by minimizing the function [6.34]:

f (c 1 , c 2 )=

Eh^3

3 l^2

(

c^21

4

+

c 22

10

+

c 1 c 2

5

)

ρhl

(

c^21

30

+

c 22

280

+

2 c 1 c 2

105

)

with respect toc 1 andc 2 , wheref=ω^2 , Eis Young’s modulus, andρis the density.

Plot the graph of 3fρl^3 /Eh^2 in(c 1 , c 2 )space and identify the values ofω 1 andω 2.

6.3 The Rayleigh’s quotient corresponding to the three-degree-of-freedom spring–mass sys-

tem shown in Fig. 6.19 is given by [6.34]

R(X)=

XT[K]X

XT[M]X

where

[K]=k





2 −1 0

−1 2 − 1

0 −1 1



, [M]=





1 0 0

0 1 0

0 0 1



, X=







x 1

x 2

x 3







It is known that the fundamental natural frequency of vibration of the system can be

found by minimizingR(X). Derive the expression ofR(X)in terms ofx 1 ,x 2 , andx 3 and

suggest a suitable method for minimizing the functionR(X).

Figure 6.17 Bar subjected to an axial load.