Engineering Optimization: Theory and Practice, Fourth Edition

774 Practical Aspects of Optimization

14.10 The eigenvalue problem for the stepped bar shown in Fig. 14.11 can be expressed as

[K]Y=λ[M]Ywith the mass matrix, [M], given by

[M]=

[

( 2 ρ 1 A 1 l 1 + ρ 2 A 2 l 2 ) ρ 2 A 2 l 2

ρ 2 A 2 l 2 ρ 2 A 2 l 2

]

whereρi, Ai, andlidenote the mass density, area of cross section, and length of the seg-

menti, and the stiffness matrix, [K], is given by Eq. (2) of Problem 14.9. IfA 1 =2 in^2 ,

A 2 =1 in^2 ,E 1 =E 2 = 30 × 106 psi, 2l 1 =l 2 =50 in., andρ 1 g=ρ 2 g= 0 .283 lb/in^3 ,

determine

(a)Eigenvaluesλiand the eigenvectorsYi, i= 1 , 2

(b)Values of∂λi/∂A 1 , i= 1 ,2, using the method of Section 14.5

(c)Values of∂Yi/∂Y 1 , i= 1 ,2, using the method of Section 14.5

14.11 For the stepped bar considered in Problem 14.10, determine the following using the

method of Section 14.5.

(a)Values of∂λi/∂A 2 , i= 1 , 2

(b)Values of∂Yi/∂A 2 , i= 1 , 2

14.12 A cantilever beam with a hollow circular section with outside diameterdand wall

thicknesst(Fig. 14.12) is modeled with one beam finite element. The resulting static

equilibrium equations can be expressed as

2 EI

l^3

[

6 − 3 l

− 3 l 2 l^2

] {

Y 1

Y 2

}

=

{

P 1

P 2

}

whereIis the area moment of intertia of the cross section,Eis Young’s modulus, and

lthe length. Determine the displacements,Yi, and the sensitivities of the deflections,

∂Yi/∂dand∂Yi/∂t (i= 1 , 2 ), for the following data:E= 30 × 106 psi,l=20 in.,d= 2

in.,t= 0 .1 in.,P 1 =100 lb, andP 2 =0.

14.13 The eigenvalues of the cantilever beam shown in Fig. 14.12 are governed by the equation

2 EI

l^3

[

6 − 3 l

− 3 l 2 l^2

] {

Y 1

Y 2

}

=

λρAl

420

[

156 − 22 l

− 22 l 4 l^2

] {

Y 1

Y 2

}

0

A

A

Y 2

P 2

P 1

Y 1

x

l

y

t

Section A-A

d

Figure 14.12 Hollow circular cantilever beam.