Engineering Optimization: Theory and Practice, Fourth Edition

748 Practical Aspects of Optimization

14.5.2 Derivatives ofYi

The differentiation of Eqs. (14.39) and (14.45) with respect toxjresults in

[Pi]

∂Yi

∂xj

= −

∂[Pi]

∂xj

Yi (14.47)

2 YTi[M]

∂Yi

∂xj

= −YTi

∂[M]

∂xj

Yi (14.48)

where∂[Pi]/ x∂jis given by Eq. (14.44). Equations (14.47) and (14.48) can be shown

to be linearly independent and can be written together as

[

[Pi]

2 YTi[M]

]

(m+ 1 )×m

∂Yi

∂xj

m× 1

= −









∂[Pi]

∂xj

YTi

∂[M]

∂xj









(m+ 1 )×m

Yi

m× 1

(14.49)

By premultiplying Eq. (14.49) by

[

[Pi]

YTi[M]

]T

=

[

[Pi] [M]Yi

]

we obtain

[[Pi][Pi] + 2 [M]YiYTi[ ]M]

m×m

∂Yi

∂xj

m× 1

= −

[

[Pi]

∂[Pi]

∂xj

+[M]YiYTi

∂[M]

∂xj

]

m×m

Yi

m× 1

(14.50)

The solution of Eq. (14.50) gives the desired expression for the derivative of the

eigenvector,∂Yi/∂xj, as

∂Yi

∂xj

= − [[Pi][Pi] + 2 [M]YiYTi[ ]M]−^1

×

[

[Pi]

∂[Pi]

∂xj

+[M]YiYTi

∂[M]

∂xj

]

Yi (14.51)

Again it can be seen that only the eigenvalue and eigenvector under consideration are

involved in the evaluation of the derivatives of eigenvectors.

Y 1 Y 3

Y 2 Y 4 Y 6

Y 5

1 in. 1 in. 1 in.

x 1 • x 2 • x 3

xi = 0.25′′ (i = 1, 2, 3), r = 0.283 lb/in^3 ,

E = 30 × 106 psi

Figure 14.4 Cylindrical cantilever beam.