14.7 Sensitivity of Optimum Solution to Problem Parameters 751
derivative,∂yj/∂xi, can be computed using Eq. (14.53) as
∂yj
∂xi
=
∑r
k= 1
j k
∂qk(t)
∂xi
, i= 1 , 2 ,... , n (14.61)
where, for simplicity, the elements of the matrix [] are assumed to be constants
(independent of the design vectorX). Note that for higher accuracy, the derivatives
ofj kwith respect toxi(sensitivity of eigenvectors, if the mode shapes are used as
the basis vectors) obtained from an equation similar to Eq. (14.51) can be included in
finding∂yj/∂xi.
To find the values of∂qk/∂xirequired in Eq. (14.61), Eq. (14.54) is differentiated
with respect toxito obtain
[M]
∂q ̈
∂xi
+[C]
∂q ̇
∂xi
+[K]
∂q
∂xi
=
∂F
∂xi
−
∂[M]
∂xi
q ̈−
∂[C]
∂xi
q ̇−
∂[K]
∂xi
q , i= 1 , 2 ,... , n (14.62)
The derivatives of the matrices appearing on the right-hand side of Eq. (14.62) can be
computed using formulas such as
∂[M]
∂xi
=[]T
∂[M]
∂xi
[] (14.63)
where, for simplicity, [] is assumed to be constant and∂[M]/∂xiis computed using
afinite-difference scheme. In most cases the forcing functionF(t) will be known to
be independent ofXor an explicit function ofX. Hence the quantity∂F/∂xican be
evaluatedwithout much difficulty. Once the right-hand side is known, Eqs. (14.62) can
be integrated numerically in time to find the values of∂q ̈/∂xi, ∂ q ̇/∂xi, and ∂q/∂xi.
Using the values of∂q/∂xi= { ∂qk/∂xi} t the critical pointa tc, the required sensitivity
of transient response can be found from Eq. (14.61).
14.7 Sensitivity of Optimum Solution to Problem Parameters
Any optimum design problem involves a design vector and a set of problem param-
eters (or preassigned parameters). In many cases, we would be interested in knowing
the sensitivities or derivatives of the optimum design (design variables and objective
function) with respect to the problem parameters [14.25, 14.26]. As an example, con-
sider the minimum weight design of a machine component or structure subject to a
constraint on the induced stress. After solving the problem, we may like to find the
effect of changing the material. This means that we would like to know the changes
in the optimal dimensions and the minimum weight of the component or structure due
to a change in the value of the permissible stress. Usually, the sensitivity derivatives
are found by using a finite-difference method. But this requires a costly reoptimization
of the problem using incremented values of the parameters. Hence, it is desirable to
derive expressions for the sensitivity derivatives from appropriate equations. In this
section we discuss two approaches: one based on the Kuhn–Tucker conditions and the
other based on the concept of feasible direction.
