Engineering Optimization: Theory and Practice, Fourth Edition

750 Practical Aspects of Optimization

Note that if the undamped natural modes of vibration are used as basis functions and if

[C] is assumed to be a linear combination of [M] and [K] (calledproportional damp-

ing), Eqs. (14.54) represent a set ofruncoupled second-order differential equations

which can be solved independently [14.10]. Onceq(t)is found, the displacement solu-

tionY(t)can be determined from Eq. (14.53).

In the formulation of optimization problems with restrictions on the dynamic

response, the constraints are placed on selected displacement components as

|yj( X,t)| ≤ymax, j= 1 , 2 ,... (14.59)

whereyjis the displacement at locationjon the machine/structure andymaxis the

maximum permissible value of the displacement. Constraints on dynamic stresses are

also stated in a similar manner. Since Eq. (14.59) is a parametric constraint in terms

of the parameter time (t), it is satisfied only at a set of peak or critical values ofyj

for computational simplicity. Once Eq. (14.59) is satisfied at the critical points, it will

be satisfied (most likely) at all other values oftas well [14.11, 14.12]. The values

ofyi at which dyj/ td =0 or the values ofyiat the end of the time interval denote

local maxima and hence are to be considered as candidate critical points. Among the

several candidate critical points, only a select number are considered for simplifying

the computations. For example, in the response shown in Fig. 14.5, peaksa, b, c,... , j

qualify as candidate critical points. However, peaksa, b, f, andjcan be discarded

as their magnitudes are considerably smaller (less than, for example, 25%) than those

of other peaks. Noting that peaksdande(organdh) represent essentially a single

large peak with high-frequency undulations, we can discard peake(org), which has

a slightly smaller magnitude thand(orh). Thus finally, only peaksc, d, h, andineed

to be considered to satisfy the constraint, Eq. (14.59).

Once the critical points are identified at a reference designX, the sensitivity of the

response,yj( X,t)with respect to the design variablexiat the critical pointt=tccan

be found using the total derivative ofyjas

dyj( X,t)

dxi

=

∂yj

∂xi

+

∂yj

∂t

dtc

dxi

, i= 1 , 2 ,... , n (14.60)

The second term on the right-hand side of Eq. (14.60) is always zero since∂yj/∂t= 0

at an interior peak (0< tc < tmax) and dtc/d xi= at the boundary ( 0 tc=tmax). The

0

yj(t)

a

b

c

de

f

gh

i

j

t t

max

Figure 14.5 Critical points in a typical transient response.