1. Upper and lower bounds on the design variables also may be imposed such that blower 5 bi _< bypper.Mathematically and numerically,these areusually treated like inequality constraints. The SAPOP program (Structural Analysis Program and Optimization Procedure) [21]was chosen for this problem due to its proven efficiency and utility. This package features a number of optimizer algorithms which can be applied interchangeably to suit the particular demands of a given problem. The particular optimizer used in this study used the hybrid SQP-GRG algorithm [22], which employs the SQP (Sequential Quadratic Programming) search direction and the GRG (Generalized Reduced Gradient) line search. This algorithm features good convergence and is applicable to non-linear problems with large numbers of variables and/or constraints. Other optimizers in the SAPOP package include a SLP (Sequential Linear Programming), SQP, and GRG. For performance comparison, the CONMIN program [23] was also used for a few cases.
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3. Each case was designed to be dynamically similar,with the initialdesigns all having the same natural frequency of 108 Hz. For the broad band cases, a 90-900 Hz frequency band was used, which encompasses the first five symmetric modes of each plate.
4. The broad band power from 90-900 Hz was then minimized using two different ranges of lower and upper bounds, 1:10 (.894-8.94mm) and 1:4 (1.97-7.87mm). The results are displayed in Figure 4 and demonstrate that wider bounds typically result in lower optimums (13.2 vs 3.4 dB improvement). These cases resulted in an increases in the plate weight of 30% and 27%.