To illustrate the minimum-weight design of one-dimensional, elastic structures under dynamic excitation, methods from optimal control theory are applied to the cantilever bar driven sinusoidally by an axial force at its tip. Other directly analogous problems are identified, and closely related cases are discussed along with practical applications. Realistic constraints are enforced during the optimizations: maximum allowable stress amplitude at any point along the bar and the minimum cross-sectional area. In the absence of damping, the design space may contain many disjoint feasible regions, and multiple optima can exist. This novel feature is examined, first by reference to a simple example with only two element areas to determine. Solutions are worked out in detail for continuous bars, with the excitation frequency less than, then greater than, the fundamental free-vibration frequency. The latter results overcome a limitation inherent in previous analyses. Above a certain excitation frequency, two or more arcs with different constraints characterize the optimal designs; a concentrated tip mass is also needed in some cases. Free vibrations of the optimal designs are analyzed. Background on the difficulties of going to even higher frequencies than those covered by the exact solutions is covered by a discussion of forced-response mode shapes, by reference to finite element results, and by a simplified two-design-variable example.
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