  Designing lightweight yet stiff vibrating structures has been a long-standing
objective in structural optimization. Here, we consider the optimization of a
structural design subject to forces in the form of harmonic oscillations. We
develop a unifying framework for the minimization of compliance and peak-power
functions, while avoiding the assumptions of single-frequency and in-phase
loads. We utilize the notion of semidefinite representable (SDr) functions and
show that for compliance such a representation is immediate, while for peak
power, the SDP representation is non-trivial. Subsequently, we propose convex
relaxations of the minimization of an SDr function subject to the equilibrium
equation as a constraint, and show promising computational results on
well-known instances of Heidari et al. and others.
