<p>We investigate the shape of the symmetric potential energy function that maximizes the ability of a globally-stable mechanical oscillator to capture energy from a band-limited noise source. We show that, in the absence of any constraints, maximum power levels are always attained when the potential energy function is quadratic (mono-stable) and the center frequency of the noise is tuned to the natural frequency of the oscillator (linear resonance tuning). On the other hand, a bi-stable potential function yields maximum power levels in the presence of realistic constraints that prevent linear resonance tuning or place a maximum allowable limit on the mean square value of the displacement (size limit) and strain (longevity).</p>