This paper outlines a mathematical framework necessary to determine the optimal transducer force for a given vibration input. This relationship, between input vibration parameters and transducer force gives a critical first step in determining the optimal transducer architecture for a given vibration input. This relationship also yields a theoretical maximum energy output for a system with a given proof mass and parasitic mechanical losses, modeled as linear viscous damping.
This relationship is then applied to three specific vibration inputs; a single sinusoid, the sum of two sinusoids, and a single sinusoid with a time dependent frequency (chirp). For the single sinusoidal case, the optimal transducer is found to be a linear spring, resonant with the input frequency, and a linear viscous damper, with matched impedance to the mechanical damping. The resulting transducer force for the input as a sum of two sinusoids is found to be inherently time dependent. This time dependency shows that an active system (not only dependent on the states of the system) can outperform a passive system (dependent only on the states). The final application, for a swept sinusoidal input, results in a transducer of a linear viscous damper, with matched impedance to the mechanical damping, as well as a linear spring with a time dependent coefficient.