||A standard lumped parameter model for an inertial vibration energy harvester consists of a proof mass, spring, and damper(s). This model can also be described with a proof mass, viscous damping element for parasitic mechanical losses, and a generalized transducer that applies some force to the mass damper system. The transducer may contain restorative spring elements and energy extraction elements to harvest power. Currently the framework to relate vibration input to an optimal transducer architecture does not exist. Previous work has shown that for some inputs nonlinear transducer architectures can result in an increased power output. This paper outlines a mathematical framework needed in order to find the optimal transducer architecture for a given vibration input. This framework defines the theoretical upper limit that any inertial transducer can harvest from a given vibration input in the presence of viscous mechanical damping. This framework is then applied to three cases of standard input types. The first application is a single sinusoid input. The transducer architecture found is the expected result, a linear spring with matched resonance to the input, and an energy extraction element, that behaves as a linear viscous damper, with matched impedance to the mechanical damping. The second application of this framework is an input of two sinusoids both having equal magnitude but different frequencies. The resulting optimal transducer is dependent on the difference in the frequencies of the two signals. This optimal transducer is often not realizable with a passive system, as it is inherently time dependent. For all cases of frequency separation between the two sinusoidal inputs, the upper limit for the energy generated is found to be twice that of a linear harvester tuned to the lower of the two frequencies. The third application is for an input whose frequency changes linearly in time (i.e. a swept sinusoid). The optimal transducer architecture for this input is found to be completely time dependent. However for the case when the change in the input frequency is much slower than the period of the system, the transducer can be approximated by a linear spring whose stiffness changes in time.