Effectiveness of Energy Transfer versus Mixing Entropy in Coupled Mechanical–Electrical Oscillators

Electrostatic energy harvesters convert kinetic energy into electrical energy via variable capacitors. Efforts to improve their power output are hampered by a lack of understanding of the fundamental limit for energy conversion efficiency. In heat engines, the theoretical limit of conversion efficiency is intrinsically related to entropy and the second law of thermodynamics. Laying the foundation for similar concepts for kinetic energy harvesters may be necessary for establishing a conversion efficiency limit. Thus, the mixing entropy concept is borrowed from statistical mechanics and is adapted here, for the first time, to characterize the energy transfer between coupled mechanical–electrical oscillators. The investigated system is composed of a spring-mass coupled to an inductance-capacitor circuit via a variable capacitor. Combining the two subsystems (electrical and mechanical) generates entropy, referred to as mixing entropy. A non-dimensional study of the governing equations of the systems and their energy terms is carried out. Trends in mixing entropy are compared with trends in the total energy of the system, assuming a conservative system, weak coupling between electrical and mechanical domains, and identical natural frequency of the two oscillators. It is found that mixing entropy can predict the peak in effectiveness of the energy transfer between the two domains. For the cases studied, the maximum mixing entropy and effectiveness values occur when the ratio of the mechanical domain energy to the total energy of the system is 67%. The maximum effectiveness is independent of the initial conditions and depends on the squared ratio of the natural frequency of the nominal coupling capacitor to the natural frequency of the mechanical system.