The benefits of using a non-linear stiffness in an energy harvesting device comprising a mass-spring-damper system are investigated. Analysis based on the principle of conservation of energy reveals a fundamental limit of the effectiveness of any non-linear device over a tuned linear device for such an application. Two types of non-linear stiffness are considered. The first system has a non-linear bi-stable snap-through mechanism. This mechanism has the effect of steepening the displacement response of the mass as a function of time, resulting in a higher velocity for a given input excitation. Numerical results show that more power is harvested by the mechanism if the excitation frequency is much less than the natural frequency. The other non-linear system studied has a hardening spring, which has the effect of shifting the resonance frequency. Numerical and analytical studies show that the device with a hardening spring has a larger bandwidth over which the power can be harvested due to the shift in the resonance frequency.

This work is concerned with Mathieu's equation—a classical differential equation, which has the form of a linear second-order ordinary differential equation (ODE) with Cosine-type periodic forcing of the stiffness coefficient, and its different generalizations/extensions. These extensions include: the effects of linear viscous damping, geometric nonlinearity, damping nonlinearity, fractional derivative terms, delay terms, quasiperiodic excitation, or elliptic-type excitation. The aim is to provide a systematic overview of the methods to determine the corresponding stability chart, its structure and features, and how it differs from that of the classical Mathieu's equation.

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