This multidisciplinary project is an international collaboration between Georgia Tech, Georgia State University and University of Southampton in the area of applied mathematics, nonlinear dynamics and energy harvesting with soft electroactive polymers. The project addresses gaps and opportunities in mathematical developments and in practical applications for impact-based engineering devices and systems with non-smooth dynamics, critical for understanding the harvesting of vibrational energy. The main research objective is to develop a universal suite of mathematical methodologies for the analysis and optimized performance of deterministic and stochastic non-smooth engineering systems. These approaches are pursued with a focus on practical engineering models of vibro-impacting energy harvesting systems and nonlinear dynamic dampers, within the broad area of targeted energy transfer. Integrating novel nonlinear, stochastic, and computational approaches with experimental and real-world data both for energy harvesting and for devices that mitigate vibration will pave new pathways for analysis-based design and model validation.