Nonlinear Reduced Order Modeling of Structures Near Buckling and Application to an Energy Harvester

The focus of this investigation is first on assessing the validity to structures under in-plane forces, in particular near buckling, of the reduced order modeling approach for nonlinear geometric response that has been extensively developed in the last two decades. This focus is motivated by a class of piezoelectric energy harvesters that rely on strongly nonlinear behavior, such as large amplitude responses, to achieve broadband energy harvesting. A simple, two-rigid bars linkage that approximates a buckling beam is first considered to discover the features of the nonlinear force–displacement relationship induced by an in-plane loading. It is observed that the corresponding form of this relationship is not consistent with the one derived from a reduced order model (ROM) but can be closely approximated by it over a large displacement range. This analysis emphasizes in particular the role of a group of ROM coefficients that are usually considered unimportant. A similar study is performed next for the buckled harvester modeled within NASTRAN and it is again found that a close match of the force–displacement relationship can be achieved. Based on that positive outlook, a six basis functions ROM of this beam harvester that includes piezoelectric effects is built and identified. It is found to provide a close match of NASTRAN nonlinear predictions over a broad range of transverse and in-plane loadings in static and dynamic conditions. The ROM usefulness in predicting the open-circuit voltage is demonstrated.