Theoretical analysis and adaptive high-energy orbit control of symmetric tri-stable energy harvesters with nonlinear damping

In this paper, the theoretical analysis and adaptive high-energy orbit control of the symmetric tri-stable energy harvester with nonlinear damping are mainly considered, aiming to reveal the dynamic response mechanism and improve the output of the harvester. The coupling relationship of the amplitude and frequency is obtained by the modified Lindstedt-Poincaré method, which is consistent with the multi-scale analysis results. The five-valued response is found, in which three of them are stable. At the same time, complicated multi-valued characteristics are found in the response amplitude. Combined with stability analysis, the response mechanism is revealed and verified. The complex influence of nonlinear damping on the distribution of the unstable region is discussed. To receive higher output in a wider frequency range, the higher order nonlinear coefficients and nonlinear damping are optimized. Then, by designing the adaptive sliding mode control, the trajectory of motion can be pushed from the low-energy orbit to the high-energy orbit when some parameters are unknown and the controller only needs to work for a short time. The error of the controlled high-energy orbit motion can also be kept within the acceptable range, thus the expected purpose can be achieved.