# Response Analysis of the Tristable Energy Harvester with an Uncertain Parameter
## 1 Introduction  
With the rapid development of wireless sensor networks and portable electronics, conventional batteries have not held pace with the demands from microelectronic devices. Given these challenges, energy harvesting from available ambient vibration has received considerable attention, and various energy harvesters have been designed and experimentally tested [1; 2; 3; 4]. In the early stage, the resonant-based vibration harvesters have been widely used to generate energy, which could only achieve considerable energy harvesting performance at or near its resonant frequency [5; 6]. To remedy this problem, many structural optimization designs [7; 8] and technologies including resonance tuning technique [9; 10; 11; 12], monostable technique [13; 14; 15; 16; 17], bistable technique [18; 19; 20; 21; 22] and tristable technique [23; 24; 25; 26; 27] have been proposed. Theoretical analysis and experimental results have shown that the nonlinear harvester could improve the ability to harvest energy under certain circumstances. In addition, for low-level excitations, the tristable energy harvester (TEH) could easily achieve higher energy interwell motions, enhance the output voltage significantly and have a wider resonance-bandwidth compared with the conventional monostable or bistable one [27; 28]. Therefore, the exploration of nonlinear dynamic characteristics of TEH is quite significant. It may provide some valuable insights for the design and improvement of the energy harvester. Most studies assume that the system parameters of the energy harvester are deterministic. However, there are various inevitable uncertain errors arising from modelling, measuring, mechanical processing and manufacturing. Thus, the electromechanical models that use a deterministic form fail to accurately predict the input-output behavior of the energy harvester under physical uncertainty, and the uncertain parameter should be taken into account to discuss dynamics of the system. Over the past few decades, a growing number of researchers have shown great interest in the effect of parametric uncertainty on the energy harvesting system. [PERSON] [29] and [PERSON] [30] investigated the influence of uncertain parameter on the energy harvesting performance, including harvested power, optimal electrical time constant and coupling coefficient. [PERSON] et al. [31] evaluated the influence of uncertain parameters in the design and optimization of a cantilever piezoelectric energy harvester. The simulated results illustrated that small variations of key design parameters can lead to a significant impact on the output electrical power. Taking it one step further, a reliability analysis of the vibration energy harvesting device under physical uncertainty was proposed [32], with the aim of calculating the probability of overall system success. Moreover, the sensitivity of the harvested power to variations in the harvester's parameters was assessed for optimal design [33], which showed that the harvested power is most sensitive to variations of eccentricity. These above-mentioned studies illustrated that the uncertainty of the system parameter may lead to obvious changes of system dynamic responses and reliability, so the parametric uncertainty analysis of an energy harvesting system needs to be further discussed. In recent years, many methods have been established to analyze the response problem of the nonlinear dynamic system with uncertain parameters, such as Monte Carlo method [29; 31], stochastic perturbation method [34], fuzzy method [35; 36], interval method [37; 38] and orthogonal polynomial approximation method [39]. Among these, the orthogonal polynomial approximation method is a powerful approximate technique, which can produce the mean response and has a significant computational advantage. This method was first introduced by [PERSON] based on homogeneous chaos theory [40]. Subsequently, [PERSON] [41] constructed an approximation method based on Hermite orthogonal polynomial expansion to solve differential equation with random parameter.