Stability and bifurcation analysis of the period-T motion of a vibroimpact energy harvester

Abstract

Stability and bifurcation conditions for a vibroimpact motion in an inclined energy harvester with T-periodic forcing are determined analytically and numerically. This investigation provides a better understanding of impact velocity and its influence on energy harvesting efficiency and can be used to optimally design the device. The numerical and analytical results of periodic motions are in excellent agreement. The stability conditions are developed in non-dimensional parameter space through two basic nonlinear maps based on switching manifolds that correspond to impacts with the top and bottom membranes of the energy harvesting device. The range for stable simple T-periodic behavior is reduced with increasing angle of incline $\beta$, since the influence of gravity increases the asymmetry of dynamics following impacts at the bottom and top. These asymmetric T-periodic solutions lose stability to period doubling solutions for $\beta \ge 0$, which appear through increased asymmetry. The period doubling, symmetric and asymmetric periodic motion are illustrated by bifurcation diagrams, phase portraits and velocity time series.

Appendix A

We give the details for the calculations of the eigenvalues $\lambda _{1,2}$. The entries in the matrices in (4.2) are

$$\begin{aligned}&\frac{\partial t_{k}}{\partial t_{k-1}} = \frac{r\dot{Z}_{k-1}-\bar{g}T_u-f(t_{k-1})T_u}{\dot{Z}_{k-1}-\frac{\bar{g}T}{1+r}}, \nonumber \\&\frac{\partial t_{k}}{\partial \dot{Z}_{k-1}} = \frac{-r T_u}{\dot{Z}_{k-1} -\frac{\bar{g}T}{1+r}},\nonumber \\&\frac{\partial \dot{Z}_{k}}{\partial t_{k-1}} = \frac{\partial t_{k}}{\partial t_{k-1}}[f(t_k)+\bar{g}]-[f(t_{k-1})+\bar{g}],\nonumber \\&\frac{\partial \dot{Z}_k}{\partial \dot{Z}_{k-1}} = -r + \frac{\partial t_{k}}{\partial \dot{Z}_{k-1}} [f(t_k)+\bar{g}], \end{aligned}$$

(A.1)

and

$$\begin{aligned}&\frac{\partial t_{k+1}}{\partial t_{k}} = \frac{r\dot{Z}_{k}-\bar{g}T_d-f(t_{k})T_d}{\dot{Z}_{k}-\frac{\bar{g}T}{1+r}}, \nonumber \\&\frac{\partial t_{k+1}}{\partial \dot{Z}_{k}} = \frac{-r T_d}{\dot{Z}_{k} -\frac{\bar{g}T}{1+r}},\nonumber \\&\frac{\partial \dot{Z}_{k+1}}{\partial t_{k}} = \frac{\partial t_{k+1}}{\partial t_{k}}[f(t_{k+1})+\bar{g}]-[f(t_{k})+\bar{g}],\nonumber \\&\frac{\partial \dot{Z}_{k+1}}{\partial \dot{Z}_{k}} = -r + \frac{\partial t_{k+1}}{\partial \dot{Z}_{k}} [f(t_{k+1})+\bar{g}]. \end{aligned}$$

(A.2)

For the period-2 motion, the trace and determinant of the linearized matrix DP are

$$\begin{aligned} Det(DP)&=r^4 , \end{aligned}$$

(A.3)

$$\begin{aligned} Tr(DP)&=\left( r+\frac{rT_u(\bar{g}+f(t_k))}{\sigma _3}\right) \nonumber \\&\qquad \left( r-\frac{rT_d(\bar{g}+f(t_{k+1}))}{\dot{Z}(t_{k-1})}\right) \nonumber \\&\qquad +\frac{rT_u \left( \bar{g}+f(t_k)-\frac{(\bar{g}+f(t_{k+1}))\sigma _2}{\dot{Z}(t_{k-1})} \right) }{\sigma _3}\nonumber \\&\qquad -\frac{rT_d \left( \bar{g}+f(t_{k-1})+\frac{(\bar{g}+f(t_{k}))\sigma _1}{\sigma _3} \right) }{\dot{Z}(t_{k-1})} \nonumber \\&\qquad -\frac{\sigma _2 \sigma _1}{\dot{Z}(t_{k-1})\sigma _3}, \end{aligned}$$

(A.4)

where $\sigma _1=T_u\cdot f(t_{k-1})-r \dot{Z}(t_{k-1})+\bar{g}T_u$, $\sigma _2=T_d \cdot f(t_{k})+r \sigma _3+\bar{g}T_d$, $\sigma _3=\dot{Z}(t_{k-1})-\frac{\bar{g}T}{r+1}$.

Considering a horizontal impact pair, (A.3) and (A.4) generate the same equations for determinant and trace for symmetric period-2 motion $(q=1/2)$ and odd n from [25].