Transition Waves for Energy Trapping and Harvesting

Abstract

The presence of multiple stable states and associated nonlinear phenomena, such as hysteresis, in multistable mechanical metamaterials enables frequency-independent energy harvesting and shock absorption. This study focuses on shock absorption achieved by locking transition waves to trap energy at designed locations within a multistable metamaterial. We further demonstrate that the same system can simultaneously harvest energy from impact loading, thereby exhibiting multifunctionality. The model of the multistable metamaterial is a one-dimensional chain of bistable units whose transition wave dynamics are related to topological solitary waves governed by the $φ^4$ equation. This connection enables analytical estimation of critical design parameters required for energy trapping and also the amount of energy trapped. Numerical simulations and experiments show that trapping energy in transition waves leads to enhanced damping performance compared to corresponding linear metamaterials. We further propose design variations to increase the amount of energy trapped in the transition wave. Additionally, we identify energy splitting as a damping mechanism that arises when there are repeated impulses or a single high-amplitude impulse that generates multiple transition waves. The transition waves interact to produce localized, fast-dissipating breathers, leading to a damped response. Furthermore, experiments demonstrate that multistable metamaterials can simultaneously achieve improved energy harvesting and better damping performance compared to their linear counterparts. Together, these results highlight the use of transition waves for creating multifunctional multistable metamaterials.

Figures (7)

• Figure 1: a) Schematic representation of the multistable metamaterial composed of masses connected by intersite springs, with each mass also connected to a bistable substrate potential. A force on the first unit can trigger a transition wave. b) Displacement profile of a transition wave propagating through the structure.
• Figure 2: a) Transition wave propagation in the absence of defect. b) Energy trapping achieved by defect of $r=0.35$ from sites 250 to 290. c) Reduced tip velocity due to energy trapping in multistable metamaterial with defect compared to linear metamaterial (with defect) and multistable metamaterial without defect. d) Critical stiffness ratio ($r^*$) for energy trapping as a function of the normalized intersite stiffness ($1/\omega_0^2$).
• Figure 3: (a) Propagation of transition wave in the presence of a defect region from sites 250 to 290. b) Energy trapping using two soft defect regions, at sites 125-145 and 250-270. c) Energy trapping using a soft defect region at sites 250-290 followed by a stiff defect region of $r=2$ at sites 290-330. All soft defect regions have $r=0.5$. d) Tip velocities of the multistable metamaterial for the three cases shown in a-c (solid lines) and those of the corresponding linear metamaterial (dotted lines). Grey shaded area highlights the range of tip velocities for cases in b and c. e) Critical stiffness ratio and f) amount of strain energy trapped in the transition wave as a function of the normalized intersite stiffness $1/\omega_0^2$ for the three types of defect regions shown in a-c. $1/\omega_0^2=10$ in a-c.
• Figure 4: a) Energy trapping and splitting due to three separate impulses (at times 0, 500, and 1000). b) Tip velocity in the multistable metamaterial compared to the corresponding linear metamaterial. $r=0.4$, $\bar{b} = 0.003$, $\bar{v}_0=2$.
• Figure 5: a) Transition wave generation and trapping for $\bar{v}_0=2$. b) Energy splitting due to generation of two transition waves that combine into a moving breather for $\bar{v}_0=3$. c) Zoomed outset showing the moving breather in (b). d) Displacement profile of the breather at time $\bar{t} = 70$ (marked by dashed line in c). e) Energy splitting and energy trapping due to generation of three transition waves for $\bar{v}_0=4$. f) Transmissibility of the multistable and the corresponding linear metamaterial as a function of the normalized initial velocity $\bar{v}_0$ imparted by the impulse. $r=0.4$, $\bar{b} = 0.003$ in all plots.