Analysis of Dry Friction Dynamics in a Vibro-Impact Energy Harvester

TL;DR

This work develops a semi-analytical framework for a dry-friction–augmented vibro-impact energy harvester, modeling a bullet-in-capsule system under harmonic forcing. By deriving discrete nonlinear maps that integrate switching and impacting events, the authors reveal intricate sequences of smooth and non-smooth bifurcations (grazing, sliding, switching) that govern periodic motions and their energy-harvesting potential. Stability analyses of 1:1 and 1:1$^{\Sigma}_{s}$ solutions are conducted via Jacobians of map compositions, and extensive comparisons with numerical simulations validate the reduced-model predictions. The results show that while dry friction generally lowers impact velocities and energy, it can also stabilize high-energy 1:1-type motions and shift grazing bifurcations to sustain performance in certain parameter windows, offering design insights for robust energy harvesting in non-smooth mechanical systems.

Abstract

Vibro-impact (VI) systems provide a promising nonlinear mechanism for energy harvesting (EH) in many engineering applications. Here, we consider a VI-EH system that consists of an inclined cylindrical capsule that is externally forced and a bullet that is allowed to move inside the capsule, and analyze its dynamics under the presence of dry friction. Dry friction introduces a switching manifold corresponding to zero relative velocity where the bullet sticks to the capsule, appearing as sliding in the model. We identify analytical conditions for the occurrence of non-stick and sliding motions, and construct a series of nonlinear maps that capture model solutions and their dynamics on the switching and impacting manifolds. An interplay of smooth (period-doubling) and non-smooth (grazing) bifurcations characterizes the transition from periodic solutions with alternating impacts to solutions with an additional impact on one end of the capsule per period. This transition is preceded by a sequence of grazing-sliding, switching-sliding and crossing-sliding bifurcations on the switching manifold that may reverse period doubling bifurcations for larger values of the dry friction coefficient. In general, a larger dry friction coefficient also results in larger sliding intervals, lower impact velocities yielding lower average energy outputs, and a shift in the location of some bifurcations. Surprisingly, we identify parameter regimes in which higher dry friction maintains higher energy output levels, as it shifts the location of grazing bifurcations.

Figures (15)

• Figure 1: (a) Schematic of the VI-EH model with $F_N=mg\cos{\beta}\text{sgn}(\dot{x}-\dot{X})$ (in red) indicating the normal force. (b) Graph of the discontinuous force due to dry friction that is proportional to the normal force $F_N$.
• Figure 2: Schematic of the phase plane in relative frame. The blue and green dotted lines indicate the impact surfaces, $\Gamma^{+}$ (bottom membrane) and $\Gamma^{-}$ (top membrane), respectively. The thick black line corresponds to the switching manifold, $\Sigma$, with $\Sigma^{+}$ ($\Sigma^{-}$) indicating regions of the phase plane with $\dot{Z}>0$$(<0)$.
• Figure 3: Schematic representations of the motion from $\Gamma^{+}$ to $\Gamma^{-}$ in (from left to right) 1:1, 1:1$^{\Sigma}_{c}$, 1:1$^{\Sigma}_{s}$, and 1:1$^{\Sigma}_{cs}$ periodic motions in the $\dot{Z}$ vs $t$ plane. The red rectangle indicates the time interval over which sliding may occur determined by \ref{['eq:SlidingOnset2a']}-\ref{['eq:SlidingOnset2c']} and \ref{['eq:NonPassToSemiPassCondToOmega2_1']}-\ref{['eq:NonPassToSemiPassCondToOmega2_3']}.
• Figure 4: Periodic solutions demonstrating the different types of motion in the VI-EH model with dry friction. The parameter values are $r=0.5$, $\beta=\pi/4$, and $\mu_k=0.5$. Left column: time series in absolute frame; Middle column: times series in relative frame. Right column: $Z-\dot{Z}$ phase plane. Panels (a),(b),(c): $A=3.1$, $d=0.502$ show a non-stick motion in which the relative velocity, $\dot{Z}$, does not change sign. Panels (d),(e),(f): $A=6.9$, $d=0.2255$ show a non-stick motion in which the relative velocity, $\dot{Z}$, changes sign. Panels (g),(h),(i): $A=6.4$, $d=0.2432$ show a sliding motion in which the relative velocity, $\dot{Z}=0$ for a nontrivial period of time. The gray dashed line in the middle- and right-column panels corresponds to $\dot{Z}=0$.
• Figure 5: Bifurcation diagrams of impact velocities, $\dot{Z}_{k}$, vs $d$ for different values of $\mu_k$, with $r=0.5$, $\beta=\pi/4$, and $A$ increasing over $[3.906,14.4]$. The blue dots correspond to impact velocities on $\Gamma^{+}$ (bottom membrane), while the green dots correspond to impact velocities on $\Gamma^{-}$ (top membrane). In (d) the gray dots correspond to impact velocities in panel (a) with $\mu_k=0$ for comparison. The red circles and squares indicate PD bifurcations, while the black arrows point to grazing on $\Sigma$ and the red arrows indicate transitions via grazing on $\Gamma$. These occur at $d=d_{\Sigma}$ and $d=d_{\Gamma^{+}}$, respectively, discussed in \ref{['sec:ExamplesOfPeriodicMotions']}.

Theorems & Definitions (11)

• Remark 1
• Remark 2
• Remark 3
• Remark 4
• Remark 5
• Remark 6
• Remark 7
• Remark 8
• Remark 9
• Remark 10