The effect of friction on the dynamics of targeted energy transfer by symmetric vibro-impact dampers

TL;DR

The paper tackles nonlinear vibration control by analyzing a symmetric VI-NES with dry friction, addressing the gap in frictional effects on targeted energy transfer. It combines an extended Multiple Scales Method with an impact-map approach to capture slow, near-resonant dynamics and fast impact events, deriving closed-form design metrics for activation, dissipation, and stability in the $2$-IPP regime. Key findings include friction-induced shifts in the backbone curve, a friction-dependent activation threshold, and the coexistence of impacting and purely sliding solutions near the optimal operating range, with stability analyzed via slow-flow and map-based methods. Practically, the framework guides tuning of VI-NES parameters under realistic frictional conditions to optimize energy dissipation and robustness, with potential experimental validation planned.

Abstract

This study investigates the nonlinear dynamics of a symmetric vibro-impact nonlinear energy sink (VI-NES) subjected to dry friction, a crucial factor that remains insufficiently explored in previous research. The combined effect of impact and friction leads to intricate behaviors that require further investigation. To address this, the multiple scales method is extended to incorporate frictional effects and is complemented with a generalized impact map approach. This allows for a systematic exploration of periodic solutions, stability, and bifurcations, revealing critical transitions between impact-dominated and sliding-dominated regimes. The activation thresholds and amplitude levels for different response regimes, including stick-slip dynamics, are identified, offering new insights into friction-induced nonlinearities. The results bridge the gap between theoretical modeling and practical implementation, offering a more accurate predictive framework for VINES behavior. This improves design strategies for enhanced energy dissipation and robustness in real-world applications.

Figures (23)

• Figure 1: Linear oscillator (LO) excited by the base motion of $e(t)$ and undergoing dry friction and impacts from its coupling with a vibro-impact absorber (VI-NES).
• Figure 2: An example of the sought motion during a free response of system (\ref{['eq: eq of motion 2a']})-(\ref{['eq: eq of motion 2b']}) for $\epsilon=0.006$, $\tilde{g}=0.211$, $r=0.76$ and $\mu=0.5$.
• Figure 3: Representation of the nonsmooth sawtooth function $\Pi(z)$, its derivative $M(z)$ and the function $\left(\Pi^2(z)-\frac{\pi^2}{4}\right)M(z)$.
• Figure 4: Projection of the free resonant motion of the LO VI-NES simulated using (\ref{['eq: eq of motion 2a']})-(\ref{['eq: eq of motion 2b']}) (red line) on the SIM given by (\ref{['eq: SIM']}) (black line) for $r =0.76$, $\mu=0.3$ and $\tilde{g}=0.355$. The dotted green line represents the SIM for $\mu=0$ and the region below the blue line corresponds to the validity region of the MSM ansatz from (\ref{['eq: w ansatz']}).
• Figure 5: Position of the equilibria for a level of excitation $G=1.65$ and four values of $\sigma$. The black line represents the SIM from (\ref{['eq: SIM']}). The purple line represent the solutions of (\ref{['eq: ESIM']}). The VI-NES parameters are $r =0.76$, $\mu=0.5$ and $\tilde{g}=0.105$.