Scaling law for the slow flow of an unstable mechanical system coupled to a nonlinear energy sink

TL;DR

This work analyzes a mechanical system with one unstable mode coupled to a nonlinear energy sink (NES) and develops a scaling law for its slow flow near a fold point of the critical manifold. By combining complexification-averaging, geometric singular perturbation, and a center-manifold reduction, the authors reduce the slow-flow dynamics to a normal form of a dynamic saddle-node bifurcation, uncovering a nontrivial ε-dependence with scaling exponents 1/3 and 2/3. The zeroth-order mitigation limit is refined into a more accurate prediction ρ^U_ε, including an ε^{2/3} correction and an optimal NES damping μ^*, and the results are validated numerically on an aeroelastic wing model coupled to a single NES. The methodology provides improved predictive power for NES design, particularly in regimes where the perturbation parameter is small but finite, and demonstrates strong agreement with full-order simulations. Overall, the paper advances analytical tools for NES-based vibration mitigation by linking slow-fast dynamics, center-manifold theory, and dynamic bifurcation scaling to practical aeroelastic applications.

Abstract

In this paper one first shows that the slow flow of a mechanical system with one unstable mode coupled to a Nonlinear Energy Sink (NES) can be reduced, in the neighborhood of a fold point of its critical manifold, to a normal form of the dynamic saddle-node bifurcation. This allows us to then obtain a scaling law for the slow flow dynamics and to improve the accuracy of the theoretical prediction of the mitigation limit of the NES previously obtained as part of a zeroth-order approximation. For that purpose, the governing equations of the coupled system are first simplified using a reduced-order model for the primary structure by keeping only its unstable modal coordinates. The slow flow is then derived by means of the complexification-averaging method and, by the presence of a small perturbation parameter related to the mass ratio between the NES and the primary structure, it appears as a fast-slow system. The center manifold theorem is finally used to obtain the reduced form of the slow flow which is solved analytically leading to the scaling law. The latter reveals a nontrivial dependence with respect to the small perturbation parameter of the slow flow dynamics near the fold point, involving the fractional exponents 1/3 and 2/3. Finally, a new theoretical prediction of the mitigation limit is deduced from the scaling law. In the end, the proposed methodology is exemplified and validated numerically using an aeroelastic aircraft wing model coupled to one NES.

Figures (8)

• Figure 1: Typical example of the critical manifold in the $(s,r)$-plane given by \ref{['eq:RealCMa']} for $\mu=0.25$ and $\alpha=5$.
• Figure 2: Illustration of the normal form of the dynamic saddle-node bifurcation. Result of the numerical integration of \ref{['eq:slowRealSlow3']} with initial condition $(q_a(0)=-1,v(0)=-0.5)$ (red dashed line) compared to the analytical scaling law $q_a^\star(y)$ given by \ref{['eq:q1vAi1']} (blue line, the dashed parts are the horizontal asymptotes of $q_a^\star(y)$ corresponding to the zeros Airy function) for $\epsilon=0.01$. The first zero and the first singularity of $q_a^\star(v)$ (orange and green lines respectively) and the corresponding critical manifold $\mathcal{M}_{0}$ (attracting part $\mathcal{M}_{0,\text{a}}$ in black and repelling part $\mathcal{M}_{0,\text{r}}$ in gray) are also shown.
• Figure 3: Sketch of the two DOFs aircraft wing coupled to one NES. $z$ and $\tilde{\varphi}$ are respectively the heave and the angle of attack (pitch) of the wing and $y$ is the displacement of the NES. $A$ is the aerodynamic center, $B$ the elastic axis, $G$ the center of gravity of the aircraft wing. $e$ is the location aerodynamic center $A$ measured from $B$ (positive ahead of $B$). $K_z$ and $K_\varphi$ are the linear heave and pitch stiffnesses respectively whereas $K_z^\text{NL}$ and $K_\varphi^\text{NL}$ are the cubic heave and pitch stiffnesses. $C_z$ and $C_\varphi$ are the heave and pitch damping coefficients. $U$ is the constant and uniform flow speed around the wing and $d$ is the offset attachment of the NES to the wing, also measured from $B$.
• Figure 4: Direct numerical integration of the wing-NES system \ref{['eq:airfiolnodim']} (in blue) depicting the several response regimes described in \ref{['sec:respregimes']} with, in addition to \ref{['eq:airfoilpar']} and \ref{['eq:NESpar']}, $\epsilon=0.01$, $\zeta_h=0.3$, $\xi_h=4$, $\xi_x=4$ and $\xi_\varphi=8$. Four values of the reduced speed are used, namely: (a) $\Theta=0.94$, (b) $\Theta=0.945$, (c) $\Theta=0.97$ and (d) $\Theta=0.98$ corresponding respectively to complete suppression, mitigation through a periodic regime, mitigation through a SMR and finally no mitigation of the harmful limit cycle oscillations of the primary structure. For comparison purposes, for each situation, a direct numerical integration of the wing system without NES is also shown (in green).
• Figure 5: Illustration of the proposed analytical procedure. (a) The critical manifold $\mathcal{M}_0$\ref{['eq:CM00']} (black line) superimposed to the direct numerical integration of the slow flow \ref{['eq:slowRealSlow2']} (red line) and its scaling law near the left fold point given by \ref{['eq:srAi1']} (blue line, the dashed parts are the horizontal asymptotes of $s^\star(r)$ due to the zeros of the Airy function). The values of $r^\text{LF}=H(s^\text{LF})$ (see \ref{['eq:rNrm']}) and $r^\infty=r^\text{LF}+K^\infty\epsilon^{2/3}$ (see \ref{['eq:rinfty']}) are depicted by magenta and green horizontal lines respectively. (b) Times series of $s(\tau)$ (blue line) and $r(\tau)$ (green line) obtained from the numerical integration of the slow flow \ref{['eq:slowRealSlow2']}. The times series are focused on the first relaxation cycle. Parameters \ref{['eq:airfoilpar']} and \ref{['eq:NESpar']} are used with $\zeta_h=0.25$, $\epsilon=0.005$ and $\Theta=0.95$.

Theorems & Definitions (1)

• Definition 3.1