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

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.

💡 Research Summary

This paper investigates the slow‑flow dynamics of a mechanical structure possessing a single unstable mode when it is coupled to a nonlinear energy sink (NES). The authors first demonstrate that, in the vicinity of a fold point on the critical manifold, the slow flow can be reduced to the normal form of a dynamic saddle‑node bifurcation. By exploiting this reduction, they derive a scaling law that captures the non‑trivial dependence of the slow dynamics on the small perturbation parameter ϵ, which represents the mass ratio between the NES and the primary structure. The scaling law involves fractional exponents 1/3 and 2/3, indicating that the trajectory near the fold point evolves with a ϵ^{1/3} speed for the NES displacement and with an ϵ^{2/3} amplitude for the unstable modal coordinate.

The governing equations start from a general nonlinear second‑order system (mass, damping, stiffness matrices) with cubic nonlinearities, to which a light NES (mass m_h, damping c_h, cubic stiffness k_{NLh}) is attached via influence vectors A and B. Introducing ϵ ≪ 1, the authors rescale the NES parameters and the state variables, thereby obtaining a fast‑slow system after neglecting higher‑order terms. A modal reduction retains only the unstable mode, and a coordinate transformation (v = x + ϵB h, w = h − A x) yields a compact set of equations (5).

Complexification‑averaging is applied to separate the fast oscillatory component from the slow amplitude modulation, leading to a fast‑slow decomposition. The linear part of the primary structure is diagonalized using bi‑orthogonal right and left eigenvectors because the system matrix is non‑symmetric. After this transformation, the dynamics of the stable modes decay, leaving a reduced system for the unstable modal coordinate q₁ and the NES displacement w.

The crucial step is the application of the center‑manifold theorem to the reduced fast‑slow system. This yields a two‑dimensional differential system that, after appropriate time scaling, matches the normal form of a saddle‑node bifurcation. Solving this normal form analytically provides the scaling law: w ∼ ϵ^{1/3} and |q₁| ∼ ϵ^{2/3}. These exponents differ from the integer powers that would arise from a regular perturbation expansion, highlighting the failure of classical methods near fold points.

Using the scaling law, the authors correct the previously obtained zeroth‑order approximation of the NES mitigation limit. The new expression shows that the critical cubic stiffness of the NES scales as ϵ^{2/3} times a combination of damping, frequency, and modal coupling coefficients. Consequently, as the NES mass becomes smaller, the required stiffness for effective mitigation grows sharply, a fact that was underestimated in earlier analyses.

The theoretical developments are validated on an aeroelastic aircraft wing model that exhibits flutter‑type instability. The wing is modeled with a reduced set of modal equations, coupled to a single NES. Numerical integration over a range of ϵ values (10^{-2}–10^{-3}) confirms that the predicted fold‑point transition times and amplitude reductions align closely with the scaling law. Compared with the zeroth‑order prediction, the new law reduces the error in the estimated mitigation limit by up to 20 %, especially for the smallest mass ratios.

In conclusion, the paper provides a rigorous analytical framework for understanding and designing NES‑based vibration mitigation when the primary system is near a Hopf bifurcation. By revealing the fractional‑exponent scaling near the fold point, it offers a more accurate tool for selecting NES parameters (mass, damping, nonlinear stiffness) and suggests that future designs should explicitly incorporate this scaling to achieve optimal energy transfer. Potential extensions include multiple NES devices, systems with several unstable modes, and more complex nonlinear damping mechanisms.