Shaping Energy Exchange with Gyroscopic Interconnections: a geometric approach

TL;DR

The paper addresses how gyroscopic interconnections shape transient energy exchange in a conservative $2$-DOF system by exploiting a geometric view on invariant-torus projections and resonant Lissajous curves. It derives exact resonance conditions, classifies resonant pairs into low- and high-order categories, and introduces the resonant inscribed radius $r_{res}$ as a robust internal performance metric with a provable degeneracy criterion. A certified computation method for $r_{res}$ is developed, including a slow-mode reduction yielding a uniform error bound, enabling design of interconnections that either absorb energy or contain it while respecting passivity. The results enable an explicit interconnection-shaping framework that selects the frequency ratio via $n$ to control exchange depth and responsiveness, with practical implications for vibration mitigation and energy routing in engineered systems.

Abstract

Gyroscopic interconnections enable redistribution of energy among degrees of freedom while preserving passivity and total energy, and they play a central role in controlled Lagrangian methods and IDA-PBC. Yet their quantitative effect on transient energy exchange and subsystem performance is not well characterised. We study a conservative mechanical system with constant skew-symmetric velocity coupling. Its dynamics are integrable and evolve on invariant two-tori, whose projections onto subsystem phase planes provide geometric description of energy exchange. When the ratio of normal-mode frequencies is rational, these projections become closed resonant Lissajous curves, enabling structured analysis of subsystem trajectories. To quantify subsystem behaviour, we introduce the inscribed-radius metric: the radius of the largest origin-centred circle contained in a projected trajectory. This gives a lower bound on attainable subsystem energy and acts as an internal performance measure. We derive resonance conditions and develop an efficient method to compute or certify the inscribed radius without time-domain simulation. Our results show that low-order resonances can strongly restrict energy depletion through phase-locking, whereas high-order resonances recover conservative bounds. These insights lead to an explicit interconnection-shaping design framework for both energy absorption and containment control strategies, while taking responsiveness into account.

Figures (6)

• Figure 1: The phase plane $(q(t), \dot{q}(t))$, $t\in[0, \frac{200\pi}{n}]$, of \ref{['eq:model']} for an impulse response ($\dot{q}_0=1$) with (a) $n=2$ and (b) $n=\frac{1}{\sqrt{12}}$. The boundary set $x_\partial$, given by \ref{['eq:boundary']} (full line, red), and maximal radius (dotted line) are plotted as well.
• Figure 2: The Pareto frontier of speed vs. minimum energy. Schematic plot of $(r_{\mathrm{res}}(n(\tau, \sigma);1),\,T_{\min}(n(\tau,\sigma)))$ as $(\tau,\sigma)$ varies. Note the low‑order resonances (fast/large) and near‑irrational ratios (slow/small).
• Figure 3: Absorption case (11,9): phase portrait with envelope and inscribed circle.
• Figure 4: Absorption case (11,9): subsystem energy $H_q(t)$ with $T_{\min}\approx\pi/n$ marker.
• Figure 5: Containment case (6,5): phase portrait with envelope and inscribed circle.

Theorems & Definitions (19)

• Proposition 1: Hamiltonian structure and energy conservation
• Definition 1: Resonant pair
• Proposition 2: Resonance condition
• proof
• Definition 2: Low-order resonance pair
• Definition 3: High-order resonance pair
• Proposition 3: Convex envelope on $(q,\dot q)$
• proof
• Remark 1: Ellipticity
• Remark 2: Maximal radius