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NLD P1

Nonlinear Dynamical Systems

Areas · Nonlinear dynamics · Mechanical vibrations · Applied mechanics

Methods · Analytical and numerical continuation of nonlinear responses · Bifurcation and stability analysis · Lyapunov exponents and basins of attraction · Experimental instrumentation with MEMS sensors and microcontrollers · Fractional-order damping formulations

The question

A motor mounted on a flexible structure does not simply spin faster as more power is supplied to it. Approaching a resonance, it can stall there — the additional energy going into the structure’s oscillation instead of into shaft speed — and then jump abruptly past it. The excitation is not independent of the response it produces. Once that coupling is admitted, most of the intuition inherited from linear vibration theory stops applying. What replaces it?

Overview

This project comprises analytical, computational and experimental studies of nonlinear systems in mechanics and engineering, with emphasis on oscillations, stability, bifurcations, chaos, synchronization, vibro-impact, nonideal excitation, fractional-order effects and energy harvesting. It is the group’s foundational line and the one from which the other pillars derive their test problems.

The organizing commitment is that the three modes of inquiry are held together rather than pursued separately. A predicted bifurcation is verified numerically and then sought experimentally in an instrumented structure; when the experiment disagrees, the model is revised rather than the discrepancy attributed to noise. This is slower than any of the three alone, and it is the reason the group’s models can be used as references elsewhere — including as ground truth for the learned models in the scientific machine learning line.

Two threads deserve specific mention because they recur. Nonideal excitation treats the energy source as having limited power, so the forcing is a dynamical variable coupled to the response; the Sommerfeld effect — resonant capture and the associated jump — is the canonical signature of this coupling. Fractional-order damping introduces memory, so the dissipative term depends on the history of the motion rather than on its instantaneous rate, which changes stability boundaries in ways an equivalent viscous model does not reproduce.

Instrumentation is treated as a research activity in its own right and not as support work. Building measurement chains from MEMS accelerometers and microcontrollers makes experimental nonlinear dynamics accessible at a cost that permits student projects to be genuinely experimental — a line that has produced a substantial part of the group’s published output.

Approach and methods

Analytical treatment establishes the structure of the response: equilibria, their stability, and the parameter values at which that stability changes. Numerical continuation then traces response branches through those transitions, including the unstable branches that experiments cannot access but that determine where jumps occur.

Characterization of complex regimes uses Lyapunov exponents to distinguish chaotic from quasi-periodic motion, and basins of attraction to establish how sensitive the final state is to initial conditions — which, in systems with coexisting attractors, is frequently the more consequential result.

Experimental work uses instrumented structures — cantilever beams, portal frames, adjacent shear buildings for vibro-impact studies, magneto-piezo-elastic harvesters — with acquisition built on MEMS sensors, Arduino and Raspberry Pi platforms. Non-stationary responses are then analyzed with the group’s time-frequency methods, which is where this pillar meets P2.

Team

Collaborations

Current directions

Mechanical clock escapements are being studied as a controlled instance of a self-sustained nonlinear oscillator with impact, examined in parallel through numerical and symbolic formulations. Work also continues on nonlinear flutter, on energy harvesting under time-varying excitation frequency, and on how fractional damping alters the stability boundaries of systems driven by limited power sources.

For prospective students

This project is P1 — Nonlinear Dynamics & Mechanical Vibrations, the group’s foundational line. Useful background: mechanical vibrations or classical mechanics, differential equations, and Python or MATLAB. Interest in building and measuring physical systems is valued as much as analytical facility.

A student here typically takes one system through the full sequence: derive its equations, locate its bifurcations numerically, then instrument a physical version and find out where the model and the measurement part company. That last step is where the research actually happens, and it is the reason the instrumentation work is treated as first-class rather than preparatory.

Last updated 2026-07-30