⚡ electrical engineering

On the motion of a catamaran-type wave energy converter under periodic external forcing

This paper rigorously establishes the existence, uniqueness, and global asymptotic stability of a $T$ -periodic solution for a catamaran-type wave energy converter modeled with equivalent linear radiation damping and a nonlinear PTO damper, using Mawhin's coincidence degree theory and confirming the results through numerical simulations.

Original authors: Xinhang Li, Zhijie Cai

Published 2026-11-02

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CC BY 4.0

Original authors: Xinhang Li, Zhijie Cai

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The ocean is a vast reservoir of untapped power, holding more energy per square meter than the sun or the wind. Yet, capturing this energy is notoriously difficult because waves are chaotic, shifting in size and speed from moment to moment. Engineers have spent decades building machines to harvest this power, with one promising design resembling a catamaran boat. This device consists of two main parts floating on the water: a large outer shell that rides the waves and a smaller inner weight that moves up and down inside it. As the waves push the outer shell, the inner weight lags behind or surges ahead, creating a relative motion between the two. This movement is the key; it drives a system of springs and dampers that converts the mechanical energy of the waves into electricity.

The challenge for scientists has been predicting exactly how this machine behaves. While simple models work well for gentle, regular waves, real-world conditions often involve complex forces and friction that do not follow simple rules. Specifically, the internal mechanism that generates power often acts like a special kind of brake, where the resistance changes depending on how fast the parts are moving. For a long time, it was unclear whether a machine with such complex, non-linear braking would eventually settle into a steady, repeating rhythm when pushed by regular waves, or if it would behave erratically. Without a mathematical guarantee that the machine would find a stable rhythm, engineers could not be certain of its long-term reliability or efficiency.

In a recent study, researchers at Fudan University tackled this uncertainty by building a rigorous mathematical model of this catamaran-style energy converter. They focused on a specific scenario where the machine is subjected to regular, repeating wave forces, mimicking a steady ocean state. The team replaced the complex, history-dependent forces of water with a simplified but accurate model of resistance, and they described the power-generating brake using a rule where the drag force grows based on the speed of the relative motion, but not in a straight line. By applying advanced mathematical tools designed to handle such complex, non-linear systems, the researchers proved that the machine does indeed settle into a single, stable, repeating motion. They demonstrated that no matter how the machine starts—whether it is sitting still, moving wildly, or starting from a random position—it will eventually calm down and synchronize with the waves, following one unique path.

The study went further to show that this stable motion is not just a possibility but a certainty for this specific design. The researchers proved that there is only one such repeating pattern for a given set of wave conditions. If two identical machines started with different initial movements, their paths would eventually merge, and they would move in perfect unison. This finding is crucial because it means the device's long-term behavior is predictable and robust. The team did not stop at theory; they ran detailed computer simulations using realistic physical parameters, including a float mass of over 206,000 kilograms and an internal oscillator weighing more than 400,000 kilograms. These simulations confirmed the mathematical proof, showing that the device's motion stabilizes within a few wave cycles, regardless of whether it started from rest or from a chaotic state.

The results provide a solid foundation for designing these energy harvesters. By confirming that the system possesses a unique, stable rhythm, the study removes a major theoretical doubt about the viability of using non-linear damping in wave energy converters. The researchers found that the device naturally filters out its initial chaos and locks onto the frequency of the incoming waves, much like a pendulum eventually swinging in time with a steady push. This stability is essential for the practical deployment of such technology, as it ensures that the energy output will be consistent and reliable once the machine is in the water. The work bridges the gap between abstract mathematics and engineering reality, offering a clear assurance that these complex machines will behave as intended under the rhythmic forcing of the ocean.

Technical Summary: On the Motion of a Catamaran-Type Wave Energy Converter Under Periodic External Forcing

Problem Statement
Wave energy converters (WECs), particularly the catamaran-type point-absorption wave energy converter (PAWEC), are promising renewable energy devices. However, their dynamic behavior is governed by high-dimensional, nonlinear, non-autonomous vibration equations under periodic wave excitation. While linearized models are common, they often fail to capture critical nonlinearities such as nonlinear Power Take-Off (PTO) damping, viscous effects, and hydrodynamic coupling. A significant theoretical gap exists regarding the rigorous proof of whether such nonlinear systems possess a stable, steady-state periodic response. Without such proof, the assumption that the device settles into a predictable periodic motion under complex nonlinear conditions remains unverified.

Methodology
The authors model the catamaran-type PAWEC as a two-body system consisting of a surface floater and an internal oscillator connected by a PTO system. The dynamics are described by a system of ordinary differential equations (ODEs) incorporating:

Radiation Damping: Represented by an equivalent constant-coefficient linear damping model ( $C_a$ ), a standard engineering approximation.
PTO System: Modeled as a linear spring ( $k_s$ ) in parallel with a nonlinear damper. The damping force is proportional to the relative velocity raised to a power $\alpha$ ( $0 < \alpha \leq 1$ ), specifically $c|v_1 - v_2|^\alpha(v_1 - v_2)$ .
Excitation: Subjected to $T$ -periodic wave excitation forces ( $f_e \cos(\omega t)$ ).

The mathematical analysis proceeds in three stages:

Existence: The authors employ Mawhin's coincidence degree theory. They construct a homotopy between the original non-autonomous system and an autonomous system. By defining a Lyapunov function and establishing uniform bounds on the solutions (using energy estimates and embedding theorems in $L^p$ spaces), they verify the conditions required to prove the existence of at least one $T$ -periodic solution.
Global Attractivity: Using Lyapunov stability theory, the authors analyze the difference between an arbitrary solution and a specific periodic solution. They demonstrate that the derivative of the Lyapunov function along the error trajectory is non-positive and vanishes only when the error is zero. Applying the invariance principle for periodic systems, they prove that any solution with arbitrary initial conditions converges asymptotically to the periodic solution.
Uniqueness: By contradiction, the authors show that if two distinct periodic solutions existed, they would violate the global attractivity property established in the previous step, thereby proving the periodic solution is unique.

Key Results

Theoretical Proofs: The paper rigorously establishes that the nonlinear dynamical system of the catamaran-type PAWEC admits a unique $T$ -periodic solution. Furthermore, it proves that this solution is a global attractor, meaning the system's motion converges to this periodic state regardless of initial conditions.
Numerical Validation: Numerical simulations were conducted using MATLAB's ode45 solver with realistic physical parameters (e.g., floater mass $\approx 206$ $\approx 206$ tons, oscillator mass $\approx 402$ $\approx 402$ tons, wave period $\approx 8.06$ $\approx 8.06$ s).
- Time-domain simulations confirmed that the system settles into a periodic motion with a period matching the wave excitation.
- Phase plane projections showed trajectories converging to a single limit cycle.
- Simulations with vastly different initial conditions (including large displacements and velocities) demonstrated that the difference between trajectories rapidly decays to zero, confirming global attractivity.

Significance and Claims
The paper claims to provide a solid mathematical foundation for the dynamical design and stability analysis of nonlinear WEC systems. Specifically:

It addresses the theoretical gap regarding the existence of steady-state periodic responses in catamaran-type WECs under nonlinear PTO conditions, moving beyond linear or weakly nonlinear assumptions.
It validates the engineering practice of using equivalent linear damping models for radiation while retaining nonlinear PTO dynamics, proving that such simplified models still yield well-posed, stable periodic behaviors.
The results confirm that the device's energy harvesting state is predictable and robust, as the system naturally converges to a unique periodic orbit from any starting state.

The authors note that while the existence and attractivity are proven, future work will focus on analytically estimating the convergence rate of transient dynamics and using semi-analytical techniques to optimize mean energy absorption rates with respect to system parameters.

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