Matlab code: stability diagram for the classical linear Mathieu equation from Onset and limiting amplitude of yaw instability of a submerged three-tethered buoy. 11 November 2019 3 February 2020

In this paper the dynamics of a submerged axi-symmetric wave energy converter are studied, through mathematical models and wave basin experiments. The device is disk-shaped and taut-moored via three inclined tethers which also act as a power take-off. We focus on parasitic yaw motion, which is excited parametrically due to coupling with heave. Assuming linear hydrodynamics throughout, but considering both linear and nonlinear tether geometry, governing equations are derived in 6 degrees of freedom (d.f.). From the linearized equations, all motions, apart from yaw, are shown to be contributing to the overall power absorption. At higher orders, the yaw governing equation can be recast into a classical Mathieu equation (linear in yaw), or a nonlinear Mathieu equation with cubic damping and stiffness terms. The well-known stability diagram for the classical Mathieu equation allows prediction of onset/occurrence of yaw instability. From the nonlinear Mathieu equation, we develop an approximate analytical solution for the amplitude of the unstable motions. Comparison with regular wave experiments confirms the utility of both models for making relevant predictions. Additionally, irregular wave tests are analysed whereby yaw instability is successfully correlated to the amount of parametric excitation and linear damping. This study demonstrates the importance of considering all modes of motion in design, not just the power-producing ones. Our simplified 1 d.f. yaw model provides fundamental understanding of the presence and severity of the instability. The methodology could be applied to other wave-activated devices.

本文通过数学建模与波浪水池试验，研究了水下轴对称波浪能转换器（submerged axi-symmetric wave energy converter）的动力学特性。该装置呈盘形，通过三根倾斜系缆实现张紧系泊，同时这些系缆亦可作为功率提取系统（Power Take-off, PTO）。本文重点关注寄生偏航运动：该运动因与垂荡运动耦合而受到参数激振。本文全程采用线性水动力学假设，同时考虑线性与非线性系缆几何特性，推导得到6自由度（6 d.f.）的控制方程。基于线性化方程，研究表明除偏航运动外，其余所有运动模式均对总功率捕获有贡献。在高阶近似下，偏航控制方程可转化为经典马蒂厄方程（关于偏航的线性形式），或带有三次阻尼与刚度项的非线性马蒂厄方程。经典马蒂厄方程的稳定性图谱可用于预测偏航失稳的起始与发生情况。基于非线性马蒂厄方程，我们推导得到失稳运动幅值的近似解析解。与规则波试验的对比验证了两种模型开展相关预测的有效性。此外，本文还分析了不规则波试验结果，成功将偏航失稳与参数激励量及线性阻尼建立关联。本研究表明，在波浪能装置设计中需全面考虑所有运动模式，而非仅关注与功率产出相关的运动模式。本文提出的简化1自由度偏航模型，可为偏航失稳的存在性与严重程度提供基础理论认知。该研究方法可推广应用至其他波浪能转换装置。