Onset and limiting amplitude of yaw instability of a submerged three-tethered buoy

Abstract

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 (DOF). 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 DOF yaw model provides fundamental understanding of the presence and severity of the instability. The methodology could be applied to other wave-activated devices.

1. Introduction

Dynamic instabilities in floating wave energy converters (WECs) have recently received much attention in the literature. Such systems can exhibit large oscillations in modes of motion not directly excited by waves. These modes are parametrically excited via a time-varying restoring force/moment coefficient, which arises due to nonlinear coupling with typically the power-producing mode(s) of motion. Under the large oscillations of the power-producing modes (to maximize energy absorption), these couplings, even though higher-order nonlinear effects, are non-negligible. For this reason, wave-activated WECs which are free to move in multiple degrees of freedom (DOF) are particularly prone to such dynamic instabilities. The classical Mathieu equation can be used for interpretation of these instabilities under regular wave conditions, where the parametric forcing is harmonic. From the solution it follows that instability is usually triggered when the parametric excitation is close to twice the natural frequency of the unstable mode.

Most of the studies so far have investigated Mathieu-type pitch/roll instabilities in surface-piercing heaving devices, either single-body buoys reacting against the seabed or self-reacting multi-body systems extracting energy from the relative heave motion. The parametric excitation originates from nonlinear hydrostatic effects, and can be incorporated into numerical models by evaluating the hydrostatic restoring forces/moment using the instantaneous body position. Tarrant and Meskell [1,2] examine an axi-symmetric two-body heaving device called Wavebob which exhibits undesired roll and pitch oscillations in experiments. Numerical simulations of their model with the appropriate nonlinearities included satisfactorily predict the onset of the instabilities and to some degree also the magnitude of the unstable motions. A similar recent study by Kurniawan et al. [3] also investigates the amplified pitch and roll motion in a two-body heaving buoy. In both of these studies, the instability is associated with a reduction in absorbed power. Giorgi & Ringwood [4] study the dynamics of CorPower’s floating heaving buoy in regular waves with a highly efficient numerical model. They identify different branches of roll instability (with some similarities to the stability diagrams of the Mathieu equation), and assess the impact of the instability on power. In another study [5], the same authors also investigate roll instability in an oscillating water column spar buoy, inspired by the experimental work of [6]. Crucially, they are able to consider irregular wave conditions, unlike most other studies which focus on regular waves, thanks to the efficiency of their model for axi-symmetric devices. They find largest roll motions in sea states where the peak period is half the natural roll period, and also investigate the effect of damping on the instability. Palm et al. [7] present CFD simulations of parametric resonance in a generic heaving device (a surface piercing cylindrical buoy with either a flat or a hemispherical hull bottom).

Motion instabilities have also been studied in other WECs which do not primarily extract power from heave motion. Babarit et al. [8] report parametric roll and yaw motion in a pitching device called SEAREV. Undesired roll motion is also mentioned in work by Stansby et al. [9] on the M4 device, which is a three-body floating line absorber. Outside the wave energy field, dynamic instabilities have been reported extensively for floating offshore structures moving as single bodies (see for example [10–12]).

In this paper we study yaw instability in a three-tethered submerged WEC. We are motivated by the need to understand extreme motions in all modes of the WEC, which is crucial for WEC design. Since unstable parasitic motions can become severe, their influence on the whole system design must be considered. Furthermore, understanding of the instability mechanism is necessary for mitigation strategies aimed at reducing the unstable motions. Such understanding can be built up from reduced complexity models, the development of which we have pursued here. We derive a series of models for the yaw motion dynamics with varying degrees of geometric nonlinearity due to the mooring/power take-off (PTO). The yaw governing equation can be written as a (linear or nonlinear) Mathieu equation, which allows us to predict the occurrence, as well as the severity, of the unstable yaw motions under regular wave conditions. Model predictions are in remarkably good agreement with measurements from wave basin experiments. Unstable yaw motion in irregular waves is also studied experimentally and theoretically. The work presented in this paper builds upon previous investigations of parametric excitation in submerged wave-activated tethered WECs. In [13] we studied sway motion instability in a single-tethered submerged WEC.

The device considered is a simplified representation of the CETO wave energy converter developed by Carnegie Clean Energy. It consists of a large cylindrical buoy, which is shallowly submerged (i.e. submergence $\ll$ radius). Three inclined tethers act as an integrated taut-mooring and PTO system (see figure 1). Under wave action, the device can move in 6 DOF. (translational motions surge, sway and heave, and rotational motions roll, pitch and yaw). Due to the axi-symmetry of the device (including moorings), power can be effectively extracted from the waves irrespective of the angle of incidence. As will be shown later, when the system is linearized, all modes of motion, apart from yaw, contribute to power production. As the device is submerged, there is no hydrostatic stiffness, and any restoring force must be provided mechanically via the PTO. In operational conditions, the mechanical stiffness coefficient is chosen to maximize power capture. Three-tethered point-absorber WECs have also been studied by [14,15].

Figure 1.

Three-tethered CETO wave energy converter. (a) Diagram of a simplified prototype device. (b) Top view schematic diagram showing orientation of the tethers. Note that the incident wave propagation direction is along the positive x axis. (c) Side view schematic diagram showing the buoy, tethers, attachment and anchor points. Solid lines represent the system in static equilibrium conditions. Dashed lines represents a perturbed system. (Online version in colour.)

2. Model derivation

Let us define a fixed coordinate system centred at the initial position of the buoy’s centre of gravity X₀, with horizontal axes x and y and vertical axis z, with the incident waves assumed to propagate along the x-axis. Let X = [X Y Z]^T be the translational motions of the buoy’s centre of gravity along the fixed coordinate system axes - i.e. surge, sway and heave. The superscript ^T denotes the vector transpose. Let $\mathit{\theta }={[{\theta }_x\hspace{0.17em}{\theta }_y\hspace{0.17em}{\theta }_z]}^{\text{T}}$ denote the buoy’s rotational motions about a translating coordinate system, which is centred at the buoy’s instantaneous centre of gravity X and whose axes are parallel to the fixed coordinate system. These extrinsic rotations are referred to as roll, pitch and yaw, respectively. Aspects of the derivation that follows are analogous to the work of [13,15,16]. Figure 1 shows a diagram of the buoy under consideration. The centre of buoyancy is assumed to coincide with the centre of gravity. In the derivation below, we assume a symmetric arrangement of equal length tethers, with one tether pointing in the down-wave direction (along the positive x axis), and two tethers pointing obliquely up-wave. This arrangement offers practical simplicity and ensures directional insensitivity. The buoy and tether orientations are decoupled (i.e. universal joints are assumed at the attachment points on the buoy’s hull and the anchor points on the sea bed). The tether attachment points are denoted by position vectors A_i for i = 1, …, 3 (with A_i,0 denoting the initial/rest positions). The anchor points are denoted by position vectors S_i for i = 1, …, 3. Their definitions are

$$\begin{array}{r}\begin{array}{rlrl}\hspace{0.17em}& {\mathit{A}}_i=\mathit{X}+\mathit{R}{\mathit{A}}_{i,0},& \hspace{0.17em}& \hspace{0.17em}\\ \hspace{0.17em}& {\mathit{A}}_{1,0}={[r{\text{s}}_{\theta },0,-r{\text{c}}_{\theta }]}^{\text{T}},& \hspace{0.17em}& {\mathit{S}}_1={[L{\text{s}}_{\alpha }+r{\text{s}}_{\theta },0,-L{\text{c}}_{\alpha }-r{\text{c}}_{\theta }]}^{\text{T}},\\ \hspace{0.17em}& {\mathit{A}}_{2,0}={[-\frac{1}{2}r{\text{s}}_{\theta },-\frac{\sqrt{3}}{2}r{\text{s}}_{\theta },-r{\text{c}}_{\theta }]}^{\text{T}},& \hspace{0.17em}& {\mathit{S}}_2={[-\frac{1}{2}(L{\text{s}}_{\alpha }+r{\text{s}}_{\theta }),-\frac{\sqrt{3}}{2}(L{\text{s}}_{\alpha }+r{\text{s}}_{\theta }),-L{\text{c}}_{\alpha }-r{\text{c}}_{\theta }]}^{\text{T}},\\ \hspace{0.17em}& {\mathit{A}}_{3,0}={[-\frac{1}{2}r{\text{s}}_{\theta },\phantom{+}\frac{\sqrt{3}}{2}r{\text{s}}_{\theta },-r{\text{c}}_{\theta }]}^{\text{T}},& \hspace{0.17em}& {\mathit{S}}_3={[-\frac{1}{2}(L{\text{s}}_{\alpha }+r{\text{s}}_{\theta }),\phantom{+}\frac{\sqrt{3}}{2}(L{\text{s}}_{\alpha }+r{\text{s}}_{\theta }),-L{\text{c}}_{\alpha }-r{\text{c}}_{\theta }]}^{\text{T}},\end{array}\end{array}$$ 2.1

where s_α, c_α, s_θ and c_θ denote sinα, cosα, sinθ and cosθ, respectively. R is the standard 3 × 3 rotation matrix (a function of θ_x, θ_y and θ_z; and given in the electronic supplementary material), r and θ are respectively the length and the angle (measured from the vertical) of a line between the buoy’s centre of gravity and any of the attachment points, and similarly L and α are respectively the length and the angle (measured from the vertical) of the initial/static tethers (when X = 0, θ = 0). The parameters r, θ, L and α define the geometry of the mooring set-up. We note here that a nomenclature table with all variable symbols and definitions used in the paper is provided in the electronic supplementary material.

The instantaneous tether vectors, from the attachment point A_i to the anchor point S_i, are given by T_i = S_i − A_i. PTO extension ΔL_i and PTO velocity $\mathrm{\Delta }{\dot{L}}_i$, which represent the change in tether length and the rate of change of the tether length respectively, are thus defined as

$$\mathrm{\Delta }{L}_i=|{\mathit{T}}_i|-L,\mathrm{\Delta }{\dot{L}}_i=\frac{\mathrm{\partial }}{\mathrm{\partial }t}|{\mathit{T}}_i|,$$ 2.2

where $|\hspace{0.17em}.\hspace{0.17em}|$ represents the magnitude of the vector under consideration (i.e. the Euclidean norm). When allowing for motions in all 6 DOF, the full expressions for PTO extension and PTO velocity are lengthy nonlinear functions of the buoy displacement and velocity variables. The first-order (in buoy motion variables) approximations for PTO extension are given below in equation (2.3), whereby yaw is seen as the only mode not contributing to tether extension at first order. In subsequent analysis pursued in §2b,c, further terms will be retained. We also note in passing that by setting α = θ = 0, we recover the expressions for a single-tethered device (see [13]).

$$\begin{array}{rl}\hspace{0.17em}& \mathrm{\Delta }{L}_1\approx Z{\text{c}}_{\alpha }-X{\text{s}}_{\alpha }+{\theta }_{y}r{\text{s}}_{\alpha -\theta },\mathrm{\Delta }{L}_2\approx Z{\text{c}}_{\alpha }+\frac{1}{2}(X+\sqrt{3}Y){\text{s}}_{\alpha }-\frac{1}{2}({\theta }_y-\sqrt{3}{\theta }_x)r{\text{s}}_{\alpha -\theta },\\ \hspace{0.17em}& \mathrm{\Delta }{L}_3\approx Z{\text{c}}_{\alpha }+\frac{1}{2}(X-\sqrt{3}Y){\text{s}}_{\alpha }-\frac{1}{2}({\theta }_y+\sqrt{3}{\theta }_x)r{\text{s}}_{\alpha -\theta }.\end{array}$$ 2.3

The governing equations, in 6 DOF, are derived below. The forces considered to be acting on the WEC are buoyancy minus self-weight, hydrodynamic forces and tether forces.

The net buoyancy force F_B acts vertically upwards. For a fully submerged buoy considered here, the expression is independent of the instantaneous body position. When the centre of gravity and the centre of buoyancy coincide, as is assumed here, the buoyancy force moments M_B vanish completely. The expressions read

$${\mathit{F}}_B={[0,0,\rho Vg-mg]}^{\text{T}},{\mathit{M}}_B={[0,0,0]}^{\text{T}},$$ 2.4

where ρ is the fluid density, V and m are the buoy volume and mass, and g is the magnitude of the gravitational acceleration.

The tether, or power take-off, forces F_PTOi act along each tether (for i = 1, …, 3). The PTO implementation considered here is composed of a pre-tension force, a linear spring restoring force and a linear damping force. We note that the pre-tension forces are in vertical equilibrium with the net buoyancy force in still/initial conditions. The PTO forces give rise to moments M_PTOi. The expressions for both are given as

$${\mathit{F}}_{PTOi}=(\frac{C}{3{\text{c}}_{\alpha }}+K\mathrm{\Delta }{L}_i+B\mathrm{\Delta }{\dot{L}}_i)\frac{{\mathit{T}}_i}{|{\mathit{T}}_i|},{\mathit{M}}_{PTOi}=(\mathit{R}{\mathit{A}}_{i,0})\times {\mathit{F}}_{PTOi},$$ 2.5

where C denotes the magnitude of the net buoyancy force C = |F_B| = ρV g − m g, K is the linear spring coefficient, B is the linear damping coefficient and × denotes a vector cross product. The PTO settings, K and B values, are assumed identical for all three tethers. In the PTO force expression, the terms in the brackets represent the tether tension magnitude, which is multiplied by a unit vector of the appropriate orientation to yield the horizontal and vertical force components.

Assuming linear hydrodynamics, the radiation and diffraction problems are decoupled, and the governing equations are given by

$$\begin{array}{r}\mathit{M}\left(\begin{array}{c}\ddot{\mathit{X}}\\ \ddot{\mathit{\theta }}\end{array}\right)=\left(\begin{array}{c}{\mathit{F}}_B\\ {\mathit{M}}_B\end{array}\right)+\left(\begin{array}{c}{\mathit{F}}_{PTO}\\ {\mathit{M}}_{PTO}\end{array}\right)+\left(\begin{array}{c}{\mathit{F}}_{\text{exc}}\\ {\mathit{M}}_{\text{exc}}\end{array}\right)+\left(\begin{array}{c}{\mathit{F}}_{\text{rad}}\\ {\mathit{M}}_{\text{rad}}\end{array}\right),\end{array}$$ 2.6

where $\ddot{\mathit{X}}$ and $\ddot{\mathit{\theta }}$ are the buoy’s translational and rotational accelerations, and M is the 6 × 6 mass and moments of inertia matrix. We note that the elements of M are not time-invariant due to the translating (but not rotating) coordinate system used for definition of the buoy’s rotational motions. However, in the analysis which follows, we have effectively assumed the matrix to be constant with non-zero entries only along the main diagonal of M. This simplification can be justified due to the rather flat geometry of the buoy studied in this work, and also due to the coordinate system being centred at the buoy’s centre of gravity. In equation (2.6), the contributions from all three tethers are expressed as ${\mathit{F}}_{PTO}=\sum _{i=1}^3{\mathit{F}}_{PTOi}$ and ${\mathit{M}}_{PTO}=\sum _{i=1}^3{\mathit{M}}_{PTOi}$. F_exc and M_exc are the hydrodynamic excitation forces and moments, which are due to the incident and diffracted waves. F_rad and M_rad are the radiation forces and moments, which are due to the hydrodynamic pressure resulting from the buoy’s motion. As only planar incident waves in the x-direction are considered, and the buoy is assumed cylindrical (while neglecting hydrodynamic contributions of the tethers), F_exc(2) = 0 and M_exc(1) = M_exc(3) = M_rad(3) = 0.

(a). Linearized model

The governing equations can be linearized, by retaining only first-order terms (in buoy motion variables) from a multi-variable Taylor expansion of the power take-off forces and moments. Analysis of the linear system is important; through comparison with experimental data, it allows us to see which modes of motion behave linearly and which need to be addressed with more complex models. Here we focus on the yaw motion. However, the full 6 d.f. linearized system is extensively studied in appendix A, including derivation of natural (undamped) angular frequencies ω_n = 2πf_n in all modes.

The linearized dynamic equation of yaw motion reads

$$I_{zz}{\ddot{\theta }}_z+(\frac{C{r}^2{\text{s}}_{\theta }^2}{L{\text{c}}_{\alpha }}+\frac{Cr{\text{s}}_{\alpha }{\text{s}}_{\theta }}{{\text{c}}_{\alpha }}){\theta }_z=0,$$ 2.7

where I_zz is the buoy’s moment of inertia about its vertical axis. The yaw natural frequency ω_n6 is therefore given by

$${\omega }_{n6}=\sqrt{\frac{\frac{C{r}^2{\sin }^2\theta }{L\cos \alpha }+\frac{Cr\sin \alpha \sin \theta }{\cos \alpha }}{I_{zz}}},$$ 2.8

which is independent of the PTO stiffness coefficient K.

The linear system analysis presented in appendix A reveals that all modes of motion, apart from yaw,

• —couple to the PTO and thus contribute to the overall power production. This is different to a single-moored device, where heave is the only power-producing mode at first order (see [13]).
• —are damped linearly through radiation damping as well as via the PTO, whereas yaw is linearly undamped. There is no interaction between a perfect fluid and the yaw motion of this axi-symmetric buoy, if the tethers’ contributions are neglected (i.e. the linear hydrodynamic coefficients of excitation moment, added mass and radiation damping are all zero such that M_exc(3) = 0, a₆₆ = 0 and b₆₆ = 0).
• —have restoring force/moment dependent on the PTO stiffness coefficient K. However, the restoring moment for yaw motion is simply a function of the buoyancy/pre-tension and the geometric parameters of the mooring system. The yaw natural frequency is thus not tuneable via the PTO settings. We note in passing that for the range of experimental values of K, the pitch-dominant natural frequency f_n15− is effectively also independent of the PTO settings (full details given in appendix A).

Laboratory testing of a model-scale buoy with two different mooring/PTO arrangements was performed. The buoy was cylindrical with rounded edges. Details of the model are given in table 1. The two mooring arrangements consisted of either inner or outer attachment and anchor points (while maintaining the same tether inclination α), resulting in two different values for both r and θ. We refer to these two configurations as θ_low and θ_high, respectively. Note that due to different fixings of the tethers to the buoy, the static tether length L also changes very slightly across the two mooring/PTO configurations. More details on the experimental campaign are provided in §3.

Table 1.

Device parameters used in the experimental campaign. Full scale values can be obtained via 1 : 20 Froude scaling. Note that the two values for θ, r and L correspond to the two mooring/PTO configurations θ_low and θ_high investigated.

parameter	value	parameter	value
water depth at model (m)	1.5	buoy moment of inertia Ixx = Iyy (kg m2)	25
submergence depth (m)	0.1	buoy moment of inertia Izz (kg m2)	48
buoy thickness (m)	0.25	angle θ:θlow and θhigh (°)	57	77
buoy diameter (m)	1.25	static tether length L (m)	1.45	1.48
buoy mass m (kg)	248	distance r (m)	0.27	0.54
PTO pre-tension coeff. C (N)	560	natural frequency fn6 (Hz)	0.26	0.45
tether angle α (°)	40	limit natural frequency fn15− (Hz)	0.22	0.25

The measured (normalized) motion spectra from irregular wave runs for a buoy with the two different mooring arrangements (specified by θ_high and θ_low) are shown in figure 2. Note that, here and elsewhere in the paper, we use PSD to denote the power spectral density (i.e. the variance density spectrum) of the variable under consideration. The computed natural frequencies (from equation (2.8), (2) and (3)) are also displayed in the figure, with the hydrodynamic coefficients for a submerged truncated vertical cylinder having been calculated according to the analytical solution of [17–19]. The plots suggest that the computed values for ω_n6 and ω_n15− are reasonably accurate. From equation (2.8) it follows that smaller θ angles result in lower ω_n6, and this trend is clearly seen in figure 2. We note that for a given buoy geometry, it is not always possible to align the heave and surge-pitch natural frequencies with the incident wave frequencies (by choosing appropriate value of the mechanical stiffness coefficient K).

Figure 2.

Normalized wave and response spectra from irregular wave runs for mooring configurations θ_high (left plot) and θ_low (right plot). The model-scale incident wave conditions correspond to significant wave height H_s = 0.1 m and peak period T_p = 2.24 s. The calculated natural frequencies f_n = ω_n/2π are shown by dashed vertical lines.

In the spectral plots we note the presence of yaw motion. Although not discernible from these normalized plots, we note that the yaw rotations could become quite substantial in certain tests. More details on irregular wave tests, including magnitudes of the yaw motions, are presented in §3b. We recall that due to our axi-symmetric geometry, there is no yaw excitation moment, and as such yaw cannot be excited by the fluid alone. In the next section we therefore extend our model in order to try to explain the experimentally observed yaw oscillations. We note in passing that for the measurements shown, parts of the surge and pitch responses (around f_n15−) do not coincide with the incident wave frequencies. This nonlinearity is not investigated in this work, however.

(b). Second-order model: linear Mathieu equation

Retaining terms of up to second order in the buoy motion variables, an extended set of approximate governing equations can be derived. We note that, at second order, all the six governing equations are now coupled. When allowing for α ≠ θ, the expressions are rather lengthy, and as such are not given in full below; only the relevant terms are explicitly expressed. We have adopted the following notation. Time-varying stiffness and time-varying damping coefficients are denoted by TVS and TVD, respectively. Nonlinear restoring and nonlinear damping terms are collectively denoted by LHS, while excitation-like terms are denoted by RHS.

Here, we focus on the yaw equation, while the 6 DOF second-order system is discussed in appendix B. The second-order yaw equation, which will be shown to be equivalent to the classical linear Mathieu equation, is given in equation (2.9). The first two terms follow from the linearized equation (2.7); with the second term representing a linear spring with a constant stiffness coefficient. The third term represents a linear spring with a time-varying stiffness coefficient, due to the coupling of yaw with heave and heave velocity. On the right-hand side, we have collected all second-order terms independent of yaw. These terms can be thought of as a driving/excitation moment arising from the mooring/PTO system. We note that when the buoy motion remains confined to the x − z plane (i.e. Y = 0 and θ_x = 0), these excitation terms vanish.

$$I_{zz}{\ddot{\theta }}_z+\beta \frac{C}{{\text{c}}_{\alpha }}{\theta }_z+TV{S}_6^{(2)}(Z,\dot{Z}){\theta }_z=RH{S}_6^{(2)},$$ 2.9

where

$$\begin{array}{rl}\beta & =(\frac{r^2{\text{s}}_{\theta }^2}{L}+r{\text{s}}_{\alpha }{\text{s}}_{\theta }),TV{S}_6^{(2)}=\beta ((3K{\text{c}}_{\alpha }-\frac{C}{L})Z+3B{\text{c}}_{\alpha }\dot{Z}),\\ RH{S}_6^{(2)}& =RH{S}_6^{(2)}(X{\theta }_x,Y{\theta }_y,{\theta }_x{\theta }_y,X\dot{Y},\dot{X}Y,X{\dot{\theta }}_x,\dot{X}{\theta }_x,Y{\dot{\theta }}_y,\dot{Y}{\theta }_y,{\theta }_x{\dot{\theta }}_y,{\dot{\theta }}_x{\theta }_y).\end{array}$$

Similarly to our analysis in [13], assuming heave to be harmonic (at the wave frequency ω), the yaw governing equation may be re-written as the classical linear un-damped Mathieu equation:

$${\ddot{\theta }}_z+(\delta +2ϵ\cos (2\tau )){\theta }_z=0,$$ 2.10

where the non-dimensional time τ is given by ωt = 2τ, the non-dimensional spring coefficient is δ = 4 (ω_n6/ω)² and the amplitude of the parametric excitation term is given via $ϵ=(2/{\omega }^2)(A_Z/I_{zz})\beta \sqrt{(3K{\text{c}}_{\alpha }-{(C/L))}^2+{(3B{\text{c}}_{\alpha }\omega )}^2}$, where A_Z is the harmonic heave amplitude. For a linearly damped Mathieu equation, an additional term $2\mu {\dot{\theta }}_z$ would be present on the left-hand side of equation (2.10), with 2μ being the non-dimensional linear damping coefficient (related to dimensional linear damping coefficient D via μ = D/I_zz/ω). As we are interested in stability, we consider only the homogeneous form of the governing equation, excluding excitation from the other modes.

According to Floquet theory, the linear Mathieu equation admits both bounded and unbounded solutions (see e.g. Chapter 3 in [20]). An initial perturbation is predicted to grow exponentially (without any external excitation), if the system is within the unstable region. The calculated stability diagram (also known as Ince-Strutt diagram) is presented in figure 3, with the red and blue curves corresponding to the un-damped and damped Mathieu equation respectively. We see that the first two instability branches are most likely to be relevant, as linear damping significantly reduces the further instability regions. The first and second instability branches correspond to ω = 2ω_n6 and ω = ω_n6, respectively. The first instability branch exhibits a period-doubling (frequency halving) phenomenon, meaning that the underlying period of the unstable response/motion is twice the period of the parametric excitation. The second instability branch does not possess this period-doubling behaviour. We have indicated this in the diagram. The stability curves have been calculated using the harmonic balance method, and the code is provided in the electronic supplementary material.

Figure 3.

Stability diagram for un-damped (red curve, μ = 0) and damped (blue curve, μ = 0.1) linear Mathieu equation, where $2\mu {\dot{\theta }}_z$ would represent a linear damping term in equation (2.10), with 2μ being a non-dimensional linear damping coefficient. (Online version in colour.)

We note that in practice the small perturbations, necessary to trigger the instability, will always be present (even in well controlled laboratory experiments, and of course in the ocean). Similarly, the right-hand-side excitation-like term $RH{S}_6^{(2)}$ in equation (2.9) could provide a perturbation in yaw, ultimately leading to unstable motions. However, since for stability analysis only the homogeneous equation is considered, this term has been omitted in equation (2.10).

We are interested in whether this model is useful in predicting the occurrence of instability of yaw motion in regular wave experiments. Rather than solving the coupled equations to calculate ϵ, we use the measured heave amplitude from the experiment. Details of the experimental campaign are presented below in §3, together with analysis of the measured data. For the buoy under consideration, the growth of the unstable yaw motion will evidently not be unbounded (as predicted by the linear Mathieu equation). It will eventually saturate at a level dependent on the full nonlinear structure of the governing equations. The linear Mathieu equation (2.10) can thus predict presence of instability, but a higher-order form of the governing equation is needed to describe the severity of the instability. A model for the limiting amplitudes of the unstable motion is proposed in §2c.

For brevity, the second-order governing equations for the other modes of motion are presented in appendix B. A summary of the relevant terms and features is provided below.

• —The second-order heave equation (B 6) contains excitation-like terms arising from other modes of motion. The yaw terms proportional to ${\theta }_z^2$ and ${\theta }_z{\dot{\theta }}_z$ indicate that there is a two-way coupling between the heave and yaw modes. Heave motion can drive yaw via parametric resonance, and the resulting yaw motion can, in return, alter heave. This two-way coupling between heave and yaw will be highlighted in the regular wave tests analysis in §3a.
• —The second-order surge-pitch equations (7) and the second-order sway-roll equations (8) contain terms with time-varying damping and time-varying stiffness coefficients, both of which represent parametric excitation (see [21] or [22] for a system with time-varying dissipation). Similarly to the yaw motion discussed above, these parametric excitation terms arise due to coupling with heave (and with other modes also for sway-roll). Investigation of the Mathieu instability mechanism in surge, sway, roll and pitch will not be undertaken in this paper, however.

(c). Third-order model: Mathieu equation with cubic nonlinearities

In this section the yaw governing equation is extended further by retaining terms up to third order in the buoy motion variables. Compared to the second-order equation (2.9), we note the appearance of nonlinear damping and nonlinear restoring moments (denoted by $LH{S}_6^{(3)}$). Furthermore, additional parametric excitation terms arise at third order, such as the extra time-varying stiffness terms due to X² and Y² for example, as well as the time-varying dissipation terms (denoted by $TV{D}_6^{(3)}{\dot{\theta }}_z$). For brevity, only the homogeneous version is shown.

$$\begin{array}{rl}\hspace{0.17em}& I_{zz}{\ddot{\theta }}_z+\beta \frac{C}{{\text{c}}_{\alpha }}{\theta }_z+TV{S}_6^{(3)}(Z,\dot{Z},X^2,Y^2,Z^2,{\theta }_x^2,{\theta }_y^2,X{\theta }_y,Y{\theta }_x){\theta }_z+\cdots \\ \hspace{0.17em}& TV{D}_6^{(3)}(X^2,Y^2,{\theta }_x^2,{\theta }_y^2,X{\theta }_y,Y{\theta }_x){\dot{\theta }}_z+LH{S}_6^{(3)}({\theta }_z^3,{\theta }_z^2\dot{{\theta }_z})=0.\end{array}$$ 2.11

To proceed analytically, the additional third-order parametric excitation terms in $TV{S}_6^{(3)}$ are neglected, along with the damping terms $TV{D}_6^{(3)}$. The governing equation can then be written as a nonlinear Mathieu equation with a harmonically varying stiffness coefficient and cubic nonlinearities.

$${\ddot{\theta }}_z+(\delta +2ϵ\cos (2\tau )){\theta }_z+c{\theta }_z^3+d{\theta }_z^2\dot{{\theta }_z}=0,$$ 2.12

where

$$\begin{array}{rl}c& =-\frac{2r{\text{s}}_{\theta }(L{\text{s}}_{\alpha }+r{\text{s}}_{\theta })}{3I_{zz}{L}^3{\omega }^2{\text{c}}_{\alpha }}(C{L}^2+3C{r}^2{\text{s}}_{\theta }^2+3CLr{\text{s}}_{\alpha }{\text{s}}_{\theta }-9KL{r}^2{\text{c}}_{\alpha }{\text{s}}_{\theta }^2-9K{L}^{2}r{\text{c}}_{\alpha }{\text{s}}_{\alpha }{\text{s}}_{\theta }),\\ d& =\frac{6B{r}^2{\text{s}}_{\theta }^2{(L{\text{s}}_{\alpha }+r{\text{s}}_{\theta })}^2}{I_{zz}{L}^2\omega },\end{array}$$

and the definitions of δ, ϵ and τ are given underneath equation (2.10). Even though the expressions for the non-dimensional coefficients c and d are lengthy, they are straightforward to evaluate, as they follow from the buoy parameters (I_zz), the mooring geometry (L, r, α, θ) and the PTO settings (C, K, B). We also note their dependence on the wave/heave frequency ω which appears due to the time non-dimensionalization. Physically, c and d can be thought of as coefficients of the nonlinear spring and nonlinear damping forces, respectively.

The presence of these cubic nonlinear terms means that the unstable solutions are bounded (see [23]). A number of researchers have studied nonlinear versions of the Mathieu equation, notably [23,24] consider an undamped Mathieu equation with a cubic restoring term $c{\theta }_z^3$, whereas the linearly damped equivalent is studied by [25,26]. A comprehensive analysis of the Mathieu equation with cubic nonlinearities can be found in [27].

Here we adopt a two-scale method to find an approximate amplitude of the unstable motions in the first instability branch (centred at δ = 1) depicted in the stability diagram (figure 3). The methodology is based on the above references, and as such is only briefly described. We note that the derived solution is for a linearly damped version of equation (2.12) such that an additional term $2\mu {\dot{\theta }}_z$ is present on the left-hand side. As will be discussed later, the value of μ is experimentally derived. First, two timescales are introduced; the fast τ₀ = τ and the slow τ₁ = ϵτ; and θ_z is expanded into a power series using ϵ such that ${\theta }_z={\theta }_z^{(0)}+ϵ{\theta }_z^{(1)}$. The zeroth-order solution becomes ${\theta }_z^{(0)}=P({\tau }_1)\cos \tau +Q({\tau }_1)\sin \tau$, where P and Q represent slowly-varying envelopes of the underlying oscillations. We note in passing the frequency-halving (period-doubling) nature of the unstable response which is characteristic in the first instability branch. Returning to the solution method, the removal of resonant terms in the first-order problem results in two coupled first-order differential equations for the solution envelopes P and Q. Now setting dP/dτ₁ = 0 and dQ/dτ₁ = 0, and simultaneously solving the two resultant algebraic equations, corresponds to solving for a steady-state solution. This procedure leads to a quadratic equation for R² = P² + Q², which is the square of the steady-state amplitude.

$$({\left(\frac{3c}{4}\right)}^2+{\left(\frac{d}{4}\right)}^2)R^4+(\frac{3c(\delta -1)}{2}+d\mu )R^2+({(\delta -1)}^2-{ϵ}^2+4{\mu }^2)=0.$$ 2.13

For a quadratic equation, up to two roots R² are possible. The positive and negative root solutions for R are given by

$$R^{\pm }=2\sqrt{\frac{-3c(\delta -1)-2d\mu \pm \sqrt{12cd\mu (\delta -1)+d^2({ϵ}^2-{(1-\delta )}^2)+9c^2({ϵ}^2-4{\mu }^2)}}{9c^2+d^2}}.$$ 2.14

One can also calculate the phase ϕ of the steady-state solution whereby Pcosτ + Qsinτ = Rcos (τ − ϕ). Here we do so only for the positive root R⁺. We note that the phase is defined with reference to the phase of the parametric excitation 2ϵcos (2τ).

$$\varphi ={\tan }^{-1}(\frac{d}{3c}-\frac{\frac{d}{3c}+\frac{18c^2\mu +3cd(1-\delta )}{ϵ(9c^2+d^2)+d^2(1-\delta )+6cd\mu }}{1-\frac{6c\sqrt{{(d\mu +\frac{3c}{2}(\delta -1))}^2-\frac{1}{4}(9c^2+d^2)({(\delta -1)}^2-{ϵ}^2+4{\mu }^2)}}{ϵ(9c^2+d^2)+d^2(1-\delta )+6cd\mu }}).$$ 2.15

The above steady-state solution given by R⁺ and ϕ is applicable within the first instability branch, which to first order is defined by ϵ² > (δ − 1)² + 4μ². It turns out, however, that to the left of the instability branch, this unstable solution can also arise, provided a large enough initial perturbation. The magnitude of this initial perturbation follows from the negative root solution R⁻. This extended region to the left of the first instability branch is demarcated by

$$\delta =1-\sqrt{{ϵ}^2-4{\mu }^2}\hspace{0.17em}\text{on the right},\delta =1-ϵ\sqrt{1+\frac{9c^2}{d^2}}+6\mu \frac{c}{d}\hspace{0.17em}\text{on the left}.$$ 2.16

Even though the above analysis is tedious, the resultant solutions are trivial to evaluate. They constitute an approximate analytical model for predicting the severity of the instability in the first instability branch. These will be used for comparison with regular wave experiments in §3a. The contour plots in figures 5 and 7 are computed according to (2.14) and (2.15). A considerable extended region given by (2.16) is shown in figure 7. Code for computation of the steady-state solution is provided in the electronic supplementary material.

Figure 5.

Regular wave experiments for mooring arrangement θ_high plotted on the stability diagram, zoomed in on the first instability branch. The coloured contour plots represent the analytical predictions for the steady-state yaw amplitude (on the left) and phase (on the right), both in degrees. Green markers denote stable runs. Red markers denote unstable runs, with stars and circles corresponding to before-instability and during-instability conditions respectively. The red values represent the measured unstable yaw motion amplitudes and phases. The red and magenta lines mark the classical and the extended instability regions respectively.

Figure 7.

Regular wave experiments for mooring arrangement θ_low plotted on the stability diagram, zoomed in on the first instability branch. The coloured contour plots represent the analytical predictions for the steady-state yaw amplitude in degrees. Green markers denote stable runs. Red markers denote unstable runs, with stars and circles corresponding to before-instability and during-instability conditions respectively. The red values represent the measured unstable yaw motion amplitudes. The red and magenta lines mark the classical and the extended instability regions respectively.

It might be possible to extend the above analysis to the second instability branch (centred at δ = 4) in the stability diagram (figure 3), though a second-order ϵ² solution would have to be considered. This is not pursued in this work. Alternatively, one can also solve the governing equation (2.12) numerically via a time-stepping algorithm such as Runge–Kutta or similar. The contour plot in figure 9 for the second instability branch is generated from such converged steady-state numerical solutions. Code for numerical evaluation of the steady-state solution is provided in the electronic supplementary material.

Figure 9.

Regular wave experiments for mooring arrangement θ_high plotted on the stability diagram, zoomed in on the second instability branch. The coloured contour plot represents numerical predictions for the steady-state yaw amplitude in degrees (with navy colour corresponding to 0^°). Green markers denote stable runs. Red markers denote unstable runs. The red lines mark the classical instability region, and have been calculated via the harmonic balance method.

(d). Mathieu equation with random parametric excitation

The Mathieu equation studied above corresponds to a buoy in regular wave conditions. However, ultimately we want to understand the behaviour in seas with broad-banded frequency content. This corresponds to a system with a randomly fluctuating (as opposed to sinusoidally varying) restoring force coefficient. An asymptotic theory for the behaviour of such as system (in terms of the response amplitude and phase in the limit for large times) has been derived by Brouwers [28]. Here we simply present the main result for a system with linear damping only. The governing equation is

$$\ddot{{\theta }_z}+{\beta }_0{\dot{\theta }}_z+(1+F(\mathcal{T})){\theta }_z=0,$$ 2.17

where the non-dimensional time $\mathcal{T}$ is given via $\mathcal{T}={\omega }_{n6}t$, β₀ is the non-dimensional linear damping coefficient (related to dimensional linear damping coefficient D via β₀ = D/I_zz/ω_n6) and $F(\mathcal{T})$ represents the parametric excitation as a stationary Gaussian random process with a zero mean. The variance density spectrum of the parametric excitation is denoted by S. Stable behaviour (i.e. the response amplitude decays to zero at large times from an initial perturbation) occurs when

$$\frac{\pi }{4}{S}_2<{\beta }_0,$$ 2.18

where S₂ denotes the parametric excitation at twice the natural frequency. When this criterion is not satisfied, the solution does not decay away. Furthermore, in the absence of nonlinear damping (equation (2.17)), the solution grows unboundedly. On the other hand, in the presence of additional nonlinear damping (of the form ${\beta }_a{\dot{\theta }}_z|{\dot{\theta }}_z{|}^a$ where a > 0), a finite response arises with the amplitude dependent on the parametric excitation and the nonlinear damping term. This is analogous to the behaviour under regular waves predicted by our models derived in §2b,c, whereby linear damping determines the presence of the instability (i.e. the stability diagram in figure 3), and the nonlinear terms in the governing equations govern the severity of the unstable response (i.e. the limiting amplitude solution in equation (2.14)). Ultimately, we would like to be able to analytically predict the severity of the unstable response under irregular waves, though we do not attempt that in this work.

Brouwers’ theory will be used for comparison with irregular wave experiments in §3b. For the system considered, the randomly time-varying spring coefficient $F(\mathcal{T})$ follows from $TV{S}_6^{(2)}$ in equation (2.9) (or from $TV{S}_6^{(3)}$ in equation (2.11)), once suitably non-dimensionalized. We note that when computing the value of S₂ from the experimental measurements, the calculated variance density spectrum of the parametric excitation needs to be pre-multiplied by the Jacobian to account for the time re-scaling such that $S_2=S(2\pi /\mathcal{T}=2)=f_{n6}S(1/t=2f_{n6})$.

3. Laboratory experiments

Carnegie Clean Energy and the University of Western Australia carried out a model-scale experimental campaign in the Ocean basin at the Coastal, Ocean And Sediment Transport laboratory (COAST lab) at the University of Plymouth, UK. Figure 4 shows the experimental set-up. Parameters of the model CETO device tested are listed in table 1. As discussed previously, two different mooring arrangements, consisting of inner and outer attachment and anchor points, were tested. These are referred to as θ_low and θ_high, respectively.

Figure 4.

Experimental set-up at the COAST lab facility at University of Plymouth, UK. (a) Underwater side-view with the active tethers highlighted. In this photograph, the outer attachment and anchor points are being used. The inner attachments on the bottom of the buoy hull, and the inner pulleys, can also be seen. (b) Top-view with the Qualisys markers. (Online version in colour.)

The free surface elevation was measured by wave gauges at a number of locations around the WEC. A Qualisys motion capturing system was used to track the buoy’s instantaneous position (in 6 DOF). Three model scale power take-offs were used, each consisting of a tether, a winch and a pulley (see fig. 1 in [29]). Each pulley was attached to the bottom of the basin, and the tether fed through it onto the computer-controlled winch, which was positioned above water. The winch torque, as well as the length of the reeled and un-reeled tether (i.e. PTO extension) were continuously monitored and fed into the winch control system. As such, a prescribed tension force function (such as the one given in equation (2.5)), could be achieved. PTO extension and tether tension were also recorded. All tests used uni-directional waves, and both regular and irregular wave conditions were used.

(a). Regular wave tests

Measurements from regular wave experiments are used to check the presence of yaw instability, as well as the severity of the unstable yaw motion, predicted by our models based on the linear and nonlinear Mathieu equation. For the buoy with θ_low, the yaw natural period T_n6 = 1/f_n6 is around 3.8 s (17 s in full scale). The first instability branch (centred at $T=\frac{1}{2}{T}_{n6}$) was thus investigated, with the range of incident wave periods T of 1.5–2.7 s (7–12 s in full scale). For the buoy with θ_high, the calculated yaw natural period is 2.2 s (10 s in full scale). Both the first and the second instability branches were investigated, as the desired wave conditions (needed to probe both branches) were within the wavemaker capabilities. Table 2 lists the regular wave test conditions. Runs with different incident wave frequencies correspond to different δ values in the stability diagram. By varying the incident wave amplitudes, and thus the resultant heave motion, the vertical position ϵ in the stability diagram changes. The intention was to run multiple tests at each frequency, to cover regions above and below the instability boundary.

Table 2.

Regular wave conditions used in the experimental campaign. Full scale values are shown in brackets, assuming 1 : 20 scale. Note that not all H and T combinations were carried out.

mooring	instability	nat. period	wave period	wave height	PTO stiffness	PTO damping
config.	branch	Tn6 (s)	T (s)	H (m)	coeff. K (N m−1)	coeff. B (Ns m−1)
θlow	first	3.78	1.53–2.65	0.03–0.30	3500	1130
		(16.9)	(6.8–11.8)	(0.6–6)	(1.40 ×106)	(2.02 ×106)
θhigh	first	2.24	1.01–1.17	0.05–0.15	2500	5000
		(10.0)	(4.5–5.2)	(1.0–3)	(1.00 × 106)	(8.94 ×106)
θhigh	second	2.24	1.69–2.82	0.03–0.25	4200	3200
		(10.0)	(7.5–12.6)	(0.6–5)	(1.68 × 106)	(5.72 × 106)

First, the regular wave runs were used to obtain an estimate of the linearized damping D in the yaw mode. As no free decay tests were carried out, the value was determined by fitting a linearly damped solution to the decaying yaw motions captured at the end of runs with incident waves (after the waves and heave motion had essentially ceased). These estimates of D ≈ 1 − −3 Nms are approximately two to three times larger than our simple calculation of the damping due to (non-interacting) laminar boundary layers on the top, bottom and side cylinder surfaces. This is to be expected, as our experimentally derived estimate of damping additionally includes fluid damping due to the tethers and other losses such as friction, which have been approximated as linear in our simple estimation method.

Using the buoy parameters and the applied PTO coefficients, together with the experimental heave amplitude A_Z, the values of δ and ϵ are evaluated for each test. These are plotted in the stability diagrams in figures 5, 7 and 9. Each run has been colour-coded by examining the recorded yaw motion. Runs which exhibited noticeable yaw oscillations are displayed in red. On the other hand, runs with minimal recorded yaw motion are shown in green. We have used a mean yaw oscillation amplitude threshold of 1^° to demarcate the stable and unstable runs. If our model worked flawlessly, all red markers would lie above the predicted instability boundaries.

For runs where the yaw motion becomes unstable, we interrogate the measured heave motions before the instability develops as well as during the unstable yaw motions. As discussed in §2b and appendix B, the heave and yaw governing equations are coupled (in fact all 6 d.f. are coupled). From the second-order heave equation (6), it can be seen that yaw back-couples into heave. The excitation-like terms ${\theta }_z^2$ and ${\theta }_z{\dot{\theta }}_z$ in $RH{S}_3^{(2)}$ imply that both the mean heave position as well as the double-frequency (of yaw motion) heave harmonic components can evolve due to the yaw instability. Altering the mean vertical position of the buoy changes the yaw natural frequency and thus the value of δ. For the period-doubling yaw motions (which occur in the first instability branch), the yaw motion affects the heave component at the incident wave frequency, and thus changes the value of ϵ. In summary, due to the heave-yaw coupling, as the yaw instability develops, the (δ, ϵ) position within the instability diagram changes dynamically due to energy transfers between these two modes. For this reason each unstable run is represented with two red markers. Red star markers correspond to before-instability conditions, and their location determines whether instability will happen or not. Red circle markers correspond to during-instability conditions, and their location indicates the limiting unstable yaw behaviour. Naturally, for stable runs, this distinction is not necessary, as the mean and harmonic heave content (and thus δ and ϵ) remain constant.

Figure 5 shows the regular wave tests performed to probe the first instability branch for the mooring arrangement θ_high. The analytical predictions, for the amplitude and the phase of the unstable steady-state yaw motion according to equation (2.14) and (2.15), are shown as coloured contour plots. Note that the contour levels shown are in degrees. It can be seen that conditions further away from the stability boundary correspond to larger amplitude unstable motions. Moreover, the phase of the resulting steady-state yaw oscillations appears to be constant along lines emanating from (δ, ϵ) = (1, 0). The measured yaw motion values are displayed for each unstable run. Our predictions are in excellent agreement with the experimental values.

In figure 5 we note the importance of the two-way coupling between yaw and heave motions indicated by different (δ, ϵ) values before and during the instability. This is also clearly illustrated by the timeseries plots in figure 6. The yaw instability affects both the mean and the harmonic components of the heave motion. In principle it would be possible to solve the 2 d.f. coupled system for heave and yaw. However, linear potential flow theory is known to be of limited validity for describing wave interaction with shallowly submerged bodies (see for example [30–32]). Moreover, change of the mean heave means that the buoy is in a different position within the water column and thus exposed to different wave forcing, which linear theory would not account for. One can, however, confirm that the observed heave offset (when t > 70 T in figure 6) is induced by the yaw motion, by using the yaw measurements and the second-order heave governing equation (6). Accordingly, mean heave is simply given as the ratio of the constant forcing and the stiffness coefficient (see for example [33]). The low-pass frequency content of the heave forcing $RH{S}_3^{(2)}$ is evaluated using the measured yaw timeseries, and the heave stiffness coefficient simply follows from the mooring geometry and the PTO settings. We can confirm that, across all the regular wave experiments, such evaluated heave offsets agree very well with the mean component of the measured heave timeseries, confirming the two-way coupling identified in our model. We hypothesize that the initial heave offset (when 25 T < t < 70T in figure 6) is due to a mean vertical hydrodynamic drift force. Lastly, we note the period-doubling nature of the unstable yaw motions, as predicted by the Floquet theory for the first instability branch.

Figure 6.

Timeseries of measured heave and yaw motions from a regular wave experiment for mooring arrangement θ_high. This test corresponds to initial (δ, ϵ) = (1.03, 0.16) within the first instability branch. The x-axis has been normalized by the incident wave period T, and the heave timeseries has been normalized by the initial harmonic amplitude A_Z. (Online version in colour.)

Figure 7 displays the analytical and experimental limiting yaw amplitudes in the first instability branch for the mooring arrangement θ_low. Note that for each incident wave period, a separate prediction plot is shown. This is due to the fact that the non-dimensional coefficients c and d are frequency dependent, as per equation (2.12). For the mooring configuration θ_high, a single prediction plot in figure 5 was sufficient due to the very narrow frequency range of the incident waves investigated. In figure 7, the individual plots (a) to (f ) are arranged by incident wave period, in ascending order. This corresponds to progressively larger values of coefficients c and d which result in smaller predicted yaw amplitudes. We point out the different structure of the limiting amplitude solution, compared to figure 5. Here the yaw motion amplitudes increase with distance from the right instability boundary. The predicted extended instability region (as per equation (2.16)) is also shown. The measurements from the regular wave tests are shown with individual markers, as explained above. Apart from subplot (a) the predictions are in exceptionally good agreement with the experimental data. In the experiments, the PTO/tether tension F_PTOi was imposed as per equation (2.5), when being within approximately 25–530 N. Outside this range, in order to prevent the tether lines from going slack in cases of very low tension and to protect the equipment in case of very large tension, equation (2.5) was not followed. Inspection of the gathered data reveals that these force limits were being applied in the unstable run from subplot (a). As such the PTO did not act as a purely linear spring and damper, as is assumed in our analytical model. It is perhaps not surprising that the analytical prediction does not work well in this case. The comparison in subplot (b) is particularly encouraging, as the measured yaw motion amplitudes agree very well with the analytical steady-state solution in the extended instability region. Heave and yaw motion timeseries from one of the regular wave runs are shown in figure 8. Change in the mean vertical position of the buoy is observed, as the period-doubling yaw instability develops.

Figure 8.

Timeseries of measured heave and yaw motions from a regular wave experiment for mooring arrangement θ_low. This test corresponds to initial (δ, ϵ) = (0.99, 0.14) within the first instability branch. The x-axis has been normalized by the incident wave period T, and the heave timeseries has been normalized by the initial harmonic amplitude A_Z. (Online version in colour.)

Figure 9 shows the second instability branch for the mooring configuration θ_high. We note that in a large number of tests, the PTO/tether force deviated from equation (2.5) as explained above. As such our model cannot be expected to reliably predict yaw instability in these runs. Nonetheless, the regular wave test data is shown in the left plot using the initial (δ, ϵ) values. Our predictions of occurrence of the yaw instability are reasonably accurate. Note that the orange marker was used to denote a test in which the buoy exhibited yaw motion initially (oscillations of ±5^° during the first 30 wave periods), followed by the buoy remaining stable for the remainder of the test (≈100 wave periods). The maximum yaw motion recorded in these regular wave runs was within ±20^°, which is in fairly good agreement with the limiting amplitude predictions shown on the right. These numerically calculated limiting amplitudes (from equation (2.12)) are for c and d values corresponding to incident wave frequencies of δ ≈ 4. The numerical predictions seem to suggest an extended instability region to the left, analogous to the first instability branch. From figure 10 it can be seen that the yaw motion is of the same frequency as the parametric excitation due to heave, which is characteristic of the second instability branch.

Figure 10.

Timeseries of measured heave and yaw motions from a regular wave experiment for mooring arrangement θ_high. This test corresponds to initial (δ, ϵ) = (4.07, 0.95) within the second instability branch. The x-axis has been normalized by the incident wave period T, and the heave timeseries has been normalized by the initial harmonic amplitude A_Z. (Online version in colour.)

We note that in a number of regular wave tests a true steady state was not reached. Envelopes of the yaw and heave motions (and sometimes other modes too) would slowly oscillate. A similar behaviour was observed by Li et al. [34], who numerically studied pitch instability of a spar platform. The pitch and heave envelope were slowly varying indicating back-and-forth energy transfers between the two modes.

A small number of repeat regular wave tests were carried out. The limiting amplitude and the phase of the steady-state yaw motion were repeatable, but the time at which yaw motion was initiated was not. This is not surprising, since the onset of the unstable motion depends on random background disturbances in the wave basin, even when allowing for sufficient settling time between tests. Additional difficulty arises for cases very close to the instability boundaries, as the resultant yaw motion will be very sensitive to the exact position within the stability diagram. The left-hand-side boundaries in figures 7 and 9 are particularly troublesome due to the very steep gradient of the solution amplitude away from the boundary.

(b). Irregular wave tests

The buoy data from 200 long-crested irregular wave tests was analysed to investigate the effect of broad-banded frequency content on yaw motion. An extensive range of wave conditions was tested; with peak periods from 1.79 to 3.58 s (8–16 s in full scale) and significant wave heights from 0.1 to 0.25 m (2–5 m in full scale). For both mooring configurations, all wave conditions were run with five different PTO settings (different K and B values). Each run was over 500 s long (corresponding to about 36 min in full scale). Notable yaw motion was recorded in some of the irregular wave runs, while in other cases the buoy exhibited no yaw instability. In agreement with Brouwers’ theory, the unstable yaw motion is at the corresponding yaw natural frequency with a slowly-varying envelope.

Figure 11 displays spectral content from five irregular wave runs with different peak periods, for both mooring arrangements. Spectra of yaw motion, input waves and parametric excitation are plotted. Note that in each plot, the wave and the time-varying stiffness spectra have been scaled in order to present the four curves on the same axes. Also, the frequency axis is expressed in terms of the corresponding yaw natural frequency f_n6.

Figure 11.

Irregular wave experiments for both mooring arrangements. Spectral plots of yaw (in degrees²/Hz), parametric excitation (scaled) and input waves (scaled). Wave conditions correspond to H_s = 0.2 m (left plots) and H_s = 0.15 m (right plots) with T_p ranging from 1.79 s (top plots) to 3.58 s (bottom plots). The printed values in each subplot represent (π/4)S₂ evaluated from $TV{S}_6^{(3)}$. (Online version in colour.)

For mooring arrangement θ_low, pronounced yaw motion is observed in the two shortest period sea states, while minimal yaw was recorded in the longer period wave conditions. The occurrence of yaw instability seems correlated to the amount of parametric excitation at twice the natural frequency, as per Brouwers’ instability theory for a system with random parametric forcing. Here, the two curves for parametric excitation are indistinguishable. We recall that the second-order accurate $TV{S}_6^{(2)}$ depends purely on heave motion (as per equation (2.9)), whereas the third-order accurate $TV{S}_6^{(3)}$ also depends on higher-order terms involving surge, sway, roll and pitch (as per equation (2.11)). In this case, the higher-order terms are negligible.

For the mooring configuration θ_high, yaw motion was recorded for the middle peak periods (T_p of 2.24 and 2.68 s, equivalent to 10 and 12 s in full scale). These runs correspond to larger parametric excitation values at twice the natural frequency, again in agreement with the theory. We note the importance of the higher-order terms in the parametric excitation.

Results from all 200 irregular wave tests are collated in figure 12, which shows the measured yaw standard deviation plotted against (π/4) S₂, with S₂ calculated from $TV{S}_6^{(3)}$ as per equation (2.11). The estimated range of values of the non-dimensional linear damping coefficient β₀ have also been added onto the plots, and is shown in grey. The range corresponds to the dimensional values D = 1 − −3 Nms. For both mooring configurations, the data agrees very well with the stability criterion from equation (2.18). Runs to the left of the grey zone (with parametric excitation below the linear damping threshold) are stable, whereas runs to the right of the grey zone are unstable.

Figure 12.

Irregular wave experiments for both mooring arrangements. Yaw standard deviation as a function of (π/4) S₂. Marker shape and colour denote respectively H_s and T_p of the underlying sea state (with each sea state tested with five different PTO settings). The grey areas represent the estimated range of the non-dimensional linear damping coefficient β₀ in the experiments (corresponding to the dimensional values of 1–3 Nms). The highlighted pairs of tests represent two different realizations of the same sea state and identical PTO settings.

The buoy with θ_low is more susceptible to yaw instability across the irregular wave conditions investigated. About 70% of the runs could be classed as unstable (when using a threshold of 1^° for the yaw standard deviation). The mean yaw standard deviation across these unstable runs was 9^°. However, the unstable yaw motions could reach rather large values: the most extreme instantaneous yaw motion recorded was up to 70^°. For the θ_high mooring configuration about 40% of the runs exhibited yaw instability. The mean standard deviation across these unstable runs was about 3^°, while the largest recorded yaw motion was just under 30^°. From figure 11 it follows that increasing the yaw natural frequency could be beneficial in seeking to reduce the occurrence of the instability. The mooring configuration θ_low with f_n6 = 0.26 Hz is seen to be unfavourable as the peak of the time-varying stiffness spectrum can be very close to 2 f_n6, the critical frequency content of the parametric excitation. On the other hand, for the mooring arrangement θ_high with f_n6 = 0.45 Hz (thanks to increased r and θ due to the use of the outer attachment and anchor points), it is the tail of the parametric excitation spectrum which triggers the instability.

A small number of the irregular wave runs were repeated as a different realization (i.e. different random phases of the individual wave components). These shed light on convergence of the results presented in figure 12. Three such ‘repeat’ tests were carried out for mooring arrangement θ_high and two for θ_low. These pairs of tests have been highlighted in figure 12 with bright green marker edges. Even though the paired markers do not overlap exactly, the variation is small. This confirms the robustness of the trends identified in figure 12.

4. Conclusion

Parametrically excited yaw motion in a submerged axi-symmetric three-tethered buoy is investigated theoretically and experimentally. Rather than numerically examining the full 6 DOF mathematical model for the device dynamics, we make significant progress analytically in understanding the device behaviour, by considering various simplified governing equations. Due to the integrated mooring/PTO system, all modes of motion are coupled. The linearized governing equations are used to derive expressions for the natural frequencies of all modes. It is found that the surge and heave natural frequencies can be varied by adjusting the PTO stiffness and damping coefficients. (This fact may be used for maximizing power capture.) However, the yaw natural frequency is independent of the PTO settings, and is set by the buoy parameters and the mooring geometry. The same is essentially true for the pitch-dominant mode.

Depending on the level of geometric nonlinearity retained, the yaw governing equation can be recast into a classical linear Mathieu equation, or a Mathieu equation with cubic nonlinearities. The parametric excitation arises due to coupling with heave at lowest order, with additional product terms at higher order. The well-known stability diagram is calculated for the linear Mathieu equation. Moreover, an approximate solution for the limiting amplitude of the unstable motions (within the first instability branch) is derived by applying a multiple-scales method to the nonlinear Mathieu equation. This solution also reveals that unstable motions can occur inside a region predicted to be stable by the classical linear Mathieu equation stability diagram (but adjacent to the boundary), provided a large enough initial perturbation. Our model predictions are first compared with measurements from laboratory-scale experiments with regular waves. The agreement is very good, both for prediction of onset of the yaw instability as well as the magnitude of the unstable yaw oscillations. This suggests that the nonlinear restoring and damping terms in the yaw dynamics equation, both of which follow from the mooring/PTO, dominate over fluid damping. This is perhaps not surprising for the smooth cylindrical buoy considered here, and suggests that the small-scale measurements could be reliably scaled to full scale. Some inconsistencies remain between the model predictions and experiments, which could be due to deviations of the PTO behaviour in the experiments from the assumed form in the model, and/or due to coupling/interactions of the other modes, which we essentially neglect. Nevertheless, the analytical approach enables fundamental understanding of the instability mechanism.

Analysis of 200 irregular wave tests reveals the presence of the yaw instability in some of the wave conditions. The unstable yaw motion is at the natural frequency, with a slowly varying envelope. According to theory developed by Brouwers [28], under irregular wave conditions, the amount of parametric excitation at twice the natural frequency governs the occurrence of the instability. Accordingly, the yaw measurements are plotted against the time-varying stiffness at this critical frequency. The data align with the theory convincingly, with minimal yaw motion observed when the level of parametric excitation is below the estimated linear damping threshold. In the experimental campaign two different mooring configurations were tested. The mooring arrangement with outer tether attachment points on the buoy hull is found to be less prone to the instability and to exhibit smaller yaw motions. It seems that increasing the yaw natural frequency further, beyond the tail of the parametric excitation spectrum, could prevent the instability, though this requires attendant increases in the geometry or mean tensions. The analysis of instability mechanism and criterion for irregular wave conditions is potentially very valuable, especially given the small number of studies in the literature (with most studies focused on regular wave conditions). Ultimately we would like to be able to predict the magnitude of the unstable yaw motions under irregular wave conditions. It is unclear whether an analytical approach would be viable, though presumably the nonlinear restoring and damping terms play a similar role in limiting the solution, as has been identified for regular wave conditions. This could be investigated numerically, and is left for future work.

Dynamic instabilities appear to be a common feature of wave-activated WECs which are free to move in multiple degrees of freedom. This is perhaps not surprising, as the desired large oscillations in the power-producing modes can act as parametric excitation due to nonlinear coupling. Any weakly damped modes (likely those which do not strongly couple to the PTO) could be prone to dynamic instabilities, as larger levels of linear damping suppress the instability onset. All modes of motion need to be considered in the design process, not just the power-producing ones. Ensuring that the natural frequencies (in fact two times the natural frequency value according to [28]) of the susceptible modes are outside the range of the parametric forcing frequencies appears crucial. This of course requires some understanding of the instability mechanism. In this work we have not investigated the implications of the instability on the efficiency of the device. Such analysis cannot be carried out with our simplified 1 DOF model, which essentially decouples the yaw governing equation from the other modes. We have highlighted the two-way coupling between yaw and heave in the governing equations, and have observed the influence of yaw instability on heave motion in the regular wave tests analysis. However, for this multi-tethered device in which all the five modes of motion (apart from yaw) linearly couple to the PTO, a full 6 DOF model would need to be considered to assess the effect on absorbed power. A comprehensive investigation on this matter is beyond the scope of this paper. Alternatively, one could interrogate the experimental data. However, ascertaining the impact of yaw instability on absorbed power from the irregular wave runs is also not straightforward, due to constantly changing instantaneous conditions. With validated CFD, however, analysis of the potential power impacts could be carried out thanks to repeat simulations with certain degrees of freedom omitted in the body governing equations. This is left for future work.

Appendix A. Linearized system analysis

The linearized yaw governing equation is discussed in §2a. Here we also investigate the other modes of motion. The linearized dynamic equations of motion, incorporating linear hydrodynamic coefficients strictly valid for single-frequency motions only, are given in equation (A 1). The coupled governing equations for sway and roll have been omitted for brevity, as they are equivalent to the homogeneous version of the surge-pitch equations. In the equations, a_jk and b_jk denote the frequency-dependent added mass (or added moment of inertia) and radiation damping coefficients respectively, and I_jj are the buoy’s moments of inertia. As the buoy is axi-symmetric, and the mass matrix M has been assumed diagonal, the only linear hydrodynamic cross-mode coupling elements are a₁₅ = a₅₁ and b₁₅ = b₅₁.

$$\begin{array}{rl}\hspace{0.17em}& \left(\begin{array}{cc}m+a_{11}& a_{15}\\ a_{15}& I_{yy}+a_{55}\end{array}\right)\left(\begin{array}{c}\ddot{X}\\ {\ddot{\theta }}_y\end{array}\right)+\left(\begin{array}{c}{B}_{11}{B}_{15}\\ B_{15}{B}_{55}\end{array}\right)\left(\begin{array}{c}\dot{X}\\ {\dot{\theta }}_y\end{array}\right)+\left(\begin{array}{c}{K}_{11}{K}_{15}\\ K_{15}{K}_{55}\end{array}\right)\left(\begin{array}{c}X\\ {\theta }_y\end{array}\right)=\left(\begin{array}{c}{\mathit{F}}_{\text{exc}}(1)\\ {\mathit{M}}_{\text{exc}}(2)\end{array}\right),\\ \hspace{0.17em}& (m+a_{33})\ddot{Z}+(3B{\text{c}}_{\alpha }^2+b_{33})\dot{Z}+(3K{\text{c}}_{\alpha }^2+\frac{C{\text{s}}_{\alpha }^2}{L{\text{c}}_{\alpha }})Z={\mathit{F}}_{\text{exc}}(3),\\ \hspace{0.17em}& I_{zz}{\ddot{\theta }}_z+(\frac{C{r}^2{\text{s}}_{\theta }^2}{L{\text{c}}_{\alpha }}+\frac{Cr{\text{s}}_{\alpha }{\text{s}}_{\theta }}{{\text{c}}_{\alpha }}){\theta }_z=0,\end{array}$$ A 1

where

$$\begin{array}{rl}\hspace{0.17em}& B_{11}=b_{11}+\frac{3B{\text{s}}_{\alpha }^2}{2},B_{15}=b_{15}+\frac{3Br}{4}({\text{c}}_{2\alpha -\theta }-{\text{c}}_{\theta }),\\ \hspace{0.17em}& B_{55}=b_{55}+\frac{3B{r}^2}{2}{\text{s}}_{\alpha -\theta }^2,K_{11}=\frac{3K}{2}{\text{s}}_{\alpha }^2+\frac{C}{2L}({\text{c}}_{\alpha }+\frac{1}{{\text{c}}_{\alpha }}),\\ \hspace{0.17em}& K_{15}=\frac{3Kr}{4}({\text{c}}_{2\alpha -\theta }-{\text{c}}_{\theta })-\frac{Cr}{4L{\text{c}}_{\alpha }}(3{\text{c}}_{\theta }+{\text{c}}_{2\alpha -\theta }),\\ \hspace{0.17em}& K_{55}=\frac{3K{r}^2}{4}(1-{\text{c}}_{2\alpha -2\theta })+Cr({\text{c}}_{\theta }+\frac{{\text{s}}_{\alpha }{\text{s}}_{\theta }}{2{\text{c}}_{\alpha }})+\frac{C{r}^2}{2L}(\frac{1}{{\text{c}}_{\alpha }}+{\text{c}}_{\alpha -2\theta }).\end{array}$$

The surge and pitch motions are coupled, both hydrodynamically and through the tethers. When α = θ, the above expressions agree with those presented in [15]. In this case, the pitch motion becomes independent of the PTO. Also, the cross-coupling terms in the surge-pitch equations reduce to a₁₅, B₁₅ = b₁₅ and K₁₅ = −C r/L, and as such the surge-pitch coupling would be weak for a small thin buoy in deep water. To first order, the heave and yaw motions are uncoupled from the other modes.

Excluding the excitation and damping terms, natural (undamped) angular frequencies ω_n = 2πf_n in the above modes can be easily computed by assuming the variables are time-harmonic. The yaw natural frequency ω_n6 is by equation (2.8). The heave natural frequency ω_n3 is given by

$${\omega }_{n3}=\sqrt{\frac{3K{\cos }^2\alpha +\frac{C{\sin }^2\alpha }{L\cos \alpha }}{m+a_{33}({\omega }_{n3})}}.$$ A 2

We note that if the heave added mass a₃₃ was constant, and not frequency-dependent, a single value of the natural frequency ω_n3 would follow. However, due to the frequency dependence of this hydrodynamic coefficient, multiple resonances can arise.

For the coupled surge-pitch (and sway-roll) equations, an eigenvalue problem arises, with the eigenvalues ${\omega }_n^2$ representing the square of the natural frequencies and the eigenvectors v_n representing the associated mode shapes. The expressions are given below.

$${\omega }_{n15\pm }=\sqrt{\frac{-b\pm \sqrt{b^2-4ac}}{2a}},{\mathit{v}}_{n15\pm }=\left(\begin{array}{c}a{\omega }_{n15\pm }^2+K_{15}{a}_{15}-K_{55}(m+a_{11})\\ -K_{11}{a}_{15}+K_{15}(m+a_{11})\end{array}\right),$$ A 3

where

$$\begin{array}{rl}\hspace{0.17em}& a=(m+a_{11})(I_{yy}+a_{55})-a_{15}^2,\\ \hspace{0.17em}& b=-(m+a_{11})K_{55}-(I_{yy}+a_{55})K_{11}+2a_{15}{K}_{15},\\ \hspace{0.17em}& c=K_{11}{K}_{55}-K_{15}^2.\end{array}$$

As for heave, more than two roots for ω_n15 are possible.

A number of simplified expressions can be derived from the above.

• —
  When α = θ and the remaining cross-coupling terms a₁₅ and K₁₅ = −C r/L in equation (A 1) are assumed to be small, the expressions for uncoupled surge and pitch natural frequencies ω_n1 and ω_n5 simplify considerably, as shown in equation (A 4). We recall that using this approximation the pitch motion, and the natural frequency, become independent of the PTO stiffness coefficient K.

  $$\begin{array}{r}{\omega }_{n1}\approx \sqrt{\frac{3K{\sin }^2\alpha +\frac{C}{L}(\cos \alpha +\sec \alpha )}{2(m+a_{11}({\omega }_{n1}))}}{\omega }_{n5}\approx \sqrt{\frac{\frac{Cr(L+r)}{2L}(\cos \alpha +\sec \alpha )}{I_{yy}+a_{55}({\omega }_{n5})}}\text{when}\hspace{0.17em}\alpha =\theta .\end{array}$$ A 4
• —
  When the PTO stiffness coefficient K becomes large, the surge-pitch natural frequency ω_n15− and the mode shape v_n15− become independent of K, such that

  $$\begin{array}{rl}\hspace{0.17em}& \underset{K\to \mathrm{\infty }}{lim}{\omega }_{n15-}=\sqrt{\frac{\frac{Cr}{2L}(L{\sin }^2\alpha (\tan \alpha \sin \theta +2\cos \theta )+r{\sin }^2\theta (\cos \alpha +\sec \alpha ))}{(I_{yy}+a_{55}){\sin }^2\alpha +(m+a_{11})r^2{\sin }^2(\alpha -\theta )+2a_{15}r\sin \alpha \sin (\alpha -\theta )}}\\ \hspace{0.17em}& \underset{K\to \mathrm{\infty }}{lim}{\mathit{v}}_{n15-}=\text{unit vector of}\left(\begin{array}{c}r\sin (\alpha -\theta )\\ \sin \alpha \end{array}\right).\end{array}$$ A 5

For the range of experimental model-scale values of K, which are of $O(1000)\hspace{0.17em}{\text{N, m}}^{-1}$, this asymptotic behaviour of equation (A 5) is appropriate. As such the ω_n15− natural frequency is not easily tuneable by changing the PTO coefficients, and the coupled surge-pitch motion v_n15− does not contribute to power production. The asymptotic behaviour can be seen in figure 13, which has been calculated for buoy parameters as per table 1. The asymptotic values, for respectively θ_high and θ_low mooring arrangements, are 0.25 and 0.22 Hz for the pitch-dominant natural frequency f_n15−, [ − 0.45, 0.89]^T and [ − 0.12, 0.99]^T for the mode shape v_n15− (with surge in m and pitch in radians), while in the limit there is no dynamic extension of the tethers.

Figure 13.

Calculated surge-pitch natural frequencies f_n15± (top plots), elements of the unit eigenvector v_n15− (middle plots) and linearized PTO extension of the first tether (bottom plots), as a function of PTO stiffness coefficient K. In each plot, the limiting behaviour as K → ∞ is shown by dashed black lines, and follows from equation (A 5) (and for ΔL₁ from equation (2.3)). (Online version in colour.)

A more visual depiction of the limiting coupled surge-pitch motion modes v_n15− is shown in figure 14, for an assumed surge motion amplitude of 10% of the buoy radius. The buoy with mooring arrangement θ_low would undergo much larger amplitude pitch oscillations, as follows from the limiting mode shapes identified above. Separate animation movie files are provided in the electronic supplementary material.

Figure 14.

Asymptotic coupled surge-pitch motion as per v_n15− as K → ∞, with assumed surge amplitude of 10% of the buoy radius. Separate animation movie files are included in the electronic supplementary material. (Online version in colour.)

We now return to the laboratory experiment with irregular waves presented in figure 2. The surge-pitch unit eigenvectors for the two mooring arrangements θ_high and θ_low, calculated according to equation (A 3) with the applied PTO stiffness coefficients K = 1470 and 3000 N m⁻¹ are v_n15− = [ − 0.59, 0.81]^T and v_n15− = [ − 0.10, 0.99]^T, respectively. As already discussed above, for the mooring arrangement θ_high both surging and pitching motion would occur at ω_n15− (with surge (in m) slightly lower compared to pitch (in radians)), whereas for the mooring arrangement θ_low the motion would be highly pitch dominated. The top plots in figure 15 display the calculated phase difference between surge and pitch motions, from cross-spectral analysis of the experimental measurements. The observed behaviour at ω_n15− matches the theoretical prediction with the two modes being out of phase. The bottom plots in figure 15 display components of a unit vector of the experimental surge and pitch motion amplitudes (calculated from the motion spectra in figure 2). At ω_n15− the agreement with the theoretical unit eigenvectors is very satisfactory. We note that as the natural frequency ω_n15+ is within the incident wave spectrum, the eigen-analysis results would not be reflected in the measurements, and as such are omitted. Lastly, from figure 15, we note in passing that the approximation for the decoupled surge and pitch natural frequencies ω_n1 and ω_n5 from equation (A 4) works better for θ_low as expected.

Figure 15.

Phase difference between surge and pitch motions (top plot) and unit vector of surge (shown in blue) and pitch (shown in green) motion amplitudes (bottom plot) for the runs in figure 2. The natural frequencies f_n = ω_n/2π from equation (A 3) are shown by vertical dashed lines, and the decoupled approximations from equation (A 4) are shown by vertical dotted lines. (Online version in colour.)

Appendix B. Second-order model: other modes of motion

The second-order yaw motion equation is discussed in §2b. We now briefly discuss the second-order governing equations for the other modes of motion.

The second-order heave equation (B 6) contains nonlinear restoring and damping terms (denoted by $LH{S}_3^{(2)}$) as well as excitation-like terms arising from other modes of motion (denoted by $RH{S}_3^{(2)}$). Here it is worthwhile to point out the yaw terms proportional to ${\theta }_z^2$ and ${\theta }_z{\dot{\theta }}_z$, indicating that there is a two-way coupling between the two modes. Heave motion can drive yaw via parametric resonance, and the resulting yaw motion can, in return, alter heave. We note the inconsistency of retaining a frequency domain representation of the radiation force in a nonlinear time domain equation. A similar approach was taken by [35] for example.

$$(m+a_{33})\ddot{Z}+(3B{\text{c}}_{\alpha }^2+b_{33})\dot{Z}+(3K{\text{c}}_{\alpha }^2+\frac{C{\text{s}}_{\alpha }^2}{L{\text{c}}_{\alpha }})Z+LH{S}_3^{(2)}(Z^2,Z\dot{Z})={\mathit{F}}_{\text{exc}}(3)+RH{S}_3^{(2)},$$ B 6

where

$$\begin{array}{r}RH{S}_3^{(2)}=RH{S}_3^{(2)}(X^2,Y^2,{\theta }_x^2,{\theta }_y^2,{\theta }_z^2,X\dot{X},Y\dot{Y},{\theta }_x{\dot{\theta }}_x,{\theta }_y{\dot{\theta }}_y,{\theta }_z{\dot{\theta }}_z,X{\theta }_y,Y{\theta }_x,X{\dot{\theta }}_y,Y{\dot{\theta }}_x,\dot{X}{\theta }_y,\dot{Y}{\theta }_x).\end{array}$$

The second-order surge-pitch equations (B 7) contain nonlinear restoring and dissipation terms (denoted by ${\mathit{L}\mathit{H}\mathit{S}}_{15}^{(2)}$), excitation-like terms (denoted by ${\mathit{R}\mathit{H}\mathit{S}}_{15}^{(2)}$), as well as terms with time-varying damping and time-varying stiffness coefficients (coefficients denoted by ${\mathit{T}\mathit{V}\mathit{D}}_{15}^{(2)}$ and ${\mathit{T}\mathit{V}\mathit{S}}_{15}^{(2)}$). Similarly to yaw motion, these parametric excitation terms arise due to coupling with heave and heave velocity (see [21] or [22] for a system with time-varying dissipation). Investigation of the instability mechanism in these modes is, however, not undertaken in this work.

$$\begin{array}{rl}\hspace{0.17em}& \left(\begin{array}{cc}m+a_{11}& a_{15}\\ a_{15}& I_{yy}+a_{55}\end{array}\right)\left(\begin{array}{c}\ddot{X}\\ {\ddot{\theta }}_y\end{array}\right)+\left(\begin{array}{c}{B}_{11}{B}_{15}\\ B_{15}{B}_{55}\end{array}\right)\left(\begin{array}{c}\dot{X}\\ {\dot{\theta }}_y\end{array}\right)+\left(\begin{array}{c}{K}_{11}{K}_{15}\\ K_{15}{K}_{55}\end{array}\right)\left(\begin{array}{c}X\\ {\theta }_y\end{array}\right)+\cdots \\ \hspace{0.17em}& {\mathit{L}\mathit{H}\mathit{S}}_{15}^{(2)}+{\mathit{T}\mathit{V}\mathit{D}}_{15}^{(2)}(Z)\left(\begin{array}{c}\dot{X}\\ {\dot{\theta }}_y\end{array}\right)+{\mathit{T}\mathit{V}\mathit{S}}_{15}^{(2)}(Z,\dot{Z})\left(\begin{array}{c}X\\ {\theta }_y\end{array}\right)=\left(\begin{array}{c}{\mathit{F}}_{\text{exc}}(1)\\ {\mathit{M}}_{\text{exc}}(2)\end{array}\right)+{\mathit{R}\mathit{H}\mathit{S}}_{15}^{(2)},\end{array}$$ B 7

where

$$\begin{array}{rl}{\mathit{L}\mathit{H}\mathit{S}}_{15}^{(2)}& ={\mathit{L}\mathit{H}\mathit{S}}_{15}^{(2)}(X^2,{\theta }_y^2,X{\theta }_y,X\dot{X},{\theta }_y{\dot{\theta }}_y,\dot{X}{\theta }_y,X{\dot{\theta }}_y),\\ {\mathit{R}\mathit{H}\mathit{S}}_{15}^{(2)}& ={\mathit{R}\mathit{H}\mathit{S}}_{15}^{(2)}(Y^2,{\theta }_x^2,Y{\theta }_x,Y{\theta }_z,{\theta }_x{\theta }_z,Y\dot{Y},{\theta }_x{\dot{\theta }}_x,\dot{Y}{\theta }_x,Y{\dot{\theta }}_x,\dot{Y}{\theta }_z,Y{\dot{\theta }}_z,{\dot{\theta }}_x{\theta }_y,{\theta }_x{\dot{\theta }}_z).\end{array}$$

The second-order sway-roll equations (B 8) contain parametric excitation terms, which arise due to coupling with heave as well as surge and pitch. There is no excitation from waves in sway and roll, as only planar incident waves in the x-direction are considered. However, the second-order mooring/PTO excitation-like terms ${\mathit{R}\mathit{H}\mathit{S}}_{24}^{(2)}$ would be non-zero, if the buoy yaws, suggesting that yaw motion can drive sway and roll.

$$\begin{array}{rl}\hspace{0.17em}& \left(\begin{array}{cc}m+a_{11}& a_{15}\\ a_{15}& I_{yy}+a_{55}\end{array}\right)\left(\begin{array}{c}\ddot{Y}\\ {\ddot{\theta }}_x\end{array}\right)+\left(\begin{array}{cc}{B}_{11}& B_{15}\\ B_{15}& B_{55}\end{array}\right)\left(\begin{array}{c}\dot{Y}\\ {\dot{\theta }}_x\end{array}\right)+\left(\begin{array}{cc}{K}_{11}& K_{15}\\ K_{15}& K_{55}\end{array}\right)\left(\begin{array}{c}Y\\ {\theta }_x\end{array}\right)+\cdots \\ \hspace{0.17em}& +{\mathit{T}\mathit{V}\mathit{D}}_{24}^{(2)}(X,Z,{\theta }_y)\left(\begin{array}{c}\dot{Y}\\ {\dot{\theta }}_x\end{array}\right)+{\mathit{T}\mathit{V}\mathit{S}}_{24}^{(2)}(X,Z,{\theta }_y,\dot{X},\dot{Z},{\dot{\theta }}_y)\left(\begin{array}{c}Y\\ {\theta }_x\end{array}\right)={\mathit{R}\mathit{H}\mathit{S}}_{24}^{(2)},\end{array}$$ B 8

where

$${\mathit{R}\mathit{H}\mathit{S}}_{24}^{(2)}={\mathit{R}\mathit{H}\mathit{S}}_{24}^{(2)}(X{\theta }_z,{\theta }_y{\theta }_z,\dot{X}{\theta }_z,X{\dot{\theta }}_z,{\dot{\theta }}_y{\theta }_z,{\theta }_y{\dot{\theta }}_z).$$

Data accessibility

Electronic supplementary material is provided, which contains the rotation matrix formulation, the harmonic balance code to calculate the stability diagram, the code for computation of the steady-state solution, the animation for the coupled surge-pitch motion modes, as well as a nomenclature table. No additional data is available.

Authors' contributions

J.O. carried out the theoretical model development, performed analysis of the laboratory data and prepared the manuscript, supported by H.W., S.D. and P.H.T., all of whom also revised the manuscript. H.W. carried out critical literature review and provided essential guidance during the theoretical development. A.R., J.O. and H.W. designed and supervised the experimental campaign. All authors gave final approval for publication and agree to be held accountable for the work performed therein.

Funding

This research was supported under the Australian Research Council’s Linkage Project scheme (LP150100598). The laboratory tests performed in the Ocean basin at the Coastal, Ocean And Sediment Transport laboratory (COAST lab) at the University of Plymouth, UK were supported by MaRINET2 funding. H.W. and S.D. acknowledge the kind support of the Lloyds Register Foundation. The Lloyds Register Foundation supports the advancement of engineering-related education, and funds research and development that enhances safety of life at sea, on land and in the air. H.W. also acknowledges financial support from Shell Australia.