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In a nutshell: Cummins

The Impulse Response Function and Ship Motions, W.E. Cummins, Schiffstechnik (1962)

This paper derives the equation of motion that is most commonly used to model the dynamics of a body in water, including ships and wave energy converters (WECs). The part of the mathematical argument essential to WECs will be summarised here.

The paper starts off with an interesting bit of history. Up until the 1950s, `classical' sea-keeping research only modelled sinusoidal waves and response. The models had the form of second order differential equations. In the 1950s there was growing interest in investigating response to spectra. Initially, they kept the form of the second order differential equations, and expressed the hydrodynamic added mass and damping as frequency dependent parameters. To run such a time domain model, it would be necessary to choose `the frequency' at which added mass and damping were to be considered. There was growing disquiet about the rigor of this approach, not to mention the fact that the models did not describe experiments very well. As Cummins muses in this paper "The shoe is squeezed on, with no regard for the shape of the foot". He goes on to derive a better fitting model.

He starts deriving the equations of motion by stating that at \(t =0 \) the body is given an impulsive displacement \(\Delta x_j\) in the \(j\)th mode (assuming the usual ship-keeping convention of 6 modes of motion). This impulsive motion has constant velocity: \(\Delta x_j = v_j \Delta t\).

Next, he splits the fluid velocity potential resulting from this impulsive motion into two parts: the velocity at the instant of the impulse, and the decaying response afterwards. During the impulse, the fluid velocity potential is proportional to the velocity of the body. After the impulse, the velocity potential is proportional to the change in body displacement. The velocity potential due to an impulse is thus \(\Theta_{\Delta} = v_j\psi_j + \varphi_j \Delta x_j\), with \(\psi_j\) the velocity potential (normalised by velocity) associated with the impulsive change in wave elevation resulting from the body's impulsive motion, and \(\varphi_j\) the velocity potential (normalised by the change in displacement) caused by the decaying motion of this initial impulsive wave.

Cummins then applies the well understood principal that the response of a linear time-invariant system to an arbitrary input is given by the convolution of this input with the system's impulse response (the response when the input is an impulse). He states that in response to an arbitrary small motion \(x_j (t)\) in the \(j\)th mode, the velocity potential of the resulting fluid flow will be:

\begin{equation*}
    \Theta = \dot{x}\psi_j + \int^{t}_{-\infty} \varphi_j (t- \tau) \dot{x} (\tau) d\tau
    \label{eq:vel pot}
\end{equation*}

Then we are reminded of the usual equations for pressure: \(p = \rho \frac{\partial \Theta}{\partial t}\), and the resulting hydrodynamic force (radiation force, as the only wave is due to body motion): \(-F_{jk} = \int_S p s_k\), with \(s_k\) the usual bunch of unit, position and normal vectors. Substitution of the velocity potential yields: \(-F_{jk} = \ddot{x}_j m_{jk} + \int^{t}_{-\infty} K_{jk} (t-\tau) \dot{x}_j (\tau) d\tau\)

where

\(m_{jk} = \rho \int_S \psi_j s_k\)

\(K_{jk} (\tau) = \rho \int_S \frac{\partial \varphi_j (\tau)}{\partial t} s_k\)

The term that is a constant times acceleration describes the force that results because the body velocity has an impulsive (instantaneous) impact on the water particle velocities and wave elevation. The term that is a convolution with velocity arises because this wave caused by past body velocity dissipates over time, and the resulting fluid motions are felt by the body.

These terms are then placed into the equation of motion for an oscillating body subjected to arbitrary exciting forces \(f_k (t)\) in the \(k\)th mode:

\begin{equation*}
\sum^{6}_{j=1}\left[(m_j \delta_{jk} + m_{jk}) + \int^{t}_{-\infty} K_{jk} (t-\tau) \dot{x}_j (\tau) d\tau + c_{jk}x_j \right] = f_k (t)
\end{equation*}

Where \(m_j =\) inertia in the \(j\)th mode, \(c_{jk}x_j =\) restoring force in the \(k\)th mode due to displacement \(x_j\) in the \(j\)th mode, and \(\delta_{jk} = 1\) if \(j=k\) and \(=0\) if \(j\neq k\). Apart from the Kronecker delta notation used to set the diagonal elements of the mass matrix to zero, this is the formulation that forms the basis of most equations of motion for immersed bodies to this day. We refer to \(m_{jk}\) as infinite frequency added mass, \(m_\infty\), and to the convolution as 'the memory term', because this force depends on past values of velocity. 

Test post for blog disemination of journal paper

Phasor modelling is not frequency domain modelling. 

There, I said it. I will now describe why this is the case in four steps:

1. Background on linear WEC modelling: why phasor models are used to provide coefficients for frequency domain models; how radiation memory is encoded in the frequency domain.
2. Why frequency domain and phasor modelling appear similar: mainly, more subjectivity than desirable required to interpret the maths.
3. Differences between frequency domain and phasor models: like the difference between cave-aged cheddar and vegan cheese; or Belgium beer vs non-alcoholic lager.
4. Why this distinction is important: for making sense of the literature, not breaking the laws of physics, and ensuring we use the right tools for research and development of wave energy converters (WECs).