Vessel theory: Manoeuvring load

Manoeuvring loads are forces and moments dependent on the low frequency added mass of the vessel and the low frequency part of its translational and angular velocity. They can play an important role in the damping of slow drift motion.

You can choose whether manoeuvring loads are included, on the calculation page of the vessel data form. To include them select manoeuvring load in the included effects applied to the vessel. No further data are needed, since the manoeuvring load is based on the added mass data that are already provided for added mass purposes on the stiffness, added mass, damping page of the vessel types form.

The manoeuvring force $(\fx,\fy,\fz)$ and moment $(\mx,\my,\mz)$ arise from the classical theory of rigid body motion in an inviscid fluid (see, for example, Lamb §124), and are given by \begin{align} \fx &= \sum_{j=1}^3 A_{1j} \, \epsilon_{jkl} \, v_l \, \omega_k + \sum_{j=1}^6 (A_{2j}u_6u_j - A_{3j}u_5u_j) \label{Fx}\\ \fy &= \sum_{j=1}^3 A_{2j} \, \epsilon_{jkl} \, v_l \, \omega_k + \sum_{j=1}^6 (A_{3j}u_4u_j - A_{1j}u_6u_j) \label{Fy}\\ \fz &= \sum_{j=1}^3 A_{3j} \, \epsilon_{jkl} \, v_l \, \omega_k + \sum_{j=1}^6 (A_{1j}u_5u_j - A_{2j}u_4u_j) \\ \mx &= \sum_{j=1}^3 A_{4j} \, \epsilon_{jkl} \, v_l \, \omega_k + \sum_{j=1}^6 (A_{2j}u_3u_j - A_{3j}u_2u_j + A_{5j}u_6u_j - A_{6j}u_5u_j) \\ \my &= \sum_{j=1}^3 A_{5j} \, \epsilon_{jkl} \, v_l \, \omega_k + \sum_{j=1}^6 (A_{3j}u_1u_j - A_{1j}u_3u_j + A_{6j}u_4u_j - A_{4j}u_6u_j) \\ \mz &= \sum_{j=1}^3 A_{6j} \, \epsilon_{jkl} \, v_l \, \omega_k + \sum_{j=1}^6 (A_{1j}u_2u_j - A_{2j}u_1u_j + A_{4j}u_5u_j - A_{5j}u_4u_j)\quad \text{(but see Dangers of double-counting below)} \label{Mz} \end{align} where

$A_{ij}$ are the vessel type data added mass matrix coefficients. If frequency-dependent added mass and damping is used then these coefficients will be taken from the added mass matrix associated with the longest period (lowest frequency) given. If the vessel is in a multibody group then the matrix used will be that vessel's diagonal block in the multibody group's longest-period added mass matrix. If the vessel type length differs from the vessel length, then these data will be automatically Froude-scaled to the vessel length.

$u_j$ are the low frequency velocity (j=1…3) and low frequency angular velocity (j=4…6) of the vessel, relative to current, at the added mass and damping reference origin.

We use $v$ and $\omega$ to express a cross product using index notation in the first term of all these load component equations. Vectors $v = u_j$ for j=1...3 and $\omega = u_j$ for j=4...6 are the low frequency velocity and angular velocity parts of $u$ as above. The symbol $\epsilon_{jkl}$ is the Levi-Civita symbol, but for our purposes it is only important to note that $\epsilon_{jkl} \, v_l \, \omega_k = (\omega \times v)_j$.

The axes used when the manoeuvring load is applied are dependent on calculation mode. The primary low frequency heading frame or diffraction frame will be used.

Manoeuvring loads are a low-speed effect (viscous effects dominate at higher speeds), and at wave frequency some of these terms are already included in the second order wave load specified by the wave drift QTFs. OrcaFlex therefore calculates the manoeuvring load using the low frequency components only of the velocity (and angular velocity) of the vessel relative to the current, and correspondingly using the added mass coefficients $(A_{ij})$ for the longest period given in the data (or the single added mass matrix if constant added mass and damping is in use).

The reference theory for manoeuvring load makes use of vessel velocity relative to current, and OrcaFlex follows suit, as above. The time derivative of vessel velocity relative to current is used in the original calculations. OrcaFlex can apply a time-varying current, and so there could be a contribution to this time derivative from the acceleration of the current. However, our understanding of Lamb is that the source mathematics does not apply if the fluid is accelerating. OrcaFlex therefore does not make a contribution to manoeuvring load from any acceleration of current that might be available from our environment data.

Dangers of double-counting

Expression (\ref{Mz}) above for $\mz$ contains the following Munk moment terms that are quadratic in the surge and sway velocity components, $u_1$ and $u_2$ \begin{equation} A_{11}u_2u_1 + A_{12}u_2^2 - A_{21}u_1^2 - A_{22}u_1u_2 \label{MM} \end{equation} If the current load effect is included then these Munk moment contributions (\ref{MM}) are assumed to be already included in the load specified by the OCIMF current load data and, to avoid double-counting these terms, if manoeuvring load and current load are both included then these Munk moment terms are omitted from the manoeuvring load.