Features and References

Most of MoorDyn’s theory is described in the following publications. This page gives a very high-level overview, highlights specific theory aspects that may be important to users, and lists the papers where more detail can be found.

Features

Version 1

MoorDyn is based on a lumped-mass discretization of a mooring line’s dynamics, and adds point-mass and rigid-body objects to enable simulation of a wide variety of mooring and cabling arrangements. Hydrodynamics are included using a version of the Morison equation.

Version 2

MoorDyn v2 contains all the features of v1 with the following additional features:

  • Simulation of 6 degree of freedom objects

  • Non-linear tension

  • Wave kinematics

  • Bending stiffness

  • Bathymetry

  • Seabed friction

The main difference between MoorDyn-C and MoorDyn-F is that MoorDyn-C uses quaternions to describe the orientation of 6DOF objects, while F uses traditional Euler angles to handle 6DOF object rotations.

Orientation of 6 DOF objects:

Euler angles – MoorDyn-F

In the following figure the 6DOF object orientation angles convention is depicted:

Angles criteria schematic view

The roll and yaw angles, \(\phi\) and \(\psi\), follow the right hand criteria, while the pitch angle, \(\theta\), follows the left hand criteria. This way the classic rotation matrices can be considered,

\[\begin{split}\begin{alignat}{1} R_x(\phi) &= \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \phi & -\sin \phi \\[3pt] 0 & \sin \phi & \cos \phi \\[3pt] \end{bmatrix} \\[6pt] R_y(\theta) &= \begin{bmatrix} \cos \theta & 0 & \sin \theta \\[3pt] 0 & 1 & 0 \\[3pt] -\sin \theta & 0 & \cos \theta \\ \end{bmatrix} \\[6pt] R_z(\psi) &= \begin{bmatrix} \cos \psi & -\sin \psi & 0 \\[3pt] \sin \psi & \cos \psi & 0 \\[3pt] 0 & 0 & 1 \\ \end{bmatrix} \end{alignat}\end{split}\]

Quaternions – MoorDyn-C

The latest MoorDyn-C internally uses quaternions to describe the location and orientation of 6 DOF objects. Externally MoorDyn-C behaves the same as MoorDyn-F, using Euler angles for both inputs and outputs. Quaternions are a common alternative to Euler angles for describing orientations of 3D objects. Further description of quaternions can be found in PR #90 in the MoorDyn repository, put together by Alex Kinley of Kelson Marine: https://github.com/FloatingArrayDesign/MoorDyn/pull/90#issue-1777700494

References

The theory behind MoorDyn is available in a collection of papers, listed below by which version they were implemented in.

Version 1

The v1 lumped-mass formulation of MoorDyn as well as its validation against experiments:

Coupling with WEC-Sim or any Simulink code for wave energy converter simulation:

Version 2

Version 2 builds upon the capabilities of Version 1. The theory behind the new features is described in the following references.

Early work on seabed friction and independent fairlead points:

Preliminary comparison of seabed friction formulations:

Overview of MoorDyn v2:

Implementation of bending stiffness modeling for power cables:

The Fortran version of MoorDyn is available as a module inside of OpenFAST:

Dynamics of 6DOF objects follows a similar approach to Hydrodyn:

Quaternion references:

  1. Fossen, Thor I. Handbook of marine craft hydrodynamics and motion control. Page 25 John Wiley & Sons, 2011.

  2. https://en.wikipedia.org/wiki/Gimbal_lock

  3. https://www.ashwinnarayan.com/post/how-to-integrate-quaternions/

  4. https://en.wikipedia.org/wiki/Quaternion#Hamilton_product

MoorDyn-C Packages used: