# Reference paper for subroutine catenary

Dear all,

I am diving in the source code of Moordyn. And when initializing the mooring line shape, Moordyn will call subroutine ‘catenary’ to use the analytical solution to calculate the initial shape.

But I am not familiar with the adopted formulas. Would anyone suggest some relevant papers which derived the analytical solution for mooring line?

Below is the description for subroutine catenary:

``````      SUBROUTINE Catenary ( XF_In, ZF_In, L_In  , EA_In, &
W_In , CB_In, Tol_In, N    , &
s_In , X_In , Z_In , ErrStat, ErrMsg    )

! This subroutine is copied from FAST v7 with minor modifications

! This routine solves the analytical, static equilibrium equations
! for a catenary (or taut) mooring line with seabed interaction.
! Stretching of the line is accounted for, but bending stiffness
! is not.  Given the mooring line properties and the fairlead
! position relative to the anchor, this routine finds the line
! configuration and tensions.  Since the analytical solution
! involves two nonlinear equations (XF and  ZF) in two unknowns
! (HF and VF), a Newton-Raphson iteration scheme is implemented in
! order to solve for the solution.  The values of HF and VF that
! are passed into this routine are used as the initial guess in
! the iteration.  The Newton-Raphson iteration is only accurate in
! double precision, so all of the input/output arguments are
! converteds to/from double precision from/to default precision.
``````

Regards,
Ran

Dear @Ran.Tu,

See the section titled, “Mooring System Modelling” in my 2009 Wind Energy journal publication: https://onlinelibrary.wiley.com/doi/10.1002/we.347. Please note, however, that there was a copy-paste error in Eq. (20b), which should be:

zF(HF,VF) = HF/w*[ SQRT( 1 + (VF/HF)^2 ) - 1 ] + VF^2/(2EAw)

Best regards,

Thank you, Dr. Jason. I will study this paper.

Regards,
Ran