Winch theory

Static analysis

The winch control mode in statics may be set to either specified length or specified tension.

Specified length

The unstretched length of wire paid out, $l_0$, is set to the specified length for the static analysis, and the wire tension $t$ and winch drive force $f$ are determined from the equation \begin{equation} \label{statics} t = f = k \epsilon \end{equation} where

$k=$ wire stiffness

$\epsilon=$ wire strain $= (l-l_0)/l_0$

$l=$ total length of the winch wire path.

Specified tension

The wire tension $t$ and winch drive force $f$ are both set to the specified tension for the static analysis, and the unstretched length paid out, $l_0$, is then determined so as to satisfy the wire tension in equation (\ref{statics}) above.

Dynamic analysis

Winch control throughout the dynamic analysis may take a number of different forms: it may be given for the whole simulation or individually for each stage; it can define constant values, change in value or rate of change; and it may be constant, time-varying or defined by external function. In all cases, however, at any one time the winch is either length-controlled or tension-controlled, and we can calculate its properties accordingly.

Length-controlled

Whatever the form of the control over the winch length, we can determine at any time the unstretched length of winch wire paid out, $l_0$. The winch wire tension $t$ (which is applied to each point on the wire) is then given by the dynamic form of equation (\ref{statics}) \begin{equation} \label{dynamics} t = \max\left\{0,\ k \epsilon + c k \ODt{\epsilon} \right\} \end{equation} where

$k$ and $c$ are the wire stiffness and wire damping values given on the winch data form

$\ODt{\epsilon}=$ wire strain rate $= \frac{1}{l_0} \left(\ODt{l}-\ODt{l_0}\right)$

$l=$ total length of the winch wire path.

Tension-controlled

Tension-controlled winches may specify a tension value directly or indirectly (through a change, or rate of change); that value is the target tension $t_\textrm{target}$ which the winch drive aims to achieve.

For simple winches, the drive is always assumed to achieve the target tension instantaneously, so $t_\textrm{target}$ is the actual winch wire tension.

For detailed winches, the winch drive tries to achieve the target tension by applying a drive force to one side of the winch inertia, opposing the wire tension being applied to the other side. The drive force $f$ applied is then a function of payout rate $v$ \begin{equation} f(v) = \begin{cases} f(0) - d + c\urm{i} v - d\urm{i} v^2 & v \lt 0 \\ f(0) & v = 0 \\ f(0) + d + c\urm{o} v + d\urm{o} v^2 & v \gt 0 \\ \end{cases} \label{fv} \end{equation} where

$v=$ rate of payout $= \mathrm{d}l_0/\mathrm{d}t$

$d=$ winch drive deadband

$c\urm{i},c\urm{o}=$ winch drive damping terms for haul in and pay out

$d\urm{i},d\urm{o}=$ winch drive drag terms for haul in and pay out

$f(0) = t_\textrm{target} + s\ (l_0-l_{00})$

$s=$ winch drive stiffness

$l_{00}=$ original value of $l_0$ at the start of the simulation (set by the static analysis).

If the winch inertia $m$ is non-zero, then the winch wire tension is calculated according to equation (\ref{dynamics}) above, and the winch inertia reacts by paying out or hauling in wire according to Newton's law \begin{equation} m \DDt{l_0} = t-f \end{equation} so the wire tension $t$ therefore tends towards the winch drive force $f$ and is hence controlled by the given tension.

If the winch inertia $m$ is zero, then the winch is assumed to be instantly responsive so that \begin{equation} \label{zeroInertia} f = t \qquad \text{at all times} \end{equation} Given the current value of $l_0$, the common value of $f$ and $t$ in this case is then found by solving the simultaneous equations (\ref{dynamics}), (\ref{fv}) and (\ref{zeroInertia}) for the payout rate $v = \mathrm{d}l_0/\mathrm{d}t$. The unstretched length of winch wire out, $l_0$, is then subjected to the calculated payrate $v$ as the simulation progresses.