# Spring TYPE12 - Pulley (/PROP/SPR_PUL)

Spring TYPE12 is used to model a pulley. When used in a seat belt model, it is defined with three nodes.

Node 2 is located at the pulley, and a deformable rope is joining the three nodes (Figure 1). The spring mass is distributed on the three nodes with ¼ at node 1 and node 3 and ½ at node 2.

A Coulomb friction can be applied at node 2, taking into account the angle between the two strands. Without friction, forces are computed as:

$$\left|{F}_{1}\right|=\left|{F}_{2}\right|=K\delta $$

With,

- $\delta $
- Total rope elongation
- $K$
- Stiffness

If the Coulomb friction is used, forces are computed as:

$${F}_{fr}=\mathrm{min}\left\{\left|\text{\Delta}F\right|,\mathrm{max}\left[0,\left(\left|{F}_{1}\right|+\left|{F}_{2}\right|\right)\cdot \mathrm{tanh}\left(\frac{\beta \cdot \mu}{2}\right)\right]\right\}\cdot sig\left(\text{\Delta}F\right)$$

Where,

- $\mu ={\mathrm{f}}_{fr}\left(\frac{\text{\Delta}F}{Xscale\_F}\right)\cdot Yscale\_F$
- $\beta $
- Angle (radians unit)
- ${\mathrm{f}}_{fr}$
- Function of
`fct_ID`_{fr}

`I`_{fr}=0 (symmetrical behavior)$$\text{\Delta}F=\left|{F}_{1}-{F}_{2}\right|$$`I`_{fr}=1 (non-symmetrical behavior)$$\text{\Delta}F={F}_{1}-{F}_{2}$$

${\delta}_{1}$
is the elongation of strand 1-2 and
${\delta}_{2}$
of strand 2-3.

Time step is computed with the same equation that for spring TYPE4, but the stiffness is replaced
with twice the stiffness to ensure stability with high friction coefficients.

Note: The two strands have to be long enough to avoid node 1, or node 3 slides
up to node 2. Nodes 1 and 3 will be stopped at node 2, if there is a knot at nodes 1
and 3.

For further information, refer to the Radioss Theory Manual.