Pulley Type Spring Elements (TYPE12)

Pulley type springs are defined by TYPE12 element property. A general representation can be seen in Figure 1. It is defined with three nodes, where node 2 is located at the pulley position. Other properties such as stiffness, damping, nonlinear and plastic effects are the same as for the other spring types, and are defined using the same format.

A deformable "rope" joins the three nodes, with the mass distribution as follows: one quarter at node 1; one quarter at node 3 and one half at node 2.

Coulomb friction can be applied at node 2, which may also take into account the angle between the two rope strands.

The two rope strands have to be long enough to avoid node 1 or node 3 sliding up to node 2 (the pulley). If this occurs, either node 1 or 3 will be stopped at node 2, just as if there were a knot at the end of the rope.

Time Step

The time step is calculated using the relation:(1)

Δ t = \frac{(\sqrt{K M + C^{2}}) - C}{2 K}

This is the same as for TYPE4 spring elements, except that the stiffness is replaced with twice the stiffness to ensure stability with high friction coefficients.

Linear Spring

See Linear Spring; the explanation is the same as for spring TYPE4.

Nonlinear Elastic Spring

See Nonlinear Elastic Spring; the explanation is the same as for spring TYPE4.

Nonlinear Elasto-Plastic Spring - Isotropic Hardening

See Nonlinear Elasto-plastic Spring: Isotropic Hardening; the explanation is the same as for spring TYPE4.

Nonlinear Elasto-Plastic Spring - Decoupled Hardening

See Nonlinear Elasto-plastic Spring: Decoupled Hardening; the explanation is the same as for spring TYPE4.

Nonlinear Dashpot

See Nonlinear Dashpot; the explanation is the same as for spring TYPE4.

Friction Effects

Pulley type springs can be modeled with or without Coulomb friction effects.

Forces without Friction

Without friction, the forces are computed using:(2)

| F_{1} | = | F_{2} | = K δ + C \frac{d δ}{d t}

Where,

$δ$: Total rope elongation = $l - l_{0}$ with $l = l_{12} + l_{23}$
$K$: Rope stiffness
$C$: Rope equivalent viscous damping

Forces with Coulomb Friction

If Coulomb friction is used, forces are corrected using:(3)

F = K (δ_{1} + δ_{2}) + C (\frac{d δ_{1}}{d t} + \frac{d δ_{2}}{d t})

(4)

Δ F = \max (K (δ_{1} + δ_{2}) + C (\frac{d δ_{1}}{d t} + \frac{d δ_{2}}{d t}), F \tanh (\frac{β μ}{2}))

(5)

| F_{1} | = F + Δ F

(6)

F_{2} = - F_{1} - F_{3}

(7)

| F_{3} | = F - Δ F

Where,

$δ_{1}$: Elongation of strand 1-2
$δ_{2}$: Elongation of strand 2-3