# Multistrand Elements (TYPE28)

Multistrand elements are n-node springs where matter is assumed to slide through the nodes.

It could be used for belt modelization by taking nodes upon the dummy. Friction may be defined at all or some nodes. When nodes are taken upon a dummy in order to modelize a belt, this allows friction to be modelized between the belt and the dummy.

## Internal Forces Computation

Nodes are numbered from 1 to $n$ , and strands are numbered from 1 to n-1 (strand $k$ goes from node Nk to node Nk+1).

## Averaged Force

The averaged force in the multistrand is computed as:

Linear spring $F=\frac{K}{{L}^{0}}\delta +\frac{C}{{L}^{0}}\stackrel{˙}{\delta }$

Nonlinear spring $F=f\left(\epsilon \right)\cdot g\left(\stackrel{˙}{\epsilon }\right)+\frac{C}{{L}^{0}}\stackrel{˙}{\delta }$

or, if $g$ function identifier is 0:

$F=f\left(\epsilon \right)+\frac{C}{{L}^{0}}\stackrel{˙}{\delta }$

or, if $f$ function identifier is 0:

$F=g\left(\stackrel{˙}{\epsilon }\right)+\frac{C}{{L}^{0}}\stackrel{˙}{\delta }$

Where,
$\text{ε}$
Engineering strain: $\epsilon =\frac{L-{L}^{0}}{{L}^{0}}$
${L}^{0}$
Reference length of element

## Force into each Strand

The force into each strand $k$ is computed as:

${F}_{k}=F+\text{Δ}{F}_{k}$

Where, $\text{Δ}{F}_{k}$ is computed an incremental way:

$\text{Δ}{F}_{k}\left(t\right)=\text{Δ}{F}_{k}\left(t-1\right)+\frac{K}{{l}_{k}^{0}}\delta {\epsilon }_{k}-\frac{K}{{L}^{0}}\delta \epsilon$

with ${l}_{k}^{0}$ the length of the unconstrained strand $k$ , $\delta \epsilon =\epsilon \left(t\right)-\epsilon \left(t-1\right)$ and $\delta {\epsilon }_{k}=\delta t{u}_{k}\cdot \left({v}_{k+1}-{v}_{k}\right)$ .

Where, ${u}_{k}$ is the unitary vector from node Nk to node Nk+1.

Assuming:

$\frac{{l}_{k}}{{l}_{k}^{0}}=\frac{L}{{L}^{0}}$

Where, ${l}_{k}$ is the actual length of strand $k$ .

Therefore, Equation 3 reduces to:

$\text{Δ}{F}_{k}\left(t\right)=\text{Δ}{F}_{k}\left(t-1\right)+\frac{K}{{l}_{}^{0}}\left(\delta {\epsilon }_{k}\frac{L}{{l}_{k}}-\delta \epsilon \right)$

## Friction

Friction is expressed at the nodes: if $\mu$ is the friction coefficient at node $k$ , the pulley friction at node Nk is expressed as:

$|\text{Δ}{F}_{k-1}-\text{Δ}{F}_{k}|\le \left(2F+\text{Δ}{F}_{k-1}+\text{Δ}{F}_{k}\right)\mathrm{tanh}\left(\frac{\beta \mu }{2}\right)$

When equation Equation 6 is not satisfied, $|\text{Δ}{F}_{k-1}-\text{Δ}{F}_{k}|$ is reset to $\left(2F+\text{Δ}{F}_{k-1}+\text{Δ}{F}_{k}\right)\mathrm{tanh}\left(\frac{\beta \mu }{2}\right)$ .

All the $\text{Δ}{F}_{k}$ (k=1, n-1) are modified in order to satisfy all conditions upon $\text{Δ}{F}_{k-1}-\text{Δ}{F}_{k}$ (k=2, n-1), plus the following condition on the force integral along the multistrand element:

$\sum _{k=1,n-1}{l}_{k}\left(F+\text{Δ}{F}_{k}\right)=LF$

This process could fail to satisfy Equation 6 after the $\text{Δ}{F}_{k}\left(k=1,n-1\right)$ modification, since no iteration is made. However, in such a case one would expect the friction condition to be satisfied after a few time steps.
Note: Friction expressed upon strands (giving a friction coefficient $\mu$ along strand $k$ ) is related to pulley friction by adding a friction coefficient $\mu /2$ upon each nodes Nk and Nk+1.

## Time Step

Stability of a multistrand element is expressed as:

$\text{Δ}t\le \frac{\sqrt{{C}_{k}^{2}+\rho {l}_{k}{K}_{k}}-{C}_{k}}{{K}_{k}},\forall k$

with ${K}_{k}=\frac{Mass\text{\hspace{0.17em}}of\text{\hspace{0.17em}}the\text{\hspace{0.17em}}multistrand}{{L}^{0}}$ :

${K}_{k}=\mathrm{max}\left(\frac{K}{{l}_{k}^{0}},\frac{F}{{l}_{k}-{l}_{k}^{0}}\right)=\mathrm{max}\left(\frac{KL}{{l}_{k}{L}^{0}},\frac{FL}{{l}_{k}\left(L-{L}^{0}\right)}\right)$
${C}_{k}=\frac{\left(f\left(\epsilon \right)\frac{dg}{d\stackrel{˙}{\epsilon }}\left(\stackrel{˙}{\epsilon }\right)+C\right)}{{l}_{k}^{0}}=\left(f\left(\epsilon \right)\frac{dg}{d\epsilon }\left(\stackrel{˙}{\epsilon }\right)+C\right)\frac{L}{{l}_{k}{L}^{0}}$