# Spring Type Pretensioners (TYPE32)

Pretensioner expected behavior is as follows: before pretensioning, a piston is fixed in its initial position; when activated, the piston is pushed and cannot slide once the piston has reached the end of its slide, it is unable to slide further in any direction in the opposite direction from its actual position

## Pretensioner Model

### Linear Model

$STI{F}_{0}$ is the spring stiffness before sensor activation. At sensor activation, the 2 input coefficients among D1, STIF1, F1 and E1 determine the pretensioner characteristics. Let us recall the following relations between the 4 coefficients:(1) ${E}_{1}=\frac{{D}_{1}\cdot {F}_{1}}{2},{K}_{1}=\frac{{F}_{1}}{{D}_{1}}$

$STI{F}_{0}$ is also used as unloading stiffness before the end of the piston's slide, and as both loading and unloading stiffness at the end of the piston's slide. $STI{F}_{0}$ should be large enough to allow locking.

### Nonlinear Model

$STI{F}_{0}$ is spring stiffness before sensor activation. Depending on the input, pretensioning force is defined as $f\left(L-{L}_{0}\right)$, with either $g\left(t-{t}_{0}\right)$, or $f\left(L\left(t\right)-{L}_{0}\right)\cdot g\left(t-{t}_{0}\right)$, with ${L}_{0}$ length of the spring at sensor activation time and at ${t}_{0}$ sensor activation time.

Similar use of $STI{F}_{0}$ allows piston locking.

### Force Computation

Let the pretensioning force ${F}_{p}\left(t\right)={F}_{1}+STI{F}_{1}\cdot \left(L\left(t\right)-{L}_{0}\right)$ for a linear model, and ${F}_{p}\left(t\right)=f\left(L\left(t\right)-{L}_{0}\right)$ or $g\left(t-{t}_{0}\right)$ or $f\left(L\left(t\right)-{L}_{0}\right)\cdot g\left(t-{t}_{0}\right)$ for a nonlinear model.

The force into the pretensioner spring is computed as:

if, ${F}_{p}\left(t+dt\right)\ge 0$

${F}_{}\left(t+dt\right)=Max\left({F}_{p}\left(t+dt\right),F\left(t\right)+STI{F}_{0}\cdot L\left(t+dt\right)-L\left(t\right)\right)$

and ${F}_{}\left(t+dt\right)=F\left(t\right)+STI{F}_{0}\cdot \left(L\left(t+dt\right)-L\left(t\right)\right)$ otherwise.