### Clapper equation: force balance and motion equations

Source: https://thermosyspro.gitlab.io/documentation/_sources/src/WaterSteam/PressureLosses/DynamicReliefValve.md.txt

These equations define the clapper's motion. The first equation sets the acceleration based on total force, the second defines velocity, and the third details the force components: weight, damping, spring, and pressure forces. The velocity and acceleration are zeroed at mechanical stops.

```math
.. math::
   M \cdot \frac{\mathrm{d} \nu}{\mathrm{d}t}=\left\{\begin{array}{l} f_{\mathrm{t}} \text { if } z_{\min }<z<z_{\max } \\
   f_{\mathrm{t}} \text { if } z \leq z_{\min } \text { and } f_{\mathrm{t}}>0 \\
   f_{\mathrm{t}} \text { if } z \geq z_{\max } \text { and } f_{\mathrm{t}}<0 \\
   0  \text { else }\end{array}\right.

```

```math
.. math::
   \nu =\left\{\begin{array}{l}\frac{\mathrm{d} z}{\mathrm{d} t} 
   \text { when } z_{\min }<z<z_{\max } \\
   0 \text { when } z \leq z_{\min } \text { or } z \geq z_{\max }\end{array}\right.

```

```math
.. math::
   f_{\mathrm{t}} &= f_{\mathrm{w}}+f_{\mathrm{s}} +f_{\mathrm{d}}+ f_{\mathrm{h}} \\
   f_{\mathrm{w}} &= -M \cdot g \\
   f_{\mathrm{d}} &= -D \cdot \frac{\mathrm{d} z}{\mathrm{d} t} \\
   f_{\mathrm{s}} &= -K_{\mathrm{e}} \cdot (z - z_{\min} + \delta) \\
   f_{\mathrm{h}} &= P_{\mathrm{i}} \cdot A_{\mathrm{i}} - P_{\mathrm{o}} \cdot A_{\mathrm{o}}

```