Saturating spring (Springsat)

$$\begin{equation*} H(q) = K_0 \, \left( \frac{q^2}{2} + K_{sat} H_{sat}(q)\right), \end{equation*}$$

$$\begin{equation*} H_{sat}(q) = - \frac{8 q_{sat}}{\pi \left(4-\pi\right)} \, \left(\frac{\pi^{2} q^{2}}{8q_{sat}^{2}} + \log{\left (\cos{\left (\frac{\pi q}{2 q_{sat}} \right)} \right)}\right). \end{equation*}$$

$$\begin{equation*} f(q)= \frac{d\,H(q)}{d q} = K_{0} \left(q + K_{sat} \frac{d\,H_{sat}(q)}{d q}\right), \end{equation*}$$

$$\begin{equation*} \frac{d\,H_{sat}(q)}{d q}= \frac{4}{4- \pi} \left(\tan{\left (\frac{\pi q}{2 q_{sat}} \right )} - \frac{\pi q}{2q_{sat}} \right). \end{equation*}$$