Solution for mixed differential and algebraic equations

[WLamers]

I'm confused finding a solution (numerical integration is ok) for the following model:

http://imageshack.com/a/img27/7080/brmb.jpg

Free body diagrams:

http://imageshack.com/a/img22/9523/r0is.jpg And the equations of motion:

\begin{align}
k_{s,1}x_1+k_{s,2}x_1^3+k_m(x_1-x_2)+k_f(x_1-x_3) &= F\\
d_m\dot{x}_2-k_m(x_1-x_2) &= 0\\
F_fsign(\dot{x}_3)-k_f(x_1-x_3) &= 0
\end{align}

How can I solve these equations, preferably using a ODE solver. Problem is the lack of first derivatives. Substitution is possible when the lowest element is not present. But with this system I cannot find a solution. Help appreciated!

 

[WLamers]

Nobody?

I guess this is a DAE problem (Differential Algebraic Equation) and will not be easy to solve with numerical integration. Maybe it will be easier to just modify the problem in a way I end up with just a set of ordinary differential equations. But this requires to seprate the lower spring + friction damper (because of the $$sign(\dot{x}_3)$$, which in turn results in a different than intented behaviour. Maybe someone has a hint to end up with ordinary differential equations without having to sacrifice the sping+friction damper element? Thanks!