# Solving Hertz Impact ODE with Simulink

I have a question about modelling a nonlinear 2nd order ODE for an impact of a sphere on a thin plate in Simulink. Depending on the solver and solver options, my results in Matlab vary a lot. Some are physically reasonable, some not. Strange enough, refining the time step makes the solution less plausible from a physical point of view. Now I thought about modelling the ODE in Simulink an see which results I can get there.The problem is the dimensionless ODE for the indentation of a sphere in a plate:

$$\frac{d^2 \sigma}{d \tau^2}+\left( 1 + \lambda \frac{d}{d\tau}\right)\sigma^\frac32 = 0$$ Where $$\sigma$$ is a dimensionless displacement and $$\tau$$ a dimensionless time. $$\lambda = 110 = constant$$. The initial conditions are: $$\sigma\left(\tau = 0\right)=0$$ and $$\frac{d\sigma}{d\tau}|_{\tau=0}=1$$.

Unfortunately, I do hardly have any experience with Simulink. Does anyone have any experience/tips for modeling this kind of problem in Simulink?

• Use integrator blocks rather than derivative ones. Although you can have mathemically equivalent implementations of the same ODE, using integrators is much better from a numerical point of view. You probably also want to familiarise yourself with the different types of ODE solvers (e.g. fixed-step vs variable-step, stiff vs non-sitff, implicit vs. explicit, etc...): uk.mathworks.com/help/simulink/ug/types-of-solvers.html – am304 Mar 26 '19 at 13:58
• @am304: Thanks for your reply. I played around a little bit and the solution looked the same as in Matlab. However, I did not find a way to model the derivative of the root of sigma without using a derivative block... Anyways, the main problem appears to be that the equation itself is not valid for such large values of lambda. In the publications I found, only the range from 0 to 5 was shown. So I may need an entirely different approach anyway. – H. Fisher Apr 8 '19 at 15:46