# Modeling Vehicle Acceleration - Simulink

Hello I am modeling a vehicle that has a seperately excited DC motor as the power plant utilizing the constant-torque and constant-power regions for traction. The acceleration is governed by the following equation $$M \frac{dv}{dt} = F_t -(\beta_1+\beta_2v^{2}+\beta_3v)$$ The problem with my model is that the $$\beta_1$$ component, the rolling resistance, is a constant value and causes the velocity to go negative for the first 0.7 seconds when the model is executed . Any suggestion on how to fix this issue?

• To complement BikerDude's answer, the terms with even powers of velocity (i.e. v^0 and v^2) could also be multiplied by the "sign function" (i.e. 1 or -1 or 0) of the velocity. Mar 3, 2021 at 20:15

If this is the case, the equation either needs a domain readjustment(so, valid from t∈[0.7,∞]) or an equation redefinition.

Using a piecewise function would look like: $$M \frac{dv}{dt} = F_t -(\beta_1+\beta_2v^{2}+\beta_3v), t∈[0.7,∞]$$ $$M \frac{dv}{dt} = 0, t∈(0,0.7)$$

Friction, is just a reactionary force. For rolling resistance can be modelled as an example of a dynamic friction. Hence, until motion occurs, $$F_t = β_1$$ Then, you should get better looking profiles.

Another way to do this would be to have a criteria, where the reactionary $$β_1$$ is just as big as $$F_t$$.

Therefore, $$M \frac{dv}{dt} = F_t -(\beta_1+\beta_2v^{2}+\beta_3v), t∈[0,∞]$$ Where $$\beta_1 =F_t, F_t∈[0,\beta_{max}]$$ $$\beta_1 =\beta_{max}, F_t∈[\beta_{max},∞]$$

Here, $$\beta_{max}$$ is your initial value of $$\beta_1$$

• Thanks for your insight, the domain readjustment won't work in this situation since F_t is not constant, forgot to mention that. However, you've given me a possible solution whereby I will try to implement a conditional statement block that only allows forward motion if the F_t > B_1. Mar 3, 2021 at 21:48
• Awesome! My edit might be helpful with your mathematical model. The last equation is valid over all +ve time. Mar 3, 2021 at 22:32

I solved the issue with a switch function block, it not the best solution but it works for my basic simulation. Thanks again 