# Calculation of acceleration in the Body 3DOF Single Track Block

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Johannes Hendewerk on 6 Aug 2021
Answered: Jason Rodgers on 15 Oct 2021
Hi,
while looking further into the formulas used for the single track model, I stumbled upon the calculation for acceleration in the documentation. The x- and y-acceleration (along the vehicle-fixed x-axis, as stated in the documentation) consists of two terms, the forces acting on the tire and car as well as an additional term (now for y-acceleration): -xdot * r.
Quick link to the documentation: https://de.mathworks.com/help/vdynblks/ref/vehiclebody3dof.html
I just cannot figure out where this term is coming from. Since the tire forces itself are calculated using the steering angle, the forces already should be in the vehicle fixed coordinate system (external forces now left out), so why would there be another term?
I hope somebody can help
Johannes

Ryan Chladny on 15 Oct 2021
Hi Johannes,
I think you are reffering to the result of the derivatives of rotating velocity vectors. Specifically, the vehicle body accelerations are derived with respect to a fixed inertial frame (does not move with the vehicle). However the forces are applied to a local frame fixed to the body (body fixed frame). When one does the derivative of the velocities (to find the accelerations) of the vehicle, one must also account for the fact that the vehicle is rotating. The resulting cross product of this generalized term in a 3DOF or planar motion case is the xdot*r term you are reffering to.
There are many references available on this topic if you are interested. For now here is one that Iwas able to find that provides a detailed derivation in context of a bicycle model like this which I hope you'll find useful.
Best regards,
Ryan

Jason Rodgers on 15 Oct 2021
This is related to the Coriolis effect. Since the dynamics of the vehicle are being calculated in a rotating frame, this effect will show up. See https://en.wikipedia.org/wiki/Coriolis_force for more info.

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