Nonlinear dynamics of a skateboard model with three degrees of freedom

In this paper we continue our investigation of dynamics and stability of motion of a skateboard with a rider. In our previous papers we assumed that the rider, modeled as a rigid body, remains fixed and perpendicular with respect to the board. Hence if the board tilts through $\gamma$, the rider tilts through the same angle relative to the vertical, i.e. only one generalized coordinate $\gamma$ describes the tilt of the board and rider.
Now we make the next step in modeling complexity and we allow the board and rider to have separate degrees of freedom, $\gamma$ and $\varphi$, respectively. Here the rider is assumed to be connected to the board with a pin along the central line of the board through a torsional spring which exerts a torque on the rider and board proportional to the difference in their tilts relative to the vertical. Equations of motion of the model are derived and the problem of integrability of the obtained equations is investigated. The influence of various parameters of the model on its dynamics and stability is studied.