Is skateboard vibration model correct? My friend and I rode a skateboard yesterday. After a certain speed the board began to wobble and I fell down.
In order to avoid a similar outcome in the future, I want to model the "vibration" that occurred mathematically so that I can analyze the effect of different wheels, suspensions or even foot placement.
I have created the following model, which is a variation of the harmonic oscillator:
Model of vertical wheel displacement:
Variable declarations:
$w_n(t):=$Position of the nth wheel
$m_n(t):=$Weight of the nth wheel in kg
$F_n(t):=$Surface irregularity force on the nth wheel in newton
$k_n,v_n\in \mathbb{R}^+:=$Coefficient of nth wheel
The vertical displacement of the wheels is then modelled by:
$m_n\ddot{w}_n=-k_n\cdot w_n-v_n\cdot \dot{w}+F_n$
Picture (orange part is modelled by the above equation):

Model of board vibration
I simulate the wobbling as rotation of the board in radian. It is
dependent on the displacement of the wheels: The larger the displacement of two adjacent wheels (front view) $w_1,w_2$, the larger the wobbling of the board.
Because the rider can counteract this wobbling by exerting a force himself, we have to include him in the model as well.
This counterforce is not random but depends on the previous vibration of the board (vibration) at an earlier time $t-\alpha$, so we add the function $\lambda(b(t-\alpha)$.
Since the rotation is centered around a pivot which is also dampened, we add the dampening term $v_b\cdot \dot{b}$.
To simplify calculations, it is possible to set $\lambda(b(t-\alpha)$=$b(t-\alpha)$.
Additional Variable declarations:
$\lambda(b(t-\alpha):=$ Counterforce of the rider in newtons.
$b(t):=$ Position of the board in radian.
$\mu,\alpha\in \mathbb{R}^+:=$Coefficients
The rotation of the board is then modelled by:
$m_b\ddot{b}(t)=\lambda(b(t-\alpha)\cdot sin(b)+\mu (w_1-w_2)-v_b\cdot \dot{b}$
I have drawn a picture again, so you can see what I mean:

Question:
Do you think this is a valid model to model high speed "board wobble"?
Perhaps, you can come up with a better approach to solve the problem?
I can tape my phone under the board (so I have an accelerometer, a gyroscope, a magnetometer, a gravity sensor, a linear acceleration sensor, a rotation sensor and a pressure sensor at my disposal).
Thank you for your help.
 A: Simple mechanical models reveal that the dynamical instability known as skateboard speed wobble results from the interaction between the skater and the skateboard, and not from the only skateboard. Also, it is well-known that the speed at which the skate starts wobbling increases with the length of the board (that's why it is recommended to use a longboard). This is documented in several recent articles [1, 2, 3, 4].

[1] M. Rosatello, J.-L. Dion, F. Renaud, L. Garibaldi: "The skateboard speed wobble", Proceedings of the ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference 6 (2015). (doi:10.1115/DETC2015-47326)
[2] B. Varszegi, D. Takacs, G. Stepan, S.J. Hogan: "Stabilizing skateboard speed-wobble with reflex-delay", Journal of the Royal Society Interface 13 (2016) 20160345. (doi:10.1098/rsif.2016.0345)
[3] B. Varszegi, D. Takacs: "Downhill motion of the skater-skateboard system", Periodica Polytechnica Mechanical Engineering 60-1 (2016) 58-65. (doi:10.3311/PPme.8636)
[4] B. Varszegi, D. Takacs, G. Stepan: "Stability of damped skateboards under human control", J. Comput. Nonlinear Dynam. 12-5 (2017) 051014. (doi:10.1115/1.4036482)
