Calculating Rotations Associated with Circular Motion

Question

Suppose a motorcyclist is driving at constant speed along a circular track and, suppose further, that a local coordinate system is associated with the motorcycle, where

$x$ is tangent to the circle and oriented in driving direction
$z$ is orthogonal to the circle in the point of contact and tilted by an angle $\phi$ against the vertical
$y$ is the cross procuct of $z$ and $x$, i.e. $y=z\wedge x$

Question:

how can I calculate the angular velocities around the individual axes?

Maybe the question is to simple, but I'm no physicist and perplexed by the peculiarities of mechanics.

Please note that homework-like questions and check-my-work questions are generally considered off-topic here. We intend our questions to be potentially useful to a broader set of users than just the one asking, and prefer conceptual questions over those just asking for a specific computation. — ACuriousMind, Apr 19 '18 at 17:10

score 0 · Answer 1 · answered Apr 19 '18 at 17:50

You construct a 3×3 rotation matrix from the components of the local axes as columns.

$$R = \left( \begin{array}{c|c|c} \hat{x} & \hat{y} & \hat{z} \end{array} \right) $$

I think for your case you have

$$R = \left( \begin{array}{c|c|c} \cos \theta & -\sin\theta \cos\phi & \sin\theta\sin\phi \\ \sin\theta & \cos\theta \cos\phi & -\cos\theta \\ 0 & \sin\phi & \cos\phi \end{array} \right) $$

where $\phi$ is the inclination angle, and $\theta$ is the angle along the arc of track.

Usually you construct this from a sequence of rotations

$$ R = \mathrm{RotZ}(\theta)\,\mathrm{RotX}(\phi) $$

This is then used to transform vectors from local coordinates to world coordinates.

Since you are given that in world coordinates $\boldsymbol{\omega} = \pmatrix{0 \\ 0 \\ \dot{\theta}} $ (ground is along the xy plane), you can transform this to local coordinates with

$$ \boldsymbol{\omega}_{\rm local} = R^\top \boldsymbol{\omega} $$

But in the case where the tilt of the track changes, you can construct the rotational velocity vector, as a sequence of rotations

$$ \boldsymbol{\omega} = \hat{k} \dot{\theta} + \mathrm{RotZ}(\theta) \hat{\imath} \dot{\phi} = \pmatrix{ \dot{\phi} \cos\theta \\ \dot{\phi} \sin\theta \\ \dot{\theta} } $$

Calculating Rotations Associated with Circular Motion

1 Answers1