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I want to write a simulation of an accelerometer (with a fixed X and Y axis in this case but with respect to a world coordinate frame) on a stick, rotating in a circle around a fixed point. From that I want to figure out the tangential acceleration along the arc. I know the velocity omega but I do not know the length of the stick.

My problem is that I don't quite understand how to simulate the accelerometer, especially dealing with the rotations. Between time steps t1 and t2 the stick rotates by an angle theta and from that I can calculate the new position and rotation of the accelerometer point (one point is enough to calculate the accelerations from my understanding).

What values do I take now to calculate the acceleration in X and Y at each timestep and thereafter get the tangential acceleration?

enter image description here

Thanks for the help!

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    $\begingroup$ You know this problem can be solved analytically, right? In any case, to compute the acceleration you require three consecutive data points: $\vec{a}(t)=[\vec{x}(t+\Delta t)+\vec{x}(t-\Delta t)-2\vec{x}(t)]/\Delta t^2+\mathcal{O}(\Delta t^2)$ $\endgroup$
    – lemon
    Apr 14, 2016 at 14:56
  • $\begingroup$ accelerometers - physically there are many ways to transduce acceleration into a signal. In general , most if not all accelerometers measure in a single axis so multiple sensors are needed for multi-axis measurement. None are perfect, so they do have cross axis coupling. To resolve axial forces into the body frame in 2D or 3D use Euler or quaternion transformations. $\endgroup$
    – docscience
    Apr 14, 2016 at 16:08
  • $\begingroup$ Thanks lemon I will try out the formula, but I am not sure what the last term means: O(Δt^2), what's the O? How do I then split up the acceleration to get the tangential acceleration or is this already the tangential acceleration? $\endgroup$
    – Milchmann
    Apr 14, 2016 at 16:13
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    $\begingroup$ Re I know the velocity omega but I do not know the length of the stick. Then you are out of luck. $\endgroup$ Apr 14, 2016 at 17:24

1 Answer 1

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I don't quite understand what you are asking. In your diagram you have

$$ \begin{aligned} a_{tangential} & = \dot{\omega} \ell \\ a_{centripetal} & = \omega^2 \ell -g \end{aligned} $$

what else to do want to know? If the angle is changing (and therefore your acceleration are not aligned with X and Y) then use

$$ \begin{aligned} a_X & = \dot{\omega} \ell \cos\theta -\ell \omega^2 \sin \theta \\ a_Y & = \omega^2 \ell \cos\theta+\ell \dot{\omega} \sin\theta -g \end{aligned} $$

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