Double-Integration of Acceleration Data using cumtrapz to get Displacement

Hello everyone and greetings from Germany!

I'm new to Matlab and searching for help. I want to extract the radial displacement of my sensor inside the tire tread. The only data i have is the radial acceleration. I recreated the signal theoretically for one wheel rotation according to a paper i read because i dont have the experimental data yet. This way i was able to get some data to experiment with. The graph should look something like this:

I'm only interested in the part inside the tire contact patch, so I treated everything outside the contact patch as if it would be zero:

Then i wanted to use the function cumtrapz twice to numerically integrate my vector of data and get the displacement, which leads to the following plot:

First of all, this is not what i was expecting. I think the result should look like a parabolic figure, starting from zero and ending at zero with a peak displacement somewhere around 20 or 30mm. Therefore i plotted the velocity after the first cumtrapz:

So here is my problem. I dont know if cumtrapz is the soultion to my problem after seeing these results. But i dont know what else i could do, because i just have datapoints of acceleration and nothing else. I read a paper where they did this too, but they didnt explain how they did it.

I hope someone can help me.

Answer 1

[Walter Roberson]

In the section you trimmed to, your second figure, you have no negative accelerations. Therefore your object never slows down -- it just does not always speed up as quickly as other times. An object that never slows down cannot have a parabolic trajectory.

[Kevin Pollmer]

Hello Walter, thank you for your quick answer!

Thats right. So i tried something. I used cumtrapz on the original data i created (first figure of my question). But since there is a lot negative acceleration the result is the opposite as above.

Then i de-meaned the acceleration and used cumtrapz twice:

But what bothers me is, that i already have a growing displacement without even reaching the point where the sensor enters the contact patch area. I know that while integrating the constant value of approx. -30m/s² of the acceleration leads to an error. Is that correct?

If so, is there a solution to this or do i simply not understand the whole problem?