# Measure imbalance from within wheel

I want to measure static imbalance in a (car) wheel, using an accelerometer. This imbalance can be caused by tire, rim damage, dirt in the tire etc. The imbalance causes the center of mass to be different than the center of the hub. I want to measure where 'this' mass is (phase), and what it's weight is (magnitude).

The sensor is placed on the inside of the rim (part of a bigger system, can't change the location). The wheel will spin at a constant speed. I'm having trouble with finding the data which tells me about the magnitude and phase of the imbalance.

The acceleration vector in the zx-plane $(\sqrt{z^2+x^2}$, $z$ = vertical acceleration, $x$ = horizontal acceleration) will contain my information. It will also have a gravitation component and radial acceleration.

How can I derive the phase (i.e. location relative to my sensor) and magnitude ( i.e. weight) of my imbalance? Is it a sinusoidal component in the radial acceleration?

• $z$ and $x$ are accelerations? What do you call "phase"? What do you call "imbalance"? And in terms of which (presumably measured) quantities do you want these? – user1583209 Feb 24 '17 at 11:54
• Thanks for your answer, I'll update the question and be more specific. For now, $z$ and $x$ are indeed accelerations. A wheel with imbalance is a wheel that's not spinning around it's center of mass. The phase and magnitude of the imbalance are the location of the mass and it's weight causing the wheel to be out of balance. This can be seen from a stationairy wheel, where the accelerometer is at a certain point (let's say o degrees from the x-as). – EvanW95 Feb 24 '17 at 12:15
• I understand what you are trying to measure. Just two things are unclear to me: (1) Does your accelerometer measure acceleration in one or in two directions? In which directions? Does it measure faster than the rotation speed or slower (i.e. averageing) than the rotation speed? (2) If the wheel, and therefore the accelerometer is "spinning at constant speed" how is it going to notice any imbalance? The only accelerations would be gravity and radial ($\omega^2 r$), no? – user1583209 Feb 24 '17 at 12:39
• (1) The accelerometer measures acceleration in the radial direction and in the tangential direction. I don't really understand what you mean by 'measuring faster than the rotation speed'. If you mean the samplerate compared to the rotation speed, then it's definitely fast enough. (2) Well, that's exactly my question. I supposed the radial acceleration wouldn't constant, since the center of the hub and the center of mass are not aligned. But I'm not really an expert in this field. Do you think the accelerometer can't measure the imbalance? – EvanW95 Feb 24 '17 at 12:50
• If you are measuring the accelerations in a non-inertial coordinate system rotating with the wheel (which is how I would interpret "placed on the inside of the rim") and the wheel is rotating at constant speed, you will just measure a constant acceleration. In theory, if you measured the geometry of the wheel and the position of the sensor very accurately, you could find the point the wheel is rotating about (which not its geometrical center if it is out of balance) but in practice this isn't going to work. – alephzero Feb 24 '17 at 14:59

I agree with @alephzero. The wheel will rotate with constant angular speed and you will measure the constant radial acceleration of the accelerometer - or rather the vector combination of radial and gravitational accelerations. (There is no tangential acceleration here.)

This is because the accelerometer is a constant distance from the axis of rotation.

TS here. I rethought the problem and I think I found a way to solve it. Let's speak theoretically, leaving sensor errors, noise, practical feasibility etc. out of scope.

As @alephzer & @sammy gerbil said, there would be a radial and gravitational acceleration in my sensordata. But there won't be a tangential acceleration, because the wheel RPM is constant. There is one more acceleration in the system (I don't know how to name it). This acceleration is sinusoidal, and is in the $Y$ direction. This acceleration is caused by the imbalance of the wheel, which forces the wheel to jump up every time the imbalance is at the highest point. It can only move in the $Y$ directions, since the suspension is quit rigid in the $X$ direction, but not in the $Y$ direction (because of the helicalspring and the stiffness of the tires).

I have to filter this acceleration out of the sensor data. This can be done by:

1. Removing the radial acceleration from the sY axis (Sensor Y axis). This acceleration can be known using an gyroscope (I know the radius of the wheel).
2. Sum up the $sX$ and $sY$ vector.
3. Delete the gravitational component. I must know the location (angle between $Y$ and $sY$) of the sensor to do this. An idea to do this is by measuring the angle when the wheel isn't turning. Then measure the angle using the gyroscope and accelerometer at low speed (gravitation acceleration isn't affected by the speed of the wheel while the acceleration caused by the imbalance is). When the $Y$ direction acceleration hits in with a significant magnitude, subtract the gravitational acceleration. 