How to obtain the translational and angular accelerations of center of mass in rigid body

We had placed several acceleration sensors (e.g., 4 to 6) on a compressor at different positions. And the translational accelerations in x,y,z-directions over time for different positions on the compressor were messasured.

Assuming the compressor as Rigid Body, we would like to calculate the translational and angular accelerations of center of mass in this compressor (the x, y, z-corrdinates of center of mass are known) by using MATLAB.

Any suggestions would be greatly appreciated. Thanks. Feng

Answer 1

[William Rose]

@Feng, I have moved my comment to an answer, since that is what I intended.

To start, we should know the positions of the sensors relative to one another in x,y,z space, at rest.

To estimate what is happening at the center of mass (COM), we have to know where the COM is, relative to the sensors. This could be calculated, if you know the mass distribution of the compressor. If you do not have that information, then you can estimate the COM location, using your own intuition, or by using the centroid of the sensor locations, or the median of the sensor locations.

Are the sensors oriented with their axes aligned? If not, then we need to estimate their relative alignment, and then we need to rotate the measured linear accelerations to a common coordinate system.

Now assume that the linear accelerations are expressed in a common coordinate system. I assume that the rotation angles will be quite small, since this is a compressor which is probably mounted securely to a base that does not move. In that case, we can estimate the motions of each sensor by double integration, without having to do quaternion math associated with rotations. We will go back and check the small-rotation-assumption later.

Now you have estimated 3D positions of each sensor as a function of time. If the entire compressor were truly rigid, then the distances between sensor locations and the angles between them would all remain exactly constant. But in the real world, this will not be the case, due to non-rigidity and due to measurement error. Therefore, at each time point, we estimate the position and orientation of the "rigid" body, by least squares fit, using singular value decomposition. This will provide an estimate of the position and orientation of the "rigid" compressor at each instant.

Then we can see how much it rotates, and we can see if the positions estimated with linear acceleration plus rotation are close to the positions we originally esitmated by ignoring the rotation.

[William Rose]

@Feng,

Here are some more details:

The equation that relates motion at two locations, A and B, on a rigid body, is

where a_A, a_B are linear accelerations at locations A and B, and alpha=rotional acceleration of the body (which is the same at all locaiotns, due to rigidity), omega=angular velocity of the rigid body, and r_B/A= vector pointing from B to A. When the angular motions are small, eq. 1 simplifies to

Suppose we measure linear acceleration at 3 locations on the body. We wish to solve for the instantaneous linear and angular acceleration at the COM. Therefore we want to estimate the 6-element vector b at each instant:

where yHat is the vector of estimated linear accelerations at the sensors, and X and bHat are defined as follows. y (without the hat) is the vector of measured accelerations at each sensor. We estimate bHat as the vector that minimizes the sum squared difference beteween the mesaured y and the estimated yHat. We can find this bHat by standard multiple linear regression equations, but it turns out that singular value decomposition does a better job.

I realize that you have 4 to 6 sensors, so your y vector and X array will have 12,15, or 18 rows, instead of 9 rows, as shown above, for the case of 3 sensors.

If you would like to discuss this further, please contact me by clicking on the envelope icon which appears when you click on the "WR" circle next to my answers and comments.

[Feng]

@William Rose, thanks a lot for your informative answer, which is definitely a big help! Yes, the sensors are all oriented with their axes aligned. As you mentioned, the rotation angles of the compressor is relatively small during operation, as it is mounted securely to a base that does not move. A comprison study should be conducted to verify if the rotation could be ignored. Thanks!

[William Rose]

@Feng,

[edit: add parenthetical remark below]

You are welcome. I have analyzed similar data for three accelerometers mounted on an aircraft engine in a test stand. The results were very good, i.e. I found a linear acceleration and rotational acceleration at the COM that predicted linear accelerations at sensors, and the match between the measured and predicted accelerations at the sensors was very good. This result suggests that (a) my method worked, including the small-rotation approximation, and (b) the sensor locations and alignment were accurate, and (c) the engine plus sensors was acting like a rigid body.

The equations in my comment above, after equation 1, are from my own analysis. I will appreciate it if you credit me in any publications. Thank you.

You said "A comprison study should be conducted to verify if the rotation could be ignored." My method does not ignore rotations. (I was incorrect when I said that it did, in my initial post.) It just assumes that rotations are sufficiently small that a linear approximation can be used.