# Probing center of mass during launch

It's easy to calculate the change in center of mass due to fuel consumption.

But suppose you want to probe it experimentally (with whatever necessary calculations you might need to do on the measured data).

Could you do this with accelerometers/gyros alone, e.g., if you had an array of them? Could you do this with some other type of (off-the-shelf, easily accessible) sensor? Say maybe you want to try this on an ROV with variable mass and you don't have a good estimate for the rate of mass change handy.

• Do you mean monitor the instantaneous center of mass during powered spaceflight when main engine(s) are firing?
– uhoh
May 24 at 0:37
• Monitor them during flight, whether or not the engines are firing (e.g., after MECO, during stage reentry, say).
– user39728
May 24 at 0:39
• Having no experience in that application, I'm going to say yes. You will need a control system continually adjusting the direction of thrust in order to keep the rocket from veering off course. How it responds to thrust differences should theoretically tell you what you want to know. But I'm thinking you could probably get better precision from calculations based on how much fuel is left in the tanks.
– Greg
May 24 at 17:46
• I'm actually not asking whether you'd need an attitude control system. It's a rocket, of course you need one :D I'm only asking if you could measure the shift in center of mass without calculating it from the mass of fuel consumed (which is the only way I've done it so far).
– user39728
May 24 at 18:50
– Greg
May 25 at 19:03

With the following constraints, it should at least be theoretically possible:

1. The rocket is rotationally symmetric around an axis.
2. The rocket is spinning around some axis that is not the symmetry axis.

The first should be known a-priori. If we can then find the spin axis, the centre of gravity should be at their intersection.

Each individual accelerometer should experience a cyclic difference in acceleration, due to the centripetal acceleration it experiences on top of the general acceleration of the rocket. We can therefore find the times when the accelerometer is on the opposite or same sides of the spin axis as the direction of acceleration, and from that get a direction vector.

With an array of accelerometers, the spin axis can be found as the intersection of these vectors.

I have nothing to say about the practical feasibility of this scheme.

• It will not work. It is self evident from the equation: Moment of inertia * Angular acceleration = Torque, written for the 2D case, that if you know the angular acceleration you do not have enough data to determine the position of the center of mass. (Even if you have multiples sensors mounted on the body that spins, they all will indicate the same angular acceleration.)
– azot
May 25 at 2:43
• @NicoloFontana Hence the need for a symmetry axis. May 25 at 10:21
• Write some equations for a simple case (if you are familiar with the laws of mechanics) and solve them for the position of the center of mass. You will see that, no matter what assumptions you make, it is impossible to determine the location of the CM just based on the readings furnished by some accelerometers mounted on the rocket.
– azot
May 25 at 14:19
• @NicoloFontana What if, as the original poster said, we also have accelerometers in the nose and tail? The ratio of nose to tail acceleration is basically your answer.
– Greg
May 25 at 19:05
• The accelerometer in the nose and that in the tail will always indicate the same translational and angular acceleration. The ratio nose/tail will always be 1.
– azot
May 25 at 23:27

I do not believe it is possible to find the center of mass of a rocket using accelerometers.

We will consider the simple case of a rocket that ascends.

(Source of the image rocketmime.com)

If it is assumed that the variation of the mass is negligible and the drag is null, the second law of dynamics written for the rocket in the above drawing is:

$$T-mg=ma$$

a is given by an accelerometer.

If you know T, you can find m. The opposite is also true.

However, at least for this simple situation, there is no way to determine the position of the CM because its coordinates relative to that of the accelerometer do not appear in the equation.

If the rocket is forced somehow to rotate, it wil always rotate about its center of mass and then maybe the data from an accelerometer could be used for determining the position of the CM.

• But the acceleration in your thrust equation isn't just the pure translational acceleration of the center of mass. Since the accelerometer will generally be offset from the center of mass (because as fuel is consumed, the center of mass shifts), your acceleration reading will be distorted by the rotational velocity and rotational acceleration of the rocket about its center of mass. And those extra terms will include the distances from the accelerometers to the center of mass. So it seems if you had a cluster of accelerometers with known separation distances, then you might be able to get CM...
– user39728
May 24 at 15:29
• But thank you for taking the time to write down your thoughts :D
– user39728
May 24 at 15:31