Moments of inertia

[mardlamock]

Hello everyone, I've been trying to build a decent model for the flight of a rocket but recently I started having a lot of problems calculating the angular acceleration.

The pivot point for the rotation will be the center of mass,which changes as the fuel burns out, I found a decent way to calculate where the center of mass will be, but the other thing needed to calculate the angular acceleration is the moment of inertia. How would one go about calculating said moment if the center of mass changes as a function of time? Would an approximation using the regular 1/2*m*r^2 formula be decent enough? Anyways, sorry if its a dumb question, its just that I really need to know if my rocket will be stable. Thanks!

[paul23]

That's impossible to say without more information lol.. If the rocket is a box shape of course it is not decent enough.. If the mass is not distributed evenly neither..

The general approach to these problems is to split the rocket in several parts which can easily be calculated (at least solveable integrals, preferably rho gets outside of the integeral so it becomes a constant - and better yet if the shapes are also some of the known inertias). Calculated each of those parts around the part's centroid and then use the parallel axis theorem to get the sum.

Now as for rocketry: most of the mass is indeed fuel, so a first order approximation can be done by a cylinderical shape. However do realise that the mass is dependent on t. And for solid fuel the burn generally goes from inward to outwards, so for solid rockets a better approximation is a thin walled cylinder, where the thickness is dependent on the time (I = m/2 (r1^2 + r2^2) ). Where r1 is (in a good build solid rocket where the amount of propellant burning is constant) a square root of the time.

[K^2]

Afraid not. Moment of inertia is a functional of mass distribution, and you have mass distribution changing in this problem.

You need to recompute the moment of inertia, but there is a shortcut. You can precompute the moment of inertia of an empty rocket and only compute moment of inertia of remaining fuel dynamically. If you assume some simple geometry for your fuel, like that it's always a cylinder, but the height and CoM of that cylinder changes, then it should be a fairly simple task.

Then, on each update, you'll find new center of mass, take rocket's precomputed moment of inertia, adjust that by MR² using R = distance from empty rocket's CoM to current CoM. Then do the same for fuel using the dynamically computed moment of inertia and R being distance from fuel CoM to current CoM. Finally, add the two together to get current moment of inertia.

You can keep doing this for as many elements as you have on your rocket. For example, if you are simulating a liquid fuel rocket and want to take into account that the fuel and ox tanks are located in different places and deplete at different rates.

Finally, keep in mind that MR² is only valid in 2D or if your axis of rotation is somehow fixed. (Which makes it behave as if it was in 2D.) For a general, 3D case, you should be dealing with moment of inertia tensor, and you'll need to use the correct tensor version of the MR² offset.

Edit: Ninja'd by paul23 with essentially the same info.

[GoSlash27]

Moment of inertia is computed the exact same way pilots compute weight and balance. Directly proportional to distance and mass, and the effects of the individual components are algebraically additive.

If you compute it once with full tanks and once with empty tanks, you will know what your moment of inertia is as a function of fuel percentage.

The problem is KSP doesn't allow for precise measurements in the VAB.

Best,

-Slashy

[K^2]

Moment of inertia is computed the exact same way pilots compute weight and balance. Directly proportional to distance and mass, and the effects of the individual components are algebraically additive.

If you compute it once with full tanks and once with empty tanks, you will know what your moment of inertia is as a function of fuel percentage.

That is true with respect to fixed datum/pivot. But if you do flight dynamics with respect to a fixed pivot, your CoM is accelerating relative to the pivot. That is a huge pain to deal with.

OP is on the right track, I think. It's much easier to simulate with respect to center of mass, and just recompute the moment of inertia to account for changes.

[paul23]

Moment of inertia is computed the exact same way pilots compute weight and balance. Directly proportional to distance and mass, and the effects of the individual components are algebraically additive.

If you compute it once with full tanks and once with empty tanks, you will know what your moment of inertia is as a function of fuel percentage.

The problem is KSP doesn't allow for precise measurements in the VAB.

Best,

-Slashy

As I showed this is untrue: it would only be true if the mass distribution doesn't change. But with solid fuel (and liquid too to some extend) this isn't the case. the moment of inertia is NOT directly proportional to the distance: .

[Guest]

And fuel sloshing...

And, at some point, fairings dropped...

[mardlamock]

Afraid not. Moment of inertia is a functional of mass distribution, and you have mass distribution changing in this problem.

You need to recompute the moment of inertia, but there is a shortcut. You can precompute the moment of inertia of an empty rocket and only compute moment of inertia of remaining fuel dynamically. If you assume some simple geometry for your fuel, like that it's always a cylinder, but the height and CoM of that cylinder changes, then it should be a fairly simple task.

Then, on each update, you'll find new center of mass, take rocket's precomputed moment of inertia, adjust that by MR² using R = distance from empty rocket's CoM to current CoM. Then do the same for fuel using the dynamically computed moment of inertia and R being distance from fuel CoM to current CoM. Finally, add the two together to get current moment of inertia.

You can keep doing this for as many elements as you have on your rocket. For example, if you are simulating a liquid fuel rocket and want to take into account that the fuel and ox tanks are located in different places and deplete at different rates.

Finally, keep in mind that MR² is only valid in 2D or if your axis of rotation is somehow fixed. (Which makes it behave as if it was in 2D.) For a general, 3D case, you should be dealing with moment of inertia tensor, and you'll need to use the correct tensor version of the MR² offset.

Edit: Ninja'd by paul23 with essentially the same info.

Ok, ill try and to that. Thanks for the info, i hadnt really understood what paul had originally said. And for the time being i am not goint to get into a 3d analysis, that would be way too much for my small brain. Im hoping to use this model to try and test how self stabilization software would work and see how accurate dead reckoning could be. Thanks!

[Nuke]

i think you can take a system of point masses and come up with a moi matrix to represent them. i dont remember the math though.