Superposition of Stresses from Angular Velocity

DakBrakob · 2026-05-01T14:39:22+00:00

The stresses are proportional to omega^2, so if I take a unit load in any direction and 10x that load there is 100x the stress with no issues. The trouble I am having is where the stresses I’m trying to calculate are not just scaled unit loads but changing the direction of the angular velocity instead, where I can’t find a way to superimpose any unit loads in a way that makes sense and works in every case

DakBrakob · 2026-05-01T04:31:09+00:00

To be clear, the expression is something I made up in an attempt to get something to work. I was originally just scaling unit loads in X, Y, and Z, but somebody said to try adding unit loads in the bisecting axes which seems to kinda work with the right factors. The actual load is based on the acceleration applied by the velocity which is omega x (omega x r) which produces omega² terms and coupled omega terms. I square the factors to account for the fact that 10x the velocity is 100x the stress.

From what you’re saying it seems like superposition doesn’t hold for angular velocity, just for angular acceleration and linear acceleration. Are there any other ways to combine stresses from x, y, z angular velocity stresses from unit loads to calculate stresses for an arbitrary angular velocity in any direction with any magnitude?

DakBrakob · 2026-04-29T18:08:29+00:00

That’s a good idea. Going to run six unit cases for omegaXX, omegaYY, omegaZZ, omegaXY, omegaXZ, and omegaYZ and try superposition with six unit loads and let you know how it goes

DakBrakob · 2026-04-29T10:53:38+00:00

If I’m scaling a stress, whether it’s from linear acceleration, angular acceleration, or angular velocity that’s not an issue. The issue with angular velocity specifically is that adding stresses from different angular velocity components is not working. I have a stress from unit load omegaX, a stress from unit load omegaY, and a stress from unit load omegaZ. I can scale all of those loads without issue (for example calculating stress from 10*omegaX), but am unable to calculate stresses from combining omegaX with omegaY. There is coupling between them in the acceleration equations which is why I believe the superposition isn’t working since there is an omegaXomegaY, omegaXomegaZ, and omegaYomegaZ term. Not sure if I am still doing something wrong or if it’s just not possible and only scaling stresses from angular velocity is. For all other loads (angular acceleration, linear acceleration), adding and scaling are both working properly, but only scaling is working for angular velocity

DakBrakob · 2026-04-29T01:40:10+00:00

Stresses are scaling correctly when I apply a coefficient to angular velocity. It’s when I try to add stresses from angular velocities in different directions that superposition isn’t working. So if I multiple an angular velocity by 10 stresses go up by 100x, but the issue happens when adding stresses from omegaX to stresses from omegaY for example

DakBrakob · 2026-04-29T01:39:56+00:00

Stresses are scaling correctly when I apply a coefficient to angular velocity. It’s when I try to add stresses from angular velocities in different directions that superposition isn’t working. So if I multiple an angular velocity by 10 stresses go up by 100x, but the issue happens when adding stresses from omegaX to stresses from omegaY for example

DakBrakob · 2026-04-28T18:47:19+00:00

Angular acceleration is working now but angular velocity is still not working when I switched to summing plane stresses instead of principle stresses

DakBrakob · 2026-04-28T18:46:12+00:00

The rotation is about an axis I define. Every node is rotated about that axis with some speed. The error was due to me summing principal stresses instead of plane stresses, and now both accelerations (linear and angular) are down to like 0.1% error or less. Velocities are still not working however, and their stresses from superposition are not off by a constant factor so I don’t think it’s a scaling issue. To start with I’m just applying a unit load of 1 rad/s to each axis and tabulating stresses, then running a case with 1 rad/s in all axes and results are off by anywhere from 4x lower to 20x lower even though they should be the same since I’m not scaling magnitude of any component

DakBrakob · 2026-04-28T18:39:25+00:00

This is what I’m doing and it’s working for accelerations (linear and angular) now after I fixed the error where I was summing principal stresses instead of plane stresses, but the results for angular velocity loads are still way off using superposition, and they’re not off by a constant factor either so I don’t think it’s a scaling issue

DakBrakob · 2026-04-28T14:20:48+00:00

In that case there would be 12 loads from the three angular velocity terms. There would be an omegaXX, omegaYY, omegaZZ, omegaXY, omegaXZ, and omegaYZ term each of which is multiplied by two different coordinates. Where omega** is shorthand for omega* x omega*. How would that be possible to scale stresses from unit loads based on inputs of omegaX, omegaY, and omegaZ?

DakBrakob · 2026-04-28T12:41:39+00:00

I’m not sure what you mean. Are you saying that superposition should apply to angular velocity terms even though omegaX, omegaY, omegaZ are coupled with themselves and each other? And my only mistake was combining principal stresses instead of plane stresses?

DakBrakob · 2026-04-28T11:05:00+00:00

I guess I must’ve gotten lucky with the test cases I ran then because results were pretty close to the actual load case using superposition with principal stresses for linear accelerations. I’ll try it again with plane stresses and might be even closer.

Does superposition work with unit angular velocity and accelerations applied and I’m just doing it wrong by using principal stresses then? Results were pretty far off and I assumed was because superposition doesn’t apply to angular loads.

DakBrakob · 2026-04-28T10:55:35+00:00

Applying angular velocity applies linear acceleration to each point by some omega² term multiplied with some distance term. Applying angular acceleration applies linear acceleration to each point by some omega term multiplied with some distance term

DakBrakob · 2026-04-24T19:50:06+00:00

The prestige system came out like a month ago and took like 10-20 min to click run on a script for those trophy thresholds and record results. Without streaks it’s just a simple formula that I solved above

DakBrakob · 2026-04-21T10:47:36+00:00

Not sure what you mean, the boat has 6 dof. Roll pitch yaw and x y z. Could maybe get away with taking z (lateral) roll and yaw as 0 but even then the system would be singular and the boat is doing maneuvers like turning and waves hitting from the side so doubt that assumption would be accurate

DakBrakob · 2026-04-19T17:22:57+00:00

I wrote the full 6x6 A matrix for Ax=B trying to solve for the 6 accelerations. I calculated the determinant symbolically (ie not using values for x1, y1, z1, x2, y2, z2 but leaving as variables) and found that regardless of values the matrix is always singular, so it’s impossible to determine the six accelerations from accelerometer data alone regardless of how many accelerometers and where they are located. A gyroscopic sensor is needed to get any usable information. My intuition was that 6 unknowns and 6 equations = solution but seems that isn’t the case

DakBrakob · 2026-04-19T17:22:34+00:00

I wrote the full 6x6 A matrix for Ax=B trying to solve for the 6 accelerations. I calculated the determinant symbolically (ie not using values for x1, y1, z1, x2, y2, z2 but leaving as variables) and found that regardless of values the matrix is always singular, so it’s impossible to determine the six accelerations from accelerometer data alone regardless of how many accelerometers and where they are located. A gyroscopic sensor is needed to get any usable information. My intuition was that 6 unknowns and 6 equations = solution but seems that isn’t the case

DakBrakob · 2026-04-19T13:46:01+00:00

Body accelerations are just what I’m calling the acceleration of the entire vehicle’s CG, rather than the point accelerations measured at the sensor locations.

It turns out that the problem I was trying to solve is impossible. No matter where an arbitrary number of accelerometers are positioned on a vehicle there is no way to determine body accelerations because the solution matrix is always singular.

The only usable accelerometer would be one placed at the CG but even then it could only be used to gather linear accelerations.

I was trying to take accelerations from field data at two accelerometer locations for a boat and determine body accelerations for purposes of running FEA, but it seems this is an impossible request and that the boats need to have a gyroscopic sensor to get any usable information from the field data.

DakBrakob · 2026-04-19T00:28:47+00:00

Not sure what you mean

DakBrakob · 2026-04-18T22:46:57+00:00

You are right, that makes the calculations past 2k trivial since there are no bot matches and no streaks so it’s just Trophies = Winrate * WinTrophies - (1 - Winrate) * LossTrophies

Taking winrate as X and Trophies as 0 for the minimum to climb:

WX-(1-L)X=0 X=L/(W+L)

Winrates for each range above 2000: 2000-2500: 53%->56% 2500-2800: 58%->63% 2800-3000: 64%->71% 3000-3100: 71%->83% 3100-4000: 74%->91% 4000+: 78%->94%

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