u/DakBrakob — reddlx

I am trying to figure out how to calculate stress by superposition for different angular velocity components. As a test case I made a cantilevered aluminum beam with a 10kg mass placed at the end. Z direction is lengthwise along the beam. I ran cases for 6 unit loads with angular velocities of 1 rad/s about the X, Y, Z, axis bisecting X and Y, axis bisecting X and Z, and axis bisecting Y and Z.

I’m trying to determine the proper way to add stresses from unit loads together to get the stress produced by an arbitrary load of omegaX, omegaY, and omegaZ of any magnitude. For the test case I just set a velocity of 1 rad/s about the sum of X, Y, and Z unit vectors.

At the top of the second image is a formula I’m using to scale stresses from the unit loads. k1 and k2 are factors to scale unit loads with stresses of s_, where _ is the axis of the unit load. To start with I set k1 and k2 equal to 1. kx, ky, and kz are factors to scale the resultant load. So if kx=1, ky=2, ky=3 then there is 1 rad/s applied to z axis, 2 rad/s applied to y-axis, and 3 rad/s applied to z-axis.

The second image are correlation plots of the stress at each node. X axis is the stress using the superposition formula I mentioned, and y axis is the stress when applying the actual load. Ideally, a line could be drawn through this plot with a slope of 1 for any applied load (ie any kx, ky, kz) and that would mean that the superposition is accurate. The different color dots are different stress types. X dots are sigmaX, Y dots are sigmaY, Z dots are sigmaZ, XY dots are tauXY, XZ dots are tauXZ, YZ dots are tauYZ, V dots are von Mises stresses.

For the load case where omega in all directions is 1, setting k1 and k2 equal to 1 is off. I’ve tried different k1 and k2 and can get a good looking correlation for this load case, but it doesn’t make sense for other cases.

I originally was only using unit loads along X, Y, and Z, and the correlation was terrible, hence why I added the loads along bisecting axes. The reasoning for unit loads in primary AND bisecting axes is that the accelerations produced are coupled between omegaX, omegaY, and omegaZ.

In the accelerations stemming from the equation a=omega x (omega x r), there is an omegaX^2 term, an omegaY^2 term, an omegaZ^2 term, an omegaX * omegaY term, an omegaX * omegaZ term, and an omegaY * omegaZ term. I believe this is why some sort or term scaling stresses from omega in bisecting axes are necessary to get a good result.

This is sort of a follow up post to the one I made earlier on this subreddit, but providing more detail and focusing specifically on angular velocity since accelerations were straightforward to superimpose once I started summing plane stresses instead of principle stresses. I’m hoping that someone here has some insight on how to superimpose angular velocity stresses that’s better than just picking coefficients and hoping there’s a good correlation, without having any physics-based explanation for the coefficient values. Im looking for some method that’s universal regardless of magnitudes of each component, or any feedback on the superposition formula I’m using such as a missing term, etc.

I need to quickly estimate stresses in a structure under both linear and angular loads. The loads would be linear accelerations in three directions, angular velocities in three directions, and angular accelerations in three directions.

I am trying to take a small dataset of FEA runs and use superposition to estimate stresses in the structure. I can do this with the linear accelerations accurately (5-10% error).

For example if I have stresses in the structure for 1g in x (sigmaX), 1g in y (sigmaY), and 1g in z (sigmaZ) then stresses from 2g in x, 4g in y, 6g in z will be approximately the 2sigmaX+4sigmaY+6sigmaZ.

I have tried to use this same principle for angular loads unsuccessfully since the loads are not linear. Using superposition for angular accelerations can sometimes produce reasonable accuracy, but it is heavily dependent on where the stress is and not accurate for the entire structure. Superposition with angular velocities is so inaccurate it’s useless.

I recognize that the core issue is that I am trying to use linear methods to approximate a nonlinear angular load, and that the “best” way to do this with the highest accuracy is to just run an FEA case with the specific loading, but this is not feasible as I don’t need a high degree of accuracy but rather a reasonable approximation that is automated for thousands of load cases and very quick (<<1s) to calculate, so running FEA on a case-by-case basis is impossible.

Is there some way to quickly get an estimate of stresses for a given load provided that stresses for “unit” loads are known? The best approach I could come up with for angular loads was to just take the linear acceleration that the angular load (angular velocity and angular acceleration) is applying for a given point and then calculate the stress at that point by applying a stress delta from angular loads based on unit linear loads.

Using the example from earlier the stress would be 2sigmaX+4sigmaY+6sigmaZ+delta.

I think this approach will reasonably approximate the stresses as long as angular accelerations and angular velocities are small, but would appreciate any suggestions for a more accurate approach if there’s any methods I’m unaware of.

And I’m scaling principal stresses with superposition in case it wasn’t clear.