The more we dig, the more we find…
I simplified the model a little bit for the frequency study (the T junction was meant to verify stress only) and ended up with a cantilever beam with mass at its tip. The mass is very close to the beam end.
I tried to find a closed-form solution for this simple arrangement, but could not find it for all types of vibrations. For the axial mode, I used the Daniel J. Inman - Engineering Vibration 3rd Edition book to calculate the first mode as 148.93Hz from the following equation:
So I modeled it in Abaqus in some different ways: MPC Beam, Rigid body, Tie, mass directly attached to the end of the beam… The only mode that I had confidence in the result (the axial one) produced very good agreement: 148.8 Hz. And even though this is not the best way to go, I “trusted” the results for the other shapes.
After this, I modeled the same beam in CalculiX using 4 approaches:
- *EQUATION
- *DISTRIBUTED COUPLING
- Mass directly attached to the end node of the beam
- *RIGID BODY
And then I compared the results with the ones produced by Abaqus.
Here, for the first 4 transversal modes, the agreement is very good: up to a 3% difference in the higher-order mode. The torsional mode also produced very good agreement, with no difference in the first decimal places. However, two things caught my attention:
- The axial mode (the only one I have confidence in due to the analytical results) is very, very far off. When analyzing the deformed shape, I’m sure it is a spurious mode. It seems that the expanded nodes of the last element are vibrating on their own. I tried to use all degrees of freedom in the *EQUATION, but nothing changed…
- The *RIGID BODY produced two zero-frequency modes. I’m not sure why it happened, seems some spurious condition as well. Also, this option produced additional modes that the others did not… As I am not sure which are the correct results for the transversal modes, I can not guarantee that these modes are wrong, but since Abaqus and all other CalculiX approaches failed to find these modes, I tend to think that they are not correct.
One may ask: why is the reference node so close to the end of the structure? The answer is that I tried to model something similar to the closed-form solution that I had in hand. However, I tested what happened when I moved the reference node from 1 to 1000mm above the beam end. In CalculiX, the results were the same, even for the 6 DOF *EQUATION keyword. For Abaqus, the results (both frequency and eigenshape) changed a lot when I did this. I do not know if the results are correct, but I’m sure that CalculiX does not account for the rotary inertia component.
In the end, the proposal was to know how to attach a mass to a beam. To this, I can conclude that three approaches work:
- *EQUATION with DOF’s 1 to 3 connecting beam nodes to mass nodes
- DCOUP3D elements inside a *DISTRIBUTED COUPLING definition
- Mass directly attached to the beam nodes
If the user wants to consider the rotary inertia of the mass, more tests are needed to find something that works (maybe a very rigid beam connecting both nodes).
https://drive.google.com/drive/folders/1oHXTBtQKfM0p5iRmEgjqW-KYJByWWRA6?usp=sharing