Why is there a huge difference between theoretical and numerical results?

[Anonymous]

Hi guys, 

I was doing a vibration analysis on different type of beams. I will take cantilever beam as an example for the problem I faced. I chose modal analysis for this experiment. After I defined the material (Al 6061-T6) and the constraint (all fixed at one end), I run the simulation. 

I went to the report tab > Frequency. I am able to retrive the natural frequency of each mode. I used these values to compare with the theoretical result which was self-calculated through formula. However there is a huge difference between the numerical result (software) and theoretical result(self-calculated). The first 2 modes does not have a big differences until the 3rd mode. All the values subsititute into the formula remained the same except for the roots value. Each modes has a different value of roots. 

Value of roots 

Mode                                Roots value

1                                      1.875

2                                      4.694

3                                      7.855

4                                     10.995

5                                     14.137

Is there by any chance that the roots value are different in the software? However to my knowledge, the roots value is a universal fixed value. From books to research papers, the roots value for each mode are the same. Does anyone know why will the result deviate by so much and the percentage error is more than 10%? I would like to add one more point, my theoretical result is almost the same as a similar research paper numerical result using ansys software.

I have attached the formula and results for your reference. 

Thanks and appreciate any help or suggestion given!! 

[AstroJohnPE · Accepted solution]

First, a 6% difference in FEA is not a huge difference. I think most engineers would be happy if their analyses were within 6% of the real thing.

Sorry, I saw that the error for mode 5 was 6%. Actually, the error is more like 100*(935-338)/935=64% error. That is significant!

Higher modes of vibration include more bending. My guess is that you imported a CAD model and created a solid mesh (brick elements). Because brick elements can only approximate bending, a "coarse" mesh may give reasonable results for the lower modes and less accurate results for the higher modes.

Please try a finer mesh, or include the midside nodes in the calculation. (Midside nodes are chosen under the Element Definition for the part or parts.) Either change should give more accurate frequency at the higher modes.

The other thing that you need to confirm is that you are comparing the correct modes from Simulation Mechanical with the theoretical modes. Simulation Mechanical is calculating vibration in 3D space, so some of your results may be for bending about the "other axis", torsional vibration, or axial vibration. (Note how Sim Mech mode 4 is close to the theoretical mode 3. Is Sim Mech mode 3 for a "different" mode of vibration?) The theoretical results are only for bending about whichever axis you used to calculate I.

[Anonymous]

Thanks John for your feedback and suggestions. 

Yes I imported the CAD model and created solid mesh. I tried adding midside nodes and have a finer mesh. However the values are still far apart. Like what you had realised from the results. After mode 2, the theoretical modes are rather identical with the alternate numerical modes. After increasing the number of modes, I realised that the mode 6 value is close to the theoretical mode 5 value. 

"The other thing that you need to confirm is that you are comparing the correct modes from Simulation Mechanical with the theoretical modes. Simulation Mechanical is calculating vibration in 3D space, so some of your results may be for bending about the "other axis", torsional vibration, or axial vibration. (Note how Sim Mech mode 4 is close to the theoretical mode 3. Is Sim Mech mode 3 for a "different" mode of vibration?) The theoretical results are only for bending about whichever axis you used to calculate I."

I would appreciate if you could explain more. Do you mean that theoretical modes do not take 3D space into consideration? Because from books and research papers that i had read, I did not encounter any 3D space modeshape roots value. Aside the possibility of bending in the "other axis", is there any possibe reasons?