I want to obtain the natural frequencies of a benchmark truss which is famously known as the ten-bar planar truss. It consists of 10 truss members and 6 nodes. Two nodes are its supports and other four nodes are free nodes. The modulus elasticity is 6.89e10 and the added non-structural mass to the free nodes is 454 kg. Fig 1 shows the schematic of the truss and the obtained cross-section areas of this problem is listed in Fig 2. This figure also lists the natural frequencies obtained by different trusses. I tried to simulate the truss obtained by MC-TLBO which is listed in the last column but I faced a problem:
I didn’t reach the natural frequency in the literature and my model resulted in omega^2=1.575894e+04 and f=(omega/(2*pi))=19.98 Hz. It’s notable that I added some negligible massed on the other degree of freedom of the nodes according to OpenSEES official examples to obtain other natural frequencies.
The figures are referred from the [1] and The Tcl model is attached by myself (The units are in SI).
Any comment will be appreciated and thanks in advance for your time.
Refs.
[1] Farshchin, M., C. V. Camp, and M. Maniat. "Multi-class teaching–learning-based optimization for truss design with frequency constraints." Engineering Structures 106 (2016): 355-369.
Which mode number is 19.98Hz in your model. The frequency in the 3rd or 4th mode is almost same as your result. Did you compare the total mass is 532.051kg?
Dear Dr. Senba,
Thanks for your comments, I performed the Eigen analysis with the eigen command in the Opensees and it's the results for the first 5 eigenvalues:
omega^2=[1.575894e+04 1.629092e+04 9.292834e+04 1.113513e+05 1.916175e+15 ]
and the frequency=(omega/(2*pi))=[19.9 20.31 48.52 53.11 6966870.16].
I also compared the total mass and I gained the weight as 532.0513 kg.
In the model, I add the mass only on the first DOF of the nodes but I also added the mass on the second DOF to compare my results:
omega^2=[2.134420e+03 1.191119e+04 1.704385e+04 1.802558e+04 3.568710e+04]
and frequency=[7.35 17.37 20.78 21.37 30.07].
The latter kind of adding mass is closer to the literature results (first natural frequency=7) but it still has a significant difference with the literature. One of my friends correctly mentioned that omega=sqrt(K/M) and K is dependent on the elements which I have double checked the cross-section areas and modulus. However, M is not applied correctly and I guess it is the problem maker and I don't know still how to resolve it.
Thanks again for sparing your time on my question.
Depending on the lumped mass or the consisitent mass in FEM, the natural frequency will be different. Which did you use in your analysis and same in the paper you read ? Same value for the additonal mass =545 kg should be added on all the DOF of course. Also you can check the other literatures which give same value of the eigenvalues.
Dear Dr.Senba,
Thanks for your constructive comment. I only add the non-structural masses on the nodes (454 kg) but I have not considered any other masses for the elements. I see that if I consider some additional masses it may reduce the first natural frequency and reaches to 7.000 but I don't know how much mass I should consider for the lumped mass.
The model of the current truss can be presented as follows and it's described by some comments (all units are in SI),
model BasicBuilder -ndm 2 -ndf 2 #Two degrees of freedom and two dimensions (a typical truss)
# Nodal coordinations
node 1 0.0000 0.0000
node 2 9.144 0.0000
node 3 18.288 0.0000
node 4 0.0000 9.144
node 5 9.144 9.144
node 6 18.288 9.144
# define the nodal constraints (supports)
fix 1 1 1
fix 4 1 1
# Elasticitiy modulous
uniaxialMaterial Elastic 1 6.98e+10
# The connectivity among the nodes and cross-sectional areas
element Truss 1 4 5 37.712e-4 1
element Truss 2 5 6 9.959e-4 1
element Truss 3 1 2 40.265e-4 1
element Truss 4 2 3 16.788e-4 1
element Truss 5 2 5 11.576e-4 1
element Truss 6 3 6 3.955e-4 1
element Truss 7 2 4 25.308e-4 1
element Truss 8 1 5 21.613e-4 1
element Truss 9 3 5 11.576e-4 1
element Truss 10 2 6 11.186e-4 1
# Added non structural masses on the nodes
mass 2 454 454
mass 3 454 454
mass 5 454 454
mass 6 454 454
# Print the first natural frequency
puts [eigen 1]
Thank you so much for your continuous help.
OK. So, calculate the mass of each truss member using material density and the cross sectional area and length. Then the mass can be divided to each D.O.F where the truss member is connected. In other words, 1/2 of the mass is just added to particular nodal mass like you added non structural mass. Please check like that.
Dear Dr. Senba,
Thank you so much for your helpful comment. I did what you suggested as I calculated the weight of each truss member and applied their divided mass in half to its connecting nodes. The mass section of my model transformed as follows:
mass 2 582.44 582.44
mass 3 500.92 500.92
mass 5 582.84 582.84
mass 6 500.68 500.68
After I performed eigen analysis the obtained natural frequencies were:
f=[6.89 15.69 19.17 19.57].
However, the reported natural frequencies in the literature are f=[7.0000 16.1853 20.0000 20.0006]. As you suggested, the new natural frequencies are closer to what is reported in the literature but I still see some difference, can you guide me through this to fix this problem.
Thanks again for your incessant guides.
Please use the consistent mass if you want, which is more accurate in the modeling for any truss structures. Maybe the difference is not qutie large, so this mainner change will result in better one.
Dear Dr. Senba,
I did what you suggested and it worked like a charm, I really appreciate your persistent help for these days and I'm grateful for your tips and guides. Thank you so much.
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