Finding the natural frequencies of a truss by using OpenSEES

[assimi]

I want to obtain the natural frequencies of a benchmark truss which is famously known as 10-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.

Figs.
Fig 1.
http://3.1m.yt/iCK85c-.png

Fig 2.
http://3.1m.yt/hxKpuE0.png

TCL model:
wipe
model BasicBuilder -ndm 2 -ndf 2
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
fix 1 1 1
fix 4 1 1
uniaxialMaterial Elastic 1 6.89e+10
element Truss 1 4 5 35.8507e-4 1
element Truss 2 5 6 14.8424e-4 1
element Truss 3 1 2 35.5768e-4 1
element Truss 4 2 3 14.9305e-4 1
element Truss 5 2 5 0.645e-4 1
element Truss 6 3 6 4.6249e-4 1
element Truss 7 2 4 23.9816e-4 1
element Truss 8 1 5 24.2358e-4 1
element Truss 9 3 5 12.6977e-4 1
element Truss 10 2 6 12.3319e-4 1
mass 2 454 1e-9
mass 3 454 1e-9
mass 5 454 1e-9
mass 6 454 1e-9
puts [eigen 1]

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.

[fmk]

for a start the negligable mass values are sometimes added for the rotational masses! not your case.

if you are getting different answers your model is incorrect or they messed up. i suspect the former, though one can not always rule out the latter.

[assimi]

Dear Frank,
First of all, thanks for your fruitful comment. I thought again and again about my problem and I'm now sure about that the mass matrix is not correct. An expert suggested me that my model looks ok but I have not yet added the lumped mass. He added that "this is done by declaring a mass element with the appropriate mass/inertial properties and then creating point mass elements where they are required." It's possible that If I consider additional mass the frequency reaches to 7.0000 but I don't know how can I implement the lumped mass to reach the frequencies reported in the literature. Now my results with considering mass on both DOFs on nodes are as follows,
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].

[fmk]

the mass is the same in horizontal and vertical direction .. for case where there is a 3rd dof representing rotation of the node, the rotational mass moment of inertia is needed, most people just set this to be 0 or something small.

if the frequencies are different from what is in the literature i suggest you write a matlab script to create the K and mass matrices yourself and check the eigenvalues.

[assimi]

Dear Frank,
I considered the consistent mass and it worked like a charm. Thanks for your incessant help and suggestions. I'm also grateful for this superb OpenSEES that you and your colleagues provided for scientific use freely.