In the modal analysis of Free-Free Beam, why modes are repeated at higher frequencies?
I am performing modal analysis of a Free-Free 2D beam (1 x 0.02 m) and I am trying to compute the mode frequencies and mode shapes up to the lowest 32 frequencies. Here I observed that the mode shapes started to repeat, such as
1,2,3 are rigid modes
4,5,6,7,8,9,10,12,13,14,16,17,19,20,21,23,24,26,27,29,30,32 are elastic modes
11,15,18,22,25,28,31 modes are similar to 2,4,5,6,7,8,9 modes
I used FENICS and verified in ABAQUS and got the same results.
Is this a phenomenon or a simulation error?
I will be using this mode data to perform Fluid-structure interaction using the modal expansion method.
Can anyone suggest what is criteria for choosing the number of mode shapes for this method?
Should I include the repeated modes in the modal expansion method?
Also, can anyone refer me to some books or articles related to the modal expansion method and modal analysis?
Mode shape plots given in the image is not very clear. There are much better ways to obtain and plot the mode shapes. However, free-free beam is a well known problem and you may find mode shapes of free-free beam in transverse vibration in many books. Even though the FE results considers 3 dimensions, i.e. 2 transverse directions plus the axial direction which are coupled with each other for certain modes, it can give an idea about how the mode shapes look like. The mode shapes which you find as similar actually different from each other. The first group starting with 11 seems to be in plane bending modes. However, each one is different from the other one. also the second group is different as well.
In order to identify the modes to be included in a FSI analysis, you need to consider the range of excitation frequency. In this case, the frequency of the excitation due to the fluid medium and also any other excitation force. You need to compare the excitation frequencies and the natural frequencies of the structure. mode shapes with natural frequencies closer to the excitation frequencies need to be included in the analysis (modal expansion). Modes with natural frequencies away from the excitation frequencies can be neglected, since their effect on the response will be slight. However, their effect can be included in the analysis by determining residual stiffness and mass terms.
You should include all modes that have natural frequencies close to the excitation frequencies. there is no repeated mode in the problem you have specified. All modes are different. How the excitation forces are applied also has effect, but it is much easier to compare natural frequency of the mode and the excitation frequency for determination of which modes to be included.
All vibration books have a part on modal expansion. You can anyone available to you. However, Modal Testing: Theory, Practice and Application by D J Ewins is a nice one if you have.
Thanks a lot, Ender Cigeroglu for the quick and detailed reply. As you mentioned, the modes that I mentioned as repeating (11,15,18,22,25,28,31 modes) are actually in-plane bending modes. I observed it when I was animating the mode shapes in paraview.
In my case, the excitation loads are arbitrary loads. In such cases, can you tell me how to choose the modes (and how many) for the modal expansion method?
The load may be arbitrary, but you need to know the frequency content of the excitation. You need to estimate the frequency content from a preliminary CFD analysis. You may consider your structure rigid first to have an idea about the frequency content of the excitation or you may specify a shape for your elastic structure to do this. After this step you can identify which modes of the elastic structure will contribute to the result. This is a must step to identify the modes to be used.
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