I am working on periodic ordered and. disordered multi span beam frequencies. Papers show that, multi span beams exhibit local modes when the spacing or support stiffness of spans of periodic multi span beams changes slightly. What I want to do is to take out these disordered span and include the stiffness contribution of adjacent spans so that single span beam will have the same modal shapes and frequencies of the local modes of that span when it is placed in the multi span beam. Basically I am trying to find an analytical formula to add probably rotational springs at boundaries of single span beam so they will represent the stiffness contribution of adjacent spans.

There are several ways to solve the frequencies and mode shapes of multi span beams such as perturbation or transfer matrix method. The results show that there are pass band frequencies which are determined by stiffness of each span. For example, if there is a 4 span periodic beam, its first mode look like the first mode of each span. But this mode also looks like 4th mode of the long multispan beam if it did not have intermediate supports. And there are 4 modes at 1st and 2nd and 3rd...... bass band frequency ranges. And the last frequency of each bass band range is fixed-fixed frequency of a span. If you look at the 2nd and 3rd and 4th mode in the first band freq zone, they look like they want to look like 5th, 6th and 7th modes of the multispan beam if it did not have intermediate supports. But they can not exactly look like these modes because they should satisfy the intermediate support conditions. 8th mode obeys this condition too but it becomes the 1st frequency of 2nd pass band. So obviously the multispan beam's frequencies are characterized by frequency of each span. However, each span gets stiffer due to coupling with adjacent spans. Basically what I want to do is to be able to take out each span but add some extra restraints to it at its boundaries so that it is going to have the same frequencies and mode shapes when that beam is placed back to multi span beam.

I also could not find a paper which explains the physical meaning of vibrations. Yes analytical formulas obeys support conditions and close form solutions give the results when you take the determinant of the matrices that you build, but I am trying to comprehend the intuative meaning of the results.

I will appreciate any help.

Thank you.