Cyclic symmetry - harmonic load components

[blue24]

I have a homework problem where I have to solve for the displacements of the attached system using cyclic symmetry. To do this, I know that I have to find the harmonic load components of the system. One thing that my professor did not make clear (or if he did, I missed it) is how to determine how many harmonics a system has. Can anyone tell me how to determine this?

I am not looking for help solving for the displacements. I'll sweat through that on my own :) I just want clarification on how many harmonics this system has, and why.

Thanks for the help!

 

[AlephZero]

You have three independent forces, so you need three coefficients to represent them as a Fourier series.

The zero harmonic has one coefficient, and higher harmonics have 2 each, for the sin and cos terms (or real and imaginary parts of a complex coefficient, or phase and quadrature, depending how you want to think about the problem).

Personally I wouldn't bother with Fourier coefficients here. I would consider a unit load at one vertex of the triangle, and then rotate the solution through 120 degrees and use superposition. That is using the cyclic symmetry of the system directly. Introducing Fourier coefficients doesn't seem to make it any easier, if you are doing it by hand. If you are using computer software, maybe that will force you to use Fourier coefficients.