Problem 2

That was some nasty algebra. Insofar as simplification goes, I figured that it would not be appropriate to make assumptions about terms being "small" (such as higher order advance ratios) as it seems to alias the higher harmonics...so here goes. I've sorted by harmonic in hopes of greater clarity:

\(\beta_{3s}=q_{3s}\left\{\frac{72\nu_{\beta}^2+72\mu^2\nu_{\beta}^2-648\mu^2-648+cos(\psi)[12\mu+18\mu^3]+sin(\psi)[-1728\mu+192\mu\nu_{\beta}^2]+cos(2\psi)[648\mu^2-72\mu^2\nu_{\beta}^2]+sin(2\psi)[9\mu^4+25\mu^2]+cos(3\psi)[-18\mu^3]+sin(4\psi)[-4.5\mu^4]}{80\mu^2-4.5\mu^4+576\nu_{\beta}^4-10368\nu_{\beta}^2+46737+cos(\psi)[192\mu\nu_{\beta}^2-1728\mu]+sin(\psi)[216\mu]+cos(2\psi)[-64\mu^2]+cos(4\psi)[4.5\mu^4]}\right\}\)

\(\beta_{3c}=q_{3s}\left\{\frac{27+75\mu^2+sin(\psi)[108\mu+54\mu^3]+cos(2\psi)[-75\mu^2]+cos(3\psi)[-18\mu^3]}{-4.5\mu^4-64\mu^2+576\nu_{\beta}^4-10368\nu_{\beta}^2+46737+cos(\psi)[1728\mu-192\mu\nu_{\beta}^2]+sin(\psi)[216\mu]+cos(2\psi)[-64\mu^2]+cos(4\psi)[4.5\mu^4]}\right\}\)