That's basically it give or take a sign. What I am doing is two reference frame transformations. Quaternions and Trig functions, but single axis as I am only testing out a workflow right now on a very simple problem. The problem does not originate from a series, but the expression I'm trying to simplify is a term in a Taylor series, so it's not completely unrelated.

Mathematica doesn't automatically eliminate Abs[w0]/Sqrt[w0^2] from an expression

The elimination can be done if w0 is a real number, but not if it is complex. Tell MMA that w0 is real and it will do the elimination.

Simplify[Abs[w0]/Sqrt[w0^2], Assumptions -> Element[w0, Reals]]
(* 1 *)

We need to be able to run your code to see what you see and trace through the computation. Since you don't supply Fc, vb, and several other definitions we can't see what's going on. All we can do is guess blindly. If you really want help, you need to provide more information.

Wild guess: Are you after this?

Normal[Series[{
   {-((U^2 w0 (CD w0 + CL\[Alpha] U ArcSin[Abs[w0]/U]))/(U^2 + w0^2)^(3/2))},
   {0},
   {(U^2 w0 (CD U - CL\[Alpha] w0 ArcSin[Abs[w0]/U]))/(U^2 + w0^2)^(3/2)}},
  w0 -> 0, Assumptions -> w0 > 0 && U > 0]]

(*  {{-(((CD + CL\[Alpha]) w0^2)/U)}, {0}, {CD w0}}  *)

It's slightly different than the result your substitutions imply, so my guess may be wrong. However, asymptotic expansions are common operations, and you seem to be treating w0 as if it were small and approaching zero. I thought it worth suggesting.

In my experience, algebraic manipulation via substitution rules is tricky and bug-prone. Rules of the form variable -> expr that replace all instances of the variable by the same expr are generally safe. But I find it's more often better to ask Mathematica to calculate the mathematical operation I want. There are exceptions, though.

You need to provide the expression you're applying ReplaceAll to.