I presume that you wish to eliminate some variables in the last expression in terms of variables defined in the other expressions. Let me begin with two pieces of advice:
- Do not use subscripted variables. They may look nice, but they can cause problems.
- Simplification is in the eye of the beholder. Mathematica's idea of simplification, based on
LeafCount, may not agree with the user's.
On this basis, rewrite the expressions as equations, named for convenience
eq1 = ωh^2 == kh/m;
eq3 = μ == m/(π*ρ*b^2);
eq5 = xα == Sα/(m*b);
eq6 = E1 == (m + π*ρ*b^2)*z + kh*w + (Sα - π*ρ*b^3*a)*n + π*ρ*b^2*V*o +
2*π*ρ*b*V*c (y + V*t + b*(1/2 - a)*o);
eliminate what I presume are the unwanted variables, and simplify the result.
FullSimplify[Eliminate[{eq1, eq3, eq5, eq6}, {ρ, kh, Sα}], b != 0 && m != 0]
(* E1 μ == m ((1 + c) o V - a (b n + 2 c o V) + (2 c V (t V + y))/b +
z + μ (b n xα + z + w ωh^2)) *)
If this is not completely simplified in the way you have in mind, you may need to play around with the code a bit, but this should give you some idea of how to proceed. Good luck.