Simplifying algebraically

Question

Is there a way to derive an simplified expression from other equations. Or create an alternate form such that the main equation takes other equations and manipulates the main function. Here is what I have.

Subscript[ω, h] = Sqrt[Subscript[k, h]/m];
Subscript[ω, α] = Sqrt[Subscript[k, α]/Subscript[
  I, α]];
μ = m/(π*ρ*b^2);
Subscript[r, a] = Sqrt[Subscript[I, α]/(m*b^2)];
Subscript[x, α] = Subscript[S, α]/(m*b);
Subscript[E, 1] = (m + π*ρ*b^2)*z + 
Subscript[k, h]*w + (Subscript[S, α] - 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)

Can you describe an example simplification of E1 that you would like to see in terms of your five previous equations? If so then sometimes Simplify[complicatedexpr, var==someexpr] will sometimes be able to find someexpr in complicatedexpr and replace that with var, for example. — Bill, May 16 '20 at 02:38
A 2D aerofoil utilizing the Theodrosen Theory to find the static divergence speed. The Simplify command did not work. I've done it by hand but there could be further simplification and I have more equations which need to be simplified that are much longer and since — Aeroelasticity, May 16 '20 at 02:45
If you can provide a much more concrete specific example, something like: Given this equation in Mathematica notation it should be able to make this specific simplification by replacing this with that, then perhaps someone can ask for more details and possibly give you a method of showing you how to get where you want to go. — Bill, May 16 '20 at 06:00
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@Bill Your suggestion, embodied as FullSimplify[x + y, z == x + y], works. But, FullSimplify[x + y, a == x + y] does not work. Weird! — bbgodfrey, May 16 '20 at 20:51

bbgodfrey · Answer 1 · 2020-05-16T14:02:28.010

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.

@Aeroelasticity More advice about Subscript: https://mathematica.stackexchange.com/a/18395/4999, https://mathematica.stackexchange.com/a/20213/4999 — Michael E2, May 16 '20 at 14:59

Simplifying algebraically

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