newbie here just trying to learn the Mathematica syntax.

I have a list of equations, some of them definitions, others relationships, as follows:

Symbolize[ParsedBoxWrapper[SubscriptBox["_", "_"]]];
domain = (Subscript[R, m] | m | s | S | ω | L | α | ρ | c) ∈ Reals && (Subscript[Z, md] | Subscript[Z, m0] | p[x, t] | u[x, t]) ∈ Complexes;
driverMotion = (r + I (ω m - s/ω) + (S p[0, t])/u[0, t]) u[0, t] == f;
zmdDefinition = Subscript[Z, md] == r + I (ω m - s/ω);
zm0Definition = Subscript[Z, m0] == (S p[0, t])/u[0, t];
driverResonanceCondition = Im[Subscript[Z, md] + Subscript[Z, m0]] == 0;
zm0DampedRigidTube = Subscript[Z, m0]/(ρ c S) == (α L - I Cos[k L] Sin[k L])/(Sin[k L]^2 + (α L)^2 Cos[k L]^2);

I've organised it so that each variable (e.g driverMotion) contains an equality. Now I want to combine them and recast the driverResonanceCondition in terms of w, m, s, S, p, c, k and L. (i.e I want to get rid of the terms Zmd, Zm0, p(x,t) and u(x,t)). e.g the textbook I am following gives the condition in the following form:

$$-\frac{c \rho S \sin (k L) \cos (k L)}{(\alpha L)^2 \cos ^2(k L)+\sin ^2(k L)}+m \omega -\frac{s}{\omega }=0$$

How do I tell Mathematica to do this? It's not a hard rearrangementand I can do it quickly on paper. I'm just not sure of the correct Mathematica commands. Have tried various combinations of Solve, Reduce, ComplexExpand, Simplify etc but no luck yet.


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