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a=20
b=1
u=0.33
rn=n*Pi/a
\[Alpha]=Sqrt[rn*(rn+Sqrt[fx/d])]
\[Beta]=Sqrt[rn*(Sqrt[fx/d]-rn)]
md1={{u*rn^2-\[Alpha]^2,0},{\[Alpha]^3,\[Beta]^3-(2-u)*rn^2*\[Beta]}}
md3={{u*rn^2-\[Alpha]^2,0},{\[Alpha]^3,\[Beta]^3-(2-u)*rn^2*\[Beta]}}
md2={{-1,0,-1,0},{\[Alpha],-\[Beta],-\[Alpha],\[Beta]},{+\[Alpha]^2-u*rn^2,0,-\[Alpha]^2+u*rn^2,0},{rn^2-\[Alpha]^3,-rn^2*\[Beta]-\[Beta]^3,-\[Alpha]^3+rn^2*\[Alpha],-rn^2*\[Beta]-\[Beta]^3}}
ma1={{\[Alpha]^2-u*rn^2,-\[Beta]^2-u*rn^2},{\[Alpha]^3-(2-u)*\[Alpha]*rn^2,0}}
ma3={{\[Alpha]^2-u*rn^2,-\[Beta]^2-u*rn^2},{\[Alpha]^3-(2-u)*\[Alpha]*rn^2,0}}
ma2={{1,1,1,1},{\[Alpha],0,-\[Alpha],0},{-\[Alpha]^2+u*rn^2,\[Beta]^2+u*rn^2,\[Alpha]^2-u*rn^2,-\[Beta]^2-u*rn^2},{rn^2*\[Alpha]-\[Alpha]^3,0,rn^2*\[Alpha]-\[Alpha]^3,0}}
md=ArrayFlatten[{{md1,0,0},{0,md2,0},{0,0,md3}}]
ma=ArrayFlatten[{{ma1,0,0},{0,ma2,0},{0,0,ma3}}]
mdni=Inverse[md]
ms=mdni.ma
mp1={{0,0,-Exp[-\[Alpha]*b],0},{0,Cot[\[Beta]*b],0,-1/Sin[\[Beta]*b]},{-Exp[-\[Alpha]*b],0,0,0},{0,-1/Sin[\[Beta]*b],0,Cot[\[Beta]*b]}}
mp=ArrayFlatten[{{mp1,0},{0,mp1}}]
mr=ms.mp
mf=IdentityMatrix[8]-mr
Solve[Det[mf]==0,fx]
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