How do I solve this for w? I know that phi1 and phi2 cannot be solved for but shouldn't 'w' be solved?

This is the error I see Reduce::inex: Reduce was unable to solve the system with inexact coefficients or the system obtained by direct rationalization of inexact numbers present in the system. Since many of the methods used by Reduce require exact input, providing Reduce with an exact version of the system may help.

Trans = ( {{1, 8},{56-7*w^2, 9}});
Reduce[( {{phi1},{0}} ) == Trans.( {{phi2},{0}} ) && phi2 != 0, w]

How do I solve this for w? I know that phi1 and phi2 cannot be solved for but shouldn't 'w' be solved?

This is the error I see Reduce::inex: Reduce was unable to solve the system with inexact coefficients or the system obtained by direct rationalization of inexact numbers present in the system. Since many of the methods used by Reduce require exact input, providing Reduce with an exact version of the system may help.

    ClearAll["Global`*"]
G = 0.8*10^11;
d = 0.1;
l = 0.6;
J = Pi*d^4/32;
kt = G*J/l;

I1 = 22.6;
I2 = 5.66;
P1 = ( {
    {1, 0},
    {-I1*w^2, 1}
   } );
P2 = ( {
    {1, 0},
    {-I2*w^2, 1}
   } );
F2 = ( {
    {1, 1/kt},
    {0, 1}
   } );
U1 = P1;
U2 = P2.F2;
Trans = Rationalize[U2.U1] // MatrixForm
T1 = 0;
T2 = 0;
Simplify[Reduce[({{phi1}, {0}}) == Trans.({{phi2}, {0}}), w], 
 Assumptions -> phi2 != 0]