Two-axle auto from 2nd Edition of Edwards-Penney, 7.4-24

100  x''     = -(4000) x + (2000)(2) theta
1000 theta'' =  (2000)(2) x - (2000)(36+16) theta

The two natural frequencies of oscillation omega1, omega2
are those frequencies which appear in the atoms constructed
from the roots of the characteristic equation 

                         det(A - r^2 I)=0.
The symbols are defined by

   A = M^{-1} K
   K = Hooke's matrix == 
       matrix([[-4000,2000*2],[2000*2,-2000*(6^2+4^2)]]);
   M = mass matrix == matrix([[100,0],[0,1000]]);

There are four values for r, because the characteristic equation
is a quartic. The values are in complex conjugate pairs. Euler's
theorem implies there are 4 atoms, which are base trig atoms, sine
and cosine. The frequencies are 6.13, 10.31, approximately.

Critical speeds v1, v2 are found by solving the equations

        omega1 = 2 Pi v1/40
        omega2 = 2 Pi v2/40


Maple Answer Check:

 K:= matrix([[-4000,2000*2],[2000*2,-2000*(6^2+4^2)]]);
 M:=matrix([[100,0],[0,1000]]);
 with(linalg):
 A:=evalm(inverse(M)&*K);
 solve(det(A-r^2*diag(1,1))=0,r);