Chengjun,
If your goal is to only get the eigenvalues (which gives you the damped natural frequencies), I would suggest that you create a state space model and find the eigenvalues of the A matrix. In MMA:
In[27]:= ssm =
StateSpaceModel[{m1*x''[t] + c1*x'[t] + ks*x[t] +
klc*(x[t] - y[t]) == 0,
m2*y''[t] + c2*y'[t] + klc*(y[t] - x[t]) + ksb*y[t] ==
ff[t]}, {{x[t], 0}, {y[t], 0}}, {ff[t]}, {x[t], y[t]}, t];
In[26]:= {a, b, c, d} = Normal[ssm]
Out[26]= {{{0, 1, 0, 0}, {-((klc + ks)/m1), -(c1/m1), klc/m1, 0}, {0,
0, 0, 1}, {klc/m2,
0, -((klc + ksb)/m2), -(c2/m2)}}, {{0}, {0}, {0}, {1/m2}}, {{1, 0,
0, 0}, {0, 0, 1, 0}}, {{0}, {0}}}
In[19]:= eig = Eigenvalues[a]
Out[19]= {(1/(m1 m2^2))
Root[klc ks m1^3 m2^7 + klc ksb m1^3 m2^7 +
ks ksb m1^3 m2^7 + (c1 klc m1^2 m2^5 + c2 klc m1^2 m2^5 +
c2 ks m1^2 m2^5 + c1 ksb m1^2 m2^5) #1 + (c1 c2 m1 m2^3 +
klc m1^2 m2^3 + ksb m1^2 m2^3 + klc m1 m2^4 +
ks m1 m2^4) #1^2 + (c2 m1 m2 + c1 m2^2) #1^3 + #1^4 &, 1], (1/(
m1 m2^2))Root[
klc ks m1^3 m2^7 + klc ksb m1^3 m2^7 +
ks ksb m1^3 m2^7 + (c1 klc m1^2 m2^5 + c2 klc m1^2 m2^5 +
c2 ks m1^2 m2^5 + c1 ksb m1^2 m2^5) #1 + (c1 c2 m1 m2^3 +
klc m1^2 m2^3 + ksb m1^2 m2^3 + klc m1 m2^4 +
ks m1 m2^4) #1^2 + (c2 m1 m2 + c1 m2^2) #1^3 + #1^4 &, 2], (1/(
m1 m2^2))Root[
klc ks m1^3 m2^7 + klc ksb m1^3 m2^7 +
ks ksb m1^3 m2^7 + (c1 klc m1^2 m2^5 + c2 klc m1^2 m2^5 +
c2 ks m1^2 m2^5 + c1 ksb m1^2 m2^5) #1 + (c1 c2 m1 m2^3 +
klc m1^2 m2^3 + ksb m1^2 m2^3 + klc m1 m2^4 +
ks m1 m2^4) #1^2 + (c2 m1 m2 + c1 m2^2) #1^3 + #1^4 &, 3], (1/(
m1 m2^2))Root[
klc ks m1^3 m2^7 + klc ksb m1^3 m2^7 +
ks ksb m1^3 m2^7 + (c1 klc m1^2 m2^5 + c2 klc m1^2 m2^5 +
c2 ks m1^2 m2^5 + c1 ksb m1^2 m2^5) #1 + (c1 c2 m1 m2^3 +
klc m1^2 m2^3 + ksb m1^2 m2^3 + klc m1 m2^4 +
ks m1 m2^4) #1^2 + (c2 m1 m2 + c1 m2^2) #1^3 + #1^4 &, 4]}
Note that this gives you the four eigenvalues which correspond to the four roots of a polynomial (two complex pairs of poles). You can get the radical form with ToRadicals[] but I think it may be more useful to put numerical values in at this point and get frequencies as the radical equations are very long even if simplified.
Regards,
Neil