F(t)  Guess 
or  
roots


real and distinct  
double real  
complex  
modal: 
Find eigenvalus of 
or 
,
Rayleigh
where

where 
speed is  
phase roots and  Unstable, repelling 
phase roots and  stable, attracting 
both real, one >0 and one <0  unstable saddle point 
equal roots and >0  unstable, degenrate 
equal roots and <0  stable, degenrate 
complex, real part>0  unstabe, spiral out 
complex, real part<0  stable, spiral in 
pure complex conjugrates  marginaly stable, cirlce 
time betwen 
convert: 
, assume Plug in, rewrite as , find eigens of sys,each find 
, use 
impulse 
2 equations of motions for unbalanced: and , where eq for is
, guess , we obtain perturbation: hence . Sub in ODE, generate 2 ODE's and solve for and use result to find watch for IC and resonanse. For system ID, set up and from the spectrum, find