F(t) | Guess |
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roots
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real and distinct | ![]() |
double real | ![]() |
complex
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modal:
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Find eigenvalus of |
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![]() ![]() Rayleigh
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speed is
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phase roots
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Unstable, repelling |
phase roots
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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![]() |
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convert:
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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