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The roots of the characteristic equation for u ′′ + 2ξωu ′ + ω 2 u = 0 are given in this table<br />
ξ < 1<br />
roots<br />
{ √ √ }<br />
−ξω + jω n 1 − ξ 2 , −ξω − iω n 1 − ξ 2<br />
time constant τ<br />
1<br />
ξω<br />
1<br />
ξ = 1 {−ω, −ω}<br />
ω<br />
{ √ √ }<br />
ξ > 1 −ω n ξ + ω n ξ 2 − 1, −ω n ξ − ω n ξ 2 1<br />
− 1<br />
√ , 1√ (which to use? the bigger?)<br />
ω nξ−ω n ξ 2 −1 ω nξ+ω n ξ 2 −1<br />
32