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Let z = Re { Ze iωt} , z ′ = Re { iωZe iωt} and let y = Re { Y e iωt} , y ′ = Re { iωY e iωt} , y ′′ = Re { −ω 2 Y e iωt} ,<br />
hence the above becomes<br />
Hence |D (r, ζ)| =<br />
m Re { −ω 2 Y e iωt} + c Re { iωY e iωt} + k Re { Y e iωt} = c Re { iωZe iωt} + k Re { Ze iωt}<br />
√<br />
1+(2ζr) 2<br />
√<br />
(1−r 2 ) 2 +(2ζr) 2<br />
ciω + k<br />
Y =<br />
−ω 2 m + ciω + k Z<br />
i2ζω n mω + k<br />
=<br />
−ω 2 m + i2ζω n mω + k Z<br />
i2ζω n ω + ωn<br />
2 =<br />
−ω 2 + i2ζω n ω + ωn<br />
2 Z<br />
i2ζr + 1<br />
=<br />
(1 − r 2 ) + i2ζr Z<br />
( )<br />
and arg (D) = tan−1 (2ζr) − tan −1 2ζr<br />
1−r 2<br />
where r = ω ω n<br />
√<br />
Hence for good vibration isolation we need |Y | max<br />
|Z|<br />
to be small. i.e. |D| 1 + (2ζr) 2 to be small. This is the<br />
same TR as for force isolation above.<br />
For small |D|, we need small ζ and small k (the small k is to make r > √ 2) see plot<br />
In Matlab, the above can be plotted using<br />
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