Dynamics cheat sheet

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