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Now, u = Re { Ue iωt} , u ′ = Re { iωUe iωt} , u ′′ = Re { −ω 2 Ue iωt} , hence the above becomes<br />
m Re { −ω 2 Ue iωt} + c Re { iωUe iωt} + k Re { Ue iωt} = −m Re { −ω 2 Ze iωt}<br />
U =<br />
mω 2<br />
−ω 2 m + iωc + k Z<br />
ω 2<br />
=<br />
−ω 2 + iω2ζω n + ωn<br />
2 Z<br />
r 2<br />
=<br />
(1 − r 2 ) + i2ζr Z<br />
( )<br />
r<br />
Hence |D (r, ζ)| = √ 2<br />
and arg (D) = − 2ζr<br />
(1−r 2 )+i2ζr tan−1 1−r 2 1<br />
Now if r is very large, which happens when ω n ≪ ω, then<br />
(1−r 2 )+i2ζr ⇒ 1 since r 2 is the dominant factor.<br />
−r 2<br />
r<br />
Therefore U = 2<br />
(1−r 2 )+i2ζr Z a now becomes U ≃ −Z a therefore measuring the relative displacement U gives<br />
linear estimate of the ground motion. However, this device requires that ω n be much smaller than ω, which<br />
means that m has to be massive. So this device is heavy compared to accelerometer.<br />
3.5 Summary of vibration transfer functions<br />
For good isolation of mass from ground motion, rule of thumb: Make damping low, and stiffness low (soft spring).<br />
system equation used to derive transfer function<br />
isolate base from force<br />
transmitted by machine<br />
isolate machine<br />
from motion of base<br />
accelerometer: Measure<br />
base acc. using relative<br />
displacement<br />
f tr (t) = f spring + f damper<br />
Use absolute mass position<br />
my ′′ + cy ′ + ky = cz ′ + kz<br />
|f tr(t)| ∣ max<br />
∣ ˆF<br />
∣<br />
|Y | max<br />
|Z|<br />
√<br />
= |D| 1 + (2ζr) 2<br />
√<br />
= |D| 1 + (2ζr) 2<br />
−1<br />
U =<br />
Use relative mass position (ωn 2 − ω 2 Z a ⇒ |D (r, ζ)|<br />
) + iω2ζω n<br />
−1<br />
mu ′′ + cu ′ + ku = −mz ′′ = √<br />
(ωn 2 − ω 2 ) 2 + (2ωζω n ) 2<br />
seismometer: Measure<br />
base motion using<br />
relative displacement<br />
Use relative mass position<br />
mu ′′ + cu ′ + ku = −mz ′′ U =<br />
r 2<br />
(1−r 2 )+i2ζr Z ⇒ |D (r, ζ)| = r 2 √<br />
(1−r 2 )+i2ζr<br />
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