Dynamics cheat sheet

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