Dynamics cheat sheet

3 Transfer functions for different vibration systems
y
c
m
k
z
u y z
3.1 Force transmissibility
Let steady state
x ss = Re
{ ˆF
k D (r, ζ) eiϖt }
Then
f tr (t) = f spring + f damper
= kx + cx ′
{
= Re k ˆF
} {
k D (r, ζ) eiϖt + Re ciϖ ˆF
}
k D (r, ζ) eiϖt
{(
= Re ˆF + ciϖ ˆF
) }
D (r, ζ) e iϖt
k
Hence
|f tr (t)| max
= ∣ ˆF
√
∣ √
∣ |D| 1 + c 2 ϖ2 ∣∣
k 2 = ˆF ∣ |D| 1 + (2ζr) 2
So TR or force transmissibility is
T R = |f √
tr (t)| ∣ max
∣ ˆF
= |D| 1 + (2ζr) 2
∣
If r > √ 2 then we want small ζ to reduce force transmitted to base. For r < √ 2, it is the other way round.
3.2 vibration isolation
We need transfer function between y and z.
Equation of motion
my ′′ = −c ( y ′ − z ′) − k (y − z)
my ′′ + cy ′ + ky = cz ′ + kz
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