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