Excitation and response of dynamic systems - Building Acoustics - page 45

ω
ζ ω 1j2 +⋅

0

F ′ = ⋅ F .

(2.32)

2

ω ω

⎛⎞

1 − + ⋅ j 2 ζ

⎜ ⎝⎠

ω ω

0

If we only are interested in the ratio of the force amplitude transmitted and the exciting
force amplitude and not their phase relationship, we calculate the modulus of the ratio of
the complex quantities, i.e. the transmissibility T as

1
2

⎡ ⎤

2

⎢ ⎥

⎛ ⎞

ω
ζ ω

12 +

⎢ ⎜ ⎟ ⎥

′

′ F

F

⎝ ⎠

⎢ ⎥

0

T == .

(2.33)

⎢ ⎥

2

FF

2 2

⎛ ⎞

⎛⎞ ⎛ ⎞

⎢ ω ω ⎥

⎜ ⎟

1 − + 2 ζ

⎜⎟ ⎜ ⎟

⎢ ⎥

⎜ ω ⎟ ω

⎝⎠ ⎝ ⎠

⎢

0 0

⎥

⎝ ⎠

⎣ ⎦

Figure 2.8 shows the transmissibility for four values of the damping ratio ζ. At low
frequencies we find the force transmitted is the same as the exciting force; there will be
no vibration isolation. The force amplification at resonance will depend on the damping
and to obtain any isolation at all, the frequency must be higher than 2 1/2 ⋅ f 0 . When
designing for vibration isolation one therefore must ensure that the frequencies of the
unbalanced forces are relatively high compared to the natural frequencies of the system.

100.0

50.0

20.0

10.0 ζ=0.01

ζ=0.075
ζ=0.15
ζ=0.3

5.0

2.0

1.0

0.5

0.2

0.1

0.1

0.2 0.5

1

2 5

10

f / f 0

Figure 2.8 Transmissibility, the ratio of transmitted force (to the foundation) and the applied force, of a simple
mass-spring system with viscous damping. The damping ratio ζ for the curves is given in the legend.