Excitation and response of dynamic systems - Building Acoustics - page 46

In the above model viscous damping was assumed, which will give a
transmissibility dependent on damping also at the higher frequencies. As an alternative
model we shall assume that damping is hysteretic (see section 2.4.1.1) . Instead of
Equation (2.30) , we now have

k

( )

′ =+⋅

F k (1 j η ) x =− +⋅ η j 1 j .

(2.34)

ω

Hence

k

(1 +⋅ j η )

1j +⋅ η

ω

′ =

F ⋅ = F ⋅ .

(2.35)

k

2

⎛⎞

ω

(1 +⋅ − j ηω ) m

1 − + ⋅ j η

ω ⎜⎟

⎝⎠

ω

0

The transmissibility T h will then be given by

1
2

⎡ ⎤

⎢ ⎥

⎢ ⎥

2

1 + η

= ⎢ ⎛ ⎥

⎥

T .

(2.36)

h

2

2

⎞

⎛⎞

⎢ ω ⎥

⎜ ⎟

2

1 − + η

⎜⎟

⎢ ⎥

ω

⎜ ⎟

⎝⎠

⎢

0

⎥

⎝ ⎠

⎣ ⎦

100.0

50.0

20.0

η=0.02
η=0.15
η=0.3
η=0.6

10.0

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 2.9 Transmissibility, the ratio of transmitted force (to the foundation) and the applied force, of a simple
mass-spring system with hysteretic damping. The loss factor η is indicated on the curves.