Building Acoustics

46 Building acoustics

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

(1 j ) j ()1 j.

k

F kxη

ω

′=+⋅ =− +⋅ηv (2.34)

Hence

(^2)
0
(1 j )
1j
.
(1 j ) 1j
k
F F
k
m
η
ω η
ηω ω
ω η
ω

+⋅

′ +⋅

=⋅=

+⋅ − ⎛⎞

−+⋅⎜⎟

⎝⎠

⋅F (2.35)

The transmissibility Th will then be given by

1

2

2

h 2 2

2

0

1

.

1

T

η

ω

η

ω

⎡⎤

⎢⎥

⎢⎥

⎢+

=

⎢⎛⎞

⎢⎥⎛⎞

⎢⎥⎜⎟−+⎜⎟

⎢⎥⎜⎟⎝⎠

⎣⎦⎝⎠

⎥

⎥

(2.36)

0.1 0.2 0.5 1 2 5 10

f/f 0

0.1

1.0

10.0

100.0

0.2

0.5

2.0

5.0

20.0

50.0

ility

η=0.02

η=0.15

η=0.3

η=0.6

ib

s

is

ansm

T

r

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.