Building Acoustics

48 Building acoustics

()

11

1j

j

M

Z k

m

η

ω

ω

==

⎛⎞+⋅

⎜⎟−

⎝⎠

(2.39)

and we must solve an integral, which after some rearranging may be written

22

2

(^22)
(^02)
0

4()

d.

1

vfGfF

k

f

f

π

η

∞

⋅

=

⎛⎞⎛⎞

+−⎜⎟⎜⎟

⎜⎟⎝⎠

⎝⎠

 ∫ 2 ⋅f (2.40)

The result is certainly wholly dependent on the force power spectrum GF(f). For
simplicity we shall assume that the spectrum is white, in any case constant in a frequency
range where the response is high. Setting GF(f) equal to G 0 we may show that

2 0

3

1

4

G

v

η km

=⋅

⋅

 (2.41)

The importance of having a high loss factor η is certainly expected as we then get a low
response at resonance. The way the mass and stiffness influence the result are, however,
not easy to guess.

2.5 Systems with several degrees of freedom

We have in the derivations above used a very simple mechanical system having one
degree of freedom to illustrate the general term transfer function and its special forms
impedance and mobility. The system has just one natural frequency and one natural way
of motion, the latter usually called a mode. An extensive coverage of mechanical systems
having several degrees of freedom is outside the scope of this book. For the interested
reader there are a number of textbooks on the subject to consult, e.g. Meirovitch (1997).
Here we shall just give a short general description of systems described either as discrete
or continuous. In the former case, we are able to model the system composed of
concentrated or lumped elements, such as masses, springs and dampers. This is opposed
to continuous systems where wave motion must be taken into consideration; where the
wavelength is becoming comparable with or less than the dimensions of the system.
We shall, in this main section, give some examples of models using lumped
elements, to present a more realistic treatment of the theme vibration isolation than that
given in section 2.4.3. Examples on continuous systems will be presented in the
following chapters.