Civil Engineering Reference
In-Depth Information
1
1
M
==
⎛
(2.39)
(
)
Z
⎞
k
1j
+⋅
η
j
ω
m
−
⎜
⎟
ω
⎝
⎠
and we must solve an integral, which after some rearranging may be written
∞
2
2
4
π
f
⋅
Gf
(
)
2
∫
F
v
=
⋅
d.
f
(2.40)
2
2
k
⎛
2
⎞
⎛⎞
f
f
0
2
⎜
⎟
η
+
−
1
⎜⎟
⎜
⎟
⎝⎠
0
⎝
⎠
The result is certainly wholly dependent on the force power spectrum
G
F
(
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
G
F
(
f
) equal to
G
0
we may show that
G
1
2
0
v
=⋅
.
(2.41)
4
3
km
⋅
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 topic. 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.
