Civil Engineering Reference
In-Depth Information
For this case we have chosen to calculate the input point mobility and
Figure 2.11
shows
the result compared with the result setting the mass
m
2
infinitely large. Comparing with
the latter result, which we calculated earlier on and where the natural frequency is 100
Hz, we have now two resonances. The corresponding natural frequencies, calculated
using Equation
(2.42)
are 30.2 and 106.0 Hz, respectively. (It should be observed that the
natural frequencies are calculated for systems without damping).
0
Stiffness
0.1 kg
-20
10
5
N/m
1 kg
-40
10
6
N/m
10 kg
-60
10
7
N/m
Mass
-80
-100
10
20
50
100
200
500
1000
2000
5000
Frequency (Hz)
Figure 2.11
Input mobility of system shown in
Figure 2.10.
Dashed curve: m
1
=1kg. k
1
=4⋅10
5
N/m. c
1
=18 N⋅s/m.
m
2
=10kg. k
1
=4⋅10
5
N/m. c
1
=100 N⋅s/m. Solid curve: m
2
= ∞.
2.5.2
Vibration isolation. The efficiency of isolating systems
We shall again use the system depicted in
Figure 2.10
to give a more realistic illustration
of vibration isolation, and in particular using the efficiency
E
, defined in Equation
(2.37)
,
to characterize the effect of the elastic support. Elastic supports are normally introduced
to reduce vibrations from a machine transmitting vibrations to the foundation,
alternatively to reduce vibrations in the foundation being transmitted to e.g. sensitive
equipment. The former case is sketched in
Figure 2.12,
where we wish to isolate any
unbalanced forces in the upper structure (machine) from the lower structure (foundation
or floor). As a measure of success we shall compare the two situations marked a) and b).
In general, this may be a complicated task. We normally have to apply more than one
isolator, which implies several degrees of freedom. Furthermore, we may as a worst case
have to model the machine and foundation as continuous systems.
We shall, however, assume that each of the three structures may be characterized
by the mobility applicable for a movement in only one direction, here specifically in the
vertical direction. Using indices m, i and f indicating the mobility of structure 1
