Civil Engineering Reference
In-Depth Information
2.5.1
Modelling systems using lumped elements
The modelling of a system as a collection of lumped elements (masses, springs and
dampers) may be applied in practice both to mechanical and acoustical systems. If the
task is to calculate the motion of each element in the system we have to find all the
natural frequencies of the system as well as the
modal vectors
(
eigenvectors
). The latter
defines the natural forms of vibration, the
natural modes
. The important point is that any
possible type of motion, resulting from a given force excitation, may be described by a
linear combination of these modal vectors. This is the background for calculating the
response using so-called
modal analysis
; see section 2.5.3.1 below.
In many cases, however, we are not interested in performing a detailed analysis of
all parts of the system. The task may only be to calculate the impedance (or mobility) at a
place where the force (or moment) is attacking, making us able to control the mechanical
power transmitted to the system. Such an analysis may be performed in a simple way
dealing with systems having a small number of elements. There exists, however, several
commercial computer programs that will be helpful if one is able to model the actual
system using an analogue electrical system.
F
m
1
k
1
c
1
m
2
k
2
c
2
Figure 2.10
System with two masses, springs and dampers. On the calculation of mobility, see text.
We shall present an example using this kind of modelling applying the system
shown in
Figure 2.10
, which is a combination of two simple mass-spring systems. The
system with elements using suffix 1 has identical data as used for calculating the
impedance shown in
Figure 2.7.
In the other system, suffix 2, the mass is increased by a
factor of 10 and the damping is also increased. The total or combined system has now
two natural frequencies. For the frequencies in this case, and for several other simple
systems, one may find analytical expressions in the literature, e.g. Blevins (1979). These
are
f
i
(
i
= 1, 2) given by
1
⎧
1
⎫
2
⎡
2
⎤
⎪
2
⎪
1
kkk
⎛
kkk
⎞
kk
⎪
⎪
1
1
2
∓
⎢
1
1
2
1
2
⎥
.
(2.42)
f
=
+
+
+
+
−
4
⎨
⎬
⎜
⎟
i
3
mm m
mm m
mm
⎢
⎥
⎝
⎠
⎪
⎪
1
2
2
1
2
2
1
2
⎣
⎦
2
2
π
⎪
⎪
⎩
⎭
