Civil Engineering Reference
In-Depth Information
j
ω
t
d()
xt
Fe
⋅
0
vt
()
=
=
j
ω
xt
()
=
.
(2.28)
d
t
Z
m
The quantity
Z
m
is the mechanical impedance, the driving point impedance at the point
where the force is applied. In this case, this will be the sum of the impedances for the
three elements because they all have the same velocity and the force is the sum of the
forces on the elements. Alternatively, expressing the impedance by its modulus and
phase as shown in Equation
(2.2)
, we write
2
k
⎛ ⎞
=+ −
2
Z
m
()
ω
c
m
ω
ω
and
⎜
⎟
⎝
⎠
(2.29)
k
m
ω
ω
−
tan
θω
(
)
=
.
c
It should be noted that the phase changes sign when the frequency ω of the applied force
is equal to the fundamental frequency (eigenfrequency) ω
0
of the system, i.e. when we
excite the system into
resonance
. One possible method of mapping the resonance
frequencies of a system is therefore by detecting the frequencies where one finds the
maximum relative phase changes.
We shall use an impedance diagram to depict the modulus given in Equation
(2.29)
.
Figure 2.7
shows an example using the following data for the system components: the
mass weight
m
is 1.0 kg, the spring stiffness
k
is 4⋅10
5
N/m and the damping coefficient
c
is equal to 18 kg/s. It should be fairly obvious why one normally divides the response to
the force into three main ranges, which are called
stiffness-controlled
,
damping-
controlled
and
mass-controlled
ranges.
100
80
10
7
N/m
10 kg
60
10
6
N/m
1 kg
40
10
5
N/m
0
.1 kg
20
Mass
Stiffness
0
10
20
50
100
200
500
1000
2000
5000
Frequency (Hz)
Figure 2.7
Modulus of the impedance of a simple mass-spring system. The mass weight is 1 kg, the spring
stiffness is 4⋅10
5
N/m and the damping coefficient is 18kg/s.
