Civil Engineering Reference
In-Depth Information
ζ > 1
shows that the system is more than critically damped giving a solution
⎡
⎤
2
2
−
ξω
t
ωζ
−
1
t
−
ωζ
−
1
t
xt
()
=
e
A
e
+
B
e
⎦
.
(2.17)
0
0
0
⎢
⎥
⎣
2.4.1.1
Free oscillations with hysteric damping
As stated in section
2.3.3.1,
using viscous damping is not appropriate in modelling a
system with elastic components such as rubber, plastics etc. The damping is better
described as hysteretic, normally using the
loss factor
η as a characteristic quantity. In
our simple mass-spring system we shall remove the viscous damper and introduce
damping through a complex spring stiffness
k
k
k
η
=
(1
+⋅
j
).
(2.18)
The loss factor η will always be much less than one. For metals one will find η
in the
the following
2
d
x
m
⋅
++⋅
k
(1
j
η
)
x
=
0.
(2.19)
2
d
t
We now assume that the solution of this equation will have the same form as Equation
(2.15)
but we shall express it using the complex form,
x
(
t
) =
A
⋅exp(jγ
t
). By insertion
into Equation
(2.19)
we easily solve for the exponent γ
k
k
η
η
⎛
⎞
⎛
⎠
γ
=
1j)
+ ⋅
η
≈
1j
+
=
ω
1j .
+
(2.20)
⎜
⎟
⎜
0
m
m
2
2
η
<<
1
⎝
⎠
⎝
Hence, we obtain
⎛
η
⎞
η
j
ω
1j
2
+
−
ω
t
⎜
⎟
o
0
⎝
⎠
x t
( )
≈
A
e
having a real solution:
xt
( )
=
A
e
2
⋅
cos(
ω
t
).
(2.21)
0
Compared with the solution
(2.15)
the damping ratio ζ is replaced by η/2 in the
exponential term. It should also be mentioned that other quantities are in use for
expressing the damping, such as the
logarithmic
decrement
δ and the
Q factor
. Assuming
that the damping is small the relationship between all these quantities is as follows
δ
1
2
ζ
===
η
Q
.
(2.22)
The
reverberation time T
is used in building acoustics to express the damping of sound
in rooms. However, the concept is useful when dealing with vibration as well. By
definition the reverberation time
T
is the time elapsed before the energy in an oscillating
system is reduced to 10
-6
of the initial value. As the energy is proportional to the square
of the vibration amplitude we may represent the definition by
