Civil Engineering Reference
In-Depth Information
ma cv k x
x
⋅+⋅+⋅=
++⋅ =
0 r
(2.12)
2
d
d
x
mc
k
x
0,
2
d
t
d
t
where
x, v
and
a
are displacement, velocity and acceleration, respectively. The damping
coefficient
c
will be the prime factor in the solution of the differential equation
describing the transient motion when the system is left to vibrate freely. A suitable
variable characterising the damping is the
damping ratio
ζ. This quantity gives the
damping coefficient
c
relative to the
critical damping coefficient c
critical
of the system,
c
c
c
ζ
=
=
=
.
(2.13)
c
2
m
ωπ
4
m
f
critical
0
0
f
0
is the fundamental frequency (
eigenfrequency
), the natural frequency of oscillation for
the undamped system, i.e. when the damping coefficient
c
is equal to zero. This
frequency is given by
ω
ππ
1
k
0
f
==
.
(2.14)
0
22
m
We shall in the following use both the frequency
f
and the angular frequency ω, choosing
the one most suitable in each case, but without changing the name of quantities such as
natural frequency etc. Inserting the damping ratio into Equation
(2.12)
we may solve the
equation for the following three cases:
ζ < 1
will give us a damped oscillatory motion where the displacement
x
may be
expressed as:
[
]
−
ζω
t
xt
()
=
e
A
sin(
ω
t
)
+
B
cos(
ω
t
)
or
0
d
d
(2.15)
−
ζω
t
xt
()
=⋅
C
e
⋅
cos(
ωθ
t
+
),
0
d
where
B
2
2
2
ωω ζ
=
1
−
,
CAB
=
+
and
tg
θ
=
.
d
0
A
The coefficients
A
and
B
, or alternatively
C
and θ, must be determined from the initial
conditions. In section 1.4.2.2, we used this transient motion for illustrating the
calculation of the Fourier transform.
ζ =1
indicates that the system is critically damped. The movement is no longer
oscillatory, the system is returning to its stable position in a minimum time. (Make a
comparison with the spring system of e.g. a car.) The solution is now
−
ξω
t
xt
()
=+⋅ ⋅
(
A B t
) e
.
(2.16)
0
