Civil Engineering Reference
In-Depth Information
cx
•
k
c
kx
m
x
(a)
(b)
(c)
Figure 10.13
(a) A spring-mass system with damping. (b) The schematic
representation of a dashpot piston. (c) A free-body diagram
of a mass with the damping force included.
Figure 10.13a depicts a simple harmonic oscillator to which has been added a
dashpot
.A
dashpot
is a damping device that utilizes a piston moving through a
viscous fluid to remove energy via shear stress in the fluid and associated heat gen-
eration. The piston typically has small holes to allow the fluid to pass through but
is otherwise sealed on its periphery, as schematically depicted in Figure 10.13b.
The force exerted by such a device is known to be directly proportional to the
velocity of the piston as
x
(10.123)
where
f
d
is the damping force,
c
is the damping coefficient of the device, and
x
is velocity of the mass assumed to be directly and rigidly connected to the piston
of the damper. The dynamic free-body diagram of Figure 10.13c represents a
situation at an arbitrary time with the system in motion. As in the undamped case
considered earlier, we assume that displacement is measured from the equilib-
rium position. Under the conditions stated, the equation of motion of the mass is
m
f
d
=−
c
˙
0
(10.124)
Owing to the form of Equation 10.124, the solution is assumed in exponential
form as
x
¨
+
c
x
˙
+
kx
=
Ce
st
(10.125)
where
C
and
s
are constants to be determined. Substitution of the assumed solu-
tion yields
x
(
t
)
=
(
ms
2
k
)
Ce
st
0
(10.126)
As we seek nontrivial solutions valid for all values of time, we conclude that
ms
2
+
cs
+
=
0
(10.127)
must hold if we are to obtain a general solution. Equation 10.127 is the charac-
teristic equation (also the frequency equation) for the damped single degree-of-
freedom system. From analyses of undamped vibration, we know that the natural
+
cs
+
k
=
