Civil Engineering Reference
In-Depth Information
Equation 10.5 shows that the variation of displacement of the mass as a func-
tion of time is periodic. Using basic trigonometric identities, Equation 10.5 can
be equivalently expressed as
)
(10.6)
where the constants
A
and
B
have been replaced by constants of integration
C
and
. Per Equation 10.6, the mass oscillates sinusoidally at
circular frequency
and
with constant
amplitude C
. Phase angle
is indicative of position at time 0 since
x
(0)
x
(
t
)
=
C
sin(
t
+
. Also, note that, since
x
(
t
)
is measured about the equilibrium
position, the oscillation occurs about that position. The circular frequency is
=
C
sin
k
m
=
rad/sec
(10.7)
and is a constant value determined by the physical characteristics of the system.
In this simple case, the
natural circular frequency,
as it is often called, depends
on the spring constant and mass only. Therefore, if the mass is displaced from the
equilibrium position and released, the oscillatory motion occurs at a constant
frequency determined by the physical parameters of the system. In the case
described, the oscillatory motion is described as
free vibration,
since the system
is free of all external forces excepting gravitational attraction.
Next, we consider the simple harmonic oscillator in the finite element con-
text. From Chapter 2, the stiffness matrix of the spring is
k
1
k
(
e
)
=
−
1
(10.8)
−
11
and the equilibrium equations for the element are
k
1
u
1
u
2
f
1
f
2
−
1
=
(10.9)
−
11
f
1
1
which is identical to Equation 2.4. However, the spring element is not in static
equilibrium, so we must examine the nodal forces in detail.
Figure 10.2 shows free-body diagrams of the spring element and mass,
respectively. The free-body diagrams depict snapshots in time when the system
is in motion and, hence, are dynamic free-body diagrams. As the mass of the
spring is considered negligible, Equation 10.9 is valid for the spring element. For
the mass, we have
u
1
f
2
2
u
2
f
2
mg
(b)
(a)
F
x
m
d
2
u
2
d
t
2
=
ma
x
=
=
mg
−
f
2
(10.10)
Figure 10.2
Free-
body diagrams of (a) a
spring and (b) a mass,
when treated as parts
of a finite element
system.
from which the force on node 2 is
m
d
2
u
2
d
t
2
f
2
=
mg
−
(10.11)
