LAB #6
The Swaying Building
Goal
: Determine a model of the swaying of a skyscraper; estimating parameters
Required tools
:
Matlab
routines
pplane
,
ode45
,
plot
; Mfiles; systems of differ
ential equations.
Discussion
Modern skyscrapers are built to be ﬂexible. In strong gusts of wind or in earthquakes
these buildings tend to sway back and forth to absorb the shocks. Oscillations with an
amplitude on the order of 5 to 10 seconds are common. You will analyse two different
differential equations that model the swaying of a building.
Let
y
(
t
) be a measure of how far the building is bent the displacement (in meters)
of the top of the building with
y
= 0 corresponding to the perfectly vertical position.
When
y
= 0, the building is bent and the structure applies a strong restoring force back
toward the vertical (see Figure 1).
y(t)
Figure 1
P
∆
Figure 2
This is reminiscent of a harmonic oscillator and therefore a very crude approximation
of the motion of a swaying building is the damped harmonic oscillator equation:
d
2
y
dt
2
+
p
dy
dt
+
qy
= 0
.
Here the constants
p
and
q
are chosen to reﬂect the characteristics of the particular
building being studied.
The PDelta Effect
.
Modeling the swaying building with a harmonic oscillator
equation is extremely crude.
We do not claim that the forces present in a swaying
building are identical to those of a spring. The harmonic oscillator is only a first ap
proximation of a complicated physical system. To extend the usefulness of the model,
we must consider other factors that govern the motion of the swaying building.
One
aspect of the model of the swaying building that we have not yet included is the effect
due to gravity. When the building undergoes small oscillations, gravity does not play a
very important role. However, if the oscillations become large enough, then gravity can
1
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have a significant effect. When
y
(
t
) is at its maximum value, a portion of the building
is not directly above any other part of the building (see Figure 2).
Therefore, gravity pulls downward on this portion of the building and this force tends
to bend the building farther.
This is called the “PDelta” effect (∆ is the overhang
distance and
P
is the force of gravity).
To include this effect in our model in a way
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 Spring '08
 CHO
 matlab, Equations, Velocity, Swaying Building

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