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lab06.desc - LAB#6 The Swaying Building Goal Determine a...

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LAB #6 The Swaying Building Goal : Determine a model of the swaying of a skyscraper; estimating parameters Required tools : Matlab routines pplane , ode45 , plot ; M-files; 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 P-Delta 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 “P-Delta” 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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