Problems on Periodic or Impulse Forcing Functions

1. We wish to investigate the shear stress on a structural girder of a tall building that is being designed to withstand the effects of earthquakes. (We dealt with this situation in an earlier assignment.) The girder behaves as a horizontal spring with mass m = 1, damping constant c = 5, and spring constant k = 6. The girder is normally subject to a time dependent external shear stress force induced by the swaying of the building. Assume that this girder is in equilibrium at rest when an earthquake strikes, exerting an exponentially decaying periodic shear stress of 10e^-t, which repeats every 2 seconds.

   (a)

   Using Laplace transforms, find the equation for the horizontal displacement of the girder over time.

   (b)

   What is the crosswise displacement of the girder after 1 sec? After 3 sec?¹

2. Do the Exercises from pages 2 and 5 of the Dirac Delta ``Function'' handout by Dr. Bryan.

3. A shock absorber-coil spring system for an imported automobile is designed to support 350 kg. The spring has a constant of 140000 kg/cm. The shock absorber exerts a damping force (in kg ^. cm/sec²) that is numerically equal to 3500 times the instantaneous vertical velocity of the system (in cm/sec). The system is in equilibrium at (relative) rest when it hits a pothole that exerts an impulse force of 5250 kg ^. cm/sec upward on the system.

   (a)

   Express the force exerted by the pothole in terms of the Dirac delta function.

   (b)

   Model this system with an initial value problem and solve it.

   (c)

   Graph the resulting equation for the first 3 sec of motion.

   (d)

   What is the maximum displacement from equilibrium that the system experiences?²

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Problems on Periodic or Impulse Forcing Functions

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Footnotes

... sec?¹

answers are $\approx$ 0.7350; $\approx$ 0.8604

... experiences?²

answer is $\approx$ 0.5336 cm