Problems on Discontinuous Forcing Functions

1. A shock absorber-coil spring system for an imported automobile is designed to support 350 kg. The spring has a constant of 122150 kg/cm. The shock absorber exerts a damping force (in kg ^. cm²) that is numerically equal to 3500 times the instantaneous vertical velocity of the system (in cm/sec). Beginning in equilibrium at rest, the system is tested by running it over a bumpy road surface that exerts a force of 3500e^-5tcos 2t (in kg ^. cm²) upward on the system for 3$\pi$/2 sec. After 3$\pi$/2 sec, the system encounters a flat surface.

   (a)

   Express the force exerted by the two road surfaces in terms of unit step functions.

   (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. 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. 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 stress is described by the function 10 cos t. Assume that this stress acts on the girder, which begins in equilibrium at rest, for $\pi$/2 seconds when an earthquake strikes, changing the shear stress to 4e^-tcos t for $\pi$ seconds. The stress then returns to normal.

   (a)

   Express the external shear stress on the girder in terms of unit step functions.

   (b)

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

   (c)

   What is the crosswise displacement of the girder after $\pi$/4, $\pi$, and 2$\pi$ seconds?

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Problems on Discontinuous Forcing Functions

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