The Building with Damping Devices. In addition to flexural restoring forces, all buildings possess intrinsic internal friction, due to structural and nonstructural elements, that causes the amplitude of vibrations to decay as these elements absorb vibrational energy. One of several techniques employed in modem earthquake-resistant design uses added damping devices such as shock absorbers between adjacent floors to artificially increase the internal damping of a building and improve its resistance to earthquake-induced motion (see Figure
The effects of intrinsic and artificial damping are accounted for by including damping forces of the form
And
On the right-hand sides of Eqs .(1),(2), and(3), respectively.
(a) Show that including damping devices between each pair of adjacent floors changes the system
Where
(b) Assume an input to Eq. (i)in part (a) of the form
Thus the gain function for the frequency response of the
(c) Numerically approximate the solution of Eq. (i)in part (a) subject to the initial conditions
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