Chapter 6.P1, Problem 2P

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 $6.P.2$).

The effects of intrinsic and artificial damping are accounted for by including damping forces of the form

$F_d{}^{\left(j\right)}=-\gamma \left(x_j^{\text{'}}-x_{j-1}^{\text{'}}\right)-\gamma \left(x_j^{\text{'}}-x_{j+1}^{\text{'}}\right),j=2,\mathrm{...},n-1$,

$F_d^{\left(j\right)}=-\gamma \left(x_1^{\text{'}}-f^{\text{'}}\right)-\gamma \left(x_1^{\text{'}}-x_2^{\text{'}}\right)$,

And

$F_d^{\left(1\right)}=-\gamma \left(x_n^{\text{'}}-x_{n-1}^{\text{'}}\right)$

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 $\left(i\right)$ in Problem $1$ to

$x"+2\delta Kx\text{'}+{\omega }_0^{2}Kx=[{\omega }_0^{2}f\left(t\right)+2\delta f\text{'}\left(t\right)]z$, (i)

Where $\delta =\frac{\gamma }{2m}$

(b) Assume an input to Eq. (i)in part (a) of the form $f\left(t\right)=e^{i\omega t}$ and show that the $n$-floor frequency response vector $G\left(i\omega \right)$(see Section $4.6$ and $6.6$) satisfies the equation

$\left[\left({\omega }_0^2+i2\delta \omega \right)K-{\omega }^2{I}_n\right]G\left(i\omega \right)=\left({\omega }_0^2+i2\delta \omega \right)z$. (ii)

Thus the gain function for the frequency response of the $j$ thfloor is the absolute value of the $j$ thcomponent of $G\left(i\omega \right)$, $\left|G_j\left(i\omega \right)\right|$. On the same set of coordinate axes, plot the graphs of $\left|G_j\left(i\omega \right)\right|$ versus $\omega \in \left[0,6\pi \right],j=5,10,20$ using the parameter values $n=20,{\omega }_0=41$, and $\delta =10$. Repeat using $\delta =50$ and interpret the results of your graphs.

(c) Numerically approximate the solution of Eq. (i)in part (a) subject to the initial conditions $x\left(0\right)=0,x\text{'}\left(0\right)=0$ using the parameter values $n=20,{\omega }_0=41$, and $\delta =10$. For the input function, use $f\left(t\right)=A\sin {\omega }_{r}t$, where $A=2$ in and ${\omega }_r$ is the lowest resonant frequency of the system, that is, the value of $\omega$ at which $\left|G_{20}\left(i\omega \right)\right|$ in part $\left(b\right)$ attains its maximum value. Plot the graphs of $x_j\left(t\right)$ versus $t$, $j=5,10,20$ on the same set of coordinate axes. What is the amplitude of oscillatory displacement of each of these floors once the steady-state response is attained $?$ Repeat the simulation using $\delta =50$ and compare your results with the case $\delta =10$.