Problem $11$

Consider the single$-$story shear building model shown at the right implemented with a Tuned Mass Damper $($TMD$)$ on top. Here, $u_i(t)$ is the displacement of mass $m_i$ relative to the ground, $c_1,$ and $k_1$ are the viscous damping coefficient, and lateral stiffness of the building, respectively. The TMD mass, $m_2,$ is attached to the building mass, $m_1,$ via a spring, $k_2,$ and a damper, $c_2.$ The building model is subjected to a ground acceleration $z_g^{}(t),$ as well as a wind force, which is applied only to building mass, $m_1.$

a$.$ Formulate the equation of motion for the building model with the TMD of the form:

${}^{}+C{u}^{}+Ku=G_{1}p(t)+G_2{z}_g^{}(t),$ where $u={\left[\begin{array}{cc}{u}_1& u_2\end{array}\right]}^T.$

b$.$ Assume the following numerical values and calculate the undamped natural frequencies and mode shapes for the system.

$m_1=1000kg,m_2=20kg,k_1=160k\frac{N}{m},k_2=3k\frac{N}{m},$ and

$c_1=500N-\frac{s}{m},c_2=20N-\frac{s}{m}$

Sketch both mode shapes, assuming that scaling them so that the displacement of building mass $m_1=1.$

$6$

c$.$ Show that the mass matrix $M$ and the stiffness matrix $K$ diagonalize under the transformation $u=,$ whereas the damping matrix $C$ does not.

d$.$ Write the equation of motion in state space form, i$.$e$.$

$]\}$

$\{[z_g^{}]\}$

$u_1^{}$

$\{[u_2-u_1]\}$

$\{[z_g^{}$

Note: you don't have to plug in numerical values.

e$.$ The Matlab command damp$($As$)$ results in modal damping values of ${}_1=0\mathrm{.}03,{}_2=0\mathrm{.}0309.$ Assuming proportional $($Rayleigh$)$ damping, determine the damping matrix $C_R=M+K$ that will result in the same modal damping values for the given system.

f$.$ Compare the damping matrix $C_R$ with the one derived in part $($a$).$ Comment on the differences. How do you expect the results using these two damping matrices to differ?