Consider the shear frame shown in Figure $1$a$.$ Each floor has a mass $=9000$ and

the flexural stiffness of all columns is $=2\mathrm{.}5\backslash$times $1062.$ The dynamic model of the

frame is presented in Figure $1$b $(1$

k and $2$

k are the equivalent stiffnesses of storey $1$ and

storey $2,$ respectively$).$ It is proposed to attach a rotating fan of mass $=11000$to

the first floor of the frame. The fan operates at a frequency of $=6$ rad$/$s$.$

a$.$ Construct the stiffness matrix $[$K$]$ and the mass matrix $[$M$]$ of the frame including the

rotating fan.

b$.$ Determine the natural angular frequencies $($eigenvalues$)2$

j

$$ and the mode

shapes of vibration $($eigenvectors$)$

ij $$ of the frame using the mass and stiffness

matrices determined in part $($a$)$ above. Use the stiffness based formulation of the

eigenproblem in your calculation.

c$.$ Check whether the rotating fan sets up resonant vibrations in the structure. If this is

the case, suggest solutions to reduce these resonant vibrations