Problem $28$

For the $3$ DOF system shown with stiffnesses $k_1=k_2=k_3=89\mathrm{.}6M\frac{N}{m}$ and masses $m_1=56Mg,m_2=64Mg$ and $m_3=28Mg,$ the mass and stiffness matrices are:

$M=\left[\begin{array}{ccc}56& 0& 0\\ 0& 64& 0\\ 0& 0& 28\end{array}\right]Mg$ and $K=89\mathrm{.}6\left[\begin{array}{ccc}2& -1& 0\\ -1& 2& -1\\ 0& -1& 1\end{array}\right]M\frac{N}{m}$

Then resulting natural frequencies and mode shapes are:

${}_1={400}^{\frac{1}{2}}=20,$rad$\frac{s}{s},{}_1={\left[\begin{array}{ccc}\frac{1}{2}& \frac{7}{8}& 1\end{array}\right]}^T,$

${}_2={3200}^{\frac{1}{2}}$~~$56\mathrm{.}569$rad$\frac{s}{s},{}_2={\left[\begin{array}{ccc}-1& 0& 1\end{array}\right]}^T,$

${}_3={5600}^{\frac{1}{2}}$~~$74\mathrm{.}833$rad$\frac{s}{s},{}_3={\left[\begin{array}{ccc}\frac{1}{2}& -\frac{3}{4}& 1\end{array}\right]}^T.$

So the eigenmatrix $$ and $($if you need it$)$ its inverse ${}^{-1}$ are given by

$=\left[\begin{array}{ccc}\frac{1}{2}& -1& \frac{1}{2}\\ \frac{7}{8}& 0& -\frac{3}{4}\\ 1& 1& 1\end{array}\right]$ and ${}^{-1}=\left[\begin{array}{ccc}\frac{4}{13}& \frac{8}{13}& \frac{4}{13}\\ -\frac{2}{3}& 0& \frac{1}{3}\\ \frac{14}{39}& -\frac{8}{13}& \frac{14}{39}\end{array}\right].$

If you need ${}^{T}M$ and$/$or ${}^{T}K,$ they are:

${}^{T}M=\left[\begin{array}{ccc}91& 0& 0\\ 0& 84& 0\\ 0& 0& 78\end{array}\right]Mg$ and ${}^{T}K=\left[\begin{array}{ccc}36\mathrm{.}4& 0& 0\\ 0& 268\mathrm{.}8& 0\\ 0& 0& 436\mathrm{.}8\end{array}\right]M\frac{N}{m}$

If a piece of equipment exerts harmonic force $f(t)$ on the structure as shown with an amplitude of $100kN$ and a frequency of $17$rad$\frac{s}{s},$ and if damping is neglected, determine: $($a$)$ the steady$-$state displacement of the roof, $x_3(t),$ as a function of time; $($b$)$ the amplitude of that response.