Chapter 6.6, Problem 13P

When viscous damping forces are included and the external force $F_2\left(t\right)$ is set to zero, the system of equations describing the motion of the coupled mass-spring system shown in Figure $\mathrm{6.1.1}$ of Section $6.1$ is

$x\text{'}=\left(\begin{array}{cccc}0& 0& 1& 0\\ 0& 0& 0& 1\\ -\left(k_1+k_2\right)/m_1& k_2/m_1& -\gamma /m_1& 0\\ k_2/m_2& -\left(k_2+k_3\right)/m_2& 0& -\gamma /m_2\end{array}\right)x+\left(\begin{array}{c}0\\ 0\\ F_1\left(t\right)/m_1\\ 0\end{array}\right)$

Where $\gamma$ is the damping coefficient for both $m_1$ and $m_3$. Assume a harmonic input $F_1\left(t\right)/m_1=e^{i\omega t}$ and use the parameter values $m_1=2,m_2=\frac{9}{4},k_1=1,k_2=3,k_3=\frac{15}{4}$ of Example 2, Section 6.4, to compute and plot the graphs of the gain functions $\left|G_1\left(i\omega \right)\right|$ and $\left|G_2\left(i\omega \right)\right|$ for displacements $x_1$ and $x_2$, respectively. Compare the graphs of the gain functions with the natural frequencies of the undamped system for each of the damping coefficients $\gamma =1,0.5$, and $0.1$.