A two degree of freedom spring-mass system is shown in Fig. P4.4. For harmonic free vibration, complete the following if \(k=5 \times 10^{6} \mathrm{~N} / \mathrm{m}\) and \(m=2 \mathrm{~kg}\).
(a) Draw the free body diagram showing the forces on the two masses during vibration.
(b) Write the two equations of motion in matrix form. First show the equations symbolically and then substitute the numerical values for \(m\) and \(k\).
(c) Write the characteristic equation for this system. First show the equation symbolically and then substitute the numerical values for \(m\) and \(k\).
(d) Determine the numerical roots of the characteristic equation (which is quadratic in \(\mathrm{s}^{2}\) ). What do these two roots represent?
(e) Determine the two mode shapes for this system. Normalize the mode shapes to coordinate \(x_{1}\).