Compute the first three natural frequencies and the corresponding mode shapes of the transverse vibrations of a uniform beam of rectangular cross section \((100 \mathrm{~mm} \times 300 \mathrm{~mm})\) with \(l=2 \mathrm{~m}, E=20.5 \times 10^{10} \mathrm{~N} / \mathrm{m}^{2}\), and \(ho=7.83 \times 10^{3} \mathrm{~kg} / \mathrm{m}^{3}\) for the following cases:

(a) when both ends are simply supported;

(b) when both ends are built-in (clamped);

(c) when one end is fixed and the other end is free; and

(d) when both ends are free. Plot the mode shapes.