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This transcendental equation has a denumerably infinite set of roots that cannot be found in closed form. However, as many of these roots as desired can be found using numerical procedures found in commercially available software packages such as Mathematica, Maple, or MATLAB. These roots of Eq. (2.198) will be functions of and the first four such roots are plotted versus C in Fig. 2.22. Denoting these roots by a,, with i — 1.2 one obtains the corresponding natural frequencies

From the plots (and from Eq. 2.198) we note that as C tends toward zero, tends toward tt/2, which means that the fundamental natural frequency is

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