Figure 2.

(a) Plot of Equation (10) (orange line), whose first four solutions ${\beta }_{n}L$ (indicated in the text) are the crossings with zero. The blue line shows the asymptotic approximation $\cos \left({\beta }_{n}L\right)=0$, which can be solved analytically, and that agrees with the numerical solution for $n\ge 2$; (b) mode shapes of the first four flexural modes of a cantilever.