Pre-Analysis

The following equations give the frequencies of the modes and the mode shapes and are derived from Euler-Bernoulli Beam Theory.

$\begin{gathered} w_n = \alpha_n^2 \sqrt{\frac{EI}{ml^3}} \\[1.5em] n = 1, 2, 3, \dots \\[1.5em] \alpha_n = 1.875, \, 4.694, \, 7.855, \dots \\[1.5em] m = \rho V = \rho \cdot l \cdot h \cdot w \\[1.5em] I = \frac{w \cdot h^3}{12} \\[1.5em] w_1 = 1.875^2 \sqrt{\frac{70\text{ E}9\,\frac{\text{kg}}{\text{m}\cdot\text{s}^2} \cdot \frac{0.346\text{m} \cdot (0.346\text{m})^3}{12}}{2.7\text{ E}3\,\frac{\text{kg}}{\text{m}^3} \cdot 4\text{m} \cdot 0.346\text{m} \cdot 0.346\text{m} \cdot (4\text{m})^3}} = 111.7\,\frac{\text{rad}}{\text{s}} = 17.8\text{ Hz} \\[1.5em] w_2 = 4.694^2 \sqrt{\frac{70\text{ E}9\,\frac{\text{kg}}{\text{m}\cdot\text{s}^2} \cdot \frac{0.346\text{m} \cdot (0.346\text{m})^3}{12}}{2.7\text{ E}3\,\frac{\text{kg}}{\text{m}^3} \cdot 4\text{m} \cdot 0.346\text{m} \cdot 0.346\text{m} \cdot (4\text{m})^3}} = 700.4\,\frac{\text{rad}}{\text{s}} = 111.5\text{ Hz} \\[1.5em] w_3 = 7.855^2 \sqrt{\frac{70\text{ E}9\,\frac{\text{kg}}{\text{m}\cdot\text{s}^2} \cdot \frac{0.346\text{m} \cdot (0.346\text{m})^3}{12}}{2.7\text{ E}3\,\frac{\text{kg}}{\text{m}^3} \cdot 4\text{m} \cdot 0.346\text{m} \cdot 0.346\text{m} \cdot (4\text{m})^3}} = 1961.2\,\frac{\text{rad}}{\text{s}} = 312.1\text{ Hz} \\[1.5em] y_i(x) = \cosh\left(\frac{\alpha_i x}{L}\right) - \cos\left(\frac{\alpha_i x}{L}\right) - \sigma_i \left( \sinh\left(\frac{\alpha_i x}{L}\right) - \sin\left(\frac{\alpha_i x}{L}\right) \right) \\[1.5em] \alpha_i = 1.875, \, 4.694, \, 7.855, \dots \\[1.5em] \sigma_i = 0.73409, \, 1.018647, \, 0.9992245, \dots \end{gathered}$