Solved Examples - Response Under a Periodic Force of Irregular form

Example 1 - Find the displacement of the water tank shown in Figure 1 (a) under the periodic force shown in Figure 1 (b) by treating it as an undamped single-degree.of-freedom system. Figure 1 - Water tank subjected to a periodic force. The circuar frequency \(\omega_n\) can be calculated as: \begin{eqnarray} \omega_n &=& \sqrt{\frac{k}{m}} = \sqrt{\frac{5\times 10^6}{10\times 10^3}} = 22.3607 \left[\frac{\mathrm{rad}}{\mathrm{s}}\right]\nonumber\\ \tau &=& 0.15 \left[\mathrm{s}\right]\nonumber\\ \omega &=& \frac{2\pi}{\tau} = 41.888 \left[\frac{\mathrm{rad}}{\mathrm{s}}\right]\nonumber\\ r&=& \frac{\omega}{\omega_n} = 1.87333,\nonumber \\ \zeta = 0 \end{eqnarray} Result can be wrtitten as: \begin{eqnarray} F(r) &=& 160.0 +25.5002 \cos 41.888 t + 242.627\cdot 41.888 t - 75.3884 \cos 83.776 t + 16.0237\sin 83.776 t \nonumber \\ &-& 75.3884 \cos 83.776 t + 16.0237\sin 83.776 t + 16.4806 \cos125\cdot 664 t +50.7237 \sin 125.664 t - 62.3538 \cos 167.552 t \nonumber\\ &+& 27.7604\cot \sin \cdot 167.552 \cdots.\nonumber \end{eqnarray}