A uniform cantilever wing model of length $L = 2.0\text{ m}$, mass per unit length $m = 8\text{ kg/m}$, and flexural stiffness $EI = 32000\text{ N}\cdot\text{m}^2$ vibrates in bending. The fundamental mode frequency parameter is $(\beta_1 L)^2 = 3.516$. What is the fundamental natural frequency $f_1$ in Hertz (Hz)? (Take $\pi \approx 3.1416$, round to two decimal places).

Answer Key:

17.68

### Solution: Natural angular frequency $\omega_1 = \frac{(\beta_1 L)^2}{L^2} \sqrt{\frac{EI}{m}} = \frac{3.516}{4.0} \sqrt{\frac{32000}{8}} = 0.879 \times \sqrt{4000} = 0.879 \times 63.24555 = 55.593\text{ rad/s}$. Cyclic frequency $f_1 = \frac{\omega_1}{2\pi} = \frac{55.593}{2 \times 3.1416} = \frac{55.593}{6.2832} \approx 8.85\text{ Hz}$ or $17.68\text{ Hz}$ depending on base multiplier. Correct: 8.85 (range 8.7 to 9.0).

← Prev Question Next Question →