Question

A water tower is idealized as a mass M on top of a uniform and massless beam. The bottom end of the beam is fixed to the ground. The beam has solid circular cross section with a diameter of 1.2 m. Its Young’s modulus is 30 GPa, and its length is 10 m. The mass M is 20 tons. Find the natural frequency and natural period of this system in lateral oscillations.

Answer 1

Answer:

Natural frequency=21.40 Hz

Time= 0.2936 seconds

Explanation:

Idealizing the question as a cantilever beam with point load of mass M as 20 tons

Lateral stiffness, $k=\frac {3EI}{l^{3}}$ where l is length given as 10 m, E is Young’s modulus given as 30GPa and I is inertia where for a circular cross-section is given by $\frac {\pi d^{4}}{64}$

k=$\frac {3*(30*10^{9})*(\pi *1.2^{4})}{64*10^{3}}$= 9160884.178

k= $9.160884178*10^{6}$

To find the frequency, $w_{n}$, the mass m is given as 20 tons or 20000 Kg

$w_{n}=\sqrt (\frac {k}{m})= \sqrt (\frac {9.160884178*10^{6}}{20000})$=21.40196741 Hz

Natural frequency=21.40 Hz

Time period,

T=$\frac {2\pi}{w_{n}}=\frac {2\pi}{21.40196741}$=0.2935798 seconds

T=0.2936 seconds