Question

A mass of 5 kg is suspended from a spring of stiffness $46kN/m$ . A dashpot is fitted between the mass and the support with a damping ratio of 0.3. Calculate the undamped frequency.
A. 15.56 Hz
B. 15.26 Hz
C. 15.44 Hz
D. 15.34 Hz

Answer 1

The undamped frequency is equal to the natural frequency in undamped oscillations. The natural frequency is $\dfrac{1}{{2\pi }}$ times the square root of the spring constant divided by the mass of the object.

Complete step by step answer:
A mass$\left( m \right)$ of 5 kg is suspended from a spring of stiffness or spring constant $\left( k \right)$ of $46kN/m$ i.e., $46000N/m\left( {1kN = 1000N} \right)$. A dashpot is fitted between the mass and the support with a damping ratio $\delta$of 0.3. A dashpot is a device which is used for damping shock or vibrations. A vibration is said to be a damped vibration if the amplitude of periodic vibrations decreases with time. The amplitude decreases due to the resisting or frictional force which the dashpot is exerting on the mass and support and the vibration or oscillation in which the amplitude remains constant is known as free or undamped vibration. The frequency in undamped oscillation remains constant and equal to the natural frequency. The frequency in damped oscillation is less than the natural frequency. Let the natural frequency be f.
$f = \dfrac{1}{{2\pi }}\sqrt {\dfrac{k}{m}}$
$\implies f = \dfrac{1}{{2\pi }}\sqrt {\dfrac{{46000}}{5}}$
$\implies f = \dfrac{1}{{2\pi }}\sqrt {9200}$
$\implies f = \dfrac{{7\sqrt {9200} }}{{2 \times 22}}\left[ {\pi = \dfrac{{22}}{7}} \right]$
$\implies f = \dfrac{{7 \times 95.91}}{{44}}$
$\therefore f = 15.26Hz$

So, the correct answer is “Option B”.

Note:
In undamped vibrations or oscillations, the amplitude and frequency remain constant. There is no loss of energy in free vibrations. In general, this kind of vibrations doesn’t happen in real because the surrounding medium always offers some resistance.