Question

2) A single degree of freedom system is excited by sinusoidal force. Determine the damping ratio of the system if the amplitude of displacement at resonance is 2 in, the exciting frequency is one- tenth of the natural frequency and the amplitude of displacement at resonance is 0.2 in a) 0.25 Hz b) 0.5 Hz c) 0.0025 Hz d) 005 Hz

Answer 1

When a single degree of freedom system excited by a sinusoidal force , the equation of motion ( time displacement ) is given by

$\frac{d^2x}{dt^2}+\frac{c}{m}\frac{dx}{dt}+\omega_n^2x=Fsin\omega t$

Or

dx w dt 2 Fsinwt dt2 n +
[ where $\xi=\frac{c}{m}$ ]

Where c is the damping coefficient ratio. $\omega_n$ is the natural frequency and
sinw
is applied sinusoidal force on the system. $\omega$ is the exciting frequency, m is the mass of the oscillator.

x is displacement of the system at time t>0.

The amplitude of the system is given by  $A=\frac{f}{\sqrt{(\omega^2-\omega_n^2)^2+\xi^2\omega_n^2}}$

Amplitude Resonance occurs when ................(i)

Vu2- 2/2 in= w2

By the problem statement, $\frac{\omega}{\omega_n}=0.1$

From (i) we get

$1=\frac{\omega^2}{\omega_n^2}-\frac{\xi^2}{2\omega_n^2}$

Or

$\frac{\xi^2}{2\omega_n^2}=0.99$

Or

$\frac{\xi^2}{4\omega_n^2}=0.495$

Or

$c_c=\frac{\xi}{2\omega_n}=0.70$

Where c_c is known as damping ratio.