Question

The equation of motion for a single degree of freedom system is given by

$\ddot{x}+\dot{x}+100x=\cos 10t.$
Identify the wrong statements

• A. Its damping is 5% of critical damping
• B. Its steady state amplitude is 0.1 m
• C. The system is at Resonance hence it cannot have steady state oscillations
• D. The magnification factor is 10

Answer 1

The correct option is C The system is at Resonance hence it cannot have steady state oscillations

$\ddot{x}+\dot{x}+100x=\cos 10t.$

(i) above equation is equivalent to

$\ddot{x}+\frac{c}{m}\dot{x}+\frac{S}{m}x=\frac{F_0}{m}\cos \omega t$

${\omega }_n=\sqrt{\frac{S}{m}}=\sqrt{100}=10\text{rad/s}$
(system is at resonance)

$\left(ii\right)2\xi {\omega }_n=\frac{c}{m}=1$

$\xi =\frac{1}{2{\omega }_n}=\frac{1}{20}=0.05=5\%$

$\text{(iii) Magnification factor}\left(MF\right)=\frac{1}{2\xi },$

$\omega ={\omega }_n$

$MF=10$

$\text{(iv)}\frac{F_0}{m}=1,\frac{S}{m}=100$

$\frac{F_0}{S}=\frac{1}{100}$

So amplitude for resonance is given by,

$A=\frac{\frac{F_0}{S}}{2\xi }=0.1\text{m}$

Hence, the system is at resonance and it has steady state oscillation.