A quick return mechanism is shown in below. The crank OS is drive at $$2$$ rev/s in counter clockwise direction

If the quick return ratio is $$1:2$$, then the length of the crank in mm is

The natural frequency of the system shown below is

The equation of motion of a harmonic oscillator is given by $${{{d^2}x} \over {d{t^2}}} + 2\xi {\omega _n}{{dx} \over {dt}} + \omega _n^2\,x = 0\,\,\,$$ and the initial conditions at $$t=0$$ are $$\,x\left( 0 \right) = X,{{dx} \over {dt}}\left( 0 \right) = 0.\,\,$$
The amplitude of $$x(t)$$ after $$n$$ complete cycles is

For an under damped harmonic oscillator, resonance