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

A vibratory system consists of a mass $$12.5$$ kg, a spring of stiffness $$1000$$ $$N/m,$$ and a dashpot with damping coefficient of $$15$$ $$Ns/m.$$

The value of critical damping of the system is:

A vibratory system consists of a mass $$12.5$$ kg, a spring of stiffness $$1000$$ $$N/m,$$ and a dashpot with damping coefficient of $$15$$ $$Ns/m.$$

The value of logarithmic decrement is

A machine of $$250$$ kg mass is supported on springs of total stiffness $$100$$ kN/m. Machine has an unbalanced rotating force of $$350$$ N at speed of $$3600$$ rpm. Assuming a damping factor of $$0.15,$$ the value of transmissibility ratio is