engineering recuitment HPCL Engineer Mock Test 2024 Mechanical Vibrations Damped Free Vibration Logarithmic Decrement
The equation of motion of a harmonic oscillator is given by
\(\frac{{{d^2}x}}{{d{t^2}}} + 2\zeta {\omega _n}\frac{{dx}}{{dt}} + \omega _n^2x = 0\)
and the initial conditions at t = 0 are \(x\left( 0 \right) = X,\;\;\frac{{dx}}{{dt}}\left( 0 \right) = 0\). The amplitude of x(t) after n complete cycles is
1
\(X{e^{ - 2n\pi \left( {\frac{\zeta }{{\sqrt {1 - {\zeta ^2}} }}} \right)}}\)
2
\(X{e^{2n\pi \left( {\frac{\zeta }{{\sqrt {1 - {\zeta ^2}} }}} \right)}}\)
3
\(X{e^{ - 2n\pi \left( {\frac{{\sqrt {1 - {\zeta ^2}} }}{\zeta }} \right)}}\)
4
\(X\)