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Mechanical Engineering QUESTION #2439
Question 1
The free response of an underdamped system (\(\zeta < 1\)) is:
  • \(x(t) = e^{-\alpha t}(C_1\sin\omega_d t + C_2\cos\omega_d t)\) where \(\alpha=\zeta\omega_n\)✔️
  • \(x(t) = A\sin(\omega_n t + \phi)\) (pure harmonic)
  • \(x(t) = C_1 e^{-\lambda_1 t} + C_2 e^{-\lambda_2 t}\)
  • \(x(t) = (C_1t + C_2)e^{-\alpha t}\)
Correct Answer Explanation
For the underdamped case, the solution is the exponentially decaying sinusoid: \(x=e^{-\alpha t}(C_1\sin\omega_d t+C_2\cos\omega_d t)\). Option 3 is for overdamped; option 4 is for critically damped.