For the SDOF damped forced vibration mddot x +cdot x +kx=F_0sin pt, the steady-state response lags the excitation by phase angle phi where:

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Mechanical Engineering QUESTION #2445

Question 1

For the SDOF damped forced vibration \(m\ddot{x}+c\dot{x}+kx=F_0\sin pt\), the steady-state response lags the excitation by phase angle \(\phi\) where:

\(\tan\phi = \frac{1-(p/\omega_n)^2}{2\zeta(p/\omega_n)}\)
\(\tan\phi = \frac{2\zeta(p/\omega_n)}{1-(p/\omega_n)^2}\)✔️
\(\tan\phi = \frac{p}{\omega_n}\)
\(\tan\phi = \frac{k}{cp}\)

Correct Answer Explanation

The phase angle: \(\phi=\arctan\frac{2\zeta(p/\omega_n)}{1-(p/\omega_n)^2}\). At resonance (\(p=\omega_n\)): \(\phi=90°\) regardless of damping.

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