GATE 2023 AE – Question 25
Which of the following statement(s) is/are true about harmonically excited forced vibration of a single degree-of-freedom linear spring-mass-damper system?
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Correct answer: (A) The total response of the mass is a combination of free vibration transient and steady-state response.; (B) The free vibration transient dies out with time for each of the three possible conditions of damping (under-damped, critically damped, and over-damped).; (D) The rate of decay of free vibration transient response depends on the mass, spring stiffness and damping constant.
Explanation
Take a single-degree-of-freedom system $m\ddot x+c\dot x+kx=F_0\sin\omega t$.
- **A. Total response = transient + steady state.** True. The complete solution is the complementary (free vibration) part plus the particular (forced) part. ✓
- **B. The transient dies out for under-, critically and over-damped systems.** True. For any positive damping, the free part decays to zero. ✓
- **C. The steady-state response depends on the initial conditions.** False. The initial conditions only affect the transient constants. The steady-state amplitude and phase depend on $F_0$, $\omega$, $m$, $c$ and $k$.
- **D. The decay rate depends on mass, stiffness and damping.** True. The transient decays like $e^{-\zeta\omega_nt}=e^{-ct/2m}$, and for the overdamped case the decay constants also depend on $k$. ✓
Answer **A, B and D**.