Question

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?

• 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).
• C. The steady–state periodic response is dependent on the initial conditions at the time of application of external forcing.
• D. The rate of decay of free–vibration transient response depends on the mass, spring stiffness and damping constant.

Hint:

Linear forced vibration \(=\) transient (IC–dependent) \(+\) steady state (IC–independent). Any positive damping kills the transient as \(e^{-\zeta\omega_n t}\), where \(\zeta\) depends on \(m,k,c\).

Answer 1

The Correct Option is A, B, D

Step 1: General solution structure for harmonic forcing.
For a linear SDOF under \(m\ddot x + c\dot x + kx = F_0\sin\omega t\), the solution is \[ x(t) = x_{\text{tr}}(t) + x_{\text{ss}}(t), \] where \(x_{\text{tr}}(t)\) is the homogeneous or free–vibration transient and \(x_{\text{ss}}(t)\) is the particular steady–state sinusoid at the forcing frequency. \(\Rightarrow\) (A) is true.

Step 2: Behavior of the transient with damping.
If \(c>0\) (under/critical/over–damped), \(x_{\text{tr}}(t)\) carries an exponential factor \(e^{-\zeta\omega_n t}\) (or an overdamped sum of decaying exponentials), so it decays to zero as \(t\to\infty\). \(\Rightarrow\) (B) is true. (Note: for \(c=0\) it would not decay, but that case is not listed.)

Step 3: Dependence of steady state on initial conditions.
\(x_{\text{ss}}(t)\) is determined solely by \(F_0,\omega,m,c,k\) through the FRF (magnitude \(X(\omega)\), phase \(\phi\)); it does not depend on initial displacement/velocity. Initial conditions only set the transient \(x_{\text{tr}}(t)\). \(\Rightarrow\) (C) is false.

Step 4: Parameters controlling decay rate.
The decay envelope is \(e^{-\zeta\omega_n t}\) with \(\omega_n=\sqrt{k/m}\) and \(\zeta=\dfrac{c}{2m\omega_n}=\dfrac{c}{2\sqrt{km}}\). Thus the decay rate depends on \(m,k,c\). \(\Rightarrow\) (D) is true.

Final Answer:
\[ \boxed{(A),\ (B),\ (D)} \]