A spring mass damper system (mass m, stiffness k, and damping coefficient c) excited by a force F(t) = B sin ωt, where B, ω and t are the amplitude, frequency and time, respectively, is shown in the figure. Four different responses of the system (marked as (i) to (iv)) are shown just to the right of the system figure. In the figures of the responses, A is the amplitude of response shown in red color and the dashed lines indicate its envelope. The responses represent only the qualitative trend and those are not drawn to any specific scale.

The correct option/answer is:
(P) → (ii), (Q) → (iii), (R) → (iv), (S) → (i)

Detailed Step-by-Step Explanation:

To determine the qualitative response of the spring-mass-damper system, we analyze the governing differential equation for a forced single-degree-of-freedom system:
$$m\ddot{x}+c\dot{x}+kx=Bsin\left(\omega t\right)$$
where:
• $m$ is the mass (labeled in the system diagram),
• $k$ is the stiffness of the spring (labeled in the system diagram),
• $c$ is the damping coefficient of the damper (labeled in the system diagram), and
• $F\left(t\right)=Bsin\left(\omega t\right)$ is the harmonic excitation force with frequency $\omega$.

Let us match each condition step-by-step:

1. Matching for (P): $c>0$ and $\omega =\sqrt{\frac{k}{m}}$
This condition describes a resonant system with positive damping. Under resonance, the excitation frequency matches the natural frequency of the system. The positive damping ($c>0$) limits the maximum amplitude of the response. The amplitude of the system grows from zero during the transient phase and eventually approaches a constant steady-state amplitude. Qualitatively, this behavior is represented by response (ii), showing a steady development of amplitude over time.

2. Matching for (Q): $c<0$ and $\omega \ne 0$
A negative damping coefficient ($c<0$) represents an unstable system where energy is continuously injected rather than dissipated. The amplitude grows dynamically. Under harmonic excitation, this instability leads to self-excited oscillations with periodic envelope variations. This is qualitatively represented by the amplitude-modulated envelope in response (iii).

3. Matching for (R): $c=0$ and $\omega =\sqrt{\frac{k}{m}}$
This represents pure resonance in a completely undamped system. Because there is no damper to dissipate energy ($c=0$) and the excitation frequency is exactly equal to the natural frequency, the amplitude of oscillation grows unboundedly. The response envelope expands rapidly over time, which is qualitatively represented by response (iv).

4. Matching for (S): $c=0$ and $\omega \approx \sqrt{\frac{k}{m}}$
When a system is undamped ($c=0$) and the excitation frequency is close to the natural frequency, the system exhibits the beat phenomenon. The response shows a slow modulation of amplitude. In the initial stage of this modulation, the envelope increases and levels off as it approaches the first beat peak, which is qualitatively represented by response (i).