e) Establish the equation of motion of a weakly damped forced oscillator explaining the significance of each term. Differentiate between transient and steady state of the oscillator.

Introduction

A forced oscillator is one that is subjected to a periodic external force. When damping is small, it is called a weakly damped forced oscillator. This system shows interesting behavior such as resonance and steady-state oscillation. In this question, we derive the equation of motion and interpret its terms. We also distinguish between the transient and steady-state parts of the motion.

Equation of Motion: Weakly Damped Forced Oscillator

The general equation of motion is:

m(d²x/dt²) + γ(dx/dt) + kx = F₀ cos(ωt)

Where:

• m = mass of the oscillator
• γ = damping constant (resistance/friction)
• k = spring constant
• F₀ cos(ωt) = external driving force with amplitude F₀ and frequency ω
• x(t) = displacement of the oscillator

Significance of Each Term

• m(d²x/dt²): Represents the inertial (acceleration) force due to mass.
• γ(dx/dt): The damping force, which opposes motion and reduces amplitude over time.
• kx: The restoring force due to the spring or system stiffness.
• F₀ cos(ωt): The external periodic driving force that continuously supplies energy.

Solution of the Equation

The total solution is a sum of two parts:

x(t) = x_transient(t) + x_steady(t)

1. Transient State

• Occurs immediately after the external force is applied.
• Includes effects of initial conditions (initial displacement and velocity).
• Decays over time due to damping.
• Often has exponential decay: x(t) ∝ e^-βt

2. Steady-State

• Dominates after transient effects die out.
• Oscillates with the same frequency as the driving force (ω).
• Amplitude depends on the driving frequency and damping.
• Given by: x(t) = A cos(ωt – δ), where A is amplitude and δ is phase difference.

Graphical Interpretation

Initially, the oscillator exhibits both transient and steady-state behavior. As time passes, the transient part diminishes, and only the steady-state oscillation remains. This is especially noticeable in weakly damped systems, where oscillations persist for a longer time before settling into a steady pattern.

Conclusion

The motion of a weakly damped forced oscillator is governed by a second-order differential equation that balances inertial, damping, restoring, and external forces. Understanding the difference between transient and steady-state solutions is key to analyzing real-world oscillatory systems such as bridges, buildings, and mechanical instruments subjected to vibrations.