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) = xtransient(t) + xsteady(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.