Applications to Vibrations of a Particle & Simple Harmonic Motion (SHM)

Explanation

Physical vibration models (spring-mass dampers, acoustic resonance, electrical LC/RLC oscillations) derive directly from Newton's second law F = m a. The second-order differential equation models restoring elasticity, viscous dissipation, and external harmonic forcing. Resonance occurs when energy is pumped into the system at its natural eigenvalue frequency.

Key points

• Periodic Time: T = 2π / ω; Frequency f = 1/T = ω / (2π); Amplitude A is maximum displacement.
• Velocity in SHM: v² = ω² (A² - x²); Maximum velocity v_{max} = A ω at center x = 0.
• Damped Oscillations: d²x/dt² + 2k (dx/dt) + ω² x = 0 where 2k is damping resistance per unit mass.
• Case 1 (Overdamped k > ω): Roots are real and negative; particle creeps back to rest without oscillating.
• Case 2 (Critically Damped k = ω): Fastest return to equilibrium without oscillation.
• Case 3 (Underdamped / Oscillatory k < ω): x(t) = e^{-kt} [c₁ cos(ω_d t) + c₂ sin(ω_d t)] with damped frequency ω_d = √(ω² - k²).
• Forced Vibrations & Resonance: d²x/dt² + 2k (dx/dt) + ω² x = F₀ cos(pt). When external excitation frequency p approaches natural frequency ω, amplitude reaches catastrophic resonance peak.

Example

Code examples

// Mathematical Formulation: Free Damped Particle Motion
// Equation: m (d^2x/dt^2) + c (dx/dt) + k x = 0
// Divide by m: d^2x/dt^2 + 2γ (dx/dt) + ω0^2 x = 0
//   where 2γ = c/m (damping factor), ω0 = √(k/m) (natural frequency)
// Auxiliary equation: r^2 + 2γ r + ω0^2 = 0
// Roots: r1, r2 = -γ ± √(γ^2 - ω0^2)
//
// In underdamped case (γ < ω0):
// Let damped angular frequency ω_d = √(ω0^2 - γ^2)
// Solution: x(t) = A e^{-γ t} cos(ω_d t - φ)
//
// Logarithmic Decrement δ:
// Ratio of successive maximum amplitudes x_n and x_{n+1}:
//   x_n / x_{n+1} = e^{γ T_d} where T_d = 2π / ω_d
//   δ = ln(x_n / x_{n+1}) = γ T_d = (2π γ) / √(ω0^2 - γ^2)

Exam questions

• [8 Marks] A particle vibrates according to the equation d²x/dt² + 4 (dx/dt) + 13x = 0. If it starts from x = 1 with velocity v = 2 at t = 0, find its displacement at any time t.
• [7 Marks] Define Simple Harmonic Motion. Derive the expressions for velocity, periodic time, and amplitude.
• [7 Marks] Explain resonance in forced mechanical vibrations with a differential equation model and amplitude-frequency curve.