Free, Damped and Forced Vibrations

Every elastic system with mass can vibrate. GATE ME focuses on the single degree of freedom (SDOF) spring–mass–damper system, which explains natural frequency, damping, resonance and isolation.

Free undamped vibration

Equation of motion: m ẍ + kx = 0.

• Natural frequency ω_n = √(k/m) rad/s; f_n = ω_n/(2π) Hz.
• Using static deflection δ = mg/k: ω_n = √(g/δ).
• Torsional system: ω_n = √(k_t/I). Simple pendulum: ω_n = √(g/L).
• Energy method: write T + U = constant and differentiate; the coefficient form m_eq ẍ + k_eq x = 0 gives ω_n = √(k_eq/m_eq).

Effective stiffness: springs in parallel add; in series their reciprocals add. A cantilever with an end mass has k = 3EI/L³; a simply supported beam with a central mass has k = 48EI/L³.

Free damped vibration

m ẍ + c ẋ + kx = 0.

• Critical damping c_c = 2√(km) = 2mω_n. Damping ratio ζ = c/c_c.
• ζ < 1: underdamped (oscillates); ζ = 1: critically damped (fastest return without oscillation); ζ > 1: overdamped.
• Damped frequency ω_d = ω_n √(1 − ζ²).
• Logarithmic decrement δ = ln(x₁/x₂) = 2πζ/√(1 − ζ²) ≈ 2πζ for small ζ.

Forced vibration

With a harmonic force F₀ sin ωt and frequency ratio r = ω/ω_n:

• Magnification factor X/(F₀/k) = 1/√[(1 − r²)² + (2ζr)²].
• At resonance (r = 1), the amplitude is F₀/(2ζk), limited only by damping.
• Transmissibility T = √[1 + (2ζr)²]/√[(1 − r²)² + (2ζr)²].
• Isolation (T < 1) needs r > √2. Above that, more damping makes isolation worse.

Whirling of shafts

The critical speed of a shaft carrying a rotor equals its natural frequency of transverse vibration: ω_c = √(g/δ) with δ the static deflection under the rotor's weight.

Worked example

A 10 kg mass on a spring of 4000 N/m with c = 80 N·s/m.

• ω_n = √(4000/10) = 20 rad/s. c_c = 2 × 10 × 20 = 400 N·s/m, so ζ = 0.2.
• ω_d = 20 √(1 − 0.04) = 19.6 rad/s.

Practice