Vibrations: Free and Forced

Why it matters

Understanding vibrations in mechanical systems is crucial for designing machines that operate smoothly and efficiently. It helps in predicting and mitigating issues related to noise, wear, and failure in mechanical components.

Key ideas

• Free Vibrations: Occur when a system oscillates under the action of internal forces only, after an initial disturbance. No time-dependent excitation acts after release; displacement is measured from static equilibrium, so constant loads may already be balanced.
• Forced Vibrations: Occur when a system is subjected to a continuous external force. For a stable damped linear system under sustained harmonic excitation, the transient decays and a periodic steady response remains.
• Natural Frequency: The frequency at which a system naturally oscillates in the absence of external forces.
• Damping: The effect of reducing the amplitude of vibrations over time, usually due to resistive forces like friction.
• Resonance: A phenomenon that occurs when the frequency of external forces matches the natural frequency, leading to large amplitude oscillations.

Scope

The damped-frequency formula applies to an underdamped linear system, 0 ≤ ζ < 1. The simple-pendulum formula assumes small angles and negligible damping. For a mass–spring oscillator, ω_n = √(k/m) and f_n = ω_n/(2π). Undamped forcing at ω_n produces a growing response; a damped displacement peak can occur below ω_n.

Formulas

• Natural Frequency of a simple pendulum: f_n = (1 / (2π)) * √(g / L)
  • f_n: Natural frequency (Hz)
  • g: Acceleration due to gravity (9.81 m/s²)
  • L: Length of the pendulum (m)
• Damped Natural Frequency: f_d = f_n * √(1 - ζ²)
  • f_d: Damped natural frequency (Hz)
  • ζ: Damping ratio (dimensionless)
• Equation of Motion for Forced Vibrations: m·x'' + c·x' + k·x = F_0·cos(ωt)
  • m: Mass (kg)
  • x: Displacement (m)
  • c: Damping coefficient (Ns/m)
  • k: Stiffness (N/m)
  • F_0: Amplitude of the forcing function (N)
  • ω: Angular frequency of the forcing function (rad/s)

Worked example

Given: A mass-spring-damper system with mass m = 2 kg, damping coefficient c = 3 Ns/m, stiffness k = 20 N/m, and a forcing function F_0 = 5 N at ω = 2 rad/s.

1. Calculate the natural frequency

   • Formula: f_n = (1 / (2π)) * √(k / m)
   • Calculation: f_n = (1 / (2π)) * √(20 / 2) = 0.503292 Hz

2. Calculate the damped natural frequency

   • Damping ratio: ζ = c / (2 * √(m * k))
   • Calculation: ζ = 3 / (2 * √(2 * 20)) = 0.237171
   • Formula: f_d = f_n * √(1 - ζ²)
   • Calculation: f_d = 0.503292 * √(1 - 0.237171²) = 0.488932 Hz

3. Determine the steady-state amplitude

   • Formula: X = F_0 / √((k - mω²)² + (cω)²)
   • Calculation: X = 5 / √((20 - 2*2²)² + (3*2)²) = 0.372678 m

Final Answer: The steady-state amplitude is 0.372678 m.

Common mistakes

• Confusing natural frequency with damped natural frequency.
• Ignoring the damping effect in forced vibrations.
• Miscalculating the damping ratio.

For GATE ME

Questions often involve calculating natural and damped frequencies, determining resonance conditions, and analyzing the effects of damping. Practice problems on deriving equations of motion and solving for steady-state responses.

Quick check

1. What is the natural frequency of a system with k = 50 N/m and m = 5 kg?
2. Define resonance in the context of forced vibrations.
3. How does damping affect the amplitude of vibrations?

Answers: 1. √10/(2π) = 0.5033 Hz, 2. Resonance occurs when the forcing frequency matches the natural frequency, 3. Damping reduces the amplitude of vibrations over time.