Mechanical Vibrations

Why it matters

Mechanical vibrations are crucial in civil engineering as they affect the stability and integrity of structures. Understanding vibrations helps in designing buildings and bridges that can withstand dynamic loads such as earthquakes and wind.

Key ideas

• Vibration Types: Free and forced vibrations. Free vibrations occur without external force after an initial disturbance, while forced vibrations occur due to continuous external forces.
• Damping: The process by which vibrational energy is gradually lost to the environment, reducing amplitude over time.
• Natural Frequency: The frequency at which a system naturally oscillates when not subjected to continuous external forces.
• Resonance: A phenomenon where the frequency of external forces matches the natural frequency, causing large amplitude oscillations.

Formulas

These equations describe a linear single-degree-of-freedom mass–spring system with viscous damping. f_n is the undamped natural frequency. For 0 ≤ ζ < 1 the free damped frequency is f_d = f_n√(1-ζ²); forced-response peaks depend on damping and the response quantity.

• f_n = 1 / (2π) * √(k / m)
  • f_n: Natural frequency (Hz)
  • k: Stiffness of the system (N/m)
  • m: Mass of the system (kg)
• ζ = c / (2 * √(m * k))
  • ζ: Damping ratio (dimensionless)
  • c: Damping coefficient (Ns/m)

Worked example

Given: A mass-spring system with mass m = 10 kg, stiffness k = 200 N/m, and damping coefficient c = 5 Ns/m.

1. Calculate the natural frequency (f_n)

   • Formula: f_n = 1 / (2π) * √(k / m)
   • Calculation: f_n = 1 / (2π) * √(200 / 10)
   • f_n = 1 / (2π) * √20
   • f_n ≈ 0.71 Hz

2. Calculate the damping ratio (ζ)

   • Formula: ζ = c / (2 * √(m * k))
   • Calculation: ζ = 5 / (2 * √(10 * 200))
   • ζ = 5 / (2 * √2000)
   • ζ ≈ 0.056

Final Answer: Natural frequency is 0.71 Hz and damping ratio is 0.056.

Common mistakes

• Confusing natural frequency with the frequency of external forces.
• Ignoring damping effects in real-world applications.
• Miscalculating the square root in frequency and damping ratio formulas.

For GATE CE

Questions often involve calculating natural frequencies, damping ratios, and analyzing the effects of resonance. Practice problems on both free and forced vibrations, and understand the implications of damping.

Quick check

1. What is the effect of increasing the damping coefficient on the damping ratio?
2. Define resonance in the context of mechanical vibrations.
3. How does mass affect the natural frequency of a system?

Answers: 1. Increases the damping ratio. 2. Large amplitude oscillations when external frequency matches natural frequency. 3. At fixed stiffness, frequency is proportional to 1/√m; increasing mass decreases it.