The Chartership Forum

Structural Dynamics / module 4

Free vibration and damping

Displace a structure, let go, and watch what happens — the source of the natural frequency, the three damping regimes, and the measurement that gets damping out of a real building.

What you will understand

  • Derive ωn = √(k/m) from the free-vibration equation
  • Derive the critical damping coefficient and explain what makes it critical
  • Write the free-vibration response for underdamped, critical and overdamped cases
  • Explain why the damped frequency is below the undamped one, and by how little
  • Derive the logarithmic decrement and use it to estimate damping from a decay trace
  • Explain why damping is the least reliable number in a dynamic model

Lessons

  1. Free vibration and the natural frequency

    Where ω_n = √(k/m) comes from, what critical damping means, and why the three regimes are one solution rather than three.

  2. Measuring damping from a decay trace

    The logarithmic decrement, derived and then used — and an honest account of why the number it gives is the least reliable input to any dynamic model.

Module summary
  • ωn = √(k/m) — stiffness raises it, mass lowers it, both under a square root
  • ccr = 2mωn = 2√(km), and ζ = c/ccr
  • ωd = ωn√(1 − ζ²): at 5% damping the frequency shift is 0.13%
  • Critical damping is the boundary at which oscillation ceases, not an optimum
  • Overdamped systems return more slowly than critically damped ones
  • δ = (1/n)ln(uáµ¢/uᵢ₊ₙ) and ζ = δ/√(4π² + δ²)
  • Energy goes as amplitude squared, so it is gone long before the motion looks small
  • Measured damping depends on amplitude, so a small-amplitude test underestimates
Teaching basis and review status

These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.

Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.

Read the lesson register and outstanding review notes

Check your understanding

Practice

A structure has m = 20 000 kg and k = 8.0 MN/m. What is the critical damping coefficient, in kN·s/m?

Practice

Peaks in a decay trace are 25.0 mm and 9.2 mm, eight cycles apart. What is the damping ratio as a percentage?

Check yourself

What does critical damping mean physically?

Practice

A structure with 5% damping is set vibrating. What fraction of its initial ENERGY remains after 3 cycles? Give the answer as a percentage.