The Chartership Forum

Structural Dynamics / module 5

Forced vibration and resonance

How repeated loading near a natural frequency can amplify response, and how damping affects the result.

What you will understand

  • Derive the steady-state response to harmonic forcing
  • Derive the dynamic magnification factor and the phase angle
  • Explain physically why resonance occurs and what limits it
  • Explain why the response lags the force, and by how much in each regime
  • Distinguish transient from steady-state response and say how long the transient lasts
  • Apply transmissibility to a vibration-isolation problem

Lessons

  1. Harmonic response and resonance

    The steady-state solution, the magnification factor, and the physical reason a lightly damped structure can be shaken apart by a force it could carry a hundred times over statically.

Module summary
  • Rd = 1/√[(1 − β²)² + (2ζβ)²], and at β = 1 it is exactly 1/(2ζ)
  • The peak sits at β = √(1 − 2ζ²), just below resonance, and vanishes above ζ = 1/√2
  • At resonance the applied force balances the DAMPING force alone
  • Phase is 0° well below, exactly 90° at β = 1, and 180° well above — whatever the damping
  • Resonance takes many cycles to build up: roughly 1/(2ζ) of them
  • Above resonance the response is inertia-controlled and the mass barely moves
  • Transmissibility is 1 at β = √2 for every damping ratio
  • Above β = √2 more damping makes force transmission worse, not better
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 1.5% damping. What is its resonant magnification factor?

Practice

A 5 kN harmonic force acts at β = 1.4 on a structure of stiffness 50 MN/m with 5% damping. What is the steady-state amplitude in mm?

Check yourself

A machine is to be isolated from a floor. At what frequency ratio does isolation begin?

Practice

How many cycles does a resonant response take to reach 63% of its steady-state amplitude, at 2% damping?