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Structural Dynamics / module 7

Energy in vibrating structures

Use energy balance to examine a dynamic response and identify inconsistencies in a time-history calculation.

What you will understand

  • Write kinetic, strain, dissipated and input energy for a vibrating structure
  • Derive the energy balance from the equation of motion
  • Explain why an undamped free vibration keeps a constant total energy
  • Use the energy balance residual as a check on a numerical analysis
  • Derive the energy dissipated per cycle by a viscous damper
  • Convert a measured hysteresis loop into an equivalent viscous damping ratio

Lessons

  1. The energy balance

    Multiply the equation of motion by velocity, integrate, and every term becomes an energy that has to be accounted for.

  2. Dissipation per cycle and equivalent damping

    How much a damper removes each cycle, why frequency appears in the answer, and how to turn a real hysteresis loop — which is not viscous at all — into a ζ that linear analysis can use.

Module summary
  • The energy balance is the equation of motion multiplied by velocity and integrated
  • Einput = Ekinetic + Estrain + Edamping, at every instant
  • Undamped free vibration: energy cycles between kinetic and strain, total constant
  • Energy goes as amplitude SQUARED, so it vanishes long before the motion looks small
  • A viscous damper dissipates ED = πcωX² per cycle — note the ω
  • ζeq = ED/(4πEs) converts any loop shape into an equivalent viscous damping
  • The balance residual measures integration error, and nothing in the displacement plot does
  • Isolation reduces the energy input; damping enlarges the dissipation — different strategies
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 of mass 25 000 kg is moving at 0.8 m/s as it passes through its equilibrium position in free vibration. What is its total mechanical energy, in kJ?

Practice

A damper with c = 15 kN·s/m operates at 8 rad/s with amplitude 30 mm. What is the energy dissipated per cycle, in J?

Check yourself

In a computed time-history analysis, the cumulative damping energy is seen to DECREASE at some steps. What does this indicate?

Practice

A loop encloses 220 J at an amplitude of 18 mm, on a system of stiffness 6.0 MN/m. What is the equivalent viscous damping ratio, as a percentage?

Check yourself

A linear time-history analysis closes its energy balance to within 0.0001%. What does this establish?