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Queensferry

Module 07

Energy in vibrating structures

The same problem seen through a different window — and the one check that will tell you whether a time-history analysis is right, when nothing in the displacement plot will.

What this module covers

  • 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. 7.1

    The energy balance

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

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  2. 7.2

    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.

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Module checkpoint

Check what you have taken in

5 questions

  1. Question 1

    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?

  2. Question 2

    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?

  3. Question 3

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

  4. Question 4

    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?

  5. Question 5

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