The Chartership Forum

Structural Dynamics / module 2

Dynamic modelling and degrees of freedom

Turning a physical structure into something that can be analysed — and keeping an honest record of what each simplification discarded.

What you will understand

  • Distinguish the physical structure from the dynamic model of it
  • Define a degree of freedom and count them for a given idealisation
  • Explain what a lumped-mass idealisation assumes and when it fails
  • Explain what a rigid diaphragm does to the degrees of freedom of a floor
  • State what each common simplification removes from the model's reach
  • Use kinetic and strain energy to justify a reduced model rather than asserting it

Lessons

  1. The model is not the structure

    What a degree of freedom is, why every dynamic model has far fewer than the real thing, and what is lost each time one is removed.

Module summary
  • A dynamic model is a set of decisions, and each one removes a class of behaviour
  • A degree of freedom is an independent coordinate needed to describe the deformed shape
  • Lumping mass at the floors is exact if the mass really is there, and an assumption otherwise
  • A rigid diaphragm reduces a whole floor of nodes to three coordinates
  • Ignoring axial deformation removes vertical modes entirely, not approximately
  • Reduction is written as u = Tq, with reduced mass TᵀMT and reduced stiffness TᵀKT
  • The shear-building model is right for regular buildings laterally and wrong for floor vibration
  • Record what a model cannot represent, not only what it can
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 twelve-storey building is modelled in three dimensions with rigid diaphragms and torsion included. How many modes will a complete modal analysis return?

Check yourself

A reduced model reports a first natural frequency LOWER than the full model it was reduced from. What does this tell you?