Structural dynamics / tutorial
Lumped or consistent mass
Two defensible mass models that bracket the exact answer from opposite sides.
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.
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Read the lesson register and outstanding review notesA uniform bar, fixed at one end and free at the other, is modelled with a single element. The mass can be lumped at the nodes or distributed consistently with the shape function. Both are standard; they do not agree; and knowing which way each errs is more useful than knowing either number.
All three answers below are multiples of (1/L)√(E/ρ), so only the coefficient is asked for.
- Uniform bar of length L, area A, modulus E, density ρ
- One axial element; axial stiffness EA/L
- Exact first frequency ω = (π/2L)√(E/ρ)
Practice
What is the exact coefficient — that is, ω L/√(E/ρ) for the continuous bar?
Practice
Lumping the mass puts ρAL/2 at the free node. What coefficient does that give?
Practice
The consistent mass matrix leaves an effective ρAL/3 at the free node. What coefficient does that give?
Practice
By what percentage does the lumped model under-estimate the exact frequency? Give the magnitude.
Neither model is 'the accurate one'. Lumped mass is cheaper and gives a diagonal mass matrix, which some solvers require; consistent mass is more faithful to the shape function. With one element they bracket the truth by about ±10%, and both converge as the mesh is refined. The error to avoid is assuming a single element of either kind is adequate because the geometry looks simple.