Cumulative tutorial 07
Lumped or consistent mass
Two defensible mass models that bracket the exact answer from opposite sides.
A 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.
What you are given
- 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)
Tutorial 07
Work the problem
4 questions
Question 1
What is the exact coefficient — that is, ω L/√(E/ρ) for the continuous bar?
Question 2
Lumping the mass puts ρAL/2 at the free node. What coefficient does that give?
Question 3
The consistent mass matrix leaves an effective ρAL/3 at the free node. What coefficient does that give?
Question 4
By what percentage does the lumped model under-estimate the exact frequency? Give the magnitude.
What this establishes
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.