1
Compute section properties
Use an aluminium-like beam with E = 70 GPa, ρ = 2700 kg/m³, L = 0.60 m, b = 0.030 m, and h = 0.003 m.
Compute A = b h, I = b h³/12, EI and mass per unit length ρA.
CA04 exercise · Core structural FEM exercise
Euler–Bernoulli Beam Modal Frequencies
Compute beam section properties, estimate analytical cantilever bending frequencies, assemble a small Euler–Bernoulli FEM model and interpret damping and acoustic relevance.
60–90 minLinked to FEM for Structure-Borne SoundDashboard progress enabled
Aim
From beam theory to a finite-element reference calculation
This exercise connects the structural FEM lecture to a concrete beam calculation. You first compute the beam section quantities by hand, then compare analytical cantilever frequencies with a small Hermite-beam FEM model.
2
Identify the beam element degrees of freedom
A two-node Euler–Bernoulli beam element uses transverse displacement and rotation at each node: [w1, θ1, w2, θ2]. This is different from the acoustic FEM element, where pressure was the primary scalar unknown.
Why does the Euler–Bernoulli beam element include nodal rotations?
3
Check the first cantilever frequency
Use the first cantilever factor βL = 1.875104. For this beam the first frequency is about 6.85 Hz.
Python exercise
Run the beam modal calculation
The code compares analytical cantilever bending frequencies with a small Euler–Bernoulli FEM eigenfrequency calculation.
Expected observation
The FEM frequencies should approach the analytical cantilever values. Increasing nel improves the numerical approximation.
Teaching note
This is a useful exercise before coupling structural surface velocity to acoustic radiation in later BEM or hybrid methods.
Written submission
Interpretation answers
Validation comment
How would you check whether this beam FEM model is reliable before using it in an acoustic radiation calculation?
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