Measurement methods · Inverse models
Inferring modulus from a bending resonance
How can a tap become an elastic modulus?
For a uniform slender rectangular strip, measured geometry, mass and an identified free–free bending frequency can be combined in an Euler–Bernoulli beam model.
Try the interactive demoStart with something familiar
A scale infers mass from its response to a load. A resonance test also works backwards: it observes a frequency, then uses a physical model and known dimensions to infer an elastic property.
Measure the inputs
L, b, h, m
Identify → infer
Use the matching physical model
Notice what changes
Start with a clearly defined object: a uniform slender rectangular strip. Measure its length, width, bending thickness and mass before interpreting the tap.
Understand why
The forward model predicts frequency from material, geometry and boundaries. Inversion solves the same relationship for the unknown E′; it is not a separate direct sensor for modulus.
Connect it to wood and making
A reliable result needs the right first bending mode and the right geometry. More precise frequency digits cannot repair a wrong thickness or a broad plate treated as a strip.
Put names to the ideas
The forward relation
Bending frequency depends on EI, mass per length, length and a boundary root β. For the first nonzero free–free bending mode β₁ ≈ 4.73004.
Invert only the assumed model
With I = bh³/12 and mass m, solve the same relation for E′. The formula assumes uniform geometry, small motion and negligible shear deformation and rotary inertia.
A plate is not a strip
A wedge, broad plate or arched violin plate does not automatically satisfy this model. Use a validated geometry-specific method rather than inserting its dimensions into a rectangular-strip equation.
In the inverted equation, measured mass and frequency sit beside the dimensions. Frequency is squared and thickness is cubed in the denominator, so some reading errors are strongly amplified.
A worked example
The teaching strip is 400 × 40 × 10 mm and 64 g, with E′ = 12 GPa. Its predicted first bending frequency is about 352 Hz. The calculator estimates E′ back from changed readings.
Explore it yourself
Change one reading at a time and inspect the estimated density and modulus. Values are SI-converted locally and are not saved as material measurements.
- ρ
- 400.0 kg/m³
- E′
- 12.00 GPa
- Density deviation
- 0.00%
- Modulus deviation
- 0.00%
Baseline: 400 × 40 × 10 mm, 64 g, 351.88 Hz. Only a uniform slender strip; changes represent reading errors, not a carved plate.