Uniform Cantilever Frequency and Engine-Order Coincidence
Calculate the first three bending frequencies of an ideal, nonrotating, uniform rectangular Euler-Bernoulli cantilever and solve exact equalities with user-supplied straight engine-order lines.
Controlled inputs
Reference results
Exact straight-line equalities in the entered interval
For each entered order q and each nonrotating reference mode, the worksheet solves N = 60 f / q. It does not apply a separation margin and does not decide whether resonance, acceptable response, or safe operation exists.
| Reference mode | Nonrotating frequency (Hz) | Entered order q | Equality speed (r/min) | Nghĩa |
|---|
Method and units
A = b h
I = b h³ / 12
fⱼ = βⱼ² / (2π L²) × √(E I / (ρ A))
fEO = q N / 60
Nequality = 60 fⱼ / q
- β₁ = 1.8751040687, β₂ = 4.6940911330 and β₃ = 7.8547574382 are the first three roots for an ideal fixed-free uniform Euler-Bernoulli beam.
- L, b and h are converted from millimetres to metres; E is converted from gigapascals to pascals; ρ is in kg/m³. The resulting f is in s⁻¹ (Hz).
- The rectangular bending axis matters because h is cubed. Swapping b and h generally changes the frequency.
- The equations are classical analytical mechanics, not a formula or acceptance criterion issued by ISO, API, ASME or another standard.
- Higher modes are more sensitive to shear deformation and rotary inertia; Euler-Bernoulli assumptions should be checked before using even this reference value.
Evidence and scope
Derives the continuous uniform cantilever relation, the characteristic equation 1 + cos β cosh β = 0 and the first root 1.875104 under Euler-Bernoulli assumptions.
Official MIT PDFLists the nonrotating uniform cantilever bending-frequency relation and βL values 1.875, 4.694 and 7.855 for modes 1-3.
Official NASA NTRS PDFDefines a positive slope in the stationary reference frame as the number of excitations per rotor revolution. This supports fEO = qN/60; it does not turn a constant nonrotating beam frequency into a real blade mode.
Official Ansys documentationUses alternating speed-dependent static prestress and perturbed modal solutions and identifies Coriolis handling. Those analyses are outside this worksheet.
Official Ansys documentationDiscusses a blade modal-frequency/engine-order crossing as a condition requiring forced-response analysis; response and fatigue cannot be inferred from equality alone.
Official NASA NTRS PDFSources accessed 16 July 2026. No closed standard is claimed or paraphrased as an acceptance rule.
Interpretation questions
Is this a Campbell diagram?
No. A real Campbell diagram follows modal frequencies as rotational speed changes. This worksheet holds ideal nonrotating beam frequencies constant and solves equalities with straight user-supplied order lines.
Does a listed equality prove resonance?
No. It is a screening flag. Actual response also depends on the real rotating mode, forcing distribution and amplitude, modal participation, damping and boundary conditions.
Does an empty table prove safe operation?
No. The simplified model may miss the real modal branch, and the entered order list may omit an excitation. No safety, fatigue-life or acceptance conclusion is produced.
Why is there no material database or 10% separation rule?
Material properties vary with alloy, treatment, direction and temperature, while an allowable separation rule must come from the applicable design authority and verified system model. The worksheet does not invent either.
Revision: 16 July 2026. Result classification: general-mechanics reference only.
