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Aluminium cantilever with a tip mass: exact frequency against Rayleigh's and Dunkerley's estimates

Tool	Modal Estimates
Source	Stowell, E. Z., Schwartz, E. B., Houbolt, J. C., and Schmieder, A. K. (1944). Bending and Shear Stresses Developed by the Instantaneous Arrest of the Root of a Cantilever Beam With a Mass at Its Tip. NACA Memorandum Report L4K30, National Advisory Committee for Aeronautics, Washington, DC. Eqs. (1)–(2) and the table of θ_n; Leissa, A. W. (1969). Vibration of Plates. NASA SP-160, Table 4.46 (cantilever beam eigenvalues).
Level	Undergraduate

Problem

A 30 in aluminium cantilever of 2 × 0.25 in rectangular section (E = 10 Msi, ρ = 0.100 lb/in³) carries a 1 lbm mass at its free end. Find the first three natural frequencies, the bare-beam fundamental, and compare the exact first frequency with Rayleigh's estimate from the static deflection shape and with Dunkerley's lower bound.

Inputs to Enter

Structure: beam with a tip mass, clamped–free; L = 30 in, rectangle b = 2 in, h = 0.25 in (A = 0.5 in², I = 0.0026042 in⁴)
E = 10.0 Msi, ρ = 0.100 lb/in³ (m = 0.05 lbm/in, beam mass 1.5 lbm); tip mass M_p = 1 lbm, r = M_p/(mL) = 2/3; 3 modes

Open the Modal Estimates tool (loaded with this example)

Answers

EI = 26,042 lb·in²; bare cantilever f_1 = (1.8751)² √(EI/m) / (2π L²) = 8.817 Hz (β_n L = 1.8751041, 4.6940911, 7.8547574, SP-160 Table 4.46)
Tip-mass frequency equation 1 + cos θ cosh θ + r θ (sinh θ cos θ − cosh θ sin θ) = 0 (MR L4K30 eq. 2) at r = 2/3: θ = 1.3496, 4.0736, 7.1633 → f = 4.568, 41.61, 128.7 Hz (numpy FEM, 40 elements: 4.56777 / 41.61374 / 128.6753 Hz)
Rayleigh, static shape under the tip load: ω² = 3EI / (L³ (M_p + 33 mL/140)), k_s = 2.894 lb/in, m_eff = 0.3536 lbm → 4.572 Hz (+0.10 %); gravity shape 4.573 Hz (+0.12 %)
Dunkerley: 1/ω² = 1/ω_beam² + M_p/k_s → 4.555 Hz (−0.28 %); the exact value lies between the two bounds

Notes

The tool's default inputs are this beam; press Submit. The library's transfer-matrix roots reproduce every entry of the MR L4K30 table of θ_n for r = 0 to 6 and Storch's (NASA CP N94-14623) Table 2 at m* = 5 to 20 (test_modal_estimates.php, 297 checks, with an independent numpy / scipy check in References/Modal-Estimates/checks/).