Parameters
Beam Type
Shape Function
Length L [m]1.0 m
Bending stiffness EI [N·m²]1000 N·m²
Mass/length ρA [kg/m]5.0 kg/m
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Rayleigh estimate [Hz]
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FEM reference [Hz]
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Error [%]
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Dunkerley [Hz]
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Effective mass ratio
Theory
Rayleigh Quotient:
$$\omega_n^2 \approx R[\varphi] = \frac{\displaystyle\int_0^L EI\left(\frac{d^2\varphi}{dx^2}\right)^2 dx}{\displaystyle\int_0^L \rho A\,\varphi^2\, dx}$$Cantilever assumed shape (polynomial): $\varphi(x) = x^2(3L - 2x)/(L^3)$
Dunkerley formula (with lumped masses):
$$\frac{1}{\omega_n^2} \approx \frac{1}{\omega_0^2} + \sum_i m_i \delta_{ii}$$Cantilever exact: $f_n = \dfrac{1.875^2}{2\pi L^2}\sqrt{\dfrac{EI}{\rho A}}$
Upper bound property: $R[\varphi] \geq \omega_1^2$ (equality when $\varphi = \phi_1$)
Applications: Pre-FEM hand-calculation to verify order of magnitude / Quick mass-addition sensitivity study / Bracketing true frequency with Dunkerley lower bound / Cross-checking experimental modal test data. Static deflection shape typically gives <2% error.