How this calculator works
These undamped linear models estimate a natural frequency rather than a forced response. Spring mode treats the spring mass as negligible. A static-deflection model uses deflection under the supported weight. Point-mass beam models neglect beam mass and use centre displacement for a simply supported beam or tip displacement for a cantilever. Uniform-beam modes use Euler-Bernoulli theory and include mass per unit length. The custom wave number supports a separately derived boundary or contraflexure model; it is not an automatic model of every structure.
Formula
omega = sqrt(k/m); f = omega/(2 pi). Static deflection: omega = sqrt(g/delta). Point-mass beams: k = 48EI/L^3 (central simple span), or 3EI/L^3 (cantilever tip). Uniform beam: omega = beta^2 sqrt(EI/(mu L^4)).
Worked example
A 400 N/m spring supporting 1 kg has omega = 20 rad/s and f = 3.18310 Hz. If stiffness is increased to 1600 N/m, frequency becomes 6.36620 Hz.
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Frequently asked questions
Does this include damping?
No. Damped and forced-response frequencies can differ from the undamped natural frequency.
Does a tip-mass beam include the beam mass?
No. Use the distributed model for beam mass alone, or a model combining both mass distributions when both are significant.
What modal factors are used?
The first flexural wave numbers beta L are pi for simply supported, 1.875104 for cantilever and 4.730041 for fixed-fixed uniform beams.