Natural frequency of a spring-mass system and a beam

The natural frequency and period of a single degree of freedom system, the first mode of a simply supported beam from its mass per metre, the self-weight deflection estimate, and the damped frequency.

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Natural frequency of a spring-mass system and a simply supported beam.

The calculation

m_s := 500 kg
k := 2000 kN/m
f_n := sqrt(k/m_s)/(2*pi) = Hz
T_n := 1/f_n = s
E := 210 GPa
I := 8360 cm^4
mu := 80 kg/m
L := 6 m
f_1 := pi/(2*L^2)*sqrt(E*I/mu) = Hz
delta := 5*mu*g*L^4/(384*E*I) = mm
f_est := 17.75/sqrt(delta/(1 mm)) =
zeta := 0.02
f_d := f_1*sqrt(1 - zeta^2) = Hz

Results for these inputs

Spring-mass system: 10.07 Hz, period 0.099 s.
Beam, first mode: 20.44 Hz.
Self-weight deflection 0.754 mm, and the quick estimate 17.75/√δ gives the same 20.4 Hz.
With 2 % damping the frequency is practically unchanged.

How it works

f = (1/2π)√(k/m) for a spring and mass. For a simply supported beam the first mode is f₁ = (π/2L²)√(EI/μ), with μ the mass per unit length. The 17.75/√δ rule (δ in mm under self-weight) is a handy check. Units such as kN/m, kg and Hz are checked for you.

These calculations use classical textbook mechanics, not a design code. For design, use the method, factors and checks that apply to your work and have the result checked by a competent person.

Go further

Solve a two-mass system with eigs on the stiffness and mass matrices, or step the motion in time with ode; see the equation solver.

Questions

How do I find the natural frequency of a beam?

For a simply supported beam: f₁ = π/(2L²)·√(EI/μ). Write it with units in Equation Pad and ask for = Hz.

What is the 17.75/√δ rule?

For a simply supported beam, the first natural frequency in Hz is about 17.75 divided by the square root of the self-weight deflection in millimetres.

Also used for

Equation Pad runs in your browser, with nothing to install, and we never see or store your calculations. Results must be checked by a competent person before they are relied on; see the disclaimer.

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