Beams Natural Vibration Frequency
Estimate structures natural vibration frequency.
Vibrations in a long floor span and a lightweight construction may be an issue if the strength and stability of the structure and human sensitivity is compromised. Vibrations in structures are activated by dynamic periodic forces  like wind, people, traffic and rotating machinery.
There are in general no problems with vibrations for normal floors with span/dept ratio less than 25. For lightweight structures with span above 8 m (24 ft) vibrations may occur. In general  as a rule of thumb  the natural frequency of a structure should be greater than 4.5 Hz (1/s) .
Structures with Concentrated Mass
f = (1 / (2 π)) (g / δ) ^{ 0.5 } (1)
where
f = natural frequency (Hz)
g = acceleration of gravity (9.81 m/s^{2})
δ = static dead load deflection estimated by elastic theory (m)
Note!  static dead load for a structure is load due to it's own weight or the weight of mass that is fixed to the structure.
Structures with Distributed Mass
General rule for most structures
f = a / (δ) ^{ 0.5 } (2)
a = numerical factor (in general 18)
The numerical factor a can be calculated to 15.75 for a single lumped system but varies in general between 16 and 20 for similar systems. For practical solutions a factor of 18 is considered to give sufficient accuracy.
Simply Supported Structure  Mass Concentrated in the Center
For a simply supported structure with the mass  or load due to gravitational force weight  acting in the center, the natural frequency can be estimated as
f = (1 / (2 π)) (48 E I / M L^{3} ) ^{ 0.5 } (3)
where
M = concentrated mass (kg)
Simply Supported Structure  Sagging with Distributed Mass
For a simply sagging supported structure with distributed mass  or load due to gravitational force  can be estimated as
f = (π / 2) (E I / q L^{4} ) ^{ 0.5 } (4)
Example  Natural Frequency of Beam
The natural frequency of an unloaded (only its own weight  dead load) 12 m long DIN 1025 I 200 steel beam with Moment of Inertia 2140 cm^{4} (2140 10 ^{ 8 } m^{4} ) and Modulus of Elasticity 200 10^{9} N/m^{2} and mass 26.2 kg/m can be calculated as
f = (π / 2) ((200 10^{9} N/m^{2}) (2140 10 ^{ 8 } m^{4} ) / (26.2 kg/m) (12 m)^{4} ) ^{ 0.5 }
= 4.4 Hz  vibrations are likely to occur
The natural frequency of the same beam shortened to 10 m can be calculated as
f = (π / 2) ((200 10^{9} N/m^{2}) (2140 10 ^{ 8 } m^{4} ) / (26.2 kg/m) (10 m)^{4} ) ^{ 0.5 }
= 6.3 Hz  vibrations are not likely to occur
Simply Supported Structure  Contraflexure with Distributed Mass
For a simply contraflexure supported structure with distributed mass  or dead load due to gravitational force  can be estimated as
f = 2 π (E I / q L^{4} ) ^{ 0.5 } (5)
Cantilever with Mass Concentrated at the End
For a cantilever structure with the mass  or dead load due to gravitational force  concentrated at the end, the natural frequency can be estimated as
f = (1 / (2 π)) (3 E I / F L^{3} ) ^{ 0.5 } (6)
Cantilever with Distributed Mass
For a cantilever structure with distributed mass  or dead load due to gravitational force  the natural frequency can be estimated as
f = 0.56 (E I / q L^{4} ) ^{ 0.5 } (7)
Structure with Fixed Ends and Distributed Mass
For a structure with fixed ends and distributed mass  or dead load due to gravitational force  the natural frequency can be estimated as
f = 3.56 (E I / q L^{4} ) ^{ 0.5 } (8)
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