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Beams Natural Vibration Frequency

Estimate structures natural vibration frequency

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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/s2)

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

f = (1 / (2 π)) (k / q)0.5                         (2)

where

q = uniform distributed static dead mass (kg/m)

k = E I / L = stiffness

where

E = modulus of elasticity (N/m2)

I = area moment of inertia (m4)

L = length (m)

General Rule for most Structures

f = 18 / (δ)0.5                         (3)

Simply Supported Structure - Mass Concentrated in the Center

Vibration - 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 L3)0.5                            (4)

where

M = concentrated mass (kg)

Simply Supported Structure - Sagging with Distributed Mass

Vibration - simply supported structure - sagging with mass distributed

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 L4)0.5                            (5)

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 cm4 (2140 10-8 m4) and Modulus of Elasticity 200 109 N/m2 and mass 26.2 kg/m can be calculated as

f = (π / 2) ((200 109 N/m2) (2140 10-8 m4) / (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) ((200 109 N/m2) (2140 10-8 m4) / (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

Vibration - simply supported structure - contraflexure with mass distributed

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 L4)0.5                            (6)

Cantilever with Mass Concentrated at the End

Vibration - cantilever structure - 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 L3)0.5                            (7)

Cantilever with Distributed Mass

Vibration - cantilever structure - mass distributed

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 L4)0.5                            (8)

Structure with Fixed Ends and Distributed Mass

Vibration - structure with fixed ends - mass distributed

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 L4)0.5                            (9)

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