Formula for Free Resonating Frequency of Cantilever to the Deflection

[nyxynyx]

Is there a formula relating the free resonating frequency of a cantilever to its deflection? Thanks!

 

[Gokul43201]

The resonant frequency is a function of material and geometric properties of the beam and is independent of the deflection.

For a cantilever beam of uniform cross-section,

$$f=\frac{1}{2\pi} \sqrt {\frac{8EI}{\rho Al^4}}$$

I: moment of inertia (depends on cross section)
E: Young's modulus
$\rho$: density
A: cross-section area
$l$: length

 

[ravenprp]

Yes, there is a formula that relates the free resonating frequency of a cantilever to its deflection. It is known as the Rayleigh frequency formula and is given by:

f = (1/2π) * √(E*I/mL^3)

Where:
f = free resonating frequency (Hz)
E = Young's modulus (N/m^2)
I = moment of inertia (m^4)
m = mass of the cantilever (kg)
L = length of the cantilever (m)

This formula shows that the free resonating frequency is inversely proportional to the cube of the cantilever's length and directly proportional to the square root of its stiffness (EI) and mass (m). Therefore, a longer and stiffer cantilever with a smaller mass will have a higher resonating frequency and a smaller deflection.

I hope this helps!