Natural frequency of a cantilever beam

In the previous section, we considered the static deflection of a cantilever beam caused by an external force.
Now suppose the external force is suddenly taken away, causing the beam to vibrate.
As no force is applied anymore, this is called free vibration.
The frequency at which this happens is called the resonance frequency or *natural frequency* of the cantilever beam.
When a vibration is introduced the vertical displacement *w(x)* is not only a function of *x* anymore;
the vibration causes the fragments of the beam to move vertically.
At every time instant the shape of the beam, and so the positions of the beam fragments, is different compared to the previous time instant.
Therefore the displacement *w(x)* is written as *w(x,t)* from now on.
The sequence of the beam being at rest at the maximum deflection points and then starting to move again, implies a lateral acceleration.
Now Newton’s second law of motion, *F = m a*, comes into play.
Acceleration is equal to the second derivative of the position; the lateral acceleration of the cantilever beam can be written as *d*^{2}w/dt^{2}.
The equation of motion of the beam is given by:

Once the external force is released elements of the beam move until their point of maximum deflection is reached. Once the beam is at its maximum deflection, all the kinetic energy of the vibration is converted to potential energy. As the beam is not in its equilibrium shape, it wants to return to the equilibrium shape. At the maximum deflection point the beam is at rest for an infinitely short amount of time, then it starts moving back again. Once it crosses the horizontal axis, where the deflection is zero, it doesn’t contain any potential energy anymore. When no damping is assumed all the potential energy stored in the beam at its maximum deflection is converted into kinetic energy.

Suppose the beam is divided into an infinitely large number of infinitely small slices.
Each of those slices has a volume which is equal to the cross-sectional area of the beam multiplied by the width of the slice.
Multiply this by the density of the material, and the mass of the slice is obtained ().
Notice that in the equations we assume the cross-sectional area of the beam to be the same everywhere.
If this isn’t the case, we could define a function *A(x)* which calculates the cross-sectional area of the beam at every *x* position.
Now the equation of motion looks like ().

The shear force caused by the neighboring slice at the left is called *V(x,t)*.
The shear force caused by the neighboring slice at the right is called *V(x,t) + dV(x,t)*.
Notice the sign conventions in the drawing. As we don’t assume any external force, the sum of the forces in vertical direction is given by ().

The equation of motion in lateral direction of a cantilever beam now looks like (). From this, the natural frequency can easily be obtained.