Natural Frequency of Free Transverse Vibrations

In the previous article, we have discussed how to find the Natural Frequency for Free Longitudinal Vibrations. In this article, we are going to discuss the Natural Frequency of Free Transverse Vibrations.

As we know the natural Frequency of the Free Transverse Vibrations can be determined by the following methods.

1. Equilibrium Method
2. Energy Method
3. Rayleigh’s method

All these three methods are discussed in detail in the previous article. But for now, we will determine the natural frequency with the help of the Equilibrium method only.

Equilibrium Method for determining Natural frequency of free Transverse Vibrations

Let’s consider a shaft whose one end is fixed and on the other hand, is attached to a body of weight W as shown in the below figure.

Where

• s = Stiffness of shaft
• δ = Static deflection due to the weight of the body
• x = Displacement of the body from the mean position after time t
• m = Mass of body = W/g (Shaft mass is neglected in this case)

Now, let’s see the body is in the equilibrium position, the gravitational pull given by

W = m.g

This gravitational pull will be balanced by the shaft stiffness. so that we can write

W = s. δ

Let’s give a displacement to the mass m by a distance x from its equilibrium position.

The restoring force will be

=W-s(δ+ x)

=W-sδ-sx

= – s.x ( ∵ W = s. δ)

Taking upward force as negative

and accelerating force will be the Mass × Acceleration

Taking downward force as positive

From the above two equations of motion of the body of mass m after time t is given by

From the fundamental equation of the simple harmonic motion of the body is

So equating these two similar equations will get us

∴ Time period

The natural frequency

The shape of the curve, into which the vibrating shaft deflects, is identical with the static deflection curve of a cantilever beam loaded at the end.

We have a proven formula for the static deflection of a cantilever beam loaded at the free end from the textbook on Strength of Materials is

Where

• W = Load at the free end, in newtons
• l = Length of the shaft or beam in metres
• E = Young’s modulus for the material of the shaft or beam in N/m²
• I = Moment of inertia of the shaft or beam in m⁴

Conclusion

We have derived the equation for finding the natural frequency of the Free Transverse Vibrations with Equilibrium Method. The Energy Method, Rayleigh’s methods you can try if you wanted to by taking the previous article. If you have any further thoughts on this topic, let us know in the comment section below.