**Tuning (Changing the Natural Frequency)**

Machines have a natural frequency, and if they move, then they have a forcing frequency.

The frequency ratio is the machine’s natural frequency divided by the forcing frequency.

Machines that move also have an amplification factor, which is:

If we plot the amplification factor vs the frequency ratio, then we will be able to visually see where resonance occurs within a range of natural frequencies.

Resonance occurs when the frequency ratio is equal to 1 and the natural frequency of the system is equal to the forcing frequency of the system.

Mechanical resonance is horrible; it can lead to a machine tearing itself apart in many cases.

The frequency range around resonance is also just as bad, as it causes large unwanted amplitudes within the system. So, it is best to have a system where the natural frequency is well outside this range if possible.

This is where tuning or changing of the natural frequency can come into play, along with many other methods to reduce unwanted vibration.

The natural frequency of the machine can be changed or (tuned) so that its outside the range of resonance.

The natural frequency of a single degree of freedom system can be found with this equation:

where k is the system’s stiffness, and m is the mass of the system.

**Ways to increase or decrease the natural frequency**

- Increase or Decrease the Stiffness
- Increase or Decrease the Mass

In some situations, manipulating the mass of the system is not feasible so manipulating the stiffness is a method that is used more often.

The location on where to change the stiffness of a system matters because manipulating the stiffness at any nodes of the system will not do anything. It is important to change the stiffness of a system at its antinodes.

**Vibration Isolation**

Isolation is similar to tuning and accomplishes the same goal of reducing unwanted vibration but with a different method that includes the additional components called isolators.

Isolation deals with reducing the force of a system that is being transmitted to a foundation.

Below is a video demonstration of vibration isolation:

Vibration Isolation Demonstration – YouTube

The transmissibility ratio is the force transmitted to the foundation divided by the force of the system:

For a 1 degree of freedom system with an isolator that has damping c and stiffness k , and system mass m the amount of force transmitted to a foundation is:

This equation can be simplified by utilizing the frequency ratio, damping ratio, and the critical damping coefficient.

If we plot the Transmissibility Ratio vs Frequency Ratio then we will have a graph that looks similar to this.

Adding Isolators will change the systems natural frequency. In order to achieve Isolation, the system’s new frequency ratio should be greater than sqrt(2) and preferably even greater than that for some cases.