# Creating a Vibration Transmissibility Plot

A vibration transmissibility plot is a useful way of seeing the response of a harmonic oscillator.  In this article we'll go over the basics of undamped and damped vibration as well as understanding what a vibration transmissibility plot means.

In the context of this article, we will be considering spring - mass and spring - mass - damper systems with linear motion.

## Transmissibility

Transmissibility can be thought of in two ways, depending on the situation.

• The force transmitted to the base or foundation on which the system is mounted.
• The displacement of the mass as a result of displacement of the base.

#### Force Transmitted to Base

In this situation a sinusoidal force is applied to the mass.  The force is transmitted through the spring-damper to the base.

#### Base Displacement Transmitted to Mass

In this situation the base has sinusoidal vertical movement.  The movement is transmitted through the spring - damper to the mass.

For force transmission, transmissibility is the ratio of transmitted force to the applied force.

For base displacement, transmissibility is the ratio of mass movement to base movement.

The equations for transmissibility are the same for both types.

## Natural Frequency

When a system is excited, such as pulling on a spring - mass system and letting go, the mass will oscillate at its natural frequency.  For linear spring - mass systems the following formulas are used.

## Undamped Vibration

An undamped spring - mass system is shown below.  The mass is linked to the base via the spring.  For the sake of illustration, lets assume the mass is a rotating machine with an imbalance that causes vibration.  The force created from the imbalance is transmitted via the spring to the base.

The transmissibility of an undamped linear system is given by:

### Undamped Transmissibility Plot

MEboost can create transmissibility plots within seconds.  It was used to create the plot below.  f/fn on the x-axis the ratio of forcing function to natural frequency.  Note how transmissibility spikes when the forcing frequency is near the natural frequency.

For a purely undamped system, transmissibility is infinite at the natural frequency.

## Damped Vibration

A damped spring - mass system is shown below.  The mass is linked to the base via the spring and damper.  As before, the force generated by the imbalance is transmitted to the base.  However, the force is now transmitted by the spring and damper.

### Damped Vibration Transmissibility Plot

A transmissibility plot of the damped system is shown below.  Note that maximum transmissibility is still when forcing frequency approaches the natural frequency.  However, the maximum value is less than the undamped system.

## Effect of Damping Ratio on Transmissibility

Let's see how changing the damping ratio affects transmissibility.  The plot below shows transmissibility for various damping ratios.  Maximum transmissibility occurs at the natural frequency when undamped.  As damping ratio increases, maximum transmissibility decreases and the maximum occurs at less than the natural frequency.

For all damping ratios, transmissibility is 1, at f/fn = 2^.5 = 1.41.

For f/fn > 1.41, transmissibility decreases less as damping ratio increases.

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