In order to quantify vibrations, the following descriptions will be used: vibration displacement (amplitude), velocity, frequency, acceleration, damping and transmissibility.

Displacement is the amplitude of a point on a structure. In the example, Fig. 1, the maximum displacement is D_o Displacement varies with time in a dynamic system.

When displacement is discussed, the maximum displacement is usually referenced. In a simple harmonic system, displacement can be represented mathematically as:

Velocity is the speed of a point in a system. It is the rate of change of displacement. Velocity is the first time derivative of displacement and can be expressed as follows:

Acceleration is the rate of change of velocity of a point in a system. Normally, accelerometers are used to detect accelerations. When measuring small vibration levels, an accelerometer is more effective compared to displacement, or velocity transducers. Acceleration can be expressed as follows:

Every structure has a natural frequency. Consider the mass on the spring in Figure 2. As the hand moves up and down slowly, the mass will also move. The displacement of the mass will be the same as that of the hand. In other words, 100% of the displacement going into the system (the hand's movement) is translated directly into displacement of the spring mass system. As the speed of the hand's movement up and down is increased (an increase in the frequency), the mass will begin to move with greater amplitude than that of the hand. This

condition is known as resonance. At resonance, the displacement of the mass is greater than that of the input displacement (the movement of the hand).

The input displacement is amplified at resonance. Resonance occurs when the disturbing frequency equals the natural frequency of the system. In the example of Figure 2, the hand simulates the motion of the floor, and the spring mass system simulates the motion of the system to be isolated.

The natural frequency is usually designated by Fn. Natural frequency can be written as:

For air systems, k (the spring constant) is replaced by the spring constant of the air suspension system. The mass term cancels out and the resonant frequency of an ideal air system can be represented as follows:

In a passive system, the proper interactions between the resonances of an isolation system and the environment are key parameters to successful vibration isolation. A 2 Hz passive isolation system can actually amplify ambient vibration in the 1-5 Hz range, but the resonance of the support in an active system can be removed by the control loop. As noted from our previous example of the spring mass system in Fig. 2, amplification