# An Idealized Isolator

##### 2.1 Overview

Figure 2 shows an idealized, one degree-of-freedom isolator based on a simple harmonic oscillator. It consists of three components: The isolated mass (M) represents the payload being isolated and is shown here as a single block mass with no internal resonances.

A spring (k) supports the payload and produces a force on the payload given by:

where X_{e} and X_{p} represent the (dynamic) position of the earth and payload, respectively. The third component is the damper (b), which is represented schematically as a dashpot. It absorbs any kinetic energy the payload (M) may have by turning it into heat, eventually bringing the system to rest. It does this by producing a force on the payload proportional and opposite to its velocity relative to the earth:

The presence of X_{e} in both of these equations shows that vibration of the earth is transmitted as a force to the payload by both the spring (k) and the damper (b). Rather than use the parameters (M), (k), and (b) to describe a system, it is common to define a new set of parameters which relate more easily to the observables of the mass-spring system. The first is the *natural resonant frequency* ω_{0}:

It describes the frequency of free oscillation for the system in the absence of any damping (b = 0) in *radians/second*. The frequency in cycles per second, or Hertz (Hz), is this angular frequency divided by 2π. One of two common parameters are used to describe the damping in a system: The *Quality factor Q* or the *damping ratio* ξ:

It can be shown that the transmissibility for this idealized system is:

Figure 3 plots the transmissibility of the system versus the frequency ratio ω/ω_{0} for several values of the quality factor Q. The values of Q plotted range from 0.5 to 100. Q = 0.5 is a special case called *critical damping* and is the level of damping at which the system will not overshoot the equilibrium position when displaced and released. The damping ratio ξ is just the fraction of the system’s damping to critical damping. We use Q rather than ξ because T≃Q at ω=ω_{0}, for Qs above about 2. There are several features which characterize the transmissibility shown in Figure 3:

- In the region ω<<ω
_{0}, the transmissibility for the system is ≃ 1. This simply means that the payload tracks the motion of the earth and no isolation is provided.

- In the region where ω ≃ ω
_{0}, the transmissibility is greater than one, and the spring/damper isolator amplify the ground motion by a factor roughly equal to*Q*.

- As ω becomes greater than ω
_{0}, the transmissibility becomes proportional to (ω_{0}/ω)^{2}. This is the region where the isolator is providing a benefit.

- In the region ω >> ω
_{0}, the best isolation is provided by the system with the smallest level of damping. Conversely, the level of isolation is compromised as the damping increases. Thus, there is always a tradeoff between providing isolation in the region ω> >> ω_{0}versus ω ≃ ω_{0}.

The amplitude of motion transmitted to the payload by forces directly applied to it has a slightly different form than that expressed in Equation 7. This transfer function has units of displacement per unit force, so it should not be confused with a transmissibility:

Figure 4a plots this function versus frequency. Unlike Figure 3, decreasing the *Q* reduces the response of the payload at all frequencies, including the region ω >> ω_{0}.

TMC’s MaxDamp^{®} isolators take advantage of this for applications where the main disturbances are generated on the isolated payload. Figure 4b shows the time-domain response of the payload corresponding to the curves shown in Figure 4a. This figure also illustrates the decay of the system once it is disturbed. The envelope for the decay is exp (-ω_{0}*t*/2*Q*).

There are some significant differences between real systems and the simple model shown in Figure 2, the most significant being that real systems have six degrees-of-freedom (DOF) of motion. These DOF are not independent but strongly couple in most systems. For example, “horizontal transfer functions” usually show two resonant peaks because horizontal motions of a payload drive tilt motions and vice-versa. A detailed description of this type of coupling is beyond the scope of this catalog.

##### 2.1 Pneumatic Isolators

Figure 5 shows a simplified pneumatic isolator. The isolator works by the pressure in the volume (*V*) acting on the area of a piston (*A*) to support the load against the force of gravity.

A reinforced rolling rubber diaphragm forms a seal between the air tank and the piston. The pressure in the isolator is controlled by a height control valve which senses the height of the payload and inflates the isolator until the payload is “floating.” There are many advantages to pneumatic isolators. It can be shown that the resonant frequency of the payload on such a mount is approximately:

where g is acceleration of gravity (386 *in/s ^{2}* or 9.8

*m/s*) and n is the gas constant for air and equal to 1.4. Unlike steel coil springs, this resonant frequency is nearly independent of the mass of the payload, and the height control valve always brings the payload back to the same operating height.* Gas springs are also extremely lightweight, eliminating any internal spring resonances which can degrade the isolator’s performance.

^{2}The load capacity of an isolator is set by the area of the piston and the maximum pressure the diaphragm can tolerate and is simply the product of these two numbers. It is common to rate the capacity at 80 psi of pressure. This allows a 4 in. piston to support a 1,000-lb load (for example). Though the simple isolator in Figure 5 will work, it has very little horizontal isolation and has very little damping.

*Equation 9 assumes the isolator’s pressure is high compared with atmospheric pressure. Lightly loaded isolators will exhibit a slightly higher resonant frequency.