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THEORY OF VIBRATION ISOLATION

5.1 Introduction to Vibration Isolation

Creating a working surface for optical setups that is free from vibrations is a two part problem. As discussed in the notes on optical tabletops, an optical table is designed to have zero (minimal) response to a deflective force or vibration. This, in itself, is not sufficient to ensure a vibration-free working surface. The rigid table may still vibrate without deforming because the entire tabletop can vibrate on top of the optical table supports. These types of vibrations are constrained translations and/or rotations of the optical table. The entire table system is subjected continually to vibrational impulses from the laboratory floor. These vibrations may be caused by large machinery within the building or even by wind or traffic-excited building resonances (swaying). Because of this, it is necessary to design optical table supports that prevent vibrations from being transmitted from the floor to the optical tabletop.

5.2 Seismic Mounting

The simplest model used in a theoretical treatment of mechanical vibrations is the ball and spring model shown in Fig. 19. Consider a ball of mass m that is suspended from a fixed surface with enormous mass via a spring. For now, we shall ignore the pendulum motions and consider a system having only one degree of freedom in the vertical direction.

Figure 19. A ball and spring example of a vibration isolation/transmission system with one degree of freedom.

5.3 Transmissability

Floor vibrations in a building can be divided into two basic types: vertical and horizontal. Typically, vertical components range from 10 to 50 Hz while horizontal components range from 1 to 20 Hz range. To prevent such vibrations from disturbing an experiment, it is important to support the optical table so that the optical table’s instantaneous position is independent of the periodic motions of the laboratory floor. This type of isolation is termed seismic mounting. When an object is truly seismically mounted with respect to the floor, the motions of the object and the floor are completely uncoupled. The term seismic is linked with the study of earthquakes since the magnitude of an earthquake is estimated from the motion of the ground with respect to a seismically mounted indicator.

In the absence of vibrational impulses, the ball in Fig. 19 will remain stationary at its rest position. Suppose the object from which the ball is suspended is not infinitely large or not infinitely stiff so that the point at which the spring is anchored starts to vibrate. Some of that vibrational energy may be transmitted to the ball, causing it to vibrate at the same frequency. The frequency of this motion is given by

Where fn is the resonant frequency of the oscillation, m is the mass moving during the oscillation, and k is the spring constant (related to the stiffness of the spring).

The flow of vibrational energy is expressed in terms of a transfer function. A transfer function is a method of quantifying how efficiently a forcing vibration can produce an excited vibration. The transfer function most applicable here is termed transmissibility and is defined as the ratio of the dynamic output to the dynamic input (i.e., the ratio of the amplitude of the transmitted vibration to that of the forcing vibration).

5.4 Resonance

In the example just outlined, the transmissibility of the spring is dependent on the frequency of the forcing vibration. The idealized transmissibility (T) of this system is given by

where the damping ratio is defined as

Here, f is the frequency of the forcing function and c is a parameter describing the damping properties of the system. The damping ratio will be discussed in more detail in section 5.5.

The graph below shows a plot of this idealized transmissibility as a function of frequency. Note the similarity to the compliance transfer function discussed in the chapter on The Theory of Tabletop Vibration (Section 3.2.4). As before, the curve can be divided into three distinct regions.

Figure 20. A typical transmissibility vs. frequency curve for a system with one degree of freedom

At low frequencies, the ball in Fig. 19 moves synchronously with the mass that the spring is suspended from and with the same amplitude (i.e., the transmissibility is unity as defined by Eq. 9). The system behaves as though the spring was rigid, and as a result, the ball is not isolated from the large mass.

As the frequency of the driving force increases, the momentum of the ball prevents the ball from moving in phase with the driving force (i.e., a change in the direction of the driving force does not instantly result in a change in the direction that the ball is moving due to the momentum of the ball). When the phase lag between the driving force and the vibration of the ball becomes exactly 90°, the system is vibrating at its natural (resonant) frequency fn.

Figure 21. The phase relationship between excited and forcing vibrations changes rapidly near resonance.

When the driving force frequency is much greater than the resonant frequency of the spring/ball system, the response of the ball is determined solely by the mass of the ball. In other words, the spring is relatively soft and the vibrational force travels slowly along as compression waves. This slow transmission effectively spreads out the oscillatory nature of the forcing vibration. Essentially, the ball experiences a time-averaged force due to its slow response to the fast moving vibrations, and unless the vibration involves a net displacement, the magnitude of the time-averaged force tends toward zero with increasing vibrational frequency. As the transmissibility approaches zero, the position of the ball is not affected by the vibration in the large mass. At this point the ball is seismically mounted.

5.5 Damping

Damping refers to any process that causes an oscillation in a system to decay to zero amplitude. It is a very important phenomenon in vibration suppression or isolation. Damping causes the energy to be diverted from vibration to other sinks. Damping in a system is usually defined as the ratio of actual damping to critical damping. Critical damping is the minimum amount of damping in a system necessary to prevent resonant oscillation, following application of an impulsive force.

Damping is a resonant effect in that primarily it affects the transmissibility function at or near resonance.

At resonance,

Using this relation, Eq. (9) can be simplified to read

The height of the transmissibility peak at resonance is mainly determined by the amount of damping. In the absence of damping, the peak would be infinitely high. Also, a system with no damping would not stop vibrating, even when the driving force is removed. Clearly, all real systems contain damping to some degree. Using the suspended ball as an example, the simplest form of damping would be to immerse the ball in a viscous medium. The drag on the ball created by the viscous medium would convert the vibrational energy to heat via friction.

5.6 Seismic Mounting of Tables

The ball and spring model suggests, in principle, a way in which an optical table could be seismically mounted. Simply suspend an optical table from a rigid ceiling by weak springs. Then, at frequencies much higher than the natural resonance of the spring isolation system, ambient building vibrations would not be transmitted to the table. Obviously, this solution has no practical value, but it elucidates the general principles involved.

Typically, ambient vibrations in a building tend to be in the range 10 to 50 Hz for vertical components and 1 to 20Hz for horizontal components. To seismically mount an optical table under these conditions requires a mounting system with a very low resonant frequency. Unfortunately, even with the use of composites, optical tables are relatively massive, typically 500 kg for a 1.2 m x 2.4 m table. Such a system would involve highly extended springs and a very large travel range of the table.

The ideal mounting system shown below has a rigid tabletop mounted to the floor in such a way that it can vibrate in vertical and horizontal directions, both with low resonant frequencies. Even though these resonant frequencies can be made lower than most building vibrations, it is important that the height, and as a result, the width of the resonant peak in the transmissibility curve be reduced. This is achieved by adding a damping mechanism, which ensures minimum oscillation at resonance and short settling times.

Figure 22. Ideal seismic mounting of an optical table consists of weak spring supports with added damping

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