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THEORY OF TABLETOP VIBRATION

3.1 Introduction

The primary goal of a well designed optical table is to eliminate relative motion between any two (or more) components on the surface of the optical table. However, before the design of optical tables can be discussed, it is necessary to examine the underlying theory of vibrations and the nomenclature commonly used to discuss tabletop vibrations: compliance, vibrations, resonance, and damping.

3.2 Compliance

An optical tabletop is an example of a common problem in engineering and physics, namely the deformation of a body or structure in response to external forces. These forces may be static such as the sagging of a tabletop due to a large localized mass being placed upon it. Alternatively, the forces may be dynamic: acoustic vibrations in the air, vibrations of a small motor sitting on top of the optical table, or vibration transmitted from the building to the optical table through the optical table supports.

3.2.1 Definition of Compliance

The most widely used transfer function for the vibrational response of an optical table is compliance. In the case of a constant (static) force, compliance is defined as the ratio of the linear or angular displacement to the magnitude of the applied force. In the case of a dynamically varying force (vibration), compliance is defined as the ratio of the excited vibrational amplitude (angular or linear displacement) to the amplitude of the force causing the vibration.

Any deflection of the tabletop is evident by the change in relative position of the components mounted on the table surface. Therefore, by definition, the lower the compliance value is, the closer the optical table is to meeting the primary goal of optical table design, minimized deflection. Compliance is frequency dependant and is measured in units of displacement per unit force (meters per Newton).

3.2.2 Compliance Curves

To understand compliance, consider a hypothetical structure with only one vibrational degree of freedom (i.e., a structure with only one direction of deformation). For example, this structure could be a steel bar that is firmly anchored at one end and able only to vibrate in one direction (i.e., can bend only in one plane).


Figure 4. A simple one-degree-of-freedom system: a constrained bar only vibrating in one plane

All periodic vibrations can be expressed as combinations of sine and cosine functions with the appropriate amplitude, frequency, and phase. Therefore, when a single frequency sinusoidal vibration is applied to the bar, the general equation of motion is

where the left hand side pertains to the forced system and the right hand side of the equation pertains to the forcing function.

, and are the acceleration, velocity, and displacement of the mass (m) being moved, respectively, while cand k are the damping and stiffness parameters, respectively

The force F is sinusoidally varying with frequency f and maximum amplitude F0. The general expression for compliance of a system such as this is given by

If we restate the above equation in words,

A plot of compliance versus frequency shows that the compliance of a rigid body can be separated into three parts: stiffness, resonance effects, and mass effects.

Figure 5. Compliance versus frequency for a system with one degree of freedom
3.2.3 Compliance at Low Frequencies – Stiffness

At zero and low frequencies, the stiffness term dominates the compliance equation. When a low-frequency forcing vibration is applied to the unattached end of the bar, it bends in response. The amount of deflection is determined by the stiffness of the bar, which ultimately depends on its shape, the tensile modulus of elasticity (Young's modulus) of the bar material, and the method of mounting and/or constraining the bar.

Any solid body has a fixed equilibrium, or rest geometry, which is a property that distinguishes a true solid from a liquid. The equilibrium geometry corresponds to a minimum in the potential energy of the object. When forces are applied to a solid body, it can be deformed from this equilibrium shape. The potential energy of the body rises and this results in a force that acts to restore the body to its equilibrium shape.

Consider a situation where the bar has been deflected from its equilibrium position and then released. The restoring force acts to return the bar to its equilibrium position. However, even though the restoring force on the bar will be zero when it reaches its equilibrium position, the momentum of the bar will cause it to overshoot this position. As a result, the bar will oscillate about its equilibrium position. The oscillation of the bar is an example of a simple harmonic oscillator. The oscillation occurs with a characteristic frequency fn, termed the resonant frequency, given by

where fn is the resonant frequency of the oscillation, m is the mass moving during the oscillation, and k is the spring (force) constant, which is related to the shape of the bar and the Young's modulus of the material from which the bar is fabricated. In the absence of damping, this oscillation would persist forever.

In a real system such as an optical table, resonant vibrations can rarely be approximated to harmonic vibrations with such simple mathematics; however, the arguments made above can still provide insight into understanding the issues involved.

3.2.4 Compliance at Resonance

When the forcing vibration frequency is at the same frequency as the resonant frequency of the bar (table), each maximum in the velocity of the forcing vibration coincides with a maximum in the acceleration of the excited vibration. As a result, the forcing vibration increases the bar’s acceleration, which thereby accumulates vibrational energy that in turn results in an amplification of the forcing vibration. Solving Eq. (4) for m and substituting that result into Eq. (2) yields

where

is referred to as the damping ratio.

From Eq. (5) it can be seen that when the frequency of the forcing function is close to the resonant frequency, the compliance is determined solely by the damping term, and as a result, the compliance can be quite large.

Another way to picture resonance in this example is to consider what happens as the forcing function frequency is increased slowly from zero. When the forcing function frequency is near zero, the bar bends synchronously with the forcing vibration. As the frequency is increased, the bar will lag behind the forcing vibration because it has momentum and cannot reverse direction instantaneously in response to the periodic direction changes in the applied force (i.e., vibration). The phase lag will increase with frequency. At the resonant frequency, the phase lag between the oscillation of the rod and the forcing vibration will be exactly 90° because the maximum in the bar’s acceleration always coincides with the maximum in the velocity of the forcing vibration. As a result, amplification of the vibrational input occurs as shown in Fig. 6.


Figure 6. The variation of phase lag between excited and forcing vibrations for a system with a single resonance

At forcing vibration frequencies greater than the resonant frequency, the phase difference between forcing and excited vibrations in this theoretical one-resonance system is 180 ° .

In a real structure such as an optical table, there are many possible resonant modes of vibration. The phase of an excited vibration depends on the location on the table at which the measurement is made as well as on the type of vibration being excited.

3.2.5 Compliance at High Frequencies - Mass Effects

At higher frequencies, the compliance is totally dominated by the mass (inertia) of the table. For f>>fn, Eq. (4) can be approximated as

Where m is the effective mass and f is the frequency of the forcing vibration.

In certain very simple vibrational systems, such as a single mass suspended from a spring, it is fairly easy to evaluate the mass involved in the vibration. However, in real systems the mass term in the general equation for compliance can become quite complex. For instance, the structure of an optical table is such that it has several types of bending and flexural resonant modes of vibration. Therefore, different points on the table are undergoing different amplitudes of vibration. At nodal points, there is no vibrational amplitude at all for that specific vibrational node. As a result, the effective mass involved in the vibration is a complicated function that is usually determined numerically.

3.3 Modifying a Resonant Frequency

As shown in Eq. (2), the resonance frequency is dependent on the stiffness and mass of the object. Decreasing the mass and/or increasing the stiffness shifts the resonance to a higher frequency (See Fig. 7). Alternatively, increasing the mass and/or decreasing the stiffness shifts the resonant frequency lower.


Figure 7. Shifting the resonant frequency by changing the stiffness-to-mass ratio

The balance of the mass and stiffness is a simple, yet important, concept in optical table design because this allows the resonances of the optical table to be pushed to higher frequencies with lower amplitudes. Clearly, this can be accomplished merely by increasing the stiffness of the table. However, if this is accompanied by a proportional increase in mass, then the resonances will not be shifted. In order to raise the resonant frequency, the stiffness must be maximized while minimizing the mass. This is discussed in more detail in the chapter on Tabletop Design.

3.4 Damping

Damping refers to any process that causes an oscillation in a solid body to decay to zero amplitude. It is a very important phenomenon in vibration suppression or isolation in real systems because it causes energy to be diverted from vibration to other sinks

Damping is a resonant effect in that it significantly affects the compliance function at or near resonance (i.e, when f ≅ fn). The height of the compliance peak at resonance is primarily determined by the amount of damping. In the absence of damping, the peak would be infinitely high. ¦ » ¦ n .

3.4.1 Types of Damping

Materials such as wood and rubber have a large amount of natural damping. The microstructure of a well-damped material is such that deformations cause strains in the material that rapidly convert the mechanical vibrations into other forms of energy such as heat. Metals manifest a small amount of internal damping. This is principally due to the small amount of friction present at grain boundaries.

In many instances, a structure may consist of supporting elements that are quite rigid and tend to ring with little natural damping. If the structure is being designed to avoid vibrations, additional damping mechanisms may be required. There are many methods of introducing damping into a system, many of which rely on using friction to degrade the vibrational energy into heat. This is discussed in the chapter on Tabletop Design.

3.5 Multiple Resonances

Real structures have multiple resonant modes. Often, these modes are coupled so that the form of the resonant vibration is quite complex, particularly in the case of non-symmetric structures. Also, each vibrational mode often gives rise to a whole series of overtones or harmonics. However, the fundamentals of the simple 1D model developed in this chapter are still valid.

  • At low frequencies, below the first resonant frequency, the compliance of a real structure is determined by the stiffness. Stiffer structures are less compliant (i.e., they vibrate less for a given forcing vibration).
  • At frequencies above the region of the first resonance, the compliance of a rigid structure is roughly given by 1/(mf2) although resonant effects are superimposed on this.
  • The compliance at the various resonant peaks is mainly dependent on the amount of damping at these resonant frequencies and by the effective mass associated with the vibrational mode. However, the relative height of these peaks is still approximately proportional to 1/cf
3.6 Antiresonances

In systems with several vibrational degrees of freedom, such as an optical table, the compliance curve will often show negative peaks (see Fig. 8) between the resonant peaks due to the interaction between multiple resonant peaks. The phase difference between the forcing function and the vibrational response of the system changes from 0° to 180° as the frequency of the forcing function is scanned over a resonance. This results in a peak in the compliance curve. When the tails of two resonance peaks overlap, the response of the system is determined by the phasor addition of the systems response to each resonant peak. When the phase of the response due to the first resonance is 180° out of phase with the second resonance and the magnitude of the response is roughly the same, it is possible for the net compliance at this point to be very small. The negative peak is referred to as an antiresonance in the compliance curve.


Figure 8. An antiresonance, caused by two vibrational components of equal magnitude which are 180 ° out of phase

3.7 Compliance of a Real Table

The concept of an ideal rigid body is useful when considering optical table performance. This theoretical structure does not resonate and therefore has no compliance peaks. The compliance of an ideal rigid body is proportional to 1/f2, and when plotted on a log-log scale, it is represented by a straight line with slope -2. It represents the design goal when manufacturing optical tables; the nearer the actual curve fits the straight line, the better the dynamic stiffness. A compliance curve of a real table can now be examined. Figure 9 shows a typical compliance curve created by measuring the response of the corner of a Thorlabs’ optical table as the tabletop is vibrated. The corner of the optical table is used to make the measurement because the performance of the optical table is typically the worst at the corners, and as a result, a measurement made at the corner can usually be considered a worst case scenario.


Figure 9. A typical compliance curve of a high performance Thorlabs' optical table

Conventionally, this type of data is presented on a plot with logarithmic horizontal and vertical scales; a vertical logarithmic scale is used since there are a large range of compliance values while a horizontal logarithmic scale is necessary to display the wide range of frequencies without loosing the important details of the compliance curve that exist near the low frequency resonances.

Several aspects of this curve merit special comment. The initial portion of the plot (i.e., before the first table resonance) is determined primarily by the table supports, not the table itself. Also, notice how the peaks at compliance resonances decrease in size towards higher frequency. As the frequency increases, the denominator in the compliance expression (Eq. 2) increases, and therefore, the compliance is reduced. This means that as the frequency increases, a given excitation force produces a smaller amplitude excitation in the table. The low frequency peaks are the most important for two reasons. First, these are the largest peaks, corresponding to the weakest points in the compliance spectrum. Second, typical vibrations from laboratory equipment are usually below 150 Hz. The peaks should be at the highest frequency possible while keeping the compliance in the 0 to 150 Hz region as low as possible. Finally, for this particular curve, an inverted peak occurs around 120 Hz; this is a typical antiresonance caused by phase effects, which were described in section 3.6.

Summarizing the performance of an optical table with a few specifications is difficult. Even an informative compliance curve such as that shown in Fig. 9 fails to provide a complete summary of optical table specifications since the measured compliance of the table is dependent on where the measurement device is placed on the table as well as the position of the point of contact for the forcing vibration. For instance, if the forcing vibration is applied at a node of the first resonant peak, the measured compliance would be significantly reduced because the vibrations would not be able to excite that resonance in the table. However, if the compliance of the optical table is measured in such a way that it accurately describes the performance of an optical table, then the plot can be further reduced to two numbers that can be used to quantify the performance of the optical table. The first number is the frequency of the first resonance peak. Typically, most laboratory vibrational sources will have a vibration frequency between 0 and 150 Hz; higher first resonant frequencies make is less likely that a source will be able to excite the optical table at that frequency. The second number of relevance is the height of the highest peak (usually the first peak) in the compliance curve above the compliance value of an ideal body at that frequency. Often, this height is reported as the ratio of the peak height to the compliance value of the ideal body. This ratio represents the amplification (Q value) of the vibration at the resonant frequency.

Remember that one compliance curve does not describe the performance of an optical table. A few modes may have very low, or zero amplitudes at the corner of measurement. In general, a compliance curve obtained at the corner of an optical table is a reasonable measure of the worst case performance of any part of the table surface.

3.8 Table Stiffness

The measured low frequency compliance of an optical table is entirely a function of the optical table support structure used during the test. The low frequency behavior of an optical table is usually expressed as the stiffness or deflection under a measured static load. This can be measured as either a displacement of the center of the table or as an angle of deformation of the tabletop. The actual values depend on the position of the optical table supports.

3.9 Table Thickness

Compliance also varies with table thickness. The graph below shows that for a table 1.5 m x 3 m, if the thickness is increased from 210 mm to 420 mm, then the compliance improves from 6 x 10-5 mm/N to 1.4 x 10-5 mm/N.


Figure 10. Compliance versus length for optical tables of different thicknesses

Furthermore, the deflection of a supported structure increases proportionally with length. Typically, a table 4.25 m long would need to be 460 mm thick to provide the same static deflection per unit load as a table 2.0 m long and 210 mm thick.

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