3.2.1 The Kelvin clamps

Type I and Type II Kelvin clamps are examples of fully constrained systems, that is ones with six constraints. When designed properly, these clamps are very effective where accurate re-positioning is required and can be stable to within nanometres [8], although around 0.1 µm is more usual.

Both clamps have a top plate (on which, for example, the object to be measured is placed) that has three rigid spheres (or hemispheres) spaced on a diameter. The three spheres then contact on a flat and in a vee and a trihedral hole, as in Figure 3.2(a), or in three vee-grooves, as in Figure 3.2(b). In the Type II clamp, it is easy to see where the six points of contact, that is constraints are – two in each vee-groove. In the Type I clamp, one contact point is on the flat, two more are in the vee-groove and the final three are in the trihedral hole. The Type I clamp has the advantage of a well-defined translational location based on the position of the trihedral hole, but it is more difficult to manufacture. A trihedral hole can be produced using an angled vee-shaped milling cutter (Figure 3.3(b)) or by pressing three spheres together in a flat-bottomed hole – the contacting spheres will then touch at a common tangent (see Figure 3.3(a)). For miniature structures, an anisotropic etchant can be used on a single crystalline material [9]. The Type II clamp is more symmetrical and less influenced by thermal variations. Note that the symmetrical groove pattern confers its own advantages (symmetry) but is not a kinematic requirement; any set of grooves will do, provided that they are not all parallel.

3.2.2 A single degree of freedom motion device

There are many methods for producing single degree of freedom motion (see, for example, Ref. [10]). One method that directly uses the idea of single point contacts is the prismatic slideway [3]. The contact points are distributed on two non-parallel flat surfaces as shown in Figure 3.4. In practice, the spheres would be attached to the carriage (and would usually be just parts of a sphere to give a point contact). The degrees of freedom in the system can be deduced by considering the loading necessary to keep all five spheres in contact. Firstly, the three-point support could be positioned onto the horizontal plane, resulting in a linear constraint in the z-axis and rotary constraints about the x- and y-axes. A carriage placed on this plane is free to slide in the x-direction until either of the two remaining spheres contacts the vertical face. The x-axis linear degree of freedom is then constrained. Further horizontal force would cause the carriage to rotate until the fifth sphere comes into contact, removing the rotary degree of freedom about the z-axis. This gives a single degree of freedom linear motion along the y-axis.

3.6.1 The structural loop

A structural loop is an assembly of mechanical components that maintain relative position between specified objects. Using a stylus surface texture measuring instrument as an example (see Section 6.6.1 and Figure 6.8), the structural loop runs along the base plate and up the bridge, through the probe, through the object being measured, down through the x-slideway and back into the base plate to close the loop. It is important that the separate components in the structural loop have high stiffness to avoid deformations under loading conditions – deformation in one component will lead to uncompensated dimensional change at the functional or measurement point.

3.6.2 The thermal loop

A thermal loop is described as a path across an assembly of mechanical components, which determines the relative position between specified objects under changing temperatures [6]. Much akin to mechanical deformations in the structural loop, temperature gradients across an instrument can cause thermal expansion and resulting dimensional changes. It is possible to compensate for thermal expansion by choosing appropriate component lengths and materials. If well designed, and if there are no temperature gradients present, it may just be necessary to make the separate components of an instrument from the same material. Thermal expansion can also be compensated by measuring thermal expansion coefficients and temperatures and applying appropriate corrections to measured lengths. This practice is common in gauge block metrology where the geometry of the blocks being measured is well known [23]. Obviously, the effect of a thermal loop can be minimised by controlling the temperature stability of the room, or enclosure, in which the instrument is housed.

3.6.3 The metrology loop

A metrology loop is a reference frame for displacement measurements, independent of the instrument base. In the case of many surface texture measuring instruments or CMMs, it is very similar to the structural loop. The metrology loop should be made as small as possible to avoid environmental effects. In the case of an optical instrument, relying on the wavelength of its source for length traceability, much of the metrology loop may be the air paths through which the beam travels. Fluctuations in the air temperature, barometric pressure, humidity and chemical composition of these air paths cause changes in the refractive index and corresponding changes to the wavelength of the light [24,25]. This can cause substantial dimensional errors. The last example demonstrates that the metrology and structural loops can be quite different.

3.7.1 Minimising thermal inputs

Thermal distortions will usually be a source of inaccuracy. To find a performance index for thermal distortion, consider a horizontal beam supported at both ends of length L and thickness h [26]. One face of the beam is exposed to a heat flux of intensity Q in the y-direction that sets up a temperature, T, gradient, dT/dy, across the beam. Assuming the period of the heat flux is greater than the thermal response time of the beam, then a steady state is reached with a temperature gradient given by

$Q=-\lambda \frac{\mathrm{d}T}{\mathrm{d}y},$ (3.5)

(3.5)

where λ is the thermal conductivity of the beam. The thermal strain is given by

$\epsilon =\alpha (T_0-T),$ (3.6)

(3.6)

where α is the thermal expansion coefficient and T₀ is the ambient temperature. If the beam is unconstrained, any temperature gradient will create a strain gradient, dε/dy in the beam causing it to take up a constant curvature given by

$K=\frac{\mathrm{d}\epsilon }{\mathrm{d}y}=\alpha \frac{\mathrm{d}T}{\mathrm{d}y}=\frac{\alpha }{\lambda }Q.$ (3.7)

(3.7)

Integrating along the beam gives the central deflection of

$\delta =C_1{L}^{2}Q\left(\frac{\alpha }{\lambda }\right),$ (3.8)

(3.8)

where C₁ is a constant that depends on the thermal loads and the boundary conditions. Thus, for a given geometry and thermal input, the distortion is minimised by selecting materials with large values of the performance index

$M_Q=\frac{\lambda }{\alpha }.$ (3.9)

(3.9)

In Refs. [3,27], they arrive at the same index by considering other types of thermal load. If the assumption that the period of the heat flux is greater than the thermal response time of the beam is not valid, then the thermal mass of the beam has to be taken into account [27]. In this case, thermal conductivity is given by

$\lambda =\frac{D}{\rho C_{\mathrm{p}}},$ (3.10)

(3.10)

where D is the thermal diffusivity of the beam material, ρ is its density and C_p is its specific heat capacity. In the case of a room with stable temperature and very slow heat cycling, Eq. (3.9) is normally valid.

3.7.2 Minimising mechanical inputs

There are many types of mechanical input that will cause unwanted deflections of a metrology frame. These include elastic deflections due to self-weight, loading due to the object being measured and external vibration sources. To minimise elastic deflections, a high stiffness is desirable. The elastic self-deflection of a beam is described by

$y=C_2\frac{W{x}^3}{EI},$ (3.11)

(3.11)

where W is the weight of the beam, E is the Young’s modulus of the beam material, I is the second moment of area of the cross section and C₂ is a constant that depends on the geometry of the beam and the boundary conditions. It can be seen from Eq. (3.11) that, for a fixed design of instrument, the self-loading is proportional to ρ/E – minimising this ratio minimises the deflection.

The natural frequency of a beam structure is given by

${\omega }_n=C_3\sqrt{\frac{EI}{m{l}^3}},$ (3.12)

(3.12)

where n is the harmonic number, m is the mass per unit length of the beam, l its length and C₃ is a constant that depends on the boundary conditions. Again, for a fixed design of instrument, ω_n is directly proportional to $\sqrt{E/\rho }$. For a high natural frequency and, hence, insensitivity to external vibrations it is, once again, desirable to have high stiffness. As with the thermal performance index, a mechanical performance index can be given by

$M_m=\frac{E}{\rho }.$ (3.13)

(3.13)

Insensitivity to vibration will be discussed in more detail in Section 3.9.

3.9.1 Sources of vibration

Different physical influences contribute to different frequency bands in the seismic vibration spectrum, a summary of which is given in Table 3.1 and discussed in Ref. [30].

Figure 3.8 shows measured seismic background noise spectra obtained from a worldwide network of 75 seismograph stations. The upper and lower bounds of this data set have been used to model ‘noisy’ and ‘quiet’ vibration environments. The underlying structure of the vibration spectra is determined by the dominant cause of seismic noise at each period band. Short-period (less than 1 s) vibration variations are often dominated by human activity and machinery [32].

For determining the low-frequency vibrations, a gravitational wave detector, in the form of a Michelson interferometer with 20 m arms, has been used to measure vibrations 1 km below sea level [33]. A summary of the results is given in Table 3.2.

3.9.2 Passive vibration isolation

Simple springs and pendulums can provide vibration isolation in both vertical and horizontal directions. The transmissibility of an isolator is the proportion of a vibration as a function of frequency that is transmitted from the environment to the structure of the isolator. For a single degree of freedom vibration isolation system, the transmissibility, T, is given by [33]

$T=\frac{{{\omega }_0}^2}{\sqrt{{({\omega }_0^2-{\omega }^2)}^2+4{\gamma }^2{\omega }_0^2{\omega }^2}},$ (3.14)

(3.14)

where ω₀ is the resonant frequency of the isolator and γ is the viscous damping factor. Figure 3.9 shows the transmissibility as a function of frequency ratio for various damping factors.

Vibration isolation is provided only above $\sqrt{2}$ times the natural frequency of the system, that is

$T={\left(\frac{f_0}{f}\right)}^2\mathrm{for}f\ll f_0.$ (3.15)

(3.15)

Therefore, to provide vibration isolation at low frequencies, the resonant frequency of the isolation system must be as low as possible. The resonant frequency for a pendulum is given by

$f_0=\frac{1}{2\pi }\sqrt{\frac{g}{l}}$ (3.16)

(3.16)

and by

$f_0=\frac{1}{2\pi }\sqrt{\frac{k}{m}}$ (3.17)

(3.17)

for a spring, where g is the acceleration due to gravity, l is the pendulum length, k is the spring constant and m is the mass.

Rewriting Eq. (3.17) in terms of the static extension or compression of a spring, δl, gives

$f_0=\frac{1}{2\pi }\sqrt{\frac{g}{\delta l}}$ (3.18)

(3.18)

since the static restoring force kδl=mg. Thus, for a low resonant frequency in a spring system, it is necessary to have a large static extension or compression (or use a specialised non-linear spring).

3.9.3 Damping

In vibration isolation systems, it is important to have damping, to attenuate excessive vibration near resonance. In Eq. (3.14), it is assumed that velocity-dependent (viscous) damping is being applied. This is attractive since viscous damping does not degrade the high-frequency performance of the system.

The effects at resonance due to other forms of damping can be represented in terms of an ‘equivalent viscous damping’, using energy dissipation per cycle as the criterion of equivalence [34]. However, in such cases, the value of the equivalent viscous damping is frequency dependent and, therefore, changes the system behaviour. For hysteresis or structural damping, the damping term depends on displacement instead of velocity.

3.9.4 Internal resonances

A limit to high-frequency vibration isolation is caused by internal resonances of the isolation structure or the object being isolated [35]. At low frequencies, the transmissibility is accurately represented by the simple theory given by Eq. (3.14), but once the first resonance is reached, the isolation does not improve. Typically, the fundamental resonance occurs somewhere in the acoustic frequency range. Even with a careful design, it is difficult to make a structure of an appreciable size with internal resonant frequencies above a few kilohertz.

3.9.5 Active vibration isolation

Active vibration isolation is a method for extending the low-frequency isolation capabilities of a system but is very difficult in practice. Single degree of freedom isolation systems are of little practical use because a non-isolated degree of freedom reintroduces the seismic noise even if the other degrees of freedom are isolated. Active vibration isolation uses actuators as part of a control system essentially to cancel out any mechanical inputs. An example of a six degree of freedom isolation system has been demonstrated for an interferometric gravitational wave detector [36].

3.9.6 Acoustic noise

Acoustic noise appears in the form of vibrations in a system generated by ventilators, music, speech, street noise, etc. over a frequency range from about 10 to 1000 Hz in the form of sharp coherent resonances as well as transient excitations [37]. Sound pressure levels in a typical laboratory environment are greater than 35 dB, usually due to air-conditioning systems.

Consider an enclosure that is a simple bottomless rectangular box whose walls are rigidly attached at each edge. When a panel is acoustically excited by a diffuse sound field, forced bending waves govern its sound transmission characteristics, and the sound pressure attenuation is determined by the panel mass per unit area [35]. The panel sound pressure attenuation (dB) is given by [38]

$\alpha =-10{\log }_{10}\left[1+{\left(\frac{\pi {\rho }_{\mathrm{s}}f}{{\rho }_{0}c}\right)}^2\right]+5\text{dB},$ (3.19)

(3.19)

where ρ_s is its mass per unit area, ρ₀ is the density of air at standard pressure and f is the incident acoustic field frequency. Equation (3.19) suggests that the enclosure wall should be constructed from high-density materials to obtain the largest ρ_s possible given the load-bearing capacity of any supporting structure. Note that the attenuation decreases for every 20 dB per decade increase in either ρ_s or frequency.