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Thus for a given geometry and thermal input, the distortion isminimized by selecting materials with large values of the performance indexMQ ¼l:a(3.9)References [24] and [3] arrive at the same index by considering other types ofthermal load. If the assumption that the period of the heat flux is greater thanthe thermal response time of the beam is not valid then the thermal mass ofthe beam has to be taken into account [24]. In this case thermal conductivityis given byDl ¼(3.10)rCpwhere D is the thermal diffusivity of the beam material, r is its density and Cpis its specific heat capacity.
In the case of a room with stable temperature andvery slow heat cycling equation (3.9) is normally valid.4546C H A P T ER 3 : Precision measurement instrumentation – some design principles3.7.2 Minimizing mechanical inputsThere are many types of mechanical input that will cause unwanteddeflections of a metrology frame. These include elastic deflections due to selfweight, loading due to the object being measured and external vibrationsources. To minimize elastic deflections a high stiffness is desirable. Theelastic self-deflection of a beam is described byWx3EIy ¼ C2(3.11)where W is the weight of the beam, E is the Young’s modulus of the beammaterial, I is the second moment of area of the cross-section and C2 isa constant that depends on the geometry of the beam and the boundaryconditions.
It can be seen from equation (3.11) that, for a fixed design ofinstrument, the self-loading is proportional to r/E – minimizing this ratiominimizes the deflection.The natural frequency of a beam structure is given byun ¼ C3rffiffiffiffiffiffiffiffiffiEIml3(3.12)where n is the harmonic number, m is the mass per unit length of the beam, lits length and C3 is a constant that depends on the boundary conditions.pffiffiffiffiffiffiffiffiAgain, for a fixed design of instrument, un is directly proportional to E=r.For a high natural frequency and, hence, insensitivity to external vibrations itis, once again, desirable to have high stiffness. As with the thermal performance index, a mechanical performance index can be given byMm ¼E:r(3.13)Insensitivity to vibration will be discussed in more detail in section 3.9.3.8 SymmetrySymmetry is a very important concept when designing a precision measuringinstrument.
Any asymmetry in a system normally has to be compensated for.In dynamics it is always better to push or pull a slideway about its axisof reaction otherwise parasitic motions will result due to asymmetry. Ifa load-bearing structure does not have a suitably designed centre of mass,there will be differential distortion upon loading. It would seem thatVibration isolationFIGURE 3.5 Kevin Lindsey with the Tetraform grinding machine.symmetry should be incorporated into a precision measuring instrumentdesign to the maximum extent. An excellent example of a symmetricalstructure (plus many other precision instrument design concepts) is theTetraform grinding machine developed by Kevin Lindsey at NPL [25,26].
Thesymmetrical tetrahedral structure of Tetraform can be seen in Figure 3.5.Calculations and experimental results showed that the Tetraform isextremely well compensated for thermal and mechanical fluctuations.3.9 Vibration isolationMost precision measuring instruments require some form of isolation fromexternal and internal mechanical excitations.
Where sub-nanometre accuracy is required it is essential that seismic and sonic vibration is suppressed.This section will discuss some of the issues that need to be considered whentrying to isolate a measuring instrument from vibration. The measurementof vibration is discussed in [27] and vibration spectrum analysis is reviewedin [28].3.9.1 Sources of vibrationDifferent physical influences contribute to different frequency bands in theseismic vibration spectrum, a summary of which is shown in Table 3.1 anddiscussed in [27].4748C H A P T ER 3 : Precision measurement instrumentation – some design principlesTable 3.1Sources of seismic vibration and corresponding frequencies [27]Frequency/mHzCause of vibration< 5050 to 500Atmospheric pressure fluctuationsOcean waves (60 mHz to 90 mHz fundamentalocean wave frequency)Wind-blown vegetation and human activity> 100Figure 3.6 shows measured vertical amplitude spectral densities for a vibrationally ‘noisy’ and a vibrationally ‘quiet’ area [29].
Note that the spectrumbelow 0.1 Hz is limited by the seismometer’s internal noise. The solid curverepresents the vibration spectrum on the campus of the University of Colorado,Boulder. The dashed curve is that from the NIST site. The ‘quiet’ NIST laboratory is small, remote and separated from the main complex. In addition, allfans and machinery were turned off during the measurements at the NISTsite.Most of the increased vibration in the solid line above 10 Hz in Figure 3.6can be attributed to human activity and machinery.
The low-frequency peakin the dashed line can be attributed to naturally occurring environmentaleffects such as high winds.For determining the low-frequency vibrations a gravitational wavedetector, in the form of a Michelson interferometer with 20 m arms, has beenused to measure vibrations 1 km below sea level [30]. A summary of theresults is given in Table 3.2.FIGURE 3.6 Measured vertical amplitude spectrum on a ‘noisy’ (continuous line) anda ‘quiet’ (dotted line) site [29].Vibration isolationTable 3.2Possible sources of very-low-frequency vibrationSourcePeriodAcceleration/m$s1Earth’s free seismic oscillationCore modesCore undertoneEarth tidesPost-seismic movementsCrustal movements102 – 103 s103 s103 – 104 s104 – 105 s1 – 103 days102 days106 – 10810111011106106 – 108107 – 1093.9.2 Passive vibration isolationSimple springs and pendulums can provide vibration isolation in bothvertical and horizontal directions.
The transmissibility of an isolator is theproportion of a vibration as a function of frequency that is transmitted fromthe environment to the structure of the isolator. For a single degree offreedom vibration isolation system the transmissibility, T, is given by [30]u0 2T ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðu02 u2 Þ2 þ 4g2 u02 u2(3.14)where u0 is the resonant frequency of the isolator and g is the viscousdamping factor. Figure 3.7 shows the transmissibility as a function offrequency ratio for various damping factors. pffiffiffiVibration isolation is provided only above 2 times the natural frequencyof the system, that is for f << f0. 2f0T ¼for f << f0 :(3.15)fTherefore, to provide vibration isolation at low frequencies, the resonantfrequency of the isolation system must be as low as possible.
The resonantfrequency for a pendulum is given byrffiffiffi1 g(3.16)f0 ¼2p land by1f0 ¼2prffiffiffiffiffikm(3.17)for a spring, where g is the acceleration due to gravity, l is the pendulumlength, k is the spring constant and m is the mass.4950C H A P T ER 3 : Precision measurement instrumentation – some design principlesFIGURE 3.7 Damped transmissibility, T, as a function of frequency ratio (u/u0)Re-writing equation (3.17) in terms of the static extension or compressionof a spring, dl, givesrffiffiffiffi1g(3.18)f0 ¼2p dlsince the static restoring force kdl ¼ mg.
Thus for a low resonant frequency ina spring system it is necessary to have a large static extension or compression(or use a specialized non-linear spring).3.9.3 DampingIn vibration-isolation systems it is important to have damping, to attenuateexcessive vibration near resonance.
In equation (3.14) it is assumed thatvelocity-dependent (viscous) damping is being applied. This is attractivesince viscous damping does not degrade the high-frequency performance ofthe system.The effects at resonance due to other forms of damping can be represented in terms of an ‘equivalent viscous damping’, using energy dissipationper cycle as the criterion of equivalence [31]. However, in such cases, thevalue of the equivalent viscous damping is frequency-dependent and, therefore, changes the system behaviour.
For hysteresis or structural damping, thedamping term depends on displacement instead of velocity.3.9.4 Internal resonancesA limit to high-frequency vibration isolation is caused by internal resonancesof the isolation structure or the object being isolated [32]. At low frequenciesVibration isolationthe transmissibility is accurately represented by the simple theory given byequation (3.14), but once the first resonance is reached, the isolation does notimprove. Typically the fundamental resonance occurs somewhere in theacoustic frequency range.
Even with a careful design it is difficult to makea structure of an appreciable size with internal resonant frequencies abovea few kilohertz.3.9.5 Active vibration isolationActive vibration isolation is a method for extending the low-frequencyisolation capabilities of a system, but is very difficult in practice.
Singledegree of freedom isolation systems are of little practical use because a nonisolated degree of freedom reintroduces the seismic noise even if the otherdegrees of freedom are isolated. Active vibration isolation uses actuators aspart of a control system essentially to cancel out any mechanical inputs. Anexample of a six degree of freedom isolation system has been demonstrated[29] for an interferometric gravitational wave detector.3.9.6 Acoustic noiseAcoustic noise appears in the form of vibrations in a system generated byventilators, music, speech, street noise, etc.
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