Invited article: Expanded and improved traceability of vibration measurements by laser interferometry.

Invited Article: Expanded and improved traceability of vibration measurements by laser interferometry Hans-Jürgen von Martens Citation: Review of Scientific Instruments 84, 121601 (2013); doi: 10.1063/1.4845916 View online: http://dx.doi.org/10.1063/1.4845916 View Table of Contents: http://scitation.aip.org/content/aip/journal/rsi/84/12?ver=pdfcov Published by the AIP Publishing

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REVIEW OF SCIENTIFIC INSTRUMENTS 84, 121601 (2013)

Invited Article: Expanded and improved traceability of vibration measurements by laser interferometry Hans-Jürgen von Martensa),b) Physikalisch-Technische Bundesanstalt (PTB), Bundesallee 100, 38116 Braunschweig, Germanyb)

(Received 22 July 2013; accepted 6 October 2013; published online 30 December 2013) Traceability to the International System of Units has been established for vibration and shock measurements as specified in international document standards, recommendations, and regulations to ensure product quality, health, and safety. New and upgraded laser methods and techniques developed by national metrology institutes and by leading manufacturers in the past two decades have been swiftly specified as standard methods in the ISO 16063 series of international document standards. In ISO 16063-11:1999, three interferometric methods are specified for the primary calibration of vibration transducers (reference standard accelerometers) in a frequency range from 1 Hz to 10 kHz. In order to specify the same (modified) methods for the calibration of laser vibrometers (ISO 16063-41:2011), their applicability in an expanded frequency range was investigated. Steady-state sinusoidal vibrations were generated by piezoelectric actuators at specific frequencies up to 347 kHz (acceleration amplitudes up to 376 km/s2 ). The displacement amplitude, adjusted by the special interferometric method of coincidence to 158.2 nm (quarter the wavelength of the He-Ne laser light), was measured by the standardized interferometric methods of fringe counting and sine-approximation. The deviations between the measurement results of the three interferometric methods applied simultaneously were smaller than 1 %. The limits of measurement uncertainty specified in ISO 16063-11 between 1 Hz to 10 kHz were kept up to frequencies, which are orders of magnitude greater; the uncertainty limit 0.5 % specified at the reference frequency 160 Hz was not exceeded at 160 kHz. The reported results were considered during the development of ISO 16063-41 by specifying the instrumentation and procedures for performing calibrations of rectilinear laser vibrometers in the frequency range typically between 0.4 Hz and 50 kHz—the interferometric methods may be applied within expanded frequency ranges using refined techniques and procedures. It is concluded that calibration frequencies up to 0.5 MHz are attainable in compliance with the first international document standard for the calibration of laser vibrometers. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4845916] I. INTRODUCTION

The SI units for physical quantities, such as metre per second squared (m/s2 ) for the quantity of acceleration, are realized by national metrology institutes (NMIs), or designated institutes, and disseminated to external clients through calibrations.1, 2 Requirements for establishing the traceability of measurement standards and measuring instruments to the SI units through an unbroken chain of calibrations or comparisons are specified in ISO/IEC Standard 17025.3 The calibration certificates issued by calibration laboratories have to state the measurement results, including the measurement uncertainty to be evaluated and expressed in accordance with the GUM.4 Though it may be expected that all calibrating laboratories (NMIs included) establish uncertainty budgets in compliance with the GUM, the accuracy attained can be reliably assessed only by appropriate comparison measurements. In the area of vibration measurement using laser interferometry, numerous international, regional, and national comparisons were performed and have largely contributed to recognize sysa) Electronic mail: [email protected] b) H.-J. von Martens is retired from PTB. This research was performed while

he was at PTB and at CENAM (Centro Nacional de Metrologia, Queretaro, Mexico).

0034-6748/2013/84(12)/121601/25/$30.00

tematic deviations and their sources, and to prove the uncertainty statements of any laboratory.5, 6 To ensure compliance of the units realized by the NMIs, within well-specified uncertainties in accordance with their definition in the SI, key comparisons (KCs) are carried out. Those KCs carried out in the area of vibration under the auspices of the Consultative Committee for Acoustics, Ultrasound and Vibration (CCAUV) of the International Committee of Weights and Measures (CIPM) are to specify key comparisons reference values (KCRVs) considered to represent the SI unit at the state at the art. The KCRVs specified in the first CIPM KC for the branch vibration, CCAUV-K1,7 were disseminated through various regional key comparisons to a great number of NMIs within the Regional Metrology Organizations (RMOs), in particular the Asia Pacific Metrology Programme (APMP), Euro-Asian Cooperation of National Metrology Institutions (COOMET), European Collaboration in Measurement Standards (EURAMET, formerly EUROMET), Southern African Development Community Cooperation in Measurement Traceability (SADCMET) reorganized as the Intra-Africa Metrology System (AFRIMETS), and Sistema Inter-Americano de Metrologia (SIM, InterAmerican System of Metrology). If approved for equivalence, the final KC reports including all results are available in the BIPM key comparison database, see www.bipm.com. This is

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Rev. Sci. Instrum. 84, 121601 (2013)

TABLE I. Motion quantities representing vibration and shock, and their units. Motion quantity

a(t) =

Unit

Acceleration Velocity Displacement Angular acceleration Angular velocity Rotational angle

m/s2 m/s m rad/s2 rad/s rad

Metre per second squared Metre per second Metre Radiant per second squared Radiant per second Radiant

to date valid for the CIPM KC (see Ref. 7), and the RMO KCs APMP.AUV.V-K1 (see Ref. 8), EUROMET.AUV.V-K1 (see Ref. 9), and SIM.AUV.V-K1 (see Ref. 10), and for several other KCs and supplementary comparisons (e.g., see Ref. 11); the current state is reported in Ref. 12. The physical quantities widely used to describe vibration and shock motion in different applications are rectilinear acceleration, velocity, displacement, and angular acceleration, angular velocity, and rotational angle (cf. Table I and Eqs. (1)–(8)). From the viewpoint of metrology, the units and associated scales of the six motion quantities have to be realized and disseminated with appropriate time dependencies (preferably sinusoidal and shock-shaped). To take advantage of the relationships between the parameters (amplitude and initial phase) of sinusoidal accelerations a, velocities v, and displacements s, acceleration exciters with low or no distortion are used and have been developed. Accordingly, angular exciters have been developed, that generate steady-state sinusoidal angular acceleration α with low distortion, so that the angular velocity Ω and the rotational angle Φ are practically free from distortion. For sinusoidal vibrations, the following relationships apply: s(t) = sˆ cos(ω t + ϕs ), v(t) = vˆ cos(ωt + ϕv ),

vˆ = ωˆs ,

a(t) = aˆ cos(ωt + ϕa ),

aˆ = ω2 sˆ ,

(1) π , 2

(2)

ϕa = ϕs + π,

(3)

ϕv = ϕs +

Φ(t) = Φˆ cos(ωt + ϕΦ ), Ω(t) = Ωˆ cos(ωt + ϕΩ ),

α(t) = αˆ cos(ωt + ϕα ),

ˆ Ωˆ = ωΦ,

ˆ αˆ = ω2 Φ,

(4) ϕΩ = ϕΦ +

π , 2 (5)

ϕα = ϕΦ + π,

(6)

where xˆ is the amplitude and ϕ x the initial phase angle of the respective motion quantity x; ω is the vibration radian frequency ω = 2π f with the vibration frequency f. For shock-shaped acceleration measured in the time domain, the relationships v(t) =

ds dt

and

(7)

d2 s dt 2

(8)

apply. Though the ISO standard (see Ref. 13) covers a frequency range of 0.4 Hz to 10 kHz and specifies not only the modulus but also the phase shift to be measured and calibrated, the CIPM KC (see Ref. 7) was restricted to the modulus measurements from 40 Hz to 5 kHz in view of the limitations in several laboratories which had asked for their participation. More recent KCs have included phase shift measurements and extended the frequency range to lower and higher frequencies, respectively. To date, two NMIs only (NIST and PTB) offer CMCs up to 20 kHz in the BIPM key comparison database, see http://kcdb.bipm.org/. Methods and techniques for accelerometer calibrations at higher frequencies up to 50 kHz have been reported in Refs. 15–20. Laser vibrometers are available for measuring vibrations having frequencies in the megahertz and gigahertz ranges.21, 22 Generally, a measuring instrument is applicable only if it has been calibrated for the conditions of use. ISO TC 108 (SC 3/WG 6 in particular) responded to the need for standard methods for the calibration of vibration and shock transducers (laser vibrometers included) required to ensure international traceability to the SI units in the field of measurements of accelerations and derived motion quantities. Recent progress in development and application of ISO calibration standards was reported in Refs. 23 and 24. The new international standard ISO 16063-41:2011 (see Ref. 25) specifies the instrumentation and procedures for performing primary and secondary calibrations of rectilinear laser vibrometers. Three interferometric methods are specified in consistency with the corresponding ISO standards for primary vibration calibration of accelerometers (see Ref. 13) and angular transducers (see Ref. 26). ISO 16063-11 (see Ref. 13) specifies an upper frequency limit of 800 Hz for Method 1 (Fringe-counting method, FCM), of 10 kHz for Method 2 (Minimum-point method, MPM) and of 10 kHz for Method 3 (Sine-approximation method, SAM) as well. To meet the higher requirements for laser vibrometer calibrations, H.-J. von Martens (project leader for the development of the new ISO standard, see Ref. 25) initiated investigations of the applicability of the interferometric methods at higher frequencies. In the first part of the experimental investigations, the applicability of the interferometric methods for a selected frequency of 100 kHz has been demonstrated.18, 27 In the second part, the experiments were extended and refined (see Refs. 23 and 28) in order to cover wider ranges of frequencies and acceleration amplitudes with higher accuracy. The standardized interferometric Methods 1 and 3 and a modified version of Method 1 (Signal-coincidence method, see Ref. 29) proved to be applicable up to frequencies of 350 kHz. The findings were taken into account by adequate specifications before the finalization of the ISO standard for the calibration of laser vibrometers.25 In Sec. II, an account is given of procedures and techniques to establish traceability for vibration and shock measurements, and the current state is demonstrated. As a basic


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for traceability, the establishment of KCRVs (Ref. 7) and the link-up of the NMIs and designated institutes worldwide (i.e., countries) to the KCRVs are explained. Measurement and calibration techniques developed by numerous NMIs and manufacturers, applied on different levels of calibration hierarchies, are described, supplemented by numerous references. In Sec. III, relevant standardization activities in the ISO Technical Committee TC 108 are reported and the interferometric methods specified in ISO standards for high-accuracy vibration measurements and for primary calibrations are described. In Sec. IV, the Signal coincidence method (SCM) referred to in ISO 16063-41 as a specific modification of the FCM is explained in more detail. In Sec. V, the applicability of different vibration generation methods and techniques to attain great acceleration amplitudes at high frequencies is reviewed and a modified piezoelectric actuator used up to 347 kHz is presented. Section VI describes briefly measurement set-ups, which were used in the reported investigations for intralaboratory comparisons between different interferometric methods in wide frequency ranges. Selected results of experimental investigations of different standard methods simultaneously applied at high frequencies are presented in Sec. VII. In Sec. VIII, a brief introduction to the basic procedure specified in the GUM (Ref. 4) for the expression of uncertainty in measurement is given, methodical tools to facilitate the uncertainty assessment for vibration and shock measurements are considered, and specific error sources and their effects are described, supplemented by numerous references to literature. Conclusions are drawn in Sec. IX. This invited article is a major expansion and modification of the specific Conference paper,30 which was addressed to experts in the field of vibration measurements by laser techniques. The article is, at the request of the editor for invited articles, an attempt to provide a review that is useful to both experts and novices in the field; experts may wish to gloss over Secs. II, III, VIII A, and VIII B. Several of the references (see 12, 24, 28, 102, 108, 109, 111, 112) are not generally available and for reasons of restricted access, cannot be made so. Readers for whom those references are of special importance may contact the author to discuss their needs. II. TRACEABILITY OF VIBRATION AND SHOCK MEASUREMENTS

Metrological traceability is defined (Ref. 31) as property of a measurement result whereby the result can be related to a reference through a documented unbroken chain of calibrations, each contributing to the measurement uncertainty. A metrological traceability chain is defined through a calibration hierarchy and used to establish metrological traceability of a measurement result. Figure 2 demonstrates a metrological traceability chain for vibration and shock measurements, whereby the reference is a national measurement standard from an ensemble of different national measurement standards.1, 45 The state of the realization of vibration quantities on the highest level of national calibration hierarchies was demonstrated in the CIPM KC,7 which was worldwide accom-

Rev. Sci. Instrum. 84, 121601 (2013)

plished. 12 NMIs from 5 RMOs (cf. Sec. I) measured the charge sensitivity of 2 transfer standard accelerometers (backto-back design and single-ended design) at 22 nominal frequencies ranging from 40 Hz to 5 kHz, using laser interferometry as specified in ISO 16063-11.13 For the results and other details, see Ref. 7. A representative example for the dissemination of the KCRVs throughout an RMO is the European KC.9 The objective of the European KC (see Ref. 9) was to link 11 European countries, which had not participated in the CIPM KC, to the KCRVs established in the CIPM KC. The linking of the RMO KC to the CIPM KC was based on the results of three “linking” laboratories that had participated in both KCs. For the results and other details, see Ref. 9. A linking procedure is presented and demonstrated in Refs. 32, 33, and 34. The results of the KCs have demonstrated the compliance of the NMIs with their measurement and calibration capabilities (CMCs) offered in the BIPM database (see Appendix C of the Mutual Recognition Arrangement (MRA, Ref. 35). Hierarchies of measurement standards have been established and are operated in compliance with new and upgraded ISO standards. Most of the NMIs use commercial high-quality vibration and shock generators or even complete calibration equipment. High-performance vibration and shock calibration systems equipped with laser interferometry are commercially available (e.g., see Refs. 2 and 36). Other NMIs have developed specific high-performance vibration and shock generators, laser interferometers, and primary calibration systems (national measurement standards) in compliance with the ISO Standards (see Refs. 13, 25, 26, 37, 38, and 39). References 40–86 describe various rectilinear vibration and shock exciters, angular vibration and shock exciters, translational and rotational laser interferometer systems, and primary calibration and measurement standards including those using the reciprocity method (see Ref. 38). Moreover, multiaxis motion exciters and measurement and calibration systems as well, have been developed (see Refs. 87 and 88) and considered in an ISO standard (see Ref. 89). Calibration laboratories are in most cases equipped with commercial calibration devices, which are produced by several manufacturers in compliance with the respective ISO standards. The traceability chain established in Germany (see Refs. 1, 45, 90, 91, and 92) covers the six motion quantities as described by the Eqs. (1)–(8) and demonstrated in Fig. 1. By convention, the dissemination has been carried out by primary vibration calibration of reference standard accelerometers (ISO 16063-11, Ref. 13) as are used in accredited calibration laboratories to calibrate working standards by comparison to a reference transducer (ISO 16063-21, Ref. 14). The comparison method is also used at the third level of the hierarchy of measurement standards (non-accredited calibration laboratories), to calibrate or check working transducers (e.g., accelerometers), measuring chains or measuring instruments equipped with a seismic transducer. The progress has led to the development and production of laser vibrometer standards, which meet the specifications of ISO 1606341 (Ref. 25) and are applicable within the traceability chain


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Rev. Sci. Instrum. 84, 121601 (2013)

and shock acceleration to be in compliance with the relevant ISO standards, r the establishment of traceability chains in the field of vibration and shock (measurands: acceleration, velocity, displacement, angular acceleration, angular velocity, and rotational angle). The current state of ISO standards and standardization projects applicable at different levels of a traceability chain for vibration and shock measurements has been reported periodically to the BIPM/CCAUV.24 B. Specification of interferometric methods and techniques in ISO standards FIG. 1. Demonstration of a metrological traceability chain for vibration and shock measurements (example of PTB).

(example: accredited calibration laboratory marked D-K15183-01-00 in Germany). As an alternative to laser interferometry, the reciprocity method has been used for the primary vibration calibration of reference standard transducers.38

To meet the increasing demands for the calibration of vibration and shock transducers by laser interferometry, the

r Standard for Basic Concepts ISO 5347-0:1987 was revised to ISO 16063-1:1998 (see Ref. 39),

r Standard for Primary Vibration Calibration by Laser r

III. ISO STANDARDS FOR VIBRATION MEASUREMENTS AND CALIBRATIONS

r

A. The ISO technical committee TC 108 and its activities

r

ISO TC 108 (Mechanical vibration, shock, and condition monitoring) was established in 1964 to develop documentary standards for mechanical vibration and shock, including transducer calibration. The ISO TC 108/SC 3 (Use and calibration of vibration and shock measuring instruments) and its Working Group 6 (Calibration of vibration and shock transducers) include recognized metrologists from national metrology institutes (NMIs) along with a wide range of manufacturers and users. Under the general title Methods for the calibration of vibration and shock pick-ups, a standard series, ISO 5347, was issued in the period between 1987 and 1997. A revision of the ISO 5347 series, re-numbered to ISO 16063, was started in 1995, focusing on the specification of calibration methods needed at different levels of a traceability chain for the field of vibration and shock: methods for primary vibration calibration, vibration calibration by comparison to a reference transducer, primary shock calibration and shock calibration by comparison to a reference transducer. ISO/TC 108 has responded to the need for upgraded and new standard calibration methods applicable to

r CIPM key comparisons, RMO key comparisons and supplementary comparisons in the field of vibration and shock measurements, r the reliable and uniform specification of the Calibration and Measurement Capabilities (CMCs) in the branch vibration, published in the BIPM key comparison database (cf. Appendix C of the MRA, Ref. 35) — all NMIs claim their CMCs in the field of vibration

r

Interferometry ISO 5347-1:1993 was revised to 16063-11 (see Ref. 13), Standard for Primary Shock Calibration by Laser Interferometry ISO 16063-13 (see Ref. 37) was developed, Standard for Primary Angular Calibration by Laser Interferometry ISO 16063-15 (see Ref. 26) was developed, Standard for Testing of Transverse Vibration Sensitivity ISO 5347-11:1993 was revised to ISO 16063-31 (see Ref. 89), Standard for the Calibration of Laser Vibrometers ISO 16063-41 (see Ref. 25) was developed.

C. Measurement ranges and accuracy specified for standard techniques

For primary calibrations using laser interferometry, the following measurement ranges and expanded uncertainties (for coverage factor k = 2) are specified. 1. Primary vibration calibration by laser interferometry

ISO 16063-11 (see Ref. 13) is applicable to a frequency range from 1 Hz to 10 kHz and a dynamic range (amplitude) from 0.1 m/s2 to 1 000 m/s2 (frequency-dependent). The limits of the uncertainty of measurement shall be as follows. For the modulus of sensitivity:

r 0.5 % of the measured value at reference conditions, r 1 % of the measured value outside reference conditions. For the phase shift of sensitivity:

r 0.5◦ of the measured value at reference conditions, r 1◦ of the reading outside reference conditions.


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2. Primary shock calibration by laser interferometry

ISO 16063-13 (see Ref. 37) is applicable in a shock pulse duration range 0.05 ms to 10 ms and a dynamic range (peak value) 102 m/s2 to 105 m/s2 (pulse duration-dependent). The limits of the uncertainty of shock sensitivity measurement shall be as follows:

r 1 % of reading at reference peak value of 1000 m/s2 and reference shock pulse duration of 2 ms,

r 2 % for all values of peak acceleration and shock pulse duration.

3. Primary angular vibration calibration by laser interferometry

ISO 16063-15 (see Ref. 26) is applicable to a frequency range from 1 Hz to 1.6 kHz and a dynamic range (amplitude) from 0.1 rad/s2 to 1 000 rad/s2 (frequency-dependent). The limits of the uncertainty of measurement shall be as follows: For the modulus of sensitivity:

r 0.5 % of the measured value at reference conditions; ≤1 % outside reference conditions. For the phase shift of sensitivity:

r 0.5◦ of the measured value at reference conditions; ≤1◦ outside reference conditions.

4. Calibration of laser vibrometers

ISO 16063-41 (see Ref. 25) is applicable to a frequency range typically from 0.4 Hz to 50 kHz. No limits but typical attainable uncertainties of measurement are specified: Example 1: A laser vibrometer standard is calibrated by primary means (Methods 1, 2, or 3) with documented small uncertainty. The temperature and other conditions are kept within narrow limits during the calibration as indicated in the appropriate clauses:

r 0.25 % to 1 % from 0.4 Hz to 50 kHz (frequencydependent). Example 2: A laser vibrometer is calibrated using a laser vibrometer standard calibrated according to Example 1:

r 1 % to 5 % from 0.4 Hz to 50 kHz (frequencydependent).

D. Measurement ranges and accuracy achieved with refined techniques

The ISO standards of the 16063 series are inclusive rather than exclusive. They allow refined versions of the standard methods to be applied, which lead to even smaller uncertainty and/or wider parameter ranges than that specified for standard techniques (see Sec. III C). The potential high accuracy (small measurement uncertainty) can be reached only if high-performance standard exciters are used. Only small deviations from uni-axial, purely

Rev. Sci. Instrum. 84, 121601 (2013)

sinusoidal or defined shock motion are tolerable. Relative motion between the transducer reference surface and the spot(s) sensed by the interferometer must be kept small or negligible. Special vibration isolation measures need to be taken to prevent ground motion and reaction forces of the exciter from excessively affecting the calibration results. Such disturbing motion may stimulate relative motion of the interferometer or resonances of fixtures and optical elements, which affect the laser light paths. Data acquisition with a sufficiently high sampling rate, high resolution, and large memory is required to achieve a small uncertainty of measurement. The result of measurement is determined on the basis of series of observations obtained under repeatability conditions. Measurement ranges and accuracy (uncertainty) achieved with refined techniques are demonstrated in Secs. VII and VIII.

E. Scope of the ISO standard for the calibration of laser vibrometers

ISO 16063-41 (see Ref. 25) specifies the instrumentation and procedures for performing primary and secondary calibrations of rectilinear laser vibrometers in the frequency range typically between 0.4 Hz and 50 kHz. It describes the calibration of laser vibrometer standards designated for the calibration of either laser vibrometers or mechanical vibration transducers in calibration laboratories, as well as the calibration of laser vibrometers by a laser vibrometer standard or by comparison to a reference transducer calibrated by laser interferometry. The specification of the instrumentation contains requirements on laser vibrometer standards. Rectilinear laser vibrometers can be calibrated in accordance with standard if they are designed as laser optical transducers with, or without, an indicating instrument to sense the motion quantities of displacement or velocity, and to transform them into proportional (i.e., time-dependent) electrical output signals. These output signals are typically digital for laser vibrometer standards and usually analog for laser vibrometers. The output signal or the reading of a laser vibrometer may be the amplitude and, in addition, occasionally the phase shift of the motion quantity (acceleration included). Modulus calibration is explicitly specified and phase calibration is provided in an informative Annex D of this ISO standard. A survey of the contents of the standard is given in Ref. 23.

F. Interferometric methods for stationary sinusoidal vibrations

1. General consideration

The laser interferometric methods, techniques, and procedures specified in the ISO standards have been developed in national metrology institutes (NMIs) such as NIST (former NBS) in USA, NMIA (former NML) in Australia, and PTB in Germany (including the vibration laboratory of the metrology institute of the former GDR in East Berlin.29, 90 For the measurement of stationary sinusoidal vibrations, the FCM, the MPM, and the SAM were invented and implemented in the ISO standards.13, 25, 26 The methods specified (see


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Ref. 37) to measure shock-shaped accelerations make use of the techniques and procedures applied to the SAM up to the calculation of a series of displacement values {si (ti )} from the sampled series of interferometer output values {u1 (ti )} and {u2 (ti )} in quadrature. To calculate the shock sensitivity (peak value of accelerometer output to peak value of acceleration), different versions of data processing are specified (see Ref. 37). Moreover, this ISO standard specifies procedures to calculate magnitude and phase shift of the complex sensitivity of an accelerometer from the complex frequency spectra of the sampled acceleration and accelerometer output signal at the spectral frequencies fn .60, 63 For sinusoidal motion quantities, the approximation of the obtained series of displacement values {si (ti )} by a sinusoid leads to the amplitude and phase shift of the motion quantity to be measured. This article focuses on the generation and measurement of rectilinear vibrations. For the interferometric Methods 1, 2, and 3, the ISO standards 16063-11, 16063-15, and 16063-41 state specific frequency ranges. In fact, the applicability of the particular methods mainly depends on the amplitudes of displacement or velocity (ISO16063-11 and -41), and of rotational angles and angular velocities (ISO 16063-15), respectively, which are measurable within given measurement uncertainties. The frequency ranges, however, not only depend on the measurement method itself but also on the frequency-dependent properties of the vibration exciters available. Using adequate vibration exciters to generate measurable amplitudes at higher frequencies, the upper frequency limits of all methods can be expanded to higher frequencies than that specified in the respective ISO standard. Method 1 and Method 3 may be used for any vibration measurement if the final result of the measurement is the magnitude and (Method 3 only) the initial phase of a sinusoidal motion quantity. In fact, Methods 1, 2, and 3 are specified for calibrations of vibration transducers, whereby the magnitude of the complex sensitivity is the measurement result (optionally the phase shift using Method 3). The specification of the instrumentation includes LVSs. Calibration equipment is shown in Fig. 2 as an example.

Rev. Sci. Instrum. 84, 121601 (2013)

2. Method 1: The Fringe-counting method (FCM)

This method is a vibration measurement method using a homodyne interferometer with a single output in conjunction with instrumentation for fringe counting of the interferometer signal. For the interferometer types used in Method 1, the number of signal periods (e.g., intensity maxima) for one vibration cycle, N, is given by 4ˆs , s

(9)

N s ff = , 4 f

(10)

N= so that sˆ =

where sˆ , is the displacement amplitude sensed by the laser interferometer; s, is the quantization interval, which is s = l/2 for the preferred interferometer type; f, is the frequency of the vibration exciter; ff , is the (mean) fringe frequency. Commonly, displacement amplitudes ≥4 μm can be measured with an uncertainty specified in the ISO standards.13, 25 At 800 Hz, 4 μm correspond to an acceleration amplitude 102 m/s2 that can be generated by usual electrodynamic vibration exciters. Hence, a frequency range up to 800 Hz was specified for Method 1 in both ISO standards. Method 1 may also be applied at smaller amplitudes and higher frequencies as explained in Refs. 29, 94, and 95 and in Sec. VIII. ISO 16063-41 states that the electronic fringe counting can be substituted by the Signal coincidence method (SCM) defined in Ref. 29; see Sec. IV for details. 3. Method 2: The Minimum-point method (MPM)

Method 2, the minimum-point method (MPM), is a vibration measurement method using a homodyne interferometer with a single output in conjunction with instrumentation for zero-point detection of a component of the frequency spectrum of the interferometer signal. Considering the frequency spectrum of the intensity and adjusting the vibration amplitude to the level at which the component of the same frequency as the vibration frequency is zero, the displacement amplitude can be calculated from the argument corresponding to the respective zero point of the Bessel function of the first kind and first order (first zero point occurs at the displacement amplitude 193.0 nm), using Eq. (11). An equally valid approach is to determine displacement using the arguments corresponding to the zero crossings of the Bessel function of the first kind and zero order (indicating the displacement amplitude 121.1 nm). However, this technique requires modulation of the position of the reference mirror. s , (11) 2π where xn are the arguments corresponding to the zero points of the Bessel function as given in Table II. Method 2 can be used for modulus calibration in the frequency range 800 Hz to 10 kHz with an electrodynamic vibration exciter (see Refs. 96, 97, and 98) and up to 50 kHz and higher with a piezoelectric vibration exciter (see Refs. 15, 17, and 20). At high frequencies beyond 100 kHz, it may be sˆ = xn

FIG. 2. Example of a calibration setup for laser vibrometers with digital output.


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TABLE II. Values for the arguments xn corresponding to the zero points of the Bessel function of first kind and first order. Zero point no. n

phases of the displacement are calculated: sˆ = A2 + B 2 ,

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

B ϕs = arctan . (16) A To obtain the output amplitude and the output initial phase angle of a vibration transducer or laser vibrometer to be calibrated, the sine approximation can be applied to its sampled output signal with the relationships (14) to (16) rewritten for the output of the calibration object. Equation (17),

0 3 831.70 7 015.59 10 173.46 13 323.69 16 470.63 19 615.86 22 760.09 25 903.68 29 046.83 32 189.68 35 332.30 38 474.77 41 617.09 44 759.32 47 901.46

ϕs = ϕu − ϕs ,

difficult to achieve a sufficient short-term stability of the vibration amplitude over the required measurement time of the MPM, because the generation of acceleration amplitudes in the order 100 km/s2 is usually accompanied by heating effects in the (piezoelectric) actuator. 4. Method 3: The Sine-approximation method (SAM)

Method 3 presented in Refs. 99 and 100 is a vibration measurement method using a homodyne or heterodyne interferometer with two electrical outputs in quadrature (i.e., phase-shifted by 90◦ ) in conjunction with instrumentation for signal sampling and processing. From the sampled interferometer output values {u1 (ti )} and {u2 (ti )}, a series of modulation phase values {ϕ Mod (ti )} is calculated: ϕMod (ti ) = arctan

u2 (ti ) + nπ, u1 (ti )

(12)

where an integer number n = 0, 1, 2 . . . is chosen so that discontinuities of {ϕ Mod (ti )} are avoided for nπ . Using Eq. (12), a series of displacement values {s(ti )} is calculated: l ϕMod (ti ). 4π

(13)

(18)

where uˆ Sˆx = (19) xˆ is the magnitude of the sensitivity to x, uˆ is the amplitude of the sinusoidal output u of the transducer (e.g., output voltage of an accelerometer/amplifier combination), xˆ is the amplitude of x, ϕ u is the initial phase angle of u, ϕ x is the initial phase angle of x, and (ϕ u − ϕ x ) is the phase shift ϕ x of the complex sensitivity, ϕx = (ϕu − ϕx ) .

(20)

The SAM may alternatively be applied in a modified version with time interval measurement.101 Laser vibrometers based on digital signal processing are presented in detail in Refs. 22 and 102. The ISO standards (see Refs. 13, 26, and 25) specify Method 3 for calibrations of accelerometers from 0.4 Hz to 10 kHz, of angular transducers from 1 Hz to 1.6 kHz and of laser vibrometers from 0.4 Hz to “typically” 50 kHz. i.e., higher frequencies are achievable. The SAM has been implemented with specific techniques and procedures (e.g., Refs. 93, 103–107). IV. SIGNAL COINCIDENCE METHOD A. General consideration

The obtained series of displacement values is approximated by solving the following system of equations for the three unknown parameters A, B, and C using the least-squares method: s(ti ) = A cos ωti − B sin ωti + C,

(17)

leads to the phase shift of the complex sensitivity of a displacement transducer. If the output of a laser vibrometer is the velocity signal, the Eqs. (14)–(16) are applicable in analogy. The complex sensitivity Sx of a vibration transducer to a motion quantity x is defined for sinusoidal excitation parallel to a specified axis: Sx = Sˆx ej(ϕu −ϕx ) ,

s(ti ) =

(15)

Argument xn

(14)

where A = sˆ cos ϕs , B = sˆ sin ϕs , and C is a constant not used for the measurement; i = 0, 1, 2 . . . N with N + 1 denoting the number of samples synchronously taken over the measurement period. From the parameter values A and B obtained through sine approximation, the amplitude and (if need be) the initial

The Signal coincidence method (SCM) after von Martens (see Ref. 29) is not a standard method but explained in ISO 16063-41 (see Ref. 25) as an applicable modification of the FCM (Method 1). From the other point of view, it may be considered a complement of the fringe disappearance method (Method 2). For simplification, it is referred to as a separate method. The SCM allows perceiving discrete displacement amplitudes (integer multiples of l/8) from specific shapes of the interferometer signal displayed on screen of an oscilloscope. The SCM has been used at the PTB as a check method to adjust displacement amplitude preferably of 158.2 nm (l/4 for


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the He-Ne laser interferometers used) generated by an electrodynamic vibration exciter. Recent experimental investigations (see Refs. 23 and 28) demonstrated that the typical measurement uncertainty of 1 % of the SCM was retained up to 160 kHz (highest frequency for attaining the coincidence amplitude l/4 = 158.2 nm) even in the original version using oscilloscope only (see Sec. IV B). Using data acquisition of the interferometer signal (analog-to-digital conversion, transient recorder) and mathematical functions given in Sec. IV C, a measurement uncertainty of 52 kHz). The vibration exciter with a rigid body bearing for high frequencies, shown in Fig. 8(b), is specified for the frequency range 1 kHz to 50 kHz with acceleration amplitudes up to 50 gn (about 500 m/s2 ); the specified resonance frequency is >52 kHz, too. The spring-guided acceleration exciter for high acceleration shown in Fig. 8(c) is specified for a frequency range 70 Hz to 500 Hz and maximum acceleration amplitudes 400 gn (about 4 km/s2 ).


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frequencies by metrology institutes such as NIST/ USA (see Refs. 15 and 20), VNIIM/Russia (see Ref. 110), and CIMM/China (see Central Institute of Metrology & Measurements, Beijing) (see Ref. 17). The following information on such exciters and their properties comes from bi-lateral cooperation with these institutes. Most of these piezoelectric vibration exciters have a monotonic frequency response up to an upper frequency limit for which they are designed (e.g., 20 kHz or 50 kHz) but attain great vibration amplitudes also at some resonances at higher frequencies. Some types show tolerable distortion at resonance frequencies only, and in some cases, significant transverse vibration was observed. Piezoelectric vibration exciters designed by the CIMM for a frequency range up to 50 kHz, and their application to a primary calibration system were presented (see Ref. 17). Experimental investigations of the vibration exciter of the NIST model S 101 shown in Fig. 9 (see Ref. 41) were conducted at the PTB (see Ref. 111) and later at CENAM (see Ref. 112) whereby an electric resonance circuit was implemented to amplify the input voltage of the vibration exciter. At the highest applicable resonance frequency of 65.4 kHz, maximum acceleration amplitudes of 19.5 km/s2 , velocity amplitudes of 47.6 mm/s, and displacement amplitudes of 115 nm were attained. Advanced piezoelectric exciter models developed at NIST were recently presented (see Ref. 20). The conclusion is that piezoelectric vibration exciters designed for calibrations of accelerometers up to 50 kHz attain to date vibration amplitudes of in round terms 20 km/s2 even at higher frequencies (resonances) yet not beyond 100 kHz because of their sizable dimensions required to mount an accelerometer.

D. Piezoelectric actuators applicable to calibrate laser vibrometers

A commercial piezoelectric actuator applicable to generate great vibration amplitudes at a resonance at 100 kHz was used for experimental investigations reported (see Ref. 18). To continue and extend these investigations to higher frequencies, a modified actuator of a piezoelectric

FIG. 8. Electrodynamic vibration exciters: (a) max 50 kHz, 400 m/s2 , (b) max 50 kHz, 500 m/s2 , and (c) max 500 Hz, 4 km/s2 . Source: www.spektra-dresden.com, reproduced with permission from SPEKTRA Schwingungstechnik und Akustik Dresden, Germany. Copyright 2012.

Other examples for the generation of great vibration amplitudes using resonance effects have been reported.30, 109 It is concluded that acceleration amplitudes up to about 5 km/s2 at frequencies up to 50 kHz are attainable by electrodynamic vibration exciters using resonance effects, without the risk of damage or destruction of the exciter. C. Piezoelectric vibration exciters designed for accelerometer calibration

Various types of piezoelectric vibration exciters have been developed for the calibration of accelerometers at high

FIG. 9. Piezoelectric vibration exciter developed for accelerometer calibrations to 20 kHz (Ref. 41), tested for laser vibrometer calibration (Ref. 112). Reprinted with permission from H.-J. von Martens, AIP Conf. Proc. 1457, 181 (2012). Copyright 2012, American Institute of Physics.


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FIG. 10. (a) Modified piezoelectric actuator from a linear motor; (b) and (c) simulation of the velocity distribution over a lateral surface 25 × 4 mm2 of the piezoelectric actuator at the moments of the positive and the negative velocity peak values. Photo and graphics reproduced with permission from Physik Instrumente (PI) GmbH and KG, Karlsruhe, Germany, see www.pi.ws.

micro-motor invented and manufactured by Physik Instrumente (PI, see www.pi.ws) has been applied (Fig. 10). The dimensions of the modified piezoelectric actuator shown in Fig. 10(a) are 25 × 10 × 4 mm3 . The wires are soldered to the electrodes on the 25 × 10 mm2 surface. The mirror of a shape of half of a circular disk 10 mm in diameter is fixed in the middle of a 25 × 4 mm2 surface where the largest vibration amplitude was found to occur [see Figs. 10(b) and 10(c)]. The end surfaces 4 × 10 mm2 were identified as appropriate for fixing (e.g., crowding between hard rubbers). An experimental investigation using an input voltage 10 V from a signal generator (50 Ohm output impedance) showed displacement amplitudes up to 1 micrometer between 150 kHz and 200 kHz and measurable amplitudes at higher frequencies up to 500 kHz. The tests included the investigations of different vibration modes of the actuator to identify the optimum mode of operation. Under the application conditions in the Primary Vibration Laboratory of CENAM (cf. Sec. VII), resonances at the frequencies 63.8 kHz, 159.4 kHz, and 347 kHz have been exploited for the comparison between the different interferometric methods simultaneously applied.

The conclusion is that certain types of piezoelectric actuators designed for positioning optical elements or for similar applications can be used to generate vibration amplitudes which are measurable by ISO standard methods up to frequencies of 350 kHz or higher (0.5 MHz may be attained). Experimental results of the applications of the piezoelectric actuator, shown in Fig. 10(a), at frequencies up to 347 kHz are presented in Sec. VII. Alternatively, other types of piezoelectric actuators may be applicable for calibrations of laser vibrometers, in particular converters for welding presses which attain displacement amplitudes of more than 1 μm at a specific resonance frequency, e.g., 20 kHz, 35 kHz, or 70 kHz.109 So far, these techniques have not been experimentally investigated, modified and used for calibration purposes. VI. MEASUREMENT SET-UPS USED FOR THE INVESTIGATIONS

Figure 28 (cf. Sec. VIII) shows the deviations between vibration measurement results obtained by different interferometric methods and techniques over a frequency range


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from 0.4 Hz to 347 kHz. Results from different investigations performed in 2004, 2006, 2007, and 2010 are shown. The measurement set-ups used for the experimental investigations are briefly described in this section. The results from 2004 were obtained within the framework of the extension of the accreditation of a German calibration laboratory (D-K-15183-01-00, SPEKTRA Schwingungstechnik und Akustik GmbH Dresden) to the service of primary calibration of accelerometers and laser vibrometers, among other extensions. A laser vibrometer standard of this laboratory was investigated and calibrated in the frequency range from 0.4 Hz to 20 kHz at the PTB, establishing the reference values by the Angular acceleration standard of PTB from 0.4 Hz to 40 Hz and by the High-frequency acceleration standard of PTB from 50 Hz to 20 kHz. These standard measuring devices belong to an ensemble of seven national measurement standards of Germany for the realization of the units and associated scales of acceleration and angular acceleration, respectively, with specific time-dependencies.45 Two other standard devices could have been used (Low-frequency acceleration standard: frequency range 0.1 Hz to 20 Hz, stroke 1 m of the air-borne moving table, and Medium acceleration standard: 10 Hz to 5 kHz) but were not preferred under the specific measurement conditions. The option to calibrate a translational laser vibrometer by rotational vibration results from the properties of the diffraction grating (2 400 grooves/mm) incorporated in the air-borne rotational measuring table of the angular acceleration exciter. Two laser interferometers (cf. Fig. 11) are adjusted so that the diffracted laser light beams of ±1st order travel back into the direction of the incident beams. Both measuring beams are frequency-modulated by the same tangential velocity (Doppler effect). For the purpose of calibrating angular vibration transducers, both interferometer channels are used as sub-systems of the angular acceleration standard (i.e., two-channel measuring system). In the modified arrangement for laser vibrometer calibration, channel 1 is the reference (part of the national measurement standard) and channel 2 the laser vibrometer to be calibrated. The dynamic range of laser vibrometer calibrations using an electrodynamic angular acceleration exciter with diffraction grating is limited (max. rotational angle ±1 rad for the example of the angular exciter used in 2004). An extension

FIG. 11. Arrangement used for low-frequency calibration of a (translational) laser vibrometer (interferometer channel 2) by the modified Angular acceleration standard of PTB (interferometer channel 1); for results, see Fig. 28.

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FIG. 12. Angular velocity exciter of PTB under test of metrological characteristics by optical means.

to rotational angle amplitudes, which are greater by orders of magnitude, may be attained by application of an angular velocity exciter, which is capable of generating sinusoidal angular velocities (and, thus, rotational angles) over a number of revolutions of the air-borne measuring table in conjunction with a concentric diffraction grating. Such a measurement setup could offer the possibility to generate an accurately known input for a laser vibrometer under calibration, which is the equivalent of a sinusoidal displacement with an amplitude of several meters at low frequencies (e.g., 0.1 Hz to 1 Hz). In fact it is the diffracted laser light beam of first order which is frequency modulated due to the Doppler effect, and phase-modulated by the displacement of the diffraction grating relative to the laser light spot. Such an angular velocity exciter was specially developed for the PTB (frequency range 0.1 Hz to 1.8 kHz, angular acceleration amplitude ≤103 rad/s2 , constant velocity in the rotating state max. ±180 rad/s). Figure 12 shows this angular velocity exciter equipped with a circular diffraction grating (3 000 grooves/mm) during an investigation of the constancy of the angular velocity, using optical means. The set-up was presented, among other things, in a lecture held on a specialized seminar in 2004.83 It was A. Täubner51–54 who introduced the rotational diffraction grating interferometry to measure angular motion quantities, including the option to calibrate translational laser vibrometers. Figure 13 shows a schematic of the High-frequency acceleration standard of the PTB used to provide the reference values for the calibrations of the laser vibrometer standard from 50 Hz to 20 kHz (cf. Fig. 28, results for 2004). The results of 2006 given in Fig. 28 were obtained from comparisons between different interferometric methods and techniques using both the Medium frequency acceleration standard and the High-frequency acceleration standard of the PTB. A more comprehensive presentation of the results of these investigations is given in Ref. 86. The investigations focused on the frequency range up to 20 kHz in which the PTB offers calibration and measurement capabilities (see http://www.bipm.org, BIPM Key Comparison Database). Detailed information on the current status of the realization and


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FIG. 13. Set-up for calibration and comparison of laser vibrometers using the HF acceleration standard of PTB. The light intensity measurement at the laser light spot on the reflecting adapter identified an optimal (i.e., Gaussian) light intensity distribution.

dissemination of the units and associated scales of acceleration, angular acceleration, and derived motion quantities at the PTB can be found on www.ptb.de. In 2007 (see Fig. 28, result at 100 kHz), comparisons between the three interferometric methods FCM, SAM, and SCM were performed at the National Metrology Centre CENAM (NMI of Mexico) in collaboration with H.-J. von Martens. One of the standard measuring devices used at CENAM for primary vibration calibration of vibration transducers and laser vibrometers is shown in Fig. 1 of Ref. 18. A measurement set-up with a specific piezoelectric actuator is briefly described in Ref. 18 used for the investigation of the standardized interferometric methods at 100 kHz. For detailed results, see Fig. 15, Table III, and Ref. 18.

The results presented in Fig. 28 at the frequencies 62.8 kHz, 159.4 kHz, and 347 kHz were obtained in 2010 using the modified piezoelectric actuator shown in Fig. 10(a). An appropriate measuring equipment was set-up by CENAM’s R&D team in the Primary Vibration Laboratory, including (i) the mounting and the vibration isolation of the piezoelectric actuator (Fig. 10(a)), (ii) the multi-axial adjustment needed to adjust at least 2 laser vibrometers to the mirror (perpendicularly), and (iii) the electric supply and the electronic and optical sub-systems. The motion was sensed by laser light at the mirror close to the geometric axis perpendicularly to the 25 × 4 mm2 surface. The different laser methods and techniques were applied simultaneously. To make use of the vibration exciter’s resonance, an electric circuit was used with variable capacitor and inductance, which were adjusted so that the maximum vibration amplitude could be attained with minimum energy. The clamping fixture was sufficiently rigid without destroying the piezoelectric element. An acceleration amplitude of 500 km/s2 was attained at a resonance frequency 347 kHz, without damage or destruction of the piezoelectric element. The experimental investigations performed focused on displacement amplitude of l/4 and l/8, respectively. The latter value corresponds to 376 km/s2 at 347 kHz.

VII. MEASUREMENT RESULTS OF THE METHODS FCM, SAM, AND SCM

This section presents results and findings from a thorough analysis of the variety of measurement data (see Ref. 28)

TABLE III. Table III(a)–(d) reprinted with permission from H.-J. von Martens, AIP Conf. Proc. 1457, 181 (2012). Copyright 2012, American Institute of Physics. (a) Survey of results of FCM, SAM and SCM at all frequencies used in the experiments Frequency/ mean value FCM SAM SCM

62.8 kHz

100 kHz

159.4 kHz

347 kHz

158.13 nm 158.51 nm 158.21 nm

157.53 nm 158.60 nm 158.21 nm

158.01 nm 157.90 nm 158.21 nm

79.06 nm 78.66 nm 79.1 nm

(b) Experimental standard deviation within series of repeated measurements (200 single values at 100 kHz, 25 single values at the other frequencies) Frequency/ Rel. st. deviation FCM SAM

62.8 kHz 0.14 %–0.23 % 0.11 %–0.23 %

100 kHz 0.20 % 0.20 %

159.4 kHz 0.05 %–0.22 % 0.04 %–0.37 %

347 kHz 0.18 %–0.49 % 0.48 %–0.52 %

(c) Experimental standard deviation of a total of the single values at a frequency (over 3 series at 62.8 kHz, 3 series at 159.4 kHz, and 5 series at 347 kHz) Frequency/ Rel. st. deviation FCM SAM

62.8 kHz 0.18 % 0.19 %

100 kHz ... ...

159.4 kHz 0.16 % 0.24 %

347 kHz 0.60 % 0.92 %

(d) Deviations between the mean measurement results of the different methods used simultaneously Frequency/ Rel. st. deviation FCM-SAM SCM-SAM FCM-SCM

62.8 kHz −0.24 % −0.19 % −0.05 %

100 kHz −0.67 % −0.25 % −0.42 %

159.4 kHz 0.15 % 0.20 % −0.13 %

347 kHz 0.50 % 0.56 % −0.05 %


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FIG. 14. Series of 25 repeated measurements of an acceleration amplitude 24.64 km/s2 (average of all values) at a vibration frequency 62.8 kHz. The FCM and the SAM measured the displacement amplitude simultaneously, adjusted by the SCM to the level at which signal coincidence was indicated (l/4 = 158.2 nm). Each result is the average of 3 values taken from 3 series of measurement.

obtained at frequencies of 62.8 kHz to 347 kHz and acceleration amplitudes of 24.6 km/s2 to 376 km/s2 . Figure 14 shows for 62.8 kHz the ith acceleration amplitude result in series, computed from the ith displacement amplitude result as average of 3 values taken from three series of 25 measurements (cf. Eq. (3)). The measurements were simultaneously carried out by the FCM and the SAM at the displacement amplitude l/4 = 158.2 nm indicated by the SCM after the vibration level was adjusted to the occurrence of signal coincidence (see Fig. 3). The trend of the measurement results of both the FCM and the SAM revealed that the am-

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plitude varied slightly with time during the measurement series, due to heating effects in the piezoelectric element. To hold deviations from the reference value 158.2 nm within close limits during the measurement series, re-adjustments of the vibration level were performed once the interferometer signal displayed on screen of an oscilloscope showed excessive deviations from the coincidence shape. In the experiments presented for the frequencies 62.8 kHz (Figs. 14 and 24), 159.4 kHz (Figs. 16, 17, and 26) and 347 kHz (Figs. 18 and 27), the original SCM using an oscilloscope only (see Sec. IV B) was applied with the amplitude 158.2 nm (79.1 nm) as the (nominal) measurement result. The refined version of the SCM using oscilloscope and digital data processing (see. IV C) was applied to 100 kHz as demonstrated in Fig. 15 and in Fig. 6 of Ref. 18. Figure 15 shows the results of measurements at 100 kHz of the displacement amplitude of (nominal) 158.2 nm by 4 methods and techniques applied simultaneously (FCM and SAM as standard methods, SCM as a reference method, and a commercial LV). A relative deviation of the ith FCM result from the ith SAM of 0.7 % (average of 25 measurements in series) is within the uncertainty of the experiments. Using more recently the improved vibration excitation and measuring techniques and procedures (cf. Secs. V and VI) at the frequencies 62.8 kHz, 159.4 kHz, and 347 kHz, no systematic deviation between the different interferometric methods could be identified. Figure 16 shows for 159.4 kHz the ith acceleration amplitude result in series, computed from the ith displacement

FIG. 15. Simultaneous measurements of a displacement amplitude of (nominal) l/4 = 158.2 nm at a vibration frequency 100 kHz by the standard methods FCM and SAM, by the reference method SCM and by a commercial laser vibrometer (LV). Results over a series of repeated measurements and 6th order fits.


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FIG. 16. Series of 25 repeated measurements of an acceleration amplitude 158.5 km/s2 (average of all measurement values) at a vibration frequency 159.4 kHz. The FCM and the SAM measured the displacement amplitude simultaneously, adjusted by the SCM to the level at which signal coincidence was indicated (l/4 = 158.2 nm). The ith result in series is the average of 3 values taken from 3 repeated series of measurements.

amplitude result as average of 3 values taken from 3 series of 25 measurements (cf. Eq. (3)). All results lie well within the boundaries ±0.5 % around the average of all values. The frequency is 3 orders of magnitude greater than the reference frequency 160 Hz specified in ISO 16063-11 for accelerometer calibration with the same limits of the uncertainty of measurement (0.5 %). Figure 17 shows all values of the displacement amplitude measured by FCM and SAM within 3 series of 25 repeated measurements. The displacement amplitude was adjusted by the SCM to (nominal) l/4 = 158.2 nm. Only few values lie outside the boundaries ±0.5 % around the average 157.95 nm of all values. Figure 18 shows for 347 kHz the ith acceleration amplitude result in series as average of 5 values taken from 5 series of 25 measurements (cf. Eq. (3)).

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FIG. 18. Series of 25 repeated measurements of an acceleration amplitude 376 km/s2 (average of all values) at a vibration frequency 347 kHz. The FCM and the SAM measured the displacement amplitude simultaneously, adjusted by the SCM to the level at which signal coincidence was indicated (l/8 = 79.1 nm). Each result is the average of 5 values taken from 5 series of measurements.

VIII. EVALUATION OF MEASUREMENT UNCERTAINTY A. Survey of procedures of the “GUM”

The Guidance document (see Ref. 4) “Evaluation of measurement data - Guide to the expression of uncertainty in measurement” is the GUM 1995 with minor corrections. It establishes general rules for evaluating and expressing uncertainty in measurement that are intended to be applicable to a broad spectrum of measurements. Associated Guidance documents (see Refs. 113, 114, and 115) provide more specific information and procedures. A brief survey of the GUM procedure to calculate an expanded uncertainty in a vibration or shock measurement may be given as follows. The purpose of the expanded uncertainty U is to provide an interval y − U to y + U within which the value of Y, the specific quantity subject to measurement or calibration and estimated by y, can be expected to lie with high probability. To assert confidently that y − U ≤ Y ≤ y + U, it is recommended determining the expanded uncertainty U as follows:

r Make every effort to identify each effect that significantly influences the measurement result, and to compensate for such effects by applying the estimated corrections or correction factors. r Represent each uncertainty component that contributes to the uncertainty of measurement, by a standard deviation ui termed standard uncertainty, which is equal to the positive square root of the variance u2i . r Determine the combined standard uncertainty uc , as the standard uncertainty of the measurement of Y, by combining the individual standard uncertainties (and co-variances, as appropriate) using the law of propagation of uncertainty:

FIG. 17. Values of displacement amplitude obtained in 3 series of 25 repeated measurements by the FCM and the SAM applied simultaneously, as a function of the running number of measurement. The displacement amplitude was sequentially adjusted to (nominal) l/4 = 158.2 nm by the SCM. Vibration frequency 159.4 kHz.

uc (y) N 2 N−1 N ∂f ∂f ∂f 2 = u (xi )+2 u(xi , xj ). ∂x ∂x i i ∂xj i=1 i=1 j =i+1 (24)


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This equation is based on a first-order Taylor series approximation of Y = f (X1 , X2 , . . . , XN ) ,

(25)

where Y is the measurand determined from N input quantities X1 , X2 , . . . , XN through a functional relationship f. An estimate of the measurand Y, denoted by y, is obtained from Eq. (24) using input estimates x1 , x2 , . . . , xN for the values of the input quantities. The output estimate, which is the result of measurement, is thus given by y = f (x1 , x2 , . . . , xN ) .

(26)

The symbols ∂f/∂xi in Eq. (24) are often referred to as sensitivity coefficients. They are equal to the partial derivatives ∂f/∂Xi evaluated at Xi = xi . The symbol u(xi , xj ) designates the estimated co-variance associated with xi and xj . For the case where no significant correlations are present, Eq. (24) is reduced to N ∂f 2 u2 (xi ). (27) uc (y) = ∂x i i=1

r Determine the expanded uncertainty U by multiplying uc by a coverage factor k: U = kuc .

A set of rules has been explained which allow the uncertainty components (“standard uncertainties”) of the “input quantities Xi ” to be calculated in dependence on the degree of information available about the uncertainty sources. Some of the methodical tools described in Refs. 116, 117, and 130 are demonstrated in Annex C of ISO 16063-1.39 Various investigations of specific uncertainty sources and their effects on the measurement results are reported in Refs. 117–129. A detailed procedure to assess the expanded uncertainty in vibration and shock measurements and calibration is given and demonstrated in Ref. 130.

(28)

Preferably, a value of k = 2 is to be used. If it may be assumed that the possible values of the calibration result are approximately normally distributed, with an approximate standard deviation uc , the unknown value can be asserted to lie in the interval defined by U with a level of confidence, or coverage probability, of approximately 95 %. When the result of the measurement y is reported, the expanded uncertainty and the value of the coverage factor k used, if different from k = 2, are to be stated. In addition, the approximate coverage probability, or level of confidence, of the interval may be stated.

B. Tools for uncertainty evaluations in vibration measurements and calibrations

In vibration and shock measurements and calibrations, the application of the GUM may be difficult and very timeconsuming unless some possibilities of simplification are made use of. In Refs. 116, 117, and 130, a survey is given of the problems typically encountered in uncertainty calculations when vibrations are measured or accelerometers calibrated. It is shown how a model function of simple structure can be established for the usually complex relationship between the output quantity (e.g., sensitivity of an accelerometer), the quantity to be measured (e.g., acceleration), and various influence quantities (noise, transverse motion, base strain, etc.). Among other things, nonlinear effects such as the influences of distortion, hum, and noise can be properly taken into account.

C. Randomization of systematic effects in interferometric measurements

The term measurement error is defined (see Ref. 31) as the measured quantity value minus a reference quantity value; a systematic measurement error as a component of measurement error that in replicate measurements remains constant or varies in a predictable manner. Observing these definitions, the abbreviated term “error” (symbol e) may be understood as the effect of a certain influence quantity, defined as the deviation of the value of a quantity or parameter from a (reference) value which would exist if the respective influence quantity was either zero (e.g., noise, distortion factor) or at its nominal (reference) value (e.g., temperature). Relative errors are expressed in the following by the symbol e*, specified by a subscript for the particular source of the error. A cause-and-effect diagram has been widely used as a methodical means to recognize and consider the variety of the error sources which may significantly affect the result of a specific measurement under given conditions. The specific cause-and-effect diagram shown in Fig. 19 was developed to describe the error sources and their effects in vibration measurements by laser interferometry using the FCM.94, 116, 117 The error sources (e.g., disturbing quantities such as hum and noise) affect the vibration measurement results of the amplitudes of displacement, velocity, and acceleration in different way on the basis of the relationships Eqs. (2) and (3) between the parameters of these vibration quantities.

1. Error due to the quantization interval

The distance between two fringes (intensity maxima or intensity minima) in the interferometer type used (preferably a Michelson interferometer) is the quantization interval, and the “quantization error” is the deviation, due to the quantization effect, of the measurement result obtained according to the formula: sˆmeas =

Z l M8

(29)

from the “true” value to be measured. In Eq. (29), Z is the number of interferometer signal periods (fringes) counted during an integer number of vibration periods (e.g., M = 10 000).


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FIG. 19. Cause-and-effect diagram for the measurement of the displacement amplitude by the fringe-counting method (FCM).

The following expression has been established (see Refs. 94 and 95) to describe the quantization error: ⎧ ˆs for μπ ≤ ϕ0 < (μ + 1/2) π − 2π |ˆs |/s ⎪ ⎪ ⎨ eQ = −ˆs + (sgn ˆs ) s/4 for (μ + 1/2) π − 2π |ˆs |/s ≤ ϕ0 < (μ + 1/2) π + 2π |ˆs |/s ⎪ ⎪ ⎩ ˆs for (μ + 1/2) π + 2π |ˆs |/s ≤ ϕ0 < (μ + 1) π, where

ˆs = sˆ − sˆ0 , sˆ0 = (2v + 1) s/4,

s = λ/2,

Equation (30) expresses the periodicity of the quantization error with increasing displacement amplitude and, at given amplitude, with the initial phase angle of the interferometer signal. Figure 20 demonstrates the quantization error as a function of the initial phase angle for three selected fractions of the displacement amplitude. It is obvious in the case ˆs = l/16 (Fig. 10(a)) that the resulting quantization error becomes negligible in the course of an integrating measurement over a sufficient number of vibration periods, if a uniform distribution of the initial-phase angle is achieved within an interval [ϕ01 ; ϕ01 + mπ ] ,

m = 1, 2, . . . .

(31)

A significant finding was (see Ref. 29) that this is valid for all possible values of the displacement amplitudes (i.e., residual section ˆs defined by Eq. (30)), in particular for the cases Figs. 10(b) and 10(c) in Fig. 20, though in these cases the quantization error for single vibration periods can run up to nearly +l/8 (Fig. 20(b)) or −l/8 (Fig. 20(c)), depending on the initial phase. Figure 6 (introduced in the context of the SCM) may explain the disturbing effect. The trigger level of the impulse counter (FCM) is normally set close to zero. The traces in Fig. 6 show that positive or negative deviations from l/4 = 158.2 nm may lead to one additional counting or to a missing counting per vibration period, i.e., to a quantization error of ±l/8 (see Eq. (29).

μ = 0, ±1, ±2, . . . ,

v = int

sˆ 2s

(30)

.

To attain averaging of the varying quantization error over a large number of vibration periods (e.g., M = 105 at a vibration frequency 10 kHz, i.e., fringe counting over 10 s), variations of the initial phase of the interferometer signal have been stimulated by generation of defined slow motion excited either at the reference reflector of an interferometer (by a piezoelectric actuator) or at the moving part of a vibration exciter. In the case of an electrodynamic vibration exciter, a current varying in time may be fed into the driving coil, in addition to the sinusoidal current generating the measurand. The three types of stimulated phase variations shown in Fig. 21 have been investigated and applied. A linear variation of the initial phase such as shown in Fig. 21(a) is most effective but in practice attainable only if disturbing phase variations (due to ground motion and other sources) are negligible. A sinusoidal phase variation (see Fig. 21(b)) has been applied, for example, in the reported measurements performed at 100 kHz (see Fig. 15 and Ref. 18); the frequency of the stimulated phase variation was 50 Hz (hum). Sinusoidal phase variation leads to some averaging and suppression of the quantization error. However, for this type, a theory, which describes the relationships and optimal conditions for application, was so far not developed, contrary to the type shown in Fig. 21(c). For measurement conditions in which disturbing variations of the initial phase do occur, another version of the counting method with excitation of a stochastic variation of


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FIG. 20. Normalized quantization error eQ (ϕ 0 ) for an amplitude deviation: (a) ˆs = l/16, (b) 4% > ˆs = l/8, (c) 4% < ˆs = l/8.

ϕ 0 has been developed.94, 95 An added low-frequency narrowband stochastic process with well-defined parameters (see Fig. 21(c)) dominates the disturbing phase variations and suppresses the quantization error when a suitable standard deviation σϕ0 of ϕ 0 is stimulated and the time of the integrating measurement is long enough (Figs. 22 and 23). Specific functional relationships established according to the general function es∗ˆ,Q = f (ˆs , ϕ0 , σϕ0 )

(32)

(see Refs. 94 and 95) express the expected value and the variance of the quantization error for various conditions of measurement, which are of interest in practice. Figure 22 shows that the expected value of the (resulting) quantization error can be reduced by three orders of magnitude already by relatively small intensity (normalized standard deviation σϕ0 /π = 0, 3) of a narrow band stochastic process

FIG. 21. Stimulation of a variation of the initial phase ϕ 0 to suppress the quantization error: (a) linear phase variation, (b) sinusoidal phase variation, and (c) stochastic phase variation (narrow-band low-frequency stochastic process). Slow phase variations are added to the phase oscillations due to the vibration to be measured (phase modulation).

of 2nd order with resonance rise Q = 10. These conditions were realized at the PTB by feeding the driving coil of an electrodynamic vibration exciter with a narrow-band noise from an electrical multi-function generator, in addition to the sinusoidal current generating the vibration to be measured.


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viation of the resulting quantization error was reduced only by an order of magnitude. This example demonstrates that the quantization error can be suppressed by stimulation of a stochastic variation of the initial phase ϕ 0 of interferometer signal (Fig. 21(c)) if the relationships and optimal conditions of use, described in Refs. 94 and 95 in more detail, are observed. 2. Uncertainty contributions at high vibration frequencies

Among various types of errors influencing the measurement result in the FCM, the following are particularly significant in the measurements performed at high frequencies above 50 kHz. a. Relative error due to trigger hysteresis of the fringe counter FIG. 22. Normalized expected value of the quantization error as a function ˆ xˆ defined by Eq. (30) with of the residual section xˆ of the amplitude x; xˆ = sˆ , xˆ = ˆs , x = s = l/2 for the interferometer type used; initial phase angle ϕ 0 is a narrow band stochastic process of 2nd order with resonance rise Q = 10, expected value E(ϕ 0 ) = π /4 + μπ ,μ = 0, ±1, ±2, . . . and standard deviation σϕ0 ; parameter is σϕ0 /π .

A center frequency f0 = 20 Hz of the narrow-band noise was used preferably. Figure 23 shows the normalized variance of the quantization error eQ as a function of the normalized measurement integration time. Under the above-mentioned measurement conditions, the period T0 is 50 ms. For fringe counting over an integration measurement time of 1 s (i.e., 105 vibration periods at a frequency of 100 kHz), the relation Tmeas /T0 is 20. For a standard deviation σϕ0 /π = 0.3, the normalized variance is reduced from 10−2 to 10−4 ; i.e., the normalized standard de-

FIG. 23. Normalized variance of the quantization error eQ as a function of the measurement integration time Tmeas related to the period T0 of the center frequency of the narrow band stochastic process. As for Fig. 22 but with σϕ0 = 0,3 π (trace 1), . . . , 1,0 π (trace 8).

1 l uH , (33) 8π sˆ uˆ where uH is the trigger hysteresis and uˆ the amplitude of the interferometer signal in terms of voltage. es∗ˆ,H = −

b. Relative error due to vibration distortion

1 aˆ (fn ) · cos ϕa(fn ) , n= 1, 3, 5, . . . , (34) · aˆ n2 where aˆ (fn ) is the amplitude of the nth harmonic of the acceleration spectrum and ϕ a the phase shift related to the acceleration’s fundamental harmonic. es∗ˆ,a(fn ) =

The interferometer function itself may be disturbed by the instability of the laser light wavelength, misalignment, optical nonlinearity, the reflectivity of the reflectors influenced by flatness and roughness and other causes (see Fig. 19). Both the FCM and the SCM have in common that they sense the peak-to-peak value rather than the amplitude or double amplitude of the fundamental harmonic of the vibration. Hence, Eq. (34), which describes the influence of nonlinear distortion (i.e., harmonics of the fundamental sinusoidal vibration) on the measurement result of the FCM, is valid for the SCM, too. In the same way, specific functional relationships (cf. Ref. 95) describing deviations of the peak-to-peak value from the double amplitude due to other influence quantities in the FCM, are valid for the SCM, too. For the SAM, the procedures for the evaluation of uncertainty have also been established and demonstrated for real measurement conditions.86 In the conventionally stated lower frequency range, drift (offset variation per vibration) has usually a dominating effect on the measurement result as it was demonstrated.86 Each vibration period is affected by a systematic measurement error due to the actual drift (change of zero position of displacement or of velocity) occurring during that period. Specific formulas have been established (see Ref. 86) which functionally relate the offset variation per vibration to the errors due to drift in the measurements of amplitude and c. Errors due to the interferometer


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FIG. 24. Relative deviations between the results of amplitude measurement obtained by FCM, SCM and SAM at the frequency 62.8 kHz and the displacement amplitude (nominal) l/4 = 158.2 nm. First order least-square-fits. For the measurement conditions, see caption of Fig. 14.

FIG. 26. Relative deviations between the results of amplitude measurement obtained by FCM, SCM, and SAM at the frequency 159.4 kHz and displacement amplitude (nominal) l/4 = 158.2 nm. First order least-square-fits. For the measurement conditions, see caption of Fig. 16.

initial phase of the displacement. At high frequencies (e.g., 100 kHz corresponding to the vibration period T = 10 μs), the drift per vibration period is under normal measurement conditions very small and its influence negligible.

Figure 25 shows the deviations between the values measured synchronously by SAM and FCM in a series of 200 repeated measurements of an acceleration amplitude of 62.5 km/s2 at 100 kHz. A relative systematic deviation of 0.7 % between the mean results is within the uncertainty of the experimental investigation reported (see Ref. 18). Figure 26 shows the relative deviations between measurement results as explained for Fig. 24 but at a frequency of 159.4 kHz. Figure 27 shows measurement results of displacement amplitude of nominally l/8 (79.1 nm) at 347 kHz obtained by the FCM and the SAM, including statistical characteristics (mean value and experimental standard deviation). The SCM was used to adjust the vibration exciter to the amplitude close to l/8 by an oscilloscope. A survey of the statistical analysis of the measurement results is given in Table III. In Table III(a), the mean value from measurement series of repeated measurements for each frequency is given. Table III(b) gives the experimental standard deviation of the single values of repeated measurements within a series. From the different series, the minimum and the maximum experimental standard deviations are given in this table. In Table III(c), the spread of all measurement

D. Assessment of the accuracy (uncertainty) achieved at frequencies up to 350 kHz

Figure 24 shows for the frequency 62.8 kHz the relative deviations of the measurement result of FCM and of SCM, respectively, from the SAM measurement result for each of the 25 repeated measurements (average of 3 values from 3 series of measurements). The deviation of the FCM result from the SAM result is not affected by variations of the displacement amplitude due by heating effects and repeated amplitude adjustments by the SCM at the times at which given tolerances were exceeded. However, the deviation of the SCM result from the SAM result is affected by such variations if the nominal amplitude value 158.2 nm for (ideal) coincidence is used as SCM result. That is valid for the results, which are represented in Fig. 24. The 1st order least-square-fit of the SCM-to-SAM results revealed that the relative amplitude variations during the 25 measurements were smaller than 0.3 %. The refined procedure of the SCM taking variations of the vibration amplitude into account (see Sec. IV C) is demonstrated in Fig. 15 and more comprehensively in Ref. 18.

FIG. 25. Relative deviation of FCM result from SAM result over a series of 200 simultaneous repeated measurements. Amplitudes of displacement 158.2 nm and acceleration 62.5 km/s2 , vibration frequency 100 kHz.

FIG. 27. Average of 5 values (from 5 measurement series) of acceleration amplitude ± experimental standard deviation for 25 repeated measurements by the FCM and the SAM, respectively. Vibration frequency 347 kHz. First order least-square-fits. For the measurement conditions, see caption of Fig. 18.


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FIG. 28. Deviations between measurement results of interferometric methods at frequencies from 0.4 Hz to 347 kHz. For explanations, see Sec. VI. Reprinted with permission from H.-J. von Martens, AIP Conf. Proc. 1457, 181 (2012). Copyright 2012, American Institute of Physics.

values over all series of measurements performed at the specified frequency is expressed by a total experimental standard deviation. This includes systematic effects between different series of measurements, if any. Table III(d) gives the relative deviations between the mean measurement results obtained by the different methods used simultaneously (i.e., the results given in Table III(a)). The results specified for 100 kHz were obtained in former experiments performed in 2007.18 They are less accurate than the results obtained in 2010 at 62.8 kHz, 159.4 kHz, and 347 kHz. In all cases, however, the deviations are

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