Microsoft Word C034954e doc Reference number ISO 16063 15 2006(E) © ISO 2006 INTERNATIONAL STANDARD ISO 16063 15 First edition 2006 08 01 Methods for the calibration of vibration and shock transducers[.]
General
Clause 4 gives recommended specifications for the apparatus necessary to comply with the scope of Clause 1 and to obtain the uncertainties of Clause 3
If desired, systems covering only parts of the ranges may be used, and normally different systems (e.g exciters) should be used to cover all the frequency and dynamic ranges
The apparatus outlined in Clause 4 includes all necessary devices and instruments for the six calibration methods detailed in this section of ISO 16063 Each method is clearly indicated in the accompanying figures.
8 and 10) © ISO 2006 – All rights reserved 3
Frequency generator and indicator
A frequency generator and indicator must possess specific characteristics, including a maximum frequency uncertainty of 0.05% of the reading, frequency stability of better than ±0.05% of the reading throughout the measurement duration, and amplitude stability also exceeding ±0.05% of the reading during the measurement period.
Power amplifier/angular vibration exciter combination
A power amplifier/angular vibration exciter combination having the following characteristics shall be used: a) total harmonic distortion: 2 % maximum;
NOTE 1 This specification relates to the input quantity for the transducer to be calibrated
When utilizing method 3A or method 3B, higher levels of harmonic distortion may be acceptable For interferometer type A, it is essential to maintain a transverse motion of less than 1% of the tangential motion component at the minimum rotational angle displacement In the case of interferometer type B, a maximum lateral motion of 2 µm is permissible, achievable only with a high-precision rotational air bearing for the measuring table Additionally, hum and noise levels must be at least 70 dB below full output, and the stability of angular acceleration amplitude should be better than ± 0.05% of the reading throughout the measurement period.
4.3.2 Electro-dynamic angular vibration exciter
An electrodynamic vibration exciter is based on the Lorentz force acting on electric charge carriers when these move through a magnetic field
The coil within the magnetized air gap of a magnetic circuit can be engineered to produce a dynamic torque through the Lorentz force, thereby exciting the measuring table with the angular transducer that requires calibration for angular vibration.
In the frequency range of 1 Hz to 1.6 kHz, the angular acceleration amplitude is directly proportional to the electric current amplitude flowing through the coil An example of an angular vibration exciter is illustrated in Figure 1, where the maximum rotational amplitude is capped at 30° (double amplitude).
1 rad) Another example of an angular acceleration exciter (amplitude of 60°, i.e 1 rad) is described in Reference [14]
Figure 1 — Example of an angular exciter (mode of function)
4.3.3 Angular vibration exciter based on a brushless electric motor
Special angular exciters have been designed and manufactured for angular transducer calibration using commercial electric motors
Rate tables have been developed over many years for testing inertial navigation sensors These tables typically utilize brushless, three-phase, hollow-shaft motors that are electronically commutated and servo-controlled, particularly for the angular velocity operating mode They usually generate a constant angular velocity and can achieve sinusoidal angular velocities with minimal distortion.
Recent advancements in control technology enable this type of exciter to generate angular acceleration A fundamental requirement for this application is the incorporation of an air bearing, similar to that used in the flat-coil exciter.
Increasing distortion after differentiation necessitates frequency-selective measurement of angular accelerometer transducer output signals This calibration is achieved through methods 3A or 3B, specifically utilizing sine-approximation techniques.
Seismic block(s) for vibration exciter and laser interferometer
To ensure accurate calibration results, it is essential to mount the angular vibration exciter and the interferometer on the same heavy block or on two separate heavy blocks This setup minimizes relative motion caused by ground vibrations and prevents the support structure of the vibration exciter from adversely affecting the calibration process.
For effective performance, a seismic block must possess a moment of inertia at least 2,000 times greater than the moving mass, resulting in less than 0.05% reactive angular vibration in the angular transducer and interferometer If the moment of inertia is insufficient, the motion induced by the vibrator must be considered.
To minimize the impact of ground motion, seismic blocks operating within the frequency range of 1 Hz to 1.6 kHz must be mounted on damped springs This design aims to limit the uncertainty caused by these disturbances to below 0.1%.
Laser
A red helium-neon laser or a single-frequency laser with a known wavelength will be utilized In laboratory conditions, specifically at an atmospheric pressure of 100 kPa, a temperature of 23 °C, and a relative humidity of 50%, the wavelength of the red helium-neon laser measures 0.63281 µm.
If the laser is provided with a manual or automatic atmospheric compensation device, this shall be set to zero or switched off.
Interferometer
The interferometer may be used to transform
⎯ the rotational angle, Φ(t), into a proportional phase shift, ϕ M (t), of the interferometer output signal,
⎯ the angular velocity, Ω(t), into a proportional frequency shift, f D (t) (Doppler frequency), of the interferometer output signal
For both transformations, a homodyne or a heterodyne interferometer (cf Figures 3 to 8 and 10) and a one- channel or two-channel arrangement (cf Figures 3 to 8 and 10) may be used
The initial transformation of Φ(t) to ϕ M (t) is defined as a standard procedure in ISO 16063, while the subsequent transformation of Ω(t) to f D (t) is presented as an optional process, supported by comprehensive descriptions in existing literature.
Interferometer types A and B share a common feature in that their measuring beams detect translational displacement motion, allowing for the use of conventional interferometers in rectilinear vibration measurements To facilitate this application, rotational motion is converted into a corresponding translational displacement component For interferometer type A, this is achieved using retro-reflectors as measuring reflectors, while for type B, a diffraction grating is positioned on the rotary measuring table In the case of type B, an optically reflecting diffraction grating must be placed on the lateral surface of an airborne rotary table to accommodate a tolerable eccentricity of 2 µm.
For methods 1A, 1B (see Figures 3 and 4) and Methods 2A, 2B (see Figures 5 and 6), a common Michelson interferometer with a single light detector is sufficient
The Michelson interferometer can be realized with a single measuring beam or with two measuring beams
In methods 3A and 3B, a modified Michelson interferometer featuring quadrature signal outputs and two light detectors is employed to sense the interferometer signal beams, as illustrated in Figures 7 and 8 The design, depicted in Figure 9, includes a quarter wavelength retarder that transforms the incident linearly polarized light into two measuring beams with perpendicular polarization states and a 90° phase shift Following interference with the linearly polarized reference beam, the two components with perpendicular polarization are spatially separated using suitable optics, such as a Wollaston prism or a polarizing beam splitter, and are subsequently detected by two photodiodes.
The modified Michelson interferometer must maintain output offsets within ± 5% of amplitude, relative amplitude deviations under ± 5%, and angle deviations of less than ± 5° from the nominal 90° To achieve these tolerances, suitable adjustments for offset, signal level, and the angle between the two interferometer signals are essential.
At large rotational angles, maintaining the specified tolerances for the deviations of the two outputs of the modified Michelson interferometer can be challenging To meet the measurement uncertainty outlined in Clause 3, these tolerances must be adhered to for small rotational angles of up to \$2 \times 10^{-2}\$ rad For larger amplitudes, more lenient tolerances are acceptable.
EXAMPLE For a rotational angle of 2,5 × 10 −2 rad (i.e angular acceleration amplitude of 1 rad/s 2 at a frequency of
For frequencies of 1 Hz, tolerances can be adjusted to ± 10% for offsets and relative amplitude deviations, and ± 20° for deviations from the nominal angle of 90°.
The tolerances stated above are valid without correction of quadrature fringe measurement errors in interferometer If the correction procedure after Heydemann [6] is applied, greater tolerances are permitted
For methods 1A, 1B, 2A, 2B, 3A or 3B, another suitable interferometer, e.g a (modified) Mach-Zehnder heterodyne interferometer (cf Figure 10) may be used in the place of the (modified) Michelson interferometer
An interferometer of type A or B should be utilized alongside a light detector to detect the interferometer signal bands, ensuring the frequency response encompasses the required bandwidth The maximum bandwidth, denoted as frequency \( f_{\text{max}} \), can be determined from the maximum angular velocity amplitude \( \Omega_{\text{max}} \) using the appropriate equation.
R is the effective radius (cf 4.6.2 for the definition for interferometer of type A and 4.6.3, for interferometer of type B);
∆s is the displacement quantization interval of the interferometer
In interferometer type A, the path difference is given by \(\Delta s = \frac{\lambda}{2}\) for a single measuring beam and \(\Delta s = \frac{\lambda}{4}\) for a two-beam arrangement, where \(\lambda\) represents the laser wavelength Conversely, in interferometer type B, the path difference is \(\Delta s = g\) for a single measuring beam and \(\Delta s = \frac{g}{2}\) for a two-beam arrangement, with \(g\) denoting the grating constant.
4.6.2 Interferometer type A (retro-reflector interferometer)
For methods 1A and 2A, a Michelson-type interferometer equipped with retro-reflectors will be utilized, along with a light detector to sense the interferometer signal bands and ensure a frequency response that meets the required bandwidth To mitigate the effects of disturbing motion, a two-beam arrangement will be implemented, featuring two symmetrically mounted retro-reflectors, positioned 180° apart at a distance, R, from the axis of rotation.
The laser beam is directed to a beam splitter, which divides it into two parallel components that are sent to retro-reflectors The reflected beams overlap, and the relevant portion of the combined light intensity is converted into an electrical signal by the photodetector, known as the interferometer signal.
The two-beam arrangement enhances sensitivity by compensating for disturbing motions, such as ground vibrations, effectively reducing the quantization interval to λ/4 instead of λ/2 Utilizing retro-reflectors instead of plane mirrors mitigates the tilting effect caused by rotational motion within a specific range Additionally, the interferometer is designed to tolerate transverse disturbing motions without impacting measurement uncertainty.
Method 3A utilizes a quadrature interferometer with retro-reflectors, incorporating measuring and reference reflectors In the homodyne interferometer configuration, a stabilized single-frequency laser serves as the light source, with its beam diameter expanded by lenses to minimize divergence The beam splitter divides the polarized laser beam into measuring and reference beams, with the reference beam being reflected and parallel-shifted by a retro-reflector A λ/8 retardation waveplate, traversed twice, creates a path difference of λ/4, resulting in two beams with orthogonal polarization directions and a 90° phase shift The measuring beam, reflected by a retro-reflector on the measuring table, maintains its linear polarization Upon superimposing the linearly polarized measuring beam with the circularly polarized reference beam and passing through a Wollaston prism inclined at 45°, two linearly polarized components are produced, perpendicular to each other This separation leads to two distinct interference systems with a 90° phase shift, and the photodetectors convert the light intensity variations into electrical signals that exhibit sinusoidal and cosinusoidal relationships with the displacement of the measuring reflector.
4.6.3 Interferometer type B (diffraction-grating interferometer)
A Michelson interferometer equipped with a diffraction grating as a measuring reflector will be utilized, along with a light detector to sense the interferometer signal bands The system will feature a frequency response that encompasses the required bandwidth.
For methods 1B and 2B, a modified Michelson interferometer with diffraction grating is used (cf Figures 2, 4 and 6)
The angular acceleration, the angular velocity or the rotational angle are measured by a special diffraction grating interferometer developed on the basis of a high-resolution grating (e.g a sine-phase grating of
The optical reflection grating, with grooves of either 2,400 or 3,000 grooves/mm, is positioned on the air-borne measuring table of the angular vibration exciter, aligned with the axis of rotation A frequency-stabilized single-frequency He-Ne laser emits a light beam that is divided into two parallel beams, which strike the grating symmetrically at an angle that allows the first-order diffracted beams to return in the direction of the incident beam As the moving part rotates, these beams experience a frequency change that is proportional to the tangential and angular velocity The interference of these beams results in a light intensity that varies periodically with the rotational angle.
For method 3B, a homodyne quadrature interferometer with diffraction grating is used (cf Figure 8)
In a quadrature diffraction-grating interferometer with a single measuring beam setup, the light beam is divided into a reference beam and a measuring beam The measuring beam hits the grating at a specific angle, causing the first-order beam, diffracted by reflection, to return in the direction of the incident beam This first-order beam, along with the reference beam, overlaps in the optical arrangement, resulting in interference The intensity of the combined light beams varies sinusoidally with the rotational angle.
Instrumentation for interferometer signal processing
The instrumentation employed involves demodulating the phase-modulated electric current or voltage output from the photodetectors to extract key vibration parameters, such as amplitude and initial phase of the sinusoidal rotational angle Various techniques are applied for methods 1A and 1B, as well as methods 2A and 2B, and methods 3A and 3B, each detailed in their respective sections.
4.7.2 Instrumentation for fringe counting (for methods 1A and 1B)
The counting instrumentation shall have the following characteristics: a) frequency range: 1 Hz to the maximum frequency needed (20 MHz is typically used); b) maximum uncertainty: 0,01 % of reading
The counter may be replaced by a ratio counter offering the same uncertainty
4.7.3 Instrumentation for zero-point detection (for methods 2A and 2B)
A tunable bandpass filter or spectrum analyser is required with specific characteristics: a frequency range from 800 Hz to 1.6 kHz, a bandwidth of less than 12% of the center frequency, filter slopes of at least 24 dB per octave, a signal-to-noise ratio exceeding 70 dB below the maximum signal, and a dynamic range greater than 60 dB.
For zero detection, instrumentation with a frequency range of 800 Hz to 1.6 kHz is required, as it effectively identifies output noise from the bandpass filter, eliminating the need for a spectrum analyzer.
4.7.4 Instrumentation for sine-approximation (for methods 3A and 3B)
A waveform recorder with a computer interface is essential for analog-to-digital conversion and storage of the interferometer quadrature outputs and the accelerometer output It must have adequate amplitude resolution, sampling rate, and memory to ensure calibration within the specified uncertainty range Typically, a resolution of 10 bits is required for the accelerometer output, while 8 bits is sufficient for the interferometer's quadrature signal outputs A two-channel waveform recorder can be utilized for the interferometer outputs, alongside another recorder with higher resolution and lower sampling rate for the angular transducer output Importantly, data conversion for both the interferometer and angular transducer outputs must commence and conclude simultaneously, adhering to the uncertainty requirements outlined in Clause 3.
To accurately capture the shortest period of the interferometer output signal at maximum velocity, a sufficient number of samples is essential As the frequency decreases for a specific angular acceleration amplitude, larger displacement amplitudes are observed, necessitating higher sampling rates and increased memory capacity If these capabilities are unavailable, it is necessary to reduce the angular acceleration amplitude.
To calibrate an angular accelerometer at a vibration frequency of 10 Hz with an angular acceleration amplitude of 1,000 rad/s², a memory capacity of 4 Mbytes is required when utilizing a sampling frequency of 20 kHz.
A computer with data-processing program (for methods 3A and 3B) in accordance with the procedure for the calculations stated in 10.4 shall be used.
Voltage instrumentation, measuring true r.m.s accelerometer output
Voltage instrumentation, measuring true r.m.s accelerometer output, having the following characteristics shall be used: a) frequency range: u 1 Hz to W 1,6 kHz; b) maximum uncertainty: 0,1 % of reading
The r.m.s value shall be multiplied by a factor of 2 to obtain the (single) amplitude
For methods 1A, 1B, 2A and 2B, an r.m.s voltmeter shall be used For methods 3A and 3B, special voltage- measuring instrumentation in accordance with 4.7.4 shall be used; an r.m.s voltmeter may be applied in addition (optional).
Distortion-measuring instrumentation
Instrumentation for measuring total harmonic distortion should be capable of detecting levels between < 1% and 5% Key specifications include a frequency range from 1 Hz to 1.6 kHz, allowing for measurement up to the fifth harmonic, and a maximum uncertainty of 10% of the reading within the distortion range of 0.5% to 5%.
Oscilloscope (optional)
An oscilloscope for optimizing the interferometer and for checking the waveform of the interferometer and accelerometer signals, with a frequency range from 1 Hz to 2 MHz minimum, may be used.
Other requirements
The transducer designated for calibration must possess structural rigidity It is essential to consider the base strain sensitivity, transverse sensitivity, and the stability of the angular accelerometer/amplifier combination, particularly if calibrated as a single unit, when calculating measurement uncertainty (refer to Annex A).
All effects influencing the measurement result shall be included in the uncertainty calculation
Methods 1B, 2B, and 3B are applicable for calibrating rotational laser vibrometers when the motion parameter is detected simultaneously by both the standard device and the laser interferometer being calibrated It is essential that the rotational vibration amplitude remains sufficiently stable to satisfy the uncertainty requirements outlined in Clause 3, especially if the motion sensing periods of the two measurement systems differ An illustrative example of a calibration setup for rotational laser interferometers is depicted in Figure 11.
The calibration shall be carried out under the following ambient conditions: a) room temperature: (23 ± 3) °C; b) relative humidity: 75 % max
Care should be taken that external vibration and noise do not affect the quality of the measurements
6 Preferred angular accelerations and frequencies
The angular accelerations (amplitude or r.m.s value) and frequencies equally covering the angular accelerometer range should preferably be chosen from the following series; a) angular acceleration (methods 1A, 1B, 3A and 3B):
⎯ 0,1 rad/s 2 , 0,2 rad/s 2 , 0,5 rad/s 2 , 1 rad/s 2 , 2 rad/s 2 , 5 rad/s 2 , 10 rad/s 2 , 20 rad/s 2 , 50 rad/s 2 ,
100 rad/s 2 , 200 rad/s 2 , 500 rad/s 2 , 1 000 rad/s 2 (1 000 rad/s 2 is valid for amplitude only); b) frequency:
⎯ selected from the standardized one-third-octave frequency series (in accordance with ISO 266) between 1 Hz and 1,6 kHz or the series of radian(s) frequencies evolving from ω = 1 000 rad/s
7 Common procedure for all six methods
Methods A1, B1, A2, B2, A3, and B3 utilize an interferometer to detect displacement at a distance, R, known as the "effective radius," from the axis of rotation of a circular measuring table in an angular vibration exciter The amplitude of the rotational angle, \$\hat{\Phi}\$, is derived from the displacement amplitude, \$\hat{s}\$, using the formula \$\hat{s} = R \hat{\Phi}\$ The effective radius, R, is determined through a specific calibration of the interferometer conducted prior to its use for transducer calibrations The measurement of \$\hat{s}\$ relies on a comparison with a precisely known value in the sub-micrometer range For interferometer type A, this known value is the wavelength \$\lambda = 0.63281 \, \mu m\$ of a red helium-neon laser, while for type B, it is the gating constant \$g\$, which corresponds to a groove length of \$0.33333 \, \mu m\$ from a sine-phase diffraction grating with 3,000 grooves/mm, accurately measured by the manufacturer.
If the grating constant is inaccurately known, the angular quantization interval \(\Delta\Phi\), which corresponds to one period of the interferometer signal, can be determined through calibration of a specialized diffraction grating interferometer Subsequently, the relationship \(\Delta s = R \Delta\Phi\) can be utilized to eliminate the radius \(R\), allowing for the calculation of \(\hat{\Phi}\) without its direct involvement, as indicated in Equation (2).
All six methods apply the result for the rotational angle amplitude, ˆΦ, obtained from Equation (2) to calculate the following: a) sensitivity (magnitude), S Φ , of rotational angle transducers, using Equation (3): ˆ ˆ
=Φ (3) b) sensitivity (magnitude), S Ω , of angular velocity transducers, using Equations (4) and (5): ˆ ˆ
`,,```,,,,````-`-`,,`,,`,`,,` - © ISO 2006 – All rights reserved 11 c) sensitivity (magnitude), S α , of angular accelerometers, using Equations (6) and (7): ˆ ˆ
2 2 ˆ ˆ 4 f α = π × Φ (7) where uˆ is the amplitude of the angular transducer output, u, (e.g output voltage of an angular accelerometer);
Ωˆ is the amplitude of the angular velocity, Ω; αˆ is the amplitude of the angular acceleration, α
The phase shift of the complex sensitivity of angular transducers can only be measured using methods 3A and 3B, with the standard procedures for phase shift calibrations outlined in Clause 10.
8 Methods using fringe-counting (methods 1A and 1B)
General
This method is applicable to sensitivity magnitude calibration the frequency range from 1 Hz to 800 Hz
At a frequency of 800 Hz and an angular acceleration amplitude of 1,000 rad/s², the rotational angle amplitude is 4 × 10⁻⁵ rad, resulting in a displacement amplitude of 2 µm when a retro-reflector or diffraction grating is positioned 50 mm from the rotation axis The fringe-counting method can measure displacement amplitudes as low as 2 µm with specified uncertainty, even without special quantization error suppression Additionally, Methods 1A and 1B can be utilized for smaller amplitudes if quantization errors are mitigated, enabling calibration at a defined angular acceleration amplitude, such as 1,000 rad/s², at higher frequencies.
In both interferometer types A and B (i.e in methods 1A and 1B), the number of signal periods (e.g intensity maxima), N, is given by Equation (7): ˆ
= ∆ × (9) where sˆ is the displacement amplitude sensed by the laser interferometer, required to apply Equations (2) to (7);
∆s is the quantization interval, equal to λ/2, specified by Equations (10) and (11) for the two versions of interferometer type A and by Equations (12) and (13) for the two versions of interferometer type B;
`,,```,,,,````-`-`,,`,,`,`,,` - f is the frequency of the angular vibration exciter; f f is the (mean) fringe frequency
To determine the rotational angle amplitude, \( \hat{\Phi} \), for type A and type B interferometers, relevant expressions for \( \Delta s \) are inserted using Equations (10) or (11) and Equations (12) or (13), respectively, along with Equation (2) to convert displacement into a rotational angle The angular velocity amplitude, \( \hat{\Omega} \), and angular acceleration amplitude, \( \hat{\alpha} \), are derived from Equations (5) and (7) The sensitivity of the angular transducer is calculated using Equations (3), (4), and (6).
Common test procedure for methods 1A and 1B
After optimizing the interferometer, assess the sensitivity of the transducer at the required angular vibration frequencies and angular acceleration amplitudes by measuring the fringe frequency with a counter or using a ratio counter to determine the relationship between the angular vibration frequency and the fringe frequency.
Expression of results
Calculate the displacement amplitude from the fringe frequency readings calculated in Equation (9) using Equation (10) for the two-beam arrangement shown in Figure 3:
∆s = λ/4 (10) or Equation (11) for the single measuring beam arrangement shown in Figure 7:
∆s = λ/2 (11) where λ is the wavelength λ = 0,632 81 àm of the laser of the red helium-neon type
To determine the amplitude, \$\hat{\Phi}\$, of the rotational angle \$\Phi\$, utilize Equation (2) Next, calculate the sensitivity \$S_{\Phi}\$ (magnitude) of the rotational angle transducer from Equation (3) For the angular velocity transducer, derive the sensitivity \$S_{\Omega}\$ (magnitude) using Equations (4) and (5) Finally, assess the sensitivity \$S_{\alpha}\$ (magnitude) of the angular accelerometer through Equations (6) and (7).
4 angular acceleration exciter a See also Figure 1
Figure 2 — Angular acceleration exciter in conjunction with a diffraction-grating interferometer
Calculate the displacement amplitude from the fringe frequency readings calculated in Equation (9) using Equation (12) for the two-beam arrangement shown in Figure 4:
∆s = g/2 (12) or Equation (13) for the single-beam arrangement shown in Figure 8:
∆s = g (13) where g is the grating constant of the diffraction grating (groove length, i.e grating constant of 0,333 33 àm of a sine-phase diffraction grating having 3 000 grooves/mm)
To determine the amplitude, \$\hat{\Phi}\$, of the rotational angle \$\Phi\$, utilize Equation (2) Next, calculate the sensitivity \$S_{\Phi}\$ (magnitude) of the rotational angle transducer from Equation (3) For the angular velocity transducer, derive the sensitivity \$S_{\Omega}\$ (magnitude) using Equations (4) and (5) Finally, assess the sensitivity \$S_{\alpha}\$ (magnitude) of the angular accelerometer with Equations (6) and (7).
1 frequency generator (4.2) 6 interferometer (4.6) 11 with ratio counter (4.7.2)
2 power amplifier (4.3) 7 laser (4.5) 12 counter (or ratio counter) (4.7.2)
3 angular exciter (4.3) 8 beam splitter 13 voltmeter (4.8)
4 angular transducer 9 light detector 14 distortion meter (4.9)
Figure 3 — Example of a measuring system for method 1A (retro-reflector interferometer, fringe counting) © ISO 2006 – All rights reserved 15
1 frequency generator (4.2) 6 interferometer (4.6) 11 with ratio counter (4.7.2)
2 power amplifier (4.3) 7 laser (4.5) 12 counter (or ratio counter) (4.7.2)
3 angular exciter (4.3) 8 beam splitter 13 voltmeter (4.8)
4 angular transducer 9 light detector 14 distortion meter (4.9)
Figure 4 — Example of a measuring system for method 1B (diffraction-grating interferometer, fringe counting)
9 Methods using minimum-point detection (methods 2A and 2B)
General
This method is used for the calibration of the magnitude of the sensitivity in the frequency range from 800 Hz to 1,6 kHz
This section describes a method for determining displacement by analyzing the zero crossings of the first-order Bessel function.
An alternative method for determining displacement involves using the zero crossings of the Bessel function of the first kind and zero order However, this technique necessitates the modulation of the reference mirror's position (refer to Reference [2]).
By analyzing the frequency spectrum of intensity and adjusting the angular vibration amplitude to eliminate the component matching the vibration frequency, the displacement amplitude detected by the interferometer can be calculated using Equation (14): \$\hat{n} \hat{n} 2s s s x \Delta\$.
= = × π (14) where ˆ s is the displacement amplitude sensed by the laser interferometer; x n are the arguments corresponding to the zero points of the Bessel function, as given in Table 1;
The quantization interval, ∆s, is defined by Equations (10) and (11) for interferometer type A and by Equations (12) and (13) for type B To determine the rotational angle amplitude, Equation (2) is utilized The angular velocity amplitude, ˆΩ, and the angular acceleration amplitude, ˆα, are derived from Equations (5) and (7), respectively The sensitivity of the angular transducer is calculated using Equations (3), (4), and (6).
Table 1 — Values for the arguments, x n , corresponding to the zero points of the Bessel function of first kind and first order
Common test procedure for methods 2A and 2B
After optimizing the interferometer, assess the sensitivity of the calibrated transducer at the specified angular vibration frequencies and angular acceleration amplitudes Utilize a bandpass filter to process the signal from the light detector, setting the center frequency to match that of the angular vibrator This filtered signal will exhibit several minimum points at specific vibration amplitudes, corresponding to the displacement and rotational angle amplitudes outlined in Table 1.
Set the calibration frequency and gradually increase the vibrator amplitude until the filtered light detector signal peaks and then drops to its minimum value, known as minimum point No 1 This minimum point corresponds to a specific displacement amplitude that varies based on the type and configuration of the interferometer (refer to section 9.3).
The measuring system for the minimum-point method is shown in Figures 5 and 6
The sensitivity of an accelerometer can be assessed using the Bessel function of the first kind and zero order This involves modulating the position of the reference mirror at a frequency that is significantly lower than the calibration frequency Additionally, the center frequency of the bandpass filter or frequency analyzer should be aligned with the modulation frequency of the mirror.
Adjusting the position of the reference mirror can enhance the efficiency of the minimum-point method by utilizing the Bessel function of the first kind and first order.
Expression of results
Calculate the displacement amplitude ˆs=sˆ n from the associated argument, x n , using Equation (14) where
∆s = λ/4 [cf Equation (10) for the two-beam arrangement shown in Figure 3]; or
∆s = λ/2 [cf Equation (11) for the single-beam arrangement shown in Figure 7 where the wavelength of the laser of the red helium-neon type λ = 0,632 81 àm]
NOTE Application of Equation (14) can be dispensed with if the displacement amplitude, sˆ n , associated with the nth minimum point from Table 2 is taken
In the single-beam arrangement illustrated in Figure 7, the amplitude, denoted as \$\hat{s}\$, is converted into the amplitude of the rotational angle using Equation (2) This value is equivalent to \$\hat{s}_n\$, as referenced in Table 2, which is expressed in Equation (15) as \$\hat{s} = \hat{s}_n\$.
In the two-beam arrangement depicted in Figure 3, the amplitude, denoted as \$\hat{s}\$, is converted into the amplitude of the rotational angle using Equation (2) This value is half of the amplitude \$\hat{s}_n\$ obtained from Table 2, as expressed in Equation (16): \$\hat{s} = \frac{1}{2} \hat{s}_n\$.
1 frequency generator (4.2) 7 laser (4.5) 12 bandpass filter tuned to vibration frequency (4.7.3)
2 power amplifier (4.3) 8 beam splitter 13 voltmeter
3 angular exciter (4.3) 9 light detector 14 voltmeter (4.8)
4 angular transducer 10 amplifier 15 distortion meter (4.9)
5 retro-reflector 11 spectrum analyser (4.7.3) 16 oscilloscope (4.10)
Figure 5 — Example of a measuring system for method 2A (homodyne retro-reflector interferometer, minimum-point detection)
Calculate the displacement amplitude ˆs=sˆ n from the associated argument, x n , using Equation (14) where
∆s = g/2 [cf Equation (12) for the two-beam arrangement shown in Figure 6]; or
The change in path length, denoted as ∆s, is determined by the equation ∆s = g, where g represents the grating constant of the diffraction grating For a sine-phase diffraction grating with 3,000 grooves per millimeter, the grating constant is 0.33333 µm, as illustrated in Equation (13) and Figure 8.
NOTE Application of Equation (14) can be dispensed with if the displacement amplitude, sˆ n , associated with the nth minimum point from Table 2 is taken
In the single-beam arrangement depicted in Figure 8, the amplitude, \$\hat{s}\$, converted to the amplitude of the rotational angle using Equation (2) matches the value of \$\hat{s}_n\$ from Table 2, as referenced in Equation (15) Conversely, in the two-beam arrangement illustrated in Figure 6, the amplitude, \$\hat{s}\$, converted to the amplitude of the rotational angle via Equation (2) is half the value of \$\hat{s}_n\$ from Table 2, as noted in Equation (16).
1 frequency generator (4.2) 7 laser (4.5) 12 bandpass filter tuned to vibration frequency (4.7.3)
2 power amplifier (4.3) 8 beam splitter 13 voltmeter
3 angular exciter (4.3) 9 light detector 14 voltmeter (4.8)
4 angular transducer 10 amplifier 15 distortion meter (4.9)
5 diffraction grating 11 spectrum analyser (4.7.3) 16 oscilloscope (4.10)
Figure 6 — Example of a measuring system for method 2B (homodyne diffraction-grating interferometer, minimum-point detection)
Table 2 — Displacement amplitudes for minimum points (λ = 0,623 81 àm) to calculate the amplitudes of the rotational angle, F a , of the angular velocity, W b , and of the angular acceleration, α c
30 4,785 3 a See Equation (2) b See Equation (5) c See Equation (7)
Calculate the amplitude, ˆΦ , of the rotational angle, Φ , using Equation (2); calculate the sensitivity, S Φ
The sensitivity, \( S_\Omega \), of the angular velocity transducer is derived from Equations (4) and (5), while the sensitivity, \( S_\alpha \), of the angular accelerometer is calculated using Equations (6) and (7) Additionally, the magnitude of the rotational angle transducer is obtained from Equation (3).
When the calibration results are reported, the expanded uncertainty of measurement in calibration shall be calculated and reported in accordance with Annex A
10 Methods using sine approximation (methods 3A and 3B)
General
This method is applicable to sensitivity magnitude and/or phase calibration in the frequency range from 1 Hz to 1,6 kHz
The sine-approximation measurement method can be effectively used for frequencies below 1 Hz and above 1.6 kHz when paired with a suitable angular exciter designed for the broader frequency range.
Methods 3A and 3B enable the measurement of the phase shift in the complex sensitivity of angular transducers, in addition to the calibration results derived from equations applicable to all six methods.
The initial phase, ϕ Φ , of the rotational angle is represented by the initial phase of the displacement, ˆs, as given in Equation (17): Φ s ϕ =ϕ (17)
Using the value, ϕ Φ , obtained by the laser interferometer, the amplitude initial phase, ϕ Ω , of the angular velocity, Ω, is calculated from Equation (18):
Ω Φ 2 ϕ =ϕ − π (18) and the initial phase, ϕ α , of the angular acceleration, α , is calculated from Equation (19): α Φ ϕ =ϕ − π (19)
The phase shift, ∆ϕ Φ , of rotational angle transducers is calculated from Equation (20): Φ u Φ ϕ ϕ ϕ
The phase shift, ∆ϕ Ω , of angular velocity transducers is calculated from Equation (21):
The phase shift, ∆ϕ α , of angular accelerometers is calculated from Equation (22): α u α ϕ ϕ ϕ
In Equations (20), (21) and (22), ϕ u is the initial phase of the output of the transducer being calibrated.
Procedure applied to methods 3A and 3B
Install the equipment in accordance with Figures 7, 8 and 9
The laser interferometer (for an example, see Figures 7 and 8) shall be adjusted to give output signals u 1 and u 2 in phase quadrature within the tolerances stated in 4.6
After optimizing the interferometer settings, measure the magnitude and phase shift of the angular transducer sensitivity at the designated vibration frequencies and amplitudes.
The transducer shall be vibrated sinusoidally The displacement amplitude should be large enough to give at least one full bright/dark cycle of the interferometer output
In the arrangement outlined in Note 8.1, displacement amplitudes of \$u = 0.5 \, \text{am}\$ can lead to a worst-case error of \$W = 0.3\%\$ in sensitivity magnitude measurements and \$W = 0.3^\circ\$ in phase shift measurements, due to the combined effects of disturbing parameters in the quadrature output signals within the tolerances specified in section 4.6 This error component can be mitigated through more precise adjustments than those allowed in section 4.6 or by implementing the correction procedure detailed in Reference [6].
To accurately assess the magnitude and phase shift of the complex sensitivity of angular transducers for small displacement amplitudes in the nanometer range, the sine-approximation method combined with a suitable heterodyne technique can be utilized, as detailed in References [7] and [8] This approach enables calibration at a desired moderate angular acceleration amplitude while operating at high vibration frequencies.
To improve the rejection efficiency of sine approximation and reduce the impact of disturbing signals, windowing techniques can be applied to displacement or modulation phase values, provided that the procedure complies with the uncertainty requirements outlined in Clause 3.
1 frequency generator (4.2) 6 interferometer (4.6) 10 digital waveform recorder (4.7.4)
3 angular exciter (4.3) 8 light detectors 12 distortion meter (4.9)
5 retro-reflector a Phase shift of 0° b Phase shift of 90°
Figure 7— Example of a measuring system for method 3A (homodyne retro-reflector interferometer, sine approximation)
1 frequency generator (4.2) 6 interferometer (4.6) 10 digital waveform recorder (4.7.4)
3 angular exciter (4.3) 8 light detectors 12 distortion meter (4.9)
5 diffraction grating a Phase shift of 0° b Phase shift of 90°
Figure 8 — Example of a measuring system for method 3B (homodyne diffraction-grating interferometer, sine approximation) © ISO 2006 – All rights reserved 25
Figure 9 — Modified Michelson interferometer with retro-reflector(s) and quadrature output
8 angular accelerometer exciter (air-borne) 16 measurement results (vibration and shock parameters) a 90° b y(t i ) c z(t i ) d arctan y(t i )/z(t j )
Figure 10 — Example of a measuring system using heterodyne interferometry in accordance with method 3B (Mach-Zehnder heterodyne interferometer with diffraction grating, frequency conversion, transient recorder and digital data processing)
1 diffraction grating 3 rotational laser vibrometer being calibrated
2 interferometer (standard) 4 signal-processing system
Figure 11 — Example of an arrangement for the calibration of rotational laser vibrometers
Method B is applied here in the version with a diffraction grating interferometer using a sine-phase grating with
2 400 grooves/mm It is arranged on the lateral surface of a disk forming the measuring table 100 mm in diameter.
Data acquisition
When selecting cut-off frequencies for low-pass and high-pass filters, it is essential to ensure that their impact on calibration results remains acceptable Additionally, to comply with Nyquist's theorem, the sampling rate must be established such that the highest frequency content is less than half of the sampling rate.
Analog-to-digital conversion of the angular transducer output voltage can occur at the same or a lower sampling rate compared to the interferometer output signals It is essential that all three sampling processes begin and conclude simultaneously, with the two interferometer signals synchronized by a single system clock.
The quadrature signals shall be equidistantly sampled during a measurement time t 0 <