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Abstract
To realize accurate dynamic axial clearance measurement in high-speed rotating machinery using basic Frequency-swept interferometry (FSI), a novel algorithm of Doppler error suppression is proposed. In this algorithm, we first extract the incremental phase of the FSI signal by Hilbert transform and use it to construct a virtual clearance variation. Then, using the periodic characteristics of rotor rotation, optimized initial clearance can be obtained. Eventually, the dynamic axial clearance at each sampling point can be recovered with the optimized initial clearance and the Doppler error is greatly suppressed. The feasibility of this method is tested and verified by both simulation and experiment. For a 4 kHz rotation simulating target located at 5 mm away, the experimental results show that the dynamic clearance measurement error is less than 2.057 µm, which meets the measurement accuracy required for most high-speed rotating machinery.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
For rotating machines, the axial clearance refers to the internal gap between the rotating rotor and the static stator, which influences the operation performance of rotating machines [1]. Too small axial clearance increases the collision risk between rotor and stator, while too large axial clearance decreases the efficiency of rotating machinery. To evaluate and optimize the design and manufacturing of rotating machines, it is important to measure axial clearance.
Due to the harsh environments, such as high-speed rotation, confined space, high temperature, strong vibration, and strong magnetic field inside rotating machines, there are few effective methods for directly measuring dynamic axial clearance at present, especially for high-speed rotating machines, and it is highly desired to find an effective method for dynamic axial clearance measurement in industry fields.
Frequency-swept interferometry (FSI) is an effective and widely used static ranging technique [2–6]. In FSI, the absolute static distance is determined by estimating the beat frequency of the FSI signal [6]. If a fiber optic probe is adopted with its reference path coaxial to the measuring path [7,8], FSI is also considered to be an ideal tool for measuring the static axial clearance in harsh environments. However, in the dynamic measurement of the high-speed rotor-stator axial clearance, the FSI faces the extremely serious problem of the Doppler effect. According to the principle of FSI, if the rotor rotates at high speed during the frequency sweeping process, the Doppler effect occurs and results in speed-related Doppler error, which can be hundreds of times larger than the actual clearance variation [5]. Many efforts have been made to suppress the Doppler error in FSI to achieve accurate dynamic measurement. In 2007, Yang, H. J. et al. proposed a setup with two frequency swept lasers operating at different sweep speeds in opposite directions, and the Doppler error can be eliminated with the phase shift difference of two reverse-sweep FSI signals [3]. In 2014, Deljohn et al. developed a device with a gas absorption pool and two reverse sweeping lasers, and using the additional information provided by these devices accurate dynamic measurement of low-speed moving objects is obtained [9]. Also, the setup of heterodyne interferometry using Acoustic-Optical Modulator (AOM) is proposed, and the Doppler error can be compensated with the help of Doppler-free saturation spectroscopy [10]. Furthermore, Cheng Lu et al. reported dynamic FSI methods with AOM-based heterodyne SFI in 2016 and 2018 respectively. The progress of these methods is that they can not only eliminate Doppler error [11], but also be immune to the laser sweeping nonlinearity by using phase-locked loop (PLL) [12]. Different from the heterodyne-based interferometry, Shao Bin et al. presented a dynamic clearance measurement method using FSI and homodyne interferometry to recover the dynamic clearance, which does not require the AOM [13].
All the above-mentioned methods address the Doppler error problem by adding auxiliary devices in different ways, but greatly increase the complexity of system and the difficulty of implementation. Liu, Z.G. et al. proposed a simple system that adopts only one frequency selection & tuning laser combined with time-invariant Kalman filtering in the dynamic FSI [14–17]. But Liu’s model will be effective only when the state transition matrix matches the current motion mode.
For high-speed rotation machinery, its motion mode is regular. With the prior knowledge of the rotation motion mode, it is possible to realize high-rate dynamic axial clearance measurement using the basic FSI without any auxiliary devices. Aiming at this purpose, we propose a novel algorithm to suppress the Doppler error and recover the dynamic axial clearance of high-speed rotating machinery at each measurement point.
2. Principle
2.1 Axial clearance of rotor in rotating machinery
Rotating machinery is a type of special machine with fixed stator and rotating rotor as shown in Fig. 1(a). The existence of tilt angle in the rotor is inevitable due to complex factors [18]. When the rotor rotates at high speed, the tilt angle will cause a high-speed clearance variation. The clearance variation can be simplified as an equivalent model as shown in Fig. 1(b) when the fiber optic probe is parallel to the rotor axis. The Cartesian coordinate system xyz is set as shown in Fig. 1(b), with the origin o (0, 0, 0) located at the center of the rotor surface. The o′ (0, L0, r) is at the end surface of the probe. Assuming that the rotor rotates around the y-axis at an angular velocity Ω, the normal vector of rotor plane $\overrightarrow {{on}} $ at time t can be written as $\overrightarrow {{on}} $ = [sinα·sin(Ω·t), cosα, sinα·cos(Ω·t)]. Correspondingly, the vector $\overrightarrow {{oM}} $ = (0, yM, r). Because $\overrightarrow {on} $⊥$\overrightarrow {oM} $, the dot product of the two vectors is zero ($\overrightarrow {on} \cdot\overrightarrow {oM} $ = 0) and can be represented as follows:
Fig. 1. (a) Schematic of the axial clearance L(t) of rotor and stator in rotating machinery; (b) The equivalent geometric model of dynamic axial clearance. (Ω: Angular velocity of the rotor; α: Tilt angle of the rotor; M: Measurement point; L(t): The axial dynamic clearance; L0: The ideal clearance between rotor and stator without the tilt angle; ΔL(t): The dynamic clearance variation caused by tilt angle α and rotating angular velocity Ω; o: Center point of the measuring surface; o′: Launching point of the fiber probe; r: Distance from o′ to the axis; $\overrightarrow {\textrm{on}} $: Normal vector of the rotor plane; ${\vec{v}}$: The velocity vector of target; ${\vec{v}}$(t) and $\overrightarrow {v} $c(t): The Axial component and tangential component of the target velocity vector.)
From Fig. 1(a), one can see that the measured dynamic axial clearance L(t) = L0 + ΔL(t), where L0 is the ideal clearance between rotor and stator and ΔL(t) is the dynamic clearance variation caused by tilt angle α and rotating angular velocity Ω. By solving Eq. (1), ΔL(t) can be obtained as:
According to Eq. (2), the amplitude and the frequency of the axial clearance variation ΔL(t) can be written as A = r·tanα and fL=Ω/2π respectively, so the dynamic axial clearance can be described as:
It can be seen that the variation of axial clearance can be considered as a sinusoidal function with an amplitude A and a frequency fL.
2.2 Optical Doppler effect
According to the principle of the optical Doppler effect, the frequency shift fD occurs in output light when the target moves. As shown in Fig. 2, fD is determined by the moving direction of the target and the propagation direction of the light beam.
Fig. 2. (a) Schematic of Doppler laser velocimetry with different incidents and receiving directions. (b) Schematic of Doppler laser vibrometer in the coaxial configuration of the incident and receiving path where β+θ=π. (f0: The optical frequency of incident laser beam; f1: The optical frequency of outgoing laser beam; ${\vec{v}}$: The velocity vector of target; ${\vec{v}}$(t) and $\overrightarrow {v} $c(t): The Axial component and tangential component of target velocity vector)
According to the principle of the optical Doppler effect, the frequency f1 of the receiving beam can be expressed as [19–21]
2.3 Frequency-swept interferometry and the Doppler error induced by target motion
According to the schematic of FSI shown in Fig. 3, if the target is stationary with a clearance L from the probe, the normalized output FSI signal S(t) sampled by PD can be written as [12,13]
Fig. 3. Schematic of the coaxial FSI for ranging. (FSL: Frequency-Swept Laser; SMF: simply Single model fiber; OC: Fiber circulator; PD: Photodetector; DAQ: Data acquisition system; L: The measuring distance; ΔL: The measuring distance variation caused by target motion.)
However, when the target moves along the axis of the probe, i.e. L becomes $L(t) = {L_0} + \Delta L(t )= {L_0} + \int_0^t {v(t )dt} $, where L0 is the initial clearance and v(t) is the instantaneous velocity of the target along the axis, one can find fB becomes unstable. By taking the derivate of the phase of S(t), one can obtain the instantaneous beat frequency fB(t) shown by the following equation:
Substituting the fB in Eq. (6) with fB(t), we can get the measured distance LM(t) as follows:
In Eq. (8), the combination of the first two terms is the real axial dynamic distance L(t), and the third term ED is the Doppler error. As Doppler error ED is proportional to the axial motion speed v(t) of the target and magnified by [(T/B)$\cdot$fI+t], it is extremely sensitive to target motion and must be suppressed.
3. Proposed algorithm and system for the dynamic axial clearance measurement of rotating machinery
3.1 Algorithm
Figure 4 demonstrates the axial clearance measurement scheme of rotating machinery using the basic FSI system. The laser beam coming from FSL is split into transmitted light and reflected light at the end face of the fiber probe. The reflected light is used as a reference beam, while the transmitted light is reflected from the surface of the rotor as the measuring beam. It should be pointed out that, although the rotor surface may be slant, due to the diffuse reflection, a portion of reflected light whose direction along the axis of the probe will return into the probe. This portion of reflected light contributes to the formation of S(t). So the impact of rotation on the FSI signal is the Doppler effect caused by the axial clearance variation. In fact, we measured the FSI signal under a slow rotating rotor in the previous study [18], and the measured FSI signal corresponds with the Eq. (8) which indicates the impact of rotation on S(t) is the Doppler effect.
Fig. 4. Schematic of the fiber-optic FSI for axial clearance measurement of rotating machinery. (FSL: Frequency-Swept Laser; OC: Optical circulator; SMF: Single model fiber; PD: Photodetector; DAQ: Data acquisition system; L: the measuring axial clearance of rotating machinery.)
Assuming the dynamic clearance caused by the rotation is L(t), the FSI signal S(t) can be rewritten as
With Δφ(t), we can use a variable Lc to construct a clearance variation △Lcons(t) as below:
Because 0<(t/T) < 1 and fI is typically tens or even hundreds of times greater than B, Ev can be approximated as
Because rotor motion can be regarded as a sinusoidal vibration, we can plot the variation trends of $\int_0^t {v(t )dt} $and Ev as shown in Fig. 5. From Fig. 5, one can imagine that:
- (2) When Lc≠L0, Ev is nonzero and the ${t^{\prime}_k}$s (k=1, 2, 3…) becomes timestamps where the △Lcons(t) = 0. Obviously, ${t^{\prime}_k}$s are not equally spaced, the function V will not be zero.
Fig. 5. Variation trends of $\int_0^t {v(t)dt}$ and − Ev. (t1∼t9: timestamps where the △Lcons(t) = 0 in the case of Lc=L0; t′1∼t′9: Timestamps where the △Lcons(t) = 0 in the case of Lc≠L0)
In fact, only when Lc is close to the true initial clearance L0, the function V converges to zero (By the way, we will demonstrate the optimization space of Lc in the simulation and experiment parts). Taking the V as an objective function, the L0 can be estimated by finding the optimized Lc-optimal. With the optimized Lc-optimal and Δφ(t), we can construct the dynamic clearance as
As a result, the dynamic absolute clearance can be recovered from just one FSI using the periodic motion information of the rotor. Although the optimal Lc-optimal could be error contained, the Doppler induced error can be suppressed significantly.
3.2 Applicable bandwidth of the algorithm
When applying this algorithm, the following conditions, which determines its applicable bandwidth, need to be satisfied:
- (1) To obtain successfully the incremental phase Δφ(t) shown in Eq. (10), the FSI signal S(t) must satisfy the Nyquist theorem. According to the Nyquist theorem, for sampling rate fs, the frequency fB(t) should satisfy 0< fB(t)<fs/2. Thus, the following relation can be obtained:
Substituting the clearance of the rotating machinery shown in Eq. (3), Eq. (15) can be rewritten as
$$- \frac{B}{T}{L_0} < \left[ {\frac{B}{T}[{A\cos ({2\pi {f_\textrm{L}}t} )} ]+ \left( {{f_\textrm{I}} + \frac{B}{T}t} \right)[{2\pi {f_\textrm{L}}A\sin ({2\pi {f_\textrm{L}}t} )} ]} \right] < \frac{c}{{4n}}{f_\textrm{s}} - \frac{B}{T}{L_0}.$$Since
$$\begin{aligned} &\frac{B}{T}[{A\sin ({2\pi {f_\textrm{L}}t} )} ]+ \left( {{f_\textrm{I}} + \frac{B}{T}t} \right)[{2\pi {f_\textrm{L}}A\cos ({2\pi {f_\textrm{L}}t} )} ]\\ &= A\frac{B}{T}\sqrt {1 + \frac{{{{\left( {{f_\textrm{I}} + \frac{B}{T}t} \right)}^2}{{({2\pi {f_\textrm{L}}} )}^2}}}{{{{\left( {\frac{B}{T}} \right)}^2}}}} \sin ({2\pi {f_\textrm{L}}t + \varphi } ), \end{aligned}$$where $\varphi = \arctan [{{{2\pi {f_\textrm{L}}(T{f_\textrm{I}} + Bt)} / B}} ]$, Eq. (16) is equivalent to$$\left\{ \begin{array}{l} - {L_0} < - A\sqrt {1 + \frac{{{{\left( {{f_\textrm{I}} + \frac{B}{T}t} \right)}^2}{{({2\pi {f_\textrm{L}}} )}^2}}}{{{{\left( {\frac{B}{T}} \right)}^2}}}} ,\\ A\sqrt {1 + \frac{{{{\left( {{f_\textrm{I}} + \frac{B}{T}t} \right)}^2}{{({2\pi {f_\textrm{L}}} )}^2}}}{{{{\left( {\frac{B}{T}} \right)}^2}}} < } \frac{{cT}}{{4nB}}{f_\textrm{s}} - {L_0} \end{array} \right..$$Therefore, the frequency fL should satisfy the following condition
$$\begin{aligned} {f_\textrm{L}} &< \frac{1}{{2\pi }}\min \left\{ {\frac{{\frac{B}{T}}}{{\left( {{f_\textrm{I}} + \frac{B}{T}t} \right)}}\sqrt {{{\left[ {\frac{{\min \left\{ {{L_0},\frac{{c{f_\textrm{s}}T}}{{4nB}} - {L_0}} \right\}}}{A}} \right]}^2} - 1} } \right\}\\ &= \frac{B}{{2\pi T({{f_\textrm{I}} + B} )}}\sqrt {{{\left[ {\frac{{\min \left\{ {{L_0},\frac{{c{f_\textrm{s}}T}}{{4nB}} - {L_0}} \right\}}}{A}} \right]}^2} - 1} , \end{aligned}$$ - (2) To calculate the objective function V expressed in Eq. (13), at least 4 timestamps are needed. So, the vibrational frequency fL of the clearance need to be 2 times higher than the frequency of the frequency swept laser. Thus, the following relation can be obtained:
Combing Eq. (19) and Eq. (20), the applicable bandwidth of this algorithm can be obtained as follow
$$\frac{2}{T} \le {f_L} < \frac{B}{{2\pi T({{f_\textrm{I}} + B} )}}\sqrt {{{\left[ {\frac{{\min \left\{ {{L_0},\frac{{c{f_s}T}}{{4nB}} - {L_0}} \right\}}}{A}} \right]}^2} - 1} .$$In addition, it is worth pointing out that, according to the principle of this algorithm, the objective function V is effective as long as the rotor movement is stable during a sweep period T of the swept laser. If there is a sudden load during a sweep period, the algorithm may fail. But for most situations, compared with the short sweep period, the axial holistic movement of the rotor is slow and the clearance variation can be regarded as a sinusoidal function.
4. Simulation, experiments, and results
4.1 Simulation
To verify the proposed algorithm, the numerical simulations are performed first. According to Eq. (9), the following parameters are used to simulate S(t): n = 1.00027, and c = 2.9979×108 m/s, fI = 196250 GHz, B = 4864 GHz, T = 0.4864 ms, and the dynamic clearance Lset(t) = [4.995×10−3 + 5×10−6cos(2πfLt+φ0)] m, where φ0 = π/2, and fL = 4 kHz. To better simulate the actual signal, we further add multiplicative envelope M(t), additive envelope D(t), and envelopes random noise G(t) to S(t) by the form of S(t)M(t)+D(t)+G(t). The specific forms of M(t), D(t) and G(t) are as follows: M(t) = 0.5 + (0.2/T)t, D(t) = sin(0.5πt/T), and G(t) is the Gaussian noise with SNR = 30 dB. A simulated FSI signal is shown in Fig. 6.
Fig. 6. The simulated FSI signal during six sweeping cycles and the partial enlargement of the third sweeping cycle (S3).
The details of the signal processing will be described in section 4.3, here we give the most representative illustrations and conclusions. Figure 7 shows the optimization space of Lc, and the optimal Lc-optimal can be obtained by finding the minimum Vmin. With the optimal Lc-optimal, the clearance variation can be reconstructed. Figure 8(a) shows reconstructed Lrec(t), the set Lset(t), and the optimal Lc-optimal. From Fig. 8(a), it can be seen that the reconstructed Lrec(t) matches well with the Lset(t). Errors of estimated Lc-optimal and reconstructed Lrec(t) are shown in Fig. 8(b). The reconstructed error is less than ±0.2µm. The error is relatively large at the ends of each cycle, this is caused by the Hilbert transform tail effect.
Fig. 8. (a) The reconstructed Lrec(t) with Lc-optimal of 6 sweep cycles and the set clearance vibration Lset(t) (4 kHz); (b) Errors of estimated Lc-optimal and reconstructed Lrec(t).
4.2 Experimental setup
In order to verify the effectiveness of the proposed method, an experimental system was constructed as shown in Fig. 9. The experimental system consists of an FSI system and a piezoelectric transducer (PZT) vibrator. In this FSI system, a 20 mW FSL (Arcadia Optronix, GC-760001c) working at C-band from 196250 GHz (fI) to 191386 GHz is utilized, and its sweeping period T is 0.4864 ms. After being detected by the PD, the FSI signal is sampled by DAQ with a sample rate of 5 MSa/s. The fiber probe was a cleaved single-mode fiber placed on a fiber clamp.
Fig. 9. (a) Schematic of the experimental system; (b) Photograph of the PZT setup. (FSL: Frequency-swept laser; SMF: Single model fiber; OC: Optical circulator; FC: Fiber clamps; PD: Photodetector; DAQ: Data acquisition system; MPTS: Motorized Precision Translation Stage; PZT: Piezoelectric transducer; PA: Power amplifier; SG: Signal generator.)
It is worth pointing out that, the main reason for using the PZT to simulate a high-speed dynamic axial clearance variation is because it is difficult to obtain the real axial dynamic clearance of a high-speed rotating machine. In other words, without the real dynamic clearance, it is unable to evaluate the measurement accuracy of the proposed method. To obtain the real dynamic clearance, we use a PZT vibrator (Thorlabs, PK4FTH3P2) to simulate the periodical axial dynamic clearance of rotating machinery and a laser Doppler velocimetry (LDV) to calibrate the PZT vibration. As shown in Fig. 9, an aluminum-coated reflective surface is bonded on the PZT. The drive signal for the PZT is generated by the Signal Generator (SG) (RIGOL, DG4102) and amplified by the Power Amplifier (PA) (NF Corporation, HAS4011). The clearance between the end face of the probe and the reflective surface can be controlled precisely using the motorized precision translation stage (MPTS) (SOFN Instruments, 7STA1030, 0.625 µm adjustment accuracy) with a motion controller (SOFN Instruments, 7SC4). To calibrate the vibration of PZT, an LDV (Polytec, OFV-5000, 0.1 pm displacement resolution) is used to determine the dynamic displacement caused by the PZT as shown in Fig. 10. To guarantee stability, the whole experimental system is placed on an optical vibration isolation platform.
4.3 Result and analysis
The dynamic clearance was measured by following steps. First, the PZT was placed at a distance of 4.9922 mm from the probe. The static clearance was measured by demodulating the static FSI signal. Then, a sinusoidal voltage signal was loaded on the PZT. The vibration amplitude of PZT was measured by the LDV as shown in Fig. 11, and the smoothed result is used as the reference true value. From Fig. 11, it can be seen that we created a vibrating PZT with an amplitude of 5.0135 µm and a frequency of 4 kHz. With the static clearance, we can obtain Lreal(t) = [4.9922×10−3 + 5.0135×10−6cos(8π×103t)] m.
Figure 12 shows the measured raw FSI signals during 6 sweeping cycles when the PZT vibrates. The six sweep cycles are noted as S1 to S6, and next, we will take the S3 as an example to better visualize the algorithm process by displaying the intermediate processing results step by step.
Fig. 12. Raw FSI signals sampled by DAQ under 4 kHz sine vibration during 6 sweeping cycles and the partial enlargement of the third sweeping cycle (S3).
Before the signal demodulation, the zero-phase band-pass filter [22] and Hilbert transform (HT) are successively applied to the FSI signal to eliminate the additive and multiplicative envelopes. The filtered and normalized S3(t) is shown in Fig. 13, and obviously, the envelope of S3(t) is well filtered. Note that, the initial time of S3(t) is set to 0 just for simplicity. Then, the phase φ(t) of S3(t) is obtained by HT as shown in Fig. 14(a). By phase unwrapping and subtracting the φ(0), the incremental phase Δφ(t) can be obtained as shown in Fig. 14(b). After obtaining the Δφ(t), a series of ΔLcons(t) is constructed according to Eq. (11). Next, the objective function V is used to find the optimal Lc-optimal which is closest to the real initial clearance L0. Figure 15(a) shows the optimization space of Lc, and the optimal Lc-optimal located at the minimum of V. By the way, in Fig. 15(a), the optimization range of Lc is from 4.98 mm to 5.01 mm in a step of 0.01 µm, and in practice the searching range can be reduced by a priori estimation of L0. Finally, with the Lc-optimal, the dynamic clearance can be reconstructed at each sampling point as shown in Fig. 15(b). Repeating the above steps, the dynamic clearance during S1∼S6 can be reconstructed as shown in Fig. 15(c). The error between the reconstructed Lrec(t) and the reference Lreal(t) is shown in Fig. 15(d). The maximum measurement error is less than 2.057 µm. In comparison, Fig. 16 shows the demodulation result LM(t) using Eq. (8) and the reconstructed Lrec(t). If not suppressed, the demodulation error of LM(t) is larger 5240 µm as shown in Fig. 16, which proves the effectiveness of the proposed algorithm.
Fig. 15. (a) Optimization space of Lc; (Vs: Smoothed V.) (b) The optimal Lc-optimal and the reconstructed Lrec(t) within sweeping cycle 3; (c) The reconstructed Lrec(t), the optimal Lc-optimals and the Lreal(t); (d) Measurement errors.
5. Conclusions
For dynamic clearance measurement, the Doppler error is inevitable in basic FSI system. To eliminate Doppler error, adding additional information is necessary. For irregular dynamic clearance, fusing the information from auxiliary devices is an effective way. However, auxiliary devices increase the complexity of the system. In this paper, a new idea is conveyed, that is, using prior information of the regular dynamic clearance from the high-speed rotating machinery. We can use the incremental phase of FSI signal to construct a function to approximate the dynamic clearance variation. By utilizing the periodic characteristics of rotor rotation, high-rate dynamic axial clearance can be accurately recovered by the optimization process. Both numerical simulation and experiment were carried out to verify the proposed method. Experimental results show that the Doppler error can be greatly reduced. This method satisfies the accuracy requirement of dynamic clearance measurement for most high-speed rotating machines. In the next work, we will try to solve the failure of the algorithm under unstable cases such as a sudden load on the rotor. We think the solution lies in how to construct an effective objective function using the prior knowledge of rotor movement characteristics.
Funding
National Natural Science Foundation of China (51805054); Technology Innovation Platform Project of Aero Engine Corporation of China (AECC) (SHYS-GXDPL-18); China Postdoctoral Science Foundation (2018M643405).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. B. Pang, G. Tang, and T. Tian, “Complex Singular Spectrum Decomposition and Its Application to Rotating Machinery Fault Diagnosis,” IEEE Access 7, 143921–143934 (2019). [CrossRef]
2. P. A. Coe, D. Howell, and R. B. Nickerson, “Frequency scanning interferometry in ATLAS: remote, multiple, simultaneous and precise distance measurements in a hostile environment,” Meas. Sci. Technol. 15(11), 2175–2187 (2004). [CrossRef]
3. H. Yang, J. Deibel, S. Nyberg, and K. Riles, “High-precision absolute distance and vibration measurement with frequency scanned interferometry,” Appl. Opt. 44(19), 3937–3944 (2005). [CrossRef]
4. C. a. Alexandre, “Accuracy of frequency-sweeping interferometry for absolute distance metrology,” Opt. Eng. 46(7), 073602 (2007). [CrossRef]
5. L. Hartmann, K. Meiners-Hagen, and A. Abou-Zeid, “An absolute distance interferometer with two external cavity diode lasers,” Meas. Sci. Technol. 19(4), 045307 (2008). [CrossRef]
6. R. R. Reibel, P. A. Roos, T. Berg, B. Kaylor, N. Greenfield, Z. W. Barber, C. J. Renner, and W. R. Babbitt, “Ultrabroadband optical chirp linearization for precision metrology applications,” Opt. Lett. 34(23), 3692–3694 (2009). [CrossRef]
7. H. Guo, F. Duan, G. Wu, and J. Zhang, “Blade tip clearance measurement of the turbine engines based on a multi-mode fiber coupled laser ranging system,” Rev. Sci. Instrum. 85(11), 115105 (2014). [CrossRef]
8. Iker García, Joseba Zubia, Josu Beloki, Jon Arrue, Durana, and Gaizka, “Optical Tip Clearance Measurements as a Tool for Rotating Disk Characterization,” Sensors 17(12), 165 (2017). [CrossRef]
9. J. Dale, B. Hughes, A. J. Lancaster, A. J. Lewis, A. J. H. Reichold, and M. S. Warden, “Multi-channel absolute distance measurement system with sub ppm-accuracy and 20 m range using frequency scanning interferometry and gas absorption cells,” Opt. Express 22(20), 24869–24893 (2014). [CrossRef]
10. G. Prellinger, K. Meiners-Hagen, and F. Pollinger, “Spectroscopically in situ traceable heterodyne frequency-scanning interferometry for distances up to 50 m,” Meas. Sci. Technol. 26(8), 084003 (2015). [CrossRef]
11. C. Lu, G. D. Liu, B. G. Liu, F. D. Chen, and Y. Gan, “Absolute distance measurement system with micron-grade measurement uncertainty and 24 m range using frequency scanning interferometry with compensation of environmental vibration,” Opt. Express 24(26), 30215–30224 (2016). [CrossRef]
12. C. Lu, Y. Xiang, Y. Gan, B. G. Liu, F. D. Chen, L. Xiaosheng, and L. Guodong, “FSI-based non-cooperative target absolute distance measurement method using PLL correction for the influence of a nonlinear clock,” Opt. Lett. 43(9), 2098–2101 (2018). [CrossRef]
13. B. Shao, W. Zhang, P. Zhang, and W. M. Chen, “Dynamic Clearance Measurement Using Fiber-Optic Frequency-Swept and Frequency-Fixed Interferometry,” IEEE Photonics Technol. Lett. 32(20), 1331–1334 (2020). [CrossRef]
14. L. Tao, Z. Liu, W. Zhang, and Y. Zhou, “Frequency-scanning interferometry for dynamic absolute distance measurement using Kalman filter,” Opt. Lett. 39(24), 6997–7000 (2014). [CrossRef]
15. X. Ji, Z. Liu, L. Tao, and Z. Deng, “Frequency-scanning interferometry using a time-varying Kalman filter for dynamic tracking measurements,” Opt. Express 25(21), 25782–25796 (2017). [CrossRef]
16. X. Jia, Z. Liu, Z. Deng, W. Deng, Z. Wang, and Z. Zhen, “Dynamic absolute distance measurement by frequency sweeping interferometry based Doppler beat frequency tracking model,” Opt. Commun. 430, 163–169 (2019). [CrossRef]
17. Z. W. Deng, Z. G. Liu, X. Y. Jia, W. Deng, and X. Zhang, “Dynamic cascade-model-based frequency-scanning interferometry for real-time and rapid absolute optical ranging,” Opt. Express 27(15), 21929–21945 (2019). [CrossRef]
18. B. Shao, W. Zhang, P. Zhang, and W. M. Chen, “Multi-Parameter Measurement of Rotors Using the Doppler Effect of Frequency-Swept Interferometry,” Sensors 20(24), 7178 (2020). [CrossRef]
19. X. Yang, R. Zhou, W. L. Wei, and X. Wang, “Double frequency of difference frequency signals for optical Doppler effect measuring velocity,” Proc. SPIE 6024, 60240G (2005). [CrossRef]
20. H. L. Zhou, D. Z. Fu, J. J. Dong, P. Zhang, and X. L. Zhang, “Theoretical analysis and experimental verification on optical rotational Doppler effect,” Opt. Express 24(9), 10050–10056 (2016). [CrossRef]
21. L. Fang, M. J. Padgett, and J. Wang, “Sharing a Common Origin Between the Rotational and Linear Doppler Effects,” Laser Photonics Rev. 11(6), 1700183 (2017). [CrossRef]
22. M. Feldman, “Hilbert transform in vibration analysis,” Mech. Syst. Signal Process. 25(3), 735–802 (2011). [CrossRef]
