Demodulation technique for 3-D tip clearance measurements based on output signals from optical fiber probe with three two-circle coaxial optical fiber bundles

Siying Xie; Xiaodong Zhang; Yiwei Xiong; Hongcheng Liu

doi:10.1364/OE.27.012600

1. Introduction

In order to monitor the structural health and implement the appropriate control strategy for turbines in aero engines, many studies have been carried out over the years to develop noncontact and nonintrusive measurement techniques for turbine blades such as tip clearance [1] and blade tip timing [2].To gain insight on the dynamic behavior of the turbine blades for structural health monitoring and control strategy of turbines, a new three-dimensional (3-D) tip clearance concept was proposed in [3] to investigate the variations of the 3-D tip clearance of a turbine blade subjected to different types of load. The simulation results showed that the 3-D tip clearance was significantly influenced by the applied stress load, making the 3-D tip clearance concept feasible to monitor the structural integrity of turbine blades.

However, the probe installed on the turbine casing above the blade tip should possess the following characteristics in order to realize the 3-D tip clearance measurements: (1) anti-electromagnetic interference, (2) high temperature resistance, and (3) low volume. Since the advent of optical measurement techniques, various measurement techniques have been developed to improve the measurement of 3-D characteristics of objects. For example, Schmitz [4] proposed a technique to measure the 3-D contours of a computer numerical control multiple-axis machine using a laser ball bar. Charge coupled device cameras have been used in numerous studies to scan the 3-D shapes of objects [5–8]. Several researchers have studied 3-D strain measurements using Bragg grating fiber optic sensors [9,10]. Yang and Butler [11] reported a 3-D fiber optic position sensor, where light was emitted from the light-emitting fiber towards a spherical mirror mounted on the surface of interest and the intensity of the reflected light was detected by an optical fiber array. Oiwaand Nishitani [12] developed a 3-D displacement probe with three reflective optical fiber displacement sensors to measure the motion of a touch ball. However, all of the aforementioned techniques require probes with complex structures and large processing devices and therefore, it is not practical to install such probes on the turbine casing or blade tips for 3-D tip clearance measurements. Hence, in our previous work [13], we presented a 3-D tip clearance optical fiber probe, which consists of three two-circle coaxial optical fiber bundles to acquire the light signals, which are modulated by the 3-D displacement of the surface to be measured. In our probe, the two-circle coaxial optical fiber bundles are packed into a single optical fiber with a diameter of 5 mm, which makes the probe compact and suitable to be installed on the turbine casing without causing any detriment to its mechanical structure. Furthermore, the processing circuit of our probe can be fitted within the central control system of the aero engine.

For many years, two-circle coaxial optical fiber bundles have been used as the sensing units for radial tip clearance measurements. For example, Duan [14] studied the characteristics of a two-circle coaxial optical fiber bundle for radial tip clearance measurements. Iker [15,16] proposed a more precise measurement system, which was used in ground tests of an aero engine. In our previous works [17,18], we presented a probe consisting of three two-circle coaxial optical fiber bundles to realize 3-D tip clearance measurements and we studied the output characteristics of a two-circle coaxial optical fiber bundle with regards to the 3-D tip clearance. However, our neural network-based technique used to demodulate the output signals from the three two-circle coaxial optical fiber bundles resulted in large demodulation errors of the circumferential angle, with a maximum value of 2° [19]. This made our demodulation technique impractical for 3-D tip clearance measurements. Owing to the high coupling between the light intensities of the signals received by the optical fiber bundles and the 3-D tip clearance, the demodulation technique based on a constant coefficient matrix [20] is also not suitable for 3-D tip clearance optical fiber probe.

Hence, in this work, we proposed a new technique to demodulate the output signals acquired from three two-circle coaxial optical fiber bundles used as the sensing units of the 3-D tip clearance optical fiber probe, hope to offer a realizable demodulation theory for the further study on the measurement technique aiming at the turbine blade. This demodulation technique is based on the ratio of the difference in the signal intensities between any two sensing units of the optical fiber probe. The demodulation equations were derived using the second-order Taylor expansion for a three-variable function. This demodulation technique improves the overall demodulation accuracy of the 3-D tip clearance measurements, which will be demonstrated in this paper. This paper has mainly concerned the principle and algorithm of the demodulation, so the study has been taken with assumptions of ideal plane.

2. Constructed functions based on the output characteristics of the optical fiber bundle

We have presented and discussed the output characteristics of a two-circle coaxial optical fiber bundle with regards to the 3-D tip clearance in our previous work [18]. To develop the demodulation technique for 3-D tip clearance measurements, we constructed the explicit output function based on the conclusions in [18], assuming that the average light intensity received by a receiving fiber can be approximated by the intensity at its midpoint.

The cross section of a two-circle coaxial optical fiber bundle as well as the space coordinate system of the optical fiber bundle and reflector are presented in our previous work [18] (or refer to Fig. 2). Assuming that the midpoints of receiving fibers 1–18 are O_i (i = 1, 2, …, 18), and the coordinate of O_i is in the space coordinate system denoted by (x_i, y_i, z_i), the 3-D tip clearance to be measured is equivalent to the 3-D displacement (z₀, α, β) of the point on the reflector, in which the optical fiber bundle is pointing. The reflector plane can be mathematically expressed as [18]

\tan (α) \cdot x + \tan (β) \cdot y + z - z_{0} = 0.

Assuming that the inclination angles α and β of the reflector are very small based on the physical characteristics of the turbine blade, thus, tan(α) ≈α and tan(β) ≈β. Hence, the radial displacement z₀_i between the midpoints of each fiber and reflector can be expressed as

z_{0 i} (z_{0}, α, β) = z_{0} - x_{i} \cdot α - y_{i} \cdot β .

Based on the space geometry, the equivalent height h_i between the midpoints of each fiber and virtual plane of the bundle [18] can be written as

h_{i} (z_{0}, α, β) = \frac{2 z_{0 i} (z_{0}, α, β)}{1 + α^{2} + β^{2}} .

The equivalent radius r_i of the light beam at the midpoint of each fiber can be written as

r_{i} (z_{0}, α, β) = \sqrt{{[\frac{2 z_{0 i} (z_{0}, α, β) \cdot α}{1 + α^{2} + β^{2}} + x_{i}]}^{2} + {[\frac{2 z_{0 i} (z_{0}, α, β) \cdot β}{1 + α^{2} + β^{2}} + y_{i}]}^{2}} .

We have proposed the intensity factor concept based on the division function for a two-coaxial optical fiber bundle in our previous work [18], which briefly describes the output characteristics of the light intensity received by the optical fiber bundle. These output characteristics were used in this work. Hence, the theoretical intensity factor at the midpoint of each receiving fiber can be expressed as

I_{t i} (z_{0}, α, β) = \frac{\exp [- r_{i}^{2} (z_{0}, α, β) / h_{i}^{2} (z_{0}, α, β)]}{h_{i} (z_{0}, α, β)} .

Because the receiving fibers of the optical fiber bundle may not be fully covered by the light spot [18], we constructed a simplified area compensation equation based on the equivalent radius r_i of the light beam at the midpoint and the equivalent radius ωr_i of the light spot at height h_i at the midpoint, ωr_i can be given as

ω r_{i} = a_{0} [1 + ζ {(h_{i} / a_{0})}^{1.5} \tan θ_{c}] .

where θ_c is the largest incident angle of the illuminating fiber, ζ is the normalized parameter, which indicates that the beam in the light spot is uniform and conforms to the Gaussian distribution. Hence, the area compensation equation can be expressed as

f_{1 i} (z_{0}, α, β) = \frac{1}{1 + {(\frac{r_{i}}{ω r_{i}})}^{8}} = \frac{1}{1 + {\frac{r_{i} (z_{0}, α, β)}{a_{0} [1 + ζ {(h_{i} / a_{0})}^{1.5} \tan θ_{c}]}}^{8}} .

As shown in Fig. 1, the attenuation characteristics of the light intensity received by the receiving fibers [Eq. (5)] are represented by Eq. (7).

Fig. 1 Attenuation characteristics of the light intensity received by the receiving fibers due to overlapping of the light spots for different values of the equivalent radius r_i and radial displacement z₀.

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We also considered the effect of the numerical aperture constraints on the light intensity received by the receiving fibers. Assuming that the angle of the light wave received at the midpoint of each receiving fiber is γ_i and the largest angle of the light wave that a receiving fiber can absorb is υ, the numerical aperture compensation equation can be expressed as [18]

f_{2 i} (z_{0}, α, β) = \frac{1}{1 + {[γ_{i} (z_{0}, α, β) / υ]}^{8}} .

The angle γ_i can be determined from the following equation

γ_{i} = \arccos (\frac{x_{s i} (z_{0}, α, β) \cdot 2 α + y_{s i} (z_{0}, α, β) \cdot 2 β + z_{s i} (z_{0}, α, β)}{\sqrt{x_{s i}^{2} (z_{0}, α, β) + y_{s i}^{2} (z_{0}, α, β) + z_{s i}^{2} (z_{0}, α, β)} \cdot \sqrt{1 + 4 α^{2} + 4 β^{2}}}) .

where (x_s_i, y_s_i, z_s_i) is the coordinate of the symmetry point with regards to O_i corresponding to the reflector. This coordinate can be easily determined based on the space geometry. The final constructed function of the light intensity factor for each receiving fiber is

I_{i} (z_{0}, α, β) = I_{t i} (z_{0}, α, β) \cdot f_{1 i} (z_{0}, α, β) \cdot f_{2 i} (z_{0}, α, β) .

The light intensity factors of the receiving fibers in the inner and outer circles of an optical fiber bundle can be expressed as

I_{i n} (z_{0}, α, β) = \sum_{i = 1}^{6} I_{t i} (z_{0}, α, β) \cdot f_{1 i} (z_{0}, α, β) \cdot f_{2 i} (z_{0}, α, β) .

I_{o u t} (z_{0}, α, β) = \sum_{i = 7}^{12} I_{t i} (z_{0}, α, β) \cdot f_{1 i} (z_{0}, α, β) \cdot f_{2 i} (z_{0}, α, β) .

It is apparent here that the reflected light spot of the probe is modulated by z₀, α, and β, hence it is necessary to apply a sensing structure of three bundles to realize the measurement of z₀, α, and β.

3. 3-D tip clearance optical fiber probe, processing circuit and calibration table

A 3-D tip clearance probe has been designed which has structure of isosceles right-angled triangle with two-coaxial optical fiber bundle on each vertex. The cross section of the probe and its working principle is shown in Fig. 2. Three sensing units of the probe are two-coaxial optical fiber bundles with same manufacturing parameters, when the probe is turned on, the illuminating fiber of each bundle emits light spot onto the surface in front of the probe, and the reflected light spot will be received by receiving fibers of the bundle. Here the 3-D tip clearance to be measured is equivalent to the 3-D displacement (z₀, α, β) of the point on the reflector which Unit 0 towards to.

Fig. 2 Measurement principle of 3-D tip clearance optical probe.

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For Unit 1 and Unit 2, the axial angle and circumferential angle are denoted by α and β, respectively, whereas the radial displacements z₁ and z₂ may be different from z₀ due to the inclination of the reflector. The radial displacements z₁ and z₂ can be written as [18]

z_{1} = z_{0} - T \tan (α) .

z_{2} = z_{0} - T \tan (β) .

where T is the interval between Unit 0 with Unit 1 or Unit 2. It is obvious that each reflected light spot of the probe is modulated by z₀, α, and β. Based on the output signals obtained from the three sensing units, the actual values of z₀, α, and β can be estimated by applying a certain algorithm. We have manufactured the probe with core radius: a₀ = 0.1mm, interval distance between two adjacent fibers: d = 0.23mm, aperture angle of illuminating and receiving fibers: θ_c = arcsin(0.22), and T = 2mm, thus Eqs. (13) and (14) can be written as

z_{1} = z_{0} - 2 \tan (α) .

z_{2} = z_{0} - 2 \tan (β) .

Based on the findings of past studies related to 3-D tip clearance of turbine blades, the radial tip clearance z₀ typically varies within a range of 1.5–2.5 mm [21] whereas the axial and circumferential angles α and β appear mostly unipolar at the axial and circumferential axis within a range of −0.2–2.2° [22]. The equivalent radius ωr of the reflected light spot at the maximum height z₀ = 2.5mm can be calculated by Eq. (6) with parameters of the probe listed above, and normalized parameter ζ chosen according the height based on our previous study [18], the result is 1.0568mm. Considering the interval of each unit is T = 2mm, and the diameter of single light spot is 2.1136mm, thus the area formed by reflected light spots of three units will have a diameter of 5.6569mm, and the diameter of the light spots area on the tip of blade is 3.9884mm. The common width of tip with regard to turbine blades in aero engine is around 7 mm, which means the parameters we used in this study are feasible. Hence, we will focus the discussion on the convergence of the 3-D tip clearance values based on these ranges.

The photo of the probe and its connection with the circuit and calibration table is shown in Fig. 3. We have provided the details of the 3-degrees of freedom (3-DOF) calibration table and processing devices (equipped with amplifiers to increase their sensitivity to the incoming signals) used in the experiments in our previous work [18] and therefore, we do not describe them here for brevity. The LD light resource used in the experiment has light wavelength of 820nm, and the photoelectric conversion chip OPT101 used in the circuit has responsibility of 0.6 A/W (0.6 V/ μW) at the wavelength of the light resource.

Fig. 3 Photo of the probe, calibration table and processing circuit.

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Study in this paper mainly focuses on the demodulation theory of output signals from three units, therefore the static performance of demodulation is most concerned. However, the demodulation technique is aiming at the application of dynamic measurement of 3-D tip clearance in the future, take aero engine for instance, its rotating speed could reach 10000 r/min or higher, the output function of the optical fiber bundle will not be affected by the rotating speed, but the photoelectric conversion part will be, the chip OPT101 itself has a dynamic response time of 24.2 μs, it can be reduced to 7μs to fulfill the requirement of turbine blade in aero engine with rotating speed 10000 r/min, through proper external circuit. Meanwhile, the signals to be measured during the rotation action will be picked when the tip of blade is covered by three light spots, using identification algorithm of blade tip in the program, thus the frequency of sampling in data acquisition module should be set as higher as possible to fit the range of rotating speed, at least 30000 Hz in the application.

As for the actual tip surface of the turbine blades, they can be treated as plane owing to the high stiffness coefficient determined by the material of blades. Although the surface of blade tip is rough, it is still possible to employ some compensation methods such as enhancement of light resource, using laser resource or other compensate algorithm, to make the measurement suitable for the turbine blades. The specific methods will be studied in the future research.

In this work, the output signals were acquired using Unit 0 of the probe in Fig. 3, by varying the radial displacement z₀ within a range of 1.5–2.5 mm with 0.5-mm intervals and axial angle α within a range of −10°–10° with 1° intervals. The simulations of the constructed functions [Eqs. (11) and (12)] were carried out by varying z₀ within a range of 1.5–2.5 mm with 0.1-mm intervals and α within a range of −0.17–0.17 rad with 0.01 rad intervals under the same conditions as those used in the experiments. The normalized parameter ζ in Eq. (6) for each z₀ was set based on our previous work. The results are shown in Fig. 4.

Fig. 4 Simulation and experiment results of the 3-D output characters of the bundle.

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In general, the simulation results were similar to the output characteristics of a two-circle coaxial optical fiber bundle. The deviation of the symmetry axis of experimental curves is mainly because the shifting of center of Unit 0 due to the manufacturing process. Based on the simulation results, it can be deduced that the output signal of a two-circle coaxial optical fiber bundle is influenced by the 3-D tip clearance, indicating that it is necessary to design a probe with three two-circle coaxial optical fiber bundles in order to demodulate the signals for 3-D tip clearance measurements.

4. Development of the demodulation technique

For most 3-D measurement sensors, the sensing units are typically modulated by a single variable. Thus, the common demodulation technique for these 3-D measurement sensors involves formulating a generic function for a sensing unit and calculating the variable for three sensing units. The final measurement results of the sensors are then determined [20]. Based on the output characteristics of the two-circle coaxial optical fiber bundle, the intensity of reflected light spot is coupled with three displacement values: z₀, α, and β. For this reason, it is not suitable to determine the values of z₀, α, and β by simply solving the equation for each sensing unit.

We had attempted to demodulate the output signals acquired from three two-circle coaxial optical fiber bundles using neural networks in our previous work [19], which required a large number of output signals to train the network. However, the results of the neural network-based demodulation technique were not really satisfactory because of the large demodulation errors of α and β, with a maximum error of 2°. This was probably due to the highly nonlinear characteristics of the two-circle coaxial optical fiber bundles and the calibration errors training the network [18]. Thus, in this work, we proposed a new demodulation technique based on the ratio of the difference in signal intensities between any two sensing units of the probe and we derived the demodulation equations for z₀, α, and β using the second-order Taylor expansion for a three-variable function.

It is known that the value of a target function at a particular point can be approximated by evaluating the Taylor expansion of the function at points within proximity of the target point. Assuming that the 3-D tip clearance measurement range can be divided into a large number of small regions (where each region has an equivalent 3-D expansion point) such that all of the 3-D expansion points in the region satisfy a common convergence domain. As shown in Fig. 5, if the approximate region of the 3-D tip clearance values (z₀, α, and β) to be measured can be estimated based on the output signals of the optical fiber probe, the 3-D tip clearance values can be determined with higher accuracy using the demodulation equations derived from Taylor expansion at the predetermined expansion point.

Fig. 5 Diagram of measurement scope and the approximate region of (z₀, α, β).

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We developed a searching algorithm to locate the approximate region of the 3-D tip clearance values (z₀, α, and β) to be measured by comparing and discriminating the output signals based on the calibration data. The searching algorithm does not require high accuracy unlike the demodulation technique solely based on neural networks, which significantly reduces the workload of the network training. In addition, our new demodulation technique is more feasible to identify the sources of errors if they are generated during the calibration process, because the demodulation accuracy is mainly influenced by the calibration data within the approximate region.

Based on the assumption that all of the three sensing units possess the same output characteristics, that is, the three units of the probe should have same manufacturing parameters including core radius, interval between any two fibers, aperture angle and arrangement of fibers, with luminous power and photoelectric conversion coefficient of the processing circuit, the intensity factor of the signal received by the receiving fiber in the outer circle as a function of the 3-D displacement values of the sensing unit can be denoted as I_out(z, α, β). Likewise, the intensity factor of the signal received by the receiving fiber in the inner circle as a function of the 3-D displacement values of the sensing unit can be denoted as I_in(z, α, β). The demodulation principle is based on the assumption that the area in which each sensing unit is pointing towards to is slightly inclined such that the 3-D displacement values of the three sensing units have a common convergence interval. Therefore, the sensing units will have the same Taylor expansion at a single expansion point. Consider Unit 0 as an example. Here, it is assumed that the intensity factor of the signal received by the receiving fiber in the inner circle of Unit 0 is I₀_in and its 3-D tip clearance function is I₀_in(z₀, α, β) whereas the intensity factor of the signal received by the receiving fiber in the outer circle of Unit 0 is I₀_out and its 3-D tip clearance function is I₀_out(z₀, α, β).By specifying an expansion point (C_z, C_α, C_β), the second-order Taylor expansion of I₀_out(z₀, α, β) and I₀_in(z₀, α, β) near the expansion point can be expressed as

\begin{array}{l} I_{0 o u t} (z_{0}, α, β) = k_{1 o} (z_{0} - C_{z}) + k_{2 o} (^{z_{0} - C_{z}) 2} + k_{3 o} (z_{0} - C_{z}) (α - C_{α}) + k_{4 o} (z_{0} - C_{z}) (β - C_{β}) \\ + f_{o u t} (α, β) . \end{array}

\begin{array}{l} I_{0 i n} (z_{0}, α, β) = k_{1 i} (z_{0} - C_{z}) + k_{2 i} (^{z_{0} - C_{z}) 2} + k_{3 i} (z_{0} - C_{z}) (α - C_{α}) + k_{4 i} (z_{0} - C_{z}) (β - C_{β}) \\ + f_{i n} (α, β) . \end{array}

where k₁_o, k₂_o, k₃_o, and k₄_o denote the expansion coefficients of I₀_out(z₀, α, β) corresponding to z₀, k₁_i, k₂_i, k₃_i, and k₄_i denote the expansion coefficients of I₀_in(z₀, α, β) corresponding to z₀, while f_out(α, β) and f_in(α, β) represent the expansion terms that are only related to α and β, we call the array consists of k₁_o, k₂_o, k₃_o, k₄_o, k₁_i, k₂_i, k₃_i, and k₄_i the coefficients array K. It shall be noted that the remaining terms in I₀_out (z₀, α, β) and I₀_in (z₀, α, β) will be eliminated during a subtraction operation. The second-order Taylor expansion of I₁_out (z₁, α, β) and I₁_in (z₁, α, β) can be written as

\begin{array}{l} I_{1 o u t} (z_{1}, α, β) = k_{1 o} (z_{0} - 2 α - C_{z}) + k_{2 o} (^{z_{0} - 2 α - C_{z}) 2} + k_{3 o} (z_{0} - 2 α - C_{z}) (α - C_{α}) \\ + k_{4 o} (z_{0} - 2 α - C_{z}) (β - C_{β}) + f_{o u t} (α, β) . \end{array}

\begin{array}{l} I_{1 i n} (z_{1}, α, β) = k_{1 i} (z_{0} - 2 α - C_{z}) + k_{2 i} (^{z_{0} - 2 α - C_{z}) 2} + k_{3 i} (z_{0} - 2 α - C_{z}) (α - C_{α}) \\ + k_{4 i} (z_{0} - 2 α - C_{z}) (β - C_{β}) + f_{i n} (α, β) . \end{array}

Likewise, the second-order Taylor expansion of I₂_out(z₂, α, β) and I₂_in(z₂, α, β) can be written as

\begin{array}{l} I_{2 o u t} (z_{2}, α, β) = k_{1 o} (z_{0} - 2 β - C_{z}) + k_{2 o} (^{z_{0} - 2 β - C_{z}) 2} + k_{3 o} (z_{0} - 2 β - C_{z}) (α - C_{α}) \\ + k_{4 o} (z_{0} - 2 β - C_{z}) (β - C_{β}) + f_{o u t} (α, β) . \end{array}

\begin{array}{l} I_{2 i n} (z_{1}, α, β) = k_{1 i} (z_{0} - 2 β - C_{z}) + k_{2 i} (^{z_{0} - 2 β - C_{z}) 2} + k_{3 i} (z_{0} - 2 β - C_{z}) (α - C_{α}) \\ + k_{4 i} (z_{0} - 2 β - C_{z}) (β - C_{β}) + f_{i n} (α, β) . \end{array}

Because α and β are the same for the three sensing units based on our assumption for the reflector [18], the expansion terms with respect to α and β are the same for all of the sensing units. Based on this characteristic, we propose a new form of signal, which is the ratio of the difference in the signal intensity factors in the outer and inner circles between any two sensing units of the optical fiber probe. We define DM₀₁, DM₀₂, and DM₂₁ as the ratios of the difference in the output signal intensity factors between Unit 0 and Unit 1, Unit 0 and Unit 2, and Unit 2 and Unit 1, respectively, as follows

D M_{01} = \frac{I_{0 o u t} - I_{1 o u t}}{I_{0 i n} - I_{1 i n}} = \frac{4 k_{2 o} z_{0} + (2 k_{3 o} - 4 k_{2 o}) α + 2 k_{4 o} β + (2 k_{1 o} - 4 k_{2 o} C_{z} - 2 k_{3 o} C_{α} - 2 k_{4 o} C_{β})}{4 k_{2 i} z_{0} + (2 k_{3 i} - 4 k_{2 i}) α + 2 k_{4 i} β + (2 k_{1 i} - 4 k_{2 i} C_{z} - 2 k_{3 i} C_{α} - 2 k_{4 i} C_{β})} .

D M_{02} = \frac{I_{0 o u t} - I_{2 o u t}}{I_{0 i n} - I_{2 i n}} = \frac{4 k_{2 o} z_{0} + (2 k_{4 o} - 4 k_{2 o}) β + 2 k_{3 o} α + (2 k_{1 o} - 4 k_{2 o} C_{z} - 2 k_{3 o} C_{α} - 2 k_{4 o} C_{β})}{4 k_{2 i} z_{0} + (2 k_{4 i} - 4 k_{2 i}) β + 2 k_{3 i} α + (2 k_{1 i} - 4 k_{2 i} C_{z} - 2 k_{3 i} C_{α} - 2 k_{4 i} C_{β})} .

D M_{21} = \frac{I_{2 o u t} - I_{1 o u t}}{I_{2 i n} - I_{1 i n}} = \frac{4 k_{2 o} z_{0} + (2 k_{3 o} - 4 k_{2 o}) α + (2 k_{4 o} - 4 k_{2 o}) β + (2 k_{1 o} - 4 k_{2 o} C_{z} - 2 k_{3 o} C_{α} - 2 k_{4 o} C_{β})}{4 k_{2 i} z_{0} + (2 k_{3 i} - 4 k_{2 i}) α + (2 k_{4 i} - 4 k_{2 i}) β + (2 k_{1 i} - 4 k_{2 i} C_{z} - 2 k_{3 i} C_{α} - 2 k_{4 i} C_{β})} .

For mere convenience, let D_in and D_out be the sum of constant terms with respect to the denominator and numerator terms in Eqs. (23)–(25), as follows

D_{i n} = 2 k_{1 i} - 4 k_{2 i} C_{z} - 2 k_{3 i} C_{α} - 2 k_{4 i} C_{β} .

D_{o u t} = 2 k_{1 o} - 4 k_{2 o} C_{z} - 2 k_{3 o} C_{α} - 2 k_{4 o} C_{β} .

Therefore, demodulation equations for z₀, α, and β can be derived from Eqs. (23)–(27) as

\begin{array}{l} z_{0} = (4 k_{2 i} D_{o u t} - 4 k_{2 o} D_{i n}) [(4 k_{2 i} - 2 k_{3 i}) (D M_{21} - D M_{02}) (4 k_{2 o} - 4 k_{2 i} D M_{01}) - 2 k_{4 i} (D M_{21} - D M_{01}) \\ (4 k_{2 o} - 4 k_{2 i} D M_{02}) - (4 k_{2 o} - 4 k_{2 i} D M_{02}) (4 k_{2 o} - 4 k_{2 i} D M_{21})] / {4 k_{2 i} [(4 k_{2 o} - 4 k_{2 i} D M_{01}) \\ (D M_{21} - D M_{02}) (4 k_{2 o} \cdot 2 k_{3 i} - 2 k_{3 o} \cdot 4 k_{2 i}) + (4 k_{2 o} - 4 k_{2 i} D M_{01}) (4 k_{2 o} - 4 k_{2 i} D M_{02}) \\ (4 k_{2 o} - 4 k_{2 i} D M_{21}) + (4 k_{2 o} - 4 k_{2 i} D M_{02}) (D M_{21} - D M_{01}) (4 k_{2 o} \cdot 2 k_{4 i} - 2 k_{4 o} \cdot 4 k_{2 i})]} - D_{i n} / 4 k_{2 i} . \end{array}

\begin{array}{l} α = (4 k_{2 i} D_{o u t} - 4 k_{2 o} D_{i n}) (D M_{21} - D M_{02}) (4 k_{2 o} - 4 k_{2 i} D M_{01}) / [(4 k_{2 o} - 4 k_{2 i} D M_{01}) \\ (D M_{21} - D M_{02}) (4 k_{2 o} \cdot 2 k_{3 i} - 2 k_{3 o} \cdot 4 k_{2 i}) + (4 k_{2 o} - 4 k_{2 i} D M_{01}) (4 k_{2 o} - 4 k_{2 i} D M_{02}) \\ (4 k_{2 o} - 4 k_{2 i} D M_{21}) + (4 k_{2 o} - 4 k_{2 i} D M_{02}) (D M_{21} - D M_{01}) (4 k_{2 o} \cdot 2 k_{4 i} - 2 k_{4 o} \cdot 4 k_{2 i})] . \end{array}

\begin{array}{l} β = (4 k_{2 i} D_{o u t} - 4 k_{2 o} D_{i n}) (D M_{21} - D M_{01}) (4 k_{2 o} - 4 k_{2 i} D M_{02}) / [(4 k_{2 o} - 4 k_{2 i} D M_{01}) \\ (D M_{21} - D M_{02}) (4 k_{2 o} \cdot 2 k_{3 i} - 2 k_{3 o} \cdot 4 k_{2 i}) + (4 k_{2 o} - 4 k_{2 i} D M_{01}) (4 k_{2 o} - 4 k_{2 i} D M_{02}) \\ (4 k_{2 o} - 4 k_{2 i} D M_{21}) + (4 k_{2 o} - 4 k_{2 i} D M_{02}) (D M_{21} - D M_{01}) (4 k_{2 o} \cdot 2 k_{4 i} - 2 k_{4 o} \cdot 4 k_{2 i})] . \end{array}

It shall be noted that all of the elements in Eqs. (28)–(30) obtained by calibration will be constant except for DM₀₁, DM₀₂, and DM₂₁. This means that if this method is feasible, as long as the values of DM₀₁, DM₀₂, and DM₂₁ from the output signals of the optical fiber probe are known, the signals will be demodulated inside the processor and the values of z₀, α, and β can be determined.

5. Partition of the expansion points within the 3-D tip clearance measurement range

In this work, the most important aspect of the demodulation technique is to determine the approximate 3-D expansion point whose convergence interval consists of three points on the reflector in which the sensing units are pointing towards to during the measurements. To determine the equivalent expansion point (C_z, C_α, C_β) of the 3-D tip clearance measurements, the radial displacement C_z appears to be the most significant parameter because the three midpoints of the light spots have equivalent values of α and β. There are two ways by which we can determine the approximate expansion point. First, we can determine the point on the reflector that has the same distance to the three midpoints of the light spots, alternatively, we can determine the point on the reflector whose radial displacement is equal to the weighted mean of the radial displacements of the three sensing units.

Assuming that the coordinate of the equivalent expansion point O_e is (x_e, y_e, z_e), the equation set for O_e based on the same distance method can be written as

{\begin{matrix} 4 x_{e} - 4 α \cdot z_{e} + 4 α (z_{0} - α) - 4 = 0. \\ 4 y_{e} - 4 β \cdot z_{e} + 4 β (z_{0} - β) - 4 = 0 \\ α \cdot x + β \cdot y + z - z_{0} = 0. \end{matrix} .

The equation for O_e based on the weighted mean method can be written as

z_{e} = \frac{ω r_{0} \cdot z_{0} + ω r_{1} \cdot z_{1} + ω r_{2} \cdot z_{2}}{ω r_{0} + ω r_{1} + ω r_{2}} .

where ωr₀, ωr₁, ωr₂ are the equivalent radii of light spots of Unit 0, Unit 1, Unit 2 on the reflector. Thus, the coordinate of O_e in terms of 3-D displacement values is (z_e, α_e, β_e).

Based on our observations of the output curves I_in (z₀, α, β) and I_out(z₀, α, β), we found that I_in and I_out will reach zero as z₀, α, and β tend to infinity. However, the constructed 3-D function for the two-circle coaxial optical fiber bundle is too complex to precisely calculate the radius of convergence using Taylor expansion. Therefore, it is more practical to use the error curves obtained from the simulations in order to estimate the radius of convergence. Furthermore, we intended to minimize the number of calibration points and therefore, we collected several typical equivalent expansion points such that these points were representative of the larger 3-D measurement region. The Taylor expansion coefficients array K were calculated through “Taylor” and “coeffs” function in MATLAB program using output function with respect to z₀, α, and β.

Assuming that the 3-D tip clearance values (z₀, α, and β) to be measured were 2.03 mm, 0.59°, and 0.35°, respectively, and using the output signals of the constructed function as the input signals for the demodulation, the values of z_e at the equivalent expansion point O_e determined using Eqs. (31) and (32) were found to be 2.01 and 2.03 mm, respectively.

We simulated the demodulation error curves of z₀, α and β at (2.03 mm, 0.59°, 0.35°) at equivalent expansion points O_e(z_e, 0.5°, 0.5°) and (z_e, 0.6°, 0.4°), where z_e was varied within a range of 1.97–2.06 mm with 0.01-mm intervals. The simulation results are shown in Fig. 6.

Fig. 6 Demodulation error curves of z₀, α and β at the equivalent expansion point O_e of (z_e, 0.5°,0.5°) and (z_e, 0.6°,0.4°).

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It is evident that z_e has a more pronounced effect on the demodulation accuracy compared to α_e and β_e. The error curves of z₀, α, and β reach a minimum value at z_e = 2.03 mm based on the weighted mean technique [Eq. (32)]. We further performed the simulations to investigate the effects of α_e and β_e on the demodulation errors of z₀, α and β at (2.03 mm, 0.59°, 0.35°) and at the equivalent expansion points O_e (2.01 mm, α_e, β_e) and (2.03 mm, α_e, β_e). To facilitate the analysis, we assumed that α_e = β_e within a range of 0.1–1.0° with 0.1° intervals. The simulation results are shown in Fig. 7.

Fig. 7 Demodulation error curves of z₀, α and β at equivalent expansion point O_e(2.01mm, α_e, β_e) and (2.03mm, α_e, β_e).

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It can be observed that the demodulation errors of z₀, α and β are lower when α_e and β_e are close to the true values of α and β to be measured. Because the demodulation method is simulated under ideal conditions, while the environment and reflector in the actual measurement are much more severe and therefore, the error tolerances for z₀, α, and β are set to be 0.01 mm, 0.1°, and 0.1°, respectively. In general, the demodulation errors of z₀, α, and β for the values of z_e, α_e, and β_e considered in the simulations fulfill the error tolerance requirements.

Based on the extreme condition where the position of the blade tip approaches the maximum inclination angle of 2° and assuming that the 3-D tip clearance values (z₀, α, and β) were 2 mm, 1.7°, and 0.8°, respectively, the z_e values at the expansion point O_e determined from Eqs. (31) and (32) were found to be 1.95 and 1.99 mm, respectively. Similarly, we simulated the demodulation error curves of z₀, α, and β at the equivalent expansion point O_e (z_e, 1.7°, 0.8°), where z_e was varied within a range of 1.94–2.00 mm with 0.01-mm intervals. The simulation results are shown in Fig. 8.

Fig. 8 Demodulation error curves of z₀, α and β at equivalent expansion point O_e (z_e, 1.7°, 0.8°).

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The results showed a different trend, where the error curve of z₀ reached a minimum value at z_e = 1.99 mm based on the weighted mean technique [Eq. (32)]. In contrast, the demodulation error curves of α and β reached a minimum value at z_e = 1.95 mm based on the equidistance technique [Eq. (31)]. In general, the error curves z₀, α, and β fulfill the error tolerance requirements. It can be deduced that the weighted mean technique used to determine z_e have a positive effect on the demodulation accuracy based on the error tolerances for z₀, α, and β.

Thus, we proposed an appropriate partition solution for the expansion points as follows. The range of z₀ (1.5–2.5 mm) was divided into 0.05-mm intervals (noting that the same interval was used for a larger range of z₀) whereas the measurement ranges of α and β (−0.2–2.2°) were divided into 0.6° intervals. This resulted in a total of 21 × 4 × 4 expansion points. We defined the photoelectric conversion output voltage of the receiving fibers in the inner circle for Unit 0, Unit 1, and Unit 2 as V₀_in, V₁_in, and V₂_in, respectively. Likewise, we defined the voltage of the receiving fibers in the outer circle for Unit 0, Unit 1, and Unit 2 as V₀_out, V₁_out, and V₂_out, respectively. The demodulation procedure proposed in this work is outlined as follows:

1) Search for the approximate region of z₀ using the output ratio of Unit 0 (V_0out/V_0in) based on the static calibration data using a suitable comparison algorithm. Determine z_e.
2) Search for the approximate regions of α and β using the difference signals V_0out /V₀_in-V₁_out/V₁_in, V_0out /V_0in- V₂_out/V₂_in based on the static calibration data using a suitable comparison algorithm. Determine α_e and β_e.
3) The coefficients array K is computed based on the calibration data at the typical equivalent expansion points in the measurement range. Because (C_z, C_α, C_β) = (z_e, α_e, β_e), while each (C_z, C_α, C_β) point and its corresponding coefficients K are stored in a whole struct in the internal storage of the processor, hence we could extract the coefficients of the demodulation equations [Eqs. (28)–(30)] with respect to (C_z, C_α, C_β), through “find” function in the MATLAB script program to locate the position of struct involving target (C_z, C_α, C_β) .
4) Use the ratio of the difference in the output signal intensities of the probe (DM₀₁, DM₀₂, and DM₂₁) as the input signals and demodulate z₀, α, and β using Eqs. (28)–(30).

6. Calibration of coefficients and experiment analysis

Because the coefficients in demodulation equation set are all the expansion coefficients obtained from Taylor expansion, the expansion coefficients at a particular expansion point (C_z, C_α, C_β) could not be determined based on the calibration data obtained from experiments. Hence, we could only determine these expansion coefficients based on the output signals at several points nearby the 3-D tip clearance according to the procedure presented in Section 5.

We estimated the approximate convergence of the 3-D tip clearance values (z₀, α, and β), considering the output characteristics of the optical fiber bundle. We could derive the expansion coefficients from I_in(z₀, α, β) and I_out(z₀, α, β) using the calibration data at (C_z + Δz, C_α, C_β), (C_z + Δz, C_α + 0.1°, C_β), and (C_z + Δz, C_α−0.1°, C_β), where Δz is the difference between z₀ and C_z based on the results obtained in Section 5, using the weighted mean method. It is necessary to formulate a set of equations to estimate the expansion coefficients of (C_z, C_α, C_β) and the differences between the signal intensity factors received by the receiving fibers in the outer circles of Unit 0 and Unit 1, inner circles of Unit 0 and Unit 1, outer circles of Unit 0 and Unit 2, and inner circles of Unit 0 and Unit 2. The set of equations can be written as

{\begin{matrix} I_{0 o u t} - I_{1 o u t} = k_{p} (V_{0 o u t} - V_{1 o u t}) = α [4 k_{2 o} z_{0} + (2 k_{3 o} - 4 k_{2 o}) α + 2 k_{4 o} β + (2 k_{1 o} - 4 k_{2 o} C_{z} - 2 k_{3 o} C_{α} - 2 k_{4 o} C_{β})] . \\ I_{0 i n} - I_{1 i n} = k_{p} (V_{0 i n} - V_{1 i n}) = α [4 k_{2 i} z_{0} + (2 k_{3 i} - 4 k_{2 i}) α + 2 k_{4 i} β + (2 k_{1 i} - 4 k_{2 i} C_{z} - 2 k_{3 i} C_{α} - 2 k_{4 i} C_{β})] . \\ I_{0 o u t} - I_{2 o u t} = k_{p} (V_{0 o u t} - V_{2 o u t}) = β [4 k_{2 o} z_{0} + (2 k_{4 o} - 4 k_{2 o}) β + 2 k_{3 o} α + (2 k_{1 o} - 4 k_{2 o} C_{z} - 2 k_{3 o} C_{α} - 2 k_{4 o} C_{β})] . \\ I_{0 i n} - I_{2 i n} = k_{p} (V_{0 i n} - V_{2 i n}) = β [4 k_{2 i} z_{0} + (2 k_{4 i} - 4 k_{2 i}) β + 2 k_{3 i} α + (2 k_{1 i} - 4 k_{2 i} C_{z} - 2 k_{3 i} C_{α} - 2 k_{4 i} C_{β})] . \end{matrix}

where k_p is the photoelectric conversion and amplifying coefficient of the processing circuit. If we take the outer circle of the receiving fibers as an example, the output voltage signals at the points nearby the 3-D tip clearance, i.e., (C_z + Δz, C_α, C_β), (C_z + Δz, C_α−0.1°, C_β), (C_z + Δz, C_α, C_β−0.1°), will be calibrated as follows

{\begin{matrix} k_{p} (V_{0 o u t} - V_{1 o u t}) |_{(C_{z} + △ z, C_{α}, C_{β})} = C_{α} (4 k_{2 o} \cdot △ z + 2 k_{1 o} - 4 k_{2 o} \cdot C_{α}) . \\ \begin{array}{l} k_{p} (V_{0 o u t} - V_{1 o u t}) |_{(C_{z} + △ z, C_{α} - 0.1 °, C_{β})} = (C_{α} - 0.1 °) [4 k_{2 o} \cdot △ z + 2 k_{1 o} - 4 k_{2 o} \cdot (C_{α} - 0.1 °) - 2 k_{3 o} \cdot 0.1 °] . \\ k_{p} (V_{0 o u t} - V_{2 o u t}) |_{(C_{z} + △ z, C_{α} - 0.1 °, C_{β})} = C_{β} [4 k_{2 o} \cdot △ z + 2 k_{1 o} - 4 k_{2 o} \cdot (C_{α} - 0.1 °) - 2 k_{3 o} \cdot 0.1 °] . \end{array} \\ k_{p} (V_{0 o u t} - V_{2 o u t}) |_{(C_{z} + △ z, C_{α}, C_{β} - 0.1 °)} = (C_{β} - 0.1 °) [4 k_{2 o} \cdot △ z + 2 k_{1 o} - 4 k_{2 o} \cdot (C_{β} - 0.1 °) - 2 k_{4 o} \cdot 0.1 °] . \end{matrix}

The coefficient k_pwill be eliminated in the subsequent demodulation operation. Therefore, we set k_p = 1 in Eq. (34) to facilitate the calculations.

It is crucial to ensure that the demodulation technique is consistent for the three sensing units of the optical fiber probe. So the probe must be manufactured precisely, especially the arrangement of the receiving fibers around the illuminating fiber of each units, because the probe we used has a significant random feature of the arrangement of the receiving fibers in [18], therefore a new probe has been made to improve the consistence of three units, shown in Fig. 3. It can be seen from photo of the cross section of the optical fiber probe that although the receiving fibers in each sensing unit are arranged regularly, there is deviation of Unit 0 from the very center determined by three units, which will lead to slight deviations in the measurements.

The 3-DOF calibration table described in [18] has an accuracy of 0.1° in both the axial and circumferential directions and therefore, we could only testify the 3-D tip clearance values within a precision of 0.1°. Because the demodulation algorithm is based on the Taylor expansion of output function of three units in the probe, and z₀, z₁, z₂ of three units have larger difference when inclination angle of reflector increases using Eqs. (15) and (16), which will affect the convergence at the 3-D expansion points in the region. Therefore we will verify the demodulation technique under three varying degrees of the inclination angle at z₀ = 2.00 mm.

In following experiments, the output signals obtained from the three sensing units at several 3-D tip clearance points at z₀ = 2.00 mm with probe and calibration system shown in Fig. 3 were demodulated using the demodulation technique described above. However, the most difficult aspect of the calibration procedure was to align the optical fiber probe perfectly when we installed the probe on top of the 3-DOF calibration table. In addition, there may be deviations between the surface of the probe and reflector considering that the probe was aligned manually. Based on the symmetrical features of the output characteristics of the optical fiber bundle, the offsets of α and β were estimated to be 0.3° and 0.4°, respectively, from experiments. The 3-D points in the following part are all true value in consideration of the offsets of α and β unless otherwise noted.

First, we assumed that the expansion point was (2 mm, 0.6°, 0.4°), and we calibrated the expansion factor based on the output signals at (2 mm, 0.6°, 0.4°), (2 mm, 0.5°, 0.4°), and (2 mm, 0.6°, 0.3°). The demodulation errors at the nearby points (where z₀ = 2 mm) are summarized in Table 1.

Table 1. Demodulation Errors at the Nearby Points of (2 mm, 0.6°, 0.4°) with Same Value of z₀ but Different Values of α and β

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The results showed that the demodulation errors of z₀, α and β all fulfill the error tolerance requirements. However, the demodulation errors of z₀, α and β at (2mm, 0.5°,0.2°) are larger than those at (2mm, 0.6°,0.6°) and (2mm, 0.5°,0.4°), and all the demodulation errors are larger than those in the simulation, the reason is probably because (2mm, 0.5°,0.2°) is further away from the expansion point (2 mm, 0.6°, 0.4°), and the probe calibration errors due to the align procedure of the probe towards reflector, together with the deviation error of Unit 0 when assembling the probe.

Then, we demodulated the voltage signals of three units at several 3-D clearance tip points with larger α and β to investigate their demodulation effects, where the expansion point was (2 mm, 1°, 0.7°). The simulation results are summarized in Table 2.

Table 2. Demodulation Errors at the Nearby Points of (2 mm, 1°, 0.7°) with Same Value of z₀ but Different Values of α and β

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The results also showed that the demodulation errors of z₀, α, and β all fulfill the error tolerance requirements with larger inclination angles. In order to verify the effect of demodulation on z₀, we used the same expansion point (2 mm, 1°, 0.7°) and a couple of points in which the values of α and β were the same but the value of z₀ was different. The demodulation errors of z₀, α, and β at these expansion points are tabulated in Table 3.

Table 3. Demodulation Errors at the Nearby Points of (2 mm, 1°, 0.7°) with the Same Values of α and β but Different Value of z₀

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The results showed that the demodulation errors of z₀ around C_z were larger than those of the previous simulation results. In addition, the demodulation errors of z₀ showed a similar trend to the demodulation errors of α and β. It can be deduced that the demodulation had a negligible effect on the z₀ obtained from the signals of probe shown in Fig. 3. This is because the demodulation effect highly relies on the calibration accuracy, but the center of Unit 0 in the probe (Fig. 3) has a deviation from the very center determined by the axial axis and circumferential axis, thus leads to calculation errors of z₀, z₁ and z₂ with regard to three units.

In order to investigate how the calibration errors affect the demodulation, we calculated the expansion coefficients using both calibration data and the output function described in Eq. (10), the results are listed in Table 4, where k₁, k₂, k₃ and k₄ of inner circle refer to k₁_i, k₂_i, k₃_i and k₄_i, while k₄_i, k₁, k₂, k₃ and k₄ of outer circle refer to k₁_o, k₂_o, k₃_o and k₄_o.

Table 4. Expansion Coefficients Estimated Using Calibration Data and Calculated Using Output Function

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Through observation on Eqs. (17)–(22) and Eqs. (28)–(30), it can be deduced that the demodulation result of z₀ is affected mainly by coefficients k₁ and k₂, while the coefficients k₁, k₂, k₃ and k₄ have equal performance in the demodulation process of α and β. It can be seen from Table 4, the orders of k₃ and k₄ calculated using calibration data are much larger than k₃ and k₄ using output function, due to the calibration errors. In that case, errors in estimating the Taylor expansion coefficients lead to imbalance among k₁, k₂, k₃, and k₄, then cause insensitivity to demodulation for z₀, this can be improved by enhancement of manufacturing accuracy of optical fiber bundles in the probe.

Next, we simulated the demodulation errors for even larger values of α and β and we fixed the value of z₀ (i.e., z₀ = C_z). Here, the expansion point was assumed to be (2 mm, 1.6°, 2.5°). The simulation results are presented in Table 5.

Table 5. Demodulation Errors at the Nearby Points of (2 mm, 1.6°, 2.5°) with the Same Value of z₀ but Different Values of α and β

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The results showed that the demodulation errors of z₀, α and β all fulfill the error tolerance requirements except error of β at (2mm, 1.9°,2.6°), with a value of ~0.136°, which was not desirable with regards to the error tolerance. This can also be attributed to errors in estimating the Taylor expansion coefficients due to deviation in center of Unit 0.

Overall, the demodulation errors appear to be satisfying regarding the error tolerance at most points, still, the demodulation accuracy need to be improved, this suggests that a high precision accuracy is needed when fabricating the optical fiber probe as well as stable processing circuit.

7. Conclusion

In this paper, we have presented a new demodulation technique for 3-D tip clearance measurements based on the output signals of an optical fiber probe, consisting of three two-circle coaxial optical fiber bundles as the sensing units. The demodulation equations were derived using the second-order Taylor expansion for a three-variable function. Based on the simulation results, the largest demodulation error of the 3-D tip clearance was (0.005mm, 0.02°, 0.02°) within a partition interval of (0.05mm, 0.6°, 0.6°), indicating that the demodulation performance was satisfactory. The experimental results proved the practicability of our demodulation technique for the optical fiber probe, However, based on the experimental results, the demodulation errors at several 3-D tip clearance points were higher than those obtained from simulations, which can be primarily attributed to the inconsistencies between the signals acquired from the three sensing units, leading to errors in estimating the Taylor expansion coefficients. Therefore, there is a critical need to fabricate the probe with a high level of precision to ensure consistency in the output signals from the three two-circle coaxial optical fiber bundles. Moreover, the demodulation technique has been verified using only ideal reflector in this paper, the compensation techniques should be studied to fulfill the requirements of measurement for actual turbine blade, and these will be the focus of our research in the future.

Funding

National Natural Science Foundation of China (NSFC) (51575436).

Acknowledgments

We wish to thank Mr. Bing Wu for his helpful advice.

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Measurement points		(2mm, 0.6°,0.6°)	(2mm, 0.5°,0.4°)	(2mm, 0.5°,0.2°)
Demodulation errors of z₀, α and β	z₀(mm)	−0.00562	0.00011	0.00546
	α (°)	−0.00959	0.00485	−0.07656
	β(°)	0.00832	0.00171	0.03374

Measurement points		(2mm, 1.1°,0.9°)	(2mm, 0.9°,0.6°)	(2mm, 0.9°,0.8°)
Demodulation errors of z₀, α and β	z₀(mm)	−0.00749	0.003056	−0.002493
	α(°)	−0.07885	−0.047334	0.041622
	β(°)	0.06782	0.000281	−0.011521

Measurement points		(2mm, 0.9°,0.7°)	(1.99mm, 0.9°,0.7°)	(1.98mm, 0.9°,0.7°)
Demodulation errors of z₀, α and β	z₀(mm)	0.000232	0.009319	0.019356
	α(°)	−0.00199	−0.002605	−0.003033
	β(°)	−0.00786	−0.012415	−0.015507

Coefficients		k₁	k₂	k₃	k₄
Experiment	Inner circle	13.0875	452.072	1019.74	1843.56
Experiment	Outer circle	−11.4362	−255.074	−338.187	−422.215
Simulation	Inner circle	−8.16843	3.86434	14.37472	10.0161
Simulation	Outer circle	7.96311	−19.7775	33.95518	31.09215

Measurement points		(2mm, 1.9°,2.6°)	(2mm, 1.5°,2.5°)	(2mm, 1.6°,2.5°)
Demodulation errors of z₀, α and β	z₀(mm)	−0.00619	0.000227	0.0004740
	α(°)	−0.030463	−0.017603	−0.020250
	β(°)	0.136298	0.029660	−0.0200523

Demodulation technique for 3-D tip clearance measurements based on output signals from optical fiber probe with three two-circle coaxial optical fiber bundles

Abstract

1. Introduction

2. Constructed functions based on the output characteristics of the optical fiber bundle

3. 3-D tip clearance optical fiber probe, processing circuit and calibration table

4. Development of the demodulation technique

5. Partition of the expansion points within the 3-D tip clearance measurement range

6. Calibration of coefficients and experiment analysis

7. Conclusion

Funding

Acknowledgments

References

Cited By

Figures (8)

Tables (5)

Equations (34)

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