Eddy Current Sensor System for Blade Tip Clearance Measurement Based on a Speed Adjustment Model

Blade tip clearance (BTC) measurement and active clearance control (ACC) are becoming crucial technologies in aero-engine health monitoring so as to improve the efficiency and reliability as well as to ensure timely maintenance. Eddy current sensor (ECS) offers an attractive option for BTC measurement due to its robustness, whereas current approaches have not considered two issues sufficiently. One is that BTC affects the response time of a measurement loop, the other is that ECS signal decays with increasing speed. This paper proposes a speed adjustment model (SAM) to deal with these issues in detail. SAM is trained using a nonlinear regression method from a dynamic training data set obtained by an experiment. The Levenberg–Marquardt (LM) algorithm is used to estimate SAM characteristic parameters. The quantitative relationship between the response time of ECS measurement loop and BTC, as well as the output signal and speed are obtained. A BTC measurement method (BTCMM) based on the SAM is proposed and a geometric constraint equation is constructed to assess the accuracy of BTC measurement. Experiment on a real-time BTC measurement during the running process for a micro turbojet engine is conducted to validate the BTCMM. It is desirable and significative to effectively improve BTC measurement accuracy and expand the range of applicable engine speed.


Introduction
With increasing applications of turbomachinery in various industrial fields, the requirement for multi-objective design methods, such as thermal efficiency, aerodynamic performance and structural life, has considerably increased in the last few decades [1][2][3]. As one of the key constraints in blade design, the blade tip clearance (BTC) changes in a complicated manner under various working statuses of rotors. Actually, reasonable BTC is beneficial for the performance and reliability of turbomachinery. The relatively large BTC will enhance the heat transfer in the blade tip region of turbomachinery, exacerbate the oxidation and erosion of the material for blade tip, and cause performance deterioration or even failure, which accounts for about a third [1] of the failure probability of high pressure turbine; conversely, the relatively small BTC may bring a friction risk between the blade and the cabinet [4]. In addition, unreasonable BTC decreases the thermal efficiency of turbomachinery, for performs a turbojet-engine test to validate the proposed method. This is followed by the concluding remarks in Section 5.

Speed Adjustment Model
In this section, the principle and method of BTC measurement using ECS are introduced. Combined with the structure of the turbomachinery, the characteristics of ECS output signal are analyzed, then the speed adjustment model (SAM) and characteristic parameters of the sensor signal are proposed. The SAM is trained using nonlinear regression from a dynamic training data set obtained by the experiment. To solve the nonlinear regression problem, the Levenberg-Marquardt (LM) algorithm is used to estimate the characteristic parameters of the model.

Principle of BTC Measurement by ECS
In the BTC measurement system based on ECS , the sensor is generally mounted in the casing along the normal direction of the blade tip surface. The sensor probe whose head is parallel to the inner surface of the casing ( Figure 1a) has a current of 1-2 MHz in the built-in coil, which forms a primary magnetic field near the tip of the blade. When the rotating blade sweeps over the eddy current probe, a variation of magnetic flux induces eddy currents in the metal surface of the blade tip, which forms a secondary magnetic field and changes the current in the sensor coil ( Figure 1b). Because the mutual inductance between the tip of the blade and the coil is related to BTC, the BTC can be measured by detecting and capturing the current change in the coil. In general, the sensor outputs a voltage signal pulse (Figure 1c) when the blade tip rotates past the ECS.
Sensors 2019, xx, 5 3 of 17 Section 4 performs a turbojet-engine test to validate the proposed method. This is followed by the concluding remarks in Section 5.

Speed Adjustment Model
In this section, the principle and method of BTC measurement using ECS are introduced. Combined with the structure of the turbomachinery, the characteristics of ECS output signal are analyzed, then the speed adjustment model (SAM) and characteristic parameters of the sensor signal are proposed. The SAM is trained using nonlinear regression from a dynamic training data set obtained by the experiment. To solve the nonlinear regression problem, the Levenberg-Marquardt (LM) algorithm is used to estimate the characteristic parameters of the model.

Principle of BTC Measurement by ECS
In the BTC measurement system based on ECS , the sensor is generally mounted in the casing along the normal direction of the blade tip surface. The sensor probe whose head is parallel to the inner surface of the casing ( Figure 1a) has a current of 1-2 MHz in the built-in coil, which forms a primary magnetic field near the tip of the blade. When the rotating blade sweeps over the eddy current probe, a variation of magnetic flux induces eddy currents in the metal surface of the blade tip, which forms a secondary magnetic field and changes the current in the sensor coil ( Figure 1b). Because the mutual inductance between the tip of the blade and the coil is related to BTC, the BTC can be measured by detecting and capturing the current change in the coil. In general, the sensor outputs a voltage signal pulse (Figure 1c) when the blade tip rotates past the ECS. The amplitude of each signal pulse represents the corresponding BTC size of the blade. However, it is not a simple linear relationship between the signal pulse amplitude (SPA) and the BTC. In static state, the BTC-SPA curve can be obtained by measuring the values of SPA under different BTCs The amplitude of each signal pulse represents the corresponding BTC size of the blade. However, it is not a simple linear relationship between the signal pulse amplitude (SPA) and the BTC. In static state, the BTC-SPA curve can be obtained by measuring the values of SPA under different BTCs  Figure 1d). In practical measurement, according to the collected SPA value, the BTC can be calculated by an interpolation algorithm.The performance of the precision of above methods are satisfactory in the low-speed range. However, the relative-high speed results in that the output signal may deviate from the expected value, which reduces the measurement accuracy.

Speed Adjustment Model
The mutual inductance M between the ECS and the measured blade varies continuously with relative position between them. Assume that M remains unchanged, and then analyze the equivalent circuit composed of a simple ECS circuit and the measured blade, as shown in Figure 2. Here, U 0 is the excitation voltage, U e is the output voltage of detection circuit, R 0 is the voltage dividing resistance, R e is the equivalent resistance of the ECS, R b is the equivalent resistance of the measured blade surface, L e the equivalent inductance of the ECS, L b is the equivalent inductance of the measured blade surface, C is the parallel resonant capacitor, I e is the current in the sensor coil, and I b is the equivalent current of the measured blade surface.  Figure 1d). In practical measurement, according to the collected SPA value, the BTC can be calculated by an interpolation algorithm.The performance of the precision of above methods are satisfactory in the low-speed range. However, the relative-high speed results in that the output signal may deviate from the expected value, which reduces the measurement accuracy.

Speed Adjustment Model
The mutual inductance M between the ECS and the measured blade varies continuously with relative position between them. Assume that M remains unchanged, and then analyze the equivalent circuit composed of a simple ECS circuit and the measured blade, as shown in Figure 2. Here, U 0 is the excitation voltage, U e is the output voltage of detection circuit, R 0 is the voltage dividing resistance, R e is the equivalent resistance of the ECS, R b is the equivalent resistance of the measured blade surface, L e the equivalent inductance of the ECS, L b is the equivalent inductance of the measured blade surface, C is the parallel resonant capacitor, I e is the current in the sensor coil, and I b is the equivalent current of the measured blade surface. By consideration of the influence of the measured blade on the ECS, the coupled equivalent resistance and inductance of the ECS are as follows [31]: f is the excitation frequency. When the ECS is in service, the sensor loop is essentially a second-order LC oscillation circuit. Based on Kirchhoff's law, we can obtain Supposing that the current in the loop is zero when t = 0, U e = U, then where β= arccos( δ Mark the average of time constant ( 1 δ(M) ) as 1 δ ; then, the amplitude of U e can be approximated as By consideration of the influence of the measured blade on the ECS, the coupled equivalent resistance and inductance of the ECS are as follows [31]: f is the excitation frequency. When the ECS is in service, the sensor loop is essentially a second-order LC oscillation circuit. Based on Kirchhoff's law, we can obtain Supposing that the current in the loop is zero when t = 0, U e = U, then where β = arccos δ ω 0 , 0 < β < π 2 , δ = R * e ; then, the amplitude of U e can be approximated as The period of interaction between ECS and the blade tip is t m = d e v tip , and the frequency of the interaction is defined (Figure 1c), the measuring frequency f m can be expressed as: and blade passing frequency [18] f p is given by where t m is the measuring time (the duration of the blade tip rotating past the sensor probe), d e represents the coil diameter of a ECS probe, v tip is blade tip velocity, r tip is blade tip radius, n is the rotor speed, and n b is the number of blades. f m is the reciprocal of t m , and f p is the reciprocal of time interval between adjacent pulses ( Figure 1c). In general, the response frequency of the ECS used for BTC measurement is matched to f p , but, in fact , f m is much higher than f p . Under the condition that the BTC is an unchanged value c i , the SPA u i varies with the rotational speed n i . According to Equation (5), the SAM of signal amplitude u i can be expressed as: where u s is the component irrelevant to the rotational speed, u r is the component related to the rotational speed, and τ is the response time of the sensor loop. The constant 1 × 10 4 in the formula is configured to make the order of magnitude of ∂u/∂u s , ∂u/∂u r , ∂u/∂τ more close. As three characteristic parameters of SAM, u s , u r , τ are pretty essential to describe the variation of SPA with the rotational speed.
The nonlinear regression method [34] can be used to solve the characteristic parameters of SAM, provided that a dynamic training data set D has been obtained. When the BTC is equal to c i , the dynamic training data set can be expressed as where i = 1, 2, 3, · · · k, and then the regression strategy iŝ The vector x = [u s , u r , τ] T contains all the characteristic parameters. Here, the LM algorithm [35] can be used to solve the extreme value of Equation (10).

Dynamic Training Data Set Obtained by Experiments
D can be obtained by numerical simulation or experimental data. However, in practice, it is difficult to accurately simulate the sensor signals due to some uncertainty factors, such as the sensor coil winding and its package manufacturing process, the machining accuracy of the tip surface and the circuit design. Therefore, this paper adopts the experimental method ( Figure 3, Step 1 to 6) to obtain the dynamic training data set D under different BTC conditions. Sensors 2019, xx, 5 6 of 17 ECS is applied to measure the inlet section of centrifugal compressor in a turbojet engine. Many factors affect the rotor system vibration in normal operation of turbojet engine, and the BTC of centrifugal compressor inlet section cannot be controlled as required. In order to obtain accurate and comprehensive dynamic training data set D during the engine test, the compressor needs to be separated from the engine and test separately. Meanwhile, in order to facilitate driving, the centrifugal compressor is made into an equivalent model ( Figure 3, which keeps the blade shape, material and surface treatment consistent with the original compressor, ensuring that the SPA of ECS under the same BTC is the same as the original engine environment). The equivalent device includes the permanent magnet brushless motor, the outer cover, ECS, and the equivalent model of compressor. The finite element model of the rotor system [36] based on the Timoshenko beam element is used to analyze the rotor system [37][38][39]; the results show that the first critical speed of the rotor system is 200,000 r/min, which far exceeds the subsequent tests. The rotor has been well balanced, the maximum radial vibration during the experiment is 0.3 g (g = 9.8 m/s 2 ), so its lateral vibration can be ignored. When the rotor system is static, the clearance between the equivalent model blade tip and the ECS head is adjusted to 0.5 mm (it changes when the turbomachinery rotates at high speed because the blade will be stretched by centrifugal force). The rotor system is slowly accelerated to the specified speed and then similarly slowed down to the initial speed. The output SPA (Figure 4a) of the sensor corresponding to each blade during the two processes is measured. Then, using the same way, when the static BTC is another value, the SPA sequence u i and rotation speed sequence n i of each blade can be obtained, respectively.  ECS is applied to measure the inlet section of centrifugal compressor in a turbojet engine. Many factors affect the rotor system vibration in normal operation of turbojet engine, and the BTC of centrifugal compressor inlet section cannot be controlled as required. In order to obtain accurate and comprehensive dynamic training data set D during the engine test, the compressor needs to be separated from the engine and test separately. Meanwhile, in order to facilitate driving, the centrifugal compressor is made into an equivalent model ( Figure 3, which keeps the blade shape, material and surface treatment consistent with the original compressor, ensuring that the SPA of ECS under the same BTC is the same as the original engine environment). The equivalent device includes the permanent magnet brushless motor, the outer cover, ECS, and the equivalent model of compressor. The finite element model of the rotor system [36] based on the Timoshenko beam element is used to analyze the rotor system [37][38][39]; the results show that the first critical speed of the rotor system is 200,000 r/min, which far exceeds the subsequent tests. The rotor has been well balanced, the maximum radial vibration during the experiment is 0.3 g (g = 9.8 m/s 2 ), so its lateral vibration can be ignored. When the rotor system is static, the clearance between the equivalent model blade tip and the ECS head is adjusted to 0.5 mm (it changes when the turbomachinery rotates at high speed because the blade will be stretched by centrifugal force). The rotor system is slowly accelerated to the specified speed and then similarly slowed down to the initial speed. The output SPA (Figure 4a) of the sensor corresponding to each blade during the two processes is measured. Then, using the same way, when the static BTC is another value, the SPA sequence u i and rotation speed sequence n i of each blade can be obtained, respectively.  ECS is applied to measure the inlet section of centrifugal compressor in a turbojet engine. Many factors affect the rotor system vibration in normal operation of turbojet engine, and the BTC of centrifugal compressor inlet section cannot be controlled as required. In order to obtain accurate and comprehensive dynamic training data set D during the engine test, the compressor needs to be separated from the engine and test separately. Meanwhile, in order to facilitate driving, the centrifugal compressor is made into an equivalent model ( Figure 3, which keeps the blade shape, material and surface treatment consistent with the original compressor, ensuring that the SPA of ECS under the same BTC is the same as the original engine environment). The equivalent device includes the permanent magnet brushless motor, the outer cover, ECS, and the equivalent model of compressor. The finite element model of the rotor system [36] based on the Timoshenko beam element is used to analyze the rotor system [37][38][39]; the results show that the first critical speed of the rotor system is 200,000 r/min, which far exceeds the subsequent tests. The rotor has been well balanced, the maximum radial vibration during the experiment is 0.3 g (g = 9.8 m/s 2 ), so its lateral vibration can be ignored. When the rotor system is static, the clearance between the equivalent model blade tip and the ECS head is adjusted to 0.5 mm (it changes when the turbomachinery rotates at high speed because the blade will be stretched by centrifugal force). The rotor system is slowly accelerated to the specified speed and then similarly slowed down to the initial speed. The output SPA (Figure 4a) of the sensor corresponding to each blade during the two processes is measured. Then, using the same way, when the static BTC is another value, the SPA sequence u i and rotation speed sequence n i of each blade can be obtained, respectively.  There are certain differences in the output SPA sequences of the sensor corresponding to various blades ( Figure 4a) due to manufacturing variations. Nevertheless, during the process of rotor acceleration and deceleration, the SPA curves corresponding to various blades are almost the same, indicating that the operation of the acceleration has no influence on signal measurement. The signal amplitude corresponding to each rotation speed can be considered as the measured value at this uniform speed. In addition, the SPA curves corresponding to various blades remain separated and clear, implying that the measurement accuracy is sufficient, and the rotor system has no whirling and is always in the state of synchronous vibration.
The average value of the data which are obtained during the acceleration and deceleration process is calculated to get the sequence of the rotation speed n i and the pulse amplitude output u i of the sensor, corresponding to each blade when the static BTC is 0.5 mm or other value. The output SPA curves of the sensor corresponding to various blades in the time domain are arranged with the rotational speed as the abscissa, so the SPA-speed curves corresponding to various blades are obtained (Figure 4b).
The SPA decreases with increasing speed, and the larger the BTC, the faster the amplitude decreases. There are certain differences in the output SPA sequences of the sensor corresponding to various blades (Figure 4a) due to manufacturing variations. Nevertheless, during the process of rotor acceleration and deceleration, the SPA curves corresponding to various blades are almost the same, indicating that the operation of the acceleration has no influence on signal measurement. The signal amplitude corresponding to each rotation speed can be considered as the measured value at this uniform speed. In addition, the SPA curves corresponding to various blades remain separated and clear, implying that the measurement accuracy is sufficient, and the rotor system has no whirling and is always in the state of synchronous vibration.
The average value of the data which are obtained during the acceleration and deceleration process is calculated to get the sequence of the rotation speed n i and the pulse amplitude output u i of the sensor, corresponding to each blade when the static BTC is 0.5 mm or other value. The output SPA curves of the sensor corresponding to various blades in the time domain are arranged with the rotational speed as the abscissa, so the SPA-speed curves corresponding to various blades are obtained (Figure 4b).
The SPA decreases with increasing speed, and the larger the BTC, the faster the amplitude decreases.  As mentioned above, the rotating bladed disk will be stretched by centrifugal force. When rotating speed is n, the blade tip displacement δ satisfies the relation δ ∝ n 2 [40], let δ = k ω n 2 . Use the finite element method (FEM) (Figure 6) to get the result of k ω . The calculation formula of dynamic BTC c (dynamic) i when the bladed disk rotates is given by The SPA-speed curves of static BTC are corrected with centrifugal force deformation where f int is a spline interpolation [41] function. Then, the final pulse amplitude-speed curves for the different BTCs are obtained ( Figure 5, solid line). Thus, the dynamic training data set is obtained by using the method mentioned above. As mentioned above, the rotating bladed disk will be stretched by centrifugal force. When rotating speed is n, the blade tip displacement δ satisfies the relation δ ∝ n 2 [40], let δ = k ω n 2 . Use the finite element method (FEM) (Figure 6) to get the result of k ω . The calculation formula of dynamic BTC c (dynamic) i when the bladed disk rotates is given by The SPA-speed curves of static BTC are corrected with centrifugal force deformation where f int is a spline interpolation [41] function. Then, the final pulse amplitude-speed curves for the different BTCs are obtained ( Figure 5, solid line). Thus, the dynamic training data set is obtained by using the method mentioned above.

Calculation of Model Characteristic Parameters
According to the dynamic training data set D obtained by the test, the characteristic parameters of the SAM can be calculated using the LM algorithm. The calculated SAM curves match very well with the test data (Figure 7), and the goodness of fit (R 2 ) can reach R 2 ≥ 0.999 (Table 1).  The characteristic parameter u s in the SAM indicates the SPA value as t m approaches ∞. It also matches quite well (Figure 8) with the sensor characteristic curve measured when the engine rotor is static ( it means that t m is large enough ).

Calculation of Model Characteristic Parameters
According to the dynamic training data set D obtained by the test, the characteristic parameters of the SAM can be calculated using the LM algorithm. The calculated SAM curves match very well with the test data (Figure 7), and the goodness of fit (R 2 ) can reach R 2 ≥ 0.999 (Table 1).

Calculation of Model Characteristic Parameters
According to the dynamic training data set D obtained by the test, the characteristic parameters of the SAM can be calculated using the LM algorithm. The calculated SAM curves match very well with the test data (Figure 7), and the goodness of fit (R 2 ) can reach R 2 ≥ 0.999 (Table 1).  The characteristic parameter u s in the SAM indicates the SPA value as t m approaches ∞. It also matches quite well (Figure 8) with the sensor characteristic curve measured when the engine rotor is static ( it means that t m is large enough ). The characteristic parameter u s in the SAM indicates the SPA value as t m approaches ∞. It also matches quite well (Figure 8) with the sensor characteristic curve measured when the engine rotor is static ( it means that t m is large enough ). The characteristic parameters τ means response time of electromagnetic interaction between ECS and tip. The response frequency of ECS is related to BTC. As shown in Table 1, τ increases rapidly with increasing BTC, partial derivative of τ with respect to BTC (∂τ/∂c) increases with increasing BTC. When BTC changes from 0.4 mm to 0.6 mm, the response time becomes nine times. When the BTC is equal to 0.5 mm, the speed varies from 0 to 20,000 r/min, and the SPA is reduced to 67%.
The measurement accuracy of ECS is related to both BTC and speed range (Figure 7). When the speed is less than 5000 r/min, partial derivative of SPA with respect to BTC (∂u/∂c) increases with increasing BTC (Table 1), it means that SPA reflects more sensitively to BTC change with increasing BTC. When the speed is higher than 15,000 r/min, the signal amplitude reflects less sensitively to BTC change with increasing BTC. These are significant references to designing appropriate sensors according to the measured object.

BTC Measurement and Geometric Evaluation Method
In this section, firstly, a method to construct a set of the SPA-speed curves based on the SAM is introduced. Then, based on spline interpolation, we calculate BTC according to the experimental data. Next, the geometric constraint equation of the BTC distribution corresponding to all blades when the rotor deviates from the vibration equilibrium position is established. Finally, a method based on the equation to solve the shaft position vector and evaluate the accuracy of the measured BTC data is presented.

BTC Measurement Method
As mentioned in Section 2, when the BTC is a certain value c i , the dynamic training data set D can be obtained by experiments; then, the value of x = [u s , u r , τ] T of the characteristic parameter in u = v(n, x) can be calculated by nonlinear regression. Given a monotonically increasing sequence of BTC as c = {c 1 , c 2 , c 3 , . . . , c k }, we can get a set of SPA-speed curves (Figure 9).
Assuming that a signal amplitude of a blade in this experiment is u test , the measured speed of rotor is n test . According to the definition of G, when the rotational speed is n test , the signal amplitude sequence u (n test ) corresponding to all the BTC values c = {c 1 , c 2 , c 3 , . . . , c k } can be obtained, The characteristic parameters τ means response time of electromagnetic interaction between ECS and tip. The response frequency of ECS is related to BTC. As shown in Table 1, τ increases rapidly with increasing BTC, partial derivative of τ with respect to BTC (∂τ/∂c) increases with increasing BTC. When BTC changes from 0.4 mm to 0.6 mm, the response time becomes nine times. When the BTC is equal to 0.5 mm, the speed varies from 0 to 20,000 r/min, and the SPA is reduced to 67%.
The measurement accuracy of ECS is related to both BTC and speed range (Figure 7). When the speed is less than 5000 r/min, partial derivative of SPA with respect to BTC (∂u/∂c) increases with increasing BTC (Table 1), it means that SPA reflects more sensitively to BTC change with increasing BTC. When the speed is higher than 15,000 r/min, the signal amplitude reflects less sensitively to BTC change with increasing BTC. These are significant references to designing appropriate sensors according to the measured object.

BTC Measurement and Geometric Evaluation Method
In this section, firstly, a method to construct a set of the SPA-speed curves based on the SAM is introduced. Then, based on spline interpolation, we calculate BTC according to the experimental data. Next, the geometric constraint equation of the BTC distribution corresponding to all blades when the rotor deviates from the vibration equilibrium position is established. Finally, a method based on the equation to solve the shaft position vector and evaluate the accuracy of the measured BTC data is presented.

BTC Measurement Method
As mentioned in Section 2, when the BTC is a certain value c i , the dynamic training data set D can be obtained by experiments; then, the value of x = [u s , u r , τ] T of the characteristic parameter in u = v(n, x) can be calculated by nonlinear regression. Given a monotonically increasing sequence of BTC as c = {c 1 , c 2 , c 3 , . . . , c k }, we can get a set of SPA-speed curves (Figure 9).
Assuming that a signal amplitude of a blade in this experiment is u test , the measured speed of rotor is n test . According to the definition of G, when the rotational speed is n test , the signal amplitude sequence u(n test ) corresponding to all the BTC values c = {c 1 , c 2 , c 3 , . . . , c k } can be obtained, Then, the BTC value c x corresponding to the SPA u test satisfies ε is a given relative precision (10 −6 ). Then, the corresponding BTC value c x of the blade can be solved.
ε is a given relative precision (10 −6 ). Then, the corresponding BTC value c x of the blade can be solved.

Geometric Evaluation Method
When the rotor system is running at high speed, the turbomachinery vibrates laterally due to mass imbalance or other non-axisymmetric forces. With the establishment of the polar coordinate system attached to a rotor shaft, the rotor equilibrium position O 1 is taken as the origin; the first blade is taken as 0-pole angular position, and the phase of the mass imbalance force is denoted as θ 0 ( Figure 10). Assuming that the rotor has no lateral vibration, the trajectory of the blade tips is a circle, with O 1 as the origin and r tip as the radius. The inner wall of the casing is a circle, with radius of r casing , so r casing -r tip is BTC. In general, the BTC is approximately equal to one percent of r tip . When the rotor exhibits lateral vibration, the phase of the vibration is θ 0 , and the amplitude is e. At this time, the position of the rotating shaft is O 2 . Thus, the amount of BTC reduction on the i-th blade position can be expressed as Mark the BTC measurement value corresponding to the i-th blade as c (m) i , and the reduction of the BTC corresponding to the i-th blade due to the rotor lateral vibration calculated from the experimental data can be expressed as: The BTC reduction of the i-th blade ∆d obtained from the experimental data. Therefore, the measured data require satisfying the geometric constraint equation

Geometric Evaluation Method
When the rotor system is running at high speed, the turbomachinery vibrates laterally due to mass imbalance or other non-axisymmetric forces. With the establishment of the polar coordinate system attached to a rotor shaft, the rotor equilibrium position O 1 is taken as the origin; the first blade is taken as 0-pole angular position, and the phase of the mass imbalance force is denoted as θ 0 ( Figure 10). Assuming that the rotor has no lateral vibration, the trajectory of the blade tips is a circle, with O 1 as the origin and r tip as the radius. The inner wall of the casing is a circle, with radius of r casing , so r casing -r tip is BTC. In general, the BTC is approximately equal to one percent of r tip . When the rotor exhibits lateral vibration, the phase of the vibration is θ 0 , and the amplitude is e. At this time, the position of the rotating shaft is O 2 . Thus, the amount of BTC reduction on the i-th blade position can be expressed as Mark the BTC measurement value corresponding to the i-th blade as c (m) i , and the reduction of the BTC corresponding to the i-th blade due to the rotor lateral vibration calculated from the experimental data can be expressed as: The BTC reduction of the i-th blade ∆d obtained from the experimental data. Therefore, the measured data require satisfying the geometric constraint equation Let the rotor axis position vector be p = (e, θ). Then, it is estimated bŷ in order to solve the SAM characteristic parameters, and the LM method can also be used to solvep. In addition, R 2 p (Goodness of Fit) is used to evaluate the measurement accuracy of BTC: Sensors 2019, xx, 5 11 of 17 Let the rotor axis position vector be p = (e, θ). Then, it is estimated bŷ in order to solve the SAM characteristic parameters, and the LM method can also be used to solvep. In addition, R 2 p (Goodness of Fit) is used to evaluate the measurement accuracy of BTC: (20) Figure 10. Geometric distribution of BTC.

Engine Test
The aforementioned method of measuring BTC using ECS is applied to measure the BTC of the inlet section of the centrifugal compressor (Table 2), and the geometric constraint equation is used to evaluate the measured data during the turbojet engine running process. The test of measuring BTC using ECS is performed on the turbojet engine test rig (Figure 11). The point to be measured locates at the inlet section of the centrifugal compressor, with a radius of about 60 mm. In order to make the measurement results more obvious, without dynamic-balance adjustment for the compressor side, and the mass imbalance is set to be five times the allowable value specified in the standards, whose phase is 40 • (Figure 12, make the No. 1 blade position be 0 • phase). The laser-reflecting patch is mounted on the 0 • phase. The laser sensor sends a pulse signal every one revolution of the engine to establish a one-to-one correspondence between the ECS output pulse and the compressor blade.

Engine Test
The aforementioned method of measuring BTC using ECS is applied to measure the BTC of the inlet section of the centrifugal compressor (Table 2), and the geometric constraint equation is used to evaluate the measured data during the turbojet engine running process. The test of measuring BTC using ECS is performed on the turbojet engine test rig ( Figure 11). The point to be measured locates at the inlet section of the centrifugal compressor, with a radius of about 60 mm. In order to make the measurement results more obvious, without dynamic-balance adjustment for the compressor side, and the mass imbalance is set to be five times the allowable value specified in the standards, whose phase is 40 • (Figure 12, make the No. 1 blade position be 0 • phase). The laser-reflecting patch is mounted on the 0 • phase. The laser sensor sends a pulse signal every one revolution of the engine to establish a one-to-one correspondence between the ECS output pulse and the compressor blade.      A segment of the sensor output signal data is analyzed here. The engine running speed ranges from 8000 r/min to 21,000 r/min in this section. The SPAs of ECS signal corresponding to various blades are extracted (Figure 13a), and the BTC values of various blades are solved using the SAM (Figure 13b). When the engine runs between 8000 r/min and 16,000 r/min, the relative position of BTC curves corresponding to various blades are almost unchanged. No. 1, No. 2 and No. 3 blades correspond to smaller BTCs whose values decrease with increasing speed, while No. 7 and No. 8 blades correspond to larger BTCs whose values increase with increasing speed. The deflection direction of the rotor points to No. 2 blade, which is consistent with the phase of the mass imbalance force (phase 40 • , Figure 12). Because the aerodynamic force is small at low speed, the lateral vibration of the rotor system is synchronized with the rotor mass imbalance force. When the engine speed is above 20,000 r/min, the BTC curves corresponding to various blades are obviously staggered, and the lateral vibration of the rotor system is no longer synchronized with the rotor mass imbalance force, which leads to low frequency whirl.
blades are extracted (Figure 13a), and the BTC values of various blades are solved using the SAM (Figure 13b). When the engine runs between 8000 r/min and 16,000 r/min, the relative position of BTC curves corresponding to various blades are almost unchanged. No. 1, No. 2 and No. 3 blades correspond to smaller BTCs whose values decrease with increasing speed, while No. 7 and No. 8 blades correspond to larger BTCs whose values increase with increasing speed. The deflection direction of the rotor points to No. 2 blade, which is consistent with the phase of the mass imbalance force (phase 40 • , Figure 12). Because the aerodynamic force is small at low speed, the lateral vibration of the rotor system is synchronized with the rotor mass imbalance force. When the engine speed is above 20,000 r/min, the BTC curves corresponding to various blades are obviously staggered, and the lateral vibration of the rotor system is no longer synchronized with the rotor mass imbalance force, which leads to low frequency whirl.   blades are extracted (Figure 13a), and the BTC values of various blades are solved using the SAM (Figure 13b). When the engine runs between 8000 r/min and 16,000 r/min, the relative position of BTC curves corresponding to various blades are almost unchanged. No. 1, No. 2 and No. 3 blades correspond to smaller BTCs whose values decrease with increasing speed, while No. 7 and No. 8 blades correspond to larger BTCs whose values increase with increasing speed. The deflection direction of the rotor points to No. 2 blade, which is consistent with the phase of the mass imbalance force (phase 40 • , Figure 12). Because the aerodynamic force is small at low speed, the lateral vibration of the rotor system is synchronized with the rotor mass imbalance force. When the engine speed is above 20,000 r/min, the BTC curves corresponding to various blades are obviously staggered, and the lateral vibration of the rotor system is no longer synchronized with the rotor mass imbalance force, which leads to low frequency whirl.   After having solved the BTCs corresponding to various blades of the compressor, the shaft position vector p = (e, θ) can be parsed according to the geometric constraint equation. Firstly, as the rotational speed increases (Figure 14, from 11,000 r/min to 17,000 r/min), the shaft lateral vibration amplitude e gradually increases, and R 2 p gradually approaches 1. At a lower speed of 11,000 r/min (Figure 14a), the shaft lateral vibration amplitude is 0.019 mm. At this time, there is a significant error (R 2 p = 0.936) between the measured data and the geometric constraint equation. At a higher speed of 17,000 r/min (Figure 14d), the shaft lateral vibration amplitude is 0.053 mm, and the measured data matches very well with the geometric constraint equation (R 2 p = 0.993). Then, when the rotor reflects a slight whirl and the rotational speed is almost unchanged (Figure 15, 20,500 r/min), the rotor vibration amplitude varies between 0.122 and 0.123 mm while the phase changes from 46 • to 84 • (46 • , 59 • , 72 • , 84 • ), the measured data and the geometric constraint equation matches very well (R 2 p > 0.98). Therefore, it can be considered that the accuracy of BTC measurement is 0.01 mm. position vector p = (e, θ) can be parsed according to the geometric constraint equation. Firstly, as the rotational speed increases (Figure 14, from 11,000 r/min to 17,000 r/min), the shaft lateral vibration amplitude e gradually increases, and R 2 p gradually approaches 1. At a lower speed of 11,000 r/min (Figure 14a), the shaft lateral vibration amplitude is 0.019 mm. At this time, there is a significant error (R 2 p = 0.936) between the measured data and the geometric constraint equation. At a higher speed of 17,000 r/min (Figure 14d), the shaft lateral vibration amplitude is 0.053 mm, and the measured data matches very well with the geometric constraint equation (R 2 p = 0.993). Then, when the rotor reflects a slight whirl and the rotational speed is almost unchanged (Figure 15, 20,500 r/min), the rotor vibration amplitude varies between 0.122 and 0.123 mm while the phase changes from 46 • to 84 • (46 • , 59 • , 72 • , 84 • ), the measured data and the geometric constraint equation matches very well (R 2 p > 0.98). Therefore, it can be considered that the accuracy of BTC measurement is 0.01 mm.

Conclusions
BTC measurement technology is one of the key technologies in engine health management and BTC active control. This article focuses on BTC measurement using ECS; on the basis of elaborating the principle of measuring BTC using ECS, a speed adjustment model (SAM) is proposed, and the measurement method of BTC is developed based on this model. Combined with the structural characteristics of turbomachinery, a method for evaluating measurement results by using a geometric constraint equation is proposed. The key results are summarized as follows: • The relationship between response time of the ECS measurement loop and BTC is quantitatively studied by an experimental method, and response time increases rapidly with increasing BTC. When BTC changes from 0.4 mm to 0.6 mm, the response time becomes nine times. The relationship between the output signal and the rotational speed is also quantitatively studied. As the speed increases, the signal amplitude decreases. When the BTC is equal to 0.5 mm, the speed varies from 0 to 20,000 r/min, the SPA is reduced to 67%.

Conclusions
BTC measurement technology is one of the key technologies in engine health management and BTC active control. This article focuses on BTC measurement using ECS; on the basis of elaborating the principle of measuring BTC using ECS, a speed adjustment model (SAM) is proposed, and the measurement method of BTC is developed based on this model. Combined with the structural characteristics of turbomachinery, a method for evaluating measurement results by using a geometric constraint equation is proposed. The key results are summarized as follows:

•
The relationship between response time of the ECS measurement loop and BTC is quantitatively studied by an experimental method, and response time increases rapidly with increasing BTC. When BTC changes from 0.4 mm to 0.6 mm, the response time becomes nine times. The relationship between the output signal and the rotational speed is also quantitatively studied. As the speed increases, the signal amplitude decreases. When the BTC is equal to 0.5 mm, the speed varies from 0 to 20,000 r/min, the SPA is reduced to 67%.
• A high-precision SAM for measuring BTC using ECS is proposed. The characteristic parameters of the model are solved by the experimental method. The goodness of fit between the SAM and the dynamic training data set is greater than 0.999. The explicit physical meaning of the model characteristic parameters is given, which is of great significance for the rational design and accurate usage of ECS.
• A BTCMM based on the SAM is proposed and applied to the BTC measurement of the compressor inlet during the engine test. The experimental results show that this method has the repeatability precision of 0.01 mm in the actual test.