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Design of high accuracy cylindrical profile measurement model for low-pressure turbine shaft of aero engine
Aerospace Science and Technology 95 (2019) 105442
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Aerospace Science and Technology www.elsevier.com/locate/aescte
Design of high accuracy cylindrical profile measurement model for low-pressure turbine shaft of aero engine Chuanzhi Sun a,b , Baosheng Wang a,b , Yongmeng Liu a,b,∗ , Xiaoming Wang a,b , Chengtian Li a,b , Hongye Wang a,b , Jiubin Tan a,b a
Center of Ultra-precision Optoelectronic Instrument Engineering, Harbin Institute of Technology, Harbin 150080, China Key Lab of Ultra-precision Intelligent Instrumentation Engineering (Harbin Institute of Technology), Ministry of Industry and Information Technology, Harbin 150080, China b
a r t i c l e
i n f o
Article history: Received 20 April 2019 Accepted 27 September 2019 Available online 30 September 2019 Keywords: Aero engine Cylindrical profile measurement LPT shaft
a b s t r a c t Coaxiality and cylindricity are the important geometric parameters of the low-pressure turbine (LPT) shaft. And the measurement accuracy of coaxiality and cylindricity directly affect the rotary characteristics of the aero engine. Therefore, a cylindrical profile measurement model with five systematic errors is designed to improve the coaxiality and cylindricity measurement accuracy of the low-pressure turbine shaft in this paper, in which eccentricity, probe offset, probe radius, geometric axis tilt and guide rail tilt are considered. Besides, the influence of systematic error and the shaft radius on the residual error for the stepped low-pressure turbine shaft is analyzed as well. The evaluation results of coaxiality and cylindricity are obtained based on different measurement strategies and models. In order to verify the effectiveness of the cylindrical profile measurement model with five systematic errors in the paper, a rotary measuring instrument with high precision is built. Compared with the traditional cylindrical profile measurement model with two systematic errors, the measurement accuracy of the coaxiality and cylindricity by the cylindrical profile measurement model with five systematic errors proposed in this paper are improved by 2.9 μm and 8.18 μm, respectively in the condition of the optimal measurement strategy for the LPT shaft with large radius. The proposed method is suitable for small probe radius and large eccentricity error, probe offset error, geometric axis tilt error and guide rail tilt error, especially for the LPT shaft with large radius. The proposed method can be applied to error separation and tolerance allocation for multistage rotor. © 2019 Elsevier Masson SAS. All rights reserved.
1. Introduction Low-pressure turbine shaft is the key rotating component of aero engine and it runs through the core machine. The quality of the LPT shaft directly affects the rotational characteristics of the aero engine [1,2]. The LPT shaft has the mechanical structural characteristics of the stepped rotary shaft. The coaxiality and cylindricity are two important geometric parameters for the LPT shaft, which respectively reflect the rotation characteristics and surface contact characteristics. The LPT shaft with low machining precision for coaxiality and cylindricity will become curved and the contact characteristics will be worse, and then will cause a large amount of vibration [3–5]. The improvement of the design accuracy of aero engine puts higher demands on the measurement accuracy of the
*
Corresponding author. E-mail address: [email protected] (Y. Liu).
https://doi.org/10.1016/j.ast.2019.105442 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.
low-pressure turbine shafts. The LPT shaft of precision turbofan aero engine, for example, its coaxiality and cylindricity error are required to be controlled within 1 μm [6,7]. Although the measuring instrument has a high accuracy, the existence of the systematic errors will bring a great influence on the measurement accuracy of the coaxiality and cylindricity for the LPT shaft, and the results may contain the systematic errors that have not been separated [8]. Therefore, in order to improve the rotation characteristics of the aero engine, the measurement accuracy of the coaxiality and cylindricity need to be improved for the LPT shaft. In recent years, researchers have done a lot of work on the measurement of coaxiality and cylindricity. The Limacon measurement model was proposed which considers the effect of the eccentricity error caused by the offset of the workpiece geometry axis relative to the rotation axis of the turntable on the circular profile measurement [9,10]. The probe offset error also has the influence on the circular profile measurement [11,12], which is caused by the sensor measurement line not pass the measured
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Fig. 1. The cylindrical profile measurement model of low-pressure turbine shaft with five systematic errors: (a) The cylindrical profile measurement model; (b) The jth measurement section model.
center of rotation. When the measured object is the stepped rotation shaft, the tilt error of the workpiece can cause noncoincidence between the workpiece geometry axis and the rotation axis of the turntable, which brings the secondary eccentricity error of the measured section. The traditional cylindrical profile measurement model with two systematic errors was proposed to improve the accuracy of cylindrical profile measurement, and the initial eccentricity and geometric axis tilt are considered in the model [13, 14]. In addition, many scholars have begun to make new attempts to improve the cylindrical profiles measurement accuracy from the perspective of error separation [15–17]. Zhao et al. proposed a new single-step spatial rotation error separation technique (SSEST). In this model, the measurement accuracy of cylindricity is improved by eliminating the spatial rotation error of the instrument spindle [18]. Yang et al. proposed the improved harmony search algorithm for cylindricity error evaluation [19]. Lai et al. applied adaptive genetic algorithms and heuristic genetic algorithms to cylindrical profile measurement to improve the accuracy of measurement results [20]. The above methods introduced the influence of eccentricity, sensor probe offset and the secondary eccentricity of the workpiece on the cylindrical profile measurement. However, when there are both eccentricity and probe offset, and the tip head of probe cannot be processed into an ideal point, the probe radius will bring new measurement errors into the cylindrical profile measurement results. Geometric axis tilt of the workpiece will cause the secondary eccentricity and it will also cause the measuring section ovalization. And in the precision measurement of coaxiality and cylindricity, the axis of motion of the guide rail cannot be completely parallel to the axis of rotation [21,22], which not only leads to the errors in the measurement of the cylindrical profile, but also causes a secondary offset of the sensor probe. Therefore, this paper designs a cylindrical profile measurement model with five systematic errors: eccentricity, probe offset, probe radius, geometric axis tilt and guide rail tilt. Based on the proposed model in this paper, different measurement strategies on coaxiality and cylindricity are proposed to improve the coaxiality and cylindricity measurement accuracy of the low-pressure turbine shaft. The structure of this paper is as follows: In Section 2 of this paper, the cylindrical profile measurement model with five systematic errors is established and three coaxiality measurement strategies and two cylindricity measurement strategies are designed. In Section 3, in order to analyze the advantages of the cylindrical profile measurement model with five systematic errors in the measurement of coaxiality and cylindricity, the influence of systematic errors and workpiece radius on the measurement results
is analyzed. The influence of systematic error on the coaxiality and cylindricity measurement results under different measurement strategies for the LPT shaft is also analyzed. In Section 4, the effectiveness of the proposed method for coaxiality and cylindricity measurement is verified by experiments on a rotary measuring instrument with high precision. Finally, the discussions and conclusions are given in Section 5. 2. The cylindrical profile measurement method of the low-pressure turbine shaft 2.1. Establishment of the cylindrical profile measurement model with five systematic errors The traditional cylindrical profile measurement model with two systematic errors considers the influence of initial eccentricity error and geometric axis tilt error. The probe offset, the probe radius, the guide rail tilt and the sectional ovalization caused by the geometric axis tilt are also considered in this paper based on the traditional model. The cylindrical profile measurement model with five systematic errors shown in Fig. 1 is established. In Fig. 1(a), for the jth measurement section, O j1 is the rotation center of the turntable, O j2 is the geometric center of the LPT shaft. roj and rlj are the lengths of semi-minor and -major axis of the fitting ellipse. Ψ is the angle between the guide rail and the rotation axis. The initial eccentricity and eccentric angle are e 0 and α0 , respectively. The jth measurement section model of the LPT shaft is shown in Fig. 1(b). e j is the amount of eccentricity, α j is the eccentric angle, r is the probe radius, and d j1 is the probe’s own offset. ε is the angle between the inclination error of the guide rail and the measurement direction. t j is the moving distance of the probe center caused by the tilt of the guide rail and its vertical measurement direction component is d j2 . d j is the total offset of the probe, O j3 is the instantaneous rotation center of the measurement line and O j4 is the probe center. β j is the angle between the major axis direction of the fitting ellipse and the X -direction. z j is the height of the jth measurement section. For the ith sample point, ρi j is the distance between the probe and O j3 . r i j is the distance between the fitting ellipse and O j2 . Δr i j is the LPT shaft machining error. θi j is the sample angle of O j1 . ϕi j is the sample angle of O j2 . τi j is the angle between the measurement direction and the O j2 O j4 direction. The geometric axis tilt angle γ not only affects the eccentricity of each measurement section, but also causes the section profile to be ovalized. For the jth measurement section, the define of the geometric axis tilt error is shown in Fig. 2.
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Equation (4) is brought into equation (5)
⎛
⎞ ⎜ d j1 + z j tan ψ sin ε + e j sin θi j − α j ⎟ ⎟ ϕi j = θi j + arcsin ⎜ ⎝ ⎠ cos2 ϕi j −β j 2 + Δri j + roj + sin ϕ − β r i j j 2 cos γ (6) In the actual measurement, the inclination angle γ is small (γ < 3◦ ), and the angle deviation caused by it is very small and can be ignored. The equations (4) and (6) can be brought into equation (2) and power series expansion is performed Fig. 2. Geometric axis tilt error of the jth measurement section.
As shown in Fig. 2, the relationship between the semi-minor and -major axis of the fitting ellipse can be seen as
rlj =
roj
(1)
cos γ
The direction vector of the geometric axis of the LPT shaft caused by the tilt angle γ is (l , m , n ). The measurement equation can be obtained in the model from Fig. 1.
⎧ ρi j = ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ Δri j + ri j + r − e j sin θi j − α j + d j1 + z j tan ψ sin ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + e j cos θi j − α j + z j tan ψ cos ε − r ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ e j = e 0 cos α0 + lz j + e 0 sin α0 + mz j
e sin α0 + mz j ⎪ ⎪ α j = arctan 0 ⎪ ⎪ e 0 cos α0 + lz j ⎪ ⎪ ⎪ ⎪ ⎪ l m ⎪ ⎪ ⎪l = , m = ⎪ ⎪ n n ⎪ ⎪ √ ⎪ ⎪ ⎪ l 2 + m 2 ⎪ ⎩ tan γ = n
⎧ ρi j = e j cos θi j − α j + z j tan ψ cos ε + Δri j + ξi j ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ e j sin θi j − α j + d j1 + z j tan ψ sin ε ⎪ ⎪ − ⎪ ⎪ ⎪ 2(Δr i j + ξi j + r ) ⎪ ⎪ ⎪ ⎪ ⎨ cos2 ηi j ξi j = roj + sin2 ηi j 2 ⎪ cos γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d j1 + z j tan ψ sin ε + e j sin θi j − α j ⎪ ⎪ ⎪ ηi j = θi j + arcsin ⎪ ⎪ Δri j + roj + r ⎪ ⎪ ⎪ ⎩ − βj (7) Where i = 1, 2, ..., k, j = 1, 2, ..., p, k is the number of sample points and p is the number of measurement sections. The traditional cylindrical profile measurement model with two systematic errors can be expressed as
ρiLimacon = Δri j + roj + e j cos θi j − α j j
(8)
In order to further analyze the advantages of the cylindrical profile measurement model with five systematic errors, the equations (7) and (8) are subtracted, and the residual error is shown in equation (9)
δi j = ρi j − ρiLimacon j
Taking O j2 as the coordinate center, the fitting ellipse equation is expressed as
= z j tan ψ cos ε + ξi j − roj 2 e j sin θi j − α j + d j1 + z j tan ψ sin ε − 2(Δr i j + ξi j + r )
Then the maximum residual error is shown in equation (10)
(2)
xi j = rlj cos y i j = roj sin
ϕi j − β j
ϕi j − β j
(3)
Where xi j and y i j are the horizontal and vertical coordinates of the ith sample point in the fitting ellipse. From equations (1) and (3), the distance r i j between the fitting ellipse and O j2 is expressed as
ri j =
x2i j
+
y 2i j
= roj
ϕi j − β j + sin2 ϕi j − β j 2 cos γ
cos2
(4)
As shown in Fig. 1(b), the relationship between the sample angle ϕi j of the geometric center O j2 and the sample angle θi j of the rotation center O j1 is expressed as
τi j = ϕi j − θi j
Δri j + ri j + r sin τi j = d j1 + z j tan ψ sin ε + e j sin θi j − α j
(5)
δmax = max δi j , i = 1, 2, · · · , k; j = 1, 2, · · · , p
(9)
(10)
If the guide rail tilt error and the geometric axis tilt error are very small, the sum of the first three terms of the equation (9) tends to be zero. And the fourth term dominates the residual error, which is mainly influenced by eccentricity, probe offset and probe radius. As the eccentricity and probe offset increase and with the decreasing probe radius, the residual error increases. With the increase of the guide rail tilt error and the geometry axis tilt error, the proportion of the first three items of the equation (9) increases, the residual error is mainly caused by the guide rail tilt error and the geometry axis tilt error. 2.2. The measurement strategy of coaxiality and cylindricity for the low-pressure turbine shaft According to the mechanical structure and working principle of the low-pressure turbine shaft of aero engine, three measurement strategies of coaxiality are proposed in Fig. 3. In Figs. 3(a) and (b), we can see both strategy 1 and strategy 2 select the lower cylinder
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Fig. 3. Coaxiality evaluation strategies: (a) Strategy 1; (b) Strategy 2; (c) Strategy 3.
Fig. 4. Cylindricity evaluation strategies: (a) Strategy 1; (b) Strategy 2.
as the reference element and the upper cylinder is chosen as the measured element. Strategy 1 uses the least squares central line of the reference element as the reference axis, and strategy 2 uses the least squares central line of the two regions of the reference elements as the reference axis. As shown in Fig. 3(c), the upper and lower regions of the LPT shaft are selected as the reference elements in strategy 3, with the least square central line of the two regions as the reference axis, and the remaining region of the LPT shaft as the measured element. This paper proposes two kinds of the LPT shaft cylindricity measurement strategies as shown in Fig. 4. Fig. 4(a) shows the least squares method for strategy 1. The least squares method is to find the ideal cylindrical surface that minimizes the sum of the squares of the points from the actual measured cylindrical surface to the ideal cylindrical surface. The minimum radius difference between the two coaxial ideal cylindrical surfaces which contain the actual measured cylindrical surface is the cylindricity error. Fig. 4(b) shows the minimum area method for strategy 2. When the minimum area method is used to evaluate the cylindricity error, the two coaxial ideal cylindrical surfaces are used to contain actual measured cylindrical surface. And when the difference in radius between the two ideal cylindrical surfaces is the smallest, the smallest difference in the radius is the cylindricity error. 3. Simulations 3.1. Influence of systematic error on profile measurement of the LPT shaft In order to further analyze the advantages of cylindrical profile measurement model with five systematic errors, compared with
the traditional model with two systematic errors, the influence of the multi-systematic error on the measurement results at different sections is analyzed. Assumption that the profile deviation caused by imperfect manufacturing Δr i j is the standard normal distribution, which is 0.001 mm, the standard deviation is one-third of Δr i j , the eccentric angle α is π /3, the angle β is π /6, the length of semi-minor axis of fitting ellipse ro is 10 mm, and the angle ε between the probe offset caused by the guide rail tilt and the measuring direction is π /6. The heights of the workpiece sections are taken as 0 mm, 50 mm, 100 mm and 200 mm, respectively, and the simulation is applied according to the equation (9). The results are shown in Fig. 5. It can be seen in Fig. 5 that under the same height of section, the residual error increases with the increase of eccentricity, probe offset, geometric axis tilt, and guide rail tilt errors and the decreases of probe radius. And in Fig. 5(a), as the systematic error level increases from Level 1 to Level 5, the maximum residual error increases from 0.1 μm to 13.4 μm. As shown in Figs. 5(a)-(d), under the same systematic error level, as the height of the measurement section increases, the residual error increases. If the systematic error level is Level 5, as the section height increases from 0 mm to 200 mm, the maximum residual error increases from 13.4 μm to 60.2 μm. And the maximum residual error position is near the vertical direction of the eccentric angle. 3.2. Influence of systematic error and the LPT shaft radius on profile measurement In this paper, according to equation (10), the influence of systematic error on the profile measurement of workpieces with different radii is analyzed. The simulation results on different height sections are shown in Fig. 6. For the sake of clearer description of the relationship between the residual error and the radius of the LPT shaft, the paper replaces the length of semi-minor axis of fitting ellipse with the radius of cylindrical component in the following description. As shown in Fig. 6, under the same component radius and measured height, the maximum residual error increases with the increases of the systematic error. As it is shown in Fig. 6(d), if the radius is 50 mm and the height is 200 mm, the maximum residual error increases from 2.5 μm to 19.9 μm as the systematic error increases from Level 1 to Level 5. The maximum residual error increases as the measured height increases at the same component radius and systematic error level. As shown in Figs. 6(a)-(d), if the component radius is 50 mm, the maximum residual error increases by 17.1 μm as the measured height increases by 200 mm in Level 5. The maximum residual error tends to decrease first and then increase as the component radius increases. In the falling phase, the maximum
C. Sun et al. / Aerospace Science and Technology 95 (2019) 105442
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Fig. 5. Residual error at different heights of object with multi-systematic error: (a) z = 0 mm; (b) z = 50 mm; (c) z = 100 mm; (d) z = 200 mm. Level 1: e = 1 μm, d = 50 μm, γ = 1 , r = 2.5 mm, Ψ = 1 ; Level 2: e = 5 μm, d = 100 μm, γ = 3 , r = 2 mm, Ψ = 3 ; Level 3: e = 10 μm, d = 200 μm, γ = 5 , r = 1.5 mm, Ψ = 8 ; Level 4: e = 20 μm, d = 300 μm, γ = 8 , r = 1 mm, Ψ = 12 ; Level 5: e = 30 μm, d = 500 μm, γ = 12 , r = 0.5 mm, Ψ = 15 .
Fig. 6. Simulation relationship diagram of systematic error and workpiece radius on maximum residual error: (a) z = 0 mm; (b) z = 50 mm; (c) z = 100 mm; (d) z = 200 mm. Level 1: e = 1 μm, d = 50 μm, γ = 10 , r = 2.5 mm, Ψ = 3 ; Level 2: e = 5 μm, d = 100 μm, γ = 30 , r = 2 mm, Ψ = 5 ; Level 3: e = 10 μm, d = 200 μm, γ = 50 , r = 1.5 mm, Ψ = 10 ; Level 4: e = 20 μm, d = 300 μm, γ = 100 , r = 1 mm, Ψ = 18 ; Level 5: e = 30 μm, d = 500 μm, γ = 150 , r = 0.5 mm, Ψ = 25 .
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Fig. 7. Coaxiality evaluation results under different measurement strategies: (a) Strategy 1; (b) Strategy 2; (c) Strategy 3. Level 1: e = 3 μm, d = 5 μm, γ = 5 , r = 2.5 mm, Ψ = 3 ; Level 2: e = 5 μm, d = 10 μm, γ = 10 , r = 2 mm, Ψ = 5 ; Level 3: e = 10 μm, d = 20 μm, γ = 15 , r = 1.5 mm, Ψ = 10 ; Level 4: e = 20 μm, d = 30 μm, γ = 20 , r = 1 mm, Ψ = 18 ; Level 5: e = 30 μm, d = 50 μm, γ = 25 , r = 0.5 mm, Ψ = 25 .
Fig. 8. Cylindricity evaluation results under different measurement strategies: (a) Strategy 1; (b) Strategy 2. Level 1: e = 3 μm, d = 5 μm, γ = 5 , r = 2.5 mm, Ψ = 3 ; Level 2: e = 5 μm, d = 10 μm, γ = 10 , r = 2 mm, Ψ = 5 ; Level 3: e = 10 μm, d = 20 μm, γ = 15 , r = 1.5 mm, Ψ = 10 ; Level 4: e = 20 μm, d = 30 μm, γ = 20 , r = 1 mm, Ψ = 18 ; Level 5: e = 30 μm, d = 50 μm, γ = 25 , r = 0.5 mm, Ψ = 25 .
residual error is mainly affected by eccentricity, probe offset, and probe radius. As the component radius increases, the influence of eccentricity, probe offset, probe radius on the maximum residual error decreases while the influence of the geometric axis tilt and guide rail tilt on the maximum residual error increases. 3.3. Simulation analysis of coaxiality and cylindricity for the LPT shaft In order to analyze the advantages of the measurement model with five systematic errors M5 on the coaxiality and cylindricity evaluation for the LPT shaft. We compare it with the traditional cylindrical profile measurement model with two systematic errors T2, the improved model with two systematic errors I2, the model with three systematic errors M3 and the model with four systematic errors M4 (the T2 model considers the initial eccentricity and geometric axis tilt, the I2 model adds the measurement sectional ovalization caused by geometric axis tilt, the M3 model adds probe offset error and the M4 model adds probe radius), and coaxiality and cylindricity errors are evaluated under different measurement strategies. The results of the coaxiality and cylindricity evaluation are shown in Figs. 7 and 8. As shown in Fig. 7, under the same measurement strategy and systematic error, the cylindrical profile measurement model with five systematic errors is the most accurate model for the coaxiality and cylindricity evaluation, compared with other measurement models. In Figs. 7(a) and 8(a), if the measurement strategy 1 is applied, compared with other models (the T2 model, the I2 model, the M3 model, the M4 model), the M5 model proposed in this paper improve the measurement accuracy of coaxiality and cylin-
dricity by 7.04 μm and 27.96 μm, 7.01 μm and 25.54 μm, 2.94 μm and 20.99 μm, 2.93 μm and 20.98 μm, respectively in Level 5. As the magnitude of systematic error increases, the measurement accuracy of the model with five systematic errors increases. As shown in Figs. 7(b) and 8(b), if the measurement strategy 2 is applied and as systematic error increases from Level 1 to Level 5, the measurement accuracy of the coaxiality and cylindricity by the M5 model increase from 0.11 μm and 2.74 μm to 6.89 μm and 26.21 μm, respectively, compared with the T2 model. The model with five systematic errors has the least dispersion of coaxiality and cylindricity measurement results. Comparing the different measurement strategies of the coaxiality from Figs. 7(a) to (c), the evaluation results of the coaxiality shown by strategy 1 and strategy 2 are almost the same. For the first four models (the T2 model, the I2 model, the M3 model, the M4 model), the evaluation results of the coaxiality by strategy 3 are much smaller than those by strategy 1 and strategy 2. For example, the coaxiality of the T2 model evaluated by strategies 1, 2, and 3 are 7.14 μm, 7.01 μm, and 3.26 μm, respectively in Level 5. The first four models do not separate the guide rail tilt error. And strategy 3 has an averaging effect on the guide rail tilt error, which is less affected by the guide rail tilt error. 4. Experiments In order to verify the effectiveness of the cylindrical profile measurement method with five systematic errors proposed in this paper for the low-pressure turbine shaft, the coaxiality and cylindricity measurement experiments were performed on a rotary measuring instrument with high precision as shown in Fig. 9. High
C. Sun et al. / Aerospace Science and Technology 95 (2019) 105442
Fig. 9. Experimental setup for coaxiality and cylindricity measurement: (a) rotary measuring instrument; (b) LPT shaft; (c) probe with different radii.
precision turntable includes the air bearing turntable and centering and tilt worktable. The air bearing turntable provides a rotational measurement reference with the radial and axial accuracy of 38 nm. The centering and tilt worktable is used to adjust the eccentricity and axial tilt of the workpiece, and the displacement and angle adjustment range are ±3 mm and ±0.5◦ , respectively. The displacement of the vertical air bearing guide rail is 2500 mm. There are two low-pressure turbine shafts with different radii. The upper radii are 50 mm and 80 mm, respectively. The lower radii are 60 mm and 100 mm, respectively. The inductive sensor is used to measure the profile data of the low-pressure turbine shaft with a resolution of 5 nm. When the high precision turntable rotates at a constant speed, the sample data is obtained by adjusting the vertical air bearing guide rail so that the sensor probe contacts the low-pressure turbine shaft at different height sections. The number of measurement sections is 20, and the number of the sample points for each section is 1024. The results of coaxiality and cylindricity are shown in Tables 1–4. T2S1, T2S2, T2S3, I2S1, I2S2, I2S3, M3S1, M3S2, M3S3, M4S1, M4S2, M4S3, M5S1, M5S2, M5S3 represent traditional two-parameter model, improved two-parameter model, three-parameter model, four-parameter model, proposed five-parameter model by evaluating strategies 1, 2, 3, respectively.
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It can be seen from Tables 1–4 that if the systematic error magnitude is small, the results of coaxiality and cylindricity evaluated by five measurement models have little difference, so the model proposed in this paper is reasonable. With the increases of the systematic error, the measurement accuracy by the M5 model is improved compared with other measurement models. As shown in Tables 1 and 3, under the strategy 2 and as the systematic error is increased from Level 1 to Level 4, the measurement accuracy of coaxiality and cylindricity by the proposed model are increased from 0.76 μm and 1.06 μm to 9.13 μm and 12.46 μm, respectively, compared with the traditional model with two systematic errors. Comparing the different strategies of coaxiality, it can be found that strategy 3 makes the coaxiality of the low-pressure turbine shaft more accurate compared with strategy 1 and strategy 2. As shown in Table 1, if the systematic error is Level 4, the measurement accuracy of strategy 3 are increased by 1.9 μm and 1.75 μm, respectively. While comparing the different strategies of cylindricity, there are only small differences between strategy 1 and strategy 2. When the radius of the workpiece increases, the measurement accuracy of the M5 model is improved and the measurement dispersion is reduced. As shown in Tables 1–4, under the strategy 1 and in Level 4, the measurement accuracy of the coaxiality and cylindricity by the M5 model are improved by 2.24 μm and 3.98 μm, respectively, with the increase of the radius of the LPT shaft. As shown in Tables 2 and 4, compared with the T2 model, under the optimal strategy of the LPT shaft with large radius, the measurement accuracy of the coaxiality and cylindricity by the proposed M5 model are respectively increased by 2.9 μm and 8.18 μm, and the evaluation results are closer to the true value. Comparing the results of cylindricity by the T2S1 model and the I2S1 model in Tables 3 and 4, the difference is more obvious as the LPT shaft radius increases. If the LPT shaft radius increase from 50 mm to 80 mm, the measurement accuracy of cylindricity is improved from 0.48 μm to 0.92 μm. That is because the influence of the sectional ovalization phenomenon is obvious when the low-pressure turbine shaft with large radius, indicating that the geometric axis tilt error has a great influence on the measurement results for the large low-pressure turbine shaft. The probe offset error has a great influence on the coaxiality and cylindricity for the LPT shaft, while the probe radius has little effect on it. As shown in Figs. 4 and 5, the maximum difference of coaxiality and cylindricity between the I2S1 model and the M3S1 model in Level 4 are 2.81 μm and 2.83 μm, respectively, and the value between the M3S1 model and the M4S1 model are only 0.02 μm and 0.05 μm, respectively. Meanwhile, the result is affected greatly by the guide rail tilt error, because it not only directly affects the measurement results, but also affects the measurement results by chang-
Table 1 Coaxiality for LPT shaft with small radius. No. Level Level Level Level
1 2 3 4
T2S1
I2S1
M3S1
M4S1
M5S1
T2S2
I2S2
M3S2
M4S2
M5S2
T2S3
I2S3
M3S3
M4S3
M5S3
7.11 10.82 15.23 19.65
7.10 10.82 15.21 19.63
6.47 8.10 9.94 13.70
6.46 8.01 9.87 13.65
6.26 6.52 7.11 9.72
7.01 10.38 14.12 18.70
6.99 10.38 14.11 18.69
6.35 7.65 9.20 12.67
6.35 7.65 9.19 12.67
6.25 6.39 6.89 9.57
6.50 7.84 9.93 12.04
6.49 7.84 9.92 12.03
6.29 6.70 7.50 9.64
6.28 6.69 7.49 9.63
6.24 6.35 6.68 7.82
The LPT shaft with coaxiality of 6.2 μm. Table 2 Coaxiality for LPT shaft with large radius. No. Level Level Level Level
1 2 3 4
T2S1
I2S1
M3S1
M4S1
M5S1
T2S2
I2S2
M3S2
M4S2
M5S2
T2S3
I2S3
M3S3
M4S3
M5S3
3.80 5.43 7.62 10.59
3.80 5.43 7.61 10.58
3.69 4.46 5.55 7.77
3.68 4.46 5.54 7.75
3.53 3.81 4.01 4.78
3.74 5.20 7.15 10.23
3.74 5.19 7.14 10.20
3.62 4.24 5.29 7.42
3.62 4.23 5.27 7.40
3.52 3.76 3.92 4.51
3.62 4.22 5.41 6.91
3.62 4.20 5.40 6.89
3.56 3.84 4.37 5.05
3.55 3.83 4.37 5.04
3.52 3.62 3.79 4.01
The LPT shaft with coaxiality of 3.5 μm.
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Table 3 Cylindricity for LPT shaft with small radius. No. Level Level Level Level
1 2 3 4
T2S1
I2S1
M3S1
M4S1
M5S1
T2S2
I2S2
M3S2
M4S2
M5S2
7.82 13.86 18.17 24.20
7.81 13.74 18.08 23.72
7.38 10.11 12.04 17.50
7.32 10.07 12.01 17.47
6.73 8.29 9.47 11.88
7.27 13.43 18.02 24.18
7.27 13.32 17.84 23.69
6.75 9.51 11.69 17.43
6.75 9.51 11.68 17.41
6.21 8.03 9.11 11.72
The LPT shaft with cylindricity of 6.2 μm. Table 4 Cylindricity for LPT shaft with large radius. No. Level Level Level Level
1 2 3 4
T2S1
I2S1
M3S1
M4S1
M5S1
T2S2
I2S2
M3S2
M4S2
M5S2
6.84 9.23 11.37 14.53
6.71 8.89 10.66 13.61
6.45 8.23 9.70 10.78
6.43 8.23 9.66 10.73
4.85 5.01 5.52 6.20
6.28 8.57 10.42 14.04
6.18 8.21 9.84 13.13
5.95 7.54 8.94 10.61
5.94 7.54 8.91 10.60
4.55 4.89 5.23 5.86
The LPT shaft with cylindricity of 4.5 μm.
ing the probe offset error. In Tables 2 and 4, compared with the M4S1 model, the measurement accuracy of coaxiality and cylindricity are improved 2.97 μm and 4.53 μm by the proposed M5S1 model in this paper, respectively. Therefore, the proposed model is effective for the coaxiality and cylindricity measurement of the low-pressure turbine shaft with large radius, which includes eccentricity, probe offset, probe radius, geometric axis tilt and guide rail tilt. The model is suitable for small probe radius and large eccentricity error, probe offset error, geometric axis tilt error and guide rail tilt error, especially for the low-pressure turbine shaft with large radius. 5. Discussions and conclusions The cylindrical profile measurement model with five systematic errors considering eccentricity, probe offset, probe radius, geometric axis tilt and guide rail tilt proposed in this paper improves the measurement accuracy of the coaxiality and cylindricity for the LPT shaft. And this paper provides three coaxiality evaluation strategies and two cylindricity evaluation strategies. Based on the proposed model and measurement strategies, the influence of systematic error and radius on the LPT shaft profile measurement results is analyzed, and the coaxiality and cylindricity are evaluated. In order to verify the effectiveness of the cylindrical profile measurement model with five systematic errors for the lowpressure turbine shaft in this paper, the coaxiality and cylindricity measurement experiments are built on a high precision rotary measuring instrument. Compared with the traditional model with two systematic errors, under the optimal strategy of the LPT shaft with large radius, the measurement accuracy of the coaxiality and cylindricity by the proposed model with five systematic errors are respectively increased by 2.9 μm and 8.18 μm, and the evaluation results are closer to the true value. As the radius of the LPT shaft increases, the results are more accurate, which proves that the proposed model is more suitable for the LPT shaft with large radius. In addition, the sectional ovalization has a great influence on the cylindricity of the large radius low-pressure turbine shaft. And the probe offset greatly affects the coaxiality and cylindricity of the low-pressure turbine shaft as well while the probe radius has less influence on them. And the guide rail tilt error also has a great influence on the result, because it not only directly affects the measurement result, but also directly affects the probe offset error which indirectly affects the measurement result. The cylindrical profile measurement model with five systematic errors proposed in this paper can be applied effectively to the case of small probe radius and large eccentricity error, probe offset error, geometric axis tilt error and guide rail tilt error. And it is especially suitable for the evaluation of the coaxiality and cylin-
dricity for the low-pressure turbine shaft with large radius. The proposed method can be used for error separation and tolerance allocation of multistage rotor. Declaration of competing interest We declare that there are no conflicts of interest to this work. Acknowledgements This research was supported by the National Natural Science Foundation of China (grant number 51805117), the National Natural Science Foundation of China (grant number 61575056), the Fundamental Research Funds for the Central Universities (grant number HIT.NSRIF.2019019), the Heilongjiang Postdoctoral Fund (grant number LBH-Z18078) and the Natural Science Foundation of Heilongjiang Province (grant number F2016012). References [1] Z. Li, X. Zheng, Review of design optimization methods for turbomachinery aerodynamics, Prog. Aerosp. Sci. 93 (2017) 1–23. [2] P.P. Xi, Y.P. Zhao, et al., Least squares support vector machine for class imbalance learning and their applications to fault detection of aircraft engine, Aerosp. Sci. Technol. 84 (2019) 56–74. [3] R. Corral, O. Khemiri, C. Martel, Design of mistuning patterns to control the vibration amplitude of unstable rotor blades, Aerosp. Sci. Technol. 80 (2018) 20–28. [4] X. Tang, J.Q. Luo, F. Liu, Adjoint aerodynamic optimization of a transonic fan rotor blade with a localized two-level mesh deformation method, Aerosp. Sci. Technol. 72 (2018) 267–277. [5] Y. Cui, S. Deng, W. Zhang, The impact of roller dynamic unbalance of highspeed cylindrical roller bearing on the cage nonlinear dynamic characteristics, Mech. Mach. Theory 118 (2017) 65–83. [6] H.T. Qi, G.H. Xu, et al., A study of coaxial rotor aerodynamic interaction mechanism in hover with high-efficient trim model, Aerosp. Sci. Technol. 84 (2019) 1116–1130. [7] L. Pawsey, D.J. Rajendran, V. Pachidis, Characterisation of turbine behaviour for an engine overspeed prediction model, Aerosp. Sci. Technol. 73 (2018) 10–18. [8] E. Pescini, M.G. De Giorgi, et al., Separation control by a microfabricated SDBD plasma actuator for small engine turbine applications: influence of the excitation waveform, Aerosp. Sci. Technol. 76 (2018) 442–454. [9] D.J. Whitehouse, Some theoretical aspects of error separation techniques in surface metrology, J. Phys. E, Sci. Instrum. 9 (1976) 531–536. [10] D.G. Chetwynd, Roundness measurement using limacons, Precis. Eng. 1 (1979) 137–141. [11] D.G. Chetwynd, High-precision measurement of small balls, J. Phys. E, Sci. Instrum. 20 (1987) 1179–1187. [12] J.B. Tan, Error Compensation Technology for Precision Measurement, Harbin Institute of Technology Press, Harbin, 1995. [13] T.S. Murthy, A comparison of different algorithm for cylindricity evaluation, Int. J. Mach. Tool Des. Res. 22 (1982) 283–292. [14] C. Kirsten, P. Ferreira, Verification of form tolerances part II: cylindricity and straightness of a median line, Precis. Eng. 17 (1995) 144–156.
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[15] S. Adamczak, D. Janecki, K. Stepien, Cylindricity measurement by the V-block method - theoretical and practical problems, Measurement 44 (2011) 164–173. [16] X.C. Zhang, X. Jiang, P.J. Scott, A reliable method of minimum zone evaluation of cylindricity and conicity from coordinate measurement data, Precis. Eng. 35 (2011) 484–489. [17] Y.N. Zheng, Z.P. Qiu, Uncertainty propagation in aerodynamic forces and heating analysis for hypersonic vehicles with uncertain-but-bounded geometric parameters, Aerosp. Sci. Technol. 77 (2018) 11–24. [18] W.Q. Zhao, Z. Xue, et al., SSEST: a new approach to higher accuracy cylindricity measuring instrument, Int. J. Mach. Tools Manuf. 46 (2006) 1869–1878.
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[19] Y. Yang, M. Li, et al., Cylindricity error evaluation based on an improved harmony search algorithm, Sci. Program. 2018 (2018) 1–13. [20] H.Y. Lai, W.Y. Jywe, et al., Precision modeling of form errors for cylindricity evaluation using genetic algorithms, Precis. Eng. 24 (2000) 310–319. [21] E. Qi, Z.Y. Fang, et al., A method for predicting hydrostatic guide error averaging effects based on three-dimensional profile error, Tribol. Int. 95 (2016) 279–289. [22] H. Tang, J.A. Duan, Q. Zhao, A systematic approach on analyzing the relationship between straightness & angular errors and guideway surface in precise linear stage, Int. J. Mach. Tools Manuf. 120 (2017) 12–19.
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Aerospace Science and Technology www.elsevier.com/locate/aescte
Design of high accuracy cylindrical profile measurement model for low-pressure turbine shaft of aero engine Chuanzhi Sun a,b , Baosheng Wang a,b , Yongmeng Liu a,b,∗ , Xiaoming Wang a,b , Chengtian Li a,b , Hongye Wang a,b , Jiubin Tan a,b a
Center of Ultra-precision Optoelectronic Instrument Engineering, Harbin Institute of Technology, Harbin 150080, China Key Lab of Ultra-precision Intelligent Instrumentation Engineering (Harbin Institute of Technology), Ministry of Industry and Information Technology, Harbin 150080, China b
a r t i c l e
i n f o
Article history: Received 20 April 2019 Accepted 27 September 2019 Available online 30 September 2019 Keywords: Aero engine Cylindrical profile measurement LPT shaft
a b s t r a c t Coaxiality and cylindricity are the important geometric parameters of the low-pressure turbine (LPT) shaft. And the measurement accuracy of coaxiality and cylindricity directly affect the rotary characteristics of the aero engine. Therefore, a cylindrical profile measurement model with five systematic errors is designed to improve the coaxiality and cylindricity measurement accuracy of the low-pressure turbine shaft in this paper, in which eccentricity, probe offset, probe radius, geometric axis tilt and guide rail tilt are considered. Besides, the influence of systematic error and the shaft radius on the residual error for the stepped low-pressure turbine shaft is analyzed as well. The evaluation results of coaxiality and cylindricity are obtained based on different measurement strategies and models. In order to verify the effectiveness of the cylindrical profile measurement model with five systematic errors in the paper, a rotary measuring instrument with high precision is built. Compared with the traditional cylindrical profile measurement model with two systematic errors, the measurement accuracy of the coaxiality and cylindricity by the cylindrical profile measurement model with five systematic errors proposed in this paper are improved by 2.9 μm and 8.18 μm, respectively in the condition of the optimal measurement strategy for the LPT shaft with large radius. The proposed method is suitable for small probe radius and large eccentricity error, probe offset error, geometric axis tilt error and guide rail tilt error, especially for the LPT shaft with large radius. The proposed method can be applied to error separation and tolerance allocation for multistage rotor. © 2019 Elsevier Masson SAS. All rights reserved.
1. Introduction Low-pressure turbine shaft is the key rotating component of aero engine and it runs through the core machine. The quality of the LPT shaft directly affects the rotational characteristics of the aero engine [1,2]. The LPT shaft has the mechanical structural characteristics of the stepped rotary shaft. The coaxiality and cylindricity are two important geometric parameters for the LPT shaft, which respectively reflect the rotation characteristics and surface contact characteristics. The LPT shaft with low machining precision for coaxiality and cylindricity will become curved and the contact characteristics will be worse, and then will cause a large amount of vibration [3–5]. The improvement of the design accuracy of aero engine puts higher demands on the measurement accuracy of the
*
Corresponding author. E-mail address: [email protected] (Y. Liu).
https://doi.org/10.1016/j.ast.2019.105442 1270-9638/© 2019 Elsevier Masson SAS. All rights reserved.
low-pressure turbine shafts. The LPT shaft of precision turbofan aero engine, for example, its coaxiality and cylindricity error are required to be controlled within 1 μm [6,7]. Although the measuring instrument has a high accuracy, the existence of the systematic errors will bring a great influence on the measurement accuracy of the coaxiality and cylindricity for the LPT shaft, and the results may contain the systematic errors that have not been separated [8]. Therefore, in order to improve the rotation characteristics of the aero engine, the measurement accuracy of the coaxiality and cylindricity need to be improved for the LPT shaft. In recent years, researchers have done a lot of work on the measurement of coaxiality and cylindricity. The Limacon measurement model was proposed which considers the effect of the eccentricity error caused by the offset of the workpiece geometry axis relative to the rotation axis of the turntable on the circular profile measurement [9,10]. The probe offset error also has the influence on the circular profile measurement [11,12], which is caused by the sensor measurement line not pass the measured
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Fig. 1. The cylindrical profile measurement model of low-pressure turbine shaft with five systematic errors: (a) The cylindrical profile measurement model; (b) The jth measurement section model.
center of rotation. When the measured object is the stepped rotation shaft, the tilt error of the workpiece can cause noncoincidence between the workpiece geometry axis and the rotation axis of the turntable, which brings the secondary eccentricity error of the measured section. The traditional cylindrical profile measurement model with two systematic errors was proposed to improve the accuracy of cylindrical profile measurement, and the initial eccentricity and geometric axis tilt are considered in the model [13, 14]. In addition, many scholars have begun to make new attempts to improve the cylindrical profiles measurement accuracy from the perspective of error separation [15–17]. Zhao et al. proposed a new single-step spatial rotation error separation technique (SSEST). In this model, the measurement accuracy of cylindricity is improved by eliminating the spatial rotation error of the instrument spindle [18]. Yang et al. proposed the improved harmony search algorithm for cylindricity error evaluation [19]. Lai et al. applied adaptive genetic algorithms and heuristic genetic algorithms to cylindrical profile measurement to improve the accuracy of measurement results [20]. The above methods introduced the influence of eccentricity, sensor probe offset and the secondary eccentricity of the workpiece on the cylindrical profile measurement. However, when there are both eccentricity and probe offset, and the tip head of probe cannot be processed into an ideal point, the probe radius will bring new measurement errors into the cylindrical profile measurement results. Geometric axis tilt of the workpiece will cause the secondary eccentricity and it will also cause the measuring section ovalization. And in the precision measurement of coaxiality and cylindricity, the axis of motion of the guide rail cannot be completely parallel to the axis of rotation [21,22], which not only leads to the errors in the measurement of the cylindrical profile, but also causes a secondary offset of the sensor probe. Therefore, this paper designs a cylindrical profile measurement model with five systematic errors: eccentricity, probe offset, probe radius, geometric axis tilt and guide rail tilt. Based on the proposed model in this paper, different measurement strategies on coaxiality and cylindricity are proposed to improve the coaxiality and cylindricity measurement accuracy of the low-pressure turbine shaft. The structure of this paper is as follows: In Section 2 of this paper, the cylindrical profile measurement model with five systematic errors is established and three coaxiality measurement strategies and two cylindricity measurement strategies are designed. In Section 3, in order to analyze the advantages of the cylindrical profile measurement model with five systematic errors in the measurement of coaxiality and cylindricity, the influence of systematic errors and workpiece radius on the measurement results
is analyzed. The influence of systematic error on the coaxiality and cylindricity measurement results under different measurement strategies for the LPT shaft is also analyzed. In Section 4, the effectiveness of the proposed method for coaxiality and cylindricity measurement is verified by experiments on a rotary measuring instrument with high precision. Finally, the discussions and conclusions are given in Section 5. 2. The cylindrical profile measurement method of the low-pressure turbine shaft 2.1. Establishment of the cylindrical profile measurement model with five systematic errors The traditional cylindrical profile measurement model with two systematic errors considers the influence of initial eccentricity error and geometric axis tilt error. The probe offset, the probe radius, the guide rail tilt and the sectional ovalization caused by the geometric axis tilt are also considered in this paper based on the traditional model. The cylindrical profile measurement model with five systematic errors shown in Fig. 1 is established. In Fig. 1(a), for the jth measurement section, O j1 is the rotation center of the turntable, O j2 is the geometric center of the LPT shaft. roj and rlj are the lengths of semi-minor and -major axis of the fitting ellipse. Ψ is the angle between the guide rail and the rotation axis. The initial eccentricity and eccentric angle are e 0 and α0 , respectively. The jth measurement section model of the LPT shaft is shown in Fig. 1(b). e j is the amount of eccentricity, α j is the eccentric angle, r is the probe radius, and d j1 is the probe’s own offset. ε is the angle between the inclination error of the guide rail and the measurement direction. t j is the moving distance of the probe center caused by the tilt of the guide rail and its vertical measurement direction component is d j2 . d j is the total offset of the probe, O j3 is the instantaneous rotation center of the measurement line and O j4 is the probe center. β j is the angle between the major axis direction of the fitting ellipse and the X -direction. z j is the height of the jth measurement section. For the ith sample point, ρi j is the distance between the probe and O j3 . r i j is the distance between the fitting ellipse and O j2 . Δr i j is the LPT shaft machining error. θi j is the sample angle of O j1 . ϕi j is the sample angle of O j2 . τi j is the angle between the measurement direction and the O j2 O j4 direction. The geometric axis tilt angle γ not only affects the eccentricity of each measurement section, but also causes the section profile to be ovalized. For the jth measurement section, the define of the geometric axis tilt error is shown in Fig. 2.
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Equation (4) is brought into equation (5)
⎛
⎞ ⎜ d j1 + z j tan ψ sin ε + e j sin θi j − α j ⎟ ⎟ ϕi j = θi j + arcsin ⎜ ⎝ ⎠ cos2 ϕi j −β j 2 + Δri j + roj + sin ϕ − β r i j j 2 cos γ (6) In the actual measurement, the inclination angle γ is small (γ < 3◦ ), and the angle deviation caused by it is very small and can be ignored. The equations (4) and (6) can be brought into equation (2) and power series expansion is performed Fig. 2. Geometric axis tilt error of the jth measurement section.
As shown in Fig. 2, the relationship between the semi-minor and -major axis of the fitting ellipse can be seen as
rlj =
roj
(1)
cos γ
The direction vector of the geometric axis of the LPT shaft caused by the tilt angle γ is (l , m , n ). The measurement equation can be obtained in the model from Fig. 1.
⎧ ρi j = ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ Δri j + ri j + r − e j sin θi j − α j + d j1 + z j tan ψ sin ε ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ + e j cos θi j − α j + z j tan ψ cos ε − r ⎪ ⎪ ⎪ ⎪ ⎪ 2 2 ⎪ ⎪ ⎨ e j = e 0 cos α0 + lz j + e 0 sin α0 + mz j
e sin α0 + mz j ⎪ ⎪ α j = arctan 0 ⎪ ⎪ e 0 cos α0 + lz j ⎪ ⎪ ⎪ ⎪ ⎪ l m ⎪ ⎪ ⎪l = , m = ⎪ ⎪ n n ⎪ ⎪ √ ⎪ ⎪ ⎪ l 2 + m 2 ⎪ ⎩ tan γ = n
⎧ ρi j = e j cos θi j − α j + z j tan ψ cos ε + Δri j + ξi j ⎪ ⎪ ⎪ 2 ⎪ ⎪ ⎪ e j sin θi j − α j + d j1 + z j tan ψ sin ε ⎪ ⎪ − ⎪ ⎪ ⎪ 2(Δr i j + ξi j + r ) ⎪ ⎪ ⎪ ⎪ ⎨ cos2 ηi j ξi j = roj + sin2 ηi j 2 ⎪ cos γ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ d j1 + z j tan ψ sin ε + e j sin θi j − α j ⎪ ⎪ ⎪ ηi j = θi j + arcsin ⎪ ⎪ Δri j + roj + r ⎪ ⎪ ⎪ ⎩ − βj (7) Where i = 1, 2, ..., k, j = 1, 2, ..., p, k is the number of sample points and p is the number of measurement sections. The traditional cylindrical profile measurement model with two systematic errors can be expressed as
ρiLimacon = Δri j + roj + e j cos θi j − α j j
(8)
In order to further analyze the advantages of the cylindrical profile measurement model with five systematic errors, the equations (7) and (8) are subtracted, and the residual error is shown in equation (9)
δi j = ρi j − ρiLimacon j
Taking O j2 as the coordinate center, the fitting ellipse equation is expressed as
= z j tan ψ cos ε + ξi j − roj 2 e j sin θi j − α j + d j1 + z j tan ψ sin ε − 2(Δr i j + ξi j + r )
Then the maximum residual error is shown in equation (10)
(2)
xi j = rlj cos y i j = roj sin
ϕi j − β j
ϕi j − β j
(3)
Where xi j and y i j are the horizontal and vertical coordinates of the ith sample point in the fitting ellipse. From equations (1) and (3), the distance r i j between the fitting ellipse and O j2 is expressed as
ri j =
x2i j
+
y 2i j
= roj
ϕi j − β j + sin2 ϕi j − β j 2 cos γ
cos2
(4)
As shown in Fig. 1(b), the relationship between the sample angle ϕi j of the geometric center O j2 and the sample angle θi j of the rotation center O j1 is expressed as
τi j = ϕi j − θi j
Δri j + ri j + r sin τi j = d j1 + z j tan ψ sin ε + e j sin θi j − α j
(5)
δmax = max δi j , i = 1, 2, · · · , k; j = 1, 2, · · · , p
(9)
(10)
If the guide rail tilt error and the geometric axis tilt error are very small, the sum of the first three terms of the equation (9) tends to be zero. And the fourth term dominates the residual error, which is mainly influenced by eccentricity, probe offset and probe radius. As the eccentricity and probe offset increase and with the decreasing probe radius, the residual error increases. With the increase of the guide rail tilt error and the geometry axis tilt error, the proportion of the first three items of the equation (9) increases, the residual error is mainly caused by the guide rail tilt error and the geometry axis tilt error. 2.2. The measurement strategy of coaxiality and cylindricity for the low-pressure turbine shaft According to the mechanical structure and working principle of the low-pressure turbine shaft of aero engine, three measurement strategies of coaxiality are proposed in Fig. 3. In Figs. 3(a) and (b), we can see both strategy 1 and strategy 2 select the lower cylinder
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Fig. 3. Coaxiality evaluation strategies: (a) Strategy 1; (b) Strategy 2; (c) Strategy 3.
Fig. 4. Cylindricity evaluation strategies: (a) Strategy 1; (b) Strategy 2.
as the reference element and the upper cylinder is chosen as the measured element. Strategy 1 uses the least squares central line of the reference element as the reference axis, and strategy 2 uses the least squares central line of the two regions of the reference elements as the reference axis. As shown in Fig. 3(c), the upper and lower regions of the LPT shaft are selected as the reference elements in strategy 3, with the least square central line of the two regions as the reference axis, and the remaining region of the LPT shaft as the measured element. This paper proposes two kinds of the LPT shaft cylindricity measurement strategies as shown in Fig. 4. Fig. 4(a) shows the least squares method for strategy 1. The least squares method is to find the ideal cylindrical surface that minimizes the sum of the squares of the points from the actual measured cylindrical surface to the ideal cylindrical surface. The minimum radius difference between the two coaxial ideal cylindrical surfaces which contain the actual measured cylindrical surface is the cylindricity error. Fig. 4(b) shows the minimum area method for strategy 2. When the minimum area method is used to evaluate the cylindricity error, the two coaxial ideal cylindrical surfaces are used to contain actual measured cylindrical surface. And when the difference in radius between the two ideal cylindrical surfaces is the smallest, the smallest difference in the radius is the cylindricity error. 3. Simulations 3.1. Influence of systematic error on profile measurement of the LPT shaft In order to further analyze the advantages of cylindrical profile measurement model with five systematic errors, compared with
the traditional model with two systematic errors, the influence of the multi-systematic error on the measurement results at different sections is analyzed. Assumption that the profile deviation caused by imperfect manufacturing Δr i j is the standard normal distribution, which is 0.001 mm, the standard deviation is one-third of Δr i j , the eccentric angle α is π /3, the angle β is π /6, the length of semi-minor axis of fitting ellipse ro is 10 mm, and the angle ε between the probe offset caused by the guide rail tilt and the measuring direction is π /6. The heights of the workpiece sections are taken as 0 mm, 50 mm, 100 mm and 200 mm, respectively, and the simulation is applied according to the equation (9). The results are shown in Fig. 5. It can be seen in Fig. 5 that under the same height of section, the residual error increases with the increase of eccentricity, probe offset, geometric axis tilt, and guide rail tilt errors and the decreases of probe radius. And in Fig. 5(a), as the systematic error level increases from Level 1 to Level 5, the maximum residual error increases from 0.1 μm to 13.4 μm. As shown in Figs. 5(a)-(d), under the same systematic error level, as the height of the measurement section increases, the residual error increases. If the systematic error level is Level 5, as the section height increases from 0 mm to 200 mm, the maximum residual error increases from 13.4 μm to 60.2 μm. And the maximum residual error position is near the vertical direction of the eccentric angle. 3.2. Influence of systematic error and the LPT shaft radius on profile measurement In this paper, according to equation (10), the influence of systematic error on the profile measurement of workpieces with different radii is analyzed. The simulation results on different height sections are shown in Fig. 6. For the sake of clearer description of the relationship between the residual error and the radius of the LPT shaft, the paper replaces the length of semi-minor axis of fitting ellipse with the radius of cylindrical component in the following description. As shown in Fig. 6, under the same component radius and measured height, the maximum residual error increases with the increases of the systematic error. As it is shown in Fig. 6(d), if the radius is 50 mm and the height is 200 mm, the maximum residual error increases from 2.5 μm to 19.9 μm as the systematic error increases from Level 1 to Level 5. The maximum residual error increases as the measured height increases at the same component radius and systematic error level. As shown in Figs. 6(a)-(d), if the component radius is 50 mm, the maximum residual error increases by 17.1 μm as the measured height increases by 200 mm in Level 5. The maximum residual error tends to decrease first and then increase as the component radius increases. In the falling phase, the maximum
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Fig. 5. Residual error at different heights of object with multi-systematic error: (a) z = 0 mm; (b) z = 50 mm; (c) z = 100 mm; (d) z = 200 mm. Level 1: e = 1 μm, d = 50 μm, γ = 1 , r = 2.5 mm, Ψ = 1 ; Level 2: e = 5 μm, d = 100 μm, γ = 3 , r = 2 mm, Ψ = 3 ; Level 3: e = 10 μm, d = 200 μm, γ = 5 , r = 1.5 mm, Ψ = 8 ; Level 4: e = 20 μm, d = 300 μm, γ = 8 , r = 1 mm, Ψ = 12 ; Level 5: e = 30 μm, d = 500 μm, γ = 12 , r = 0.5 mm, Ψ = 15 .
Fig. 6. Simulation relationship diagram of systematic error and workpiece radius on maximum residual error: (a) z = 0 mm; (b) z = 50 mm; (c) z = 100 mm; (d) z = 200 mm. Level 1: e = 1 μm, d = 50 μm, γ = 10 , r = 2.5 mm, Ψ = 3 ; Level 2: e = 5 μm, d = 100 μm, γ = 30 , r = 2 mm, Ψ = 5 ; Level 3: e = 10 μm, d = 200 μm, γ = 50 , r = 1.5 mm, Ψ = 10 ; Level 4: e = 20 μm, d = 300 μm, γ = 100 , r = 1 mm, Ψ = 18 ; Level 5: e = 30 μm, d = 500 μm, γ = 150 , r = 0.5 mm, Ψ = 25 .
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Fig. 7. Coaxiality evaluation results under different measurement strategies: (a) Strategy 1; (b) Strategy 2; (c) Strategy 3. Level 1: e = 3 μm, d = 5 μm, γ = 5 , r = 2.5 mm, Ψ = 3 ; Level 2: e = 5 μm, d = 10 μm, γ = 10 , r = 2 mm, Ψ = 5 ; Level 3: e = 10 μm, d = 20 μm, γ = 15 , r = 1.5 mm, Ψ = 10 ; Level 4: e = 20 μm, d = 30 μm, γ = 20 , r = 1 mm, Ψ = 18 ; Level 5: e = 30 μm, d = 50 μm, γ = 25 , r = 0.5 mm, Ψ = 25 .
Fig. 8. Cylindricity evaluation results under different measurement strategies: (a) Strategy 1; (b) Strategy 2. Level 1: e = 3 μm, d = 5 μm, γ = 5 , r = 2.5 mm, Ψ = 3 ; Level 2: e = 5 μm, d = 10 μm, γ = 10 , r = 2 mm, Ψ = 5 ; Level 3: e = 10 μm, d = 20 μm, γ = 15 , r = 1.5 mm, Ψ = 10 ; Level 4: e = 20 μm, d = 30 μm, γ = 20 , r = 1 mm, Ψ = 18 ; Level 5: e = 30 μm, d = 50 μm, γ = 25 , r = 0.5 mm, Ψ = 25 .
residual error is mainly affected by eccentricity, probe offset, and probe radius. As the component radius increases, the influence of eccentricity, probe offset, probe radius on the maximum residual error decreases while the influence of the geometric axis tilt and guide rail tilt on the maximum residual error increases. 3.3. Simulation analysis of coaxiality and cylindricity for the LPT shaft In order to analyze the advantages of the measurement model with five systematic errors M5 on the coaxiality and cylindricity evaluation for the LPT shaft. We compare it with the traditional cylindrical profile measurement model with two systematic errors T2, the improved model with two systematic errors I2, the model with three systematic errors M3 and the model with four systematic errors M4 (the T2 model considers the initial eccentricity and geometric axis tilt, the I2 model adds the measurement sectional ovalization caused by geometric axis tilt, the M3 model adds probe offset error and the M4 model adds probe radius), and coaxiality and cylindricity errors are evaluated under different measurement strategies. The results of the coaxiality and cylindricity evaluation are shown in Figs. 7 and 8. As shown in Fig. 7, under the same measurement strategy and systematic error, the cylindrical profile measurement model with five systematic errors is the most accurate model for the coaxiality and cylindricity evaluation, compared with other measurement models. In Figs. 7(a) and 8(a), if the measurement strategy 1 is applied, compared with other models (the T2 model, the I2 model, the M3 model, the M4 model), the M5 model proposed in this paper improve the measurement accuracy of coaxiality and cylin-
dricity by 7.04 μm and 27.96 μm, 7.01 μm and 25.54 μm, 2.94 μm and 20.99 μm, 2.93 μm and 20.98 μm, respectively in Level 5. As the magnitude of systematic error increases, the measurement accuracy of the model with five systematic errors increases. As shown in Figs. 7(b) and 8(b), if the measurement strategy 2 is applied and as systematic error increases from Level 1 to Level 5, the measurement accuracy of the coaxiality and cylindricity by the M5 model increase from 0.11 μm and 2.74 μm to 6.89 μm and 26.21 μm, respectively, compared with the T2 model. The model with five systematic errors has the least dispersion of coaxiality and cylindricity measurement results. Comparing the different measurement strategies of the coaxiality from Figs. 7(a) to (c), the evaluation results of the coaxiality shown by strategy 1 and strategy 2 are almost the same. For the first four models (the T2 model, the I2 model, the M3 model, the M4 model), the evaluation results of the coaxiality by strategy 3 are much smaller than those by strategy 1 and strategy 2. For example, the coaxiality of the T2 model evaluated by strategies 1, 2, and 3 are 7.14 μm, 7.01 μm, and 3.26 μm, respectively in Level 5. The first four models do not separate the guide rail tilt error. And strategy 3 has an averaging effect on the guide rail tilt error, which is less affected by the guide rail tilt error. 4. Experiments In order to verify the effectiveness of the cylindrical profile measurement method with five systematic errors proposed in this paper for the low-pressure turbine shaft, the coaxiality and cylindricity measurement experiments were performed on a rotary measuring instrument with high precision as shown in Fig. 9. High
C. Sun et al. / Aerospace Science and Technology 95 (2019) 105442
Fig. 9. Experimental setup for coaxiality and cylindricity measurement: (a) rotary measuring instrument; (b) LPT shaft; (c) probe with different radii.
precision turntable includes the air bearing turntable and centering and tilt worktable. The air bearing turntable provides a rotational measurement reference with the radial and axial accuracy of 38 nm. The centering and tilt worktable is used to adjust the eccentricity and axial tilt of the workpiece, and the displacement and angle adjustment range are ±3 mm and ±0.5◦ , respectively. The displacement of the vertical air bearing guide rail is 2500 mm. There are two low-pressure turbine shafts with different radii. The upper radii are 50 mm and 80 mm, respectively. The lower radii are 60 mm and 100 mm, respectively. The inductive sensor is used to measure the profile data of the low-pressure turbine shaft with a resolution of 5 nm. When the high precision turntable rotates at a constant speed, the sample data is obtained by adjusting the vertical air bearing guide rail so that the sensor probe contacts the low-pressure turbine shaft at different height sections. The number of measurement sections is 20, and the number of the sample points for each section is 1024. The results of coaxiality and cylindricity are shown in Tables 1–4. T2S1, T2S2, T2S3, I2S1, I2S2, I2S3, M3S1, M3S2, M3S3, M4S1, M4S2, M4S3, M5S1, M5S2, M5S3 represent traditional two-parameter model, improved two-parameter model, three-parameter model, four-parameter model, proposed five-parameter model by evaluating strategies 1, 2, 3, respectively.
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It can be seen from Tables 1–4 that if the systematic error magnitude is small, the results of coaxiality and cylindricity evaluated by five measurement models have little difference, so the model proposed in this paper is reasonable. With the increases of the systematic error, the measurement accuracy by the M5 model is improved compared with other measurement models. As shown in Tables 1 and 3, under the strategy 2 and as the systematic error is increased from Level 1 to Level 4, the measurement accuracy of coaxiality and cylindricity by the proposed model are increased from 0.76 μm and 1.06 μm to 9.13 μm and 12.46 μm, respectively, compared with the traditional model with two systematic errors. Comparing the different strategies of coaxiality, it can be found that strategy 3 makes the coaxiality of the low-pressure turbine shaft more accurate compared with strategy 1 and strategy 2. As shown in Table 1, if the systematic error is Level 4, the measurement accuracy of strategy 3 are increased by 1.9 μm and 1.75 μm, respectively. While comparing the different strategies of cylindricity, there are only small differences between strategy 1 and strategy 2. When the radius of the workpiece increases, the measurement accuracy of the M5 model is improved and the measurement dispersion is reduced. As shown in Tables 1–4, under the strategy 1 and in Level 4, the measurement accuracy of the coaxiality and cylindricity by the M5 model are improved by 2.24 μm and 3.98 μm, respectively, with the increase of the radius of the LPT shaft. As shown in Tables 2 and 4, compared with the T2 model, under the optimal strategy of the LPT shaft with large radius, the measurement accuracy of the coaxiality and cylindricity by the proposed M5 model are respectively increased by 2.9 μm and 8.18 μm, and the evaluation results are closer to the true value. Comparing the results of cylindricity by the T2S1 model and the I2S1 model in Tables 3 and 4, the difference is more obvious as the LPT shaft radius increases. If the LPT shaft radius increase from 50 mm to 80 mm, the measurement accuracy of cylindricity is improved from 0.48 μm to 0.92 μm. That is because the influence of the sectional ovalization phenomenon is obvious when the low-pressure turbine shaft with large radius, indicating that the geometric axis tilt error has a great influence on the measurement results for the large low-pressure turbine shaft. The probe offset error has a great influence on the coaxiality and cylindricity for the LPT shaft, while the probe radius has little effect on it. As shown in Figs. 4 and 5, the maximum difference of coaxiality and cylindricity between the I2S1 model and the M3S1 model in Level 4 are 2.81 μm and 2.83 μm, respectively, and the value between the M3S1 model and the M4S1 model are only 0.02 μm and 0.05 μm, respectively. Meanwhile, the result is affected greatly by the guide rail tilt error, because it not only directly affects the measurement results, but also affects the measurement results by chang-
Table 1 Coaxiality for LPT shaft with small radius. No. Level Level Level Level
1 2 3 4
T2S1
I2S1
M3S1
M4S1
M5S1
T2S2
I2S2
M3S2
M4S2
M5S2
T2S3
I2S3
M3S3
M4S3
M5S3
7.11 10.82 15.23 19.65
7.10 10.82 15.21 19.63
6.47 8.10 9.94 13.70
6.46 8.01 9.87 13.65
6.26 6.52 7.11 9.72
7.01 10.38 14.12 18.70
6.99 10.38 14.11 18.69
6.35 7.65 9.20 12.67
6.35 7.65 9.19 12.67
6.25 6.39 6.89 9.57
6.50 7.84 9.93 12.04
6.49 7.84 9.92 12.03
6.29 6.70 7.50 9.64
6.28 6.69 7.49 9.63
6.24 6.35 6.68 7.82
The LPT shaft with coaxiality of 6.2 μm. Table 2 Coaxiality for LPT shaft with large radius. No. Level Level Level Level
1 2 3 4
T2S1
I2S1
M3S1
M4S1
M5S1
T2S2
I2S2
M3S2
M4S2
M5S2
T2S3
I2S3
M3S3
M4S3
M5S3
3.80 5.43 7.62 10.59
3.80 5.43 7.61 10.58
3.69 4.46 5.55 7.77
3.68 4.46 5.54 7.75
3.53 3.81 4.01 4.78
3.74 5.20 7.15 10.23
3.74 5.19 7.14 10.20
3.62 4.24 5.29 7.42
3.62 4.23 5.27 7.40
3.52 3.76 3.92 4.51
3.62 4.22 5.41 6.91
3.62 4.20 5.40 6.89
3.56 3.84 4.37 5.05
3.55 3.83 4.37 5.04
3.52 3.62 3.79 4.01
The LPT shaft with coaxiality of 3.5 μm.
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Table 3 Cylindricity for LPT shaft with small radius. No. Level Level Level Level
1 2 3 4
T2S1
I2S1
M3S1
M4S1
M5S1
T2S2
I2S2
M3S2
M4S2
M5S2
7.82 13.86 18.17 24.20
7.81 13.74 18.08 23.72
7.38 10.11 12.04 17.50
7.32 10.07 12.01 17.47
6.73 8.29 9.47 11.88
7.27 13.43 18.02 24.18
7.27 13.32 17.84 23.69
6.75 9.51 11.69 17.43
6.75 9.51 11.68 17.41
6.21 8.03 9.11 11.72
The LPT shaft with cylindricity of 6.2 μm. Table 4 Cylindricity for LPT shaft with large radius. No. Level Level Level Level
1 2 3 4
T2S1
I2S1
M3S1
M4S1
M5S1
T2S2
I2S2
M3S2
M4S2
M5S2
6.84 9.23 11.37 14.53
6.71 8.89 10.66 13.61
6.45 8.23 9.70 10.78
6.43 8.23 9.66 10.73
4.85 5.01 5.52 6.20
6.28 8.57 10.42 14.04
6.18 8.21 9.84 13.13
5.95 7.54 8.94 10.61
5.94 7.54 8.91 10.60
4.55 4.89 5.23 5.86
The LPT shaft with cylindricity of 4.5 μm.
ing the probe offset error. In Tables 2 and 4, compared with the M4S1 model, the measurement accuracy of coaxiality and cylindricity are improved 2.97 μm and 4.53 μm by the proposed M5S1 model in this paper, respectively. Therefore, the proposed model is effective for the coaxiality and cylindricity measurement of the low-pressure turbine shaft with large radius, which includes eccentricity, probe offset, probe radius, geometric axis tilt and guide rail tilt. The model is suitable for small probe radius and large eccentricity error, probe offset error, geometric axis tilt error and guide rail tilt error, especially for the low-pressure turbine shaft with large radius. 5. Discussions and conclusions The cylindrical profile measurement model with five systematic errors considering eccentricity, probe offset, probe radius, geometric axis tilt and guide rail tilt proposed in this paper improves the measurement accuracy of the coaxiality and cylindricity for the LPT shaft. And this paper provides three coaxiality evaluation strategies and two cylindricity evaluation strategies. Based on the proposed model and measurement strategies, the influence of systematic error and radius on the LPT shaft profile measurement results is analyzed, and the coaxiality and cylindricity are evaluated. In order to verify the effectiveness of the cylindrical profile measurement model with five systematic errors for the lowpressure turbine shaft in this paper, the coaxiality and cylindricity measurement experiments are built on a high precision rotary measuring instrument. Compared with the traditional model with two systematic errors, under the optimal strategy of the LPT shaft with large radius, the measurement accuracy of the coaxiality and cylindricity by the proposed model with five systematic errors are respectively increased by 2.9 μm and 8.18 μm, and the evaluation results are closer to the true value. As the radius of the LPT shaft increases, the results are more accurate, which proves that the proposed model is more suitable for the LPT shaft with large radius. In addition, the sectional ovalization has a great influence on the cylindricity of the large radius low-pressure turbine shaft. And the probe offset greatly affects the coaxiality and cylindricity of the low-pressure turbine shaft as well while the probe radius has less influence on them. And the guide rail tilt error also has a great influence on the result, because it not only directly affects the measurement result, but also directly affects the probe offset error which indirectly affects the measurement result. The cylindrical profile measurement model with five systematic errors proposed in this paper can be applied effectively to the case of small probe radius and large eccentricity error, probe offset error, geometric axis tilt error and guide rail tilt error. And it is especially suitable for the evaluation of the coaxiality and cylin-
dricity for the low-pressure turbine shaft with large radius. The proposed method can be used for error separation and tolerance allocation of multistage rotor. Declaration of competing interest We declare that there are no conflicts of interest to this work. Acknowledgements This research was supported by the National Natural Science Foundation of China (grant number 51805117), the National Natural Science Foundation of China (grant number 61575056), the Fundamental Research Funds for the Central Universities (grant number HIT.NSRIF.2019019), the Heilongjiang Postdoctoral Fund (grant number LBH-Z18078) and the Natural Science Foundation of Heilongjiang Province (grant number F2016012). References [1] Z. Li, X. Zheng, Review of design optimization methods for turbomachinery aerodynamics, Prog. Aerosp. Sci. 93 (2017) 1–23. [2] P.P. Xi, Y.P. Zhao, et al., Least squares support vector machine for class imbalance learning and their applications to fault detection of aircraft engine, Aerosp. Sci. Technol. 84 (2019) 56–74. [3] R. Corral, O. Khemiri, C. Martel, Design of mistuning patterns to control the vibration amplitude of unstable rotor blades, Aerosp. Sci. Technol. 80 (2018) 20–28. [4] X. Tang, J.Q. Luo, F. Liu, Adjoint aerodynamic optimization of a transonic fan rotor blade with a localized two-level mesh deformation method, Aerosp. Sci. Technol. 72 (2018) 267–277. [5] Y. Cui, S. Deng, W. Zhang, The impact of roller dynamic unbalance of highspeed cylindrical roller bearing on the cage nonlinear dynamic characteristics, Mech. Mach. Theory 118 (2017) 65–83. [6] H.T. Qi, G.H. Xu, et al., A study of coaxial rotor aerodynamic interaction mechanism in hover with high-efficient trim model, Aerosp. Sci. Technol. 84 (2019) 1116–1130. [7] L. Pawsey, D.J. Rajendran, V. Pachidis, Characterisation of turbine behaviour for an engine overspeed prediction model, Aerosp. Sci. Technol. 73 (2018) 10–18. [8] E. Pescini, M.G. De Giorgi, et al., Separation control by a microfabricated SDBD plasma actuator for small engine turbine applications: influence of the excitation waveform, Aerosp. Sci. Technol. 76 (2018) 442–454. [9] D.J. Whitehouse, Some theoretical aspects of error separation techniques in surface metrology, J. Phys. E, Sci. Instrum. 9 (1976) 531–536. [10] D.G. Chetwynd, Roundness measurement using limacons, Precis. Eng. 1 (1979) 137–141. [11] D.G. Chetwynd, High-precision measurement of small balls, J. Phys. E, Sci. Instrum. 20 (1987) 1179–1187. [12] J.B. Tan, Error Compensation Technology for Precision Measurement, Harbin Institute of Technology Press, Harbin, 1995. [13] T.S. Murthy, A comparison of different algorithm for cylindricity evaluation, Int. J. Mach. Tool Des. Res. 22 (1982) 283–292. [14] C. Kirsten, P. Ferreira, Verification of form tolerances part II: cylindricity and straightness of a median line, Precis. Eng. 17 (1995) 144–156.
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