1. Introduction
The assembly error of an aeroengine rotor assembly (e.g., compressor and turbine) refers to the relative deviation or deflection between the specific structures on different rotor parts after the connecting processes [
1]. Manufacturing enterprises have set strict upper limits for rotor assembly errors, because they significantly affect the wear, vibration, and aerodynamic performance of the final assembly [
2,
3]. However, manual operation using simple mechanisms is still widely used. Each assembly error is guaranteed to be lower than the corresponding upper limit through multiple trial connecting processes and repeated measurements, because the qualification rate of a single trial connecting process is only lower than 50%. To address the low quality and efficiency of the traditional manual method, an automated six-axis numerical control (NC) mechanism for rotor position and pose adjustment and connection, as well as a three-axis coordinate measuring machine (CMM), can be applied in modern aeroengine assembly lines. Servo linkage axes can completely replace manual measuring and connecting motions on multiple spatial degrees of freedom with higher motion accuracy [
4]. However, a key issue is introduced: to verify the applicability of a newly designed multi-axis measuring and connecting mechanism to rotors with different accuracy levels, an accurate prediction method of rotor assembly errors is highly necessary.
Studies have been devoted to proposing prediction methods for assembly errors using various mathematical models [
5,
6,
7,
8,
9,
10]. Sun et al. used a particle swarm optimization neural network to propose a prediction method for the concentricity and perpendicularity of aeroengine multistage rotors [
11]. Ding et al. predicted the overall concentric performance of rotors with optimized connection angles based on the Jacobian Torsor model [
12]. Zhang et al. predicted assembly errors by mainly considering the plane geometric form errors of multiple parts based on the kernel function [
13]. Their studies achieved reasonable accuracy in predicting assembly errors, but only considered a specific type of main error source in the assembly process.
However, owing to the use of complex multi-axis measuring and connecting mechanisms, different types of error sources are introduced into the assembly errors. The characterization and propagation of the error sources should be clarified in the prediction method to improve the prediction accuracy. The main error sources include: (1) the shape and position errors of the rotors [
14,
15,
16,
17]—the errors at the flange end faces and rabbet joint structures affect the relative position and pose between adjacent rotors. The errors at the positioning structures for other functional parts (e.g., bearings, couplings, and sealing discs) have direct impacts on the assembly errors; (2) measurement errors [
18,
19,
20,
21]—there are errors in the fitted geometric features owing to the deviations between the measured points and their nominal locations when measuring the connection structures and positioning structures; (3) mechanism errors [
22,
23,
24]—there are relative motion and static errors between each two adjacent axes in the six-axis mechanism for adjusting and connecting rotors, which cause the position and pose error of the end fixture relative to the fixed base after the connection motion; and (4) mounting errors [
25,
26,
27]—there are errors of the clamped rotor relative to its nominal position and pose in the clamping fixture on the six-axis mechanism.
Thus, the rotor assembly error prediction algorithm should integrate and simulate multiple types of error sources. The construction of the prediction algorithm faces two difficulties. First, a unified mathematical model should be used to characterize different errors, as well as to describe the error propagation among them. Second, various error sources include a lot of random error items. The initial conditions of connecting processes are also random. The prediction algorithm should be able to calculate different statistical inputs and outputs. In addition, the prediction algorithm flow should be highly consistent with the actual connection processes.
To solve the first difficulty, general homogeneous coordinate transformation matrices could be used as the unified mathematical model to represent different types of errors. A general 4 × 4 homogeneous coordinate transformation matrix
clarifies the relative spatial relationship between the coordinate systems A and B in two geometric features [
28,
29]. The column matrices
,
, and
in
represent the pose of B relative to A, and the column matrix
represents the corresponding relative position. The error propagation can be described using the error transformation matrix
. In addition, the homogeneous matrices are also convenient for the operation in a complex algorithm [
30].
In previous studies on modeling different types of errors, homogeneous coordinate transformation matrices show universality. (1) With regard to shape and position errors, Whitney et al. applied homogeneous coordinate transformation matrices to represent both the nominal relations between parts and variations caused by shape and position errors allowed by the tolerances [
31]. Wu et al. used homogeneous coordinate transformation matrices for the position calculation of geometric features in assembly tolerance analysis [
32]; (2) With regard to measurement errors, Zhang et al. established an error transfer model of an aeroengine rotor measuring machine using homogeneous coordinate transformation matrices [
33]. Zhou et al. used homogeneous coordinate transformation matrices to illustrate the common points and control points for measuring aerospace components [
34]. Similarly, the measured and fitted geometric features relative to their nominal position and pose can also be expressed by homogeneous coordinate matrices [
35,
36,
37]. Calvo et al. proposed a new minimum zone fitting algorithm for sphericity according to the measuring points containing measurement errors using homogeneous coordinate transformation matrices [
38]; (3) With regard to mechanism errors, the relative position and pose with errors between each two adjacent axes in the multi-axis mechanism can be expressed by the multiplications of homogeneous coordinate transformation matrices under a specific law, and the actual position and pose of the terminal fixture containing all mechanism errors is obtained after multi-axis transmissions. The spatial mechanical error can be accurately determined using the modified D-H method, which comprises homogeneous coordinate transformation matrices with multiple parameters [
39,
40]. Zhou et al. proposed a general mathematical model for analyzing the forward kinematics and errors of a six-axis platform based on the D-H method with multiple geometric parameters [
41]. The error differential matrix
, based on the D-H method in the multi-axis mechanism, clarifies the value and direction of the synthetic error between the two axes [
42,
43]. Chen et al. regarded the error of a single motion axis as a differential movement and then calculated the tool pose error in a four-axis mechanism using error differential matrices [
44]; (4) With regard to mounting errors, the error of the rotor relative to its clamping fixture on the six-axis mechanism is a static position and pose error, which can also be calculated using homogeneous coordinate transformation matrices [
45,
46,
47]. In general, it can be seen that a specific type of error could be described using the corresponding modified homogeneous coordinate transformation matrices. However, only a few studies have applied them to model various errors in an assembly process. Homogeneous coordinate transformation matrices and error differential matrices have the potential to compose the unified error characterization and propagation mathematical models in the rotor assembly prediction method, when using a complex multi-axis measuring and connecting mechanism.
To solve the second difficulty, the Monte Carlo method is applicable to numerical simulations of various types of statistical inputs and outputs of errors [
48,
49,
50]. There are system and random error items in each error source during the rotor-connecting processes. In practical applications, system errors are determined through repetitive experiments, and removed through accuracy adjustment or NC compensation [
51,
52]. However, random errors fluctuate within certain ranges according to the accuracy of the corresponding processes or mechanisms [
53,
54]. An accurate error value and direction cannot be determined before each connecting process. Therefore, the Monte Carlo method can be used for the statistical characterization and simulation of random errors [
55,
56]. Tang et al. introduced the positioning, straightness, pose, and axial errors of each axis of a six-axis mechanism into the Monte Carlo method to analyze the error sensitivity [
57]. Liu et al. presented an error propagation analysis method to estimate the final integrated error of an H-drive stage using the Monte Carlo method, in which a variety of error sources including straightness, thermal, deformation, and bearing gap change errors are sampled from their distributions as the inputs respectively [
58]. Thus, the Monte Carlo method could be the main process framework of the prediction algorithm for rotor connection processes, in which several types of errors are input, calculated and output. Furthermore, in contrast to existing studies, the algorithm process should simulate the actual measuring and connecting processes when using an automated multi-axis mechanism. For example, the new position and pose of the last connected rotor are measured by the CMM to guide the connecting motion of the next rotor. The flow of the prediction algorithm should be consistent with these corresponding actual processes to ensure simulation accuracy.
In this study, a systematic investigation was carried out to propose a prediction method of the rotor assembly error considering multiple error sources, based on the use of a novel multi-axis measuring and connecting mechanism. First, four main error sources that affect rotor assembly errors were investigated. The error propagation among them was indicated by the general homogeneous coordinate transformation matrices and error transformation matrices. Second, the characterization models for the four error sources were established. The rotor shape and position errors at the flange end faces, rabbet joint structures and positioning structures were modeled, and they were related to the corresponding tolerances. The measured point coordinates affected by the measurement errors were generated when measuring the annular end faces and cylindrical faces on the rotor connection and positioning structures. Then, the position and pose matrices representing the measured geometric features were derived. The errors of the six-axis rotor connecting mechanism were derived based on the modified D-H method with five parameters, and the mounting errors were modeled using the differential matrix. Subsequently, the rotor assembly position and pose errors were derived. Third, a rotor assembly error prediction algorithm was constructed based on the abovementioned error models using the Monte Carlo method. The algorithm procedure was highly consistent with the actual connection processes. The measured errors, or errors sampled from their distributions, as well as the initial conditions, were input into the algorithm. The outputs were used to analyze the applicability of the new multi-axis mechanism to the rotors with different accuracy levels. Finally, connection experiments were performed using three aeroengine rotor assemblies to verify the proposed prediction method.
This paper is organized as follows. In
Section 2, the propagation of the error sources of rotor assembly errors is derived; in
Section 3, the specific mathematical models of the different error sources are established; in
Section 4, a rotor assembly error prediction algorithm is constructed, and the results are analyzed; in
Section 5, the verification experiments are performed; and in
Section 6, the conclusions are presented.
2. Classification and Propagation of the Error Sources in Rotor Connecting Processes
The main error sources that affect the final assembly errors in the rotor connecting processes include: shape and position errors, which are related to the connection structure of the rotors; measurement errors when measuring rotor initial position and pose, as well as final assembly errors; mechanism errors of the six-axis mechanism for adjusting and connecting rotors; and mounting errors of the rotor clamping fixture on the six-axis mechanism, as shown in
Figure 1.
With regard to the propagation of these four types of errors, the shape and position errors of the rotors are direct influencing factors. This is because assembly errors are defined by the relative positions and poses of some specific positioning structures on different rotors. The shape and position errors directly determine the deviations and deflections of these specific positioning structures relative to the geometric references. In addition, the shape and position errors on rotor connecting structures, for example, flanges and rabbets, lead to deviations and deflections between adjacent rotors. At the beginning of the connecting process, the initial position and pose of the rotor should be measured, that is, the flange end face, center of the rabbet joint structure, and center of the datum connection hole (for threaded fasteners, dowel pins, etc.). In the measurement processes, owing to errors in the acquisition of point cloud coordinates, the fitted geometric features deviate and deflect from the actual features. These fitted geometric features with measurement errors are input into the NC system of the six-axis mechanism to guide the clamped rotor to make the adjusting and connecting motions. At the end of the connecting process, the relative positions and poses of the specific structures of the different rotors should be measured. The measurement errors are introduced into the assembly errors again. When the six-axis mechanism receives the linkage motion NC instruction, the static and motion errors between adjacent axes are transmitted from the fixed base of the mechanism to the end fixture. In addition, there is a mounting error in the clamped rotor relative to its nominal mounting position and pose. The mounting error and six-axis mechanism error together cause a synthetic error between the final actual position and pose of the clamped rotor and its target position and pose determined by the NC instruction.
Error propagation is described by position and pose representation matrices and error transformation matrices based on homogeneous coordinate transformations. The assembly error can be expressed using the relative position and pose transformation matrix
, which represents the transformation from the
jA-th positioning structure on the aft
iA-th rotor to the
jF-th positioning structure on the fore
iF-th rotor.
where
is the transformation matrix from B to E
ij, superscript B represents the basic coordinate system; and subscript E
ij represents the
j-th positioning structure on the
i-th rotor.
is derived as follows:
where the subscripts (or superscripts) FJ
i and AJ
i represent the fore and aft rabbet joint structures on the
i-th rotor, respectively.
Shape and position errors of rotors are introduced into Equation (2).
where
and
represent the shape and position error transformation matrices from AJ
i to FJ
i and E
ij, respectively.
The matrix in Equation (2) indicates the transformation in the connecting process of adjacent rotors. Several errors are included in .
First, the influence of measurement error is analyzed. The reference coordinate system is converted from the basic coordinate system B, to measurement coordinate system M, using
. There is a measurement error transformation matrix
when measuring the initial position and pose of FJ
i. After measurement and fitting, the obtained transformation matrix from M to FJ
i is shown as Equation (7).
The measured and fitted position and pose of FJ
i are transformed by rotation with the connecting circumferential angle
and translation with the anti-collision clearance
.
can be determined by several circumferential connecting methods, for example, random angle connection with high efficiency, high- and low-point matching that takes into account the connection quality, or
solved by a global optimization algorithm [
59].
was used to prevent the rotor end faces from colliding. Subsequently, the NC instruction guiding the (
I + 1)-th rotor mounted on the six-axis mechanism is generated, that is, the target matrix
.
where
is the rotation transformation matrix of
around the coordinate axis
; and
is the translation transformation matrix of
along the new coordinate axis
after the last transformation.
Second, owing to the influences of the six-axis mechanism error transformation matrix
and mounting error transformation matrix
, the actual position and pose of the rotor clamped on the six-axis mechanism deviate from its target position and pose.
where subscript γ represents the γ-axis clamping fixture on the six-axis mechanism.
Subsequently,
is derived as:
After the connecting process, the measurement error is introduced again when measuring the positioning structures for assembly errors, as shown in Equation (11). The final matrix
contains the information of assembly errors.
4. Construction and Application of the Rotor Assembly Error Prediction Algorithm
The Monte Carlo method is used to propose a rotor assembly error prediction algorithm based on the models of rotor shape and position errors, measurement errors, mechanism errors and mounting errors. In the prediction algorithm, the initial condition parameters, system errors, and random errors (sampled according to their related processes or mechanism accuracy) are input. Subsequently, multiple virtual rotor assemblies are simulated based on the actual connecting processes. Finally, the distributions of the assembly errors are output and analyzed.
4.1. Procedure of the Prediction Algorithm
The procedure for the rotor assembly accuracy prediction algorithm based on the Monte Carlo method is illustrated in
Figure 7. In this procedure,
N determines the total number of virtual assemblies under the same connecting conditions. An increase in
N improves the prediction reliability while reducing the calculation efficiency. In each virtual assembly, the random initial position and pose of the 1st rotor are sampled, that is,
and
in Equation (14) are sampled to constitute
. There are two methods to determine the shape and position errors of each rotor in Equations (12) and (13). In the first method, the distribution of each shape and position error is constructed according to the input rotor accuracy level, thereafter, the error value is sampled from its normal distribution, and the error direction angle is sampled from its uniform distribution. The other method is that all shape and position errors are directly input under the condition that they have been obtained through measurements.
The actual position and pose of the fore rabbet joint structure of the connected or fixed i-th rotor is determined by substituting Equations (12) and (14) into Equations (5) and (6). Subsequently, the virtual measuring points on the fore flange end face are generated using Equations (17)–(21), which are influenced by the CMM comprehensive measuring accuracy . The pose column matrix , representing the normal vector of the fore flange end face, is derived using Equations (23)–(25). The virtual projected points and of the measuring points on the fore rabbet joint structure and datum connection hole on the fitted plane OFFi-xy(fit) are generated using Equations (26)–(31). The position column matrix representing the fore rabbet joint structure center and the other two pose column matrices and are derived using Equations (33)–(35). The transformation matrix containing the measurement errors is obtained using Equations (15) and (36). The target matrix for the (i+1)-th rotor clamped on the six-axis mechanism is derived using Equation (8), where the connecting circumferential angle is calculated using its selection strategy.
The mounting error matrix
for the (
I + 1)-th rotor is derived using Equation (39). The matrix
for generating the six-axis mechanism NC instruction excludes the effect of the mounting error. Thereafter, the motion variables
dZ,
dX,
θβ,
dY,
θα, and
θγ for controlling the six axes are obtained through inverse solutions from
. Owing to the influence of static and motion errors in the six-axis mechanism, the actual position and pose
is derived by substituting
Table 1 into Equations (37) and (38). The random motion errors Δ
dZ, Δ
dX, δ
θβ, Δ
dY, δ
θα, and δ
θγ are sampled from their normal distributions, which are related to the positioning accuracy of the six axes. The uncompensated items in the other 24 static errors are substituted into
. Subsequently, the mounting error
is substituted back into
using Equation (40). Similarly, the actual position and pose of the (
I + 2)-th rotor are derived through the above iterative calculations based on the determined
of the (
I + 1)-th rotor.
When the virtual connection of the n-th assembly containing I rotors is completed, the position and pose of the positioning structures, including their shape and position errors are derived by substituting and Equation (13) into Equation (2). Similarly, measurement errors are introduced again to fit the normal vector of the annular positioning faces and the centers of the cylindrical faces of the positioning structures through Equations (17)–(25) and Equations (26)–(33), respectively. The assembly errors are calculated by substituting into Equations (1) and (42)–(51). Thereafter, the virtual connection of the (n + 1)-th assembly is performed according to the above cyclic procedures. Finally, the assembly errors , , , and of the N assemblies are output for subsequent statistical analysis.
4.2. Inputs of the Prediction Algorithm
An aeroengine low-pressure turbine rotor simulator is applied, which has three rotors in an assembly. Considering the balance of prediction reliability and calculation efficiency, the total number N of virtual assemblies under the same connecting conditions is set to 100,000.
The initial position and pose of the 1st rotor are random in each virtual assembly. The corresponding matrix elements in
obey uniform distributions, as shown in Equations (52) and (53). The matrix elements representing the datum positions are
= 613.5,
= 0,
= 820 mm.
There are two methods to input the shape and position errors in each rotor. One is to sample the shape and position errors according to the designed tolerance and process capability of the parts. The process capability index C
P is introduced into the prediction algorithm, which influences the distribution of the rotor shape and position error [
61], as shown in Equation (54). The C
P is set to 1.00, 1.33, and 1.67 (i.e., shape and position error qualification rates η = 99.73%, 99.9968%, and 99.999971%) respectively for presenting different accuracy levels of rotors. The standard deviation
of the normal distributions of the rotor shape and position errors derived from their designed tolerance
are listed in the second rows in
Table 2,
Table 3 and
Table 4. In the prediction algorithm, assuming that the system errors in the shape and position errors of each rotor were corrected during the machining processes, the mathematical expectation is
. The other matrix elements representing the error directions obey uniform distributions.
In the other input method, the measured values of the rotor shape and position errors are directly input into the corresponding matrices. The measured concrete errors of the three rotor assemblies used in the experiments are listed in the third to fifth rows in
Table 2,
Table 3 and
Table 4.
The other matrix elements representing datum dimensions of the 1st rotor are dAN1 = Φ140, dFN1 = Φ237.4, hN1 = 151, and hE11 = 54 mm.
The other matrix elements representing datum dimensions of the 2nd rotor are dAN2 = Φ299.4, dFN2 = Φ308.7, hN2 = 11, and hE21 = −24 mm.
The other matrix elements representing datum dimensions of the 3rd rotor are: dFN3 = Φ308.7, hN2 = 11, hC31 = 68, and hC32 = 1037 mm.
According to the applied CMM, the comprehensive measuring accuracy
= 0.002 mm. The measurement system errors
because they are pre-compensated in the NC system. The matrix elements representing the location of the measurement coordinate system M are
= 889.5,
= −624, and
= 67 mm. The number and nominal locations of the measuring points are listed in
Table 5.
The connecting circumferential angle is derived using the high- and low-point matching strategy which is commonly used in connecting processes, and it is also convenient for calculation in the prediction algorithm. The anti-collision clearance is set to 0.005 mm.
The static errors and motion system errors were reduced through the mechanism precision adjustment processes and the NC system, respectively. Hence, they are assumed to be zero in the prediction algorithm. The motion random errors are determined by the positioning accuracy of the drive axes and then converted to the corresponding motion execution axes. The motion error standard deviations
σΔ obtained through the repeated measurement tests are listed in
Table 6. The other D-H matrix elements representing the datum distance between the axes are
lz1 = 107,
lz2 = 206,
lx1 = 97, and
lx2 = 131 mm.
The elements in the mounting error matrix
in the prediction algorithm obey normal distributions, as shown in Equations (55) and (56). In the inverse and forward solutions of the six axes in the prediction algorithm, the measured mounting errors are excluded and introduced, respectively.
4.3. Results of the Prediction Algorithm
The distributions of the predicted six types of assembly errors under the conditions that the rotor C
P = 1.00, 1.33, and 1.67 are shown in
Figure 8. The rotor assembly errors are distributed with kurtosis
and skewness
. When C
P improves from 1.00 to 1.33, and then to 1.67, the decline rates of the average value
are 20.8% and 16.1%, respectively, the decline rates of
are 22.8% and 18.9%, respectively, the decline rates of
are 18.2% and 12.7% respectively, and the decline rates of
are 17.8% and 12.9%, respectively. However, the decline rates of
and
are small. This is because the distance between the corresponding positioning structures is large (
). The distance deviation between the center of the middle positioning structure and the long common reference line
is insensitive to variations in rotor errors.
With regard to the rotor connection qualification rate, the qualification rate of each assembly error is higher than 93%. The connection qualified condition of a rotor assembly is that all six errors are lower than their qualified upper limits. The final connection qualification rates of the rotor assemblies are 88.9%, 97.4%, and 99.3% when CP = 1.00, 1.33, and 1.67, respectively. It can be seen that even if the rotors have relatively low shape and position accuracy (CP = 1.00), the connection qualification rate is still significantly higher than that less than 50% when using traditional manual connection processes. If the rotors have general or high accuracy (CP ≥ 1.33), the connection qualification rate is higher than 97%. In general, the prediction results show that the novel mechanical system (three-axis CMM, six-axis rotor connecting mechanism, and clamping fixture) has high applicability for rotors with different accuracies.
5. Verification Experiment
To verify the proposed prediction method, three different rotors satisfying the shape and position accuracy requirements (C
P ≥ 1.00) are applied in the connection experiments. The rotor shape and position errors are presented in
Table 2,
Table 3 and
Table 4. The motion errors of the pre-compensated six-axis mechanism are listed in
Table 6. The comprehensive measuring accuracy of the applied three-axis CMM equipped with a contact probe (RENISHAW OMP40-2) is
= 0.002 mm. The number of connection experiments for each rotor assembly is 30.
The experimental setup is illustrated in
Figure 9. The experimental process of each assembly is consistent with the prediction algorithm flow, which includes the following steps: Step 1: the 1st rotor is mounted on the aft adjustable fixture separated from the six-axis mechanism. The initial position and pose are adjusted randomly according to Equations (52) and (53) using the adjustable fixture. Step 2: the 2nd rotor is mounted on the clamping fixture on the six-axis mechanism. Thereafter, the six axes move to their calibrated zero points, where the mechanism errors were corrected through precision adjustment. The current position and pose of the 2nd rotor are measured by the CMM to obtain the
. The mounting error
is derived using Equation (41). Step 3: the initial position and pose of the fore rabbet joint structure of the 1st rotor are measured using the parameters in
Table 5, and
is fitted using Equations (22)–(25) and (31)–(36). Step 4: the target matrix
for the 2nd rotor is obtained using Equation (8) with
derived using the high- and low-point matching strategy, and anti-collision clearance
= 0.005 mm. To exclude mounting errors, the motion parameters of the six axes are solved inversely according to
. Subsequently, the NC instructions of the six axes are generated to adjust the position and pose of the 2nd rotor, and connect it to the 1st rotor. Local gaskets and a few bolts are used to fasten the two rotors to minimize the impact of fastening on the assembly errors. Step 5: repeat Steps 2-4 for the connection of the 3rd rotor. Step 6: the position and pose matrices
,
,
and
of the four positioning structures are obtained according to the measuring and fitting methods similar to Step 3. The assembly errors are then derived using Equations (1), (15) and (42)–(51).
The experimental results are shown in
Figure 10. The measured shape and position errors of the rotors in the three assemblies are also input into the prediction algorithm for comparison. It can be seen that the distribution ranges of the six assembly errors obtained from 30 experiments of each assembly are within the corresponding min-max ranges of 100,000 predicted results. Compared with the corresponding average values of the six assembly errors of each assembly in the predictions, the deviation rates of the corresponding experimental results are all lower than 14%.
There are still small deviations between the distribution ranges of the experimental assembly errors and the 95% confidence intervals (CIs) of the corresponding predicted results. The deviations may be caused by the following factors: (1) In the characterization model for rotor shape and position errors, the flange SRO is based on a typical geometric feature with a single high point caused by turning or milling processes. This is not applicable to specially processed rotors with multiple high points; (2) The system errors among the six axes of the rotor connecting mechanism are pre-compensated through the mechanism precision adjustment processes and NC system. However, there may be non-zero values that are not input into the mechanism error model because of the mutual interference of a large number of errors; (3) The system errors in the three-axis CMM are not completely compensated. The fitting methods have fitting errors; (4) The effect of the force is not considered in the prediction method, for example, the connection contact force, fastening force, and gravity. In general, the proposed prediction method considers the influences of the main error sources and is highly consistent with the actual connection processes. The deviations between the predicted assembly errors and corresponding experimental results are acceptable. The proposed rotor assembly error prediction method has high accuracy and practical significance.