Clearance Analysis of Rotor–Stator Coupled Structures Under Maneuver Flight Conditions Considering Multi-Physical Fields

Abstract

In the previous studies on the clearance between rotors and stators, only a single physical field or part of the physical fields are considered. In fact, multi-physical fields (inertial moment, temperature, centrifugal, and aerodynamic load) significantly affect the clearance analysis results, especially the inertial load caused by maneuver flight. However, few studies have been found in clearance studies that can comprehensively consider the influence of various loads. To reflect the actual service environment and accurately calculate the clearance, a clearance analysis method is proposed which can consider the inertia moment, temperature, centrifugal, and aerodynamic load simultaneously. Firstly, the dynamic model of the rotor–stator coupled structure is created by using the finite element method. The coupling between blade tenons and disk grooves is realized based on the friction contact way. The bolt connection at the stator flange is simulated by the beam element, and the bearing supports at the rotor are simulated by the spring element. Furthermore, the clearance experiment platform of the rotor–stator coupled structure is constructed, and the validity of the proposed method is verified based on the clearance test results. Finally, the influence of multi-physical fields on the clearance of the rotor–stator coupled structure is investigated, and some new clearance variation rules are found.

1. Introduction

The blade-tip clearance (clearance between rotors and stators) is one of the important factors that affect the performance of aeroengines, and the clearance can directly affect the service efficiency and safety of aeroengines [1]. If the blade-tip clearance is too large, the energy consumption of the aeroengine will increase, and the energy efficiency of the aeroengine will be reduced. On the contrary, if the clearance is too small, it will reduce the service safety of the aeroengine. In serious cases, the aeroengine will have a wear-out failure, which may cause the aeroengine to be out of service, and even lead to overall damage of the aircraft. The rotor–stator coupled structure in aeroengines is often used in the multi-physical fields environment (inertial moment, temperature, centrifugal, and aerodynamic load, etc.). How to accurately consider the influence mechanism of each physical field and propose an analysis method to accurately study the clearance between rotors and stators under multi-physical fields is the key to improve the service performance of aeroengines.

Clearance analysis is a hot topic concerned by scholars, and the accurate establishment of a structural dynamic model is an important prerequisite for accurate clearance calculation.

Semi-analytical methods are often used to establish dynamic models of structures such as the plates [2,3,4], shells [5], and pipelines [6,7,8] of aeroengines. However, the finite element method is more suitable for establishing the dynamic model of complex structures [9,10,11]. The finite element method is widely used in the modeling of rotor–stator coupled structures because of its ability to establish complex models. Qu et al. [12] established the finite element model of rotors and stators by using the SOLID185 element in ANSYS, and the bearing between rotors and stators was simulated by the Conbin14 element. This modeling method has been widely used in recent studies [13,14]. The modeling method based on the solid element greatly guarantees the solution accuracy of simulation calculation, which makes the finite element model closer to the actual structure, and the calculation results are more accurate. But it brings the problem of low solving efficiency. To solve this problem, some scholars use simpler elements to establish the finite element model of rotor–stator coupled structures. For example, Mokhtar et al. [15] established a coupled rotor–stator subsystem model in which both rotor and stator structures are simulated by beam elements and coupled by bearings. Li et al. [16] and Fu et al. [17] used beam elements to simulate the rotor shaft segment and disc, and the finite element model of the coupled system was established by utilizing spring elements to simulate the bearing. Ao et al. [18] established a dual rotor–bearing–stator finite element model by simulating the shaft, disc, and stator with a beam element and shell element. Jin et al. [19] used the same modeling method to study the friction characteristics between the blade and stator. Moreover, some scholars also simplified the finite element model of the rotor–stator coupling structure to conduct the corresponding study works [20,21,22]. The above method using the beam element and shell element to model rotors and stators greatly improves solving efficiency, but also reduces the solving accuracy, which may not be suitable for the analysis and design of the clearance between rotors and stators in aeroengines.

In terms of clearance analysis, some scholars mainly focused on the influence of temperature load on blade-tip clearance. For example, Kumar et al. [23] calculated the influence of transient heat on blade-tip clearance by using a transient thermal analysis module and structural analysis module in the ANSYS Workbench software. Zhang et al. [24] analyzed the influence of high-temperature creep on the blade-tip clearance of aeroengines. Luberti D et al. [25] analyzed the influence of disk temperature distribution on blade-tip clearance. Moreover, many scholars also carried out studies on the influence of temperature load on blade-tip clearance [26,27,28,29,30]. Yepifanov et al. [31] proposed an improved IRM method, which overcomes the shortcomings of existing methods and can more accurately simulate engine transients to adapt to the dynamic changes in tip clearance caused by heating engine components. Some scholars also focused on the influence of centrifugal load on blade-tip clearance. For example, He et al. [32] conducted numerical studies on blade-tip clearance, back clearance, and back cavity clearance flow in turbines under the influence of centrifugal load. Zhao et al. [33] adopted the fluid–solid interaction method to consider blade deformation caused by centrifugal load. In addition, there are also some studies on the effect of aerodynamic loads on blade-tip clearance. For example, Manar et al. [34] explored the deformation of rotating wings under aerodynamic loads. Chen et al. [35] studied the clearance variation of a subsonic high-speed axial compressor under aerodynamic load. In addition to the above studies, there are still some scholars using different methods to explore the relationship between blade-tip clearance and aerodynamic load [36,37,38,39,40].

Furthermore, some scholars have also studied the change in blade-tip clearance under inertial load. For example, the adverse effects of the friction failure caused by the inertia load reducing the blade-tip clearance on the aeroengine have been studied in some works [41,42,43]. All the above studies provide important technical support for the clearance analysis. However, in the actual situation, aeroengines always work in a complex multi-physical fields coupling environment, and it is difficult to accurately obtain the change law of blade-tip clearance when only considering a single physical field. Therefore, it is necessary to carry out clearance analysis under multi-physical fields.

Some scholars have carried out clearance analysis considering different physical fields. For example, Fei et al. [44] and Han et al. [45] explored the change in blade-tip clearance when temperature load and centrifugal load acted at the same time. Li et al. [46] carried out a study on the change in blade-tip clearance under the simultaneous action of centrifugal load and inertial load. Moreover, the simultaneous effects of temperature load, centrifugal load, and aerodynamic load on blade-tip clearance were completed in some studies [47,48,49]. However, although some scholars have carried out clearance analysis under multi-physical fields, there are still some differences from the actual situation. In practice, the aeroengine is subjected to more loads during operation, including centrifugal load, aerodynamic load, temperature load, and inertial moment. During the flight, the flight state of the aircraft often changes. During the changing process, the inertial moment generated by the aeroengine has a significant influence on the blade-tip clearance of the aeroengine, and in severe cases, the aeroengine may suffer from serious failures such as grinding. Therefore, among the loads affecting the blade-tip clearance of the aeroengine, the inertial moment caused by maneuver flight is particularly important. At present, few findings have been found on clearance analysis under the influence of inertial moments. Therefore, the proposed clearance analysis method that can consider the coupling effect of inertial moments and other load environments (centrifugal load, aerodynamic load, temperature load) can effectively make up for the shortcomings in the current clearance analysis field and lay an important foundation for the accurate clearance prediction of aeroengines.

In view of the above research gaps, the effects of inertial moment, centrifugal, aerodynamic, and temperature load on blade-tip clearance caused by maneuver flight are fully considered in this paper. Based on the finite element method, a general clearance analysis method under multi-physical fields is proposed, and a systematic and comprehensive experimental study is carried out to verify the accuracy of the proposed method. Furthermore, the influence mechanism of load parameters and geometric parameters on blade-tip clearance is explored. The framework of this paper is as follows: In Section 2, the finite element model of the rotor–stator coupled structure is established by using the modeling method of beam–solid–shell element coupling, and the key modeling techniques are emphatically described, including the coupling method of blade tenons and disk grooves, the simulation of the bolt connection, and the coupling simulation of the rotor and stator. In Section 3, the static analysis methods of rotor–stator coupling structures under centrifugal load, aerodynamic load, temperature load, and inertial moment are given, and the models used to calculate the effects of these loads on blade-tip clearance are also described. Further, the clearance analysis methods under multi-physical fields are proposed. In Section 4, finite element modeling is carried out with the actual rotor–stator coupled structure as the research object, and the accuracy of the model is verified based on the experimental results. Then the clearance analysis is carried out. In Section 5, important conclusions are given.

2. Finite Element Modeling Method of Rotor–Stator Coupled Structure

According to the characteristics of the rotor–stator coupled structure, the finite element modeling idea of the rotor–stator coupled structure in ANSYS software is described in this section. Moreover, the modeling process and key technologies in the modeling process (the coupling method of blade tenons and disk grooves, the simulation of bolt connection, and the coupling simulation of rotor and stator) are also given.

2.1. Finite Element Modeling Idea of Rotor–Stator Coupled Structure

The simulated rotor–stator coupled structure is mainly composed of a rotor and stator. To ensure the high solving accuracy and efficiency of the finite element model, different element types are used to create the finite element model according to the structural characteristics and mechanical properties. The modeling method of beam–solid–shell element coupling is adopted here. The rotor–stator coupled structure shown in Figure 1 is used as an example to illustrate this modeling idea. It should be noted that the geometric structure in Figure 1 has been simplified. It can be seen from the model shown in Figure 1 that the rotor blades are relatively long. The model in this paper mainly refers to low-pressure compressors. The rotor–stator coupled structure in actual aeroengines is extremely complex. If the actual structure is completely simulated, the efficiency of finite element analysis will be significantly reduced. This paper mainly proposes the gap analysis method of the rotor–stator coupled structure under multi-physical fields and clarifies the influence law of different fields on the gap. To enhance the analysis efficiency, this paper simplifies the rotor–stator coupled structure. However, the gap analysis method under multi-physical fields proposed in this paper is applicable to real structures, and the influence laws of different fields on the gap obtained in this paper are still applicable to real structures.

It should be noted that this paper primarily focuses on the introduction methods for multi-physical fields and the calculation method for clearances under the action of multi-physical fields. Although the influence patterns of different physical fields on the clearance obtained are correct, the calculated clearance values are not the actual real clearances of the rotor–stator coupled structure in aeroengines. To obtain real clearance values, it is necessary to refer to the actual rotor–stator coupled structure in aeroengines, create a precise the finite element model of aeroengines, and then use the clearance analysis method proposed in this paper to calculate the clearance values under multi-physical fields.

During the finite element modeling, the solid element and shell element with higher calculation accuracy are used for components with a large deformation area or complex structural characteristics, and the beam element with higher calculation efficiency is used for components with a small deformation area or simple structural characteristics. Then, the rotor is coupled with the stator through the bearing, and the finite element model of the rotor–stator coupled structure is created.

Under the action of unbalanced excitation, the region with large deformation appears in the blade and spoke region, as shown in Figure 2. The main reason is that under the action of unbalanced excitation, the rotor vibration is caused, and the vibration is transmitted to the stator through the bearing, so that the stator vibration and deformation occur. At the same time, the rotor will be subjected to centrifugal force due to rotation, and the rotor blade will also produce large deformation under the combined action of unbalanced excitation and centrifugal force. Therefore, elements with higher computational accuracy are needed in the blade and spoke plate region. Moreover, the stator is a cylindrical shell, and the mechanical properties of the stator can be expressed by the shell element. The structure of the blade disk is more complex, so it is necessary to use solid element modeling. Finally, to improve calculation efficiency, the shaft stop ring, bearing end cover, and fixed base, which only play a fixed role but do not participate in vibration and deformation, are simplified. In finite element modeling, the shaft stop ring and bearing end cover are not modeled, and the fixed base is simplified to apply full constraints on the corresponding position of the stator shell.

Considering the above factors, the finite element modeling idea of the rotor–stator coupled structure based on the beam–solid–shell element coupling method can be described in Figure 3. It can be expressed as follows: the blade and disk are modeled by the solid element in the finite element model of the rotor; the shaft is modeled by the beam element; the shell element is used to model the spoke plate, shell, and flange in the finite element model of the stator.

2.2. Dovetail Connection Between Blade Tenons and Disk Grooves

The coupling between blade tenons and disk grooves is called the dovetail connection.

The rotor blade and the blade disk are connected through tenons and grooves, as shown in Figure 4. When the rotor is in a static state, the circumferential side of tenons does not contact the side of grooves, and there is a certain gap. When the rotor rotates, the blade will be thrown out along the radial direction due to the centrifugal force. At this time, the side of tenons and the side of grooves will fit together, and with the increase in the rotating speed, this fit will be closer.

In this paper, the friction contact pair is established to complete the coupling between tenons and grooves. The contact element and the target element are used to establish a friction contact pair between the side of tenons and grooves, as shown in Figure 4. In addition, to avoid the relative sliding between tenons and grooves in the axial direction, the axial freedom degree of tenons is constrained. It is also necessary to establish a friction contact between the two axial sides of tenons and the blade stop ring, as shown in Figure 5. The contact element between the side of the tenons and grooves (and the axial side of tenons and the blade stop ring) is selected as the CONTA174 element, and the target element is selected as the TARGE170 element.

2.3. Simulation of Bolt Connection

For the stator, the finite element modeling is mainly carried out on the three-segment cylindrical shell and the spoke plate. The cylindrical shell and the spoke plate are connected by bolts, as shown in Figure 6. During the modeling process, shell elements are used to model the cylindrical shell and the spoke plate. To improve calculation efficiency, the bolts which play the role of connection can be simplified. The beam element with higher computational efficiency is used to simulate the bolt connection between the flange on the cylindrical shell and the spoke plate. The connection mode is shown in Figure 6. The specific operation is as follows: a beam element is established between the shell elements representing the flange and the spoke plate, and the node in the beam element is used as the main node. Moreover, a connection is established between the main node and the shell element node representing the influence zone of bolts.

2.4. Coupling Between Rotor and Stator

The rotor and stator are mainly coupled through bearings. Spring elements are used to simulate bearings during modeling in this section. The COMBI214 spring element is specially used to represent the bearing, and the node freedom of the element can be set according to requirements. The schematic diagram of the spring element simulating the bearing is shown in Figure 7. A node on the beam element representing the shaft is used as the rotating end of the bearing, and a node on the shell element representing the bearing seat is used as the fixed end. The spring element is established between these two nodes. Moreover, spring elements should be established between the spoke plate and the shaft, and only one spring element need be established in the radial direction at each place.

2.5. Simulation of Unbalance

In the actual structure, due to the existence of machining deviation, the neutrality of the whole structure will be reduced, so that the unbalanced factors will have an obvious impact on the structure. To make the finite element model of the coupled structure closer to the actual structure, the method of applying force F to the model in the radial direction is adopted to simulate the unbalance factors, as shown in Figure 8. Moreover, the Coriolis effect is activated, and the excitation frequency is synchronized with the speed frequency of the structure.

3. Clearance Analysis Method Under Multi-Physical Fields

3.1. Introduction of Multi-Physical Fields

The operating conditions of aircraft including cruising conditions and maneuvering flight conditions are considered in this paper. Under cruising conditions, the rotor–stator coupled structure is subjected to centrifugal load, aerodynamic load, and temperature load. Under maneuvering flight conditions, the rotor–stator coupled structure also bears additional inertial moment. The specific situation is listed in

Table 1. The force situation of the structure under different operating conditions.

Operating Conditions	Cruising condition	Maneuvering flight condition
Force Situation	Including centrifugal load, aerodynamic load, and temperature load	Including centrifugal load, aerodynamic load, temperature load, and inertial moment

. According to the different loads on the rotor–stator coupled structure, how to apply the corresponding loads in the finite element modeling is described in this section, including centrifugal load, aerodynamic load, temperature load, and inertial moment.

3.1.1. Introduction of Centrifugal Load

During the operation of the aeroengine, the rotor rotates, and the centrifugal force generated by the rotor will mainly act on the rotor blade and disk, which will cause the blade and disk to produce radial deformation. The blade-tip clearance will also change. Moreover, the connection between the blade and the disk is realized by means of grooves and tenons. Under the action of centrifugal force, tenons will exert a radial tension on grooves, which will result in the radial deformation of grooves. In addition, the displacement of the blade along the radial direction will also cause changes in the blade-tip clearance. In this section, the methods of introducing centrifugal force into the finite element models of rotor and rotor–stator coupled structures are presented, respectively.

(1)

Introduction of centrifugal load in the rotor

Centrifugal load is introduced by applying rotational speed to the rotor in ANSYS, as shown in Figure 9. After the finite element model is established, the model itself has a mass, so the centrifugal load can be introduced by directly applying the rotational speed to the rotor. Here the “OMEGA” command is used to apply the rotational speed to the finite element model of the rotor.

(2)

Introduction of centrifugal load in the rotor–stator coupled structure

In the rotor–stator coupled structure, only the rotor does the rotating motion, and the stator is stationary. So, the centrifugal force applied to the coupled structure cannot be used by the “OMEGA” command, but by the “CMOMEGA” command to introduce centrifugal load. The function of this command “CMOMEGA” is to apply a rotation speed around a user-defined rotation axis to one of the components in the complex structure.

3.1.2. Introduction of Aerodynamic Load

In the compressor of the aeroengine, the gas flowing into the aeroengine is continuously compressed, and the air pressure inside the aeroengine is constantly increasing, which makes the various components inside the aeroengine bear the pressure generated by the compressed gas. This pressure is called the aerodynamic load, and the aerodynamic load mainly acts on the stator casing. The aerodynamic load acts radially outward on the stator casing. This tends to expand the casing radially.

Aerodynamic force is often applied in the form of uniformly distributed surface loads. Figure 10 illustrates the schematic diagram for applying aerodynamic load to the casing. The procedure involves selecting the target surface on the casing and then applying the surface load to simulate the aerodynamic load effect on the stator. Note that the surface load is applied radially, directed outward from the center of the rotor–stator coupled structure. Since the stator in this study is modeled using shell elements, the effective direction of the load depends not only on the sign (positive or negative) of the load value but also on the orientation of the shell element’s thickness direction. By convention, a positive load value typically corresponds to a force acting along the positive thickness direction of the shell element. The specific direction must be verified based on the actual finite element model configuration.

3.1.3. Introduction of Temperature Load

When a low-pressure fan and high-pressure compressor compress air, they not only increase the air pressure inside the aeroengine but also increase the temperature at the parts of the low-pressure fan and high-pressure compressor. As a result, the stators and rotors at these two positions are heated and expanded, thus changing the blade-tip clearance. At the same time, the high temperature generated by the combustion chamber will be transmitted forward and backward to the high-pressure compressor air outlet, high-pressure turbine, and low-pressure turbine, which will cause the expansion of the blades, disk, and stator at the high-pressure compressor outlet stage, high-pressure turbine, and low-pressure turbine, resulting in changes in the blade-tip clearance at these parts.

Due to the large influence of temperature load on blade-tip clearance, it is necessary to calculate the influence of temperature load on the clearance in a way with higher calculation accuracy. Therefore, the radial deformation of the stator and rotor are calculated by using the finite element model of the stator and rotor, respectively, and then the change in blade-tip clearance is calculated. In this section, the method of applying temperature loads to the finite element models of the stator and rotor are described, respectively.

(1)

Application of temperature load in the stator

Due to the difference between the temperature when the gas enters the aeroengine casing and the temperature when it leaves the aeroengine casing, the temperature before and after the aeroengine casing is also different. Therefore, the thermal balance of the aeroengine casing needs to be calculated before the radial thermal deformation of the casing is calculated. For this finite element model, the calculation of its radial deformation under temperature load needs to be divided into two steps: the first step is to calculate the thermal balance of the stator; the second step is to calculate the radial thermal deformation of the stator. In the calculation for the thermal balance of the stator, the SHELL181 element used in the establishment of finite element model needs to be converted into the SHELL131 element specially used for thermal analysis. Then, different temperatures are applied to the front node of the first shell, the rear node of the third shell, and the nodes on the spoke plate between the first shell and the second shell in the finite element model of the stator, as shown in Figure 11. Furthermore, the thermal deformation calculation is further carried out on the basis of heat balance calculation.

(2)

Application of temperature load in the rotor

In the rotor, the temperature at the rotor blade is higher and the temperature at the center of the blade disk is lower, which means that for the calculation of radial thermal deformation of the rotor as well as the calculation of stator radial thermal deformation, there is a process of calculating thermal balance. Therefore, when calculating the radial deformation of the rotor under temperature load, it is also necessary to divide it into two steps: the first step is to calculate the thermal balance of the rotor; the second step is to calculate the radial thermal deformation of the rotor. When performing thermal balance calculations, the SOLID186 entity element needs to be transformed into the corresponding SOLID90 entity element dedicated to thermal analysis, and the degrees of freedom of the contact element at the tenon and groove need to be changed from UX, UY, and UZ to TEMP. In addition, the thermal expansion coefficient of the rotor needs to be defined. Furthermore, the thermal balance calculation is carried out. After the thermal balance calculation is completed, the result of the thermal balance calculation can be used as the input load for the radial thermal deformation calculation, and then the radial thermal deformation can be obtained. The initial temperature load distribution in the rotor is shown in Figure 12.

(3)

Application of temperature load in the rotor–stator coupled structure

The temperature load applied to the finite element model of the rotor–stator coupled structure is also divided into two steps: thermal balance calculation and radial thermal deformation calculation; and the temperature application method is similar to the temperature application method in the finite element model of the rotor. When calculating thermal balance, the SHELL181 shell element and SOLID186 solid element need to be converted into the SHELL131 element and SOLID90 element. Then, different temperatures are applied to the surface at the corresponding position. On the basis of defining the thermal expansion coefficient, the thermal balance of the rotor–stator coupled structure is calculated. Finally, based on the thermal balance calculation results, the radial thermal deformation is calculated.

3.1.4. Introduction of Inertial Moment

The maneuvering flight condition is more common in the aircraft service process, which mainly includes temporary climb and hover flight, and the overall movement of the aircraft is revolution around a point in space. In maneuvering flight conditions, the whole aeroengine will bend due to the action of centrifugal force, but the bend degree of the stator and rotor is different. For the stator, because it does not rotate, it is only affected by centrifugal force during maneuver flight, and its bending and radial deformation mainly depends on the magnitude of centrifugal force. However, for the rotor, due to its rotating motion, it is not only affected by centrifugal force, but also affected by Coriolis force when doing maneuver flight, so the radial deformation of the rotor is mainly related to the joint action of centrifugal force and Coriolis force. As shown in Figure 13, when the rotor is at rest and subjected to centrifugal action, the rotor disk displaces in the radial direction. When the rotor is in the rotating state, the rotor disk will not only displace in the radial direction, but also deflect to one side. Because the stator and rotor have different radial deformation under inertial moment, the blade-tip clearance will change.

Maneuvering flight similar to aircraft turning or climbing is considered in this paper. Maneuvering flight will cause the structure to bear inertial moment. Because the inertial moment acts on both the stator and rotor, the finite element model of the rotor–stator coupled structure is used to calculate the effect of inertial moment on blade-tip clearance. The schematic diagram of the specific application method is shown in Figure 14. Since the inertial moment is generated when the rotor is rotating and the coupled structure is rotating around a revolution center, the rotating speed of the rotor must be applied, and then the rotor–stator coupled structure is applied with a rotating speed around the revolution center, and the Coriolis force should be taken into account. In Figure 14, the revolution of the structure is completed around the y-axis in the xoz plane.

3.2. Clearance Calculation Method

In this section, the calculation method of the blade-tip clearance under each physical field and when different physical fields exist at the same time is given. In Section 3.2.1, the calculation method of clearance is given in the presence of centrifugal, aerodynamic, and temperature load simultaneously or partially. In Section 3.2.2, a calculation method is given to consider the clearance in the presence of the inertial moment and some or all of the other loads (centrifugal, aerodynamic, and temperature load).

3.2.1. Clearance Calculation Under Centrifugal, Aerodynamic, and Temperature Load

(1)

Clearance calculation under centrifugal load

Since the stator of the rotor–stator coupled structure does not undergo deformation under centrifugal load, and only the rotor is affected by centrifugal load, the clearance change in the rotor–stator coupled structure under centrifugal load is equal to the radial blade-tip deformation of the rotor under centrifugal load. The clearance calculation method under centrifugal load can be described as

$$X_{\mathrm{c}}=X_0-{\delta }_{\mathrm{c}}$$

(1)

where $X_0$ represents the initial clearance; ${\delta }_{\mathrm{c}}$ represents the radial blade-tip deformation of the rotor under centrifugal load.

(2)

Clearance calculation under aerodynamic load

The aerodynamic load mainly acts on the stator of the rotor–stator coupled structure, so the radial deformation of the stator under the aerodynamic load is calculated. The clearance change in the rotor–stator coupled structure under aerodynamic load is equal to the radial deformation of the stator under aerodynamic load. The clearance calculation method under aerodynamic load can be given by

$$X_{\mathrm{a}}=X_0+{\delta }_{\mathrm{a}}$$

(2)

where ${\delta }_{\mathrm{a}}$ represents the radial blade-tip deformation of the stator under aerodynamic load.

(3)

Clearance calculation under temperature load

The temperature load acts on the rotor and stator at the same time and has a great influence on the rotor and stator. Therefore, when calculating the influence of temperature load on the clearance, it is necessary to consider the radial deformation of the rotor and the radial deformation of the stator at the same time. The radial deformation of the rotor–stator coupled structure under temperature load is equal to the difference between the radial deformation of the stator and the radial deformation of the rotor. Specifically, the clearance calculation method under temperature load can be written as

$$X_{\mathrm{t}}=X_0-\left({\delta }_{\mathrm{tr}}-{\delta }_{\mathrm{ts}}\right)$$

(3)

where ${\delta }_{\mathrm{ts}}$ represents the radial blade-tip deformation of the stator under temperature load, and ${\delta }_{\mathrm{tr}}$ represents the radial blade-tip deformation of the rotor under temperature load.

In the above Equations (1)–(3), the calculation methods of blade-tip clearance under centrifugal, aerodynamic, and temperature load are given, respectively. The calculation method of the blade-tip clearance when considering simultaneously centrifugal, aerodynamic, and temperature load is expressed as

$$X=X_0+{\delta }_{\mathrm{a}}-{\delta }_{\mathrm{c}}+{\delta }_{\mathrm{ts}}-{\delta }_{\mathrm{tr}}$$

(4)

3.2.2. Clearance Calculation Considering Inertial Moment

In this section, the calculation method of clearance when the inertial moment is considered only and the influence of other loads is excluded is first given, and the calculation method of the clearance when considering simultaneously the influence of inertial moment and other loads is also given. Similar to the temperature load, inertial moment acts on both the rotor and stator and has great influence on both the rotor and stator. Therefore, when calculating the influence of inertia load on blade-tip clearance, the radial deformation of the rotor and stator should be considered at the same time, and the clearance change can be obtained by making the difference between them.

It should be noted that when inertial load is applied to the rotor–stator coupled model in Section 3.1.4, rotating velocity is also applied to the rotor. Therefore, centrifugal load is also introduced to calculate the effect of inertial load on the clearance. In addition, considering that the rotor–stator coupled structure is subject to inertial moment only in the case of maneuvering flight, at this time, the rotor–stator coupled structure is not only subject to the action of inertial moment and centrifugal load, but also to the action of temperature and aerodynamic load. The effect of inertial moment on blade-tip clearance under the action of temperature and aerodynamic load is different from that of inertial moment at normal temperature and pressure. Therefore, it is necessary to consider the influence of temperature and aerodynamic load on the clearance of the rotor–stator coupled structure when introducing inertial moment. The calculation method of clearance when inertial moment, centrifugal, aerodynamic, and temperature load are considered simultaneously can be expressed as

$$X_4=X_0+{\delta }_{\mathrm{a}}-{\delta }_{\mathrm{c}}+{\delta }_{\mathrm{ts}}-{\delta }_{\mathrm{tr}}+{\delta }_{\mathrm{is}}-{\delta }_{\mathrm{ir}}$$

(5)

where ${\delta }_{\mathrm{is}}$ and ${\delta }_{\mathrm{ir}}$ represent the radial blade-tip deformation of the stator and rotor under inertial moment, respectively.

It is valuable to study the influence of the inertial moment on the clearance of the rotor–stator coupled structure without other loads (centrifugal, aerodynamic, and temperature load) in the mechanism study of the clearance analysis. As mentioned above, other loads need to be considered when introducing the inertial moment, to exclude the influence of other loads. In this paper, the clearance value $X_4$ is obtained when the inertial moment and other loads exist at the same time, and the clearance change value ${\delta }_3$ of the structure is calculated when the centrifugal load, aerodynamic load, and temperature load exist and inertial moment does not exist. By differentiating $X_4$ and ${\delta }_3$, the blade-tip clearance of the structure can be obtained when only the inertial moment is considered. The specific expression for calculating the clearance when only considering the inertial moment can be expressed by

$$X_{\mathrm{i}}=X_4-{\delta }_3$$

(6)

where

$${\delta }_3={\delta }_{\mathrm{c}}-{\delta }_{\mathrm{a}}-{\delta }_{\mathrm{ts}}+{\delta }_{\mathrm{tr}}-{\delta }_{\mathrm{is}}+{\delta }_{\mathrm{ir}}$$

(7)

4. Case Study

4.1. Experimental Structure Description

Taking a concrete rotor–stator coupled structure as an example, inertial moment, centrifugal load, aerodynamic load, and temperature load are applied to the finite element model of the rotor–stator coupled structure. The specific composition of the rotor–stator coupled structure is described in Figure 15, which is mainly divided into two parts: stator and rotor. The inner diameter, wall thickness, and flange thickness of the stator shell are 321 mm, 2 mm, and 3 mm, respectively. The length of the three shells is 157 mm, 124.5 mm, and 53 mm, respectively. The diameter of the hole used to measure the gap is 20 mm. The thickness and diameter of the spoke plate are 3 mm and 345 mm, respectively. The thickness of the fixed base is 20 mm. In the rotor, the outer diameter, inner diameter, and thickness of the blade disk are 60 mm, 20 mm, and 10 mm, respectively. The length, width, and thickness of the blade are 110 mm, 10 mm, and 2 mm, respectively. The specific type and parameters of blades are listed in

Table 2. Specific type and parameters of blades.

Type	Mean Camber Angle	Length	Width	Thickness
Non-swirled blade	0	110 mm	10 mm	2 mm

. The height, thickness, and bottom width of the tenon are 5 mm, 10 mm, and 9.5 mm, respectively, and the angle between the hypotenuse and the bottom is 63°. The length of the shaft is 277 mm. The depth, thickness, and bottom width of the groove are 5 mm, 10 mm, 10 mm, respectively, and the angle between the hypotenuse and the bottom is also 63°. The materials of each part of the system are aluminum alloy. In the following, the correctness of the model will be verified first, and then the blade-tip clearance analysis of the rotor–stator coupled structure is conducted.

4.2. Model Verification

4.2.1. Finite Element Model of Rotor–Stator Coupled Structure

The finite element model of the concrete rotor–stator coupled structure provided in Section 4.1 is shown in Figure 16. The finite element model of the stator is shown in Figure 16a, in which the SHELL181 element is used to model the shell, spoke plate, and bearing seat, and the BEAM188 element is used to simulate the bolt. The finite element model of the stator has 97,548 elements and 99,885 nodes. The finite element model of the rotor without the blade is shown in Figure 16b. The SOLID186 element is used to model the middle disk, the BEAM188 element is used to model the shaft, and the COMBI214 element is used to simulate the bearing. Two virtual nodes are established on the outside of the axis to connect with the nodes on the axis, full constraints are applied, and axial constraints are applied to the nodes on the axis. The finite element model of the rotor has 33,338 elements and 141,887 nodes. The finite element model of the coupled structure is shown in Figure 16c. The SOLID186 element is used to model the blade, and the CONTA174 element and TARGE170 element are used to simulate the friction contact between the tenon and groove. The finite element model of the coupled model has 130,922 elements and 241,808 nodes.

4.2.2. Modal, Thermal Deformation, and Clearance Test

The experimental tests in this section mainly include the modal test, the thermal deformation test, and the clearance test of the rotor–stator coupled structure. The modal test experiment of the rotor and stator is mainly used to modify the finite element model by the experimental results, and the thermal deformation experiment of the rotor and stator is mainly used to verify the correctness of the temperature load introduction method. The clearance test experiment of the rotor–stator coupled structure is used to verify the correctness of the model modification and the accuracy of the clearance analysis results considering multi-physical fields in this paper.

(1)

Modal test of stator

The modal test experiment of the stator is mainly used to modify the geometric and material parameters of the beam element used to simulate the bolt in the model of the stator. The modal test belongs to the basic experiment work in the dynamic research field, and the experimental system is shown in Figure 17. The stator is fixed on the test bench, and there are 20 M8 bolts on the fixed base used to fix the stator. The flange of each cylindrical shell and spoke plate are connected by 36 M5 bolts. Then, the natural characteristics of the stator are tested by the hammer method. At the same time, a PCB SNLW246108 three-way acceleration sensor fixed on the cylindrical shell is used to pick up vibration, and the LMS is used to collect data.

(2)

Modal test of rotor

The modal test of the rotor is mainly used to modify the stiffness K_x and K_y of the spring element used to simulate the bearing, and the experimental system is shown in Figure 18. The rotor is fixed on the test bench through the fixture, and there is a total of 12 M8 bolts with a preload of 18 Nm on the two clamps and bearing seats. The motor is connected to the shaft by coupling, and the motor shaft is locked by a PLC controller during modal testing to fix the circumferential freedom of the rotor. Then, the natural frequency and modal shape of the rotor are measured by the hammer method.

(3)

Clearance change test

The experimental system is shown in Figure 19. The rotor–stator coupled structure is fixed on the test bench through a fixed base. The motor is fixed on the test bench through the fixture, and the motor speed is controlled by the PLC controller (Yiwu, Zhejiang, China: Zhejiang Zhicheng Technology Co., Ltd.). An RP660607-01-04-50-00 Eddy current displacement sensor (Shanghai, China: Shanghai Ranpu Electronic Technology Co., Ltd.) is inserted through a small hole on the stator, and the distance from the blade tip is 1 mm. The speed of the motor is adjusted by the PLC controller to change the speed of the blade sweeping through the eddy current sensor, so that the clearance change can be measured at different speeds. During the test, when the blade passes the eddy current sensor, electromagnetic induction will be generated, so that the electrical signal obtained by the eddy current sensor will change. According to this change, the distance change between the blade tip and the eddy current sensor can be obtained, and the change in the clearance can be obtained according to the initial clearance and the distance between the eddy current sensor and the stator. It should be noted that when the blade tip is attached with the eddy current sensor, the value measured by the experiment is 0.75 mm. Therefore, when testing the distance from the blade tip to the eddy current sensor, it is necessary to subtract 0.75 mm from the value obtained by the test. In summary, the formula for measuring the change in blade-tip clearance can be described as

$$\Delta d=\left[L+\left(\Delta l-0.75\right)\right]-d$$

(8)

where Δd represents the change value of blade-tip clearance; d represents the initial clearance; L represents the distance between the eddy current sensor and the stator; Δl represents the distance between the blade tip and the eddy current sensor obtained by the experiment.

(4)

Thermal deformation test of stator

The experimental system for measuring stator thermal deformation is shown in Figure 20. The stator is fixed on the experimental bench, and the stator is heated by the CDN-RT06TQL heater (Mianyang, Sichuan, China: Sichuan Changhong Electric Co., Ltd.). The temperature of the stator is measured by a UT306S infrared thermometer (Guangdong, China: Ulide Technology (China) Co., Ltd.), and the deformation of the stator is measured by a laser displacement sensor. In the process of the experiment, five temperature-measuring points (T1–5) are arranged on the stator, and the average temperature of these five points is taken as the actual temperature of the stator. At the same time, two deformation measuring points (D1–2) are arranged on the three o’clock direction (deformation measuring point 1) and nine o’clock direction (deformation measuring point 2) of the stator, and the deformation values of these two points are tested. Here, after the KEYENCE IL-600 laser displacement sensor (Shanghai, China: Keyence Co., Ltd.) is fixed, the value displayed is 0. When the temperature of the stator is stable, the value displayed by the laser displacement sensor is the deformation value of the stator.

(5)

Thermal deformation test of rotor

The experimental system for measuring rotor thermal deformation is shown in Figure 21. The rotor is fixed on the test bench, and the shaft is connected with the motor through the coupling. The motor is fixed on the test bench through the fixture, and the motor speed is controlled by the PLC controller. The rotor is heated by a CDN-RT06TQL heater (Mianyang, Sichuan, China: Sichuan Changhong Electric Co., Ltd.), the temperature of the rotor is measured by a UT306S infrared thermometer (Guangdong, China: Ulide Technology Co., Ltd.), and the deformation of the rotor is measured by a KEYENCE IL-600 laser displacement sensor (Shanghai, China: Keyence Co., Ltd.).

During the experiment, the tips of six blades are taken as the temperature-measuring points of the experiment, and the average temperature of these six measuring points is taken as the actual temperature. When the temperature of the rotor tends to be stable, the blade deformation at the three o’clock direction (blade 1) and nine o’clock direction (blade 4) is measured.

4.2.3. Model Modification Based on Modal Test Results

According to the modeling method described in Section 2, there are some uncertain parameters in the finite element model of the rotor–stator coupled structure, including the elastic modulus E, density ρ, Poisson’s ratio ν, and the diameter D of the beam element used to simulate bolts; the stiffness K_x and K_y of the spring element are used to simulate the bearing. These parameters will affect the correctness of the finite element model of rotor–stator coupled structure and also affect the accuracy of the analysis results obtained by using this model. Therefore, these parameters need to be modified. Here, the finite element model of the structure is modified based on the modal experiment results. The initial values of parameters to be modified in the simulation calculation of natural characteristics are shown in

Table 3. Initial value of the parameters to be modified.

Parameters	Initial Value
Elastic modulus of beam element E	2 × 1011 Pa
Density of beam element ρ	7850 kg·m−3
Poisson’s ratio of beam element ν	0.3
Diameter of the beam element D	2.5 mm
Stiffness of the spring element Kx	2 × 1011 N·m−1
Stiffness of the spring element Ky	2 × 1011 N·m−1

.

(1)

Modification of finite element model of stator

Based on the initial values of elastic modulus E, density ρ, Poisson’s ratio ν, and diameter D of the beam element in Table 3, the natural frequencies and modal shapes of the stator obtained by simulation are compared with the experimental results in

Table 4. Comparison between natural frequencies obtained by experiment and simulation.

Order	Experiment Results/Hz	Simulation Results/Hz	Difference/%
1	594.2	549.95	7.4
2	695.031	649.22	1.5
3	809.314	797.07	1.5
4	978.431	937.19	4.2
5	1035.018	1109.124	7.2

and

Table 5. Comparison between modal shapes obtained by simulation and experiment.

Order	1	2	3	4	5
Experiment results
Simulation results

. It can be seen from Table 5 that the modal shapes of the stator obtained by experiment and simulation are basically consistent. As can be seen from Table 4, the deviation between the simulation results of the second- and third-order natural frequencies and the experimental results is small, while the simulation results of the first and last two natural frequency orders are significantly different from the experimental results. Therefore, it is necessary to modify the parameters of the beam element used to simulate bolts to reduce the deviation between the simulation results and the experimental results and ensure the rationality of the finite element model of the stator.

During the process of model modification, when there are many parameters to be corrected, it will lead to a decrease in the modification accuracy. To ensure the accuracy of parameter modification, the four most critical parameters requiring identification are identified by using sensitivity analysis. The sensitivity analysis of the material and geometric parameters of the beam element is carried out before implementing the modification algorithm. The elastic modulus E, density ρ, Poisson ratio ν, and diameter D of the beam element are taken as random input parameters, and the square of the deviation between the natural frequencies obtained by simulation and experiment is taken as random output parameters. The sensitivity analysis results are shown in Figure 22. The elastic modulus E and density ρ of the beam element have a great influence on the natural frequency of the stator, and the influence of Poisson’s ratio ν and diameter D of beam element on the natural frequency of the stator is not significant. Therefore, these two parameters (E and ρ) are selected as the design variables of the correction algorithm.

After the sensitivity analysis, the elastic modulus E and density ρ of the beam element are brought into the modification study based on the genetic algorithm as the design variables. Finally, the modified parameters of the beam element are shown in

Table 6. Parameters of beam element before and after modification.

Parameters	Before Modification	After Modification
Elastic modulus of beam element E/Pa	2 × 1011	1.967 × 1012
Density of beam element ρ/kg·m−3	7850	17,500

, and the comparison between natural frequencies obtained by simulation and experiment is shown in

Table 7. Comparison between natural frequencies obtained by simulation and experiment.

Order	Experiment Results/Hz	Simulation Results/Hz	Difference
1	594.2	577.84	2.80%
2	695.031	665.17	0.89%
3	809.314	807.22	0.26%
4	978.431	989.69	1.15%
5	1035.018	1038.90	0.38%

. It can be seen from Table 7 that after the model modification, the differences between the simulation results and the experimental results of the first five natural frequencies orders are significantly reduced, where the maximum and minimum differences are 2.80% and 0.26%, respectively.

(2)

Modification of finite element model of rotor

Based on the initial values of stiffness K_x and K_y of the spring element used to simulate the bearing, the natural frequencies and modal shapes of the rotor obtained by simulation are compared with the experimental results in

Table 8. Comparison between natural frequencies of rotor obtained by experiment and simulation.

Order	Experiment Results/Hz	Simulation Results/Hz	Difference
1	511.125	544.51	7.02%
2	3085.554	3505.3	11.90%

and

Table 9. Comparison of modal shapes between simulation and experiment of rotor.

Order	Experiment Results	Simulation Results
1
2

. It can be seen from Table 9 that the modal shapes of the rotor obtained by experiment and simulation are consistent. As listed in Table 8, the difference between the first two orders of natural frequencies for the rotor obtained by the simulation and experiment is large, and the maximum is 11.90%. Therefore, it is necessary to modify the parameters of the spring element to ensure the rationality of the finite element model for the rotor.

In the finite element model of the rotor, the stiffness K_x in the x direction and stiffness K_y in the y direction of the spring element simulating the bearing are corrected, and K_x and K_y are brought into the correction algorithm based on the genetic algorithm as design variables. Finally, the modified parameters of the spring element are listed in

Table 10. Parameters of spring element before and after modification.

Parameters	Before Modification	After Modification
Stiffness Kx/N·m−1	2 × 1011	8.87 × 107
Stiffness Ky/N·m−1	2 × 1011	1.58 × 109

, and after the model modification, the comparison between natural frequencies of rotor obtained by simulation and experiment is shown in

Table 11. Comparison between natural frequencies of rotor obtained by simulation and experiment.

Order	Experiment Results/Hz	Simulation Results/Hz	Difference
1	511.13	523.40	2.40%
2	3085.55	3045.00	1.31%

. It can be seen from Table 8 and Table 11 that the differences between the natural frequencies obtained by the experiment and simulation are significantly reduced by the model modification, and the maximum difference is 2.40%.

4.2.4. Model Verification Based on Clearance Test Results

In this section, before verifying the finite element model based on the clearance test results, the correctness of the method of introducing temperature loads is verified, and the thermal deformation test of the rotor and stator is carried out. According to the method of applying temperature load to the finite element model described in Section 3.1.3, the radial thermal deformation of the stator and rotor is calculated in ANSYS software. The comparison between the simulation results of stator thermal radial deformation and the experimental values is shown in

Table 12. Comparison between simulation results and experimental values of stator thermal deformation.

Measuring Point	Experimental Result/mm	Simulation Result/mm	Error
1	0.2	0.209	4.5%
2	0.2	0.209	4.5%

, and the comparison between the simulation results of rotor thermal radial deformation and the experimental values is shown in

Table 13. Comparison between simulation results and experimental values of rotor thermal deformation.

Measuring Point	Experimental Result/mm	Simulation Result/mm	Error
Blade 1	0.1	0.093	7%
Blade 4	0.1	0.093	7%

. It can be seen from Table 12 that the error between the simulation results and the experimental values of the stator radial thermal deformation is 4.5%, and the error between the simulation results and the experimental values of the rotor radial thermal deformation is 7%. The error between the simulation results and the experimental values of the rotor and stator radial thermal deformation is small, which proves the rationality of the thermal environment method introduced in this paper.

Further, the finite element model is verified based on the clearance test results. The test results of the distance Δl from the blade to the eddy current sensor at 400 rpm obtained by the clearance test are shown in Figure 23. The initial clearance d is 20.5 mm, and the distance L between the eddy current sensor and the stator is 19.34 mm. The clearance change can be obtained by taking the experimental results, the initial gap d, and the distance L into Equation (8). The maximum clearance change (ΔXmax) is 0.45 mm and the minimum clearance change (ΔXmin) is −0.48 mm.

At the same time, the clearance of the rotor–stator coupled structure is calculated based on the finite element model established in this paper. Because of the unbalanced force, the rotor shaft will bend. The blade-tip clearance with the smallest angle to the plane where the shaft is bent changes the most. Because the rotor has a total of six blades, there are two blades (blades 1 and 4) with the smallest angle from the plane where the axis is bent at the same time, and the change in blade-tip clearance for blades 1 and 4 is obtained in the simulation calculation. The experimental result at 400 rpm is used as the standard value, and the comparison between the clearance change obtained by the simulation calculation and the experimental is shown in

Table 14. Comparison between simulation results and experimental results of clearance change.

Clearance Change	Experimental Results	Simulation Results	Difference
Blade 1/mm	0.45	0.47	4.44%
Blade 4/mm	−0.48	−0.49	2.08%

. As can be seen from Table 14, the simulation results of clearance change after are accurate, and the maximum difference between the simulation results and the experimental results is 4.44%, which proves the accuracy of the model established in this paper.

4.3. Clearance Analysis Under Multi-Physical Fields

4.3.1. Clearance Analysis Under Centrifugal, Aerodynamic, and Temperature Load

In the clearance study, the clearance variation is often used to describe the change trend of clearance. When the clearance variation is negative, the clearance is reduced, and when the clearance variation is positive, the clearance is increased.

(1)

Clearance analysis under centrifugal load

As described in Section 3.1.1, the centrifugal load only acts on the rotor. In this section, centrifugal load is applied to the finite element model of the rotor, and the change in clearance under centrifugal load is analyzed. The radial deformation cloud diagram of the rotor under centrifugal load is shown in Figure 24. It can be seen from Figure 24 that the maximum radial deformation of the rotor occurs at the blade tip of the rotor under the centrifugal load. This is because under the action of centrifugal load, the blade disk will first undergo radial deformation, and the blade will also undergo radial deformation under the action of centrifugal load. The displacement at the blade tip contains these two kinds of radial deformation at the same time, so the radial deformation at the tip is the largest. The subsequent analysis is mainly concerned with the clearance at the blade with the largest clearance change.

Firstly, the clearance analysis at 250 rpm, 500 rpm, 750 rpm, 1000 rpm, 1250 rpm, and 1500 rpm is carried out. The clearance change in the structure at different speeds is shown in

Table 15. Blade-tip clearance under different rotating speeds.

Rotating speed/rpm	250	500	750	1000	1250	1500
Clearance change/mm	−2.12 × 10−5	−8.50 × 10−5	−1.91 × 10−4	−3.40 × 10−4	−5.31 × 10−4	−7.65 × 10−4

, and the change curve of clearance change at different speeds is shown in Figure 25. As can be seen from Table 15 and Figure 25, the clearance change at each speed is negative, and the value gradually decreases with the increase in rotor speed. Therefore, it can be seen that the blade-tip clearance gradually decreases with the increase in rotational speed.

Secondly, the length and thickness of the blade at different positions are different in the actual engine structure, and the radial deformation of these blades with different lengths and thicknesses under the same centrifugal load will be different, resulting in different clearance changes at different positions. The following will discuss the clearance change under different lengths and thicknesses of blade. Blades with lengths of 130 mm, 135 mm, 140 mm, 145 mm, and 150 mm and blades with thicknesses of 1 mm, 1.5 mm, 2 mm, 2.5 mm, and 3 mm are taken as examples to calculate the clearance change in these blades under centrifugal load.

Clearance variations under different speeds and blade lengths are shown in

Table 16. Clearance variation (10⁻⁵ mm) under different rotating speeds and blade lengths.

	Rotating Speed/rpm	250	500	750	1000	1250	1500
Blade Length/mm
130		−1.69 × 10−5	−6.76 × 10−5	−1.52 × 10−4	−2.70 × 10−4	−4.22 × 10−4	−6.08 × 10−4
135		−1.90 × 10−5	−7.59 × 10−5	−1.71 × 10−4	−3.04 × 10−4	−4.75 × 10−4	−6.83 × 10−4
140		−2.12 × 10−5	−8.50 × 10−5	−1.91 × 10−4	−3.40 × 10−4	−5.31 × 10−4	−7.65 × 10−4
145		−2.37 × 10−5	−9.47 × 10−5	−2.13 × 10−4	−4.09 × 10−4	−5.92 × 10−4	−8.53 × 10−4
150		−2.63 × 10−5	−1.05 × 10−4	−2.37 × 10−4	−4.21 × 10−4	−6.58 × 10−4	−9.47 × 10−4

and Figure 26. It should be noted that clearance variations under different lengths are based on the initial clearance without deformation under corresponding lengths. It can be seen from Table 16 and Figure 26 that at the same speed, the clearance variation gradually decreases when the blade length increases, and the clearance variation is negative. Moreover, the change trend of clearance variation of blades under different speeds is roughly the same as that shown in Figure 25. The above reason is that when the blade length increases, the mass of the blade and the radius where the blade tip is located will increase. According to the calculation formula of centrifugal force, the centrifugal force is proportional to the mass, speed, and radius of the object’s circular motion. So, the centrifugal force will increase when the blade length increases, resulting in an increase in radial deformation. Therefore, the clearance variation is negative and its value gradually decreases with the increase in blade length.

The clearance variations under different thicknesses of blade and rotating speeds are shown in

Table 17. Clearance variation (10⁻⁵ mm) under different rotating speeds and thicknesses of blade.

	Rotating Speed/rpm	250	500	750	1000	1250	1500
Blade Length/mm
1		−1.92 × 10−5	−7.69 × 10−5	−1.73 × 10−4	−3.08 × 10−4	−4.81 × 10−4	−6.92 × 10−4
1.5		−2.03 × 10−5	−8.12 × 10−5	−1.83 × 10−4	−3.25 × 10−4	−5.08 × 10−4	−7.31 × 10−4
2		−2.12 × 10−5	−8.50 × 10−5	−1.91 × 10−4	−3.40 × 10−4	−5.31 × 10−4	−7.65 × 10−4
2.5		−2.25 × 10−5	−8.99 × 10−5	−2.02 × 10−4	−3.60 × 10−4	−5.62 × 10−4	−8.09 × 10−4
3		−2.34 × 10−5	−9.36 × 10−5	−2.11 × 10−4	−3.74 × 10−4	−5.85 × 10−4	−8.42 × 10−4

and Figure 27. As can be seen from Table 17 and Figure 27, at the same speed, when the thickness of the blade increases, the clearance variation under centrifugal load will decrease, and the clearance variation is negative. Moreover, the change trend of clearance variation under different speeds is roughly the same as that shown in Figure 25. The above reason is that when the thickness of the blade increases, the mass of the blade will increase, resulting in a larger centrifugal force generated by the blade in the rotating process, so that the clearance change is negative and gradually decreases.

(2)

Clearance analysis under aerodynamic load

According to the method of introducing aerodynamic load described in Section 3.1.2, the aerodynamic load is applied to the finite element model, and the clearance variation caused by aerodynamic load is analyzed. The cloud diagram of radial deformation for the finite element model under aerodynamic load is shown in Figure 28. It can be seen from Figure 28 that the cylindrical shell structure deforms along the radial direction under the action of aerodynamic load, and the radial deformation is the largest at the position 180° from the constraint end below.

The radial deformation of the structure under 0.5, 1, 1.5, 2, 2.5, and 3 atmospheric pressures is calculated, and corresponding clearance analysis is also conducted. The clearance variation of the structure under different pressures is shown in

Table 18. Radial deformation of stator under aerodynamic load and clearance change.

Air pressure/0.101 MPa	0.5	1	1.5	2	2.5	3
Clearance variation/mm	0.020	0.041	0.062	0.082	0.103	0.124

, and the change curve of clearance variation under different pressures is shown in Figure 29. It can be seen from Table 18 and Figure 29 that the clearance variation of the structure increases with the increase in air pressure.

In the actual structure, the radius and thickness of the casing (shell) in different structures are different, the radial deformation under aerodynamic load will also be different, and the clearance will also change. The clearance variation of the shell at different thicknesses (1 mm, 1.5 mm, 2 mm, 2.5 mm, and 3 mm) and different radii (150.5 mm, 155.5 mm, 160.5 mm, 165.5 mm, and 170.5 mm) is discussed below.

The clearance variation under different thicknesses of shell and aerodynamic loads is shown in

Table 19. Clearance variation (mm) under different air pressures and thicknesses of shell.

	Air Pressure/0.101 MPa	0.5	1	1.5	2	2.5	3
Thicknesses/mm
1		0.037	0.074	0.111	0.149	0.186	0.223
1.5		0.027	0.046	0.069	0.092	0.115	0.138
2		0.018	0.035	0.053	0.070	0.088	0.105
2.5		0.014	0.029	0.043	0.057	0.072	0.086
3		0.012	0.024	0.037	0.049	0.061	0.073

and Figure 30. It can be seen from Table 19 and Figure 30 that under the same air pressure, clearance variation decreases with the increase in shell thickness. Moreover, when thickness is the same, the change trend of clearance variation under different air pressures is roughly the same as that shown in Figure 29. The above reason is that when the thickness of the shell increases, the overall stiffness of the structure will also increase, and when the air pressure remains unchanged, the radial deformation of the structure will decrease, resulting in a decrease in the clearance variation.

The clearance variation under different radii of shell and aerodynamic loads are shown in

Table 20. Clearance variation (mm) under different radii of shell and air pressures.

	Air Pressure/0.101 MPa	0.5	1	1.5	2	2.5	3
Radius/mm
150.5		0.016	0.033	0.049	0.064	0.082	0.097
155.5		0.017	0.034	0.051	0.067	0.085	0.101
160.5		0.018	0.035	0.053	0.070	0.088	0.105
165.5		0.019	0.039	0.055	0.073	0.091	0.109
170.5		0.020	0.041	0.057	0.076	0.094	0.114

and Figure 31. It can be seen from Table 20 and Figure 31 that under the same air pressure, the clearance variation will increase with the increase in the radius, and when the radius is the same, the change trend of the clearance variation under different air pressures is roughly the same as that shown in Figure 29. The above reason is that when the radius of the shell increases, the overall stiffness of the structure will decrease, and when the air pressure remains unchanged, the radial deformation of the structure will increase, resulting in an increase in the clearance variation.

(3)

Clearance analysis under temperature load

According to the method of introducing temperature load described in Section 3.1.3, the temperature load is applied to the finite element model of the structure, and the clearance analysis is carried out for the clearance variation caused by temperature load. The radial deformation cloud rotor of the finite element model of the stator and rotor under temperature load is shown in Figure 32a and Figure 32b, respectively. As can be seen from Figure 32a, the radial inward contraction deformation occurs at the position near the lower constrained end in the stator, and the radial outward expansion deformation occurs at the position away from the constrained end. Moreover, the radial inward contraction deformation is smaller than that of outward expansion, so under the action of temperature load, the position of the maximum radial deformation of the stator appears at the position of 180° from the constraint end. It can be seen from Figure 32b that the radial deformation of the rotor under temperature load presents a symmetrical distribution, and its maximum radial deformation is located at the blade tip.

The radial deformation of the structure is calculated and the clearance analysis is carried out with the temperature of the first-stage stator and blade at 50 °C, 100 °C, 150 °C, 200 °C, 250 °C, and 300 °C as an example. The clearance variation and radial deformation of the stator and rotor under different temperatures are shown in

Table 21. Clearance variation and radial deformation of stator and rotor under different temperatures.

Temperature/°C	50	100	150	200	250	300
Deformation of stator/mm	0.39	0.78	1.18	1.57	1.96	2.36
Deformation of rotor/mm	0.13	0.19	0.23	0.34	0.43	0.52
Clearance variation/mm	0.26	0.59	0.95	1.23	1.53	1.84

, and the change curve of clearance variation with temperature is shown in Figure 33. It can be seen from Table 21 and Figure 33 that the radial deformation of the stator and rotor increases with the increase in temperature, and the clearance variation also increases.

In the actual structure, the difference in thicknesses of the shell and blade will also affect the clearance variation under temperature load. The influence of temperature load on clearance variation when the thicknesses of the shell and blade are different is discussed. Here, the thickness of the shell varies from 1 mm to 3 mm, the thickness of the blade varies from 1 mm to 3 mm, and the temperature varies from 50 °C to 300 °C.

The clearance variation of the structure under different temperatures and different thicknesses of shell is shown in

Table 22. Clearance variation under different thicknesses of shell and temperature.

	Temperature/°C	50	100	150	200	250	300
Thickness/mm
1		0.32	0.64	0.96	1.28	1.60	1.93
1.5		0.31	0.61	0.92	1.22	1.54	1.85
2		0.26	0.59	0.95	1.23	1.53	1.84
2.5		0.25	0.58	0.94	1.22	1.52	1.83
3		0.21	0.51	0.83	1.13	1.44	1.74

and Figure 34. It can be seen from Table 22 and Figure 34 that under the same temperature, the clearance variation is inversely proportional to the thickness of the shell, which is consistent with the conclusion in Figure 30, and the reason is also caused by the increase in the structural stiffness generated by the increase in the thickness of the shell. Moreover, the clearance variation is proportional to the temperature under different thicknesses of shell, because the radial deformation of the shell is greater than that of the rotor after the temperature is raised, so the clearance variation is increased.

The clearance variations of the structure under different thicknesses of blade and different temperatures are shown in

Table 23. Clearance variation under different thicknesses of blade and different temperatures.

	Temperature/°C	50	100	150	200	250	300
Thickness/mm
1		0.27	0.61	0.99	1.31	1.61	1.94
1.5		0.27	0.61	0.99	1.31	1.61	1.94
2		0.26	0.59	0.95	1.23	1.53	1.84
2.5		0.26	0.59	0.95	1.23	1.53	1.84
3		0.25	0.57	0.91	1.17	1.45	1.74

and Figure 35. It can be seen from Table 23 and Figure 35 that the clearance variation is inversely proportional to the thickness of the blade and proportional to the temperature load. The reasons are similar to those in Figure 34 and will not be repeated here.

4.3.2. Clearance Analysis Under Multi-Physical Fields Including Inertial Moment

In this section, according to the method of introducing inertial moment in Section 3.1.4, inertial load is applied to the finite element model of the rotor–stator coupled structure, and the radial deformation of the rotor–stator coupled structure under inertial load is calculated. Based on the clearance analysis method considering inertial load described in Section 3.2.2, the clearance analysis of the rotor–stator coupled structure under inertial load is carried out. The amplitude of the inertial load is related to the speed of the object around an external axis (the revolution speed) and the speed of the rotor (the rotation speed). The clearance analysis of the rotor–stator coupled structure is completed in the following cases where the revolution speed is 1 rpm, 2 rpm, 3 rpm, 4 rpm, and 5 rpm and the rotating speed is 250 rpm, 500 rpm, 750 rpm, 1000 rpm, 1250 rpm, and 1500 rpm.

Moreover, centrifugal, aerodynamic, and temperature load are also considered in the clearance analysis under inertial load in this section. The amplitude of centrifugal load varies with the rotation speed, while the value of aerodynamic load and temperature load is fixed. The aerodynamic load is 0.5 atmospheric pressure and the temperature is 50 °C. Figure 36a shows the radial deformation cloud image of the rotor–stator coupled structure under the simultaneous action of four loads, and Figure 36b shows the radial deformation cloud image under the action of centrifugal load, aerodynamic load, and temperature load. As can be seen from Figure 36a, considering the action of inertial moment, the maximum radial deformation position of the stator appears at the front end of the constraint end, and the maximum radial deformation position of the rotor appears at the tip position of the blade at the six o’clock direction. The six blades of the rotor all have radial torsional deformation. It can be seen from Figure 36b that without considering the action of inertial moment, the maximum radial deformation of the stator occurs at the position of 180° from the constraint end, and the maximum radial deformation of the rotor occurs at the tip position of the blade at the 12 o’clock direction. It can be seen that the inertial moment has a significant effect on the clearance of the rotor–stator coupled structure.

When the revolution speed is constant and the rotation speed changes, the clearance variation of the rotor–stator coupled structure is shown in

Table 24. Clearance variation of the rotor–stator coupled structure with the rotation speed.

Revolution speed/rpm	2
Rotation speed/rpm	250	500	750	1000	1250	1500
Clearance variation/mm	9.7 × 10−6	−3.25 × 10−5	−1.08 ×1 0−4	−2.17 × 10−4	−3.63 × 10−4	−5.44 × 10−4

and Figure 37. In this paper, the influence of rotation speed on clearance variation is analyzed under the revolution speed = 2 rpm. As can be seen from Table 24 and Figure 37, when the rotation speed gradually increases, the clearance variation gradually decreases, but the absolute value of the clearance variation gradually increases. This indicates that the blade-tip clearance between the rotor and the stator gradually decreases when the rotation speed increases. As can be seen from Figure 37, the decreasing rate of blade-tip clearance increases with the increase in the rotation speed. It can be inferred that when the rotation speed increases to a certain value, the tip clearance will decrease rapidly, resulting in friction failure between the rotor and the stator.

When the rotation speed is constant and the revolution speed changes, the clearance variation of the rotor–stator coupled structure is shown in

Table 25. Clearance variation of the rotor–stator coupled structure with the revolution speed.

Rotation speed/rpm	1000
Revolution speed/rpm	1	2	3	4	5
Clearance variation/mm	−2.54 × 10−4	−2.17 × 10−4	−1.77 × 10−4	−1.30 × 10−4	−7.70 × 10−5

and Figure 38. When the rotation speed is 1000 rpm, the influence of the revolution speed on the clearance variation is analyzed. It can be seen from Table 25 and Figure 38 that with the continuous increase in the revolution speed, the clearance variation gradually increases. This shows that the clearance between the rotor and stator increases with the increase in revolution speed. The reason is that the inertial moment has the effect of inhibiting the bending of the shaft. As the revolution speed increases, the inhibition of the inertial moment on the bending of the shaft increases, so the clearance increases.

When the rotation speed and revolution speed change at the same time, the clearance variation of the rotor–stator coupled structure is shown in

Table 26. Clearance variation of the rotor–stator coupled structure under different revolution speeds and rotation speeds.

	Rotation Speed/rpm	250	500	750	1000	1250	1500
Revolution Speed/rpm
1		−8.15 × 10−6	−4.86 × 10−5	−1.28 × 10−4	−2.54 × 10−4	−3.93 × 10−4	−5.78 × 10−4
2		9.70 × 10−6	−3.25 × 10−5	−1.08 × 10−4	−2.17 × 10−4	−3.63 × 10−4	−5.44 × 10−4
3		4.38 × 10−5	1.83 × 10−5	−4.36 × 10−5	−1.77 × 10−4	−2.72 × 10−4	−4.38 × 10−4
4		5.52 × 10−5	7.6 × 10−5	1.01 × 10−5	−1.30 × 10−4	−2.00 × 10−4	−3.57 × 10−4
5		9.59 × 10−5	1.00 × 10−4	7.73 × 10−5	−7.70 × 10−5	−1.22 × 10−4	−2.70 × 10−4

and Figure 39. It can be seen from Table 26 and Figure 39 that when the rotation speed and revolution speed increase at the same time, the clearance variation gradually decreases. On the contrary, the clearance variation increases when the rotation speed and revolution speed decrease at the same time. When the rotation speed increases and revolution speed decreases, the clearance variation decreases. On the contrary, the clearance variation increases when the rotation speed decreases and the revolution speed increases.

5. Conclusions

In this paper, the effects of multi-physical fields (inertial moment, centrifugal load, aerodynamic load, and temperature load) are considered comprehensively, and the clearance analysis of the rotor–stator coupled structure is carried out. Based on the finite element method, the dynamic model of the rotor–stator coupled structure is established, and the introduction of multi-physical fields is given. The accuracy of the finite element model and the rationality of the clearance analysis method are verified by experimental tests. The influence of different physical fields on clearance is systematically analyzed, and some new clearance variation rules are given. The main conclusions are as follows:

(1)

A general clearance analysis method of the rotor–stator coupled structure under multi-physical fields is proposed. To ensure high solving accuracy and efficiency, the modeling method of beam–solid–shell element coupling is employed to establish the finite element mode. The modal tests of the rotor and stator are used to modify the established model, and the clearance test bench of the coupled structure is set up. The maximum error between the clearance test results and the simulation results is 4.44%, which verified the accuracy of the model modification and the clearance analysis method.

(2)

The blade-tip clearance is inversely proportional to the centrifugal force and the thickness of the shell and blade. The reason is that the increase in the thickness of blade will cause the increase in centrifugal force, which results in the increase in the radial deformation of the rotor, and then the clearance decreases. The increase in shell thickness will increase its stiffness, and the radial deformation of the shell will be reduced, so the clearance will decrease. Moreover, the decreasing rate of clearance increases with the increase in the centrifugal force, which indicates the clearance will decrease rapidly when the rotation speed increases to a certain value, resulting in friction failure.

(3)

The blade-tip clearance of the rotor–stator coupled structure is proportional to the aerodynamic load and temperature load. The aerodynamic load has great influence on the stator and can cause the radial deformation of the stator, so the blade-tip clearance increases as the aerodynamic load increases. Moreover, the temperature load can cause the radial deformation of both the rotor and stator, but the influence of temperature load on the deformation of the stator is greater than that of the rotor, so the blade-tip clearance increases with the increase in the temperature.

(4)

The blade-tip clearance is proportional to the inertial moment generated under maneuvering flight conditions. The reason is that the inertial moment will inhibit the bending of the rotating shaft, so the clearance will also increase when the inertial moment increases. In addition, the clearance decreases and increases, respectively, when the rotation speed and the revolution speed are increased separately, while the clearance decreases when the rotation speed and the revolution speed are increased simultaneously, which indicates that the centrifugal force has a greater effect on the blade-tip clearance than the inertial moment.