A Thermomechanical Model for Time-Varying Deformations of Spigot Interference Connection under Shrink-Fitting Assembly

Abstract

The interference fit connection structure is widely employed in various industries. Different from the conventional connection structure, the aero-engine rotor connection has a spigot-bolt structure. The quality of the shrink-fitting assembly process directly affects the final assembly performance of the rotor. The complexity of the connection structure and the time-varying thermal deformation bring great challenges in analyzing the formation patterns of connection performance. However, existing methods of interference connection analysis are mainly used in the wide range of interference connection structures, which makes them difficult to apply in low height interference connection for aero-engine rotors. This paper introduces a thermomechanical interference fit pressure model. First, a theoretical model for interference fit pressure considering temperature-variable loads is established to obtain the time-varying pressure during the assembly process. Second, a finite element model is established to explore the influence of interference and temperature loads on the spigot pressure and the mounting edge deformation. Finally, the experiments validate the accuracy of both the theoretical model and the finite element analysis. The results indicate that during the shrink-fitting assembly process, the interference fit pressure exhibits a nonlinear evolution trend, and the warping deformation of the mounting edge is a result of the combined influence of temperature and interference fit pressure. The law found in this paper has an application prospect in the process parameter setting of shrink-fitting assembly for special structures.

1. Introduction

Interference fit connection structures, due to their excellent load-bearing and torque transmission capabilities, are widely utilized in critical junctions of aero-engine rotors. Shrink-fitting assembly has the advantages of causing less damage to the connection interface, easy operation, etc., and it is applied to the assembly process of aero-engine rotor spigot interference connection structures [1,2]. Assembly, as the final step in the engine manufacturing chain, directly influences the initial state of geometric and stiffness characteristics of aero-engines [3], and the quality of shrink-fitting assembly processes directly affects the final assembly performance of the rotor [4,5].

During shrink-fitting assembly, it is necessary to create a gap fit at the spigot connection by either heating or cooling the rotor components [6]. Uneven heat transfer between the connecting parts leads to the nonlinear growth of the spigot interference pressure and warping deformation of the mounting edge, which will lead to an initial non-fitting of the mounting edge and will seriously affect the application of the pressing force [7], which also leads to significant randomness and poor uniformity of the rotor connection state, with problems such as a low success rate of one-time assembly and poor product reliability [8,9]. Therefore, it is of great significance to explore the time-varying law of the interference pressure and the thermal deformation during shrink-fitting assembly to improve the final assembly quality of the rotor.

For the interference fit connection, the numerical value and distribution of the interference fit pressure have always been the focus of research. For the calculation of the interference fit pressure, Campos et al. [10] proposed a simplified form of Lame’s equation, which provides a method for quickly calculating the contact pressure and hoop stress between two contact surfaces. Ozturk F et al. [11] established a two-dimensional model for the interference fit assembly based on Abaqus, and determined the stress distribution state of the solid shaft. Wang G P et al. [12] used the special solution of the bending equation of the thin-walled cylindrical shell to replace the special solution in the state without a bending moment, and obtained a new general solution. Güven et al. [13] used the finite element method to analyze the model with different edge thicknesses, and showed that the increase in tangential stress is very important to the fatigue life of gear. Persson et al. [14] used the Persson contact mechanics theory to explain the effect of surface roughness on the press fit design. Shang et al. [15] used the classic elastic plane stress theory, and the exact solutions of the displacements of the hub and shaft were derived. In addition to the analytical aspect of the interference force, the press-fitting curve and the interference junction are also important. Wang X et al. [16,17] proposed an analytical method based on TCT and the resistance calculation method, which can accurately predict the interference fit curve. Shu et al. [18] considered the cumulative effect of fretting wear in the sliding zone, and studied the variation in wear width and depth with the number of load cycles. Marshall et al. [19] used ultrasonic reflection to determine contact conditions in interference fit to reduce the risk of fatigue failure. Radmir et al. [20] proposed an alloy power driver with shape memory effect (SME) to improve the joint quality of the interference fit. Benuzzi et al. [21] established a method to predict intermediate pressure fit curves for railway axle and wheel assemblies using friction measurements as inputs. Madej et al. [22] compared the calculated joint bearing capacity with Lame’s formula and concluded that the pressing process may be elastoplastic. Lei G et al. [23] analyzed the interference fit problem of the drive axle housing of the tripper using a nonlinear finite element model. Zhou H et al. [24] could accurately simulate the stress concentration at the contact edge by using the finite element method. Zhang S et al. [25] simulated the influence of fit clearance on the contact pressure on the interference fitting surface using the elastic–plastic contact finite element. However, research on the interference pressure has mostly focused on the calculation of the contact structure pressure and the connection joint of the interference fit. In fact, the assembly of the interference connection structure is a dynamic process, especially the shrink-fitting assembly process. The impact of temperature variation on the interference pressure cannot be ignored, and the interference pressure shall be predicted according to the actual temperature.

Shrink-fitting assembly is actually a process of pressure and deformation changes caused by temperature transfer. It includes the problem of thermo-mechanical coupling. Regarding the heat transfer between objects, Singh et al. [26] investigated the effects of two temperatures on the displacements and stresses and for the two-dimensional thick plate axis without energy dissipation. Youssef et al. [27] established a new thermomechanical theory and proved the uniqueness theorem of two-temperature dissipation without energy. Meyer et al. [28] proposed an energy conversion method to describe plastic deformation, which can reduce the time of coupled thermal calculation. Xu et al. [29] established the transient thermomechanical sliding model of elastoplastic micro-convex contact, and analyzed the micro-convex contact behavior in the sliding process. Hamidi et al. [30] proposed a sinusoidal plate theory for the thermomechanical bending analysis of functionally graded sandwich panels, and studied the thermomechanical response of functionally graded sandwich panels under various factors. Liu et al. [31] constructed a dynamic nonlinear thermomechanical coupled finite element algorithm, and obtained the conclusion that the displacement and stress in the plate are more sensitive to the nonlinear material than the temperature. Abdelsalam et al. [32] evaluated and compared the fatigue life of a multilayer cylinder under cyclic heat and pressure loading and developed a new shrink-fitting assembly process. Bae J H et al. [33] derived the correction coefficient of the tooth top and root dimensional deformation error in a gear temperature shrink-fitting assembly process of automobile transmission parts based on finite element analysis. Kim T J et al. [34] present a closed-form equation for the prediction of contact pressure and fitting load, which can be used to optimize the shrink-fitting assembly of automobile transmission. Cui Y et al. [35] used the finite element method to accurately predict stress evolution and stress distribution during the heat shrink-fitting process. Sun M Y et al. [36] optimized the heat shrink-fitting process of the crankshaft from the aspects of heating mode, contact behavior and structural deformation, effectively improving the bonding strength. Li R Q et al. [37] established a two-dimensional nonlinear thermomechanical coupling model to study the thermal shrink-fitting process of the reactor coolant rotor. However, the shrink-fitting assembly process has a great impact on the deformation of the connector structure. At present, there are many studies on the temperature transfer characteristics of the shrink-fitting assembly process, but the research object is basically a simple hub mating structure. In fact, the thermal deformation caused by temperature and pressure cannot be ignored for the spigot-bolt structure facing the rotor of the aero-engine, which shall be the focus of the assembly process.

To sum up, although many scholars have studied interference pressure and shrink-fitting assembly, there is little research on the change law of the spigot interference pressure with the characteristics of the time-varying temperature. In addition, existing methods of interference connection analysis are mainly used in the wide range of interference connection structures, which makes it difficult to apply to the spigot-flanged connection structure for aero-engine rotors. The complexity of the structure and the time-varying thermal deformation pose significant challenges in analyzing the mode of formation of the joint behavior. In this paper, a thermomechanical model for the time-varying deformation of the spigot interference connection is established, which considers the dual load of temperature and pressure. The theory and the finite element model are used to analyze the deformation causes of the mounting edge of the spigot-flanged connection structure, exploring the deformation law of the mounting edge under the temperature and pressure load, and this can provide theoretical support for the shrink-fitting assembly process of the aero-engine.

The organizational structure of this paper is as follows: The second section is theoretical modeling, where we established a time-varying model for interference fit pressure considering thermomechanical loading and obtained the law of mechanical evolution at the spigot connection. The third section is the finite element model, where numerical simulations of the deformation patterns of the mounting edge are conducted. The fourth section is the results and discussion, where the two key assembly time points of “Initial fitting time” and the “Pressure stabilization time” are defined; the causes of deformation of the mounting edge are analyzed, and the influence law of temperature load, the interference value, and other factors are studied. The fifth section contains the experimental design to verify the accuracy of the theory and finite element analysis. The sixth section summarizes the content of the entire paper.

2. Theoretical Model

In this section, based on the calculation theory of interference force (TCT), the pressure equation of the spigot with time considering temperature variation is derived. Combined with the thin shell theory, the deformation of the mounting edge is analyzed. The evolution law of the pressure at the spigot of the shrink-fitting assembly and the influence law of the pressure on the deformation of the mounting edge are revealed.

2.1. Theoretical Model of Temperature Transfer and Interference Pressure

Related scholars have given the calculation equation of the interference pressure with temperature difference [38], as shown in Formula (1).

$$p=-\frac{\delta -\frac{2r_{c1}\alpha }{r_{c1}^2-r_a{}^2}{\displaystyle \underset{r_a}{\overset{r_{c1}}{\int }}T(x)xdx}+\frac{2r_{c2}\alpha }{r_b^2-r_{c2}{}^2}{\displaystyle \underset{r_{c2}}{\overset{r_b}{\int }}T(x)xdx}}{r_{c2}(\frac{r_{c2}^2+r_b{}^2}{E\left(r_b^2+r_{c2}{}^2\right)}+\mu )+r_{c1}(\frac{r_a^2+r_{c1}{}^2}{E\left(r_b^2+r_{c1}{}^2\right)}-\mu )}$$

(1)

where $E$ is Young’s modulus, $\alpha$ is linear expansion coefficient, $\delta$ is interference value, $r_a$, $r_b$, $r_{c1}$, $r_{c2}$ is the basic structure size of the connector, $T(x)$ is the temperature distribution function in the radial direction of the connector, and $p$ is the contact pressure.

Although the equation takes into account the basic structural dimensions of the connector and the radial temperature information, it represents the pressure at a specific moment in a stationary state. Shrink-fitting assembly is a dynamic process that evolves over time. To obtain an equation describing the variation in spigot pressure with time, it is necessary to analyze the temperature transfer between the connectors.

Figure 1 shows the dimensional parameters associated with the theoretical model, where $u_1$, $u_2$ are the deformation values between two contacts, and the calculation method is shown in Formulas (2) and (3). It can be seen from Formulas (2) and (3) that temperature and pressure are the key factors affecting the radial deformation of the connector.

$$u_1=\frac{2r_{c1}\alpha }{r_{c1}{}^2-r_a^2}{\displaystyle \underset{r_a}{\overset{r_{c1}}{\int }}T(x)xdx-\frac{(1+\mu )r_a{}^2{r}_{c1}+(1-\mu )r_{c1}{}^3}{E(r_{c1}{}^2-r_a^2)}}p$$

(2)

$$u_2=\frac{2r_{c2}\alpha }{r_b{}^2-r_{c2}^2}{\displaystyle \underset{r_{c2}}{\overset{r_b}{\int }}T(x)xdx+\frac{(1+\mu )r_a{}^2{r}_{c2}+(1-\mu )r_{c2}{}^3}{E(r_b{}^2-r_{c2}^2)}}p$$

(3)

Figure 1, above, shows the heat transfer of the connection, the heat conduction formula of which is shown in Formulas (4) and (5), below.

$$\frac{{\partial }^2{T}_a\left(x,t\right)}{\partial x^2}=\frac{1}{{\alpha }^t}\frac{\partial T_a\left(x,t\right)}{\partial t},\left(t>0\right)$$

(4)

$$\frac{{\partial }^2{T}_b\left(x,t\right)}{\partial x^2}=\frac{1}{{\alpha }^t}\frac{\partial T_b\left(x,t\right)}{\partial t},\left(t>0\right)$$

(5)

where ${\alpha }^t$ is the thermal diffusivity coefficient, $T_a^0$, $T_b^0$ is the initial temperature of the connector, and $T_a\left(x,t\right)$, $T_b\left(x,t\right)$ is the radial temperature of the connection.

Assume equal heat flux at the contact and that the connecting parts at both ends of the connector are adiabatic, and the following condition Equations (6)–(10) can be obtained:

$$T_a\left(x,0\right)=T_a^0$$

(6)

$$T_b\left(x,0\right)=T_b^0$$

(7)

$$\frac{\partial T_a\left(-\infty ,t\right)}{\partial x}=\frac{\partial T_b\left(+\infty ,t\right)}{\partial x}=0$$

(8)

$$\frac{\partial T_a\left(0,\tau \right)}{\partial x}=\frac{\partial T_b\left(0,\tau \right)}{\partial x}$$

(9)

$$-\lambda \frac{\partial T_a\left(0,t\right)}{\partial x}=h[T_a\left(0,t\right)-T_b\left(0,t\right)]$$

(10)

where $\lambda$ is the thermal conductivity and $h$ is the thermal resistivity.

According to the above conditions, the temperature transfer equations of the two connectors can be solved as Formulas (11) and (12).

$$T_a\left(x,t\right)=T_a^0+\frac{\left(T_b^0-T_a^0\right)}{2}\left[\begin{array}{l}erfc\left(\frac{x}{2\sqrt{{\alpha }^t\cdot t}}\right)-\exp \left(\frac{2h}{\lambda }x+\frac{4h^2\alpha }{{\lambda }^2}t\right)\\ \cdot erfc\left(\frac{2h\sqrt{{\alpha }^t\cdot t}}{\lambda }+\frac{x}{2\sqrt{{\alpha }^t\cdot t}}\right)\end{array}\right]$$

(11)

$$T_b\left(x,t\right)=T_b^0-\frac{\left(T_b^0-T_a^0\right)}{2}\left[\begin{array}{l}erfc\left(\frac{x}{2\sqrt{{\alpha }^t\cdot t}}\right)-\exp \left(\frac{2h}{\lambda }x+\frac{4h^2\alpha }{{\lambda }^2}t\right)\\ \cdot erfc\left(\frac{2h\sqrt{{\alpha }^t\cdot t}}{\lambda }+\frac{x}{2\sqrt{{\alpha }^t\cdot t}}\right)\end{array}\right]$$

(12)

where $erfc\left(\mathrm{x}\right)$ is the residual error function of the error function $erf\left(\mathrm{x}\right)$. It can be seen that the heat load is mainly composed of the temperature of the connector $T_a$ and the temperature of the initial temperature difference $T_b^0-T_a^0$, driving the heat transfer during the heat conduction process of the shrink-fitting assembly.

By integrating the temperature transfer functions (11) and (12) with Formula (1), the time variation equation of the spigot interference pressure, considering the temperature transfer, can be obtained in Formula (13).

$$p\left(t\right)=-\frac{\delta -\frac{2r_{c1}\alpha }{r_{c1}^2-r_a{}^2}{\displaystyle \underset{r_a}{\overset{r_{c1}}{\int }}{T}_a\left(x,t\right)xdx}+\frac{2r_{c2}\alpha }{r_b^2-r_{c2}{}^2}{\displaystyle \underset{r_{c2}}{\overset{r_b}{\int }}{T}_b\left(x,t\right)xdx}}{r_{c2}(\frac{r_{c2}^2+r_b{}^2}{E\left(r_b^2+r_{c2}{}^2\right)}+\mu )+r_{c1}(\frac{r_a^2+r_{c1}{}^2}{E\left(r_b^2+r_{c1}{}^2\right)}-\mu )}$$

(13)

The equation can express the change in interference pressure with time under the condition of contact temperature transfer of the connector, including the temperature information and structure information, which is more consistent with the actual shrink-fitting assembly process conditions.

2.2. Deformation Analysis of the Mounting Edge

The above analysis can obtain the pressure change curve of the contact surface with time under the condition of temperature transfer of the connector. In fact, for the spigot connection structure with the mounting edge, the increase in spigot pressure will cause a warping deformation of the mounting edge, which will have a serious impact on the assembly results. Therefore, it is necessary to obtain the relation between the spigot pressure and the deformation of the mounting edge.

Figure 2 shows the process by which the interference force acts on the spigot contact part, causing the contact surface to undergo bending deformation, and then causing the formation of the mounting edge angle. Assuming that the deformation angle at the root of the cylindrical section is in harmony with the deformation angle at the connection part of the mounting edge, the height of the cylindrical section is far greater than the thickness of the mounting edge, the material is always in a linear elastic state when the cylindrical section bears load, and the thickness of the cylindrical section is constant and far less than its diameter, then the cylindrical coordinate deflection Formula (14) of the lower cylindrical shell can be obtained:

$$D\frac{d^4\omega }{d{z}^4}+\frac{Et}{R^2}\omega =P$$

(14)

where D is bending stiffness, E is material elastic modulus, P is distributed interference pressure, stands for bending deformation, R is the radius, z is the axial coordinate, and w is the radial displacement.

The general solution formula of Formula (14) is Formula (15). The expression of interference force Q and installation corner angle $\theta$ are shown in Formulas (16) and (17).

$$\omega =e^{\lambda z}(C_1\cos \lambda z+C_2\sin \lambda z)+e^{-\lambda z}(C_3\cos \lambda z+C_4\sin \lambda z)$$

(15)

$$Q=D{\left.\frac{d^3\omega }{d{x}^3}\right|}_{x=0}$$

(16)

$$\theta ={\left.\frac{d\omega }{dx}\right|}_{x=0}$$

(17)

3. Finite Element Analysis Model

Based on the above theoretical model, the time-varying property of the temperature is very important to the change trend in the spigot pressure. However, for the spigot interference connection structure with the mounting edge, the theoretical analysis is far from enough. In this section, the finite element model is established to simulate the shrink-fitting process of the spigot interference connection structure, realizing the quantitative control of the thermal load.

In consideration of the calculation efficiency and symmetry of the connection structure, the spigot interference connection structure is equivalent to 1/8 of the sector element, as shown in Figure 3. To simulate the actual working condition, the symmetric boundary and frictionless constraint are set on both sides of the sector, fixed support is applied on the inner circular wall of the intermediate plate, and the interference values are set via parameterization at the contact of spigot, and the interference values can be adjusted. The friction is set as 0.1 coulomb friction contact, the extended Lagrange algorithm is adopted to solve the contact, the detection of the contact point is set at the Gaussian point, and the specific boundary condition is set as shown in Figure 3a.

The basic dimensions of the finite element model are shown in Figure 3b, where the spigot height H is 7 mm, the mounting edge height ${\mathrm{h}}_1$ is 5 mm, the middle hub height is 10 mm, the upper shaft shell thickness t is 4 mm, and the mounting edge length is 25 mm. The properties of the material are as follows: Young’s modulus: 198 GPa, Poisson’s ratio: 0.3, coefficient of thermal expansion: 1.3 × 10⁻⁵/°C, and thermal conductivity: 0.0114 W/(mm·°C).

Under the above geometric parameters, the eight-node hexahedral mesh sol-id185 was selected as the mesh element type, and the model was parametrically divided into finite element mesh elements. In the grid division setting, first specify the cell size in the contact area, and other cells will maintain a certain mapping relationship with it. In order to select the appropriate mesh size of the contact area and analyze the mesh quality, the element size of the contact area is set to be 100~900 μm, successively, under a set of typical geometric parameters (interference of 0.05 mm, cooling temperature of −160 °C). Convergence analysis tests can detect the change in two key parameter values with the change in element size, including spigot pressure and mounting edge deformation, and the results are shown in Figure 4a.

The above results show that when the unit size is reduced, the spigot pressure and the mounting edge deformation are gradually reduced, and their change rates are gradually reduced. When the cell size is reduced below 500 μm, the change rates of the two key parameter values drop to almost zero. In conclusion, the influence of element size change on the convergence accuracy can be ignored below 500 μm. Therefore, in order to ensure convergence accuracy and efficiency at the same time, the mesh of the contact area is set as 500 μm, and the mesh quality is above 0.9, as shown in Figure 4b, the finite element model has a high accuracy and fast solution speed with the overall element number of 84,397 and node number of 378,715. The finite element model has high accuracy and fast solution speed in the contact area.

4. Results and Discussion

This section validates the accuracy of the finite element model. It contains an analysis of the nonlinear growth phenomenon of the spigot pressure and the fundamental causes of deformation at the mounting edge. Two critical time points in the growth of spigot pressure were identified: the “Initial fitting time” and the “Pressure stabilization time”. The discussion further explores the effects of cooling temperature and interference values on pressure and deformation. It summarizes the patterns of influence of temperature and pressure on spigot pressure and mounting edge deformation.

4.1. Finite Element Analysis Results

Using a set of parameters (cooling temperature = −160 °C, interference fit = 0.05 mm) as a case study, we analyzed the time-varying characteristics of the spigot.

Figure 5a displays the temperature distribution cloud map at 85 s, showing that the upper shaft, due to immersion in the cooling fluid, reached extremely low temperatures, with the lowest recorded at −106 °C. In the spigot contact region, there was a temperature trend from the spigot towards the mounting edge, which was consistent with the theoretical results. Figure 5b depicts the curve illustrating the variation in spigot pressure over time. It is observable that during the initial stages of temperature transfer, the spigot pressure remained at zero, which can be attributed to the lack of adhesion between the components at the start of the temperature transfer. Importantly, the observed trend in spigot pressure growth closely aligns with the theoretical predictions, confirming the accuracy of the finite element analysis.

4.2. Time-Varying Characteristics of Spigot Interference Pressure

Using the finite element analysis with an interference fit of 0.05 mm and a cooling temperature of −160 °C, an analysis of the temperature transfer between the components and the variation in spigot interference pressure yielded the following results.

Figure 6a illustrates the temperature variation at the spigot interference between the two components. It is observable that, during the initial 0–30 s, the temperature of the upper shaft rapidly increased. In the subsequent 30–180 s, the temperature of the upper shaft rose gradually. At 180 s, a temperature inflection point appeared at the middle hub, where the temperature began to transition from a decrease to an increase. This temperature variation indicates that before 30 s, the temperature of the upper shaft was exceptionally low and air convection had a significant influence. For 30–180 s, there was a temperature transfer between the two connecting components, characterized by mutual compression. After 180 s, the temperatures between the two connecting components became roughly uniform, and they returned to room temperature together, and the compression phenomenon weakened. Therefore, it can be concluded that temperature variations between the two components induced the thermal deformation of the components, subsequently influencing the interference pressure.

In Figure 6b, it is evident that there was no pressure at the spigot interface before 30 s. This was primarily due to the excessive initial deformation of the upper shaft, leading to a lack of initial adhesion. This phase is aptly defined as “Gap growth”. Between 30 and 180 s, the pressure increased rapidly, which was a consequence of mutual compression between the upper shaft and the middle hub. This phase is suitably defined as “Fast growth”. After 180 s, the pressure gradually increased. During this phase, both the upper shaft and the middle hub returned to room temperature together. This phase is appropriately defined as “Steady growth”.

Based on this non-linear growth trend of spigot interference pressure, two critical assembly time points can be defined: “Initial fitting time” and “Pressure stabilization time”. These two time points are pivotal in the context of shrink-fitting assembly, and the three growth phases effectively capture the non-linear characteristics of spigot interference pressure.

After establishing the non-linear growth characteristics of spigot interference pressure, finite element analysis was conducted with varying interference values and cooling temperatures, yielding the following results.

Figure 7a displays the spigot pressure curves under different cooling temperature conditions with an interference fit of 0.05 mm. It is evident that as the cooling liquid temperature decreased, the initial fitting time shifted earlier. This shift was primarily due to excessive radial deformation at the spigot caused by the lower cooling temperature. Figure 7b displays the impact of different interference fit values on the spigot pressure curve under a cooling temperature of −160 °C. It can be observed that as the interference fit value increased, the initial fitting time shifted earlier, and larger interference fit values led to a rapid increase in seal pressure.

The above results indicate that both cooling temperature and interference fit values are critical factors influencing the time-varying characteristics of spigot interference pressure. Therefore, “Initial fitting time” and “Pressure stabilization time” at the spigot should be comprehensively considered based on the combination of cooling temperature and interference fit values.

4.3. Analysis of Factors Affecting Mounting Edge Deformation

In this set of finite element analyses, different cooling conditions and interference values were configured to explore the influence patterns on mounting edge deformation. The results obtained are as follows.

Figure 8 illustrates the deformation of the mounting edge under different cooling temperatures and interference values. It is noticeable that the curves exhibited a pronounced “rebound” trend. Deformation did not continuously increase or decrease, but rather showed a pattern of initially decreasing and then increasing. When the cooling temperature was the same, the initial buckling deformation of the mounting edge was consistent, measuring 0.065 mm. However, when the cooling temperature differed, the initial buckling deformation exhibited a decreasing trend with increasing temperature. Moreover, when the cooling temperature was consistent, a larger interference value led to greater deformation.

The conclusion can be drawn that, under different cooling temperatures and interference fit values, the “Initial fitting time” and the turning point of the “rebound” phenomenon are essentially consistent. The buckling deformation of the mounting edge appears to be the result of the combined influence of temperature and interference pressure.

4.4. Analysis of the Causes of the “Rebound” Phenomenon

In this set of finite element analyses, aimed at investigating the “rebound” phenomenon in the mounting edge, two control experiments were conducted. One experiment involved no interference fit to explore the influence of temperature on mounting edge deformation, while the other experiment excluded any temperature load to examine the impact of interference fit on the buckling deformation of the mounting edge. The results obtained are as follows.

In Figure 9a, which represents the results without temperature but with increased interference fit values, it can be observed that as the interference fit value increased, the deformation of the mounting edge increased. Furthermore, the deformation exhibited nearly linear growth up to 0.08 mm, after which it increased slowly. This behavior aligned with theoretical expectations, indicating that interference fit values indeed influence the buckling deformation of the mounting edge. Figure 9b, on the other hand, represents the results without interference fit values. In this case, the deformation initially decreased slowly, without any “rebound” phenomenon. As the temperature transfer process continued, the deformation gradually approached zero.

From this, we can conclude that the initial buckling deformation of the mounting edge is primarily caused by temperature. Before the “Initial fitting time”, in the absence of interference pressure, the initial buckling deformation of the mounting edge decreased, with temperature being the main influencing factor. After the “Initial fitting time”, interference pressure rapidly increased, leading to an enlargement of the mounting edge deformation and the occurrence of the “rebound” phenomenon, at which point pressure became the main influencing factor.

5. Experiments

In this section, a shrink-fitting assembly experiment was conducted using the high-pressure compressor structure of the aero-engine rotor. Temperature sensors and high-precision dial indicators were used to validate the correctness of the finite element analysis and theoretical models. Additionally, thin-film pressure sensors were incorporated to measure the press forces at different time points. The results indicated that the initial press forces varied at different moments, and the deformation of the mounting edge affected the application of press forces. Therefore, analyzing the deformation of the mounting edge is necessary.

5.1. Experimental Setup

In Figure 10, a vertically adjustable three-dimensional coordinate platform was constructed. The dial indicator used is manufactured by Japan’s Mitutoyo Corporation, model C112XB, with a range of 12.700 mm and precision to 0.001 mm. The three-dimensional coordinate platform served the purpose of positioning and clamping the dial indicator. The dial indicator head was placed against the mounting edge to measure its deformation data.

Figure 11 illustrates the positioning of the RFP thin-film pressure sensors. These sensors were strategically arranged between the mounting edges to measure pressure values at the contact interface. The RFP thin-film pressure sensors had a measurement range of 0–50 MPa and a temperature tolerance range from −100 °C to 260 °C, ensuring compliance with the measurement requirements. The data collected by these sensors were transmitted to a PC through a dedicated data acquisition system. Customized software was employed to process, store, and analyze the pressure data.

5.2. Experimental Methods and Results

After cooling the upper shaft and inserting it into the middle hub, the pressure between the mounting surfaces was measured using RFP thin-film pressure sensors. The pressing torque was gradually applied, and the relationship curve between the pressing and mounting surface pressure was recorded. This provided data on the initial fitting pressure at different time points. At the same time, the temperature sensor was used to detect the temperature, and the accuracy of the heat transfer theoretical model was proved by using the dial indicator to record the deformation value of the installation side, verifying the initial fitting time in the theoretical model, and comparing with the simulation model.

Figure 12 shows the comparison between the experimental deformation of the mounting edge and the finite element model. It can be seen that the deformation trend was basically the same, and the accuracy of the finite element model was verified. The experiment and finite element showed the “rebound” characteristics of the deformation of the mounting edge. The reduction in the deformation of the mounting edge in the early stage was caused by the temperature rise, while the increase in the deformation in the later stage was caused by the increase in the spigot pressure, which was also the reason for the “rebound” phenomenon. The “rebound” phenomenon should occur during the initial fitting time.

In order to verify the accuracy of the thermomechanical theoretical model, the experimental and theoretical temperature transfer curves of the rotor were compared to obtain the curve shown in Figure 13a. It can be seen that the temperature change in the upper shaft was basically consistent with the theoretical model, and the accuracy of the temperature transfer theoretical model in 2.1 was verified. Figure 13b shows the theoretical calculated spigot interference pressure curve and the experimental “rebound” curve. It can be seen that the initial fitting time obtained by the theoretical model was the time when the “rebound” phenomenon occurred, which also proves the accuracy of the initial fitting time predicted by the thermomechanical theoretical model.

As shown in Figure 14, the initial fitting torque varied at different time intervals. It is evident that the initial fitting torque at 30 s was higher than that at 60 s, indicating that during the initial heat transfer phase, the initial fitting torque decreased. However, as time progressed, the initial fitting torque gradually increased, particularly during the later stages of heat transfer. This demonstrates that as the interference pressure increased with time, the initial fitting pressure also rose. The deformation of the mounting edge affects the application of the initial fitting torque. The “rebound” phenomenon of the mounting edge can have a significant impact on assembly quality.

6. Conclusions

This paper has proposed an interference pressure equation that considers temperature-variable characteristics that can be used to predict the pressure in spigot interference connection structures under shrink-fitting assembly conditions. The theory revealed the mechanisms of nonlinear growth of interference pressures in shrink-fitting assembly processes and has identified two important time points in the process: the “Initial fitting time” and the “Pressure stabilization time”. Based on the finite element model, an analysis was conducted on a special interference connection structure with a mounting edge, exploring the underlying causes of mounting edge deformation. The time-varying patterns of interference pressure and mounting edge deformation under the influence of temperature and pressure loads were determined, laying the foundation for improving assembly quality. From this work, the following conclusions were obtained.

(1) Temperature and interference fit value are closely related to the spigot interference pressure. This paper combined thermal elastic deformation theory with heat transfer theory to establish a time-varying equation for pressure and temperature. Based on the trend of interference pressure, two critical time points for shrink-fitting assembly were proposed. The paper also discussed the influence of interference fit value and temperature on these two time points. The theoretical model showed good agreement with the finite element analysis.

(2) The deformation of the mounting edge can be attributed to two main factors: temperature and interference pressure. Temperature primarily determines the initial deformation of the mounting edge, while the interference pressure dictates the trend of deformation growth. The “rebound” phenomenon in the deformation of the mounting edge indicates that, before the initial fitting time, temperature plays a dominant role, whereas after the initial fitting time, the pressure at the interface becomes the dominant factor.

(3) The experiment used RFP thin-film pressure sensors to measure the fitting pressure at the mounting edge. It was observed that the pressure sensor only registered readings when the torque reached a certain value. The trend of the initial fitting pressure was found to be consistent with the trend of mounting edge deformation. Based on this, applying an appropriate press force according to the mounting edge deformation trend is a subject that requires further research in the design of the shrink-fitting assembly process.