Mathematical Problems in Engineering
Volume 2019 (2019), Article ID 4789030, 15 pages
https://doi.org/10.1155/2019/4789030

Research Article

Multiobjective Intelligent Cooperative Design for the Multilayer Interference Fit

Ke Ning,1,2 Jianmei Wang,1,2 Hongwei Jiang,³ Dan Xiang,⁴ and Dingbang Hou²

¹Engineering Research Center Heavy Machinery Ministry of Education, Taiyuan University of Science and Technology, Taiyuan, China
²School of Mechanical Engineering, Taiyuan University of Science and Technology, Taiyuan, China
³Technology Center, Taiyuan Heavy Machinery Group Co., Ltd., Taiyuan, China
⁴Oil-Film Bearing Branch, Taiyuan Heavy Machinery Group Co., Ltd., Taiyuan, China

Correspondence should be addressed to Jianmei Wang; wjmhdb@163.com

Received 20 June 2019; Revised 9 August 2019; Accepted 23 September 2019; Published 13 October 2019

Academic Editor: Francesco Pellicano

Copyright © 2019 Ke Ning et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Interference fit has extensively been applied in different mechanical fields due to its advantages such as compactness and high rigidity. With the actual needs of engineering, the multilayer interference fit has also been widely used. It is significant for the design and manufacture of multilayer interference fit to consider stress concentration problem and achieve multiobjective and multivariable collaborative optimization in the optimization algorithm. To achieve this goal, the mechanical model and reliability mathematical model of the multilayer interference fit are established. Giving an evaluation method of reliability, the recommended range of interference amount is obtained. Considering the lightweight design and stress concentration problems, a multiobjective intelligent cooperative design (MOICD) method for the multilayer interference fit is proposed, and multiparameter analysis is done. Taking the typical wind turbine’s shrink disk of three-layer interference fit structure as an application example, the key design parameters of the shrink disk are determined, and its sample data are obtained using the orthogonal experimental method. Considering yield strength, torque, and mass, the approximate mathematical model of response surface using the Kriging algorithm is given. Nondominated sorting genetic algorithm-II (NSGA-II) is selected as the optimal algorithm to realize MOICD of the shrink disk. Numerical analysis and analytical calculation proved that the optimization index of the MOICD method is improved compared with the traditional method. The sample is tested on a specific test bench, and the test results meet the design requirements, which verifies the feasibility of MOICD.

1. Introduction

Interference fit has advantages such as precision positioning and high load-carrying capacity, comparing with key connection and pin connection. It has widely been used in heavy machinery, new energy sources, and general machinery, such as semipermanently connected gears, wheels, flanges, and similar mechanical components [1].

Many studies on the interference fit have been performed over the years. Sun et al. [2] researched the process of a pin interference fit and the subsequent fatigue damage evolution, and the result shows that the effect of the interference fit on the fatigue life improvement is beneficial. Wu et al. [3, 4] deepened the study of the interference fit to improve fatigue life and used the spring-mass model to explore the fatigue enhancing mechanism of interference. The study of Zeng et al. [4] on an interference fit between hole and rivet shows that an interference fit can not only improve fatigue life but also enhance strength effect by changing size. Bahloul et al. [5] used the interference fit process as a technique to arrest existing crack in mechanical structures, improving the fatigue life and investigating the effects of different hole diameters and sizes to optimize the interference fit process parameters. Zou et al. [6] established the general mechanical model during the interference fit pin installation process, predicting the delamination onset at different interference fit percentages, the insertion load, and stress. Strozzi et al. [1] addressed the stress concentrations developed from the static press-fit of the shaft-hub interference fit, and various design charts have been compiled that report the elastic stress concentrations within the hub versus the proposed normalizing parameter, including fillet radius, shaft radius, and interference. Wang et al. [7] deduced the stress distribution equation of the interference fit based on the stress concentration problem, and a new theoretical model of stress distribution prediction was established. Most of the above research was aimed at single-layer interference fit, but with the actual needs of engineering, many use double or multilayer interference fit structure to achieve load transfer. Wang et al. [8–10] deduced the mechanical model of the multilayer interference fit, analyzed its bearing capacity by the finite element method, and proposed a reliable and robust design method. Qiu and Zhou [11] investigated the temperature effect of the multilayer interference fit. Bai et al. [12] established a theoretical model of the multilayer interference fit considering the effect of centrifugal force. You et al. [13, 14] developed an analytical solution for the mechanical behaviors associated with a three-dimensional anisotropic multilayered medium under moving loads, and the dynamic response analysis of multilayer structures was realized. However, until now, there has been little research on the design algorithm of the multilayer interference fit considering stress concentration problem, as well as little research on optimization design based on multiobjective and multivariable interaction.

To solve the above problem, this work gives the reliability mathematical model of the multilayer interference fit based on the mechanical model. The recommended range of interference is determined. Considering the lightweight design and stress concentration problems, the multiobjective and multivariable synergistic relationship is established, and the multiobjective intelligent cooperative design (MOICD) method for the multilayer interference fit is proposed. Constructing an approximate mathematical model of the response surface, the prediction of load transfer, material yield strength, and equipment weight is realized. Nondominated sorting genetic algorithm-II (NSGA-II) is selected as the optimal algorithm to optimize the multiobjective model composed of the response surface mathematical model. The typical three-layer interference fit of the wind turbine’s shrink disk is taken as an example to design. Based on finite element analysis and mechanical model calculation, the optimized shrink disk is validated to meet the requirements of optimization, and the sample is verified by a special test bench, which meets the operational requirements.

2. Reliability Model of Multilayer Interference Fit

2.1. Theoretical Deduction

According to the thick-walled cylinder theory, the multilayer interference fit can be simplified as a multilayer cylinder, as shown in Figure 1, and the cylinder is numbered S₁, … , Si+1 from inside to outside. pi is defined as the contact pressure of the interference layer between Si and Si+1, and the interference amount between Si and Si+1 is δi.

Figure 1: The diagram of a multilayer interference fit.

The matrix expression between contact pressure and interference amount of the multilayer interference fit can be derived as

where i = 1,2, … , n, , , d1,i is the inner diameter of Si, d2,i is the outer diameter of Si, and and Ei are Poisson’s ratio and modulus of elasticity of Si, respectively.

The boundary conditions of the multilayer interference fit with clearance fit can be expressed as follows:

where are the radial displacements of inner and outer surfaces after compression of Si. and ; are the assembly clearance between S₁ and S₂, S₂ and S₃, … , Si−1 and Si.

According to equations (1) and (3), the contact pressure p₁ of first-layer interference is expressed as

where i = 1, 2, … , n and N₁, … , Ni is the relational expression among K1,i, … , K4,i. [15].

Taking the three-layer interference fit as an example, some formulas of equation (4) are as follows:

Combining equations (3) and (4), the contact pressure of other interference layers can be deduced as

where and G, … , J is certain coefficient related to boundary conditions, which satisfies , , and .

2.2. Mechanical Model

Considering that the primary function of the multilayer interference fit is to transfer torque, the mechanical model is constructed. The first-layer interference, i.e., the contact surface between S₁ and S₂, meets the rated torque transfer. The minimum design contact pressure is as follows:

where , Mt is the rated transfer torque, μ₁ is the friction coefficient between S₁ and S₂, d₁ and l₁ are the combined diameter and contact length of first-layer interference, respectively.

To ensure the reliability of the multilayer interference fit, the maximum contact pressure of each interference layer should be less than the plastic deformation or fracture limit of the material. Making S₁ as an enveloped part of first-layer interference, the maximum contact pressure between S₁ and S₂ allowed for no plastic deformation or fracture should be as follows.

When cylinder S₁ is a plastic material,

where , , and is the yield strength of S₁.

When cylinder S₁ is a brittle material,

where is the tensile strength of S₁ and c is the safety factor for the strength of brittle material.

The enveloping part of first-layer interference takes as a whole; the maximum contact pressure between S₁ and S₂ allowed for no plastic deformation or fracture should be as follows.

When the cylinder is a plastic material,

where , , and is the equivalent yield strength of .

When the cylinder is a brittle material,

where is the equivalent tensile strength of .

The maximum contact pressure between cylinders S₁ and S₂ is , and the mechanical model of the multilayer interference fit can be deduced by combining equations (7)∼(11):

where i = 1,2, … , n, is the design contact pressure of i-th layer interference (cylinder Si and Si+1), is the maximum permissible contact pressure, i.e., , is the maximum permissible contact pressure for the enveloped part, and is the maximum permissible contact pressure for the enveloping part.

2.3. Reliability Mathematical Model

According to the classical interference fit design method [16], the basic random variables X include friction coefficient , assembly clearance of first-layer to i-th layer interference, the design interference amount δi of i-th layer interference, and the yield strength of cylinder Si; . The basic random variables are assumed to obey normal distribution and mutual independence.

The reliability mathematical model of the multilayer interference fit is established based on the mechanical model, and several mechanical conditions must be satisfied at the same time so that the overall reliability can be guaranteed. It has distinct series characteristics. The reliability mathematical model is as follows [8]:

where are the corresponding state functions, is the reliability index to satisfy , is the reliability index to satisfy , and is the reliability index to satisfy .

According to equations (4), (6), (12), and (13), the state function is as follows:

where is the relational expression between .

The reliability index corresponding to the state function is expressed as

where is the mean value of the state function and is the standard deviation of the state function.

So, the reliability mathematical model of the multilayer interference fit can be obtained as

where , is a function conforming to the normal distribution. And normal distribution can be calculated by using function normcdf in MATLAB.

The wind turbine’s shrink disk is the crucial connecting part between the shaft and gearbox. If it fails, the whole transmission system of the wind turbine will be completely invalid. Therefore, it requires a very high reliability. The shrink disk can be regarded as a three-layer interference fit model. This model consists of a shaft (S₁), sleeve (S₂), inner ring (S₃), and outer ring (S₄), as shown in Figure 2. It forms the interference fit on the conical surface between the inner ring and outer ring by tightening the bolts, the cone angle of the conical surface is β, and the interference amount is δ₃. Because the interference between inner ring and outer ring causes compaction, the assembly clearance between inner ring and sleeve, and sleeve and shaft are counteracted to form the three-layer interference fit.

Figure 2: Schematic diagram of the wind turbine’s shrink disk.

According to equation (16), the reliability mathematical model of the shrink disk is established as

A certain type of the shrink disk in Table 1 is selected as the application example to get the basic random variable X, including the friction coefficients between shaft and sleeve, sleeve and inner ring, and inner ring and outer ring; the assembly clearance between shaft and sleeve, and sleeve and inner ring; and the yield strength of shaft, sleeve, inner ring, and outer ring. The mean and standard deviation are obtained by the empirical formula [16], and the results are shown in Table 2. In the design process, taking the interference amount δ₃ between inner ring and outer ring as the primary design variable, the mean value of δ₃ is given by design, and the standard deviation is based on the empirical formula with a value of 0.021 mm.

Table 1: The basic parameters of a certain type of the wind turbine’s shrink disk.

Parameter
Numerical value

Inner diameter of shaft d1,1 (mm)
60

Outer diameter of shaft d2,1 (mm)
520

Outer diameter of sleeve d2,2 (mm)
640

Average diameter of long conical surfaces of inner ring d2,3 (mm)
663.006

Outer diameter of outer ring d2,4 (mm)
1020

Length of contact surface between shaft and sleeve l₁ (mm)
280.5

Elastic modulus of material for shaft, inner ring, and outer ring (GPa)
210

Elastic modulus of material for sleeve (GPa)
180

Poisson’s ratio
0.3

Assembly clearance between shaft and sleeve
0.079

Assembly clearance between sleeve and inner ring
0.16

Table 2: The value of random variable and standard deviation.

Random variable
Mean value
Standard deviation

μ₁
0.15
0.02

μ₂
0.15
0.02

μ₃
0.045
0.006

R₁ (mm)
0.079
0.019

R₂ (mm)
0.16
0.027

(MPa)
930
35.70

(MPa)
835
30.03

(MPa)
930
35.70

(MPa)
930
35.70

In order to verify the accuracy of the reliability mathematical model, the result is verified by Monte Carlo simulation. Its reliability is equivalent to the product of probability:

In random variables , satisfies the following inequalities:

The number of random variables satisfying equation (19) is U, and the total number of random variables is N, N = 10⁴. According to the Monte Carlo method, the reliability problem is solved as [8]

Figure 3 shows the result contrast between reliability algorithm and Monte Carlo simulation. The results of both methods have the same law and good coincidence. When δ₃ is within [1.5, 2.2] mm, the reliability increases gradually, and the growth rate slows down gradually; when δ₃ is within [2.2, 2.8] mm, the reliability basically remains unchanged; and when the δ₃ is within [2.8, 3.2] mm, the reliability decreases rapidly. The initial increase in reliability is due to the increase in the interference amount to transmit more torque, and the final decrease in reliability is due to excessive interference amount, thus resulting in strength failure of the material. The recommended range of δ₃ is [2.2, 2.8] mm.

Figure 3: Reliability comparison of different methods.

3. Multiobjective Intelligent Collaborative Design (MOICD)

Based on the reliability mathematical model, the MOICD of the multilayer interference fit is proposed by introducing a lightweight design and considering the synergistic relationship between multiobjective and multivariable. Firstly, according to the design experience, the key variables of the multilayer interference fit design are determined, and the sample data of each key variable are obtained by the orthogonal experimental method [17]. Then, the Kriging method with good fitting effect on high-dimensional nonlinear problems is selected as a response surface approximation mathematical model to realize prediction [18]. And the approximation model can meet the prediction requirements of load transfer requirements, material strength, and equipment weight.

Among them, considering the stress concentration problem, equivalent stress is regarded as the indicator of material strength. Next, NSGA-II is selected as the optimal algorithm to realize MOICD of the multilayer interference fit, comparing the algorithm of NSGA-II [19], multiobjective simulated annealing (MOSA), and multiobjective particle swarm optimization (MOPSO). Figure 4 shows the flow chart of the MOICD method. di is the combined diameter of cylinders Si and Si+1.

Figure 4: Flow chart of the MOICD method.

The MIOCD of the multilayer interference fit is implemented by taking the wind turbine’s shrink disk as an application example. The design indicators corresponding to the approximation model is determined, including the minimum contact pressure p1,min between shaft and sleeve, the maximum equivalent contact stress Emax, and the minimum mass mmin. Then, related indicators are converted to optimized size parameters [16], including the minimum diameter d3 min of long cone surface of the inner ring (consistent with , where is the average diameter of long conical surfaces of the inner ring and b is a coefficient related to size), the diameter of the outer ring, and the maximum design interference amount δ3 max between inner ring and outer ring; the relationship between these size parameters is shown in Figure 2.

Using the orthogonal experimental method, the corresponding result data of design indicators and size parameters are determined as shown in Table 3, and the relevant results are obtained from finite element ABAQUS simulation. The value range of d3min and d2,4 is given by combining with the production process of the shrink disk. The recommended range of the interference amount for the reliability mathematical model is [2.2, 2.8] mm; in order to get more data points, on this basis, extend the scope of δ3max to [2, 3] mm.

Table 3: Data obtained by the orthogonal experimental method.

Sample points
d2,4 (mm)
δ3 max (mm)
d3 min (mm)
Emax (MPa)
p1,min (MPa)
mmin (kg)

1
1000
2
648
657.019
117.279
960.970

2
1000
2
652
638.443
114.912
960.970

3
1000
2
656
622.318
112.473
960.970

4
1000
2.5
648
850.578
154.505
960.970

5
1000
2.5
652
820.793
151.672
960.970

6
1000
2.5
656
802.022
148.636
960.970

7
1000
3
648
1039.589
191.921
960.970

8
1000
3
652
1003.183
188.497
960.970

9
1000
3
656
980.801
184.872
960.970

10
1020
2
648
664.741
120.915
1026.727

11
1020
2
652
641.577
118.686
1026.727

12
1020
2
656
625.665
116.272
1026.727

13
1020
2.5
648
854.312
159.347
1026.727

14
1020
2.5
652
826.184
156.546
1026.727

15
1020
2.5
656
804.984
153.539
1026.727

16
1020
3
648
1044.206
197.857
1026.727

17
1020
3
652
1009.640
194.471
1026.727

18
1020
3
656
984.447
190.882
1026.727

19
1040
2
648
668.134
124.451
1093.787

20
1040
2
652
644.916
122.243
1093.787

21
1040
2
656
628.195
119.969
1093.787

22
1040
2.5
648
860.380
163.912
1093.787

23
1040
2.5
652
829.261
161.141
1093.787

24
1040
2.5
656
809.633
158.162
1093.787

25
1040
3
648
1045.534
203.590
1093.787

26
1040
3
652
1013.402
200.104
1093.787

27
1040
3
656
988.631
196.678
1093.787

An approximate response surface mathematical model based on design indicators (objective function) and size parameters is constructed by the Kriging method, as shown in equations (21)∼(23), and this approximate model is achieved by using DACE toolbox of MATLAB:

where is the maximum equivalent stress function, is the function of minimum contact pressure between shaft and sleeve, is the minimum mass function, is residual, the expectation is zero, and the covariance is not zero.

The coefficient of determination R² is used to test the accuracy of the Kriging method, and its expression is as follows:

where h is the number of data sample points, is the predicted value of the approximate model, yi is the true value, and is the average value. The larger the decision coefficient R² is in the range of [0, 1], the higher the accuracy of the model. The correlation coefficient of , … , is 0.999954, 0.999999, and 1. It shows that the Kriging method meets the accuracy requirement.

According to the main causes of the interference fit failure such as insufficient load transfer and the maximum equivalent stress exceeding the material yield limit [7], the multiobjective model of the shrink disk first meets the requirements of torque and yield strength, and then meets the minimum mass. The mathematical expression of the multiobjective model is as follows:

The maximum equivalent stress Emax cannot exceed the material yield limit Q, the minimum contact pressure p1,min must be greater than the pressure required to transmit the rated torque, and d3 min, d2,4, and δ3 max meet the upper and lower limit requirements. According to design experience [9, 10], Emax and p1,min need to be as large as possible; that is, the negative number of its objective function −f₁ (x) and −f₂ (x) need to be as small as possible, and at the same time, f₃ (x) is as small as possible to reduce the mass of the shrink disk and lower the manufacturing cost. The priority of the above three functions is as follows: −f₂ (x) > −f₁ (x) > f₃ (x).

Equations (21)∼(25) are solved using NSGA-II, MOSA, and MOPSO algorithm. Among these three algorithms, NSGA-II setting: the population number is 1000, the variation rate is 0.1, and the crossing rate is 0.85; MOSA setting: the Boltzmann coefficient is 1, the maximum internal loop number is 1000, and the cooling coefficient is 0.99; MOPSO setting: the population number is 1000 and the nearest population is 2. The certain type of the shrink disk is designated as an example, and the three algorithms are used to run 20 times independently [20]. The minimum mass obtained by these three algorithms is the same . For a more intuitive comparison of the three algorithms, −f₁ (x) and −f₂ (x) are taken as the research objects, and the contrast is shown in Figure 5.

Figure 5: Comparison of the optimal solution sets of −f₁ (x) and −f₂ (x) in three algorithms.

Figure 5 shows the optimal solution sets of −f₁ (x) and −f₂ (x). The results revealed that NSGA-II has significant advantages over other algorithms in optimal performances [20]. Specifically, the negative number of maximum equivalent stress and minimum contact pressure from NSGA-II is smaller than that from MOSA and MOPSO. That is to say, the maximum equivalent stress and minimum contact pressure obtained by NSGA-II are the largest; besides, the computational optimization time required for NSGA-II is less than that for the other two algorithms, as shown in Table 4. Table 4 is a comparison of the average computational time required by the three algorithms when calculating the results in Figure 5. Finally, NSGA-II is selected as the optimal algorithm.

Table 4: The mean time use of three algorithms.

Time
Algorithm

NSGA-II
MOSA
MOPSO

Mean time of use (s)
15.07
44.07
22.31

In the calculation result of NSGA-II, the minimum value of −f₁ (x) and −f₂ (x) is chosen as the best solution of optimal solution sets. Then, d3 min and d2,4 are combined to update the overall structure size. Simultaneously in equations (1) and (6), the minimum design interference amount δ3min between inner ring and outer ring is as follows:

According to equation (7), the minimum contact pressure p1 min is obtained, where p2 min is the minimum contact pressure between sleeve and inner ring and p3 min is the minimum contact pressure between inner ring and outer ring. Kij is the relational expression for the known parameters of the shrink disk.

The optimized size parameters d3 min, d2,4, δ3 min, and δ3 max are determined by the MOICD method and compared with the dimension data obtained by the traditional design method, as shown in Table 5. The traditional design method combines equations (4) and (12) to achieve the design of main dimensions by calculating contact pressure, without considering the problems of stress concentration and lightweight [21].

Table 5: Comparison of design results between the traditional and MOICD methods.

Design results
Traditional method
MOICD method

d2,4 (mm)
1020
1000

d3 min (mm)
652
653.8

δ3 min (mm)
1.841
1.907

δ3 max (mm)
3.310
2.820

mmin (kg)
1026.727
960.970

Mt (kN·m)
2800.2
2800.7

As can be seen from Table 5, d2,4 of the MOICD method is less than that of the traditional method, which reduces the mass of the shrink disk by 6.7%. With δ3 min as the input parameter, the minimum contact pressure p1 min is obtained by using equation (4), which proves that the rated torque Mt designed by the MOICD method is improved compared with the traditional method. With δ3 max as the input parameter, the results of the two methods are compared using finite element ABAQUS analysis. The feasibility of the finite element model is verified by the stress test experiment. The test method is selected to fit the triaxial strain rosette. The schematic diagram of the experiment is shown in Figure 6. In the experimental scheme, two CS-1 dynamic resistance strain gauges and one XHCDSP data acquisition instrument are used. The experimental stress value is the average value of four strain rosette tests, and the error with the finite element simulation is 2.31%.

Figure 6: The schematic diagram of the experiment.

The modeling data of the finite element are shown in Table 1. The friction coefficient of the contact surface between the shaft and sleeve, and the sleeve and inner ring is set to 0.15, and that between the inner ring and outer ring (coated with molybdenum disulfide grease) is 0.045. This finite element model belongs to the problem of the multicontact surface with tangential slip, so the contact mode is selected as general contact, the contact surface is defined by the penalty function friction formula, and the contact pair is used as the finite sliding of surface-to-surface contact. Considering the structural symmetry of the shrink disk, in order to reduce the calculation cost, the two-dimensional axisymmetric reduced integral element CAX4R is selected as the finite element model cell type [22, 23]. The total number of units is 7828. The mesh of the model is divided into a quadrilateral, and the mesh sizes of the outer ring, inner ring, sleeve, and shaft are 4 mm, 2 mm, 4 mm, and 4 mm, respectively [9]. The layout seeds are determined by the length of edges, which optimizes the aspect ratio of the mesh and improves the mesh quality. The average aspect ratio of the shaft, sleeve, inner ring, and outer ring meshes is 1.02, 1, 1.41, and 1.03, respectively [24]. Setting the boundary conditions, the shaft UX = UY = 0, the sleeve UX = UY = 0, the inner ring UX = 0, UY = δ₃/2 sin β cos β, and the outer ring is no boundary conditions. The type of analysis step is set to static general. Because the assembly process is a large displacement problem, the geometric nonlinearity is set to On. At the same time, the incremental step is set to a fixed value, and the maximum incremental step is defined as 0.02. Figure 7 is the stress distribution diagram of the finite element model of the traditional method and MOICD method.

Figure 7: Stress distribution diagram based on two methods.

Figure 7 shows that the maximum equivalent stress of the shrink disk based on the traditional method is 1103.646 MPa, which is much higher than the maximum yield strength 930 MPa, and plastic deformation occurs. According to the conventional design theory [21], plastic deformation has a negative impact on the strength of mechanical structures, resulting in a reduction in service life. The maximum equivalent stress based on the MOICD method is 915.313 MPa, which is less than the yield limit of the material. Comprehensive analysis proves that the shrink disk based on the MOICD method has been improved in the design objectives of minimum mass, torque transfer, and yield strength, which meets the multiobjective design requirements.

4. Multiparameter Analysis of Multilayer Interference Fit

4.1. The Influence of Outermost Diameter of Enveloping Part on the Contact Pressure

Multiparameter analysis of the multilayer interference fit is carried out using finite element software ABAQUS. Figures 8∼10 show the impact of the outermost diameter of the enveloping part (outer ring diameter d2,4) and axial contact position on contact pressure of the shrink disk. d2,4 takes 1000 mm, 1020 mm, and 1040 mm as research objects.

Figure 8: The influence of d2,4 and axial contact position on p₁.

Figure 9: The influence of d2,4 and axial contact position on p₂.

Figure 10: The influence of d2,4 and axial contact position on p₃.

In Figures 8∼10, the contact pressure of each interference layer increases with d2,4, and the change of contact pressure from the innermost interference layer between shaft and sleeve to the outermost interference layer between inner ring and outer ring decreases continuously. d2,4 is smaller due to the lower contact pressure, which could not meet the torque transmission requirements. A larger d2,4 is conducive to increase the contact pressure and help the shrink disk to undertake more torque, but it will increase the mass and manufacturing cost. Therefore, the multiobjective optimization of d2,4 can not only meet the load transfer requirements but also help the shrink disk to reduce the mass and achieve the design goal of lightweight.

4.2. The Influence of Subouter Diameter of Enveloping Part on Contact Pressure

Figures 11∼13 show the influence of suboutermost diameter of enveloping part (the minimum diameter d3 min in the inner ring long cone) and axial contact position on contact pressure of the shrink disk. d3 min takes 648 mm, 652 mm, and 656 mm as research objects.

Figure 11: The influence of d3 min and axial contact position on p₁.

Figure 12: The influence of d3 min and axial contact position on p₂.

Figure 13: The influence of d3min and axial contact position on p₃.

In Figures 11∼13, the contact pressure of each interference layer decreases with d3 min, and the change of contact pressure from the innermost interference layer to the outermost interference layer decreases continuously. In addition, the pressure distribution of the contact layer between inner ring and outer ring is also affected by d3 min. A smaller d3 min could increase the contact pressure and help the shrink disk transmit more load, but it will increase the difficulty of machining inner ring. Therefore, in the design process of the inner ring, smaller d3 min should be selected as the design size based on considering appropriate processing technology.

4.3. The Influence of Interference Amount on Contact Pressure

Figures 14∼16 show the impact of interference amount δ₃ and axial contact position on contact pressure of the shrink disk. δ₃ takes 2 mm, 2.5 mm, and 3 mm as research objects, and there is a positive correlation between δ₃ and contact pressure; the change of contact pressure from the innermost interference layer to outermost interference layer is decreasing. Too small δ₃ design results in insufficient contact pressure which could not transfer rated torque, and excessive δ₃ design results in excessive contact pressure which is straightforward to produce plastic deformation; therefore, it is necessary to strictly control the interference amount.

Figure 14: The influence of δ₃ and axial contact position on p₁.

Figure 15: The influence of δ₃ and axial contact position on p₂.

Figure 16: The influence of δ₃ and axial contact position on p₃.

The rules shown in Figures 8∼16 verify the correctness of equations (4) and (6). In addition, these figures show that there is a large contact pressure singularity at both ends of the interference layer, and the singularity of the contact surface between inner ring and outer ring is the largest, which is caused by the stress concentration of the interference fit. Stress concentration is often the leading cause of damage, so it must be considered in the design.

5. Test Verification

In order to test whether the optimized wind turbine’s shrink disk meets the requirements of the application, based on the optimized design size in Table 5, shrink disk samples are manufactured, and the experimental verification is completed by a specific test bench. This test bench exerts torque using a hydraulic cylinder and supporting bracket, as shown in Figure 17. According to experimental experience, the torque retention time is 3 minutes. It is determined that no circumferential slip occurs in the three interference fit layers of the sample, which proves to meet the requirements of torque design. The test data are shown in Table 6, M = 2800 kN·m, which is illustrated in Table 1.

Figure 17: Test bench of the wind turbine’s shrink disk.

Table 6: Static torque detection data.

Torque value (kN·m)
Circumferential slip (mm)
Holding pressure time (min)

0.4 M
0
3

0.5 M
0
3

0.65 M
0
3

0.8 M
0
3

M
0
3

1.2 M
0
3

6. Conclusions

(i)The mechanical model and the relationship of contact pressure among each interference layer are obtained, the reliability mathematical model on the reliability evaluation method is built, and the range of recommended interference amount is given.(ii)Considering lightweight design and stress concentration, the MOICD method for the multilayer interference fit is proposed. Taking the Kriging model as the prediction method of the response surface, comparing with NSGA-II, MOSA, and MOPSO, NSGA-II is proved to be the optimal algorithm for multilayer interference fit design.(iii)Using the equivalent stress as the evaluation index of the stress concentration problem, the design defects due to stress concentration in traditional design can be avoided [7]. Comparing finite element analysis and mechanical calculation, the maximum equivalent stress on the shrink disk is less than the yield strength of the material, and at the same time, the transmission torque is increased and the weight is reduced, ultimately the optimization goal of the MOICD method is achieved.(iv)The rules of different parameters on contact pressure are obtained using multiparameter analysis on outermost layer diameter, subouter layer diameter, and interference amount, which guides the design and manufacturing optimization of the multilayer interference fit.(v)According to the design sizes from the MOICD method, the sample is developed, and the torque is exerted gradually on the shaft by the test platform. Experiments show that the shrink disk can meet the requirements of torque design.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this article.

Acknowledgments

This research was supported by the National Natural Science Foundation of China (No. 51875382), Shanxi Provincial Key Research and Development Plan (No. 201803D421103), Shanxi Provincial Key Science and Technology Special Projects (No. 20181102023), Postgraduate Innovation Project of Shanxi Province (2018SY089), and the Coordinative Creation Center of Taiyuan Heavy Machinery Equipment (1331 Project).

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