Contact Pressure Algorithm of Multi-Layer Interference Fit ... - MDPI

applied sciences Article

Contact Pressure Algorithm of Multi-Layer Interference Fit Considering Centrifugal Force and Temperature Gradient Zebing Bai, Jianmei Wang *

ID

, Ke Ning and Dingbang Hou

Engineering Research Center Heavy Machinery Ministry of Education, Taiyuan University of Science and Technology, Taiyuan 030024, China; [email protected] (Z.B.); [email protected] (K.N.); [email protected] (D.H.) * Correspondence: [email protected]; Tel.: +86-186-3514-9106 Received: 29 March 2018; Accepted: 23 April 2018; Published: 5 May 2018







Abstract: In order to optimize the algorithm on contact pressure and interference magnitude of multi-layer interference cylinders, the effects of centrifugal force and temperature gradient on the performance of multi-layer interference cylinders are considered. The mathematical matrix model of contact pressure and interference magnitude of multi-layer interference cylinder is constructed under centrifugal force and temperature gradient. Four kinds of interference cylinder models are established, and the numerical solutions using finite element methods are compared with the analytical solutions. The results show that the contact pressure of each contact surface of multi-layer interference cylinder decreases with speed increase; the slip rate of contact pressure increases with speed increase; the maximum radial displacement of an interference cylinder under centrifugal force occurs at the outermost layer. The contact pressure of a multi-layer interference cylinder decreases linearly with the increase of temperature gradient. The analytical solution is basically consistent with the numerical solution, and the maximum relative error of midpoint is 11.62% and −7.69% under two factors. It means that the algorithm can provide a theoretical guide for the design of a multi-layer interference cylinder considering centrifugal force and temperature gradient. Keywords: centrifugal force; temperature gradient; multi-layer interference cylinder; contact pressure

1. Introduction The interference fit has the advantages of simple structure, good neutrality, strong load-carrying capacity, good impact resistance, and so on, so it has a wide application in mechanical engineering [1]. Many scholars have done a lot of research on the algorithm and influence of interference fit. Sogalad [2] studied the influence of cylindricity and surface modification on the load-carrying capacity of interference fit components. Zhang et al. [3] analyzed the influence of different influence factors on the dynamic performance of oblique cone interference fit. Compared with the equal diameter cylindrical interference fit, the influence of interference magnitude on the local contact stress is obvious. Wang et al. [4,5] proposed a variety of algorithms for multi-layer interference fit based on thick-wall cylinder theory and analyzed the effects of many factors on the performance of multi-layer interference fit. McMillan [6] studied the phenomenon of torque increase during the loading cycle of the interference specimen. By establishing a cycle wear test, the reason for the torque increase is related to the friction coefficient between interference fit. The research provides a basis for the optimization of interference fit design. Based on the geometric defects and surface roughness of the interference specimen, Boutoutaou H et al. [7,8] studied its influence on the strength of interference connection. Li J et al. [9,10] studied the effect of interference size on the fretting fatigue life of carbon fiber reinforced polymers and titanium alloy bolts. Appl. Sci. 2018, 8, 726; doi:10.3390/app8050726

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surface roughness of the interference specimen, Boutoutaou H et al. [7,8] studied its influence on the strength of interference connection. Li J et al. [9,10] studied the effect of interference size on the Appl. Sci. 2018, 8, 726 2 of 12 fretting fatigue life of carbon fiber reinforced polymers and titanium alloy bolts. The interference components usually work at high speed and under a temperature field, various rotational speeds and temperatures different onand the performance of interference The interference components usuallyhave work at higheffects speed under a temperature field, components. At present, sometemperatures scholars have made someeffects research in performance these fields. of For example, various rotational speeds and have different on the interference Kurvinen [11] studied the dynamic characteristics and mechanical properties of ball bearings with or components. At present, some scholars have made some research in these fields. For example, without force. Guo [12]characteristics established aand displacement model basedwith on Kurvinencentrifugal [11] studied the dynamic mechanical mathematical properties of ball bearings centrifugal and temperature bearing inner ring, coupled with the bearing or without force centrifugal force. Guo in [12]the established a displacement mathematical model dynamic based on model of bearing, andtemperature studied change of bearing interference and influence on spindle centrifugal force and in therule bearing inner ring, coupled fit with thethe bearing dynamic model system dynamics. The results that the initial interference magnitude will on increase with the of bearing, and studied changeshow rule of bearing interference fit and the influence spindle system combined influence factors of temperature increase and centrifugal force, and eventually lead to the dynamics. The results show that the initial interference magnitude will increase with the combined increase bearing stiffness. Ding [13]and analyzed the force, calculation method lead of the interference influenceof factors of temperature increase centrifugal and eventually to the increase of magnitude and stress the rotor and the permanent establishing a finiteand element bearing stiffness. Dingbetween [13] analyzed the calculation method ofmagnet, the interference magnitude stress model forthe stress analysis, the results were compared. Zhou [14]element established a three-dimensional between rotor and theand permanent magnet, establishing a finite model for stress analysis, single-layer interference cylinder model and analyzed influence of centrifugal force on the and the results were compared. Zhou [14] established a three-dimensional single-layer interference interference fit. Qiu al. [15] proposed calculating contact pressure andet stress of cylinder model andetanalyzed influencean of algorithm centrifugalfor force on the interference fit. Qiu al. [15] multi-layer fit under multicontact contactpressure and temperature. engine crankshaft bearing case proposed aninterference algorithm for calculating and stress An of multi-layer interference fit under was established using finite element method to verifybearing the correctness its algorithm. multi contact and temperature. An engine crankshaft case was of established using finite element However, thethe research on the factors of interference fit is mainly concentrated in a method to verify correctness ofinfluencing its algorithm. singleHowever, layer. Currently, multi-layer interferencefactors fit is of widely used infit engineering, for example the research on the influencing interference is mainly concentrated in low-speed coupling of wind turbine (Figure 1), which be used typical application of a single layer. Currently, multi-layer interference fit iscan widely usedas in the engineering, for example multi-layer [16]. Here, the 1), effects of centrifugal force temperature gradient low-speed interference coupling of cylinder wind turbine (Figure which can be used as and the typical application of on the contact pressure cylinder of multi-layer interference are analyzed. matrix model of multi-layer interference [16]. Here, the effectscylinder of centrifugal force andThe temperature gradient contact pressure and interference magnitude of multi-layer interference cylinder under centrifugal on the contact pressure of multi-layer interference cylinder are analyzed. The matrix model of contact force andand temperature gradient is derived. Four kinds of interference cylinder models areforce builtand to pressure interference magnitude of multi-layer interference cylinder under centrifugal respectively of centrifugal force and temperature gradient thetoperformance temperatureanalyze gradientthe is influence derived. Four kinds of interference cylinder models are on built respectively of multi-layer interference cylinder. By and comparing the gradient calculation results of the finite element analyze the influence of centrifugal force temperature on the performance of multi-layer numerical method and method, some conclusions multi-layer interference cylinder interference cylinder. Byanalytic comparing the calculation results of of thethe finite element numerical method and are obtained. analytic method, some conclusions of the multi-layer interference cylinder are obtained.

Figure1.1.Low Lowspeed speedcoupling couplingof ofwind windturbine. turbine. Figure

2.Algorithm Algorithmfor forMulti-Layer Multi-LayerInterference InterferenceCylinder Cylinder 2. According to to the the Lame Lame equation, equation, the the radial radial displacement displacementof ofany anypoint pointin in the the cylinder cylinder can can be be According obtained[17]: [17]: obtained 1 − µ a2 p − b2 p2 1 + µ a2 b2 ( p − p2 ) 1 (1) u ϕ = 1  · a 2 p12  b22 p · ρ +1   a 2·b2 ( p 2 1p )2 1 · E ρ b1 −2a  u E  b 21− a 2 2    (1) 2 2 E E  b a b a where a is the inner radius of the cylinder, b is the outer radius of the cylinder, ρ is the radius of any point ofacylinder, p1 is radius the internal of the p2 radius is the external pressure  of the cylinder, Where is the inner of thepressure cylinder, b iscylinder, the outer of the cylinder, is the radius E is the modulus of elasticity, and µ is Poisson ratio. of any point of cylinder, p1 is the internal pressure of the cylinder, p2 is the external pressure of the Figure 2 is the schematic diagram of force on two cylinders with interference fit. Ci is defined as cylinder, E is the modulus of elasticity, and  is Poisson ratio. cylinder i; Ci+1 is cylinder i + 1; pi−1 is the internal pressure of Ci ; pi is the contact pressure between

Figure 2 is the schematic diagram of force on two cylinders with interference fit. Ci is defined Appl. Sci. 2018, 8, REVIEWi  1 ; as cylinder i ;x FOR C PEER is cylinder i 1

3 of 12 pi 1 is the internal pressure of Ci ; pi is the contact pressure

between Ci and Ci 1 ; pi 1 is the external pressure of Ci 1 ; r1,i is the inner radius of Ci ; r2,i is schematic diagram of force on two cylinders with interference fit. Ci is defined 3 of 12 the outer radius of Ci ; r2,i 1 is the outer radius of Ci 1 ; S is contact surface between Ci and Ci 1 . as cylinder i ; Ci 1 is cylinder i  1 ; pi 1 is the internal pressure of Ci ; pi is the contact pressure Figure the Appl. Sci. 2018,28,is 726

between Ci and Ci 1 ; pi 1 is the external pressure of Ci 1 ; r1,i is the inner radius of Ci ; r2,i is Ci and Ci+1 ; pi+1 is the external pressure of Ci+1 ; r1,i is the inner radius of Ci ; r2,i is the outer radius r2,i 1 is of the radius of Ci ; radius theCouter radius of Ci 1 ; S between is contactCsurface between Ci and Ci 1 . of Couter i ; r2,i +1 is the outer i +1 ; S is contact surface i and Ci +1 .

Figure 2. The force diagram of interference cylinder.

According to Equation (1), the relation between the radial displacement of contact surface of Figure The force diagram interference multi-layer interference cylinder obtained. Based on thecylinder. radial displacement relationship Figure 2. 2.can Thebe force diagram of of interference cylinder. between Ci and Ci 1 , a matrix model of multi-layer interference cylinder can be obtained as: According the radial displacement of contact surface of According to to Equation Equation (1), (1), the the relation relation between between δ = Kp the radial displacement of contact surface(2)of multi-layer interference cylinder can be obtained. Based displacement relationship multi-layer interference cylinder can be obtained. Based on on the the radial radial displacement relationship T T between C and C , a matrix model of multi-layer interference cylinder can be obtained i i  1 between Ci and Ci+δ1 ,=amatrix model of multi-layer interference cylinder can be obtained as:as: Among them, 1 ,  2 ,,  n  , p =  p1 , p2 ,, pn  , δ = Kp (2)  K1,2  K 4,1    K 2,2 0  δ = Kp 0 0 0 (2)   T T Among them, δ =  ,  ,   ,  , p = p , p ,  , p ,  K K  K  K 0  0 0  1 2    1,3 1 4,22  3,2 3 n 2， n   Among them, δ = [δ1 , δ2 , · · · · · · , δn ]T , p =[ p1 , p2 , · · · · · · , pn ]T , 0  K 3,3 0  0  K1,4  K 4,3   K 2,4    K1,2  K 4,1   K 2,2 0  0 0 0 K   (K1,2 + K4,1 )     −K2,2 0 ··· 0 0 0   K1,3(K K+  K− 0 0  ··· 0 0    K03,2    4,2K 2， 3K − K 0 0 )  0  K K  K  K 0 3,2 2,3 1,3 4,2  1,n1 4,n 2  3, n  2 2, n 1     0 0 −K K 0   K 3,3 K (K1,4K+4,3K4,3 )  K − 0 0  ··· 0   0 3,3  1,4  02,4 . 2,4  K 3, n 1 . K1, n  K 4, n 1   K 2, n .   0 .  .   ..  K K =  ..  ..  ..  ..    0 0 0  0  K 3, n K1, n 1  K 4, n      0 0 · · · 0 − K K + K − K 0   0  K 3, n  23,n−2K1,(n 11,n −K1 4, n  2 4,n−2 )  K 2, n 1 2,n−1 0  0 0 ··· 0 −K3,n−1 −K2,n (K1,n + K4,n−1 )   0 0 0 0  0 0 ···  K 3, n 1 0 (KK2, n + K )  K1, n  K 4,−nK13,n  1,n+1 4,n  3. Algorithm Forceand Temperature Gradient 0 Considering 0 Centrifugal 0 0  K 3, n  K1,n1  K 4,n  

           

3. Algorithm Considering Centrifugal Force and Temperature Gradient 3.1. Centrifugal Force 3.1. CentrifugalConsidering Force 3. Algorithm Centrifugal Force and Temperature Gradient Multi-layer interference cylinder is often used to transmit torque. Under high speed, the Multi-layer interference cylinder is often used to transmit torque.interference Under high speed, thecannot influence influence of centrifugal force on contact pressure of multi-layer cylinder be 3.1. Centrifugal Force of centrifugal force on contact pressure of multi-layer interference cylinder cannot be ignored. ignored. Multi-layer cylinder is often used unit to transmit torque. high speed, the According to tointerference elastic mechanics, mechanics, taking the small small unit as aa micro micro unit polar radial According elastic taking the as unit in inUnder polar coordinate coordinate radial influence of centrifugal force on contact pressure of multi-layer interference cylinder cannot be plane force diagram is is shown in in Figure 3. 3. plane of of arbitrary arbitrary radius radius ρ,its , its force diagram shown Figure ignored. According to elastic mechanics, taking the small unit as a micro unit in polar coordinate radial plane of arbitrary radius  , its force diagram is shown in Figure 3.

Figure unit. Figure 3. 3. A A schematic schematic diagram diagram of of the the centrifugal centrifugal force force of of aa micro micro unit. Figure 3. A schematic diagram of the centrifugal force of a micro unit.

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It is assumed that the density of micro unit is ρ0 , the centrifugal force of the micro unit is: Kρ = ρω 2 ρ0

(3)

The equilibrium differential equation of the micro unit is [18]: σρ − σϕ dσρ + + ρω 2 ρ0 = 0 dρ ρ

(4)

According to the geometric equation, radial strain ε ρ and circumferential strain ε ϕ are given as: ερ =

du u , εϕ = dρ ρ

(5)

Radial stress and circumferential stress of the micro unit can be obtained:
σρ = 1−Eµ2 ε ρ + µε ϕ
σϕ = 1−Eµ2 ε ϕ + µε ρ

(6)

The equilibrium differential equation expressed by displacement can be obtained according to Equations (4)–(6). u 1 − µ2 2 d2 u 1 du + − 2 =− ρω ρ0 (7) dρ ρ dρ E ρ The Equation of the displacement of a free cylinder at any point under centrifugal force can be obtained by solving Equation (7).   i ρ 1 − µ2
ω 2 ρ3 (3 + µ ) ρ0 ω 2 h 0 2 2 2 2 2 us = (1 − µ ) a + b ρ + (1 + µ ) a b − 8Eρ 8E

(8)

It is assumed that Ln,i is radial displacement caused by centrifugal force, n= 1 or 2, i= 1, 2, 3, . . . ; n = 1 means inner surface, n = 2 means outer surface; i means cylinder number. Transforming Equation (8) into diameter direction, the radial displacement of outer surface of cylinder Ci based on centrifugal force is shown as:  L2,i

=

(3+ µ i ) ρ0 32Ei d2,i




(1 − µi ) d1,i 2 + d2,i 2 d2,i 2 + (1 + µi )d1,i 2 d2,i 2 −

ρ0 (1−µi 2 )d2,i 3 32Ei



ω2

(9)

= G2,i ω 2 Similarly, radial displacement of the inner surface of cylinder Ci+1 based on centrifugal force is shown as: L1,i+1 = G1,i+1 ω 2 (10) According to Equations (2), (9), and (10), the matrix model of interference magnitude of multi-layer interference cylinder under the centrifugal force can be obtained: ffi = Kp + Gω 2

(11)

Among them, G =[ G1,2 − G2,1 , G1,3 − G2,2 , · · · · · · , G1,n+1 − G2,n ]T . 3.2. Temperature Gradient Temperature distribution function is shown as [19]: t = ta +

tb − t a ln(ρ/r1 ) ln(r2 /r1 )

(12)

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When the cylinder is expanded by heat, the displacement of any point is shown as: ui =

Z ρ 0

(13)

αtdρ

According to Equations (12) and (13), the displacement due to the thermal expansion of cylinder can be obtained: t − ta ρ ui = αρ[t a + b (ln − 1)] (14) ln(b/a) a where α is the coefficient of thermal expansion, t a is the temperature of inner surface of cylinder, and tb is the temperature of outer surface of cylinder. Assuming that Qn,i is radial displacement due to temperature, tn,i is cylinder surface temperature. Substituting Ci parameter into Equation (14), the radial displacement of outer surface of cylinder Ci under the temperature can be obtained: Q2,i

  t1,i − t2,i 1 = αd2,i t2,i + 2 ln d2,i − ln d1,i

(15)

Similarly, the radial displacement of the inner surface of cylinder Ci+1 under the temperature is shown as:   t1,i+1 − t2,i+1 1 Q1,i+1 = αd1,i+1 t1,i+1 + (16) 2 ln d2,i+1 − ln d1,i+1 The displacement of contact surface of multi-layer interference cylinder under temperature gradient is shown as: ∆Qi

= Q1,i+1 −Q2,i = 21 αd1,i+1 ln d

∆ti+1 2,i +1 −ln d1,i +1

−

∆ti ln d2,i −ln d1,i

(17)



According to Equations (2) and (17), the matrix model of interference magnitude of multi-layer interference cylinder under the temperature gradient is shown as: δ = Kp + ∆Q

(18)

And, ∆Q =[ Q1,2 − Q2,1 , Q1,3 − Q2,2 , · · · · · · , Q1,n+1 − Q2,n ]T . 4. Finite Element Model Here, five cylinders named by C1 , C2 , C3 , C4 , C5 are constructed with steel material, and the multi-layer interference cylinder model is consists of four interference cylinders named by M1 , M2 , M3 , M4 . The according parameters of models are listed in Table 1. Table 1. Model parameters. Cylinder No

C1

C2

Internal diameter (mm) External diameter (mm) Height (mm) Elastic modulus Poisson ratio Density Coefficient of linear expansion

100 200

200 300

C3 300 400 150 206 GPa 0.3 7800 kg/m3 1.1 × 10−5

C4

C5

400 500

500 600

Elastic modulus Poisson ratio Appl. Sci. 2018, 8, x FOR PEER REVIEW Density Appl. Sci. 2018, 8, 726 modulus CoefficientElastic of linear expansion

206 GPa 0.3 7800 kg/m3 206 1.1GPa × 10−5

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Poisson ratio 0.3 Density 7800 kg/m3 According totoNational National Standard GB/T1800.4-1999, the maximum interference magnitude According Standard GB/T1800.4-1999, the maximum Coefficient of linear expansion 1.1 × 10−5 interference magnitude between

between C1 and C2 is 0.195 mm, C2 and C3 is 0.272 mm, C3 and C4 is 0.330 mm, C4 and C 1 and C2 is 0.195 mm, C2 and C3 is 0.272 mm, C3 and C4 is 0.330 mm, C4 and C5 is 0.400 mm. Model M1 0.400 mm. Model Min M , M 4 is as shown Figure 4. interference magnitude 5 2is 1 , 2 ,4. M 3 GB/T1800.4-1999, to National Standard theinmaximum ,CM ,M M Figure 3 ,According 4 is as shown between C1

and C2

is 0.195 mm, C2 and C3 is 0.272 mm, C3 and C4 is 0.330 mm, C4 and

C4 in Figure 4. C1 Model C2 M 1 , M 2 ,C3M 3 , M 4 is as shown C5 is 0.400 mm. C3

C2

C1

M1

M2 M1

M2

C4

C5 C5

M3 M3

M4 M4

Figure 4. Multi-layer cylinder model. model. Figure 4. Multi-layer interference interference cylinder Figure 4. Multi-layer interference cylinder model.

By Equations (11) and (18), the contact pressure of each surface of multi-layer interference By Equations (11)(11) andand (18), the ofeach eachsurface surface multi-layer interference By Equations (18), thecontact contact pressure pressure of of of multi-layer interference cylinder based on centrifugal force and temperature gradient can be respectively obtained. To verify cylinder on centrifugal force temperaturegradient gradient can can be To To verify cylinder basedbased on centrifugal force andand temperature be respectively respectivelyobtained. obtained. verify its its correctness and and analyze effects ofofcentrifugal force and andtemperature temperature gradient oncontact the contact its correctness analyze effects centrifugal force gradient on the correctness and analyze effects of centrifugal force and temperature gradient on the contact pressure, pressure, three-dimensional models ofof MM1 1 ,, M M33 , ,and andMM established by finite element pressure, three-dimensional M 22 are ,, M are established by software finite element 4 are 4 by three-dimensional models of M1models , M2 , M3 , and M established finite element ABAQUS 4 software ABAQUS 6.14. A “penalty-type” formulation was applied tothe each interface, and the ABAQUS 6.14. A “penalty-type” friction formulation applied to each interface, and the 6.14. Asoftware “penalty-type” friction formulation friction was applied to eachwas interface, and friction coefficient is friction coefficient is set as 0.15. Figure5a 5aisis boundary boundary condition forfor M 4 Mas4 an example, whichwhich is friction coefficient is set as 0.15. Figure condition as an example, is set as 0.15. Figure 5a is boundary condition for M4 as an example, which is used to fix displacement used to fix displacement = U2 = U3= =0)0)of of cylinder cylinder CC apply rotational bodybody force,force, then the 1 and used=to (U1(U1 =C U2 = U3 apply then the 1 and (U1 U2fix= displacement U3 = 0) of cylinder force, thenrotational the different speed gradients 1 and apply rotational body different speed gradients are set up to obtain numerical solution of contact pressure of multi-layer different gradients are set up to of obtain numerical of contact pressurecylinder of multi-layer are set upspeed to obtain numerical solution contact pressuresolution of multi-layer interference under interference cylinder under centrifugal force. Figure 5b is a model with three-dimensional meshes [20]. interferenceforce. cylinder under 5b is a model with three-dimensional [20]. centrifugal Figure 5bcentrifugal is a modelforce. withFigure three-dimensional meshes [20]. In Figuremeshes 6, gradient In Figure 6, gradient temperature is applied to the inner and outer surfaces of each cylinder taking In Figure 6, gradient temperature is applied to the inner and outer surfaces of each cylinder taking temperature is applied to the inner and outer surfaces of each cylinder taking M as an example. 4 M 4 as an example. M 4 as an example.

Figure 5. Three-dimensional model of multi-layer interference cylinder， (a) Boundary condition Figure 5. Three-dimensional model of multi-layer interference cylinder, (a) Boundary condition diagram; (b) Mesh partition diagram diagram; Mesh partition diagram. Appl. Sci. 2018, 8,(b) x FOR PEER REVIEW 7 of 12 Figure 5. Three-dimensional model of multi-layer interference cylinder， (a) Boundary condition diagram; (b) Mesh partition diagram

Figure 6. The temperature gradient of M 4 . Figure 6. The temperature gradient of M4 .

5. Results and Discussion

5.1. Influences of Centrifugal Force Under the condition of high speed, the multi-layer interference cylinders must ensure that

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Figure 6. The temperature gradient of M 4 .

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5. Results 5. Results and and Discussion Discussion

5.1. Influences of of Centrifugal Centrifugal Force Force 5.1. Influences Under the must ensure Under the condition condition of of high high speed, speed, the the multi-layer multi-layer interference interference cylinders cylinders must ensure that that contact the minimum contact pressure to transmit required torque [21]. contact pressure pressureisisalways alwaysgreater greaterthan than the minimum contact pressure to transmit required torque According to Equation (11), the pressure between the contact surfaces can becan obtained under [21]. According to Equation (11),contact the contact pressure between the contact surfaces be obtained four gradients of 0, 200, TheThe result is shown in in Figure 7. 7. underrotation four rotation gradients of 0,400, 200, and 400, 600 andrad/s. 600 rad/s. result is shown Figure

Figure the centrifugal centrifugal force. force. Figure 7. 7. Contact Contact pressure pressure of of each each model model under under the

As shown in Figure 7, the contact pressure of multi-layer interference cylinder decreases with As shown in Figure 7, the contact pressure of multi-layer interference cylinder decreases with the increase of rotational speed. For the same layer, the contact pressure slip rate is slower before 400 the increase of rotational speed. For the same layer, the contact pressure slip rate is slower before rad/s. When the rotational speed reaches 600 rad/s, the contact pressure will decrease significantly. 400 rad/s. When the rotational speed reaches 600 rad/s, the contact pressure will decrease significantly. Analyzing the four models, with the increase of layer number of interference, the contact pressure Analyzing the four models, with the increase of layer number of interference, the contact pressure between the same contact surfaces rises significantly, and the contact pressure gradually decreases between the same contact surfaces rises significantly, and the contact pressure gradually decreases from inside to outside [22], that is, p1  p2  p3  p4 . It indicates that the contact pressure at the from inside to outside [22], that is, p1 > p2 > p3 > p4 . It indicates that the contact pressure at the outermost surface is the minimum. Under centrifugal force, force, the the outermost outermost contact contact outermost surface is the minimum. Under the the influence influence of of centrifugal pressure is more easily reduced to below the critical value, which results in the loosening the pressure is more easily reduced to below the critical value, which results in the loosening of theof outer outer interference components and the failure of torque transmission. interference components and the failure of torque transmission. In order to further explore the effect of centrifugal on thepressure contact ofpressure of the In order to further explore the effect of centrifugal force onforce the contact the multi-layer multi-layer interference cylinder, the contact pressure on contact surfaces of model M is analyzed 4 interference cylinder, the contact pressure on contact surfaces of model M4 is analyzed separately. separately. In Figure 8, comparing with contact pressure p1 , p2 , p3 , p4 of M4 , the drop of contact pressure In Figure comparing with contact pressure p1 , p2p1, ispfrom M 4 to, 134 the drop decreases from 8, inside to outside with the increase of speed. MPa; of p2 contact is from 3 , p176 4 ofMPa pressure decreases from the increase speed. from Combining 176 MPa towith 134 153 MPa to 116 MPa; p3 isinside from to 92 outside MPa to with 65 MPa; p4 is fromof44 MPa top129isMPa. Table slope of contact pressure of the same surface increases gradually with the increase of speed. MPa; 2,pthe 2 is from 153 MPa to 116 MPa; p3 is from 92 MPa to 65 MPa; p4 is from 44 MPa to 29 For example, the slope p1 is2,−the 0.024 when the rotational speed is 0same rad/s, when the rotational speed MPa. Combining with of Table slope of contact pressure of the surface increases gradually reaches 600 rad/s, the slope is increased to −0.119.

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with the increase of speed. For example, the slope of p1 is −0.024 when the rotational speed8 of is 12 0 Appl. Sci. 2018, 8, 726 rad/s, when the rotational speed reaches 600 rad/s, the slope is increased to −0.119. Appl. Sci. 2018, 8, x FOR PEER REVIEW -0.024

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-0.048 -0.095

175

with the increase of speed. For example, the slope of p1 is −0.024 when the rotational speed is 0

-0.024

125

-0.048 -0.095

175

100

-0.119

150

Contact pressure (MPa)

Contact pressure (MPa)

-0.119 rad/s, when the rotational speed reaches 600 rad/s, the slope is increased to −0.119. 150

75 125

p1

50 100

25

p2 p3

75

50

p4 0

100

200

300

400

500

p1

600

p2

Rotational speed (rad/s)

p3 p4

25

Figure pressure of M at400 different rotational speed. 0 100pressure 200 of 300 500 rotational 600 Figure8.8.Contact Contact M44 at different speed. Rotational speed (rad/s)

Table of with theincrease increaseofofspeed. speed Table 2.The Theslope slope of of pp11 with the Figure 8. 2. Contact pressure M 4 at different rotational speed.

Comparison parameters numerical value Comparison Parameters Numerical Table 2. The slope of p1 with the increase of Value speed Rotational Speed (rad/s)

0

200

400

600

Rotational Speed (rad/s) 0 200 400 600 Comparison parameters value Contact pressure (MPa) 176176 numerical 172 158 134134 Contact pressure (MPa) 172 158 Rotational Speed (rad/s) 0 200 400 600 Slope −0.024 −0.095 Slope −0.024 −−0.048 0.048 −0.095 −0.119 −0.119 Contact pressure (MPa) 176 172 158 134 Slope −0.024 −0.048 −0.095 −0.119

Under the centrifugal force, the contact surface of multi-layer interference cylinder will have Under the centrifugal force, the contact surface of multi-layer interference cylinder will have different deformations, which force, is thethe fundamental reason for the change of contact pressure. With Under the centrifugal contact surface of multi-layer interference cylinder will have different deformations, which is the fundamental reason for the change of contact pressure. With the the increase of deformations, speed, the interference state betweenreason contact surfaces gradually decreases. In Figure different which is the fundamental for the change of contact pressure. With increase of speed, the interference state between contact surfaces gradually decreases. In Figure 4, the increase speed, the interference state between contact surfaces gradually so decreases. In Figure 4, cylinder C1of of model M 4 is squeezed by the four cylinders, the interference thethe cylinder C of model M is squeezed by the other fourother cylinders, so the interference state is thestate best. 1

4, the cylinder C

4

of model M

is squeezed by the other four cylinders, so the interference state

4 1 is thecylinder best. The Cat is located at the outermost withcontact the smallest contact pressure, The C5 cylinder is located outermost layer, with thelayer, smallest pressure, leading to the 5 the is the best. The cylinder C5 is located at the outermost layer, with the smallest contact pressure, worst interference state. This shows that theshows outermost surfacecontact is mostsurface likely toisbe detached leading to the worst interference state. This that contact the outermost most likely leading to the worst interference state. This shows that the outermost contact surface is most likely when the speedwhen is large to be detached theenough. speed is large enough. to be detached when the speed is large enough. In order order to verify the accuracy ofofmatrix for multi-layer interference cylinders under In to verify the accuracy of matrix model for multi-layer multi-layer interference cylinders In order to verify the accuracy matrix model model for interference cylinders underunder centrifugal force, taking model M4 M as4M anasexample, comparing the analytical solution with the numerical centrifugal force,force, taking model an example, comparing the analytical solution with the centrifugal taking model 4 as an example, comparing the analytical solution with the solution under 600 rad/s, the results are shown in Figure 9. numerical solution under 600600 rad/s, thetheresults inFigure Figure numerical solution under rad/s, resultsare are shown shown in 9. 9.

Numericalsolution solution Numerical 200

Contact pressure (MPa)

Contact pressure (MPa)

150

200

Analytical solution Analytical solution

Middle point relative error:-2.84% Middle point relative error:-2.84%

150

p1

p1

172

Middle point relative error:3.70%

172

129

p2

Middle point relative error:3.70%

129

99 66 57

99

Middle point relative error:5.88%

66

Middle point relative error:5.88%

p3

p3

57

Middle point relative error:11.62%

p4

38

Middle point relative error:11.62% 38

-20

0

20

40

60

80

100

120

p2

160p4

140

Axial distance (mm) -20

0

20

40

60

80

100

120

140

160

Figure 9. Comparison betweenAxial numerical distancesolution (mm) and analytical solution.

In Figure 9, the numerical solutions of contact pressures p1 , p2 , p3 , and p4 under centrifugal force are gradually reduced, which is in agreement with the analytical results. Without considering the stress concentration on both sides of the cylinder, the numerical solution and the Appl. Sci. 2018, 8, 726 9 of 12 analytic solution are in good agreement with the axial distance of 25–115 mm. The maximum relative error of the midpoint is 11.62%, which is within the allowable range of design.

In Figure 9, the numerical solutions of contact pressures p1 , p2 , p3 , and p4 under centrifugal force 5.2. Influence of Temperature Gradient are gradually reduced, which is in agreement with the analytical results. Without considering the stressAccording concentration on both sides of(15) the and cylinder, solution and the analytic to the Equations (16),the thenumerical contact pressure of each contactsolution surface are of in good agreement with the axial distance of 25–115 mm. The maximum relative error of the midpoint multi-layer interference cylinder is influenced by temperature of inner and outer surfaces. In Figure is 11.62%, whichofis temperature within the allowable design.pressure of multi-layer cylinder is analyzed 6, the influence gradientrange on theofcontact by applying the temperature on inner and outer surfaces, then the matrix model is verified by 5.2. Influencethe of numerical Temperaturesolution Gradientwith the analytical solution. comparing According tothe Table 3, the temperature is applied the inner and outer surfaces According to Equations (15) and (16),gradient the contact pressurebetween of each contact surface of multi-layer respectively, and the temperature gradient is gradually increased. Ultimately, the influence of the interference cylinder is influenced by temperature of inner and outer surfaces. In Figure 6, the influence same temperature gradient contact pressure of multi-layer interference cylinder is simulated of temperature gradient on on thethe contact pressure of multi-layer cylinder is analyzed by applying the and analyzed. temperature on inner and outer surfaces, then the matrix model is verified by comparing the numerical solution with the analytical solution. 3. Temperature gradient parameters of model According to Table 3, Table the temperature gradient is applied between the inner and outer surfaces respectively, and the temperature gradient is gradually increased. Ultimately, the Surfaces influence of the Temperature of Inner and Outer C C Inner Surface of /Outer Surface of i i same temperature gradient on the contact pressure of multi-layer interference t / t + Δt cylinder is simulated and analyzed. T (°C) 0 10 20 30 40 50 Table 3. Temperature gradient parameters of model.

Figure 10 is the contact pressure diagram of model M 1 , M 2 , M 3 , and M 4 under different temperature gradients. It indicates that the contact pressureofofInner multi-layer interference cylinder Temperature and Outer Surfaces Inner Surface of Ci /Outer Surface of Ci t/t + ∆t decreases with increase of temperature gradient, which is consistent with the centrifugal force. ◦ C) 11, the contact pressures 0 10p1 , p220, p3 , 30 50 M 4 As shown in ∆T( Figure and p4 40of model decreases linearly with the increase of temperature gradient, and the rate of change decreases from p1 Figure to p4 in indicates thatdiagram temperature gradient interference magnitude 10 turn, is thewhich contact pressure of model M1 ,will M2 ,reduce M3 , and M4 under different

temperature It indicates the contact of multi-layer interference cylinder of multi-layergradients. interference cylinder, that thereby affectingpressure the performance of torque transmission of decreases with increase of temperature gradient, which is consistent with the centrifugal force. multi-layer interference components.

Figure the temperature gradient. Figure 10. 10. Contact Contact pressure pressure of of each each model model under under the temperature gradient.

As shown in Figure 11, the contact pressures p1 , p2 , p3 , and p4 of model M4 decreases linearly with the increase of temperature gradient, and the rate of change decreases from p1 to p4 in turn, which indicates that temperature gradient will reduce interference magnitude of multi-layer interference cylinder, thereby affecting the performance of torque transmission of multi-layer interference components.

Appl. Sci. 2018, 8, 726 Appl. Sci. 2018, 8, x FOR PEER REVIEW

10 of 12 10 of 12

Contact pressure (MPa)

180

p1

p3

p2

p4

40

50

150

120

90

60

30

0

10

20

30

t/℃

Figure and the the same same temperature temperature gradient. gradient. Figure 11. 11. Relationship Relationship between between contact contact pressure pressure and

In order to validate the correctness of the relationship between contact pressure and In order to validate the correctness of the relationship between contact pressure and interference interference magnitude of any multi-layer interference cylinder under temperature gradient, the magnitude of any multi-layer interference cylinder under temperature gradient, the relative error relative error between the numerical solution and the analytical solution under the temperature between the numerical solution and the analytical solution under the temperature gradient ∆t = 10 ◦ C gradient t  10 °C is compared and analyzed. The result is shown in Table 4. is compared and analyzed. The result is shown in Table 4. Table 4. The result of analytic solution and numerical solution Table 4. The result of analytic solution and numerical solution.

p1

Contact Pressure (MPa) Contact Pressure (MPa)

Analytical solution Analytical solution solution Numerical Numerical solution Relative error Relative error

p1

169.374

p2 p2

147.059

p3 p3

88.408

p4 p4

40.801

169.374 172.561 147.059 146.768 88.408 95.783 40.801 41.929 172.561 146.768 95.783 41.929 −1.85% 0.198% −7.69% −2.69% −1.85% 0.198% −7.69% −2.69%

Table 4 shows the analytical solution is basically consistent with the numerical solution. The Table 4 showserror the analytical solution is basically with range the numerical maximum relative is only −7.69%, which is withinconsistent the allowable of design.solution. So, the The maximum relative error is only − 7.69%, which is within the allowable range of design. So, the matrix matrix model of multi-layer interference cylinder under temperature gradient is of high accuracy. model of multi-layer interference cylinder under temperature gradient is of high accuracy. 6. Conclusions 6. Conclusions The multi-layer interference cylinder matrix model and the finite element model under The multi-layer interference cylinder matrix model and the finite element model under centrifugal centrifugal force and temperature gradient are built, the influences of both factors on the contact force and temperature gradient are built, the influences of both factors on the contact pressure of the pressure of the multi-layer interference cylinder are analyzed, and the algorithm is validated to be multi-layer interference cylinder are analyzed, and the algorithm is validated to be suitable for actual suitable for actual design. The specific conclusions are as follows: design. The specific conclusions are as follows: (1) The matrix models of the relationship between contact pressure and interference magnitude of (1) aThe matrix models of the relationship between contact pressure and interference multi-layer interference cylinder under centrifugal force and temperaturemagnitude gradient of area multi-layer interference under andtheory. temperature gradient are deduced deduced based on elasticcylinder mechanics andcentrifugal thick-wall force cylinder based on elastic mechanics and thick-wall cylinder theory. (2) The contact pressure of multi-layer interference cylinder under centrifugal effect decreases (2) with The contact pressure multi-layer interference cylinder effectthe decreases with the increase of ofrotational speed, and the curve under slope centrifugal increases with increase of the increase of rotational speed, and the curve slope increases with the increase of rotational rotational speed. The maximum radial displacement of contact surface is at the outmost layer speed. The maximum radial of contact at the outmost of interference of interference cylinder. In displacement the same model, the surface radial is displacement oflayer contact surface is cylinder. Inincreased the same model, the radial surfacemagnitude is graduallyshould increased gradually from inside to displacement outside. Theof contact interference be from inside to outside. The interference magnitude should be compensated according to the compensated according to the centrifugal force during the design of multi-layer interference centrifugal force during the design of multi-layer interference fit. fit. (3) Without considering considering the the stress stress concentration, concentration, the the numerical numerical solution solution is is in in good good agreement agreement with (3) Without with the analytical maximum relative error of midpoint is 11.62%, it is kept within the analytical solution. solution.Even Evenif ifthethe maximum relative error of midpoint is 11.62%, it is kept the allowable range of range design,ofwhich shows that shows the matrix of multi-layer fit within the allowable design, which thatmodel the matrix model interference of multi-layer under centrifugal force can be used for design reference. interference fit under centrifugal force can be used for design reference. (4) The contact pressure of multi-layer interference cylinder decreases linearly with increase of temperature gradient. The maximum relative error of analytical solution and numerical solution under temperature gradient is −7.69%, which is in the allowable design range. It

Appl. Sci. 2018, 8, 726

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The contact pressure of multi-layer interference cylinder decreases linearly with increase of temperature gradient. The maximum relative error of analytical solution and numerical solution under temperature gradient is −7.69%, which is in the allowable design range. It indicates that the matrix model of multi-layer interference fit under the same temperature gradient can be used for reference.

Author Contributions: Z.B. and J.W. conceived the research direction and wrote the paper; K.N. designed finite element simulation; Z.B. and K.N. analyzed the data; D.H. collected relevant information. Acknowledgments: This work was supported by the National Science Foundation of China (no. U1610109 and 51505475), the Shanxi Provincial Natural Science Foundation of China (no. 201601D011049), the Shanxi Provincial Key Research and Development Project (no. 201603D111017), and the Provincial Special Fund for Coordinative Innovation Center of Taiyuan Heavy Machinery Equipment (1331 Project). Conflicts of Interest: The authors declare no conflict of interest.

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