# Wear Analysis of Support Spring of Sprag Clutch during State of Overrunning

Authors: Hongzhi Yan; Bo Wen; Zhibiao Wang; Chu Zhu; De Ni; Minghao Lin
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

This paper analyzes the wear of the support spring of sprag clutches for different structural parameters during overrunning state. Specifically, the stiffness model of the support spring is established based on the characteristics of the variable helix angle, and also the supporting force model is established according to the energy conservation law during the working process of the sprag clutch. Meanwhile, the relative rotation speed between the support spring and the wedge is analyzed by dynamic simulation method. Furthermore, the wear model of the support spring in process of overrunning is established based on Archard wear theory. The results show that there is a linear relationship between supporting force and spring radial displacement compression. In addition, increasing free length and wire diameter of springs will aggravate the wear of springs, whereas enlarging pitch diameter of springs will reduce it. When free length is increased from 270 to 290 mm, the wear time is reduced by 20.88%; when wire diameter is increased from 0.5 to 0.7 mm, the wear time is reduced by 42.75%; when pitch diameter is increased from 2 to 2.4 mm, the wear time is increased by 3.35%. The degree of influence of the structural parameters on the spring wear from large to small is free length, wire diameter, and pitch diameter. Exactly speaking, when wear time is 100 hr, the relative sensitivity of structural parameters affecting the wear of the support spring is in order of free length (2.2), wire diameter (0.97), and pitch diameter (0.08). The research provides a reference for the design and optimization of the support spring for highly reliable sprag clutches.


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## Body

## 1. Introduction

The positive continuous engagement (PCE) sprag clutch can bear a large load and is one of the key components of the helicopter transmission system. The structure of the PCE-type sprag clutch is shown in Figure1. In the PCE-type sprag clutch, the support spring pretightens the wedge providing radial supporting forces. During the operating process of the clutch, friction occurs between the support spring and wedge. Different support spring parameters affect the wear of the spring, and thus affect the performance and reliability of the sprag clutch.Figure 1
Schematic diagram of PCE-type sprag clutch.In recent years, many scholars have carried out relevant researches on sprag clutches. Nevertheless, most of the researches mainly focused on the wedge of sprag clutches. Xu and Lowen [1, 2] analyzed the contact mode between the wedge and the inner and outer rings of the sprag clutch, established a complete model considering the dynamic contact nonlinearity during the transmission process, and carried out an experimental verification. Liu et al. [3] established a dynamic model of the engagement process for the sprag clutch based on the Hunt–Crossley collision theory. It was studied that the indexes of the impact force between the wedge and the inner and outer rings and the engagement time of the inner and outer rings. The aforementioned mathematical model and dynamic model provide an important reference for the wear of wedge. Vernay et al. [4] analyzed the mechanical properties of the sprag clutch and the dynamic friction in contacts between the wedge and inner and outer rings under an instantaneous overload. Li et al. [5] studied the wear of wedge for sprag clutch during state of overrunning, proposed the calculation equation of wedge wear depth based on Archard wear theory, and discussed the effects of temperature, speed, and lubrication conditions on wedge wear through experimental tests. Huang et al. [6] proposed an alternate friction model considering stationary and dynamic friction, and it was applied in the contacts of the outer and inner race.In addition, many researchers have also conducted a lot of research on wear mechanics. Yu et al. [7] presented the wear-life analysis model of 6000 deep groove ball bearing under axial load, confirmed the allowable wear extent based on archard theory, and analyzed the effects of axial loads, rotating speed, and structural parameters on bearing wear characteristics. Liu et al. [8] proposed a modified wear model based on commonly used Archard wear model, which explained friction-induced dynamic variations in mechanical properties, but it was suitable for predicting some certain materials. Lee et al. [9] evaluated the wear life of the punch in notching process based on Archard wear model and numerical analysis. Maruda et al. [10, 11] analyzed the tool wear of P25 cemented carbide inserts in finish turning of AISI 1045 carbon steel for different cooling conditions, proved the MQCL + EP/AW method can significantly reduce tool wear and shown MQCL medium can improve cutting tool wear rate.In the engineering practice, support springs provide radial supporting forces for wedges, affecting the service performance of the sprag clutch. However, rear investigations are reported on the support spring of the sprag clutch. Yan et al. [12] analyzed the wear and fatigue life of the support spring under the joint state of the sprag clutch. Through theoretical calculation and finite-element simulation, it was concluded that the life of the support spring depends on wear. However, Yan et al. neglect the factor of centrifugal force, thus the model established is not accurate enough. Gao et al. [13] analyzed the causes of fatigue failure of support spring. According to the analysis of fracture surface, mechanical and chemical performance and the microstructure of the spring material, it was noted that the stress concentration on the spring internal surface and oxide inclusions led to the fracture of springs. Similarly, Nazir et al. [14] studied the stiffness of helical springs with different structure parameters including free lengths, wire diameter, pitch diameter, and spring coil, and obtained different stiffness values for particular applications by varying structure parameters. Yasar et al. [15] analyzed the impact of brush spring pressure on the wear performance. By experiment tests performed on a novel wear-test machine, they found that specific wear rate increased with the increase of brush spring pressure in a certain pressure range.In recent years, the previous investigations lie in the wear of support springs on the joint state, whereas few investigations are reported on that during state of overrunning. The primary contribution of this work is to establish a complete wear model to analyze the wear of the support spring during state of overrunning. Through a mechanical and deformation analysis of the support spring of the sprag clutch, the stiffness and the supporting force model of the support spring to the wedge are established. Based on Archard wear theory, the wear model of the support spring during the overrunning state is established, and the influence of the structural parameters of the support spring on the overrunning wear is studied, providing a reference for the design of a long-life spring of a PCE-type sprag clutch.


## 2. Stiffness and Mechanical Model of Support Spring

### 2.1. Stiffness Model of Support Spring

#### 2.1.1. Establishment of Helix Equation of Support Spring

The free length of the support spring of the sprag clutch isL (the expanded length of the spring in a free state), the pitch is p, the pitch diameter is D, the radius of the spring coil is RL, the number of effective coils of the spring is n, and the wire diameter is d. According to the method for establishing the helix equation of circumferential spring [16], the process of establishing the helix equation of a support spring is as follows: Set point Q to rotate uniformly with frequency w1 along the circle with radius R (R = D/2) and rotate uniformly with frequency w2 along the circle with radius RL. The helix curve of the support spring is shown in Figure 2, and the helix equation of the support spring can be expressed as follows:(1)x=Rcosw1t,y=RL1−cosw2t−Rsinw1tcosw2t,z=RLsinw2t+Rsinw1tsinw2t.Figure 2
Helical curve of support spring: (a) geometric parameters of support spring and (b) spiral trajectory of support spring.
(a)(b)Setting upθ1 = w1t and θ2 = w2t, when the number of support springs is n, let θ1/θ2 = n; in addition, substituting into equation (1), the helix equation of the support spring of the sprag clutch is obtained as follows:(2)x=Rcosθ1,y=RL1−cosθ1n−Rsinθ1cosθ1n,z=RLsinθ1n+Rsinθ1sinθ1n.


#### 2.1.2. Analysis of Support Spring Deformation

The expression of the helix angle of an ordinary cylindrical helical spring is(3)α=arctanpπD.According to the change law of the helical curve of the support spring of the sprag clutch, the expression of the spring pitchp is obtained as follows:(4)p=2πRL−Rsinθ1n.Substituting equation (4) into equation (3), the helix angle expression of the support spring can be obtained as follows:(5)α=arctanRL−Rsinθ1nR.According to the spring manual, the relationship between the end deformationε and load F is as follows:(6)ε=LFD24cos2αGIp+sin2αEI.According to the arc length integral, the length expression of the single-coil support spring is obtained as follows:(7)ΔL=∫02πdL=∫02πx˙2+y˙2+z˙2dθ1.The deformation of single-coil spring ∆ε can be expressed as(8)Δε=ΔLFD24cos2αGIp+sin2αEI.For cylindrical helical springs, the moment of inertiaI = πd4/64, polar moment of inertia Ip = πd4/32, and the shear modulus G of the material can be expressed using the elastic modulus E and Poisson’s ratio v as G = E/2(1 + v), which can be obtained by substituting it into equation (8).(9)Δε=16ΔLFD2πEd41+vcos2α.The total deformation of the support springε0 is(10)ε0=n⋅Δε.By combining the above equations, the spiral equation of the support spring is substituted into the deformation model, and the integral can be obtained after simplification.(11)ε0=16D2FGd41+vRL2+nR2+vnR2RL2+nR2.


#### 2.1.3. Stiffness Calculation of Support Spring

The stiffness expression of the support spring is(12)K=Fε0.According to L =2πRL = np, by substituting equation (11) into equation (12) and simplifying it, the expression of the stiffness K of the support spring of the sprag clutch can be obtained:(13)K=Gd41+v16D2L/2π2+LD/2p2+vLD/2p2/L/2π2+LD/2p2.


### 2.2. Mechanical Model of Support Spring

The support spring is installed in the wedge groove of the sprag clutch, which is under compression by the wedge and is pressed along the radial direction by the wedge. The force of the support spring on the wedge under different working states of the sprag clutch is shown in Figure3, where Fr is the radial supporting force of the support spring on the wedge, and f is the friction force between the wedge and support spring.Figure 3
Force of support spring on wedge in different states: (a) Installation state and (b) working state.
(a)(b)During the installation, owing to the small radial deformation of the spring and the large number of wedges, the support spring bears the radial compression force exerted evenly by the wedges. According to the conservation of energy, the work done by the radial compression force of the wedges on the spring is converted into the elastic potential energy of the spring, and the energy is obtained as follows:(14)mFrΔs=12KΔx2.Under a working state, considering the radial compression force and tangential friction force, according to the conservation of energy, the work done by the wedge radial compression force and the friction force on the spring is transformed into the elastic potential energy of the spring, and the elastic potential energy of the spring is obtained as follows:(15)μFrΔx+mFrΔs=12KΔx2,where m is the number of wedges and pieces, Δs is the change in radial displacement (mm), Fr is the radial supporting force (N), K is the spring stiffness (N/mm), Δx is the axial deformation of the spring (mm), and μ is the friction coefficient between the spring and wedge.The rotation angleϕ under the working state of the wedge can be obtained through a coordinate transformation. According to the rotation angle ϕ, the radius of the support spring coil can be obtained under different positions. The position coordinates of the contact point between the support spring and wedge are shown in equation (16). The radius of the working state of spring Rd is expressed by equation (17). The points (xo1, yo1) of the equation represent the contact points between the spring and wedge in the wedge design coordinate system, and the points (xo2, yo2) represent the contact points between the spring and wedge in the clutch coordinate system. Schematic diagrams for the different clutch coordinate systems are shown in Figure 4.(16)xo2yo201=cosϕxo1−sinϕyo1sinϕxo1+cosϕyo1+Ro¯−ro¯01,(17)Rd=xo22+yo22−12D+d.Figure 4
Schematic diagram of sprag in different coordinate systems.The axial deformation Δx and radial displacement variation Δs of the support spring can be expressed through equation (18). According to the geometric relationship, the radial displacement variation Δs can be expressed by the free length L, as shown in equation (19).(18)Δx=2π⋅Δs,(19)Δs=L2π−Rd.Combined with the aforementioned equations, the expression of the radial contact force between the support spring and a single wedge is obtained as follows:(20)Fr=πKL−2πRdm.The mass expression of the support spring is as follows:(21)mt=14ρπ2d2Dn,where m is the mass of the spring (kg) and ρ is the density of the spring material (kg/mm3).When the clutch is working, a centrifugal force is acted upon the support spring, the expression of which is given as follows:(22)Ft=mtRLπnt302×10−3,where nt is the spring speed (r/min).In the working state, considering the centrifugal force of the support spring, the expression of the radial contact force between the spring and a single wedge is obtained as follows:(23)Fr=πKL−2πRd2πμ+m+1mFt.TakingJ = 9.5 (where J is the difference in the radius of the inner and outer rings) and 33 wedges size sprag clutch as examples, the basic parameters of the support spring and the working conditions of the sprag clutch are shown in Table 1. The data are substituted into equations (20) and (23) to calculate, and the radial supporting force of the support spring to the wedge under different states of the clutch are shown in Table 2.Table 1
Basic parameters of support spring and working conditions of sprag clutch.
Basic parametersValueClutch speed (r/min)7200Clutch power (kW)1000Wire diameterd (mm)0.6Pitch diameterD (mm)2.2Free lengthL (mm)280Number of effective coilsn (r)298Spring pitchp (mm)0.94Table 2
Radial supporting force of support spring on wedge under different clutch states.
Clutch stateInstallationOverrunningJointSupporting forceFr (N)0.884.444.22As shown in Table2, due to the centrifugal force, the spring has a greater force on the wedge under the working state, the supporting force under an overrunning state is the largest, and thus the wear between the spring and wedge is greater under such a state.


## 2.1. Stiffness Model of Support Spring

### 2.1.1. Establishment of Helix Equation of Support Spring

The free length of the support spring of the sprag clutch isL (the expanded length of the spring in a free state), the pitch is p, the pitch diameter is D, the radius of the spring coil is RL, the number of effective coils of the spring is n, and the wire diameter is d. According to the method for establishing the helix equation of circumferential spring [16], the process of establishing the helix equation of a support spring is as follows: Set point Q to rotate uniformly with frequency w1 along the circle with radius R (R = D/2) and rotate uniformly with frequency w2 along the circle with radius RL. The helix curve of the support spring is shown in Figure 2, and the helix equation of the support spring can be expressed as follows:(1)x=Rcosw1t,y=RL1−cosw2t−Rsinw1tcosw2t,z=RLsinw2t+Rsinw1tsinw2t.Figure 2
Helical curve of support spring: (a) geometric parameters of support spring and (b) spiral trajectory of support spring.
(a)(b)Setting upθ1 = w1t and θ2 = w2t, when the number of support springs is n, let θ1/θ2 = n; in addition, substituting into equation (1), the helix equation of the support spring of the sprag clutch is obtained as follows:(2)x=Rcosθ1,y=RL1−cosθ1n−Rsinθ1cosθ1n,z=RLsinθ1n+Rsinθ1sinθ1n.


### 2.1.2. Analysis of Support Spring Deformation

The expression of the helix angle of an ordinary cylindrical helical spring is(3)α=arctanpπD.According to the change law of the helical curve of the support spring of the sprag clutch, the expression of the spring pitchp is obtained as follows:(4)p=2πRL−Rsinθ1n.Substituting equation (4) into equation (3), the helix angle expression of the support spring can be obtained as follows:(5)α=arctanRL−Rsinθ1nR.According to the spring manual, the relationship between the end deformationε and load F is as follows:(6)ε=LFD24cos2αGIp+sin2αEI.According to the arc length integral, the length expression of the single-coil support spring is obtained as follows:(7)ΔL=∫02πdL=∫02πx˙2+y˙2+z˙2dθ1.The deformation of single-coil spring ∆ε can be expressed as(8)Δε=ΔLFD24cos2αGIp+sin2αEI.For cylindrical helical springs, the moment of inertiaI = πd4/64, polar moment of inertia Ip = πd4/32, and the shear modulus G of the material can be expressed using the elastic modulus E and Poisson’s ratio v as G = E/2(1 + v), which can be obtained by substituting it into equation (8).(9)Δε=16ΔLFD2πEd41+vcos2α.The total deformation of the support springε0 is(10)ε0=n⋅Δε.By combining the above equations, the spiral equation of the support spring is substituted into the deformation model, and the integral can be obtained after simplification.(11)ε0=16D2FGd41+vRL2+nR2+vnR2RL2+nR2.


### 2.1.3. Stiffness Calculation of Support Spring

The stiffness expression of the support spring is(12)K=Fε0.According to L =2πRL = np, by substituting equation (11) into equation (12) and simplifying it, the expression of the stiffness K of the support spring of the sprag clutch can be obtained:(13)K=Gd41+v16D2L/2π2+LD/2p2+vLD/2p2/L/2π2+LD/2p2.


## 2.1.1. Establishment of Helix Equation of Support Spring

The free length of the support spring of the sprag clutch isL (the expanded length of the spring in a free state), the pitch is p, the pitch diameter is D, the radius of the spring coil is RL, the number of effective coils of the spring is n, and the wire diameter is d. According to the method for establishing the helix equation of circumferential spring [16], the process of establishing the helix equation of a support spring is as follows: Set point Q to rotate uniformly with frequency w1 along the circle with radius R (R = D/2) and rotate uniformly with frequency w2 along the circle with radius RL. The helix curve of the support spring is shown in Figure 2, and the helix equation of the support spring can be expressed as follows:(1)x=Rcosw1t,y=RL1−cosw2t−Rsinw1tcosw2t,z=RLsinw2t+Rsinw1tsinw2t.Figure 2
Helical curve of support spring: (a) geometric parameters of support spring and (b) spiral trajectory of support spring.
(a)(b)Setting upθ1 = w1t and θ2 = w2t, when the number of support springs is n, let θ1/θ2 = n; in addition, substituting into equation (1), the helix equation of the support spring of the sprag clutch is obtained as follows:(2)x=Rcosθ1,y=RL1−cosθ1n−Rsinθ1cosθ1n,z=RLsinθ1n+Rsinθ1sinθ1n.


## 2.1.2. Analysis of Support Spring Deformation

The expression of the helix angle of an ordinary cylindrical helical spring is(3)α=arctanpπD.According to the change law of the helical curve of the support spring of the sprag clutch, the expression of the spring pitchp is obtained as follows:(4)p=2πRL−Rsinθ1n.Substituting equation (4) into equation (3), the helix angle expression of the support spring can be obtained as follows:(5)α=arctanRL−Rsinθ1nR.According to the spring manual, the relationship between the end deformationε and load F is as follows:(6)ε=LFD24cos2αGIp+sin2αEI.According to the arc length integral, the length expression of the single-coil support spring is obtained as follows:(7)ΔL=∫02πdL=∫02πx˙2+y˙2+z˙2dθ1.The deformation of single-coil spring ∆ε can be expressed as(8)Δε=ΔLFD24cos2αGIp+sin2αEI.For cylindrical helical springs, the moment of inertiaI = πd4/64, polar moment of inertia Ip = πd4/32, and the shear modulus G of the material can be expressed using the elastic modulus E and Poisson’s ratio v as G = E/2(1 + v), which can be obtained by substituting it into equation (8).(9)Δε=16ΔLFD2πEd41+vcos2α.The total deformation of the support springε0 is(10)ε0=n⋅Δε.By combining the above equations, the spiral equation of the support spring is substituted into the deformation model, and the integral can be obtained after simplification.(11)ε0=16D2FGd41+vRL2+nR2+vnR2RL2+nR2.


## 2.1.3. Stiffness Calculation of Support Spring

The stiffness expression of the support spring is(12)K=Fε0.According to L =2πRL = np, by substituting equation (11) into equation (12) and simplifying it, the expression of the stiffness K of the support spring of the sprag clutch can be obtained:(13)K=Gd41+v16D2L/2π2+LD/2p2+vLD/2p2/L/2π2+LD/2p2.


## 2.2. Mechanical Model of Support Spring

The support spring is installed in the wedge groove of the sprag clutch, which is under compression by the wedge and is pressed along the radial direction by the wedge. The force of the support spring on the wedge under different working states of the sprag clutch is shown in Figure3, where Fr is the radial supporting force of the support spring on the wedge, and f is the friction force between the wedge and support spring.Figure 3
Force of support spring on wedge in different states: (a) Installation state and (b) working state.
(a)(b)During the installation, owing to the small radial deformation of the spring and the large number of wedges, the support spring bears the radial compression force exerted evenly by the wedges. According to the conservation of energy, the work done by the radial compression force of the wedges on the spring is converted into the elastic potential energy of the spring, and the energy is obtained as follows:(14)mFrΔs=12KΔx2.Under a working state, considering the radial compression force and tangential friction force, according to the conservation of energy, the work done by the wedge radial compression force and the friction force on the spring is transformed into the elastic potential energy of the spring, and the elastic potential energy of the spring is obtained as follows:(15)μFrΔx+mFrΔs=12KΔx2,where m is the number of wedges and pieces, Δs is the change in radial displacement (mm), Fr is the radial supporting force (N), K is the spring stiffness (N/mm), Δx is the axial deformation of the spring (mm), and μ is the friction coefficient between the spring and wedge.The rotation angleϕ under the working state of the wedge can be obtained through a coordinate transformation. According to the rotation angle ϕ, the radius of the support spring coil can be obtained under different positions. The position coordinates of the contact point between the support spring and wedge are shown in equation (16). The radius of the working state of spring Rd is expressed by equation (17). The points (xo1, yo1) of the equation represent the contact points between the spring and wedge in the wedge design coordinate system, and the points (xo2, yo2) represent the contact points between the spring and wedge in the clutch coordinate system. Schematic diagrams for the different clutch coordinate systems are shown in Figure 4.(16)xo2yo201=cosϕxo1−sinϕyo1sinϕxo1+cosϕyo1+Ro¯−ro¯01,(17)Rd=xo22+yo22−12D+d.Figure 4
Schematic diagram of sprag in different coordinate systems.The axial deformation Δx and radial displacement variation Δs of the support spring can be expressed through equation (18). According to the geometric relationship, the radial displacement variation Δs can be expressed by the free length L, as shown in equation (19).(18)Δx=2π⋅Δs,(19)Δs=L2π−Rd.Combined with the aforementioned equations, the expression of the radial contact force between the support spring and a single wedge is obtained as follows:(20)Fr=πKL−2πRdm.The mass expression of the support spring is as follows:(21)mt=14ρπ2d2Dn,where m is the mass of the spring (kg) and ρ is the density of the spring material (kg/mm3).When the clutch is working, a centrifugal force is acted upon the support spring, the expression of which is given as follows:(22)Ft=mtRLπnt302×10−3,where nt is the spring speed (r/min).In the working state, considering the centrifugal force of the support spring, the expression of the radial contact force between the spring and a single wedge is obtained as follows:(23)Fr=πKL−2πRd2πμ+m+1mFt.TakingJ = 9.5 (where J is the difference in the radius of the inner and outer rings) and 33 wedges size sprag clutch as examples, the basic parameters of the support spring and the working conditions of the sprag clutch are shown in Table 1. The data are substituted into equations (20) and (23) to calculate, and the radial supporting force of the support spring to the wedge under different states of the clutch are shown in Table 2.Table 1
Basic parameters of support spring and working conditions of sprag clutch.
Basic parametersValueClutch speed (r/min)7200Clutch power (kW)1000Wire diameterd (mm)0.6Pitch diameterD (mm)2.2Free lengthL (mm)280Number of effective coilsn (r)298Spring pitchp (mm)0.94Table 2
Radial supporting force of support spring on wedge under different clutch states.
Clutch stateInstallationOverrunningJointSupporting forceFr (N)0.884.444.22As shown in Table2, due to the centrifugal force, the spring has a greater force on the wedge under the working state, the supporting force under an overrunning state is the largest, and thus the wear between the spring and wedge is greater under such a state.


## 3. Finite-Element Model Validation

To verify the correctness of the theoretical calculation model of the support spring on the wedge radial supporting force, the compression process of the spring installed in the wedge groove is simulated. The 3D model of the support spring compression is established using 3D modeling software and is composed of a support spring and 33 simplified wedges. The 3D model is imported into the commercial finite-element simulation software, and the finite-element model of the support spring compression state is obtained, as shown in Figure5.Figure 5
Finite-element model of support spring under compression.Each part of the support spring compression model is meshed independently, and the C3D8R element type is used. A hexahedral neutral axis algorithm is used for the mesh. The mesh size of the 33 simplified wedges is 1 mm, and the spring mesh size is 2 mm.The material properties of each component in the support spring finite-element model are listed in Table3. The contact type between the wedge and spring adopts a surface contact and penalty function control algorithm.Table 3
Material parameters of each part of sprag clutch.
PartsMaterialsDensity (kg/m3)Elastic modulus (GPa)Poisson’s ratioWedgeGCr1578102120.29Inner and outer rings18CrNi4A78502020.28Cage18CrNi4A78502020.28Support spring7078101970.25The analysis step is set to 0.01 s. In this analysis step, a radial displacement Δs of 3.68 mm is applied to 33 wedges to simulate the process of compression of the support spring by the wedges. The loading curve of the radial displacement of the 33 wedges is shown in Figure 6.Figure 6
Loading curve of sprag radial displacement.The compression process of the support spring is simulated using commercial finite-element software, and the von Mises stress nephogram after spring compression is obtained, as shown in Figure7. The pitch of the support spring decreases, and the radial deformation is uniform during the compression process.Figure 7
Mises stress nephogram of support spring after compression.The supporting force obtained from the finite-element model simulation is output, and the primary curve is obtained by fitting the simulation data points, which are compared with the theoretical calculation results. The variation curve of the radial supporting force with radial displacement compression between the simulation model and theoretical calculation is shown in Figure8.Figure 8
Curve of the radial supporting force with radial displacement compression.As shown in Figure8 and Table 4, during the compression of the support spring, the theoretical calculation shows that there is a linear relationship between the supporting force and the spring radial displacement compression, with a linear slope of 0.4783. The model simulation shows that there is a linear relationship between the supporting force and the spring radial displacement compression. The simulation data are fitted, and the slope of the fitting line is 0.4846. The slopes of the curve obtained using a theoretical calculation and model simulation are approximately equal, and the variation law of the supporting force is the same. After the compression of the support spring, the theoretical value of the supporting force is 1.76 N, and the simulation value of the model is 1.70 N. The theoretical and simulation values of the supporting force are basically the same, which verifies the accuracy of the theoretical calculation model of the supporting force.Table 4
Comparison of theoretical calculation and model simulation results of radial supporting force.
Theoretical calculationModel simulationSupporting force after compression (N)1.761.70Slope of curve0.47830.4846


## 4. Analysis of Support Spring Parameters on Self-Wear under Overrunning State

When the sprag clutch is in an overrunning state, the radial supporting force of the support spring to the wedge is larger, and the wedge is not wedged tightly in the inner and outer rings, which makes the wedge slide in the inner and outer rings, and the difference in the rotation speed between the spring and wedge is larger, and thus the wear of the support spring is larger during an overrunning state.


### 4.1. Wear Model of Support Spring Based on Archard Theory

From the Archard wear calculation model, the expression of the wear depth per unit time of the two parts is as follows:(24)dh=kvPHSadt,where dh is the wear depth per unit time (mm), dt is the unit time (s), H is the surface hardness of the material, Sa is the nominal contact area (mm2), k is the coefficient of adhesion wear, v is the sliding speed (mm/s), and P is the contact force (N).When the sprag clutch is in an overrunning state, the support spring is affected by the radial compression and friction forces of the wedge, a force diagram of which is shown in Figure9.Figure 9
Stress diagram of support spring under an overrunning state.When the wear depth of the support spring of the sprag clutch ishs, the contact surface between the spring and wedge is elliptical. According to the spring parameters, the expression is(25)Sa=πhsD+d−hsd−hs.According to the wear length of the support spring in the wedge groove, by substituting equation (25) into equation (24), the expression of the wear depth and wear time of the support spring can be obtained as follows:(26)dh=ml2πRL⋅kvFrπHhsD+d−hsd−hsdt,where RL is the radius of the spring ring (mm), m is the number of wedges, and l is the length of the wedge groove (mm).According tov=ΔnoπRL/30, the expression of the wear time of the support spring obtained by its substitution into (26) and integration is as follows:(27)t=∫0h0πHhsD+d−hsd−hs60∗mlkΔnoFrdhs,where t is the wear time (h), ho is the wear depth of the spring (mm), and Δno is the relative rotation speed of the spring and wedge (r/min).


### 4.2. Influence of Support Spring Parameters on Spring Wear

In the wear model of the support spring, the relative rotation speed∆no of the spring and the wedge is obtained through a simulation of the clutch dynamics model under the overrunning state established by the dynamic analysis software, and the contact force Fr between the spring and wedge is calculated using the mechanical model of the support spring.


#### 4.2.1. Dynamics Simulation of Sprag Clutch

TakingJ = 9.5 in a sprag clutch as an example, the 3D model of the clutch established using 3D modeling software is imported into the dynamic analysis software for a dynamic simulation. The material parameters of each part of the sprag clutch are listed in Table 3.The contact condition and geometry of each part of the sprag clutch are simplified and assumed: (1) It is assumed that no parts produce errors during the processing, manufacturing, and assembly. (2) The wedge is evenly subjected to the force of the support spring. According to the typical working conditions of the sprag clutch, the inner and outer rings are applied with a speed and load to simulate the clutch. A dynamic simulation model of the sprag clutch is shown in Figure10.Figure 10
Dynamic simulation model of sprag clutch.The boundary conditions and loads are applied as follows: (1) The rotating pair of the inner and outer rings of the sprag clutch are set. (2) The contact constraints between the wedge and inner and outer rings, support spring, and cage are imposed, and the contact parameters are defined. (3) In the overrunning state of the sprag clutch, the outer ring rotates counterclockwise relative to the inner ring, a counter-clockwise constant speedno is added to the outer ring, and the auxiliary torque T0 of the spring is applied to the 33 wedges. The loading curves are shown in Figure 11. (4) The gravity acceleration is 9.8 m/s2.Figure 11
Loading curve of rotational speed and auxiliary torque under overrunning state.The torque of the support spring to the rotation center of the wedge is calculated according to the force of the wedge and the position of the action point of the support spring under the overrunning state of the sprag clutch. The auxiliary torque of the wedge under different free lengths, pitch diameters, and spring wire diameters of the support spring are listed in Table5.Table 5
Auxiliary torque under different parameters of support spring.
Free lengthsL (mm)Auxiliary torqueT0 (N·mm)Pitch diametersD (mm)Auxiliary torqueT0 (N·mm)Wire diametersd (mm)Auxiliary torqueT0 (N·mm)270−3.32.0−7.50.5−2.8280−5.82.2−5.80.6−5.8290−8.42.4−4.60.7−11.0The inner and outer rings are loaded according to the conditions shown in Figure11, and the auxiliary torque of the wedge is loaded according to Table 5. The dynamic models of the sprag clutch with different structure parameters of support spring are established, and the time-varying curves of the speed of the wedge and support spring of the sprag clutch under the overrunning state are obtained, as shown in Figure 12.Figure 12
Rotational speed curves of support spring and sprag.By processing the data and taking the average relative rotation speed of the spring and wedge under a stable overrunning state, it is determined that the relative rotation speed of the support spring and wedge with different parameters under a stable overrunning state is approximately 10 r/min.


#### 4.2.2. Application of the Present Model and Discussions

This paper aims at analysizing the stiffness, the supporting force, and the wear of support springs. The applied approach is based on theoretical curve equations and the characteristic of periodic change of the helix angle, the law of energy conservation, and Archard theory. The calculated results are more accurate and easier to obtain, thereby more convenient for engineering application. However, the tiny variations of supporting forces with structure parameters in the wear analysis are ignored, which may be a limitation in this approach.The supporting force under different parameters of the support spring and the relative rotation speed between the support spring and wedge∆no are substituted into the wear model of the support spring for calculation, and the relationship between the wear time and wear depth of the support spring under different free lengths, pitch diameters, and wire diameters is obtained, as shown in Figures 13 to 15, for which the wear depth is 0.2 mm.Figure 13
Relationship between spring wear time and wear depth under different spring free lengths.Figure 14
Relationship between spring wear time and wear depth under different spring pitch diameters.Figure 15
Relationship between spring wear time and wear depth under different wire diameters.It can be seen from Figure13 that with the increase of the free length, the wear time of the support spring decreases at the same wear depth, the wear of the support spring intensifies, and thus the reliability of the sprag clutch decreases. One can see that when free length is increased from 270 to 290 mm, the wear time is reduced by 20.88%. It can be seen from Figure 14 that with the increase of the pitch diameter, the wear time of the support spring increases slightly at the same wear depth, the wear of the support spring decreases, and thus the reliability of the sprag clutch increases. One can see that when pitch diameter is increased from 2 to 2.4 mm, the wear time is increased by 3.35%. It can be seen from Figure 15 that with the increase of wire diameter, the wear time of the support spring decreases at the same wear depth, the wear of the support spring intensifies, and thus the reliability of the sprag clutch decreases. One can see that when wire diameter is increased from 0.5 to 0.7 mm, the wear time is reduced by 42.75%.It can be seen that structural parameters of support springs have great effect on spring wear. The phenomenons can be explained as follows: the increase of free length and pitch diameter leads to the decrease of spring stiffness, whereas the increase of wire diameter leads to the increase of spring stiffness [14]; according to the supporting force model, it can be seen that the supporting force increases with the increase in the free length and the stiffness, the wear of the support spring thus increases [7]. Therefore, we can draw the conclusion that in order to improve the wear resistance of support springs, it is necessary to reduce the values of free length and wire diameter properly, whereas appropriately increase the value of pitch diameter.


#### 4.2.3. Relative Sensitivity Analysis of Spring Wear to Various Parameters

The aforementioned analysis considers the depth of the spring wear under different parameters of the support spring. The relative sensitivity of the spring wear to the change in each parameter is as follows [17]:(28)SWt=Wt−W1/W1Pt−P1/P1,where SWt is the relative sensitivity, Wt is the wear depth at a given wear time t, W1 is the wear depth at the initial time, Pt is the spring parameter at a given wear time t, and P1 is the spring parameter at the initial time.The relative sensitivity regarding the amount of wear of the support spring to the changes in the parameters can be calculated under different wear times, as shown in Table6, where t indicates the wear time.Table 6
Relative sensitivity of wear in support spring to changes in the various parameters at different times.
Spring parametersSensitivity att = 100 hSensitivity att = 300 hSensitivity att = 500 hFree lengthL2.202.012.15Pitch diameterD0.080.080.07Wire diameterd0.971.021.03It can be seen from the results in Table6 that the degree of influence of the structural parameters on the spring wear from large to small is free length, wire diameter, and pitch diameter. Exactly speaking, when wear time is 100 hr, the relative sensitivity of structural parameters affecting the wear of the support spring is in order of free length (2.2), wire diameter (0.97), pitch diameter (0.08).


## 4.1. Wear Model of Support Spring Based on Archard Theory

From the Archard wear calculation model, the expression of the wear depth per unit time of the two parts is as follows:(24)dh=kvPHSadt,where dh is the wear depth per unit time (mm), dt is the unit time (s), H is the surface hardness of the material, Sa is the nominal contact area (mm2), k is the coefficient of adhesion wear, v is the sliding speed (mm/s), and P is the contact force (N).When the sprag clutch is in an overrunning state, the support spring is affected by the radial compression and friction forces of the wedge, a force diagram of which is shown in Figure9.Figure 9
Stress diagram of support spring under an overrunning state.When the wear depth of the support spring of the sprag clutch ishs, the contact surface between the spring and wedge is elliptical. According to the spring parameters, the expression is(25)Sa=πhsD+d−hsd−hs.According to the wear length of the support spring in the wedge groove, by substituting equation (25) into equation (24), the expression of the wear depth and wear time of the support spring can be obtained as follows:(26)dh=ml2πRL⋅kvFrπHhsD+d−hsd−hsdt,where RL is the radius of the spring ring (mm), m is the number of wedges, and l is the length of the wedge groove (mm).According tov=ΔnoπRL/30, the expression of the wear time of the support spring obtained by its substitution into (26) and integration is as follows:(27)t=∫0h0πHhsD+d−hsd−hs60∗mlkΔnoFrdhs,where t is the wear time (h), ho is the wear depth of the spring (mm), and Δno is the relative rotation speed of the spring and wedge (r/min).


## 4.2. Influence of Support Spring Parameters on Spring Wear

In the wear model of the support spring, the relative rotation speed∆no of the spring and the wedge is obtained through a simulation of the clutch dynamics model under the overrunning state established by the dynamic analysis software, and the contact force Fr between the spring and wedge is calculated using the mechanical model of the support spring.


### 4.2.1. Dynamics Simulation of Sprag Clutch

TakingJ = 9.5 in a sprag clutch as an example, the 3D model of the clutch established using 3D modeling software is imported into the dynamic analysis software for a dynamic simulation. The material parameters of each part of the sprag clutch are listed in Table 3.The contact condition and geometry of each part of the sprag clutch are simplified and assumed: (1) It is assumed that no parts produce errors during the processing, manufacturing, and assembly. (2) The wedge is evenly subjected to the force of the support spring. According to the typical working conditions of the sprag clutch, the inner and outer rings are applied with a speed and load to simulate the clutch. A dynamic simulation model of the sprag clutch is shown in Figure10.Figure 10
Dynamic simulation model of sprag clutch.The boundary conditions and loads are applied as follows: (1) The rotating pair of the inner and outer rings of the sprag clutch are set. (2) The contact constraints between the wedge and inner and outer rings, support spring, and cage are imposed, and the contact parameters are defined. (3) In the overrunning state of the sprag clutch, the outer ring rotates counterclockwise relative to the inner ring, a counter-clockwise constant speedno is added to the outer ring, and the auxiliary torque T0 of the spring is applied to the 33 wedges. The loading curves are shown in Figure 11. (4) The gravity acceleration is 9.8 m/s2.Figure 11
Loading curve of rotational speed and auxiliary torque under overrunning state.The torque of the support spring to the rotation center of the wedge is calculated according to the force of the wedge and the position of the action point of the support spring under the overrunning state of the sprag clutch. The auxiliary torque of the wedge under different free lengths, pitch diameters, and spring wire diameters of the support spring are listed in Table5.Table 5
Auxiliary torque under different parameters of support spring.
Free lengthsL (mm)Auxiliary torqueT0 (N·mm)Pitch diametersD (mm)Auxiliary torqueT0 (N·mm)Wire diametersd (mm)Auxiliary torqueT0 (N·mm)270−3.32.0−7.50.5−2.8280−5.82.2−5.80.6−5.8290−8.42.4−4.60.7−11.0The inner and outer rings are loaded according to the conditions shown in Figure11, and the auxiliary torque of the wedge is loaded according to Table 5. The dynamic models of the sprag clutch with different structure parameters of support spring are established, and the time-varying curves of the speed of the wedge and support spring of the sprag clutch under the overrunning state are obtained, as shown in Figure 12.Figure 12
Rotational speed curves of support spring and sprag.By processing the data and taking the average relative rotation speed of the spring and wedge under a stable overrunning state, it is determined that the relative rotation speed of the support spring and wedge with different parameters under a stable overrunning state is approximately 10 r/min.


### 4.2.2. Application of the Present Model and Discussions

This paper aims at analysizing the stiffness, the supporting force, and the wear of support springs. The applied approach is based on theoretical curve equations and the characteristic of periodic change of the helix angle, the law of energy conservation, and Archard theory. The calculated results are more accurate and easier to obtain, thereby more convenient for engineering application. However, the tiny variations of supporting forces with structure parameters in the wear analysis are ignored, which may be a limitation in this approach.The supporting force under different parameters of the support spring and the relative rotation speed between the support spring and wedge∆no are substituted into the wear model of the support spring for calculation, and the relationship between the wear time and wear depth of the support spring under different free lengths, pitch diameters, and wire diameters is obtained, as shown in Figures 13 to 15, for which the wear depth is 0.2 mm.Figure 13
Relationship between spring wear time and wear depth under different spring free lengths.Figure 14
Relationship between spring wear time and wear depth under different spring pitch diameters.Figure 15
Relationship between spring wear time and wear depth under different wire diameters.It can be seen from Figure13 that with the increase of the free length, the wear time of the support spring decreases at the same wear depth, the wear of the support spring intensifies, and thus the reliability of the sprag clutch decreases. One can see that when free length is increased from 270 to 290 mm, the wear time is reduced by 20.88%. It can be seen from Figure 14 that with the increase of the pitch diameter, the wear time of the support spring increases slightly at the same wear depth, the wear of the support spring decreases, and thus the reliability of the sprag clutch increases. One can see that when pitch diameter is increased from 2 to 2.4 mm, the wear time is increased by 3.35%. It can be seen from Figure 15 that with the increase of wire diameter, the wear time of the support spring decreases at the same wear depth, the wear of the support spring intensifies, and thus the reliability of the sprag clutch decreases. One can see that when wire diameter is increased from 0.5 to 0.7 mm, the wear time is reduced by 42.75%.It can be seen that structural parameters of support springs have great effect on spring wear. The phenomenons can be explained as follows: the increase of free length and pitch diameter leads to the decrease of spring stiffness, whereas the increase of wire diameter leads to the increase of spring stiffness [14]; according to the supporting force model, it can be seen that the supporting force increases with the increase in the free length and the stiffness, the wear of the support spring thus increases [7]. Therefore, we can draw the conclusion that in order to improve the wear resistance of support springs, it is necessary to reduce the values of free length and wire diameter properly, whereas appropriately increase the value of pitch diameter.


### 4.2.3. Relative Sensitivity Analysis of Spring Wear to Various Parameters

The aforementioned analysis considers the depth of the spring wear under different parameters of the support spring. The relative sensitivity of the spring wear to the change in each parameter is as follows [17]:(28)SWt=Wt−W1/W1Pt−P1/P1,where SWt is the relative sensitivity, Wt is the wear depth at a given wear time t, W1 is the wear depth at the initial time, Pt is the spring parameter at a given wear time t, and P1 is the spring parameter at the initial time.The relative sensitivity regarding the amount of wear of the support spring to the changes in the parameters can be calculated under different wear times, as shown in Table6, where t indicates the wear time.Table 6
Relative sensitivity of wear in support spring to changes in the various parameters at different times.
Spring parametersSensitivity att = 100 hSensitivity att = 300 hSensitivity att = 500 hFree lengthL2.202.012.15Pitch diameterD0.080.080.07Wire diameterd0.971.021.03It can be seen from the results in Table6 that the degree of influence of the structural parameters on the spring wear from large to small is free length, wire diameter, and pitch diameter. Exactly speaking, when wear time is 100 hr, the relative sensitivity of structural parameters affecting the wear of the support spring is in order of free length (2.2), wire diameter (0.97), pitch diameter (0.08).


## 4.2.1. Dynamics Simulation of Sprag Clutch

TakingJ = 9.5 in a sprag clutch as an example, the 3D model of the clutch established using 3D modeling software is imported into the dynamic analysis software for a dynamic simulation. The material parameters of each part of the sprag clutch are listed in Table 3.The contact condition and geometry of each part of the sprag clutch are simplified and assumed: (1) It is assumed that no parts produce errors during the processing, manufacturing, and assembly. (2) The wedge is evenly subjected to the force of the support spring. According to the typical working conditions of the sprag clutch, the inner and outer rings are applied with a speed and load to simulate the clutch. A dynamic simulation model of the sprag clutch is shown in Figure10.Figure 10
Dynamic simulation model of sprag clutch.The boundary conditions and loads are applied as follows: (1) The rotating pair of the inner and outer rings of the sprag clutch are set. (2) The contact constraints between the wedge and inner and outer rings, support spring, and cage are imposed, and the contact parameters are defined. (3) In the overrunning state of the sprag clutch, the outer ring rotates counterclockwise relative to the inner ring, a counter-clockwise constant speedno is added to the outer ring, and the auxiliary torque T0 of the spring is applied to the 33 wedges. The loading curves are shown in Figure 11. (4) The gravity acceleration is 9.8 m/s2.Figure 11
Loading curve of rotational speed and auxiliary torque under overrunning state.The torque of the support spring to the rotation center of the wedge is calculated according to the force of the wedge and the position of the action point of the support spring under the overrunning state of the sprag clutch. The auxiliary torque of the wedge under different free lengths, pitch diameters, and spring wire diameters of the support spring are listed in Table5.Table 5
Auxiliary torque under different parameters of support spring.
Free lengthsL (mm)Auxiliary torqueT0 (N·mm)Pitch diametersD (mm)Auxiliary torqueT0 (N·mm)Wire diametersd (mm)Auxiliary torqueT0 (N·mm)270−3.32.0−7.50.5−2.8280−5.82.2−5.80.6−5.8290−8.42.4−4.60.7−11.0The inner and outer rings are loaded according to the conditions shown in Figure11, and the auxiliary torque of the wedge is loaded according to Table 5. The dynamic models of the sprag clutch with different structure parameters of support spring are established, and the time-varying curves of the speed of the wedge and support spring of the sprag clutch under the overrunning state are obtained, as shown in Figure 12.Figure 12
Rotational speed curves of support spring and sprag.By processing the data and taking the average relative rotation speed of the spring and wedge under a stable overrunning state, it is determined that the relative rotation speed of the support spring and wedge with different parameters under a stable overrunning state is approximately 10 r/min.


## 4.2.2. Application of the Present Model and Discussions

This paper aims at analysizing the stiffness, the supporting force, and the wear of support springs. The applied approach is based on theoretical curve equations and the characteristic of periodic change of the helix angle, the law of energy conservation, and Archard theory. The calculated results are more accurate and easier to obtain, thereby more convenient for engineering application. However, the tiny variations of supporting forces with structure parameters in the wear analysis are ignored, which may be a limitation in this approach.The supporting force under different parameters of the support spring and the relative rotation speed between the support spring and wedge∆no are substituted into the wear model of the support spring for calculation, and the relationship between the wear time and wear depth of the support spring under different free lengths, pitch diameters, and wire diameters is obtained, as shown in Figures 13 to 15, for which the wear depth is 0.2 mm.Figure 13
Relationship between spring wear time and wear depth under different spring free lengths.Figure 14
Relationship between spring wear time and wear depth under different spring pitch diameters.Figure 15
Relationship between spring wear time and wear depth under different wire diameters.It can be seen from Figure13 that with the increase of the free length, the wear time of the support spring decreases at the same wear depth, the wear of the support spring intensifies, and thus the reliability of the sprag clutch decreases. One can see that when free length is increased from 270 to 290 mm, the wear time is reduced by 20.88%. It can be seen from Figure 14 that with the increase of the pitch diameter, the wear time of the support spring increases slightly at the same wear depth, the wear of the support spring decreases, and thus the reliability of the sprag clutch increases. One can see that when pitch diameter is increased from 2 to 2.4 mm, the wear time is increased by 3.35%. It can be seen from Figure 15 that with the increase of wire diameter, the wear time of the support spring decreases at the same wear depth, the wear of the support spring intensifies, and thus the reliability of the sprag clutch decreases. One can see that when wire diameter is increased from 0.5 to 0.7 mm, the wear time is reduced by 42.75%.It can be seen that structural parameters of support springs have great effect on spring wear. The phenomenons can be explained as follows: the increase of free length and pitch diameter leads to the decrease of spring stiffness, whereas the increase of wire diameter leads to the increase of spring stiffness [14]; according to the supporting force model, it can be seen that the supporting force increases with the increase in the free length and the stiffness, the wear of the support spring thus increases [7]. Therefore, we can draw the conclusion that in order to improve the wear resistance of support springs, it is necessary to reduce the values of free length and wire diameter properly, whereas appropriately increase the value of pitch diameter.


## 4.2.3. Relative Sensitivity Analysis of Spring Wear to Various Parameters

The aforementioned analysis considers the depth of the spring wear under different parameters of the support spring. The relative sensitivity of the spring wear to the change in each parameter is as follows [17]:(28)SWt=Wt−W1/W1Pt−P1/P1,where SWt is the relative sensitivity, Wt is the wear depth at a given wear time t, W1 is the wear depth at the initial time, Pt is the spring parameter at a given wear time t, and P1 is the spring parameter at the initial time.The relative sensitivity regarding the amount of wear of the support spring to the changes in the parameters can be calculated under different wear times, as shown in Table6, where t indicates the wear time.Table 6
Relative sensitivity of wear in support spring to changes in the various parameters at different times.
Spring parametersSensitivity att = 100 hSensitivity att = 300 hSensitivity att = 500 hFree lengthL2.202.012.15Pitch diameterD0.080.080.07Wire diameterd0.971.021.03It can be seen from the results in Table6 that the degree of influence of the structural parameters on the spring wear from large to small is free length, wire diameter, and pitch diameter. Exactly speaking, when wear time is 100 hr, the relative sensitivity of structural parameters affecting the wear of the support spring is in order of free length (2.2), wire diameter (0.97), pitch diameter (0.08).


## 5. Experimental Verification

To verify the correctness of the aforementioned analysis results, friction and wear tests are carried out on the CFT-I multifunctional material surface performance tester, the test equipment of which is shown in Figure16. As the underlying principle of the equipment, under the drive of the motor, the spindle drives the upper sample steel ball to achieve a linear reciprocating motion; in addition, the upper sample can provide a normal load, the lower sample disc is fixed, and the reciprocating motion frequency and stroke are adjustable.Figure 16
CFT-I multifunctional material surface comprehensive performance tester: (1) lifting motor, (2) lifting platform, (3) left-right motor, (4) left-right moving platform, (5) left-right moving knob, (6) sensor, (7) gripper, (8) sample fixture, (9) reciprocating friction components, and (10) instrument main table.The upper sample of the friction pair (GCr15 steel ball with a Ø3 mm diameter) is used to replace the wedge in the sprag clutch; the lower sample of the friction pair has 70 steel discs with a size of 50 mm × 5 mm, with an average hardness of 59.9HRC after heat treatment and is used to replace the support spring in the sprag clutch. An aviation lubricant oil bath is used for lubrication during the test.Under the overrunning state of the sprag clutch, the supporting force obtained by theoretical calculation is 4.44 N. To increase the efficiency of wear test, the selected loads are 44.4 N, 88.8 N, and 133.2 N, respectively, during the test. The relative rotation speed∆no between the support spring and wedge is 10 r/min, and the sliding speed is 42.8 mm/s. In addition, combined with the theoretical calculation and finite-element simulation results, the test parameters of the wear are set as shown in Table 7.Table 7
Parameters of tribological tests.
Test groupNormal load (N)Loading time (s)Sliding speed (mm/s)144.4180042.8288.8180042.83133.2180042.8The test is carried out according to the test parameters listed in Table7. The three-dimensional profile of the support spring steel after the wear test is shown in Figure 17, and the section curves of the wear tracks for the support spring steel under different normal loads is shown in Figure 18.Figure 17
3D profile of 70 steel support spring after wear test: (a) 3D profile with the normal load of 44.4 N, (b) 3D profile with the normal load of 88.8 N, and (c) 3D profile with the normal load of 133.2 N.
(a)(b)(c)Figure 18
Cross-sectional curves of the wear tracks for the support spring steel under different normal loads.From Figure18, it is observed that the maximum wear depth values for the spring steel sample are 1.82 μm, 3.02 μm, and 4.46 μm, respectively. According to the Archard wear model, the adhesive wear coefficient under test conditions is calculated, and the calculation equation is as follows:(29)k=HhPvt,where k is the adhesive wear coefficient, h is the wear depth (mm), H is the Brinell hardness of the material surface, P is the contact pressure (MPa), v is the sliding speed (mm/s), and t is the wear time (s).The average contact pressureP [18] is expressed as follows:(30)P=1π6WE∗2R21/3,where P is the average contact pressure (Pa), W is the load (N), R is the radius of the steel ball (m), and E∗ is the equivalent elastic modulus of the material (Pa).The calculation equation of equivalent elastic modulusE∗ is as follows:(31)1E∗=1−v12E1+1−v22E2,where E1 and v1 are the elastic modulus and Poisson’s ratio of the first sample materials, and E2 and v2 are the elastic modulus and Poisson’s ratio of the other sample materials, respectively.According to equations (29) to (31), the calculated adhesive wear coefficient k are 4.299 × 10−9, 5.662 × 10−9, and 7.305 × 10−9, respectively. In addition, comparing the theoretical values of wear depth with the test ones, the results can be listed in Table 8. It is noted that the errors between theoretical and test values are less than 20%, which verifies the accuracy of the wear model of the support spring under an overrunning state.Table 8
Comparison of theoretical and experimental wear depth results with different normal load.
Test groupTheoretical wear depth (μm)Test wear depth (μm)Error (%)12.271.8219.8223.733.0219.0335.204.4614.23


## 6. Conclusion

In this study, stiffness model and supporting force model of the support spring are established according to theoretical analysis and finite-element method. On this basis, a wear model of the support spring in process of overrunning is established based on Archard wear theory, and the effect of structural parameters of the support spring on the wear is discussed. Based on the present results, the conclusions can be summarized as follows:(1)
The stiffness of the support spring is related to structural parameters such as free length, wire diameter, pitch diameter, and spring pitch.(2)
There is a linear relationship between supporting force and spring radial displacement compression.(3)
Increasing free length and wire diameter of springs will aggravate the wear of springs, whereas enlarging pitch diameter of springs will reduce it. When free length is increased from 270 to 290 mm, the wear time is reduced by 20.88%; when wire diameter is increased from 0.5 to 0.7 mm, the wear time is reduced by 42.75%; when pitch diameter is increased from 2 to 2.4 mm, the wear time is increased by 3.35%.(4)
The degree of influence of the structural parameters on the spring wear from large to small is free length, wire diameter, and pitch diameter. Exactly speaking, when wear time is 100 hr, the relative sensitivity of structural parameters affecting the wear of the support spring is in order of free length (2.2), wire diameter (0.97), and pitch diameter (0.08).



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*Source: 3417760-2022-03-09.xml*