# Non-Hertzian Elastohydrodynamic Contact Stress Calculation of High-Speed Ball Screws

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Analysis of Force and Motion

_{A}and Q

_{B}denote the normal contact force of the ball at the screw contact point A and the nut contact point B, respectively. F

_{SA}and F

_{SB}indicate the friction of the ball at the screw contact point A and the nut contact point B, respectively. F

_{IH}is the inertia force. M

_{SA}and M

_{SB}are the friction torque of the ball at the screw contact point A and the nut contact point B, respectively. M

_{IH}is the inertia torque. The parameters of DNBSs in the operating state, such as β

_{A}, β

_{B}, Q

_{A}, Q

_{B}, F

_{SA}, and F

_{SB}, are obtained by Equation (2) using the Newton iteration method. The specific solution process of the parameters of DNBSs can be observed in the reference [20]. The structural parameters of DNBSs are shown in Table 1.

#### 2.2. Non-Hertzian Normal Contact Stress Calculation

**c**is the contact surface gap matrix. c

_{i}

_{,j}is the contact surface gap distribution in the mesh region, which is the element of the contact surface gap matrix

**c**. d

_{0}denote the approaching distance of the contact surface.

**c**is the contact surface geometry matrix. c

_{g}_{gi}

_{,j}indicates the contact surface geometry in the mesh region, which is the element of the contact surface geometry matrix

**c**.

_{g}**c**is the contact surface roughness matrix. c

_{r}_{ri}

_{,j}is the contact surface roughness distribution, which is the element of the contact surface roughness matrix

**c**.

_{r}**V**is the surface elastic deformation matrix. V

_{i}

_{,j}is the surface elastic deformation distribution in the mesh region, which is the element of the surface elastic deformation matrix

**V**. I and J indicate the mesh grid number in the x and y directions, respectively.

_{i}

_{,j}is the element of the contact stress matrix

**P**. K

_{i − k}

_{,j − l}is the element of deformation influence coefficient matrix

**K**.

_{i}

_{,j}= 0, p

_{i}

_{,j}≥ 0; Out of the contact region, the boundary condition of the contact equation is c

_{i}

_{,j}> 0, p

_{i}

_{,j}= 0. According to the above two conditions, c

_{i}

_{,j}, and p

_{i}

_{,j}cannot be both zero, thus

- The initial normal contact stress
**P**is given, p_{i}_{,j}≥ 0. The normal contact stress distribution p_{i}_{,j}is also satisfied in the mesh region. - The discretized normal contact deformation
**V**is determined by Equation (4). - The contact gap
**c**between the two contact surfaces is calculated by Equations (3)–(6), and the contact gap is satisfied with the boundary condition. - According to the variational principle, the contact in Equation (7) can be transformed by the conditional extremum of the quadratic function, the normal contact stress p
_{i},_{j}is modified by Equation (10) satisfying the boundary condition in Equation (11). - The current relative error is determined by:$$\epsilon ={d}_{s}{Q}_{A}{}^{-1}{\displaystyle \sum _{i,j}\left|{p}_{i,j}-{p}_{i,j}^{old}\right|}$$
- If ε ≥ ε
_{0}, the iteration solution process of the non-Hertzian contact stress continues from steps (2)–(5), otherwise the iteration solution process ends.

#### 2.3. Non-Hertzian Elastohydrodynamic Contact Stress Calculation

- The lubricant and the solid are isothermal, and the contact stress and the film thickness do not vary with time.
- The curvature radius of the solid surface is much larger than the lubricant film thickness.
- No relative sliding between the lubricant film and solid surface at the common interface.
- Compared with the film shear stress, the inertial force and other bulk forces of the lubricant film are negligible.
- Due to the lubricant film thickness being very thin, it can be assumed that the film contact stress remains constant along the film thickness direction, namely no squeeze film effect.
- The inlet of the contact region is flooded fully by the lubricant film.
- The lubricant is the Newtonian fluid and obeys Newton’s law of viscosity.

_{s}could be accurately determined by the Reynolds equation [24], which is written as

_{1}and u

_{2}indicate the linear velocity of the ball and the raceway in the tangential direction (x-direction) of the helix raceway track, respectively. The entrainment velocity of the lubricant film is u

_{s}= (u

_{1}+ u

_{2})/2. The output parameters of point A and point B calculated by Equation (2), such as Q

_{A}, β

_{A}, Q

_{B}, and β

_{B}, are substituted into the Reynolds Equation (14), the elastohydrodynamic characteristics of point A and point B can be determined, respectively.

_{0}indicates the central film thickness in the contact region. η

_{0}and ρ

_{0}denote the initial value of the film viscosity and density, respectively. ROU(x,y) indicates the roughness of the contact surface calculated by the fractal theory [28] shown in Figure 4a. The major parameters of the rough fractal rough surface are the fractal dimension D and scale parameter G, D = 2.47, G = 0.41. The rough fractal surface is the contact region of the raceway. The hardness range of the ball is 62–64 HRC higher than that of the screw raceway 58–62 HRC; The processing technology of the working ball is mature, and the surface of the working ball is smooth. According to the above two reasons, the ball contact surface is regarded as a smooth surface. The surface roughness of the working ball is much less than that of the screw raceway. Therefore, the surface roughness of the ball has little effect on the calculation efficiency of this model.

_{o}and y

_{o}are the entrance coordinates of the contact region of BSs, respectively, and x

_{e}and y

_{e}are the exit coordinates of the contact region of BSs, respectively. The values of these coordinate parameters have been given: x

_{o}= y

_{o}= −277 μm, x

_{e}= y

_{e}= 277 μm.

## 3. Experiments and Verification

#### 3.1. Experiments Procedures

_{pre}= 1500 N, constant lubricating oil viscosity η

_{0}= 0.05 Pa·s, and the different screw speeds. The simulated friction torque of the BSs can be expressed as

_{SB}

_{t}and F

_{SB}

_{b}represent friction components at the directions of t and b in the Frenet–Serret coordinate system Otnb at point B, respectively [31]. n

_{bb}denotes the number of working balls.

#### 3.2. Experimental Results

_{f}, and the asperity bearing force Q

_{a}. The ratio of the asperity bearing stress is R

_{S}, R

_{S}= Q

_{a}/Q. μ

_{a}is the dry friction coefficient and μ

_{a}= 0.1. τ

_{f}is the film shear stress proposed by the Eyring model [33]. A

_{h}is the film hydrodynamic contact area, η is the film viscosity, Δu is the relative sliding velocity, and Δu = u

_{1}− u

_{2}. u

_{1}and u

_{2}are the same as that of Equation (14). h

_{0}is the film central thickness.

## 4. Results Analysis and Discussion

#### 4.1. Comparison of the Normal Contact Stress between Hertzian Solution and Non-Hertzian Solution

_{pre}is 1500 N, and the helix angle α is 21.7°. The contact stress comparison between the Hertzian solution and non-Hertzian solution of BSs is shown in Figure 7. As shown in Figure 7, the difference contour maps between the Hertzian solution and non-Hertzian solution at contact point A and contact point B indicate the great difference between the two calculation methods. The difference of the contact stress in the edge of the contact region is the largest, and the one in the center of the contact region is the smallest. Thus, there is a big error of contact stress distribution calculated by the Hertzian contact theory, and the contact stress distribution calculated by the non-Hertzian solution is more accurate.

#### 4.2. Analysis of the Non-Hertzian Contact Stress with Different Helix Angles

_{A}at the contact point A generates a deflection angle θ

_{s}shown in Figure 8a. The contact stress distribution difference P

_{A}-P

_{AH}between the non-Hertzian solution P

_{A}and the Hertzian solution P

_{AH}in the edge of the contact region is largest, and the one in the center of the contact region is smallest shown in Figure 8b.

#### 4.3. Analysis of the Elastohydrodynamic Contact Stress with Different Helix Angles

#### 4.4. Effect of the Elastohydrodynamic Contact Stress under Different Screw Speeds

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Glossary

a | Major axis of the elliptical contact region/mm |

b | Minor axis of the elliptical contact region/mm |

c | Contact gap between the two contact surfaces/mm |

d | Approaching distance of the contact surface/mm |

h | Lubricant film thickness/μm |

n | Number of the working ball |

p | Normal contact stress/MPa |

r | Radius/mm |

u | Entrainment velocity/(mm/s) |

x, y, z | Coordinates in ball screws |

A | Area/(μm^{2}) |

D | Fractal dimension |

E | Elasticity modulus/MPa |

F | Force/N |

G | Scale parameters |

K | Deformation influence coefficient/(mm/N) |

I, J | Total number in the x or y direction |

L | Pitch of the screw/mm |

M | Torque/(N/mm) |

Q | Load/N |

R | The ratio of the asperity bearing stress |

ROU | Profile of the rough surface/μm |

V | Contact deformation/mm |

W | Quadratic function |

α | Helix angle/° |

β | Contact angle/° |

ε | Iteration precision |

θ | Deflection angle of the contact region/° |

υ | Poisson ratio |

τ | Lubricant film shear stress/MPa |

ρ | Lubricant film density/(kg/m^{3}) |

η | Lubricant film viscosity/(Pa·s) |

μ | Friction coefficient |

Ω’ | Screw speed/(mm/s) |

## Nomenclature

0 | Initial value |

1, 2 | The two contact surfaces |

a | Asperity bearing |

bb | Ball |

f | Lubricant film bearing |

g | Contact gap |

h | Mesh region |

i, j | Column or row number |

pre | Preload |

t, n, b | Coordinates in O_{H}tnb |

o, e | Entrance and exit coordinates of the contact region |

A | Contact point between ball and screw raceway |

AH | Contact point between ball and screw raceway with Hertzian contact |

B | Contact point between ball and nut raceway |

BH | Contact point between ball and nut raceway with Hertzian contact |

IH | Inertia |

S | Screw |

SA | Friction between ball and screw raceway |

SB | Friction between ball and nut raceway |

N | Nut |

## Abbreviations

BSs | Ball screws |

DNBSs | Double-nut ball screws |

EHL | Elastohydrodynamic lubrication |

NC | Numerical control |

## Appendix A

#### The Finite Difference Method

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**Figure 1.**Force analysis of ball screws. (

**a**) Force balance analysis. (

**b**) Contact mechanics analysis.

**Figure 3.**A comparison of the non-Hertzian and Hertzian solutions for the contact stress distribution between the ball and the screw raceway.

**Figure 4.**Diagram of the elastohydrodynamic contact mechanics. (

**a**) The fractal rough surface. (

**b**) Elastohydrodynamic contact mechanics.

**Figure 5.**Experimental setup of the DNBSs. 1. Motor. 2. Coupling. 3. Base. 4. Guideway. 5. Table. 6. Screw. 7. Nut. 8. Clamping device. 9. Dowel bar. 10. Tension–compression sensor.

**Figure 6.**Results of the friction torque test. (

**a**) A schematic Stribeck curve [34]. (

**b**) Friction coefficient curve. (

**c**) Comparison of the simulated and experimental friction torque.

**Figure 7.**The contact stress distribution comparison between the Hertzian solution and the non-Hertzian solution. (

**a**) Contact stress solution at the ball/screw–raceway point A. (

**b**) Contact stress solution at the ball/nut raceway point B.

**Figure 8.**The variation of the non-Hertzian contact stress with different helix angles. (

**a**) Contour map of non-Hertzian contact pressure distribution P

_{A}(Mpa). (

**b**) Distribution difference between non-Hertzian contact pressure and Hertzian contact pressure P

_{A}-P

_{AH}(Mpa).

**Figure 9.**Comparison of the Hertzian and non-Hertzian contact stress. (

**a**) The central contact stress of the point A. (

**b**) The central contact stress of the point B. (

**c**) The contact stress error. (

**d**) The contact region deflection angle.

**Figure 10.**Elastohydrodynamic characteristics using the non-Hertzian contact stress calculation method. (

**a**) Elastohydrodynamic contact stress distribution. (

**b**) Film thickness distribution.

**Figure 11.**Comparison of the elastohydrodynamic characteristics between the Hertzian solution and non-Hertzian solution. (

**a**) Central elastohydrodynamic contact stress of point A. (

**b**) Central elastohydrodynamic contact stress of point B. (

**c**) Central elastohydrodynamic film thickness of point A. (

**d**) Central elastohydrodynamic film thickness of point B.

**Figure 12.**Comparison of the elastohydrodynamic stress between the Hertzian solution and non-Hertzian solution under different screw speeds. (

**a**) Central elastohydrodynamic contact stress of point A. (

**b**) Central elastohydrodynamic contact stress of point B.

**Figure 13.**Comparison of the asperity contact stress between the Hertzian solution and non-Hertzian solution under different screw speeds. (

**a**) Central asperity contact stress of point A. (

**b**) Central asperity contact stress of point B.

Screw radius, r (mm) | 16 | Ball radius, r_{b} (mm) | 2.9765 |

Radius of the normal section on the screw raceway, r _{S} (mm) | 3.215 | Radius of the normal section on the nut raceway, r _{N} (mm) | 3.215 |

Helix angle, α (°) | 5.68 | Pitch, L_{0} (mm) | 10 |

Initial contact angle β_{0} (°) | 40.26 | Preload, F_{pre} (N) | 1500 |

Screw speed, Ω’ (mm/s) | 83 | Number of working balls, n_{bb} | 168 |

Young’s modulus, E_{1} = E_{2} (GPa) | 200 | Density ρ_{0} (kg/m^{3}) | 970 |

Poisson’s ratio, υ_{1} = υ_{2} | 0.3 | Viscosity η_{0} (Pa·s) | 0.05 |

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**MDPI and ACS Style**

Sun, T.; Wang, M.; Gao, X.; Zhao, Y.
Non-Hertzian Elastohydrodynamic Contact Stress Calculation of High-Speed Ball Screws. *Appl. Sci.* **2021**, *11*, 12081.
https://doi.org/10.3390/app112412081

**AMA Style**

Sun T, Wang M, Gao X, Zhao Y.
Non-Hertzian Elastohydrodynamic Contact Stress Calculation of High-Speed Ball Screws. *Applied Sciences*. 2021; 11(24):12081.
https://doi.org/10.3390/app112412081

**Chicago/Turabian Style**

Sun, Tiewei, Min Wang, Xiangsheng Gao, and Yingjie Zhao.
2021. "Non-Hertzian Elastohydrodynamic Contact Stress Calculation of High-Speed Ball Screws" *Applied Sciences* 11, no. 24: 12081.
https://doi.org/10.3390/app112412081