A calculation method for friction coefficient and meshing efficiency of plastic line gear pair under dry friction conditions

A calculation method for the friction coefficient and meshing efficiency of plastic line gear (LG) pair under dry friction conditions was studied theoretically and experimentally, taking a polyoxymethylene parallel line gear pair (POM PLGP) as an example. Firstly, the geometric and mechanical models of PLGP were built by considering the effects of misalignment and loaded deformation under the actual operating condition. Then, the friction coefficient of POM specimens was obtained via the ball-on-disk experiment, of which the value varies between 0.35 and 0.45 under the experimental conditions. The calculation formula for the friction coefficient of POM LG pair was obtained by fitting the friction coefficient of the POM specimens, and the meshing efficiency of POM LG pair was calculated based on the calculation formula for friction coefficient and the meshing efficiency calculation approach. Finally, the meshing efficiency of POM PLGP specimens was measured using a homemade gear meshing efficiency test rig. The experimental results validated the feasibility of the proposed calculation method for the friction coefficient and meshing efficiency of the plastic LG pair. This study provides a method for the calculation of the friction coefficient and meshing efficiency of plastic gear pairs under dry friction conditions. It also provides the basis for the wear calculation of plastic LG pair under dry friction conditions.


Introduction
The advantages of plastic gears, such as lightweight, low noise, self-lubrication, and low cost, have led to their wide use in gearing systems applied under conditions of dry friction, low rotational speed, and light load, such as measuring instruments, food machinery, and mobile communication equipment [1][2][3][4]. The remote control unit (RCU) is the core equipment of a base station antenna, which can realize the tilt angle adjustment through a gear transmission system. Polyoxymethylene (POM) is free from electromagnetic interference and has the advantages of low hygroscopicity, high strength, and good wear resistance [5][6][7]. Thus, POM gear pairs are commonly used in RCU. However, relative sliding between meshing surfaces occurs throughout the meshing process (except for the pitch point) of commonly used POM gear pairs, such as spur gear, helical gear, and worm gear pairs, which results in low meshing efficiency and restricts the application of POM gear pairs under dry friction conditions [3,8,9].
Line gear (LG), a novel gear mechanism invented based on the space curve meshing theory proposed by Chen et al. [10,11], achieves transmission through the continuous point-contact meshing of a pair of space conjugate curves called driving and driven Nomenclature Subscripts i = 1 and i = 2 represent the parameters of the driving and driven LGs, respectively; subscript kj represents the parameters of the meshing point kj of tooth pair j.  [12]. For example, when a cylindrical helix is selected as the driving contact curve of a parallel LG pair (PLGP), the sliding rates can be equal to zero throughout the meshing process [13]. This indicates that high meshing efficiency can be achieved even under | https://mc03.manuscriptcentral.com/friction dry friction conditions. Therefore, PLGP is a good choice for gearing applications under dry friction conditions. However, misalignment and loaded deformation under the actual operating condition could change the actual meshing positions of the LG pair. This leads to relative sliding between the meshing surfaces, which influences the meshing efficiency. Thus, it is necessary to study the meshing efficiency of LG pairs under the actual operating condition. Frictional loss caused by relative sliding between meshing surfaces is the main reason for low meshing efficiency under dry friction conditions. It is a function of load, relative sliding velocity, and friction coefficient of the gear pair [9,14,15]. Thus, determining the friction coefficient of the gear pair is the key to studying the meshing efficiency. Larson and Timpe [16], Ziemianski and Capanidis [17], and Ginzburg et al. [18] measured the friction coefficient of POM against steel under various loads, relative sliding velocities, and lubrication conditions, however, they did not derive any calculation formulas for the friction coefficient. Xiong et al. [19] conducted a pin-ondisk experiment and obtained a calculation formula for the friction coefficient of POM under dry friction conditions. However, the formula is not associated with the operating conditions of any gear pair. Besides, the face-contact model of the pin and disk specimens is completely different from the point-contact model of the LG pair. Thus, the formula based on the pin-on-disk experiment is unsuitable to be used to calculate the friction coefficient of the LG pair. Miler et al. [20] proposed a calculation formula for the friction coefficient of the POM spur gear pair. However, this formula is only valid for POM spur gear with polytetrafluoroethylene (PTFE) lubricant and had not been validated by the gear transmission experiment. In short, we have not found any calculation methods or formulas for the friction coefficient of plastic gear pairs under dry friction conditions.
There are three existing methods to determine the friction coefficient of the gear pair. The first method assumes a constant friction coefficient throughout the meshing process [21,22]. However, the friction coefficient varies with changes in operating conditions. The second method determines the friction coefficient using the models based on friction mechanism and lubrication theory. Xu and Kahraman [14,15] proposed an effective method for calculating the friction coefficient based on the elastohydrodynamic lubrication (EHL) model when studying the meshing efficiency of steel spur and helical gear pairs. The third method determines the friction coefficient by the empirical formulas based on the measured friction coefficient data of the materials of the gear pair. Marjanovic et al. [23] measured the friction coefficient of steel specimens and established a calculation formula for the friction coefficient of steel spur gear pairs under oil lubrication. For the third method, a certain number of friction coefficient tests are needed, and the types and conditions of the tests can be selected according to the types, materials, and operating conditions of the gear pairs. Thus, the third method can be applied to determine the friction coefficient of plastic gear pairs under dry friction conditions.
In this paper, the friction coefficients of POM specimens were measured through the ball-ondisk experiment by considering the point-contact model of the LG pair, and the calculation formula for the friction coefficient of the POM LG pair is obtained. On this basis, the calculation formula for the meshing efficiency of the POM LG pair was derived. The feasibility of the calculation formulas for the friction coefficient and meshing efficiency of the POM LG pair was then validated by comparing the calculated and measured meshing efficiency values of the POM LG pair specimens.

Geometric and mechanical models of PLGP
Meshing efficiency is a function of load, relative sliding velocity, and friction coefficient. The friction coefficient is influenced by the contact pressure and relative sliding velocity. In this section, the normal force, contact pressure, and relative sliding velocity are deduced based on the geometric and mechanical models of PLGP under the actual operating condition.  (1) and (2), and 1  and 2  are expressed as Eqs. (3) and (4), where i m is the helix radius of i r , m 1 > 0, and m 2 > 0; i n the screw parameter of i r , n 1 > 0, and n 2 < 0.
i 12 is the transmission ratio, and m m Driving and driven contact curves begin to mesh when t i equals to t is , and begin to separate when t i equals to e i t . i  is the parameter indicating the scope of i P . i R is the radius of i P . zi  is the modification angle of the tooth profile. Misalignments under the actual operating condition may change the actual meshing positions of the LG pair, which may influence the normal force, contact pressure, and relative sliding velocity, followed by the friction coefficient and meshing efficiency. As shown in Fig. 2, misalignments of the PLGP are classified as follows: center distance deviation, i.e., an extra displacement a  in the are presented as Eqs. (5) and (6), respectively. n .
The condition of continuous tangential contact between the tooth surfaces at the meshing points is presented by Eq. (9) [24].
A meshing period from meshing start to meshing end of tooth pair j is divided evenly into K meshing positions, and the rotation angle of the driving LG is given as 1kj Eq. (9), the other five unknown parameters 1 1 ( ), The coordinates of the meshing point kj on 1  and 2  are 1 1

Normal force and contact pressure on the tooth surfaces
In this section, the normal force on the tooth surfaces is deduced first, followed by the contact pressure on the tooth surfaces according to the radii of principal curvatures and the angle between the principal directions at the meshing points.

Normal force on the tooth surfaces
Taking the driven LG, the normal force n 2 1 ( ) kj  F is applied perpendicularly on the tooth surface 2  at the meshing point kj of the tooth pair j , as shown in Fig. 3.
. The contact ratio of an LG pair is usually greater than 1, i.e., the number of tooth pairs meshing simultaneously is at least one during the meshing process. Thus, the total torque on the driven LG is the sum of torques on the tooth pairs meshing simultaneously: where J is the number of the tooth pairs meshing simultaneously; t2 1 ( ) is the meshing radius at the point kj of tooth pair j of the driven LG; and 2 T is the output torque. The load distribution coefficient among the loaded tooth pairs K akj is introduced to simplify the calculation of the normal force, where K akj is equal to 1 in the case of single-tooth meshing, 0.5 in the case of double-tooth meshing, and so on. Then, the magnitude of n2 1 ( ) where n2  cos

Contact pressure on the tooth surfaces
Large contact pressure may occur at the meshing points due to the point-contact model of the LG pair. Contact deformation is the main type of loaded deformation. Meshing surfaces of tooth pair j are regarded as two elastic bodies that are initially in contact at the meshing point kj , as indicated by the solid curves in Fig. 4. When the The influence coefficient method is an efficient method to solve for the contact pressure and deformation [25]. A calculation domain containing the actual contact region is discretized into N x N y elements denoted as (g, h) ( 1, 2,..., , 1, 2,..., y N Equation (12) is satisfied within the calculation domain.
where , K   g e h f is the influence coefficient of the deformation of element (g, h) caused by contact pressure p kj (e, f) on the element (e, f). For additional information on the influence coefficient, please refers to Ref. [27].
Contact pressure ( , ) kj p g h and mutual approach kj  are solved through the method described above.
In addition, the average contact pressure p akj on the tooth surfaces can be obtained and used to calculate the friction coefficient.

Relative sliding velocity between the tooth surfaces
Relative sliding velocity is determined by the misalignment and loaded deformation under the actual operating condition. The rotation angle 2 1 ( ) where L f1 and L f2 are the transmission matrices. As shown in Fig. 1 Therefore, the relative sliding velocity between the tooth surfaces at the point kj is presented as Eq. (17).

Testing and fitting formula for the friction coefficient of POM LG pair
In this section, a friction experiment is conducted to determine the friction coefficient of the POM LG pair under dry friction conditions. POM specimens in the shape of the ball and disk are prepared for the friction experiment to simulate the point-contact model of the LG pair. Several normal loads and relative sliding velocity levels are set according to the operating conditions of the plastic LG pair. The calculation formula for the friction coefficient of the POM LG pair under dry friction conditions is obtained by the measured friction coefficient data of the POM specimens.

Preparation of specimens and experimental setup
Several types of friction experiments, including ball-on-disk, ring-on-block, twin-disk, and pin-ondisk experiments [20,[28][29][30], can be conducted to determine the friction coefficient for different contact models under dry friction conditions. The ball-ondisk experiment can simulate the point-contact model of the LG pair. Thus, the ball-on-disk experiment was conducted to determine the friction coefficient of the POM specimens in this study. POM specimens in the shape of a ball and disk are shown in Fig. 5. The diameter of the POM ball specimen is 9.5 mm, and that of the POM disk specimen is 46 mm. All the specimens were cleaned with alcohol before the friction experiment. The Universal Mechanical Tester (UMT Tribolab) was used in the ball-on-disk test, as shown in Fig. 6(a).   sensor, and frictional moment sensor. The schematic diagram of the ball-on-disk experiment is shown in Fig. 6(b). The POM ball specimen is fixed at the end of the fixture and the POM disk specimen below the ball specimen is connected to the rotation motor and rotates around its axis at a constant speed. The relationship between the rotational speed s n of the rotation motor and the relative sliding velocity s v of the POM specimens is as

Experimental approach
External environmental factors, such as temperature and humidity, were not considered in this study. Thus, the main factors affecting the friction coefficient are the load and relative sliding velocity [19]. The normal load and relative sliding velocity were selected as two variables in the ball-on-disk experiment. The ranges of these two factors were determined according to the operating conditions of plastic LG pairs. Contact pressure may be large at the meshing points owing to the point-contact model of the LG pair. The upper limit of the maximum contact pressure of the POM specimens is set at approximately 80 MPa according to the compression strength of POM, i.e., the maximum average contact pressure is approximately 50 MPa. The maximum normal load is set at approximately 25 N according to the dimensions of the POM specimens. The relative sliding velocity between the tooth surfaces is influenced by the design parameters, misalignment, and loaded deformation of the LG pair. As the sliding rates of PLGP are equal to zero under ideal conditions, and the rotational speeds and loads of plastic gear pair are small, the relative sliding velocity of the plastic LG pair should be relatively low, despite the misalignment and loaded deformation. The maximum relative sliding velocity was set as 2 mm/s. When s r was set as 2 mm, the maximum rotational speed was approximately 9.55 rpm.

Experimental results and calculation formula for the friction coefficient
As shown in Fig. 7, when the normal load is increasing relative sliding velocity. The friction coefficient of POM against steel varies from approximately 0.2 to 0.4 when the contact pressure is 5-15 MPa, and the relative sliding velocity is 50-900 mm/s [19]. The friction coefficient of POM against POM using PTFE lubricant varies from approximately 0.03 to 0.7 when the load is 15-450 N, and the relative sliding velocity is 50-2,700 mm/s [20]. In short, the friction coefficient of POM is influenced by the operating conditions, such as load, relative sliding velocity, and lubrication conditions.
An approximate functional relation between the friction coefficient and the average contact pressure and relative sliding velocity of the POM specimens was obtained by fitting the measured friction coefficient data. First Goodness-of-fit was evaluated by the coefficient of determination and root-mean-squared error. As shown in Table 1, Eq. (19a) has the largest coefficient of determination and the smallest root-mean-squared error, i.e., Eq. (19a) is the most appropriate to describe the measured friction coefficient data. As shown in Fig. 8, the measured friction coefficient data are indicated by the dot-dash curves, and the calculated friction coefficient data by Eq. (19a) are indicated by the solid curves. It can be seen that the calculated result shows good agreement with the measured result.  (20) where akj p (MPa) and 12 1 ( ) kj  v (mm/s) are the average contact pressure and relative sliding velocity at the point kj , respectively.
It should be noted that Eq. (20) can be applied to calculate the friction coefficient of the POM LG pair under dry friction conditions within the operating conditions in this study. However, the extrapolation of Eq. (20) to its application is not recommended as this is beyond the scope of the operating conditions.

Meshing efficiency of POM LG pair
Taking POM PLGP as an example, the meshing efficiency of the plastic LG pair was studied by comparing the calculated and experimental results of the POM PLGP specimens.

Calculation method and formula for meshing efficiency
The calculation method for the meshing efficiency of the LG pair is shown as the flowchart in Fig. 9. A meshing period of tooth pair j was divided evenly into K meshing positions, i.e., the range of rotation angle 1 j  was discretized evenly as into Eq. (20). Assuming that there are J pairs of tooth meshing simultaneously, the expression of instantaneous frictional power losses ins P of the LG pair is presented as Eq. (21): The instantaneous meshing efficiency ins  of the LG pair is presented as Eq. (22): Assuming that the meshing process of each tooth pair is identical, the average meshing efficiency was obtained by superimposing the instantaneous meshing efficiency for all discrete meshing positions and then calculating the average value for a meshing period of tooth pair j . The calculation formula for the average meshing efficiency cal  of the LG pair is presented as Eq. (23):

Calculation examples and results
The parameters of the POM PLGP specimens are listed in Table 2. Computer numerical control (CNC) form milling of LG [31] is a commonly used manufacturing method that was adopted to manufacture the POM PLGP specimens, as shown in Fig. 10.
The test results of misalignment as well as the input speeds and output torques of the POM PLGP specimens are summarized in Table 3. For a meshing period from meshing start to meshing end of a tooth pair, the range of 1 j  is [0,π/2], which is evenly discretized as      (t si , t ei ) (-π，-π/2) (0，-π/6) i 12 3 Contact ratio 1.5

Young's modulus (MPa) 2750
Poisson's ratio 0.37 The friction coefficient As shown in Fig. 11(a), when the output torque 2 T is 300 N·mm, decreased with increasing input rotational speed, and the calculated results under other output torques are similar. As shown in Fig. 11(b), when the input rotational speed 1 n is 300 rpm, 1 ( ) kj   first increased with increasing output torque when the output torque 2 T is 100-300 N·mm, and then it slightly increased to the peak value 0.45 when 2 T is 300-500 N·mm. After T 2 reaches 600 N·mm,  As shown in Fig. 12, the calculated meshing efficiency cal  under the same 2 T increased slightly with increasing 1 n when 1 n is 250-350 rpm. cal  under the same 1 n first decreased and then increased with increasing 2 T when 2 T is 100-600 N·mm. cal  varied with the output torque and input rotational speed similar to the variation of friction coefficient. As the meshing efficiency under a certain operating condition is influenced by the load, rotational speed, and friction coefficient, fluctuations of the calculated results are inevitable, but since the frictional losses are small under the operating conditions, the fluctuation is very slight, and the average calculated meshing efficiency is around 99.88%.

Experimental setup and conditions
A homemade gear meshing efficiency test rig was developed to measure the meshing efficiency of the POM PLGP specimens. As shown in Fig. 13, the test rig consists of gearbox 1, rotational speed, and torque sensors 2 and 6, servo motor 3, powder brake 4, and flexible couplings 5. Servo motor 3 provides sufficient input torque and constant input rotational speed for gearbox 1. The powder brake 4 applies constant output torque onto gearbox 1. The rotational speed and torque sensor 2 located between gearbox 1 and servo motor 3 collects the real-time input rotational speed and torque data of gearbox 1. The rotational speed and torque sensor 6 located between gearbox 1 and powder brake 4 collects the real-time output rotational speed and torque data of gearbox 1.
Input rotational speed levels are set as 250, 300, and 350 rpm. Under each input rotational speed condition, the output torque applied by powder brake 4 increased gradually from 0 to approximately 600 N·mm.

Experimental results and analysis
The total power losses of the gearbox are expressed as Eq. (24): where in T and out T are the measured input and output torques; and in  and out  are the measured input and output angular velocities, respectively.
Total power losses total P consist of the loaddependent power losses load P and load-independent power losses spin P . spin P was measured under the non-loading condition, and the measured results The calculated and measured meshing efficiency under different input rotational speed conditions are shown in Figs. 14(a)-14(c) respectively. Measured meshing efficiency is indicated by the dashed curves and the calculated meshing efficiency is indicated by the solid curves. Under all set rotational speed and loading conditions, the measured meshing efficiency values ranged between 99.5% and 100% and correspond with the calculated meshing efficiency values with maximal deviation of less than 1%. This confirms the validity of the proposed calculation method for the friction coefficient and meshing efficiency of plastic LG pair under dry friction conditions. The fluctuation of the measured results and the deviations between measured and calculated results were attributed to the errors in data collection and processing, manufacturing errors of the POM PLGP specimens, and rolling friction power losses, which were neglected in the calculation procedure.

Discussion
Metal gears cannot be used in some practical situations. For example, since the electromagnetic interference caused by metallic materials can deteriorate the communication quality, plastic gear pairs must be utilized in the RCU transmission system to adjust the tilt angle of the base station antenna instead of metal gear pairs. POM has the advantages of high strength, low hygroscopicity, good wear resistance, and is capable of adapting to complicated outdoor operating conditions. Therefore, POM gear pairs are commonly used in the current RCU transmission system.
Temperature is regarded as an important factor influencing the meshing efficiency of the POM gear pair. Frictional and hysteresis losses during the meshing process are the two sources causing temperature rise of the POM gear pair without considering of impact of ambient temperature, and temperature rise mainly generates from frictional losses since the latter is negligible [5,32]. However, both the calculated and measured results of the meshing efficiency of the POM PLGP indicate that the frictional losses are extremely small. Thus, there is no significant temperature rise compared to other types of POM gear pairs during the low power meshing process. The meshing efficiency of POM involute gear pair is around 93% when rpm 1 300 n  and N mm 2 1,000 T   [33], that of POM sine-curve gear pair is around 94% when  1 n rpm 300 and N mm 2 1,000 T   [33], but that of POM PLGP is higher than 99% when rpm 1 300 n  and N mm 2 500 T   . This paper focuses on the meshing efficiency of POM PLGP under normal use which mainly correlates to load, rotational speed, and misalignment. It should be noted that significant wear and deformation may occur on the tooth surfaces under a long-term operation that increases the relative sliding between the meshing surfaces, leading to frictional losses and temperature rise. Therefore, the effect of temperature must be considered in further studies of wear and fatigue failure of POM PLGP under long-term operation. For higher precision of the meshing efficiency calculation, the effects of the manufacturing errors of tooth surfaces must be considered in building the geometric and mechanical models of plastic LG pair.

Conclusions
Taking POM PLGP as an example, this paper proposed calculation methods and formulas for the friction coefficient and meshing efficiency of plastic LG pair under dry friction conditions. The main conclusions can be summarized as follows: 1) From the results of the ball-on-disk experiment, it was found that the friction coefficient of the POM specimens varied with the average contact pressure and relative sliding velocity, and ranged between 0.35 and 0.45 under the experimental conditions studied.
2) The calculated meshing efficiency is almost in coincidence with measured meshing efficiency, which validated the feasibility of the calculation methods and formulas for the friction coefficient and meshing efficiency of POM LG pair under dry friction conditions.
3) The study can serve as a reference for the calculation of the friction coefficient and meshing efficiency of plastic gear pairs under dry friction conditions. It also provides a basis for future studies of the wear calculation and lifetime prediction of plastic LG pairs.