Properties of Elasto-Hydrodynamic Oil Film in Meshing of Harmonic Drive Gears

Among the essential issues facing designers of strain wave gears, the provision for correct lubrication should be of paramount importance. The present paper presents the results of research on elasto-hydrodynamic oil film in meshing of a harmonic drive with an involute tooth profile. The research was carried out based on theoretical models developed by Dowson and Higginson. For the discussed structural problem, results of the study are presented graphically in the form of static characteristics of the oil film. Correct operation regimes were determined for two different oils. The paper also provides a review of information concerning the design and principle of operation of strain wave transmission.


Modern Drive System Design Solutions
Dynamic development of technology has resulted in the automation and robotization of not only manufacturing processes, but also many aspects of everyday life. A good example here is a device for neurorehabilitation, the prototype of which together with the results of its experimental examination was presented by Yamine et al. [1]. The device, equipped with two drives and transmissions providing a reduction rate of 49:1, was designated to support the process of rehabilitation in patients with physical motor disabilities, which made it difficult for them to carry out everyday activities. It is also worth mentioning other devices intended for operation in outer space or on other planets [2].
In view of the above, it is advisable to improve and adapt existing transmission designs and develop new ones, capable of meeting the sophisticated requirements demanded from modern drive systems. Such requirements include, among other things: high accuracy and precision in the transfer of motion; the possibility to obtain large transmission ratios at small overall dimensions of the gear; high efficiency; operability in extreme conditions (e.g., low temperatures [3]); and the overall high culture of drive train operation. The following may serve as examples of research projects aimed at the development of existing solutions: -The use of duplex worm gears with an adjustable backlash in the joints of robot manipulator arms [4]. The paper presents the results of research on the improvement of motion transfer precision in the drive and minimization of vibrations which were obtained as a result of the reduction in play in meshing by means of the use of a special housing; -Modal analysis of a planetary transmission gear [5]. The paper quotes results of numerical analyses carried out in the ANSYS Workbench environment aimed at the determination of frequencies and forms of natural vibrations of components making up a planetary transmission. The possibility of occurrence of the teeth profiles interference effect, because the gears have small modules with a small backlash in the meshing.
The subject of the present study is a harmonic drive with a double-wave elliptical cam generator. For the assumed input quantities such as the load and contact geometry, the theoretical model proposed by Dowson and Higginson was used to determine properties of the elasto-hydrodynamic oil film occurring in meshing of toothed wheels. The results presented in this paper were obtained as part of a wider research project concerning the properties of harmonic drives. Trajectories of the displacement of points representing the position of a flexspline tooth axis relative to the tooth space of the circular spline are presented in Figure 2. The trajectories depend on, among other things, the design of the generator. In the discussed design solution, the output shaft rotation sense will be opposite to that of the input shaft. On the input shaft, a double-wave generator is fixed. In the case of a strain wave gear with double-wave generator, a single full turn of the input shaft will result in displacement of the rigid circular spline by two tooth spaces. The number of waves, denoted z w , is the number of engagement areas and is an assumed number, based on which the number the number of flexspline teeth z 1 and circular spline teeth z 2 is determined:

The Harmonic Drive Design and Principle of Operation
The drive reduction ratio i r is calculated from the relationship: Trajectories of the displacement of points representing the position of a flexspline tooth axis relative to the tooth space of the circular spline are presented in Figure 2. The trajectories depend on, among other things, the design of the generator. Trajectories of the displacement of points representing the position of a flexspline tooth axis relative to the tooth space of the circular spline are presented in Figure 2. The trajectories depend on, among other things, the design of the generator. The effect of selected harmonic drive parameters on the form of the trajectories discussed above is discussed in the studies reported in articles written by Mijał, Ostapski, Kalina et al. [11][12][13][14]. The effect of selected harmonic drive parameters on the form of the trajectories discussed above is discussed in the studies reported in articles written by Mijał, Ostapski, Kalina et al. [11][12][13][14].

Harmonic Drive Lubrication Methods
Strain wave gears can be lubricated with plastic grease or with oil. These two lubrication methods are used typically in applications of drive to manipulators and robots. Table 1 presents a summary of information concerning the grease application recommended by some manufacturers of strain wave gears (Harmonic Drive ® (Limburg an der Lahn, German) [2] and Laifual Drive (Zhejiang, China) [15]).
The compactness of the structure characterizing strain wave gears is an argument in favor of the use of plastic grease. Limited space inside the transmission and internally meshed splines allows introduction of the lubricant directly into the tooth spaces of transmission gears. Tooth space bottom function as lubrication pockets and the smallness of space creates favorable conditions for achieving proper conditions of lubrication. Moreover, grease is applied inside the flexspline body (lubrication of surfaces engaged with the outer race of the flexspline bearing) and onto non-working surfaces of the flexspline sleeve (countering of corrosion of abutment surfaces). The plastic grease quantity and application method depend on the conditions under which the drive is operated [2]. Table 2 shows the relationship between the maximum admissible input shaft speed and the lubricant used to lubricate series CFS harmonic drives [2]. Table 1. Plastic greases used for strain wave gears.

Manufacturer
Grease Operating Temperature Range It follows from the information contained in Table 2 that the drives lubricated with oil can be operated at higher input shaft speeds compared to gears lubricated with the use of plastic grease. Transmissions lubricated with grease operate in the mixed friction regime, whereas those in which oils are used as a lubricant operate in the fluid friction area. The effect of the lubricant on the coefficient of friction is shown in Figure 3 [16].  In the meshing, the lowest values of the coefficient of friction ( Figure 3) are obtained in fluid friction conditions. Properties of the oil film depend significantly on the type of the oil used. An additional advantage of lubrication of that type is the possibility to carry away the heat and contaminants from the gear engagement area together with the flowing oil. Table 3 lists some of the oils intended for the lubrication of harmonic drives [2].  the oil used. An additional advantage of lubrication of that type is the possibility to carry away the heat and contaminants from the gear engagement area together with the flowing oil. Table 3 lists some of the oils intended for the lubrication of harmonic drives [2].

Mathematical Models Used to Describe the Oil Film
The point of departure for many contemporary research projects on the issue of hydrodynamic (HD) and elasto-hydrodynamic (EHD) lubrication in gear transmissions is a study carried out by a team led by Dowson [17]. The presented model was repeatedly modified in view of its usefulness in the description of lubrication of, among other things, roller bearings and plain bearings [18,19]. Within the framework of further research, models intended for the description of lubrication conditions in the engagement area have been developed, which were also based on the results published by Dowson et al. [17]. Examples of such studies include publications concerning, among other things, lubrication in transmission gears: -With an involute teeth profile [20,21]; -Bevel gears [22,23]; -Hypoid gears [21,24]; -Spiroid gears [25]; -Worm gears [26][27][28][29].
Available in the literature are studies concerning the use of oils to lubricate strain wave transmissions which, however contrary to the references quoted above, do not present any models capable of describing the properties of the oil film. This follows from, among other things, the difficulties which are posed by the description and experimental verification of the results obtained from theoretical models. Moreover, the harmonic drives, in view of their specific structure, typically have a very small inner space which hinders direct observation of phenomena occurring in the course of engagement of toothed rims. It is also worthwhile mentioning that the design of strain wave gears and factors connected to the specificity of their application (drive systems for robots) favors the selection of plastic greases as lubricants. In view of the above, studies concerning lubrication in harmonic drive gears concern: -Research on properties of lubricants intended for strain wave gears. Special attention is deserved here to the studies concerning the lubrication of harmonic drive gears designed for operation in outer space [30,31]; -The tear and wear of the transmissions and selected tribological aspects [32,33].
Results of theoretical analysis indicate, therefore, that there are good reasons to carry out research aimed at the development of mathematical models capable of describing the parameters of the oil film generated in the area of engagement of harmonic drive gears. This paper presents the algorithm used in the authors' original model, allowing description of the properties of the oil film in the meshing of a strain wave gear together with results of analysis of the effect of input shaft speed on the oil film minimum height at the selected transmission operating point.

Assumptions and Boundary Conditions
Main assumptions adopted for the developed physical and mathematical model:

•
The problem was considered in planar reference systems; • To describe the geometry and kinematics of meshing, two reference systems were adopted, namely XOY (with the origin situated on the generator shaft axis) and designed for operation in outer space [30,31]; -The tear and wear of the transmissions and selected tribological aspects [32,33]. Results of theoretical analysis indicate, therefore, that there are good reasons to carry out research aimed at the development of mathematical models capable of describing the parameters of the oil film generated in the area of engagement of harmonic drive gears. This paper presents the algorithm used in the authors' original model, allowing description of the properties of the oil film in the meshing of a strain wave gear together with results of analysis of the effect of input shaft speed on the oil film minimum height at the selected transmission operating point.

Assumptions and Boundary Conditions
Main assumptions adopted for the developed physical and mathematical model: • The problem was considered in planar reference systems; • To describe the geometry and kinematics of meshing, two reference systems were adopted, namely XOY (with the origin situated on the generator shaft axis) and X1O1Y1 (the origin of which is translated in the direction of the Y-axis by the value of Rnu equaling the length of the radius of the neutral layer of non-deformed flexspline). The systems XOY and X1O1Y1 are depicted in Figure 4. • The immovable rigid circular spline was oriented in the reference systems XOY and X1O1Y1 in such way that the rigid spline coincided with the direction of axes Y and Y1; Figure 4. Reference systems XOY and X 1 O 1 Y 1 adopted to describe the geometry and kinematics of harmonic drive meshing.

•
The immovable rigid circular spline was oriented in the reference systems XOY and X 1 O 1 Y 1 in such way that the rigid spline coincided with the direction of axes Y and Y 1 ; • The flexspline tooth position as a function of the generator rotation angle ϕ G was determined in terms of two characteristic points M and N which define the position of the flexspline tooth axis. Point M is situated at the intersection of the flexspline tooth axis with the addendum circle, whereas N is situated at the intersection of the tooth axis with the root circle; • In view of the symmetry and cyclic nature of the curve describing the generator cam profile shape, the research was carried out for the generator position angle ϕ G varying in the range ϕ G < 0,π/2 >; • Trajectories of points M and N are symmetrical relative to the tooth space axis; • The oil viscosity function and the oil density function take into account the effect of pressure and temperature; • The oil flow in the engagement area is consistent with the peripheral direction, whereas the oil flow in both axial and radial directions is negligibly small; • The heat is carried away from the contact zone by the engaged surfaces of toothed wheels and/or the flowing oil; • The contact surfaces are considered perfectly cylindrical and smooth; • At boundary surfaces, the slip effect does not occur-velocity of the oil boundary layer and of the tooth surface was the same; • Linear contact of teeth was assumed on the whole width of the toothed rims; • The engagement areas are symmetrical and equally loaded; • Calculations were carried out for the steady-state operation of the gear transmission; • Thermal expansion of toothed rims has no effect on the shape of the flexspline tooth relative path; • The model does not take into account the effect of oil anti-wear additives; • For each generator position angle ϕ G at which the engagement of toothed rims occurs, a substitution model of the contact is constructed consistent with Figure 5.
• The heat is carried away from the contact zone by the engaged surfaces of toothed wheels and/or the flowing oil; • The contact surfaces are considered perfectly cylindrical and smooth; • At boundary surfaces, the slip effect does not occur-velocity of the oil boundary layer and of the tooth surface was the same; • Linear contact of teeth was assumed on the whole width of the toothed rims; • The engagement areas are symmetrical and equally loaded; • Calculations were carried out for the steady-state operation of the gear transmission; • Thermal expansion of toothed rims has no effect on the shape of the flexspline tooth relative path; • The model does not take into account the effect of oil anti-wear additives; • For each generator position angle φG at which the engagement of toothed rims occurs, a substitution model of the contact is constructed consistent with Figure 5.

Algorithm of the Developed Method
In order to determine the properties of the oil film in the gear engagement area of a harmonic transmission it is necessary, first of all, to define and determine values of the quantities which are required for further calculations. Complexity of phenomena involved in the specific engagement of toothed rims in strain wave transmissions was taken into account in developing the calculation models, concerning:

Algorithm of the Developed Method
In order to determine the properties of the oil film in the gear engagement area of a harmonic transmission it is necessary, first of all, to define and determine values of the quantities which are required for further calculations. Complexity of phenomena involved in the specific engagement of toothed rims in strain wave transmissions was taken into account in developing the calculation models, concerning: The above-listed quantities depend, among other things, on the generator position angle ϕ G , and change for each value of the quantity. For that reason, a method was developed allowing determination of the parameters as functions of the angle ϕ G . The corresponding algorithm is presented in Figure 6. The above-listed quantities depend, among other things, on the generator position angle φG, and change for each value of the quantity. For that reason, a method was developed allowing determination of the parameters as functions of the angle φG. The corresponding algorithm is presented in Figure 6.

Elasto-Hydrodynamic Oil Film in Meshing of Harmonic Drive Gears
The problem of elasto-hydrodynamic isothermal oil film in the meshing of harmonic drive gears can be described with the use of the following set of equations [16-18,34]: • Pressure distribution in the spline engagement area:

Elasto-Hydrodynamic Oil Film in Meshing of Harmonic Drive Gears
The problem of elasto-hydrodynamic isothermal oil film in the meshing of harmonic drive gears can be described with the use of the following set of equations [16][17][18]34]: • Pressure distribution in the spline engagement area: where: (Pa) distributed pressure in the circular spline-flexspline contact zone, u 0 (m/s) average oil stream velocity in the teeth clearance given by the formula [34]: • Stress in the flexspline tooth-circular spline tooth contact zone: where: σ Hmax (Pa) maximum normal stress, F (N/m) the load force per the linear contact length given by the formula: reduced curvature radius at the point of contact of a flexspline tooth profile with the circular spline tooth profile given by the formula: • Surface elastic deformation: where: a variable determining the position of load application per unit surface area in the adopted system of coordinates; • Quantities characterizing lubricant properties: where:  One of the quantities characterizing the properties of the oil film is the oil film minimum height. This is the least elasto-hydrodynamic film thickness which occurs at the lubrication gap constriction at the high-pressure zone end (Figure 7). The quantity is determined from the formula [17]: where u0 = (u1 + u2)/2 is the slip speed at the point of contact between the two engaging surfaces. The quantity is calculated as the difference of the tangent velocities u1 and u2 of the engaged surfaces, which are the flexspline tooth side and the circular spline tooth side, respectively. In view of the fact that in this analysis, the rigid circular spline is immobile (u2 = 0), so the slip speed will be: As already mentioned, the oil film minimum height can be used as one of the criteria for assessment of the quality of lubrication provided to the meshed toothed rims. Engagement of teeth will be better, the lower the coefficient of friction is between the teeth side surfaces. To ensure that the gears are engaged with fluid friction, the oil film at the point of its minimum height should be sufficiently thick to separate the engaged surfaces com- Figure 7. A model of elasto-hydrodynamic oil film: η 0 -dynamic viscosity; α-pressure viscosity coefficient; E -reduced Young's modulus; p(x)-pressure distribution curve; p H -Hertz contact pressure distribution curve; h(x)-oil film height; h min -oil film minimum height; F 0 -load force; u 0 -slip speed [16][17][18]34].
One of the quantities characterizing the properties of the oil film is the oil film minimum height. This is the least elasto-hydrodynamic film thickness which occurs at the lubrication gap constriction at the high-pressure zone end (Figure 7). The quantity is determined from the formula [17]: (12) where u 0 = (u 1 + u 2 )/2 is the slip speed at the point of contact between the two engaging surfaces. The quantity is calculated as the difference of the tangent velocities u 1 and u 2 of the engaged surfaces, which are the flexspline tooth side and the circular spline tooth side, respectively. In view of the fact that in this analysis, the rigid circular spline is immobile (u 2 = 0), so the slip speed will be: As already mentioned, the oil film minimum height can be used as one of the criteria for assessment of the quality of lubrication provided to the meshed toothed rims. Engagement of teeth will be better, the lower the coefficient of friction is between the teeth side surfaces. To ensure that the gears are engaged with fluid friction, the oil film at the point of its minimum height should be sufficiently thick to separate the engaged surfaces completely. Geometrical structure of teeth side surfaces can be described by means of the roughness parameter Rz (µm) (Figure 8).
where u0 = (u1 + u2)/2 is the slip speed at the point of contact between the two engaging surfaces. The quantity is calculated as the difference of the tangent velocities u1 and u2 of the engaged surfaces, which are the flexspline tooth side and the circular spline tooth side, respectively. In view of the fact that in this analysis, the rigid circular spline is immobile (u2 = 0), so the slip speed will be: = 2 (13) As already mentioned, the oil film minimum height can be used as one of the criteria for assessment of the quality of lubrication provided to the meshed toothed rims. Engagement of teeth will be better, the lower the coefficient of friction is between the teeth side surfaces. To ensure that the gears are engaged with fluid friction, the oil film at the point of its minimum height should be sufficiently thick to separate the engaged surfaces completely. Geometrical structure of teeth side surfaces can be described by means of the roughness parameter Rz (μm) (Figure 8). It is assumed that the condition of separation of the engaged surfaces is met when:

Results
Numerical simulations were carried out for a specific generator position determined by a generator position angle ϕ G = 0.2063 rad. The adopted transmission operating parameters together with data concerning material properties of the components are listed in Table 4. Table 4. Adopted parameter values.

Parameter Symbol Value
Input shaft rotational speed n in 1000 rpm Input shaft angular speed ω in 104.7198 rad/s Generator rotation angle ϕ G 0.2063 rad/11.82 deg Flexspline tooth velocity at the contact point, tangent to profile u 1 60.1193 mm/s Load force in the meshing (assumed based on the curve) [34] F 0 240 N Reduced curvature radius at the contact point (for concave surface curvature radius r c1 = 19.2194 mm and convex surface curvature radius contact r c2 = 19.2307 mm- Figure 5 The tests were carried out for oils, parameters of which are presented in Table 5. Table 5. Parameters of oils subjected to testing.

Parameter Symbol Value
Pressure viscosity coefficient     As a result of analysis of the course of function hmin (nin, η0, F0) shown in Figures 9-11, the input shaft admissible speed nadm was determined for which the condition hmin ≥ hadm was met. In Figure 12, the effect of the oil viscosity η0 and the load F0 on the value of the speed nadm is presented.

Discussion
Based on numerical simulations performed with the use of a computer program, the following conclusions can be formulated: As a result of analysis of the course of function h min (n in , η 0 , F 0 ) shown in Figures 9-11, the input shaft admissible speed n adm was determined for which the condition h min ≥ h adm was met. In Figure 12, the effect of the oil viscosity η 0 and the load F 0 on the value of the speed n adm is presented. As a result of analysis of the course of function hmin (nin, η0, F0) shown in Figures 9-11, the input shaft admissible speed nadm was determined for which the condition hmin ≥ hadm was met. In Figure 12, the effect of the oil viscosity η0 and the load F0 on the value of the speed nadm is presented.

Discussion
Based on numerical simulations performed with the use of a computer program, the following conclusions can be formulated: • The oil film minimum height increases with increasing value of the product η0·u0;

Discussion
Based on numerical simulations performed with the use of a computer program, the following conclusions can be formulated:

•
The oil film minimum height increases with increasing value of the product η 0 ·u 0 ; • The increase in the reference viscosity value from η 0 = 0.06 Pa·s to η 0 = 0.295 Pa·s results in a 5.7-fold increase in the oil film minimum height ( Figure 9); • For the oil characterized with the reference viscosity coefficient η 0 = 0.06 Pa·s, fluid friction occurs for the input shaft speed n in exceeding the value of 5670 rpm, whereas for the oil with the reference viscosity η 0 = 0.295 Pa·s, fluid friction can be observed already for n in > 1194 rpm ( Figure 9); • With the decreasing value of force F 0 , the speed value at which the condition h min ≥ h adm is met decreases accordingly. The most significant effect of value of the force F 0 can be observed for the oil with VG68 viscosity grade; • Value of the input shaft admissible speed n adm depends on, among other things: oil parameters (α, η 0 ); the transmission load (F 0 /L); materials of which the toothed wheels were made (E ); and the teeth meshing geometry (R). It should be considered, however, that the average oil stream velocity u 0 , the force F 0 , and the reduced curvature radius R depend on the generator rotation angle ϕ G . That means that the speed n adm is also a function of the angle ϕ G and its value will be different at any working point; • For transmissions operated at lower speeds and higher loads, in view of specific engagement of toothed wheels in harmonic drives, it is recommended to use oils with elevated viscosity. This follows from, among other things, the course of functions plotted in Figure 12; • Selecting the lubricant for a harmonic drive, it is necessary to take into account, among other things, the reduction rate. Transmissions with high rates are characterized by the fact that the oil stream speed values in the cam flexible bearing will be much higher than those in the meshing. It is therefore necessary to select the lubricant type and parameters in a way enabling the formation of an oil film in the cam flexible bearing and in the meshing at the same time.
The obtained results of theoretical considerations and conclusions following from weight are in favor of using oils as lubricants in strain wave gears. It is worthwhile collating the obtained oil film characteristics with those recommended by the manufacturer HarmonicDrive (Table 1) [2]. The oils are characterized by viscosity, corresponding to VG68 grade. However, as can be seen from the plot shown in Figure 12, the use of oil with such viscosity enables generation of a film with the desired height only for n in speed in the range above about 4400 rpm for the load defined by the force in meshing F = 60 N and above 5700 rpm for the force F = 240 N. As shown in Table 2, the transmissions offered by the manufacturer [2], the parameters of which are close to those of the drive examined by us, can operate within this specific range of input shaft speeds. Some thought, however, should be given to the issue as to whether the oil with VG68 viscosity grade is suitable for transmissions operating at lower speeds n in (n in < 2000 rpm). It follows from the performed research that, in such cases, it is recommendable to use oils with higher viscosity, such as VG150 and above, especially in more heavily loaded transmissions. It should also be remembered that with increasing viscosity, properties of an oil become increasingly closer to those of a plastic grease, which triggers another problem-how to select oil parameters for the transmissions with large reduction ratios (i r > 150) in which the speed of the generator is much higher than that of the flexspline. It is, therefore, possible to assume that the transmission ratio is one of the limitations for increasing the oil viscosity.
The presented results concern a single operating point determined by the generator rotation angle ϕ G . For other values of the angle, quantities subject to changes include, among other thing, the contact geometry, the force in meshing, and the average oil stream velocity in tooth clearance. The quantities have an important effect on the oil film minimum height h min . This means that there are good reasons to investigate the film oil properties within the whole range of variability of the angle ϕ G because, according to the publication [35], the resultant velocities at points M and N (Figure 2) are increasing functions of the generator rotation angle. Admittedly, the referenced publications [12,32] present results of research on harmonic transmissions lubricated with oils; however, the studies did not include examination of the effect of the angle ϕ G on the parameters F, u 0 , and R. That creates a gap in the knowledge which should be filled by ways of further research. The model worked out by us will serve, in further stages of research, as a base for the development of software which will enable the determination of oil film properties within the full range of angle ϕ G and present the results in a clear and limpid way.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
List of basic symbols a-contact area half-width (m); E-Young's modulus (Pa); E -reduced Young's modulus (Pa); F 0 -load force (N); F -load force per linear contact length (N/m); h-oil film height (m); h admminimum admissible oil film height (m); h min -minimum oil film height (m); n adm -admissible input shaft speed (rpm); n in -input shaft rotational speed (rpm); p-distributed pressure in the circular spline-flexspline contact zone (Pa); R-reduced curvature radius at the point of contact of a flexspline tooth profile with the circular spline tooth profile (m); R z1,2 -roughness of flexspline and circular spline tooth side surfaces (µm); T 0 -reference temperature ( • C); u 0 -average oil stream velocity in the teeth clearance (m/s); u 1 -flexspline tooth velocity component tangent to tooth profile at the contact point (m/s); u 2 -circular spline tooth velocity component tangent to tooth profile at the contact point (m/s); z 1 -number of flexspline teeth (-); z 2 -number of circular spline teeth (-); z w -the number of engagement areas, which is an assumed quantity (-); α-pressure viscosity coefficient (Pa-1); η-dynamic viscosity (Pa·s); ρ-oil density (kg/m 3 ); ϕ G -generator rotation angle (rad).