Influence of the movement of involute profile gears along the off-line of action on the gear tooth position along the line of action direction

movement of gears along OLOA is caused by a frictional force that can be high during tooth degradation e.g. scuffing. Results show that when the movement of gears along the OLOA direction is significant, its influence on the distance between the mating teeth should not be neglected. Highlights Abstract Algorithm for finding the exact value of normal • backlash is presented. Influence of gear parameters on normal backlash • for gear OLOA displacement is checked. Dynamic simulation is performed and several as-• pects are analyzed.


Introduction
A general trend in simulation models for mechanisms and machines is to achieve an accurate solution in acceptable time.Gears and gear units are very often analyzed because of their importance [3,26].Analytical models are efficient and based on basic principles ensuring the correctness of results for a given condition [5].Mechanisms are very often described by equations of motion that do not have an exact solution [18].In effect, numerical methods must be employed to resolve this problem, and numerical models can be called as semi-analytical.Numerical models are very popular, the dominant numerical technique being finite element analysis (FEA) [16].FEA models are usually very detailed and can include different types of phenomena.A disadvantage of these models is their very long computation time.To overcome this drawback, hybrid models [12] are used.
All model types are developed to minimize their disadvantages.In this study, analytical formulas are derived to describe spur gears in a more precise and detailed way that can be used in analytical, semianalytical and hybrid models.A common practice in modeling is to orient the axes of a coordinate system according to the line of action (LOA) and off-line of action (OLOA).This simplifies calculations because then both the normal force and the friction force do not consist of two components.Also, if the center distance is changed along LOA, the same displacement is made between the surfaces of the in-volute profile teeth that are in mesh.As far as gear movement along OLOA is concerned, its impact on the meshing tooth surface distance is neglected in most studies [4,6,17,21,24,27].In this study, the relationship between the varying distance of gears along OLOA and its effect on the distance between the teeth in mesh is established.Obtained equations are precise for involute gears and no simplifications were made to derive them.To the author's best knowledge, this is a novel solution.
In some studies, the problem of varying center distance and its influence on gear parameters is investigated.A change in the distance between the mating teeth can be considered as varying backlash.In the LOA direction it is normal backlash.In study [13] a single stage spur gear is analyzed.Geometric eccentricity and its influence on backlash are considered.The change in backlash is calculated according to the formula ∆b r α where α is the theoretical meshing angle and the theoretical meshing angle ′ α is 20°.The term "meshing angle" is ambiguous.This formula is also used in [25] and the previous study is quoted as a source.In the study, bearing deformation is the reason for changing the center distance and the influence on backlash is included.Time-varying backlash is where b ∆ is the dynamic backlash, α 0 denotes the initial pressure angle for the initial center distance 0

R R d is the actual pressure
When gears change their distance along the off-line of action (OLOA) direction, this affects the distance between the working surfaces of the meshing teeth along the line of action (LOA).This effect is usually neglected in studies.To include this effect precise equations are derived for spur gears.The analysis is carried out for the general case of spur gears with shifted profiles frequently used in the industry.The influence of OLOA gear displacement on LOA direction is also a function of gears parameters.An analysis is conducted, and the impact of parameters such as module, pressure angle, gear ratio, and the number of teeth is determined.As an example, a simulation of a 12 DOF analytical model is presented.The movement of gears along OLOA is caused by a frictional force that can be high during tooth degradation e.g.scuffing.Results show that when the movement of gears along the OLOA direction is significant, its influence on the distance between the mating teeth should not be neglected.

Highlights Abstract
Algorithm for finding the exact value of normal • backlash is presented.
Influence of gear parameters on normal backlash • for gear OLOA displacement is checked.
Dynamic simulation is performed and several as-• pects are analyzed.
Influence of the movement of involute profile gears along the off-line of action on the gear tooth position along the line of action direction

Łukasz Jedliński a
a Lublin University of Technology, ul.Nadbystrzycka 36, 20-618 Lublin, Poland angle.According to these equations, the backlash b ∆ depends on two parameters: the pressure angle and the center distance.This formula is, however, incorrect.For simplicity, let us consider the movement of one gear.Along the LOA direction, changes in center distance and pressure angle are not significant but have the greatest impact on backlash.In contrast, the movement of a gear along the OLOA direction has a significant impact on center distance and pressure angle, but its impact on backlash is minimal.This stands in contradiction with the above formula, and results will differ by several orders of magnitude, according to the author's calculations.In [7] the effect of eccentricity on backlash is investigated.Time-varying backlash for a driving gear caused by eccentricity is expressed as ∆b e = − ± ( ) α , where e and θ are the gear eccentricity and its initial phase, ϕ is the angular displacement of gear, and α is the pressure angle.This is a simplified equation.Pressure angle is maintained constant and changes in backlash do not have an exact cosine shape.According to Fig. 1 given in the reference article, the initial phase angle θ is measured from the line which connects the axes of gear rotation.Assuming that θ 1 and ϕ 1 are equal to 0, the displacement of gears will take place along a direction passing through the axes of rotation.Backlash will reach the maximum value, which is not the case with involute gears.Large bearing clearance and variations in backlash were reported in [14].The relationship between center distance and backlash was established with approximation of tooth profile.The involute curve was treated as a line.A planetary gear unit used in a turbo-fan engine is studied in [22].The model presented in the work is comprehensive and many of its parameters are made depended on time.One of them is backlash.The authors derived the formula for calculating backlash from [13], which is incorrect.
From the above paragraph, it can be concluded that information about normal backlash as a function of gear displacement is of vital importance for the model to be more accurate and comprehensive.The relationship between the movement of gears along OLOA and the distance between the mating tooth surface along LOA (changes in normal backlash) is especially important when the movement (displacement) of gears along OLOA is considerable.Suitable conditions are ensured when e.g.bearing clearance, eccentric gear movement and high friction force are considered in analysis.These three cases will be discussed for different displacement values.An analysis of a rotor-bearing-pedestal system was presented by Cao et al. [1].With the bearing fit clearance equal to 10 µm and unbalance mass on the rotor, the axis orbit can have a diameter of about 60 µm.The problem of clearance between the rolling bearing outer ring and housing was modelled and analyzed by Chen and Qu [2], who considered a fit clearance of up to 500 µm.Radial internal clearance in the rolling bearing was set to 20 µm in [19,20].In a model for analyzing the vibration behavior of a rotor-bearing system, Wang and Zhu [23] set an internal clearance of the cylindrical roller bearing at 60 µm in compliance with ISO 5753-1:2009.Grade accuracy has a great impact on the runout of gears.According to the ISO 1328-2 Cylindrical gears -ISO system of accuracy [9], the runout for gears with a diameter of up to 125 mm can amount to about 200 µm for grade 12. Variations in the center distance reported in [25] amount up to 30 µm and those reported in [22] up to 120 µm.The gear friction coefficient is high when failure occurs.Insufficient lubrication or lack thereof may be its cause.This can lead to scuffing [8,15].During this process the friction coefficient has a high value.
For these reasons, it is important to take into account the influence of gear movement along OLOA on distance between the meshing teeth along the LOA direction (normal backlash).An accurate algorithm is derived and an analysis is carried out.In the first simulation the impact of gear parameters such as module, pressure angle, gear ratio and the number of teeth on the distance between the meshing teeth is examined.The other simulation compares results obtained with and without taking into account the movement of gears along OLOA on the dynamic meshing force and bearing force.

Calculation of the dependency between the movement of gears along OLOA on the contact point on LOA
A change in the position of gear axis of rotation along the OLOA direction has impact on the distance between the meshing teeth.From the point of view of dynamic analysis, it is important to know a formula describing changes in the tooth distance along LOA depending on the movement of gears along OLOA.This affects dynamic forces.For clarity of the figure, below is given an example of the displacement of pinion axis of rotation when the center distance is increased without gear movement.Relationships will be derived for a general case describing the displacement of two gears or one gear only.
The nominal position of the gears is marked with a blue dashed line (Fig. 1).The teeth are in mesh at the pitch point C on LOA.According to Fig. 1, the pinion is displaced by a value p y in the y axis direction.In effect, the distance between the gear axes is increased.To calculate a new center distance 1 w a , it is convenient to divide the pinion displacement p y into two separate displacements, p e and p f , according to a system of coordinates with the e and f axes: where:  Changes in the center distance of gears have impact on tooth meshing conditions.The green line marks the new LOA.In the magnified image in Fig. 1 one can clearly see a clearance between the mating gear teeth (distance 1 2 PP ) at the contact point C before pinion displacement.
For convenience, the distance 1 2 PP can be divided into two sections.One is the pinion tooth distance 1 P from a new contact point 1 C , while the other is the gear tooth distance 2 P from the contact point 1 C .Fig. 2 shows the pinion along with the relationships enabling the determination of the 1 1 PC distance.It should be stressed that the figure is not drawn to scale due to the fact that actual displacements are very small.The angle ζ between the line connecting the gear center distance before displacement 1 2 OO and after displacement 11 2 OO is equal to:

C
The distance 1 1 PC is equal to the arc length  UU .This is due to the fact that the involute profile pitch is constant (involutes generated on the base circle are spaced by a constant distance measured tangentially to this circle).

PC x r py
where β is given in radians.
A similar approach can be adopted to calculate the distance 2 1 PC .By knowing the distance 1 1 PC , the distance between gear tooth profile and new contact point 1 C can be calculated as: The total displacement resulting from the displacement of gears is: Formulas ( 4), ( 5) and ( 6) describe simultaneous displacement of two gears, one gear and a pinion.The displacement can be positive or negative about the y axis (OLOA).The gear teeth in contact can take any position on LOA.In all cases, whether the gear axis displacement along OLOA causes a decrease or increase in the center distance, this always results in a clearance between the mating teeth.
The above formulas were derived for a general case in which the gears have shifted profiles.If the gears are without this correction, the following parameters simplify to: w a a = , α α w = , w r r = .

Simulation of the influence of gear parameters on the total displacement x pgy
The displacement of gears along the OLOA direction can have different effects on the resultant distance between the mating teeth along LOA pgy x .Tooth size and shape, contact ratio and center distance are the main parameters affecting pgy x and thus will be investigated in this study.The following properties of gears were selected for simulations, depending on the case under analysis: m = 3 mm, a w = a = 100 mm, z p = 20, z g = 20, α w = 20°.In all five cases (Fig. 3-7) only the pinion was displaced (y p = -200 µm ÷ 200 µm) along OLOA, which affected the nominal center distance.
Figure 3 illustrates the influence of module m.On changing this parameter, the center distance, gear diameters and tooth height change significantly too.The smaller the value of the module is, the greater the total displacement pgy x becomes.This relationship is nonlinear.The maximum value pgy x 2.7 µm is obtained for m = 1 mm and y p = − 200 µm or 200 µm.
The influence of the gear ratio u is presented in Figure 4. Different values of the gear ratio u are obtained by changing the number of gear teeth z g = 16 ÷ 105, with the number of pinion teeth maintained constant at z p = 20.By changing the number of gear teeth, the gear diameter, center distance and contact ratio change, too.The smaller value of the gear ratio is, the greater the total displacement pgy x becomes.This relationship is nonlinear.The maximum value pgy x 1.1 µm is obtained for u = 0.8 and y p = -200 µm or 200 µm.
The pressure angle α is another investigated parameter.In this case, other parameters do not change like in previous cases, the only exception being the base circle dimension.The smaller the pressure angle value is, the greater the total displacement pgy x becomes.This relationship is nonlinear (Fig. 5).The maximum value pgy x 1.8 µm is obtained for α = 11° and y p = -200 µm or 200 µm.
The last examined parameter is the number of the gear teeth z p , z g .Their number is the same (z p = z g = 18 ÷ 60) and has impact on the dimension of gears and center distance.To obtain more general results, Fig. 3. Relationship between module m, pinion displacement y p and total tooth displacement x pgy , presented in three types of diagrams two variants are considered.In the first one (Fig. 6), the center distance a has a constant value of 100 mm, hence the module m must be changed.The total tooth displacement x pgy does not depend on the number of teeth in this case.
In the second variant, the module m is maintained constant at 3 mm, therefore the center distance must vary.Under these conditions, the total tooth displacement x pgy depends on the number of gear teeth.The smaller the number of teeth is, the greater the total displacement pgy x becomes.This relationship is nonlinear (Fig. 7).The maximum value pgy x 1.1 µm is obtained for z p = z g = 18 and y p = -200 µm or 200 µm.
It can be seen that the total tooth displacement x pgy is not the same despite the identical pinion displacement y p along the positive and negative sense of the y axis, which is especially visible in Figure 5.If the pinion displacement along OLOA causes a decrease in the center distance (negative value), its impact on the total tooth displacement is greater.The same relationship can be observed for the gears.It makes no difference whether one gear or two gears move along the OLOA direction.The resulting center distance is a factor affecting the total tooth displacement x pgy .This conclusion can be drawn from Equation (1).

Simulation of spur gears for different values of tooth friction coefficient and bearing stiffness
To analyze the influence OLOA displacement of gears on their dynamics, a simulation was performed.One of the situations in which gear displacement along OLOA can be significant is when the force in a radial direction is high.This situation occurs during scuffing.The tooth friction force can achieve significant values as a result of this phenomenon.Nine cases of friction coefficient µ were considered with a step changed every 0.1, from 0.02 to 0.82.Bearing stiffness has a great impact on gear displacement, too.Four values of the bearing stiffness k b were analyzed: 1.1 • 10 8 N/m; 1.1 • 10 8,5 N/m; 1.1 • 10 9 N/m; 1.1 • 10 9,5 N/m.The stiffness values were the same for all bearings and in all directions.Parameters of gear unit and other components are presented in Table 1 and 2.
The simulation was performed on a 12 DOF model (Fig. 8).The model consisted of rigid elements.Every shaft had 5 DOF.The gears were located in the middle of the bearings.The gear unit was connected by couplings with a motor and output device.Dynamic equations were as follows: Detailed information about tooth stiffness, Coulomb friction and other details concerning the analytical model can be found in [10,11].The reaction force in Fig. 9a is slightly higher than the nominal value for the coefficient of friction ranging from 0.2 to 0.22.For a higher coefficient of friction, the reaction force decreases below the nominal value (nominal value means, that result is obtained without taking into account influence of OLOA displacement of gears on LOA direction).The situation changes when the bearing stiffness is increased (Fig. 9b).For this case, the reaction force is always higher than the nominal value.The difference in the reaction force values in Fig. 9c is very small and in Fig. 9d it is negligible.
The resultant meshing force is a load on the bearings, thus the results in Fig. 9 and Fig. 10 show very similar trends.The maximum difference between the resultant meshing force and the nominal value is 12.1% for the bearing stiffness of 1.1 • 10 8,5 N/m (Fig. 10b).For the bearing stiffness equal to 1.1 • 10 8 N/m the difference in the resultant meshing is 5.1% (Fig. 10a), while for the bearing stiffness of 1.1 • 10 9 the difference is 18.1% (Fig. 10c).In Fig. 10d

Conclusions
This study investigated the effect of varying the center distance along OLOA on the gear tooth position along LOA.An exact formula has been derived for a general case of spur gears with shifted profiles.It has been found that changes in the nominal center distance result in an increased distance between the working surfaces of the gear teeth, i.e. normal backlash.The presented method for determining the distance between the working tooth surface along LOA (normal backlash) is suitable not only for the OLOA direction, but for any other directions, too.
A simulation was performed to establish the relationship between gear parameters and total tooth displacement.It has been found that the module has the greatest impact out of all tested parameters.The second highest result was obtained for the pressure angle.Given that most gear parameters are interdependent, it is not easy to formulate general conclusions.Nonetheless, the movement of gears along the OLOA direction has a greater impact on the movement of the mating teeth along LOA for small gears with a lower gear ratio and a smaller number of teeth.
The effect of the total displacement x pgy on the dynamic behavior of gears was investigated.Based on an analytical model of reaction forces for bearings, resultant meshing force and waveforms of total displacement x pgy were presented.The reaction forces of bearings and the resultant meshing force are strictly interdependent, and the trends obtained for these two parameters are very similar.In the presented example, the reaction forces were higher by more 12 %, and the resultant meshing force was higher, too.The trends are not linear, so a higher frictional force does not always mean that the bearing reaction forces and resultant meshing force will be higher too.It has been found that bearing stiffness has a great impact on the total displacement x pgy and dynamic behavior of gears.

T
-input motor torque [Nm] d T -output device torque [Nm] m I -mass moment of inertia of the motor rotor and half of coupling [kg m 2 ] p I -mass moment of inertia of the pinion, shaft and half of coupling (pinion subassembly) [kg m 2 ] g I -mass moment of inertia of the gear, shaft and half of coupling (gear subassembly) [kg m 2 ] d I -mass moment of inertia of the device rotor and half of coupling [kg m 2 ] px I ( px py II = ) -mass moment of inertia of the pinion, shaft and half of coupling about op y axis [kg m 2 ] gx I ( gx gy II = ) -mass moment of inertia of the gear, shaft and half of coupling about the og y axis [kg m 2 ]  ϕ -angular acceleration [rad/ s 2 ]:  ϕ m -motor rotor,  ϕ p -pinion,  ϕ g -gear,  ϕ d -device rotor  θ px -angular acceleration of the pinion about the op y axis [rad/ s 2 ]  θ gx -angular acceleration of the gear about the og y axis [rad/ s2]  θ py -angular acceleration of the pinion about the op x axis [rad/ s2]  θ py -angular acceleration of the gear about the og x axis [rad/s2]  x p -linear acceleration of the pinion on plane defined by the op x axis and pinion axis of rotation [m/s 2 ]  x g -linear acceleration of the gear on plane defined by the og x axis and gear axis of rotation [m/s2] py x -distance between new contact point and pinion tooth flank along LOA (x) caused by the movement of gears along OLOA (y) [m], gy x -distance between new contact point and gear tooth flank along LOA (x) caused by the movement of gears along OLOA (y) [m],  y p -linear acceleration of the pinion on plane defined by the op y axis and pinion axis of rotation [m/s 2 ]  y g -linear acceleration of the gear on plane defined by the og y axis and gear axis of rotation [m/s 2 ] on the motor coupling from damping [Nm] on the motor coupling from stiffness [Nm] on the pinion from damping [Nm] on the pinion from stiffness [Nm] on the gear from damping [Nm] on the gear from stiffness [Nm] on the device coupling from damping [Nm] on the device coupling from stiffness [Nm] fp f fp M Fr = -torque applied on the pinion from tooth friction [Nm] fg f fg M Fr = -torque applied on the gear from tooth friction [Nm] f r -moment arm of sliding friction force [m] of bearing 1 from stiffness parallel to the x(LOA) axis [N] (Subscript 2, 3, 4 -bearing 2, bearing 3, bearing 4)  -reaction force of bearing 1 from damping parallel to the x(LOA) axis [N]

w a = 1 2 2 O==
OO -center distance of gears with shifted profiles, -point of intersection with gear axis of rotation, cosα -displacement of gear axis of rotation about the f axis, f y g g w = cosα -displacement of gear axis of rotation about the f axis, sinα -displacement of gear axis of rotation about the e axis.

Fig. 1 .
Fig. 1.Pinion displacement by a value p y about the y axis, leading to a change in pinion axis of rotation position from 1 O to 11 O pressure angle after pinion displacement (for the center distance 1 w a ) [rad], ˆw α is the contact pressure angle before pinion displacement -nominal position (for the center distance w a ) [rad].

Fig. 2 .
Fig. 2. Relationships for determining the pinion profile distance between 1 P

Fig. 8 .Fig. 9 .Fig. 10 .
Fig. 8. Analytical model of gear unit with motor M and output device D one can only observe one slight change for the highest friction coefficient value.Examples of waveforms of the total displacement x pgy are presented in Fig. 11.Different colors mark different values of the friction coefficient.The value x pgy is always positive.The maximum displacement is obtained for the bearing stiffness value equal to 1.1 • 10 8,5 N/m.A straight line in Fig. 11d means that displacement does not occur.