Elements of dynamics of geometrically-accurate crossed-axes gear pair with line contact between the tooth flanks

. The paper deals with geometrically-accurate crossed-axes gearing that features line contact between the tooth flanks of a gear, and of a mating pinion. The gearing of this design is commonly referred to « R − gearing ». R − gearing is the only possible kind of crossed-axes gearing, in which the tooth flanks of a gear, and of a mating pinion are in line contact with one another. Contact motion characteristics, and the key elements of dynamics of R − gearing are concisely outlined in the paper. The tooth profile sliding, and sliding in the lengthwise direction of the gear tooth, are covered at the beginning of the paper in the section titled Contact motion characteristics . The key elements of dynamics of R − gearing are discussed in the rest sections of the paper. Here, analytical solution to the problems under consideration is presented. The obtained results of the research are also valid for gear pairs that operate on intersected axes of rotation of a gear, and of a mating pinion. This becomes evident if one assumes the center-distance in the gear pair equal to zero. The discussed results of the research form a foundation for the further analysis of dynamics of real gear pairs that operate on crossing axes of rotation of a gear, and of a mating pinion.


Fundamentals of R-gearing
R − gearing is a novel kind of gearing. Gearing of this design was invented by Prof. S.P. Radzevich at around ~2008. Since that time, the crossed-axes gearing of the proposed design was discussed in numerous papers that are available in the public domain [1][2][3][4][5], and others.
Despite of gearing itself, the key accomplishments in the field of R − gearing are also extensively used in design of gear cutting tools for machining gears for R − gear pairs [6], and others.

Principal kinematics of R-gearing
In a crossed-axes gear pair, a rotation, p  , from a driving shaft is transformed and transmitted to a rotation,

Tooth flank geometry in R-gearing
To generate tooth flanks, G and P , of a gear, and of a mating pinion, in a crossed-axes gear pair, a desirable line of contact, des LC , is used. The desirable line of contact, des LC , is a planar curve that is entirely situated within the plane of action, PA . This curve is commonly specified in the reference system, pa pa pa X Y Z , associated with the plane of action, PA . The desirable lines of contact, des LC , of various geometries are used in present day practice: this could be a straight-line segment, a circular arc, circ LC (see Fig. 2), an arc of cycloid, and so forth.
In a particular case of desirable line of contact in the form of straight-line segment, a line des LC , of one of the following geometries can be employed: ✓ the line of contact of the base cone of the gear, and the plane of action (straight tooth gear is generated in this case); gear pair of this design is referred to as "straight-gear a C − gearing" ✓ the line of contact of the base cone of the pinion, and the plane of action (straight tooth pinion is generated in this case); gear pair of this design is referred to as "straight-pinion a C − gearing" ✓ the axis of instantaneous rotation, ln P ; (neither straight tooth gear, nor straight tooth pinion is generated in this case); gear pair of this design is referred to as "straight a C − gearing" To generate the tooth flanks, G and P , three main coordinate systems are employed. They are: (a) a All three fundamental laws of gearing are met [3] if the tooth flanks fulfil Eq. (1) and Eq. (2).

Contact motion characteristics in Rgearing
Rotation of the input shaft, and rotation of the output shaft in a crossed-axes gear pair cause several other motions. Rolling and sliding of the tooth flanks are among these motions. Sliding is inevitable in crossedaxes gearing. For better understanding of the nature of sliding, an in-detail investigation into the teeth sliding in geometrically-accurate crossed-axes gearing is required to be undertaken.

Tooth flank sliding in R-gearing
Sliding is inevitable in crossed-axes gearing. The velocity of sliding of tooth flanks in crossed-axes gearing can be viewed as a component of the velocity of the relative motion of the tooth flanks when the gears rotate.
In the particular case under consideration, it is convenient to interpret the plane of action, PA (see Fig. 4), as a driving component, and to consider both, the gear, and the pinion, as driven components.
Point m is an arbitrary point of the desirable line of contact, LC , between the tooth flanks, G and P . , can be derived. When a gear pair operates, the driving member, the driven member, and the plane of action, all of them, are rotated in a timely manner, that is, the rotations, g  , p  and pa  , are synchronized with one another.

Sliding in the lengthwise direction of gear teeth
Sliding of two different kinds are distinguished when crossed-axes gearing operates. The tooth flank sliding in the lengthwise direction of the teeth is the first kind of sliding. The tooth flank sliding in the transverse direction of the teeth is the second kind of sliding.
Coordinate system transformations are extensively used in the analysis of sliding of a gear, G , and of a mating pinion, P , tooth flanks.

Analytical solution to the problem
For the analytical description of the linear transformation, namely, for the transition from the reference system, g g g X Y Z , associated with the rotated gear, to the reference system, pa pa pa X Y Z , associated with the rotated plane of action, PA , the operator, () Ca g pa →

Rs
, of the resultant coordinate system is used [3]. A similar operator of linear transformation, () Ca p pa →

Rs
, is derive for the pinion [3].
The projection, . ZA , in a crossed-axes gear pair can be interpreted graphically.

Specific sliding in geometrically-accurate crossed-axes gearing
For the specification of sliding between tooth flanks, G and P , of a gear and of a mating pinion in crossed-axes gear pair, a dimensionless parameter is preferres to be used. This sliding parameter can be also referred to as specific sliding. The specific sliding is denoted by   . An actual value of specific sliding does not depend on rotation of the input/output shafts, and depends only on the design parameters of a gear, and of a mating pinion. The latter is especially important when optimizing the design parameters of gears in crossedaxes gear pairs. Two different parameters,   , are distinguished. First, the slide/roll ratio for tooth flank, G , of a gear. This ratio is calculated from a formula: .. . .
Second, the slide/roll ratio for tooth flank, P , of a pinion. This ratio is calculated from a formula: .. . .
to calculate the specific roll/slide ratios, .g   and . p   , in crossed-axes gearing.
The specific sliding,   , is of a positive value on the addendum portions of the tooth flanks. The parameter,   , does not exceed 1. At points within the axis of instantaneous rotation, ln P , specific sliding,   , is equal to zero, and it is equal to 1 at the base cone of the mating gear.
The specific sliding on the dedendum portion of the tooth flanks is of a negative value. It is equal to zero at points within the axis of instantaneous rotation, ln P , and it approaches a minus infinity at the base cone. The specific roll/slide ratios, .g   and . p   , can be plotted within the zone of action, ZA , as only the region, ZA , of the plane of action comes into effect when investigating the engagement of the gear teeth.

Features of specific sliding in geometricallyaccurate crossed-axes gearing
The interaction of the tooth flanks, G and P , in geometrically-accurate crossed-axes gearing features sliding in the lengthwise direction of gear teeth (see Fig. 4). Therefore, in addition to specific roll/slide ratios, It is right point to turn the readers' attention here to that that gears with a low tooth count are more vulnerable to sliding between the tooth flanks, G and P . They are also more sensitive to the variation of the roll/slide conditions within the zone of action, ZA .

Elements of dynamics of geometrically-accurate crossed-axes gearing
In a crossed-axes gear pair, the input shaft, and the output shaft, are loaded by an input torque, and by output torque, correspondingly. As the gears interact with one another, a force of the interaction is exerted from the driving member of the gear pair. An actual value of the force of interaction, as well, as the components of this force, depend on the input torque, and on the design parameters of the gear, and of the mating pinion. Considering an input torque, and an input rotation of constant values (namely, no acceleration/deceleration is taken into account in the performed below analysis), it is required to determine the forces that act between a gear, and a mating pinion in a crossed-axes gear pair

The principal assumption adopted in the load analysis of R-gearing
When a crossed-axes gear pair operates, the tooth flanks, G and P , of the gear and that of the mating pinion, interact with one another at points within the line(s) of contact, LC . The line(s) of contact is entirely situated within the plane of action, PA . This makes possible a conclusion that the force of interaction between the tooth flanks, G and P , acts along a straight line that is also entirely located within the plane of action, PA . As the line of action of the force is entirely situated within the plane of action, PA , then the following assumption seems to be reasonable in the load analysis of geometrically-accurate crossed-axes gearing.
Referring to Fig. 5, consider base cones of a gear, and of a mating pinion, along with the plane of action. The gear is rotated, g  , about its axis of rotation,  with one another so, as to fulfil the equations: The input torque, p T , is applied to the pinion shaft, and the output torque, g T , is applied to the gear shaft. The torque, pa T , is a virtual parameter. This torque is applied to the rotated plane of action, PA . The ratios of the magnitudes, g T , p T , and pa T , of the torques, g T , p T , and pa T , are inverse to the corresponding ratios of the rotations, g  , p  , and pa  :  In the best-case scenario, the slice thickness approaches to a zero ( 0 pa F → ), and the number of the slices approaches an infinity ( sl N →). In this latter case, the slice thickness is denoted by pa dF . It is assumed here and below that the torque being transmitted by each of the slices is equal to one another. Considering the plane-of-action torque, pa T , the following expression: can be composed for a torque per unit length, pa t . Similar equations: are valid with respect to the gear, as well, as to the mating pinion.
As the torque per unit length, pa t , is of a constant value, and the equality, pa pa pa F = tT , is valid, then the actual value of the plane-of-action torque, pa T (lumped torque), is proportional to the shadowed area in Fig. 6.
Equal torque share among the slices is graphically illustrated in Fig. 6.
In order to transmit a given power, it is always desirable to design and to implement gearboxes of the smallest possible size. From this perspective, the active portion of the line of contact, LC , should begin from the plane-of-action apex, pa A . Evidently, the line of contact of such a geometry is far from to be practical, as the maximum contact and bending strength of the gear teeth is restricted by physical properties of a material the gear and the mating pinion are made of.
Calculation of the design parameters of the favorable portion of the line of contact is based on the assumption that the power being transmitting by a gear pair is equally shared within active portion of the gear pair face width. With that said, under the torque of a constant value, the smaller diameter of a gear/pinion, the large the force, and vice versa. Therefore, a practical value of the smallest possible diameter of the gear/pinion is limited by the yield contact and bending stress in the gear tooth.
The above discussion makes reasonable the following assumption: This assumption is referred to as the first fundamental assumption in dynamics of crossed-axes gearing The first fundamental assumption is derived on the premis of the mandatory equilibrium of the adjacent infinitesimally narrow strips of the plane of action, PA , which must be stationary in relation to one another (and, thus, are not allowed traveling with respect to each other).
A similar assumption has been made with respect to parallel-axes gear pairs, as well, as with respect to intersected-axes gearing.

Forces of the interaction in geometricallyaccurate crossed-axes gearing
The forces that act in a geometrically-accurate crossedaxes gear pair are considered below in different reference systems. The analysis begins with the load applied within the plane of action, PA . Then, this analysis enhanced to the forces that act on the bearings, on the gear housing, and so forth.

Total force acting in geometrically-accurate crossed-axes gearing
In the plane-of-action, PA , the torque, pa T , creates the plane-of-action tangential force,   LC , that is remote from the plane-of-action apex, pa A , at distance, cg r (see Fig. 6). Transmission of a rotation from a driving shaft to a driven shaft is due to the tangential forth, pa F , that acts between the tooth flanks, G and P , of the mating gears.
When the gears rotate, friction is observed between the gear tooth flank, G , and the mating pinion tooth flank, P . The friction between the compressed gear and pinion tooth flanks, G and P , is due to the sliding that occurs between the tooth flanks, G and P . As it is already shown earlier, sliding of two types are distinguished in crossed-axes gear pairs. The profile sliding is the first kind, and the sliding in the lengthwise direction of the gear teeth is the second kind of sliding in crossed-axes gearings. Therefore, friction forces of two types have to be recognized, namely, the profile friction force, pr F , and the friction force in the lengthwise direction, In that same reference system, pa pa pa X Y Z , the friction force in the lengthwise direction, F that act on the gear and the pinion, are entered into equations for the calculation of the bending, and of the contact strength of the gear and of the pinion teeth, the bearings, the housing, the shafts, and so forth. The forces that act on the gear, have to be expressed in a stationary coordinate system, .
. . g s g s g s X Y Z , associated with the motionless gear (that is, associated with the gear pair housing).
After the total tangential force, pa F , of interaction between the tooth flanks, G and P , of a gear, and of a mating pinion, has been pre-multiplied by the operator of the linear transformation, ()   LC . This discussion is also valid with respect to the cases when multiple lines of contact, as well, as multiple portion of the lines of

Normal force acting on the gear in geometrically-accurate crossed-axes gearing
For the calculations of gear teeth for contact strength, as well, as for contact strength analysis, a component, . not allowed to migrate along the ln P , or, at least, the distance of this migration has to be minimized. In this way, the vibration generation, and noise excitation can be minimized.

Cases of multiple lines of contact
Gear pairs of all designs feature the contact ratio greater than one ( 1 m  ). The contact ratio in the range of 1.4 1.8 m = is common in preset-day parctice. As the inequalty, 1 m  , is always valid, periods of gear meshing when eithe one, or two (or more), portions of the line of contact are occur.
When a single line of contact is observed, the entire load is transmitted through this line of conact between the interacting tooth flanks, G and P .
When two (or more), portions of the line of contact are observed, the active lines of contact share the load being transmitted by the gear teeth.
The load carrying capacity of a gear pair depends on the manner the load is shared between the lines of contact.
Consider an active portion of the plane of action, PA , in a geometrically-accurate crossed-axes gear pair, as illustrated in Fig. 10