Mathematical Models, Oriented to the Synthesis of Hyperboloid Gear Drives with Eliminated Singularity on the Mesh Region

The study illustrates the mathematical models, applied to the synthesis of hyperboloid gear drives. The kinematic approach for registration and elimination singularity from the vicinity of the pitch contact point at the mesh region of the synthesized spatial transmissions is shown. Analytical dependencies are illustrated in order to provide the definition and control of the singularity of first order on the conjugated active tooth surfaces, when hyperboloid gear sets are synthesized by applying two approaches for their synthesis: “synthesis upon a pitch contact point” and “synthesis upon region of mesh”.


Background
The science, that studies the processes of rotations transformation upon a preliminary given law between non-coplanar axes, by three-link mechanisms, having one or more high kinematic joints, can be treated as an independent direction of the science "Applied Mechanics". It studies the kinematic and dynamic behavior of these systems in relation to the geometric characteristics of the elements of the constituting them -high kinematic joints, Litvin [1 -3], Abadjiev [4].
Mechanisms, in which the rotations transformation between non-coplanar axes is realized by set of high kinematic joints, which elements comes in and goes out of tangential contact, by following a certain logical sequence, are known in the books, doctor thesis and scientific studies of the following authors Litvin [1 -3], Abadjiev [4], as hyperboloid gear drives (spatial gear mechanisms with crossed axes of rotations). The rotations transformation is realized as a result of the normal forces, which acts in the places of tangential contact of the elements of high kinematic joints.
It is known, that the design of mechanical multibody systems is a complex task, which main stage is known in Applied Mechanics under the name Synthesis of Mechanisms.
In the most general case, the synthesis of the mechanisms includes the following two main tasks, treated in the Sc.D. thesis of Abadjiev [4]:  Synthesis of the structure of the designed mechanism by carrying out a structural synthesis.  Design of the kinematic scheme of the mechanism.
This task is known as kinematic synthesis of the mechanism. Through its solution it is achieved the determination of the constant geometrical parameters of the chosen structure of the mechanism. These parameters satisfy its preliminary defined kinematic characteristics and related with them specific geometric features of the mechanism. In the case of synthesis of hyperboloid gear mechanisms (spatial gear drives with crossed axes of rotation), it is not necessary to solve the problem of structural synthesis. The spatial rotations transformation, as a rule, is realized by three-links gear mechanisms, whose movable linksrotations axes are crossed in the space. The motions transformation and transmission of mechanical energy is accomplished through a system of high kinematic joints. Therefore, the synthesis of different type hyperboloid gear mechanisms is reduced to finding out a solution of the kinematic synthesis task. The registration and control of the singularity of the conjugate active tooth surfaces is accomplished through the developed adequate mathematical model for the synthesis of that class gear drives.
Hence, the synthesis of different types gear transmissions is turned into solving the task of the kinematic synthesis.
Singularity is a characteristic of a great importance for the processes of generation and conjugate action of the active tooth surfaces of the synthesized hyperboloid gear drives. The reason for this is that it defines the ordinary nodes and undercutting points, the increased friction, law lubrication, pitting and etc. on the contacting tooth surfaces, an object of study in the offered mathematical models from Abadjiev in his Sc. D thesis [4]. This characteristic is often ignored from the designers, due to the lack of knowledge of its existence. Its localization and control are complicated processes, which require a development of adequate mathematical approaches.

Methods
The aim of the task of the kinematic synthesis of every Universal Journal of Mechanical Engineering 5(3): 87-96, 2017 89 mechanical system is the definition of optimal values of so called independent parameters (parameters of the synthesis). The determination of the set of parameters of the synthesis is by complying of two sets of conditions: basic and additional ones, which ate discussed in the Sc. D. thesis of Abadjiev [4].
In order to synthesize a mechanism, that satisfies a preliminary given characteristics, it is necessary to combine many contradictory conditions, relating to the purpose of the gear mechanism, to the technology of its manufacturing, to the conditions of its exploitation and so on. One of this conditions is a basic one. For the gear mechanisms, including the hyperboloid gear drives, the basic condition of the synthesis has a kinematic character. A condition of this type is the realization with the sufficient accuracy of the transfer function of the mechanism. For prevailing in practice cases of synthesis of gear mechanisms that condition is the gear ratio (ratio of the values i (1) In this case, for the synthesis the optimum value of deviation between actual and theoretical given gear ratio is pursued.
In other cases, the synthesis of gear drives can be subjected to the ensuring of the optimal efficiency of motions transformation by reaching a preliminary given value of the efficiency, which in this case is a basic condition.
All of the others conditions are additional ones. Into this group the restrictions, regarding the size of the movable links and their strength loading, the control of the singularity of the conjugate active tooth surfaces (elements of high kinematic joints), etc. can be included.
In most cases of the practice, the main purpose of every gear mechanism is to realize with necessary accuracy, at minimal losses of the transmitted mechanical energy and optimal strength characteristics, the preliminary given law of motions transformation.
The gear drives with crossed axes of rotations are characterized with presence of large number of free parameters. When it is searched for suitable combinations among them, it is created the possibility for the synthesis of these class transmissions to obtain the desired optimum combinations of technological and exploitation characteristics.
The said up to now, determines the applied by the authors of this study, kinematic character of the approach to the synthesis and respectively the kinematic character of the created mathematical models. This approach, applied by authors, for registration of the singularity in the mesh region of the synthesized hyperboloid gear drives is a kinematically oriented. Тhe control (registration and elimination) of the singularity of the conjugate active tooth surfaces (elements of high kinematic joints), depending on the designation of the designed transmission can be treated as the basic condition of the synthesis. The mathematical model for synthesis upon a pitch contact point is based on the assumption, that the necessary quality characteristics, defining a concrete exploitation and technological requirements to the active tooth surfaces, are guaranteed for only one contact point Р (for its close vicinity, respectively) of the active tooth surfaces 1 Σ and 2 Σ (Fig. 1); the model's detailed explanation is accomplished in Abadjiev [4]. The model is applicable to the synthesis of spatial gear mechanisms, not only with linear contact but with the point contact. According to it, the common contact point Р of the conjugate tooth surfaces and w a determine the structure type and its dimensions and shape, as well as the dimensions and mutual positions of the active links in the fixed space (in coordinate systems Hence, from what is said up to here, the mathematical model for synthesis upon a pitch contact point ensures the algorithmic solution of two type basic tasks, which are an object of study in Abadjiev [4] and in his further publications Abadjiev et al. [5]: • Synthesis of the pitch configurations; • Synthesis of the active tooth surfaces. When these tasks are solved together, the necessary preliminary defined geometric characteristics of the synthesized gear mechanisms in close vicinity of the pitch contact point are ensured.
In conclusion, it should be noted that the approach, described here, for the synthesis of spatial gear drives is based on the following kinematic condition: The relative velocity vector It should be emphasized that the aimed approach to the basic synthesis upon a pitch contact point, based on solving the mentioned tasks, is characterized in that the mathematical model and based on this developed algorithm have a typical feature -universal structure for all types hyperboloid gear transmissions.

Mathematical model for synthesis upon a mesh region
When hyperboloid gear drives with linear contact between their active surfaces are synthesized, it is necessary to control the quality of mating in the entire mesh region or in a fixed, by some reasons, zone. This approach to the synthesis task requires development of an adequate mathematical model. Its common kinematic scheme is shown on Fig. 2, in accordance with the shown in the scientific work of Abadjiev [4].
The mathematical model for synthesis upon a mesh region is not a structurally universal. The reason is that concrete geometric and kinematic characteristics of the mesh region depend on its placement in the fixed space and also depend on the geometric characteristics of the instrumental tooth surfaces J Σ , which generates the active This mathematical model is suitable for application, when: • it is not possible to define pitch circles, pitch contact point, respectively; • the condition for the special orientation of the longitudinal lines of the synthesized toothed surfaces is not mandatory or cannot be fulfilled; • specific technological requirements have to be ensured. In the basis of mathematical modeling for synthesis of spatial gears upon a region of mesh is the kinematical model of the surfaces of action.
Following the approach, we will note that the optimization process is essentially a determination of the optimal geometry and the limits of the mesh region as part of the surface of action (Fig. 2). As it was mentioned, when a pair of gears with linear contact is synthesized, it is necessary to control their quality characteristics in whole mesh region. The surface of action will be defined through the active tooth surfaces of one of the movable links of the hyperboloid mechanism.
To illustrate this approach, let us accept that the 1 Σ -an active surface of one of the movable links ( 1 i = -pinion) of the three-link gear mechanism is known. The technology for the generation of the active tooth surfaces of these mechanisms is based on the second Olivier's principle, which principle is applied in the scientific studies of Litvin [1] and Abadjiev [4]. Let 1 Σ is presented with its vector equation: where: ps L is 3х3 a transformation matrix from the fixed co-ordinate system S (co-ordinate system fixed to the frame) into co-ordinate system p S of the pinion; Generally, the defined geometry, dimensions and location of the mesh region in the fixed space, as part of the action surface are optimal ones, if: • the singular points on it are registered and eliminated; • the orientation and placement of the contact lines on the mesh region are determined. This is realized in order to reach the maximum possible loading capacity and coefficient of efficiency.
92 Mathematical Models, Oriented to the Synthesis of Hyperboloid Gear Drives with Eliminated Singularity on the Mesh Region

Active tooth surfaces singularity registration
On the questions dealing with the problems of singularity of conjugate tooth surfaces, studies of many scientists, such as F. Litvin independently [1] and in collaboration Litivn et al. [5], S. Lagutin [6], K. Minkov [7], W. Nelson [8] and others are dedicated.
When mutual enveloping surfaces 1 Σ and 2 Σ are mated, for certain geometric and kinematic conditions, part of the contact points are transformed into node (singular) points, for which it is fulfilled the following the condition: relative velocity vector at one of the contacting tooth surfaces is the zero vector, as it was defined in Litvin [1] .
The equation (9)  The existence of singularity of second order in the mesh region of the studied class hyperboloid drives, which synthesis is realized in accordance with the second Olivier's principle, cause the appearance of the "undercutting phenomenon", when the process of instrumental meshing of the surfaces 1 J Σ Σ ≡ and 2 Σ is realized. The physical meaning of "undercutting" is that the transition surface, generated by the tip of the cutting tooth of the instrumental surface J Σ takes (cuts) a portion of the active tooth surface 2 Σ . As a result, there are regions, in which more often the transition surface, in the base of the tooth, and the active tooth surface 2 Σ don't have a tangent contact, but they intersect. The common line of these surfaces is locus of the undercutting points (points of the singularity of second order). As a rule, the undercutting leads to decrease of bending strength of the generated teeth. The curvatures of the tooth, close to the undercutting points, take values which are not desirable for its contact strength. In order to avoid the phenomena "undercutting" of 2 Σ , it is necessary to choose a suitable geometry. This is accomplished by optimization of the geometry of Further in the study, the character of two types undercutting points will be explained briefly, in the context of kinematically conjugate tooth surfaces, whose synthesis is realized in accordance with the second principle of T. Olivier. In other words, we will search vector and analytical dependencies, through which the conditions of undercutting point appearance in the mesh region, respectively on the mating tooth surfaces of the synthesized gear mechanism, are defined. And also we will search for possible approaches for their elimination in the process of synthesis in connection with caused by them negative effects accompanying processes of instrumental and work gearing.
After joining to (10) Σ , as a locus of the same contact lines, but in the non-fixed ( rotating) space, defined by g S . Further in the work, the indexes p and g , indicating the coordinate system, in which the concrete study is realized, will be omitted.
As a rule, the analytical expression of In other words, equation (19), together with the condition (15) (respectively (17)) provides a validation of the criteria (9), when the contact point is regular one, i.e. for the common normal vector at it, the following condition is fulfilled: The realized analysis shows, that the criteria (9)  Hence, the criteria (9) and (10) ensure the registration of all nodes in the mesh region for every synthesized gear mechanism.

Crossed orientation of the active tooth surfaces at the pitch contact point, when it is considered as an ordinary node
Here, keeping the direction of the realized analysis, it will be clarified the applied by authors analytical approach for the registration and elimination of the ordinary nodes on the meshed surfaces. In accordance with already determined conditions, defining one conjugate contact point, as an ordinary node, it will be shown the vector dependency connecting geometric and kinematic parameters of synthesized gear drives, which characterize it as arbitrary ordinary node point.
Here and further when it is taken into account the given in Fig. 3  After differentiating the above equation and taking into account specific kinematic characteristics which make a regular contact point, to become an ordinary node, the following vector equation is obtained and examined in the studies of Litvin [1] Abadjiev [4] and Minkov [7]: The values of the angle cr α affect on the size of the normal angle of the profile in the pitch contact point and through the main kinematic and strength characteristics onto the quality of the synthesized gear mechanism. The consideration of cr α , when the synthesis is realized, is the reason for non-symmetrical tooth profile of the teeth of hyperboloid gear drives with face mating gears. Hence, the normal profile of the teeth of the synthesized gear mechanisms, has two pressure angels: low-side (for low-side driving) and high-side pressure angels (for high-side driving) (see Fig. 4). This is the reason for the different values of the acting forces and coefficient of efficiency, when directions of rotation of the gears are changed.

Elimination of the singular points of first order from the mesh region
As already mentioned, one point belonging to the mesh region of spatial gear pair is singular point of the first order if the following conditions are satisfied, as it is shown in Litvin [1], Abadjiev [4] and Abadjiev [9]: , 0 Taking into account equation (24) as well equation (18), it is not difficult to prove that Formula (25) is a condition for the non-existence of singular points of firs order in the mesh region of the synthesized gear set. Using the above formula, the obtained system equalities, that includes the geometric parameters of the non-orthogonal convolute, involute and Archimedean Spiroid gears, as well as their gear ratio, ensuring the elimination of the singularity points of second order from their mesh region, will be illustrated (see Fig. 5).
The unification fixed an area of existence of convolute Spiroid gear pair, which contact lines have no ordinary nodes.
Taking into account the geometrical characteristics of the involute Spiroid gears, as it is presented in the study of Abadjiev [4,9], the inequalities  Assuming that 2 / p δ = in all the above relations, it is possible to obtain conditions for the eliminations of ordinary nodes on the contact lines of Spiroid gears with orthogonally skewed axes. The classic Spiroid gear, which is a trade mark of Illinois Tool Works, Chicago, Illinois, is an orthogonal Archimedean type Spiroid gear. The shown in Fig 4,a Helicon gear pair is a classic type and belongs to the gear transmissions, which are also trade mark of Illinois Tool Works, Chicago, Illinois. Their basic characteristic, is that the active tooth surfaces of the pinion are cylindrical Archimedean helicoids.

Conclusions
The kinematic nature of the essence of the concept of developing mathematical models for synthesis is presented. The mathematical models for synthesis of hyperboloid gear drives (model for synthesis upon a pitch contact point, model for synthesis upon a mesh region and a combined model) are illustrated. The character of the possible singularity in the region of mesh is analyzed, when their tooth surfaces are generated in accordance with the second principle of T. Olivier. A generalized vector condition for the existence of a critical (boundary) normal vector at a point of contact of the tooth surfaces is shown when this point is an ordinary node.
The angle, defining the orientation of the critical vector in the fixed space, is determined by the parameters of the pitch contact point. It is shown that the angle's formula is an invariant with respect to the index of the parameters, determining the belonging of the pitch contact point to the corresponding movable link, as well as in relation to the sign in front of the angle, that determines the longitudinal orientations of the corresponding tooth surface at the pitch contact point.
The presented analytical dependencies, which serve for elimination of the singularity of second order from the vicinity of the pitch contact point and from the mesh region of the Spiroid gears, have served for the creation of several Bulgarian patents with author Abadjiev [10][11][12][13].