Research of the Meshing Performance of New Concave-convex Ball Teeth

Aiming at the limitations of the existing hydraulic pumps, a new type of bidirectional water-hydraulic internal ball gear pump using concave-convex ball teeth meshing to transfer power is proposed. In this paper, the principle of internal gear pump is utilized to establish the model of internal ball gear pump, and its mechanical properties and dynamic meshing characteristics are numerically investigated by using the finite element method. The results show that (a) the concave-convex ball gears run smoothly without any interference, and (b) increasing the number of gear rows can effectively reduce the maximum stress and meshing impact under a given load and speed. This study provides a basis for the development of high-performance ball gear pumps.


Introduction
Water hydraulic technology with its outstanding safety and environmental protection and other advantages, in the field of industrial equipment, especially in the field of marine electromechanical equipment has been widely used [1][2].For example, water hydraulic manipulator [3], high pressure water jet propulsion system, bilge drainage system [4][5].Hydraulic pump as the core power element of the hydraulic system, its performance largely determines the working performance of the hydraulic system, is one of the most important research direction in modern hydraulic transmission [6].Internal gear structure applied to water hydraulic pumps can significantly reduce the volume and weight of the water hydraulic pump itself, the structure is simple and compact [7], but the physicochemical properties of water and mineral oil are very different, to the design of water hydraulic internal gear pumps bring a lot of difficult problems, the need for a series of key technologies such as material matching, friction and wear and processing and manufacturing of in-depth research [8].Harbin Institute of Technology to Japan Fujikoshi (NACHI) IPH oil hydraulic internal gear pump as a prototype, the pump axial and radial gap sealing device according to the physical and chemical properties of water re-design and manufacture, the development of a water-hydraulic internal gear pump, the results show that in the working pressure of 4MPa, the pump volumetric efficiency of 88%, the overall efficiency of 68%, 6.3MPa volumetric efficiency of 85%, total efficiency of 74% [9]; Zhejiang University proposed a new type of involute internal gear pair, the gear pair with full tooth profile and external gear tooth profile conjugate internal gear, and for the physicochemical properties of water, the design of the water lubrication hydrostatic support, but processing and manufacturing prototype experimental results and theoretical volumetric efficiency is a large difference, failing to meet the design requirements [10].Water-hydraulic internal gear pump related information mainly focuses on the structure and performance of the internal gear, the lack of systematic analysis and research and perfect design theory [11].
Based on the above research status of water-hydraulic internal gear pumps, this paper takes the designed and developed bidirectional water-hydraulic internal mesh ball gear pump as the carrier [12], takes the key moving part of the internal mesh ball gear vice as the research object, puts forward the theoretical design of a kind of concave-convex ball gears, and establishes its mathematical model.On this basis, the meshing characteristics of the designed new type of concave-convex ball gears are systematically studied, and the meshing characteristics under different rows are comparatively analyzed.It has certain theoretical significance and application value for the development of water hydraulic transmission technology.

Working Principle of Ball Gear Pump
Figure 1 shows the structural schematic diagram of the water-hydraulic internal ball gear pump, the working principle is consistent with the linear conjugate internal gear pump, which can realize two-way free rotation.Inner rotor 1 and outer rotor 2 adopts eccentric arrangement, inner rotor outer circle and outer rotor inner circle tangent on one side, the other side is separated by the crescent block, the outer rotor is installed with spherical outer teeth 4 (part of the ball), the inner rotor is opened corresponding to the spherical groove (concave ball teeth), the fixed crescent block 3 outer circumference corresponds to the opening of the same radius as the convex ball teeth semi-circular arc groove, to ensure that the outer rotor convex ball teeth through the normal at the same time the high and low pressure chambers on both sides.The high and low pressure chambers on both sides are separated.

Tooth Shape Parameter Analysis
Following the relevant parameters and definitions of involute gears, the key structural parameters and geometrical dimensions of the internal meshing ball gear sub-ball teeth are defined as shown in Fig. 1.
(1) Pitch Circle: The circle tangent to the inner and outer rotor when the ball teeth are engaged is the pitch circle (indexing circle), the radius of the pitch circle is r1, r2,; (2) Distribution circle: ball teeth (convex ball teeth and concave ball teeth) center of the ball along the pitch circle distribution circle for the ball teeth distribution circle, inside and outside the rotor distribution circle radius were R1, R2; (3) Radius of ball teeth: the radius of convex ball teeth and concave ball teeth, expressed in r; (4) Eccentricity: center distance between inner and outer rotors, expressed in e; (5) Number of teeth: the number of ball teeth (convex ball teeth and concave ball grooves) on the inner and outer rotors, expressed as z1, z2 respectively.Based on the above definition, the ball tooth pitch circle (indexing circle) is d, at the same time, because the ball tooth and the convex ball tooth ring for the split structure, the ball tooth is fixed in the outer ring needs a certain distance, so that the ball tooth fixed distance h.Because the ball tooth in the distribution of the circle for the distribution of the phenomenon of non-interference and overlap each other, adjacent to the two ball teeth in the indexing circle corresponding to the arc length of the l1 and the ball teeth between the radius of the existence of limiting relations.The inner rotor axis O as the origin to establish X-Y plane right-angle coordinate system, as shown in Figure 1.
The length corresponding to the spherical center distance between adjacent ball teeth on the indexing circle (pitch circle) is The radius r of the ball tooth is the length of a segment of the chord on the circumference of the distribution circle of the ball tooth of radius R1, which is shown by the red line segment in the figure.Take the midpoint of the line segment and connect it with the center of the distribution circle to get a line segment perpendicular to the chord length r, and construct two right triangles.Connect the hypotenuse of the two triangles to the two points of intersection with the spherical pitch circle to obtain the green line segment shown in the figure, which is the corresponding chord length on the spherical pitch circle.The angle of the center of the circle corresponding to the distribution of each right triangle is noted as θ.Then we have Since the fixed distance h is much smaller than the radius of the distribution circle and the radius of the pitch circle, the centroid angle corresponding to the green chord length on the pitch circle can be approximated to be equal to the centroid angle 2θ corresponding to the red chord length on the distribution circle, and the arc length corresponding to the chord length on the pitch circle is The pitch circle on the adjacent ball teeth between the interval arc length of l, according to the design requirements, the ball teeth do not occur between the interference and overlap phenomenon, that is, the adjacent ball teeth between the arc length of l> 0, then there are

Stress Analysis
In this paper, the force state at the ball tooth mesh is solved with the help of finite element analysis software.The finite element model of the internal meshing ball tooth sub is shown in Fig. 2, and the mesh is reasonably encrypted.The material of the convex ball teeth is assigned as Al2O3 ceramic, the material of the inner rotor is assigned as QAl9-4, and the material of the outer rotor is assigned as 9Cr18.A clockwise torque is added to the bore face of the inner rotor shaft as the driving force with a magnitude of 7.13 N-m, which is transmitted to the outer rotor through the ball teeth meshing.The results of finite element calculation are shown in Figs.It is found that the maximum deformation of the inner rotor occurs at the relative side of the effective moment, and the maximum deformation is 0.0037mm; the convex ball tooth is made of highperformance Al2O3 ceramics with high hardness, and the deformation is only 0.0006mm.The effective moment is concentrated in the top of the concave ball tooth and the interface between the convex ball tooth and the outer rotor, and the maximum equivalent force is about 399MPa; The maximum equivalent force at the convex ball tooth surface is about 241MPa, both of which are smaller than the permissible stresses of the respective corresponding materials and meet the design requirements.

Determination of Effective Torque Transfer Position
Considering the influence of the number of rows of ball teeth on the motion characteristics, the meshing models of single-row and double-row ball teeth are constructed respectively, as shown in Fig. 2.
In the transient dynamics analysis, the modeled kinematic pair creates two rotating pairs against the ground based on the inner surface of the inner rotor bore and the outer circular surface of the outer rotor, respectively, and then applies the load to the inner meshing ball and gear pair.In order to avoid the shock at the beginning of rotation, the loading method of speed and load is set as ramp loading.Extracting the cross-section of the single-row ball tooth perpendicular to the axis over the center of the ball and the cross-section at the same position of the double-row meshing model, the overall dynamic stress changes of the two models are obtained as shown in Fig. 4. Comparative observation reveals that the double row of ball teeth effectively reduces the stress peaks in both states, with the maximum value being about 1/2 of that in the single-row engagement state.Moreover, the double rows of ball teeth mesh alternately, which effectively reduces the stress in the process of ball teeth meshing and makes the meshing movement smoother.

Conclusions
(1) Based on the principle of the internal meshing gear pump, a model of the internal meshing ball gear pump is proposed, and it is verified that under the same load and speed conditions, the double-row ball gear design can effectively reduce the maximum stress during the ball gear meshing process (2) The operation of the concave-convex spherical gear is smooth and interference-free.Compared with the internal meshing gear pump, it effectively solves the problem of difficulty in forming effective lubrication for the internal meshing gear pump.Developed ideas for new water hydraulic pumps.
(3) The extrusion stress generated during the engagement of concave-convex ball teeth is still very large, and the structure of the ball teeth needs to be improved.

Figure 1 .
Figure 1.Working principle and key parameter diagram of internal gear pump.

Figure 4 .
Figure 4. Single and double row stress change curves and cloud diagrams.