Numerical simulation of elastoplastic contact of heavy-loaded spiroid gears

The paper presents the analysis of heavy-loaded low-speed multi-pair spiroid gears under the effect of elastic contact, bending and shearing, and elastoplastic contact interaction of spiroid gearwheel teeth and spiroid worm threads. The main stages of the developed algorithm for analysis of load distribution and plastic strain of flanks with account of elastic and elastoplastic contact conditions, multi-pair contact and macro-roughnesses on flanks are described in short. At elastoplastic contact the zones of plastic contact on flanks of spiroid gearwheel teeth are determined and the value of plastic strain is analyzed. The flanks of the spiroid gearwheel teeth and worm threads are represented in the algorithm as the set of segments (cells) with coordinates of centers determined with account of the analyzed factors that influence the load distribution in spiroid gearing. These factors are the errors of manufacturing and (or) assembly, microroughnesses, and deformations of spiroid gear supports. The paper gives the numerical results of research of the real spiroid gears of gearboxes for pipeline valves (PV) at elastic and elastoplastic contact represented as summary tables and diagrams of contact stress distribution along contact areas and of plastic strain, that show the workability of the algorithm. The presented numerical results of the algorithm functioning are well agreed with the experience of operation of spiroid gears and can be applied for analysis of load-carrying capacity of spiroid gears.


Introduction
Spiroid gears are well-proven in heavy-loaded low-speed operation of pipeline valve drives under the action of short-term increased loads and practically instant peak overloads which has been multiply confirmed by research and operation experience of gearboxes designed at the scientific school headed by Prof. V. I. Goldfarb [1][2][3][4][5]. Spiroid gears have a number of advantages [1][2][3][4][5][6][7][8][9] as compared to cylindrical worm gears that have been commonly applied [17][18][19] in the mentioned drives: • high theoretical contact ratio -up to 10% -12% of the total number of gearwheel teeth can be simultaneously meshed; • high integrity of contacting surfaces; • favorable arrangement of contact lines -practically orthogonal with regard to the vector of relative speed in the gear; • high speed of displacement of contact lines along operating flanks; • lower sensitivity to manufacturing and assembly errors; • possibility to adjust easily the backlash in the meshing.
Practice of operation of PV gearboxes (Fig. 1) shows that even at the first cycles of operation the traces of plastic strain appear on flanks that influence the stressstrain state (SSS) of the gear and, as a consequence, its load-carrying capacity. This effect irreversibly breaks the initial geometry of teeth and leads to redistribution of the load transmitted by teeth. The relevant quantitative account of plastic strain (Fig.  2) of tooth flanks is necessary when assessing the SSS of heavy-loaded gears. In [12][13] the algorithm of such an assessment is suggested. This paper considers the results of assessing the SSS of a number of real spiroid gears of pipeline valve gearboxes by means of this algorithm.

Algorithm for analysis of the stressstrain state of elastoplastic contact of heavy-loaded spiroid gears
At present time the common approach to analyze the loaded systems (including gears) is the finite-element method [14][15][16]. Practice of applying this approach shows that it is convenient in those cases when the design solution is already accepted or almost ready. Its refining by considering multiple versions of gears is therefore not suitable, since the analysis at the necessary degree of accuracy takes too much time. That is why, more efficient methods of analysis of various structural elements are developed [7,10,11,13].
Our developed algorithm applies the discrete representation of tooth flanks as the set of elementary areas (cells) with coordinates of centers (nodes) preliminary calculated with account of necessary modifications and inevitable manufacturing errors [2].
The loaded gear represents a discretely loaded system in which the joint displacement of cells of worm threads and gearwheel teeth and the condition of force equilibrium are described by the following system of equations: where w km is the elastoplastic displacement; S 0km is the initial clearance (prior to analysis of load distribution) between the km th cells of the surface,  km is the relative displacements of the km th cells as a result of approaching the gear elements at gear loading, ř 2k'm' is the arm of action of the force F k'm' applied at the k'm' th cell relative to the gearwheel axis, Т 2 is the assigned torque at the spiroid gearwheel.
The number of equations of the pointed system (1) is unknown beforehand and is specified during calculation. It is also efficient to specify all the sought parameters during calculation: the contact area D, the value   of approaching the elements and the value of discretely applied forces.
The system (1) is usually solved in the elastic statement [10,11,13]. In order to account plastic contact deformations, the following iterative algorithm of analysis is presented: 1) input of initial data: initial clearances S 0km , physical and mechanical properties of materials of elements, and torque Т 2 ; 2) assignment and calculation of data for the first approximation (the first iteration): the approach of elements, the set of loaded (embedded) cells, and forces at these cells.  The consequent steps of the algorithm are related both to the first and to all the next iterations. 3) calculation of elastic displacement of cells due to forces acting in them; 4) calculation of clearances/embedding (further -discrepancies ξ) that appear between teeth after account of elastic and elastoplastic displacements at the assigned approaching. Correction of the value of approaching depending on the average discrepancy ξ av ; 5) correction of the contact area D; those cells are added in which the value of discrepancy became negative (the design embedding took place, the cell came into contact and began transmitting the torque); 6) correction of forces in cells depending on the value of discrepancy. Cells with the zero force are excluded from the contact area with the consequent correction of the forces remaining to provide the equilibrium condition (the last equation in the system (1)); 7) analysis (correction) of plastic strain w p in cells where the contact stress σ H exceeded the allowable one [σ H ] [21], in this case, forces at these cells are corrected (decreased), that is, calculated according to [σ H ] and the area of the cell; 8) approximation of the final number of strain areas that appeared on tooth flanks; 9) checking the condition of finalizing the algorithm according to the chosen criterion of calculation accuracy and the completed task of SSS analysis. As it can be observed, during the algorithm running one can successively specify the components of displacement of cells under load -the elastic component (step 3) and the plastic one (step 7).

Results of numerical simulation
Parameters and load characteristics of spiroid gears of PV gearboxes chosen for numerical analysis are given in Table 1. The peak torque Т 2peak acting practically instantly or in statics is taken as the torque Т 2 . Numerical analysis was carried out with account of the pitch error f xp of spiroid worm threads, and the error of spiroid gearwheel profile due to the presence of faceting f fac .
Results of simulation are summarized in Tables 2-4  Note: a w -centre distance; m x1 -axial module of the worm; z 2 /z 1 -number of gearwheel teeth / worm threads; d a1 -external diameter of the spiroid worm; D e2 /D i2 -external and internal diameters of spiroid gearwheel; h a1 /h f1 -addendum/dedendum factor; α nr /α nl -profile angles of the right/left flanks.

Analysis of results
The obtained results give ground for the following conclusions: 1. The developed numerical algorithm of SSS analysis of a multi-pair spiroid gear with account of the elastoplastic character of contact is operational enough. For a multipair elastoplastically loaded spiroid gear with the cell dimensions 0.07m x (30 cells along the tooth height), the algorithm is reduced to the solution during the acceptable number of iterations (about 30 at average) with the mean discrepancy less than 0.5 mcm (about 0.0001m x ). 2. As it was expected, the relevance of account of the elastoplastic character of contact rises with the increase in the level of gear loading. Thus, for heavy-loaded lowspeed gearboxes for pipeline valves, this account is urgent under the action of extreme static torques that exceed twice the maximum operating ones. 3. The elastic model for a heavy-loaded gear can have a very limited application: the maximum contact stress calculated by this model at maximum torques exceeds the allowable one by strain for not only gears but also for rolling bearings (3500 MPa [22]). It affirms the necessity of account of the elastoplastic contact. 4. The account of the plastic strain of contacting surfaces allows for specifying the loads acting on teeth, especially at the presence of large errors and strains of the structure in gears with high (more than 3-4) overlap ratio. In most cases, the plastic strain of 10-30 mcm stipulates for unloading of the most loaded teeth (up to 10-15%). 5. The researched gears of PV gearboxes showed high tolerance to the action of two-fold overloading torques even at the presence of relatively large errors (about 9-10 degree of accuracy), having exhibited the uniform load distribution between tooth pairs.

Conclusion
The algorithm of load distribution analysis in a multipair spiroid gearing presented in this paper allows for assessing the area of the elastoplastic contact, the value of plastic strain of tooth flanks, to specify the values of loads acting on individual teeth. The calculation results must become the foundation for assessing the strength of a multi-pair heavy-loaded gear in accordance with the allowable contact strain of surfaces and the strength of worm threads and gearwheel teeth under the action of instant (including static one) overloading torques.