Research On Reduced Quadratic Programming Algorithm Of Elastic Contact Problem For Gears Engagement

Gear as one of the important transmission components, their work performancehas a direct impact on the transmission quality. In the good lubrication of geartransmission, the mostly failure mode of the tooth surface is pitting, and this is mainlycaused by the shortage of the gear contact strength, therefore, it is important for thedesign of gear transmission that do calculation and analysis of the gear engagementcontact problem.Gear engagement belongs to small elastic deformation in normal work, and thenonlinear nature mainly comes from the uncertainty of the contact state. Although thetraditional iterative method of nonlinear contact and quadratic programming methodcan solve such problems, many deficiencies in terms of convergence and computationworks: iterative method based on the convergence conditions continually iterate andmodify the contact state under the working load, its solving process is very long, andthe convergence is not guaranteed; quadratic programming method although stableconvergence, the solving efficiency will be greatly reduced with the increase of designvariable degrees of freedom.For the lack of uses the traditional method to solve the gear engagement contactproblem, this article proposes a new reduced quadratic programming method, the mainresearch work is as follows:(1) Proposed the reduced quadratic programming model and algorithm of elasticcontact problem: Based on potential energy principle, frictionless elastic contactproblem quadratic programming model is constructed by treating nodal displacementsas variable and choosing contact conditions as constraint equation, on the basis, a newmethod of reduced degree of freedoms based on singular coordinate transformation isput forward, which greatly reduces the dimension of quadratic programming, andmakes the quadratic programming has no dominance equality constraint. According tothe characteristics of the elastic contact mechanics system, and contact freedomdisplacement mode is assume by people，a simple and efficient singular modal matrixcalculation method is proposed.(2) Researched on the construction method of contact normal vector andconstraint equation about the gear engagement elastic contact problem: According tothe characteristics of gear transmission, and makes the orthogonal vector of gear lineof action as a contact normal vector, which avoids the multifarious calculation progress of the construction of contact normal vector in traditional nonlinear contactalgorithm.It is more conforms with the actual working state of gear transmission,and avoid the influence of the grid subdivision; In view of the node gridinconsistencies between the contact body, a kind of establishing contact constraintequations of virtual node based on shape function interpolation is put forward, whichimproved the adaptation of the reduced quadratic programming algorithm for thecontact area grid subdivision.(3) Developed the elastic contact reduced quadratic programming algorithmprocedures base on sparse matrix calculation: Making use of the sparse matrix tobuild stiffness matrix parameters in elastic contact reduced quadratic programmingalgorithm, and then established the contact normal vector, according to theimpenetrable condition between the contact body and generated inequality constrainsin elastic contact reduced quadratic programming algorithm, then designed the elasticcontact problem of quadratic programming algorithm procedures.By the comparative analysis of the numerical cases such as two cylindricalextrusion contact、axis holes fit contact and two dimension、three dimension gearengagement contact show that the proposed elastic contact problem of reducedquadratic programming algorithm has high precision, computation efficiency andstable convergence characteristic, which can effectively use in gear engagementcontact and the general solution of the elastic contact problem.