Eight node finite element method for gear tooth contact

The elastic contact problem is a small deformation boundary nonlinear problem, which is caused by both the change of contact area and the change of contact pressure distribution. The solution process is a repeated iterative process to search for accurate contact state. Therefore, it is necessary to assume a possible contact state first, and then bring in the definite solution conditions to obtain the contact internal force and displacement of the contact point, judge whether the contact conditions are met, modify the contact state of the contact point when the contact conditions are not met, and solve again until all contact points meet the contact conditions.

The finite element method is an approximate method for numerical calculation. Its basic idea is to discretize a continuum into finite non overlapping sub regions called elements, connect them with nodes, give the mode of approximate solution in the element body, and characterize the characteristics of the element with unknown parameters on a finite number of nodes, Then, the relations of each element are combined into the equations of these unknown parameters by an appropriate method, the linear equations are solved, and the unknown parameter values of each node are obtained, which solves the difficulty that infinite continuous particles cannot realize numerical calculation.

In the finite element analysis, a very important problem is to select the specific element and determine the displacement function of the element. After the shape of the element and the corresponding displacement are determined, the operation can be carried out according to the standard steps. The practice shows that it is troublesome to divide a spatial structure into combinations of tetrahedral elements, and like triangular elements, the segmentation must be small enough, otherwise the accuracy is too low. For the structure with curved surface boundary such as spiral bevel gear, it is more appropriate to use hexahedral element. At this time, in order to ensure the continuity of displacement on the element boundary, coordinate transformation is carried out. We choose the displacement function formula and the coordinate transformation formula to have exactly the same structure. They use the same number of corresponding node values as parameters and have exactly the same shape function. In this way, the isoparametric element is constructed.

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