Involute Cylindrical Gears and Non-Uniform Rational B-Splines
The Isogeometric Analysis (IGA) appears to be an appropriate alternative for numerical tooth modeling. It speeds up computation time by still maintaining the underlying geometry in a highly accurate manner. The power and effectiveness of this simulation strategy is attributable to the use of Non-Uniform Rational B-Splines (NURBS). For instance, this generic category of B-Splines is the common basis of computer-aided design (CAD) software and allows designers to create complex geometries. As NURBS also hold the partition of unity property, their application in structural analyses gets numerically possible in a similar way compared to the classical finite element method (FEM). The actual transition from FEM to IGA is made by exchanging the formulation of the underlying shape functions. Whereas the FEM commonly describes the geometry to be investigated mathematically by linear or quadratic polynomials, the IGA replaces this discretization step by NURBS. In Figure 1, the underlying basis functions of both approaches FEM and IGA are schematically compared against each other for a second order representation, where the parametric space is divided into five equal subspaces.
Figure 1: Quadratic shape functions of FEM (left) and IGA (right)
Figure 1 shows that NURBS, unlike the polynomials, extend beyond the boundaries of the individual subspaces and hence offer more opportunities in geometry modeling by the same number of “elements” or so called “patches”. In reverse, complex contours with smooth transition areas can be described by NURBS with a lower number of shape functions. As a result, the number of degrees of freedom and thus the computation time decreases, while a high-resolution level of the physical model is maintained.
A simple example of this characteristic is given in Figure 2. Here, a circle is described once with second order NURBS and once again with second order polynomials. Both approaches result in eight degrees of freedom and four “elements” or ”patches”, respectively. The IGA discretization is able to cover the exact shape of the circle, while the classical FEM approach shows minor deviations. However, there is one drawback to recognize since the IGA “control points” are not necessarily located directly on the physical contour as the FEM “nodes” do. As a result, the calculation approach within its interpretation of results is more elaborate from the point of view of algorithm implementation.
Figure 2: Exact NURBS/IGA representation of a circle and approximate polynomial/FEM representation of the same circle with each approach having eight degrees of freedom
Anyway, the power of IGA is demonstrated more clearly in Figure 3 as the same circle is again exactly mapped by only three degrees of freedom with NURBS. In contrast, a second order polynomial with only three nodes is not able to cover this contour.
Figure 3: Exact NURBS/IGA representation of a circle with three degrees of freedom
The application to a gear tooth is finally given in Figure 4. The complex shape of the involute tooth flank as well as the individual tooth root geometry is almost exactly captured with a low number of “control points”. A subsequent static analysis by means of the IGA delivers a stress distribution as it indicated on the right hand side and implies the basic potential of the IGA in gear industry.
Figure 4: Involute cylindrical gear tooth described by NURBS (left) and result of a corresponding IGA stress analysis (right)
Authors
Andreas Beinstingel, M. Sc.
Computational Engineer, RENK GmbH, Augsburg & PhD Candidate, Chair of Vibroacoustics of Vehicles and Machines, Technical University of Munich (TUM), Garching and Speaker at „International Conference on Gears 2022“, Garching/Munich