Abstract
In this study, the realm of three-dimensional geometrical contact problems is delved into, with a specific focus on the scenario involving two \({C}^{1}\) gear surfaces undergoing rotational motion about two fixed axes in the spatial domain. The development of a novel mathematical model for this contact problem is encompassed by our endeavor. The foundation of our model is rooted in a new parameterization technique, which enables the reduction of the conventional system of five generally non-linear equations, each with five unknowns and one independent parameter, into a more manageable system. This streamlined approach results in a system consisting of just two equations, two unknowns, and one independent parameter. Subsequently, a back-substitution methodology is employed to explicitly derive the values for the remaining three unknowns. In a bid to ascertain the efficacy of the newly proposed model, a comparative analysis is conducted with the well-established model formulated by Litvin. Our findings demonstrate that the accuracy and stability of numerical solutions in the context of this intricate geometrical contact problem are not only enhanced by our model but also that it represents a significant advancement in the understanding and computational treatment of such scenarios.
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Abbreviations
- \({C}^{\mathrm{1,2}}\) :
-
Continuous function that has derivatives to the first (or second) order at least
- \({E}_{i}\) :
-
Envelope of surfaces (\(i=\mathrm{1,2}\))
- \({S}_{\mathrm{1,2}}\) :
-
Fixed coordinate system of first and second surfaces
- \({S}_{f}\) :
-
Fixed coordinate system
- \({S}_{q}\) :
-
Additional fixed coordinate system to simulate the misalignment
- \({\Sigma }_{i}\) :
-
Surface rotating around fixed axis (\(i=\mathrm{1,2}\))
- \({\overrightarrow{n}}_{i}\) :
-
Normal vector function of first (\({n}_{1}\)) and second (\({n}_{2}\)) surfaces
- \({\overrightarrow{r}}_{i}\) :
-
Radius vector from origin to the contact point
- \({v}_{i}\) :
-
Projection of \({r}_{i}\) on the center line (\(z\)-axis)
- \({u}_{i}\) :
-
Radial distance from the axis of rotation to the point of contact (\(i=\mathrm{1,2}\))
- \({a}_{12}\) :
-
Distance between center of fixed coordinate system
- \(R\) :
-
Generic rotary translation matrix \({\overrightarrow{r}}_{i}\) rotated vector function
- \({\phi }_{i}\) :
-
Angle of rotation of each surfaces around fixed axis (\(i=\mathrm{1,2}\))
- \({w}_{i}\) :
-
Axis of rotation of first and second surfaces (\(i=\mathrm{1,2}\))
- \({f}_{i}\) :
-
Vector function of parametric surface
- \(\beta\) :
-
Helix angle
- \({\alpha }_{i}\) :
-
Angle of initial involution (\(i=\mathrm{1,2}\))
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Temirkhan, M., Spitas, C. & Wei, D. A computationally robust solution to the contact problem of two rotating gear surfaces in space. Meccanica 58, 2455–2466 (2023). https://doi.org/10.1007/s11012-023-01738-2
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DOI: https://doi.org/10.1007/s11012-023-01738-2