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Geometry
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
A circle rolling on a straight line gives a trochoid, with the cycloid as a special case when the pole P lies on the circle (Figure 4.29). If the moving circle M has radius a and the distance from the pole P to the center of M is k, the trochoid’s parametric equation is x=aϕ-ksinϕ,y=a-kcosϕ. $$ x = a\phi - k{\text{~sin~}}\phi ,~y = a - k{\text{~cos~}}\phi . $$
Fixed displacement gerotor oil pump (FDOP): a survey
Published in International Journal of Ambient Energy, 2022
V.T. Gannesh, R. Sivakumar, G. Sakthivel
Generally gerotor profiles are developed based on the development method called ‘Trochoid’. Trochoid is defined as the locus of a point on the interior of a circular disc which rolls without slipping on a fixed circle. If the disc rolls on the outside of a circle, the locus of point B is an epitrochoid (Figure 2) and if the disc rolls on the inside of the circle, the locus is a hypotrochoid. Colbourne (1974) defined a parametric equation for epitrochoid and hypotrochoid with its envelope and illustrated the limitation of trochoid profile such as crossover formation when the locus point is outside the rolling circle and cup formation when the locus point is in the circumference of the circle.
Reduction of tooth wear on asymmetric spur gear through profile correction factors
Published in Australian Journal of Mechanical Engineering, 2022
The shape of the gear tooth plays a vital role in the evaluation of the performance of gears during their service life. The involute and trochoidal curves define the shape of the asymmetric gear tooth. Kapelevich (2000) developed mathematical relations to generate the involute and trochoidal profiles for asymmetric spur gear tooth. The asymmetric spur gear was generated, and FEM-based bending stress analysis was performed, as detailed in the References section (Senthil Kumar, Muni, and Muthuveerappan 2008; Costopoulos and Spitas 2009; Prabhu Sekar and Muthuveerappan 2014, 2015). In the present work, the same procedure has been suitably modified to generate the asymmetric tooth with profile corrections. The finite element method is used to solve the real complex problems that involve complicated shapes of geometry and boundary conditions. The asymmetric pinion and gear are discretised into ‘n’ number of simple subdomains called finite elements, and the grouping of elements is called a finite element mesh. The asymmetric tooth profiles and the finite element model of the asymmetric spur pinion and gear are shown in Figure 1. The accuracy of the solution mainly depends on the geometry of the element, number and size of the elements, location of nodes, applied load and boundary conditions. A triangular plane 42 element is chosen to discretise the FE model of the asymmetric gear. The h-version of the mesh refinement study is also conducted to ensure that the number of elements required is present in the gear drive for getting accurate results. The contact and fillet areas are treated as the most critical regions in the pinion and wheel, and so these regions are discretised with a very fine mesh (3000 elements in fillet region and 60,000 elements in contact region for a single tooth); the other non-critical regions are discretised with a coarse mesh to minimise the computational time. CONTA 172 and TARG 169 elements are used to create a contact between the asymmetric pinion and wheel. Asymmetric wheel and pinion with a wider face width are taken (the face width is larger than the tooth height and thickness) so that the plane strain formulation is used for the two-dimensional FE analysis. The inner hub of the pinion is arrested in the radial direction, and rotational torque is applied in the tangential direction, whereas the gear is constrained in both radial as well as tangential directions. The tooth contact load (Fj) and the contact pressure (σcj) at the contact positions are determined through finite element based asymmetric teeth contact analysis. Furthermore, the determined values of Fj and σcj are substituted in Equations (16) and (20) to find the tooth wear of asymmetric pinion.