Calculation formula of meshing stiffness of variable hyperbolic arc gear

According to the forming principle of rotary cutter head, Zhao Fei et al. Verified mathematically that the middle section tooth profile of variable hyperbolic arc tooth profile gear is involute tooth profile, and its theoretical contact point is in the middle section of tooth width, which belongs to point contact type arc tooth profile cylindrical gear. The theoretical meshing trace of main and driven gears are shown in Fig. 1 and Fig. 2 respectively. In the theoretical meshing, the meshing trace of the tooth surface is involute, so the coincidence degree of the variable hyperbolic gear can be calculated according to the involute cylindrical gear. Because the variable hyperbolic arc gear still keeps involute meshing on the tooth section, the instantaneous transmission ratio I remains unchanged. For variable hyperbolic arc gear, the tooth line in the direction of tooth width is space arc line, which is longer than the contact line of involute spur gear, so it has better bearing capacity.

The expression of single tooth meshing stiffness is as follows:

Where FN is the normal contact of the tooth profile surface in the meshing area and UN is the comprehensive elastic deformation of a single pair of teeth in the meshing area.

The normal contact force of the tooth surface of the variable hyperbolic arc gear includes the radial, tangential and axial components. Because the curvature of each point on the tooth profile surface of the variable hyperbolic arc gear is different, it is very complex to accurately calculate the tooth surface deformation in the meshing area. Generally, the numerical solution is obtained by finite element calculation.

The comprehensive elastic deformation of gear teeth belongs to nonlinear deformation, including the contact elastic deformation uh caused by Hertz contact, the offset UB of gear contact position caused by gear bending, and the influence UF caused by the self deformation of bearing, shaft and support structure. Considering the deformation of bearing, shaft and support structure, the problem becomes too complex, and it is difficult to consider the influence of UF through finite element calculation

When multiple pairs of gear teeth contact, the coupling relationship between each pair of gear teeth is parallel, so the comprehensive meshing stiffness of gear teeth can be expressed as follows:

Where p is the number of teeth engaged at the same time.