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This paper investigates the meshing stiffness of beveloid gears in the beveloid rotate vector (BRV) transmission. It is a new kind of transmission evolved from rotate vector (RV) reducer. In the BRV transmission, the beveloid gear is a kind of involute gear with a bevel angle. The BRV transmission have high power density, large transmission ratio and high precision in geared coupled systems. However, there is rare systematic research conducted on the meshing stiffness analysis of the BRV transmission at present. Based on the loaded contact finite element analysis principle, a meshing stiffness analysis model for beveloid gears is established. The influence of different factors such as pitch cone angle, addendum coefficient, load and rim structure parameters of external gear on meshing stiffness are studied. The results show that the pitch cone angle and addendum coefficient have little effect on the shape of the meshing stiffness curve, but they have a significant influence on the amplitude of meshing stiffness. In contrast, the load can affect both the shape and the amplitude of the meshing stiffness curve obviously. Also, the size of scallop-hole and rim thickness have a great impact on the amplitude of the meshing stiffness. The prescribed piece of study can provide a better understanding for gear researchers in order to understand the influence of different parameters on dynamic characteristics analysis of the BRV transmission systems.

The mathematical models of beveloid surface with concave and convex modifications were derived. The finite element and analytical mesh models were developed considering the shaft deformation and misalignments including pinion axial position, center distance and shaft angle errors. Then, tooth contact analysis were conducted to study the impacts of loads and misalignments on contact characteristics. Results show that the higher load applied has stronger effects for enlarging contact pattern and lowering transmission error with the concave and convex modifications. Moreover, the proposed tooth modifications can decrease the sensitivity of contact characteristics to misalignments. The shaft angle error has the greatest impact on the mesh behaviors. The contact path and pattern are prone to shift the location along tooth trace direction due to shaft angle and center distance errors. The contact stress, bending stress and transmission error are seen to increase due to these two assembly errors. Pinion axial position error does not affect the location of the contact path and pattern significantly. However, it can increase the magnitude and change the location of the maximum tooth bending stress in the axis direction. Finally, it only produces minimal effect on the contact stress, contact area and transmission error.

Based on the generation concept of beveloid gears, a novel design approach with parabolic modification is introduced in this paper. The equation of normal section of parabolic modified rack-cutter is derived, and the generating procedure of beveloid gears with parabolic modification is proposed. Therefore, the mathematical model of parabolic modified beveloid gears is obtained. Thereafter, the solid models of beveloid gear pair with and without parabolic modification are developed in the CAD software. Based on the installation condition, the mesh model considering misalignments is established. Then, the loaded tooth contact analysis (LTCA) is performed, and the effects of load and misalignments on the general beveloid gear pair and parabolic modified beveloid gear pair are discussed. Also, the contact pressure, transmission error, contact pattern and tooth root stress are investigated. Results show that under the same load, parabolic modification can reduce the maximum value of contact force and tooth root stress of beveloid pinion. Meanwhile, it can reduce the peak to peak value of transmission error and change the shape of contact pattern from wave mode to parallelogram. Under shaft angle error and offset error, the location of contact pattern and maximum value of tooth root stress move to the heel and toe side, while the axial position error of pinion and gear only slightly affects the contact performance. After parabolic modification, the peak to peak value of transmission error and the maximum value of tooth root stress of beveloid gear pair reduced are reduced significantly under misalignments.

A numerical design approach with tooth modifications to improve the mesh characteristics is presented for crossed beveloid gears. The mathematical model of the beveloid tooth surface with different modifications was developed. Considering the first principle direction (FPD) angle, the effects of tooth modifications on contact ellipse, contact path, transmission error, relative curvature were investigated by theoretical tooth contact analysis. For large FPD angle, independent helix crowning modification and flank line slope modification can increase the area of contact ellipse. But the flank line slope modification tends to increase the transmission error. Profile crowning modification has a degressive effect on the meshing. The contact ellipse and relative curvature are not sensitive to the transverse profile slope modification. For small FPD angle, flank line slope modification and helix crowning modification can improve the mesh characteristics. But the transmission error decreases firstly then increases and the area of ellipse increases with the increase of helix crowning modification. Compared to the large FPD angle situation, the mesh behaviors for small FPD angle are more sensitive to the tooth modifications. Finally, the loaded tooth contact analysis were performed and it shows a good correlation with the theoretical tooth contact analysis results.

This paper presents a manufacturing method for straight internal beveloid gear pair by shaper cutter with parallel axes between the cutter and the gear blank. The cutting motions are represented using three degrees of freedom that are the rotation along its axes, translations along the axial and radial directions. With the proposed method, the beveloid gear with a linear or nonlinear variable profile shift coefficient along the tooth width direction can be generated and the tilt deviations by the inclined cutting tool shaft can be avoided. The mathematical description of shaper cutter and the coordinate systems for the cutting procedure were defined. And the mathematical model of external and internal beveloid gears were derived and the loaded tooth contact analysis were conducted. Results show that the teeth mesh in at the toe and mesh out at the heel. The length of the simultaneous contact line on tooth surface increases first and then decreases due to the cone angle. As the load increases, the mesh area, contact pressure, mean value of mesh stiffness and the peak-peak value of angular transmission error are increased due to the elastic deformation. However, the speeds of the increase tendency slow down.

Considering the gear manufacturing and meshing theory, a numerical design approach with parabolic modifications was proposed to improve the mesh behaviors of crossed beveloid gear transmission. The generating procedure with parabolic modifications for the crossed beveloid gears with small shaft angle was proposed. And the analytical mesh model with misalignments of crossed beveloid gear transmission was developed. Then the effects of parabolic modifications on contact path and contact ratio were investigated. Also, the misalignments sensitivity was analyzed for the beveloid gears with and without parabolic modifications. Results show that the design situation with a positive parabola coefficient for the pinion and a negative parabola coefficient for the gear is better than other combinations of the parabola coefficients. Both the offset error and shaft angle error tend to deviate the contact path from the theoretical middle region. The parabolic modifications can reduce the sensitivity of contact path to the offset error and shaft angle error for crossed beveloid gear transmission effectively. Both axial pinion and gear position error has little influence on the mesh behaviors for the crossed beveloid gear pair with and without parabolic modifications.

Dynamic modeling of beveloid gears is less developed than that of conventional involute gears with parallel axis because of their complicated mesh mechanism and 3-dimensional dynamic coupling. For evaluating the dynamic characteristics of a marine gearbox with crossed beveloid gears more realistically, a systematic dynamic modeling approach was proposed based on the lumped parameter and the finite element method. Considering the time-varying mesh stiffness excitation, time-varying loaded translational transmission error excitation, sliding friction excitation and backlash nonlinearity, a coupled gear–shaft–bearing–housing dynamic model for a marine gearbox with crossed beveloid gears was developed based on the proposed modeling method. Then, numerical integrations applying the explicit Runge–Kutta formula and the implicit direct integration were used to solve the nonlinear dynamic model. The dynamic transmission error, dynamic mesh force, and dynamic equivalent shaft-bearing supporting force were obtained as well as the dynamic response of the housing. Finally, a test rig was established for the marine gearbox with crossed beveloid gears and vibration tests were performed to investigate the dynamic responses. Through analyzing the responses in time domain and frequency domain, the experimental results compared well with the theoretical dynamic results.