Influence comparisons of two version tooth profile modifications on face gear dynamic behaviors

2.1. Calculation solutions and influence mechanisms of linear addendum modifications of face gears

Tooth profile modifications could reduce engagement impacts of face gear drives. Typically, pinions are driving gears, and face gears are driven gears. At engagements, face gear addendum circles would be contacted with pinion dedendum circles. Thus, in order to reduce engagement impacts of face gear drives, two version tooth profile modifications are suggested. One is linear addendum modifications of face gears, namely, addendum straight line modifications of face gears, as shown in Fig. 1.

Fig. 1A sketch of face gear teeth associated with linear addendum modifications

Meanwhile, according to the reference [5], due to point contact transmissions employed by face gear drives, a face gear tooth can be considered as a sequence in which modified involute gears are superimposed along its face width, as shown in Fig. 2.

As illustrated in Fig. 1 and Fig. 2, a linear addendum modification of face gears could be equivalent to that of involute gears, and could be simplified as shown in Fig. 3, according to Ishikawa model [15].

According to the reference [15], each part flexibility of Ishikawa model can be calculated, as listed in Table 1.

Fig. 2A sketch of an equivalent face gear tooth

Fig. 3A sketch of an equivalent linear addendum modification of face gear teeth

The symbols $q_{Br}$,$q_{Bt}$, $q_s$,$q_c$, $q_G$and $q_w$ are a bending flexibility of the rectangle part, a bending flexibility of the trapezoid part, a shear flexibility, a contact flexibility, and the flexibilities induced by base rotations of teeth and rotations of wheel bodies, respectively, $E$ is a modulus of elasticity, $\gamma$ is a Poisson ratio, $z_1$ and $z_2$ are tooth numbers, $d_{d1}$ and $d_{d2}$ are rim diameters, $d_1$ and $d_2$ are reference circle diameters of pinions and face gears, respectively, $u$ is a drive ratio, as well as ${\omega }_x$_, and $h_i$, $h_r$, $h_x$ are an acting angle and the geometry parameters of gears, which can be expressed as [15]:

where subscript $x$, $b$, $f$ and $a$ express mesh point positions, base circles, dedendum circles, and addendum circles, respectively, $s$ is a tooth thickness, $r$ is a reference circle radius, $\alpha$ is a pressure angle, and $s_f$ is a width of minimum life sections.

As given in Fig. 1 to Fig. 3, face gear addendum thicknesses would be decreased by linear addendum modifications of face gears. Meanwhile, according to Eq. (1), only geometry parameter $h_i$ could be affected by addendum thicknesses, and only two flexibilities, namely, $q_{Bt}$ and $q_s$, which occupy smaller proportions of flexibility sum values based on engineering experiences, would be affected by $h_i$, according to the flexibility calculation equations, as listed in Table 1. Thus, mesh stiffness of face gear drives is almost not to be impacted by linear addendum modifications of face gears, due to mesh stiffness being a reciprocal of flexibility sum values of face gear drives. However, static transmission errors of face gear drives would be affected, due to a mesh error increase caused by the modification, which is the component of STE and could be equivalent as shown in Fig. 4.

Fig. 4A sketch of equivalent tooth form deviations of linear addendum modifications

As illustrated in Fig. 4, a mesh error increase caused by linear addendum modifications of face gears could be defined as:

where $V_{max}$ is a maximum value of linear addendum modifications of face gears, and $F_{pb}$ is a percentage of modification lengths versus base pitches.

2.2. Calculation solutions and influence mechanisms of arcuate dedendum modifications of pinions

The other version tooth profile modification is arcuate dedendum modifications of pinions, as shown in Fig. 5.

As given in Fig. 5, minimum life sections of pinions would be affected by arcuate dedendum modifications, and the width variation of minimum life sections, as shown in Fig. 6, could be derived by:

where $\mathrm{\Delta }{s}_f$ is a width variation of minimum life sections, $\mathrm{\Delta }{r}_V$ is a fillet radius maximum variation of pinions caused by arcuate dedendum modifications.

According to the equations, as listed in Table 1, and Eq. (1), tooth flexibilities of pinions would be impacted by minimum life section width changes. Thus, both mesh stiffness and STE would be affected by arcuate dedendum modifications of pinions.

Fig. 5A sketch of pinion teeth associated with arcuate dedendum modifications

Fig. 6A sketch of variations of minimum life sections of pinions

2.3. An equivalent evaluation solution of two version tooth profile modifications

The difference of two version tooth profile modifications on face gear dynamics can be expressed, as shown in Fig. 7.

As illustrated in Fig. 7, in order to asses influences of two version tooth profile modifications on face gear dynamic behaviors, an equivalent evaluation solution of two version tooth profile modifications, based on the viewpoint of maximum fluctuation values equaled of STE, could be defined as:

where $A_{STEF}$ is a maximum fluctuation value of STE associated with linear addendum modifications of face gears, $A_{STEP}$ is a maximum fluctuation value of STE associated with arcuate dedendum modifications of pinions, $A_{SET}$ is a maximum fluctuation value of STE, $A_{max}$ is a maximum amplitude, $A_{min}$ is a minimum amplitude, $e$ is a STE, $\mathrm{\Lambda }$ is a comprehensive mesh error, as well as $D_F$ and $D_P$, which could be calculated according to the equations listed in Table 1, are tooth deformations of face gears and pinions, respectively.

Fig. 7A diagram of the difference between two version profile modifications

2.4. Four DOF dynamic model

In order to compare linear addendum modifications of face gears with arcuate dedendum modifications of pinions on dynamic behaviors of face gear drives, a four DOF dynamic model of face gear drives is formulated, as shown in Fig. 8.

Fig. 8A four DOF dynamic model of face gear drives

In Fig. 8 subscript $f$ and $p$ express face gears and pinions, respectively, $\theta$ is a torsion degree, $S$ is a bending degree, $T$ is a torsion, $k$ is a bending stiffness, $c$ is a bending damping, $k_m$ is a mesh stiffness, $c_m$ is a mesh damping, and $\gamma$ is a shaft angle.

As given in Fig. 8, the mathematic equations of the dynamic model could be derived by:

where $m$ is a quality, $R_b$ is a base circle radius, $I$ is a moment of inertia, and $F_m$ could be deduced as:

3.1. STE simulation and analysis

In order to evaluate dynamic behavior differences of face gear drives associated with two version tooth profile modifications, geometry parameters, material parameters, operating conditions, and modification parameters of linear addendum modifications of face gears, which are determined by the reference [16] and working experiences, of an example case of face gear drives are listed in Table 2.

According to the parameters listed in Table 2, the equations listed in Table 1, Eq. (2) and Eq. (4), STE without any modifications and that associated with linear addendum modifications of face gears are simulated, as shown in Fig. 9.

In the case of Fig. 9, the amplitudes of STE associated with linear addendum modifications of face gears versus engagement-in-and-out positions are less than those without any modifications, but the phenomenon is opposite versus pitch positions. According to Eq. (4), the STE maximum fluctuation values, as shown in Fig. 9, are calculated in Table 3.

A maximum fluctuation value difference between two STE simulation results could be defined as:

where subscript 1 and 2 express two STE simulations, respectively.

Fig. 9STE simulations of the example case

a) Without any modifications

b) With linear addendum modifications of face gears

According to the results listed in Table 3 and Eq. (7), the maximum fluctuation value difference between without any modifications and with linear addendum modifications of face gears of the example case of face gear drives is 1.35 %.

According to the proposed equivalent evaluation solution of two version tooth profile modifications, namely, Eq. (4), and the simulated STE maximum fluctuation value associated with linear addendum modifications of face gears, an arcuate dedendum modification value of pinions of the example case, and a STE and the STE maximum fluctuation value associated with the modification are calculated and simulated, as shown in Table 4 and Fig. 10, respectively.

Fig. 10STE simulation associated with arcuate dedendum modifications of pinions

In the case of Fig. 10, the amplitudes of STE associated with arcuate dedendum modifications of pinions versus engagement-in-and-out positions is greater than that associated with linear addendum modifications of face gears, while, the amplitudes of STE simulation results of two version tooth profile modifications versus pitch positions are the same.

3.2. Mesh stiffness and natural frequency simulation and analysis

Mesh stiffness and natural frequencies of face gear drives associated with linear addendum modifications of face gears are equal to those without any modifications, due to mesh stiffness of face gear drives not being impacted by linear addendum modifications of face gears.

According to the parameters listed in Table 2 and the equations listed in Table 1, the mesh stiffness of the example case of face gear drives without any modifications and associated with two version tooth profile modifications are simulated, as shown in Fig. 11.

Fig. 11Mesh stiffness simulations of the example case

a) Without modifications / With linear addendum modifications

b) With arcuate dedendum modifications

Introducing the mesh stiffness, as given in Fig. 11, into Eq. (5), the natural frequencies of the example case of face gear drives associated with two version tooth profile modifications are simulated, as shown in Fig. 12.

Fig. 12Natural frequency simulations of the example case

a) Without modifications / With linear addendum modifications

b) With arcuate dedendum modifications

In the case of Fig. 11 and Fig. 12, the average mesh stiffness and natural frequencies of the example case of face gear drives associated with arcuate dedendum modifications of pinions is less than those with linear addendum modifications of face gears, due to minimum life section width reductions caused by arcuate dedendum modifications of pinions.

3.3. Simulation and analysis of relationships between accelerations and velocities

The relationships between accelerations and velocities of the example case of face gear drives without any modifications and associated with two version tooth profile modifications are simulated, as shown in Fig. 13, by introducing the STE simulation results, as shown in Fig. 9 and in Fig. 10, into Eq. (5).

As illustrated in Fig. 13, whatever linear addendum modifications of face gears or arcuate dedendum modifications of pinions would not cause dynamic behavior crises of face gear drives, that is, except vibration suppressions, not any side effects of dynamic behaviors of face gear drives could be produced by such two version tooth profile modifications.

Fig. 13Relationships between accelerations and velocities of the example case simulated

a) On the pinion

b) On the face gear

3.4. Dynamic mesh force simulation and analysis

In order to evaluate the effects of two version tooth profile modifications on dynamic mesh force suppressions of face gear drives, the dynamic mesh forces of the example case of face gears without any modifications and associated with two version tooth profile modifications are simulated, as shown in Fig. 14.

In the case of Fig. 14, the suppressions of two version tooth profile modifications on dynamic mesh forces of face gear drives are effective. Meanwhile, the dynamic mesh force suppression effect differences of two version tooth profile modifications of the example case of face gear drives could be expressed as curves, which could be defined as amplitude differences between one of tooth profile modifications and without any modifications versus frequencies, as shown in Fig. 15.

According to Fig. 15, an average value of suppressions could be defined as:

where $E_s$ is an average value of suppressions, $A_{si}$ is an amplitude of suppression curves versus frequencies, $N$ is a number of frequencies.

Fig. 14Dynamic mesh forces of the example case simulated

a) Without any modifications

b) With linear addendum modifications of face gears

c) With arcuate dedendum modifications of pinions

Fig. 15Suppression effect comparisons of two version tooth profile modifications of the example case

According to the results, as shown in Fig. 15, the average values of suppressions of two version tooth profile modifications of the example of face gear drives are calculated as listed in Table 5.

Based on the limited simulation results in the issue, as shown in Fig. 15, when two version tooth profile modifications are equivalent, the suppression effect of the linear addendum modification of face gears on dynamic mesh forces of face gear drives is better than that caused by arcuate dedendum modifications of pinions at whole frequencies. While, the suppression effect of the linear addendum modification of face gears on dynamic mesh forces of face gear drives is worse than that caused by arcuate dedendum modifications of pinions at the mesh frequency. Considered the suppression average values of two version tooth profile modifications on dynamic mesh forces of the example case of face gear drives, and dynamic mesh force suppression effects at the mesh frequency, the arcuate dedendum modification of pinions is recommended for face gear drives.