Design alternative for a helicopter final stage planetary gearbox

Traditional planetary gear systems, featuring a central external sun gear encircled by planet gears and an internal ring gear, have size, weight, and load distribution limitations. A new orbicular gear mechanism (OGM) has been introduced and designed to replace the epicyclic planetary systems. The OGM includes at least two face gears, one as a sun gear and one as a ring gear, rotating around a central axis, with an external spur or helical planetary gear group connected to a carrier. The concept and the arrangements of the OGM are explained, and a case study of a B-type OGM for a final-stage helicopter gearbox has been designed and studied.

1 Introduction

Before the 1990s, face-gear drive systems were typically employed in low-power applications. However, during the 1990s, helicopter manufacturers initiated extensive research to develop lightweight and reliable main rotor drive systems with advanced torque-splitting capabilities. Consequently, face-gear drives gained popularity due to their numerous advantages [2,3,11].

A face-gear drive system consists of an involute spur or helical pinion engaged with a face gear. It is primarily employed to transmit torque between intersecting and non-parallel shafts. The face-gear drive system facilitates weight and volume reduction through load sharing or torque-splitting capabilities [6], making it particularly suitable for helicopter and marine transmissions, as exemplified in Figure 1.

Epicyclic gear mechanisms have long been pivotal in helicopter transmissions, offering advantages over fixed-center countershaft gear systems. The ANSI/AGMA 6123-C16 standard has detailed various epicyclic gear arrangements, serving as a foundation for power transmission designers to enhance existing designs and innovate new solutions [1]. This article introduces a new concept of face gear planetary systems, similar in arrangement to traditional epicyclic planetary gear systems. The new concept replaces the epicyclic sun and ring gears with ring and sun face gears, each meshing with a spur or helical pinion. The concept of the new face gear planetary system, known as the orbicular gear mechanism (OGM), is presented conceptually, followed by a case study of an OGM design for a final-stage helicopter transmission system.

The study methodology primarily uses theoretical calculations and quasistatic system-level simulations, which are then validated using finite element analysis (FEA). The main aspects of the case study demonstrate how to view the OGM system and test its conceptual validity as a new design for a final-stage helicopter transmission system.

1.1 Literature

The last three decades have seen significant advancements in the use of face gears for heavy-load application gearboxes. Face gears have shown promising benefits, including weight reduction, improved torque-splitting capabilities, high-contact ratios, various shaft angle configurations, and volume reductions [13]. However, they also have some limitations, such as the limited face width, which constrains their load- carrying capacity and, consequently, the sizing of the face gear.

Litvin et al. (2000) studied face gears extensively, focusing on profile generation, contact analysis, efficiency calculations, the effect of backlash, and various applications in helicopter transmission systems [9,14]. Basstein et al. (1993) detailed the theoretical methodology for calculating bending and contact stresses using modified coefficients for Hertzian contact stress and ISO6336 bending stress calculations [7]. Guingand et al. (2005) conducted an analytical quasistatic load-sharing analysis of the split torque face gear mechanism [8]. Li et al. (2019) investigated the effect of floating face gears on load sharing in multiple planet gears meshed with a face gear mechanism [15]. Mustafa A. (2022) has studied the split torque face gear’s nonlinear dynamics, the effects of backlash, and the effects of unequal torque and power transmitted through different paths [18]. Handschuh et al. (2013) explored new types of planetary face-gear drives as an alternative gearbox layout for helicopter transmission systems [9].

Planetary gear systems can be categorized into three primary configurations based on their application: [9]

I. Single input, single output, and one fixed element: In this configuration, the system includes one input, one output, and one stationary component. The input shaft can be the sun gear or the carrier with the output being the remaining element. This setup functions as a speed reducer or speed multiplier.

II. Single input, dual output, and no fixed element: The system comprises one input and two outputs without fixed elements. The input shaft is typically the carrier, while the output is the sun and ring gears. This arrangement operates as a differential, distributing the output torque between the two output shafts.

III. Dual input, single output, and no fixed element: This configuration involves two input shafts and one output shaft with no fixed elements. It serves as a speed combiner, where the output speed is a linear combination of the speeds of the two input shafts.

1.2 OGM Arrangements

The OGM is similar to an epicyclic layout consisting of at least two face gears, a sun, and a ring gear, rotating around a central axis, meshing with an external spur or helical planetary gear connected to a carrier, as shown in Figure 2d.

OGM could have different shaft angle configurations, as shown in Figures 2b and 2d. As explained in [16], several options and possible selections are based on each application type to find an alternative OGM compared to the traditional planetary mechanism. A schematic presentation of the different arrangements of the OGM is shown in Table 1.

The table describes the OGM arrangements similarly to the ANSI/AGMA 6123-C16 standard, which includes the formulations to calculate the gear ratios, rotating and fixed components, the direction of rotation, and the fundamental frequency of the gear meshes. More theoretical calculations considering the maximum number of planets, contact and bending stress, conditions for correct meshing calculations, backlash, and load distributions are presented in detail in the following sections.

The selection of the proper layout of the OGM is dependable on the application required. For example, the angular speed of the carrier is a limitation for high-speed applications of OGM. This limitation exists because of the centrifugal load (due to the inertia of planetary gears) that affects the bearings of the carrier, which requires having the OGM configuration with a fixed carrier.

1.2.1 Maximum Number of Planets

The primary objective of the OGM is to achieve a load distribution among the planet gears. Increasing the number of planet gears reduces the load per gear mesh, hence the bearing loads. This allows the designer to reduce the sizes of the planets and the bearings.

The sun gear’s inner diameter limits the maximum number of planets of the OGM. The tip diameter of the planets is compared with the perimeter of the sun gear (inner diameter). The maximum number of planets is calculated using Equation 2, where the equality condition in Equation 1 should be achieved with a safety clearance. This clearance is necessary for the carrier bearing assembly and lubrication considerations. Equations 1 and 2 are valid for 90° shaft angle configurations. OGM configurations with shaft angles other than 90° are limited to a lower number of planets due to the small size of the sun gear, and they are not included in this study.

 

 

Where:

n_p(max) is rounded to the smaller integer value.
a is the planet gear’s addendum length in (mm).
b₁ is the face gear face width in (mm).
b₂ is the planet gear’s face width in (mm).
d_2i is the inner diameter of the face gear in (mm).
r_p1 is the pitch radius of the planet gear in (mm).
Z₁ is the number of teeth of the sun gear in (mm).
Z₂ is the number of teeth of the sun planet in (mm).

The number of teeth of the sun and ring face gears should be correctly divisible by the number of planet shafts to have a uniform distribution of the meshed planet gear along the perimeter of the face gear, as shown in Figure 3.

Where:

Z₁ is the number of teeth of the sun gear.
Z₃ is the number of teeth of the ring gear.
n is the number of planets.

1.2.2 Centrifugal Forces

The OGM system centrifugal effects could be reduced using the helix planet gear instead of the spur gear. The axial, centrifugal force resulting from the translational motion of the planet shafts can be reduced by increasing the gear mesh force’s axial component to the opposite direction of the centrifugal force. The centrifugal force is calculated first using Equation 4 to calculate the suitable helix angle. Then, the helix angle is calculated as a function of the gear mesh tangent force, as explained in Equation 5.

The centrifugal force of the OGM is calculated using Equation 4:

Where:

F_a is the axial centrifugal force (N).
m is the mass in (kg).
w is velocity in (rad/s).
R is the distance from the center of rotation to the center of mass (cg) in (m).

Based on the tangential force at the gear mesh F_t and the axial force due to the centrifugal force, the suitable helix angle β to cancel the centrifugal force can be calculated using Equation 5:

However, there is a limit to how much axial force can be reduced using only the helix angle. This limitation is based on the maximum allowable helix angle and the amount of tangential force at the gear mesh. Similarly, the limitation is extended to the maximum velocity allowed for the planet by substituting Equation 4 into Equation 5.

1.2.3 Efficiency

Letvin et al. (2004), in their study of the planetary systems using face gears, provided a simple methodology for calculating the planetary face gear train efficiency [9]. Applying the same method, the efficiency of the orbicular gear train could be calculated using Equations 6-9.

Where:

P_a is the input power in (kW).
P_L is the power loss in (kW).
η_ab  is the gearbox efficiency.
M_b is the output torque in (N.m).
ω_a is the input in (rpm).
(M_aω_a – P_L) represents the output power, and M_aω_a represents the input power. To determine the power loss, a coefficient ψ that represents the gear mesh efficiencies is introduced in Equation 8:

Where:

η_(p) is the total efficiency of the gear meshes.

The power loss in Equation 7 is then further simplified into Equation 9:

Where:

a, b are the input and the output and can be substituted with 1 for the sun gear, 2 for the ring gear, c for the carrier and 3 for the planet gear.

The calculation does not consider the number of gear meshes. Rather, it considers the total efficiency of the gear meshes in the calculation.

1.2.4 Stress Calculation

Basstein et al. (1993) have described and experimentally validated a methodology of calculating the contact and bending stresses for face gears using the Iso6336 and Din 3991 with modified factors that consider the different geometry and contact pattern of the face gears [5, 7, 10].

1.2.4.1 Tooth Root Bending Stress

When assuming equal load distribution along the face gear contact line, the nominal and operational root bending stresses are calculated using Equations 10 and 11:

 

 

Where:

Y_fa  is tooth form factor.
Y_sa  is stress correction factor.
Y_ε is contact ratio factor.
Y_β is helix angle factor.
K_A is application factor: The application factor K modifies the nominal load to account for incremental gear loads from outside sources.
K_v is dynamic factor: The load increases caused by internal dynamic effects are considered by the dynamic factor K_v. Since the meshing is similar, K_v is calculated using the helical parallel gear approach.
K_Fα is transverse load factor: To account for the impact of the non-uniform load distribution across the gear’s face width.
K_Fβ is width factor: To account for the impact of the helix angle on width change across the face width.
b is face width in (mm).
m_n is normal module in (mm).

Where:

α_1max, α_1min are the pressure angles at the beginning and end of contact line of the face gear.

The contact line of face gears differs from that of a spur gear. It is inclined along the face width of the face gear from the toe to the heel of the teeth, which resembles the helical gear contact pattern. It is affected by the pressure angle at the most top and bottom point of contact along the teeth face. The helix angle factor considers this change and can be calculated using Equations 13, 14, 15.

 

 

 

 

 

 

 

 

 

1.2.4.2 Contact stress

The contact stress is calculated using the Hertzian stress formula with modified factors to accommodate the differences between the face gear and the parallel gears.

Assuming an equal load distribution along the contact line, the nominal and operational contact stresses are calculated using Equations 16, 17.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Where:

Z_E is elasticity factor: material properties’ impact.
Z_H is zone factor: the influence of the tooth curvature at the calculation point.
Z_ε is contact ratio factor: the impact of contact line length.
Z_β is helix angle factor: the impact of the helix angle on the inclined lines of contact’s (spur) face gears.
K_Hβ is surface face load factor.
K_Hα is flank transverse load factor.

For each point of contact, the reduced radius of curvature can be computed mathematically. These computations demonstrate that the face radius of curvature only matters at smaller face gear-pressure angles, and it can be calculated using Equation 22. Then, a more straightforward calculation approach is feasible, in which the planet gear’s radius of curvature may be taken into consideration as being crucial for the stress computation (the face gears’ (“rack”) radius of curvature being comparatively big and presumed to be infinite). The pressure angle of the planet gear is used to indicate the calculating point for contact stress.

Where:

d₁ is pitch diameter in (mm).
α_n is normal pressure angle in (degrees).
α_p is pressure angle at the point of calculation in (degrees)

Where:

L is length of the line of contact in (mm).
C_n is correction factor for the line of contact.
α_p is transverse contact ratio at the middle of face width.

1.2.5 Backlash Effect on Contact Pattern

Prior experimental research has demonstrated that correctly installing planet gear and face gears is essential for good operation and load carrying [12].

Adjusting the face gear backlash has a notable impact on the contact pattern. Displacing the face gear beyond the meshing position increased backlash and a shift in the contact pattern from the heel to the toe. Conversely, the backlash is reduced when the face gear was brought into the mesh, and the contact pattern shifted from the toe to the heel, as shown in the Figure 6.

The proper choice of the backlash considers the conditions:

• Edge contact.
• Double flank contact.
• Desired load-carrying conditions.

AGMA 916-A17 presented an analytical formula to calculate the maximum backlash based on the tooth thickness, the pressure angle, and the pitch diameter, as shown in Equation 24 [2].

Where:

j_wmax is the maximum backlash in (mm).
d_wf1max  is the max operating pitch diameter of the spur planet gear in (mm).
s_1min is the minimum tooth thickness of the planet gear in (mm).
s_2min is the face gear tooth thickness at the intersection of the reference diameter and the reference plane in (mm).
α is the operating pressure angle in (degrees).
α_wf1max  is the maximum functional operating pressure angles of the spur planet gear in (degrees).

2 A Case Study

The design of an OGM system was built and analyzed quasi-statically using Romax. A reduction ratio, size limits, and load configuration were given as the boundary conditions for the development. A trade study was first conducted to select the suitable OGM arrangement, which led to the choice of a 1-stage- B-type OGM as a suitable configuration as shown in Figure 7.

2.1 Gearbox Sizing

The initial sizing parameters of the designed gearbox are presented in Table 2. In the same table, the power circulation and efficiency of the B-type OGM formulas and their results are explained. The efficiency calculation formula is derived from Letvin’s efficiency calculation methodology, explained in [9]. While the gear mesh efficiency is an assumption, the total efficiency depends on the gear mesh efficiency and the number of teeth of the sun, ring face gears, and sun planet, ring planet gears.

The initial sizing calculations are done using the formulas given in Table 2.

During the design phase, the carrier design and the selection of the bearings were the most challenging design topics. Since the carrier is rotating, the applied torque is high, which requires a very stiff design. This has been achieved by the iterative design method, where the FE analysis of the designed carrier is checked in terms of its maximum deflection and stress concentration points. The aim was to keep the deflections below 0.4 mm as shown in Figure 8. The bearing selection and positioning were chosen based on the dominant loads on the planetary shafts to decrease the moment resulting from unequal loading on the right and left bearings.

2.2 Gear Mesh Forces

Another design perspective involves theoretically checking the bending and contact stresses using the methodology explained in [7]. This required knowledge of the gear mesh forces, which were calculated by Romax. To validate the GMF results, an FE model was used where torques and velocities were applied to the model, and the FEA results were compared with the Romax results, as shown in Figure 9.

The results from Romax and FEA were very close to each other. However, it was observed that the force magnitude differed for each gear mesh. This indicates different load carrying for each tooth, leading to non-uniform carrier and bearing loads, as shown in Figure 10. The main issue of non-uniform load sharing is that overloaded teeth and bearings will fail prematurely.

One of the main parameters that contribute to the load-carrying capacity of the gear mesh teeth is the backlash. A parametric study in Romax shows the backlash has a linear relationship with the load carried by the gear mesh, as shown in Figure 11. By adjusting the backlash for every gear mesh, the load sharing could be uniform.

2.3 Backlash and Load Sharing

According to AGMA 916, the maximum backlash for the face gear is calculated to be 0.25 mm, and a value of 0.2 mm on the ring gear and 0.23 mm on the sun gear was found suitable to load the bearings equally, as shown in Figure 12. However, the uniform load shared between all the planets has yet to be achieved.

A proposed solution for this issue suggests using floating sun planets by mounting the sun planet on the planet shaft using a spline coupling with a clearance. Since both the sun planet and ring planet gears are rigidly mounted onto the same shaft, increasing or decreasing the backlash through the radial displacement of one of them results in an opposing action on the other planet, as shown in Figure 13b.

This behavior requires a torsionally rigid and radially floating mechanism that allows both planet gears to adjust their backlash independently while meshing and rotating correctly. This was solved by floating the sun planets by mounting them on the planetary shafts using a coupling spline, as shown in Figure 13a.

2.3.1 Floating

A parametric study was conducted to analyze the behavior of a straddle-mounted planet shaft with a floating spline under varying loads. The results are as follows:

• Low torque: At low torque, the load is transmitted by four teeth on the spline, positioned at ±twice the spline pressure angle (30 degrees) from the load application point. For instance, if the load is applied at an angle of 270°, the minimum clearance occurs at 210° and 330°, but on different flanks.
• Medium torque: In the transitional region, the additional torque causes the contact to shift primarily to the right flank. However, the load is still shared over several teeth, leading to additional radial forces.
• Optimal operating torque: At optimal operating torque, the load is distributed across more teeth on the right flank, resulting from the imbalance of forces aligning the spline.

This detailed analysis highlights the planet gear’s adaptive behavior under different loading conditions, emphasizing the importance of proper spline clearance to ensure effective load distribution as shown in Figure 14.

Applying the same principle to the simulation model, a similar spline contact pattern is observed in Figure 15.

However, the backlash is calculated as an input inserted into the simulation software, making it possible to check the exact value of the backlash resulting from floating the spline. A similar principle of adjusting the backlash by changing the axial displacement of helical pinion meshing along the axis of the face gear was implemented by [9], as shown in Figure 16.

2.4 Bending and Contact Stress

Table 3 lists the factors used in the calculation. The bending stress results are validated using FEA, and the difference is shown in Table 4.

3 Conclusion

The OGM arrangements presented in this article provide a systematic process of implementing face gear planetary systems and to expand the application of face gears. This allows designers to replace the epicyclic gears with face gears to gain different benefits, such as weight and size reduction.

Besides, it paves the way for more extensive research to standardize this type of gearing arrangement. The case study showed the OGM system can be analyzed using available tools, such as modeling and analysis software, and highlighted the points that need further analysis (Table 5).

Throughout the case study, the design requirements were achieved, and some advantages were gained, such as reduced weight and size compared to the original design targets. In addition, the axial loads on the mast shaft were reduced due to the opposite-direction loads coming from the OGM. Reducing the size of the mast shaft and its bearings can lead to further weight reduction.

3.1 Future Works

The OGM needs to be thoroughly inspected for micro-geometry analysis. This study serves as a starting point for identifying the new mechanism. A more detailed analysis set needs to be built based on the allowable manufacturing tolerances for the components.

Parallel to the theoretical studies, manufacturing and validation studies are planned as future work for this study. Before finalizing the detailed design phase, a comprehensive lubrication simulation set needs to be developed, and the OGM mesh lubrication characteristics need to be thoroughly understood. The current study features a titanium OGM carrier design. Based on the stiffness requirements, developing a steel carrier is also possible for future studies, with a detailed geometry generative optimization study targeting minimum deflections under transmission and flight loads.

Dynamic modeling and characteristics of the torque split face gear drives have already been modeled and analyzed. With this new model, the nonlinear dynamic response of a face-gear drive system is sought, and the dynamic stability and limit states of this structure need to be investigated.

In addition to VTOL applications, the OGM layout can be a very promising alternative for high-speed planetary applications. Due to the minimized centrifugal effects compared to traditional planetary layouts, similar concept studies can be conducted to select the appropriate configuration, and detailed thermal models need to be developed based on system lubrication and efficiency.

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