Split-torque and power-combining transmissions represent a cornerstone of modern high-power, compact drivetrain design, particularly in demanding sectors such as aerospace and marine propulsion. Among these, the configuration featuring a coaxial dual-output arrangement utilizing cylindrical gear pairs offers a compelling blend of structural compactness, high power density, and reliability. The fundamental premise of a split-torque system lies in its inherent symmetry—dividing input power across multiple parallel paths before recombining it—which theoretically promises optimal load distribution and reduced stress on individual components. However, in practical engineering applications, this ideal state of perfect load sharing is seldom achieved. Manufacturing tolerances, assembly misalignments, and subtle dynamic interactions invariably lead to unequal load partitioning among the parallel branches. This imbalance can precipitate premature fatigue, excessive noise, vibration, and ultimately, a reduction in the system’s operational lifespan and reliability. Consequently, a profound understanding of the factors governing load distribution is paramount for the robust design of such transmissions.
This article presents a comprehensive static analysis of a coaxial dual-output cylindrical gear split-combined-torsion transmission. The primary objective is to investigate a critical, yet often nuanced, factor affecting load equilibrium: the meshing phase difference between gear pairs in parallel branches. While traditional error analysis focuses on dimensional inaccuracies, the phasing of meshing stiffness variations—a consequence of designed backlash or assembly timing—introduces a time-varying stiffness asymmetry that significantly impacts static load sharing. We develop a detailed static model that incorporates the torsional compliance of shafts, the flexibility of bearings and supports, and, most importantly, the time-varying mesh stiffness of the cylindrical gear pairs. By establishing force equilibrium and deformation compatibility equations for the entire system, we quantify the influence of phase differences at both the split-torque (first) stage and the power-combining (second) stage on the system’s static load sharing coefficients. The findings provide crucial design insights for mitigating load imbalance in this sophisticated transmission architecture.
System Description and Modeling Framework
The subject of this analysis is a two-input, coaxial dual-output transmission system. Each input drive is distributed via a bevel gear stage (not modeled in this static analysis, as the focus is on the parallel cylindrical gear stages) to two separate split-torque paths. These paths are dedicated to driving the inner and outer output shafts concentrically. For each output shaft (inner and outer), there are two independent input branches (Left and Right). Within each branch, the power flow undergoes two distinct cylindrical gear meshing stages: a split-torque stage and a combining stage. The system’s symmetry is local but complex, with multiple parallel load paths converging at each output shaft.
To analyze the static load sharing characteristics, a lumped-parameter spring-mass model is constructed. Each mechanical element—gears, shafts, bearings—is represented by its stiffness property. The gear meshes are modeled as spring connections with time-varying stiffness, \( k_{mesh}(t) \), while shaft torsional stiffness, \( K_{shaft} \), and bearing support stiffnesses in orthogonal directions (\( K_x, K_y \)) are represented by linear springs. Key coordinates and stiffness notations are defined systematically, where indices denote the output path (U=Inner, D=Outer), input branch (L=Left, R=Right), and sub-branch (j=1,2). For instance, \( K_{U,R1}^{nps} \) represents the mesh stiffness between the input pinion and the first split gear on the Right branch of the Inner output path.
Formulation of Static Equilibrium and Compatibility
The static analysis assumes the output shafts are fixed. Application of input torque causes elastic deformation throughout the system, leading to angular displacements. The governing equations are derived from two fundamental principles: force/torque equilibrium and geometric compatibility of deformations.
Torque and Force Equilibrium
The equilibrium of torques for gears on a common shaft or in mesh is given by:
$$
\begin{aligned}
&T_{kif} – T_{kip} = 0 \\
&T_{ki1s} + T_{ki2s} – i_1 T_{kip} = 0 \\
&T_{kijs} – T_{kijh} = 0 \\
&T_{kB} – i_2 \sum_{i} \sum_{j} \sum_{k} T_{kijh} = 0
\end{aligned}
$$
where \( T \) denotes torque, \( i_1, i_2 \) are the gear ratios of the split and combining stages, and indices follow the defined convention. The relationship between mesh force \( F \) and torque is \( T = F \cdot r_b \), with \( r_b \) being the base circle radius.
Force equilibrium at each gear center, considering support stiffness in the transverse plane (X-Y), accounts for the components of mesh forces. For a generic gear, the equations are:
$$
\begin{aligned}
&\sum F_x = \frac{T_{neighbor1}}{r_{b,neighbor1}} \cos(\alpha_{neighbor1}) + \frac{T_{neighbor2}}{r_{b,neighbor2}} \cos(\alpha_{neighbor2}) – K_x \cdot x = 0 \\
&\sum F_y = \frac{T_{neighbor1}}{r_{b,neighbor1}} \sin(\alpha_{neighbor1}) + \frac{T_{neighbor2}}{r_{b,neighbor2}} \sin(\alpha_{neighbor2}) – K_y \cdot y = 0
\end{aligned}
$$
where \( x, y \) are gear center displacements, \( K_x, K_y \) are support stiffnesses, and \( \alpha \) is the angle of the mesh force line of action relative to the positive X-axis.
Deformation Compatibility
Compatibility conditions ensure that the relative angular displacements between connected components sum correctly. The total angular displacement difference \( \phi_{kijp} \) experienced by an input pinion relative to its path through branch \( j \) is a superposition of several elastic and geometric angles:
$$
\phi_{kijp} = i_1 (\phi^1_{kij} + \phi^2_{kij} + \phi^3_{kij}) + \phi^4_{kij} + \phi^5_{kij} + \phi^6_{kij}
$$
The components are defined as follows:
- \( \phi^1_{kij}, \phi^2_{kij} \): Angular deflections due to tooth bending and contact deformation at the split and combining mesh, proportional to torque and inversely proportional to mesh stiffness: \( \phi = T / (r_b^2 \cdot K_{mesh}) \).
- \( \phi^3_{kij} \): Torsional wind-up of the quill shaft connecting the two large cylindrical gears in a branch: \( \phi^3 = T_{sh} / K_{sh} \).
- \( \phi^4_{kij} \): Equivalent angular displacement due to circumferential backlash \( J \): \( \phi^4 = J/(2r) \), where \( r \) is the pitch radius.
- \( \phi^5_{kij}, \phi^6_{kij} \): Angular displacements caused by relative transverse movement of gear centers at the split and combining meshes. This is a critical component as it couples transverse bearing deflections and gear errors into the load sharing problem.
The calculation of \( \phi^5 \) and \( \phi^6 \) requires analyzing the change in the operating pressure angle and line of contact due to gear center offsets \( \Delta U \) and \( \Delta V \). From the initial meshing position, the new pressure angle \( \alpha^* \) and gear rotation angles \( \lambda_1, \lambda_2 \) are computed geometrically, leading to the angular shift \( \Delta \phi \).
The fundamental compatibility condition for the two parallel branches (j=1 and j=2) within the same input line (k,i) is that the total angular displacement from the common input pinion through each branch must be equal:
$$
\phi_{ki1p} – \phi_{ki2p} = 0
$$
This equation, for all (k,i) pairs, is the key to solving for the load distribution.
Meshing Phase Difference and Time-Varying Stiffness
The core of this investigation lies in the effect of meshing phase difference. In a perfectly symmetric and simultaneously assembled system, parallel cylindrical gear pairs would mesh in perfect unison. However, intentional backlash or assembly variations cause one gear pair to enter the meshing zone slightly earlier or later than its counterpart in the parallel branch. This timing offset results in a phase shift between their time-varying mesh stiffness functions.
The mesh stiffness \( K_{mesh}(t) \) of a cylindrical gear pair is periodic with the gear mesh frequency. It is computed using the potential energy method, considering Hertzian contact, bending, shear, axial compression, and fillet foundation deflections for both the driving and driven gears. For a double-tooth contact zone, the total stiffness is the sum of the stiffnesses from the two engaged tooth pairs. The basic parameters for the cylindrical gears used in this study are summarized in Table 1.
| Parameter | Split-Torque Stage | Combining Stage | General |
|---|---|---|---|
| Module (mm) | 2.4 | 3.7 | – |
| Number of Teeth (Pinion/Gear) | 36 / 100 | 31 / 159 | – |
| Face Width (mm) | 25 | 35 | – |
| Pressure Angle (°) | 20 | 20 | 20 |
| Young’s Modulus (MPa) | 2.1e5 | 2.1e5 | 2.1e5 |
| Poisson’s Ratio | 0.3 | 0.3 | 0.3 |
Let \( K_a(t) \) and \( K_b(t) \) represent the mesh stiffness functions of two parallel gear pairs (e.g., branch 1 and branch 2) with period \( T \). If pair \( b \) begins meshing at time \( \Delta t \) after pair \( a \), the phase difference \( \gamma_{ba} \) is defined as:
$$
\gamma_{ba} = \frac{\Delta t}{T}
$$
A positive phase difference indicates that the second pair’s stiffness profile lags behind the first. Figure 1 conceptually illustrates the resulting stiffness curves. The instantaneous stiffness difference \( \Delta K(t) = K_a(t) – K_b(t + \gamma T) \) directly influences the elastic deflection \( \phi^1 \) in each branch, thereby governing how torque is shared instantaneously.
Analysis of Phase Difference Impact on Load Sharing
The system of equilibrium and compatibility equations, incorporating the phased mesh stiffness, is solved under the condition of equal input torque per input shaft. The static load sharing coefficient \( \Omega_{kij} \) for a sub-branch is defined as the ratio of its carried torque to the average torque for its two parallel sub-branches:
$$
\Omega_{ki1} = \frac{2 T_{ki1sh}}{T_{ki1sh} + T_{ki2sh}}, \quad \Omega_{ki2} = \frac{2 T_{ki2sh}}{T_{ki1sh} + T_{ki2sh}}
$$
The load sharing coefficient for an entire input line (k,i) is the maximum of these two: \( \Omega_{ki} = \max(\Omega_{ki1}, \Omega_{ki2}) \). A value of 1.0 indicates perfect load sharing; values greater than 1.0 indicate imbalance.
The analysis systematically varies the phase difference \( \gamma \) for both the split-torque stage and the combining stage, separately for the Left and Right input branches of both the Inner (U) and Outer (D) output paths. To clearly present the volatility introduced by phase difference, we define the Load Sharing Fluctuation Amplitude \( \Delta \Omega \) as the difference between the maximum and minimum value of \( \Omega_{ki} \) over one complete mesh cycle for a given phase difference setting.
The key results are synthesized in Table 2 and the subsequent discussion.
| Transmission Stage & Path | Key Influence Trend | Relative Severity of Impact |
|---|---|---|
| Combining Stage (Both Paths) | Phase difference causes larger fluctuation \( \Delta \Omega \) than the split-torque stage. | Higher Impact |
| Split-Torque Stage (Inner Path) | Right branch phase difference influences \( \Delta \Omega \) more than Left branch. | Right > Left |
| Combining Stage (Inner Path) | Right branch phase difference influences \( \Delta \Omega \) more than Left branch, regardless of sign. | Right > Left |
| Split-Torque Stage (Outer Path) | Right branch phase difference influences \( \Delta \Omega \) more than Left branch. | Right > Left |
| Combining Stage (Outer Path) | Left branch phase difference influences \( \Delta \Omega \) more than Right branch. | Left > Right |
Detailed Observations and Mechanistic Explanation
1. General Detrimental Effect: The introduction of any non-zero meshing phase difference consistently worsens the static load sharing performance compared to the in-phase condition (\( \gamma = 0 \)). While a minor fluctuation exists even at \( \gamma = 0 \) due to modeled errors, the fluctuation amplitude \( \Delta \Omega \) increases significantly once a phase offset is introduced. This is because phase difference creates moments during the mesh cycle where one branch’s cylindrical gear pair is in a high-stiffness region (e.g., two-tooth contact) while the other is in a low-stiffness region (e.g., one-tooth contact). This stiffness mismatch directly translates into a torque imbalance via the deformation compatibility equations.
2. Dominance of the Combining Stage: The phase difference in the combining (second) stage cylindrical gear pairs has a more pronounced effect on load sharing fluctuation than that in the split-torque (first) stage. This can be attributed to the power summation point. The combining stage is where the torque from two parallel branches per input line is summed. A stiffness asymmetry at this final summation node has a more direct and un-attenuated effect on the reaction forces and the relative distribution of load between the upstream branches.
3. Asymmetry Between Inner and Outer Paths: The system is not globally symmetric. The physical layout and load paths for the inner and outer output shafts differ. This leads to differing sensitivities:
- For the Inner Output Path, the phase difference in the Right input branch consistently causes larger load fluctuation for both gear stages.
- For the Outer Output Path, the phase difference in the Left input branch’s combining stage has a larger impact, while for the split-torque stage, the Right branch’s effect remains greater.
This asymmetry must be accounted for during the tolerance allocation and assembly process.
4. Design Guidance for Phase Management: If phase differences cannot be eliminated, their impact can be mitigated by strategic orientation. The analysis suggests that configuring the system so that the combining stage phase difference is negative (branch 2 leads branch 1) for the Inner path’s Right branch, and positive (branch 2 lags branch 1) for the Outer path’s Left branch, results in a configuration with slightly lower overall load sharing fluctuation amplitudes. This is deduced from evaluating the \( \Delta \Omega \) versus \( \gamma \) plots for these specific branches.
Mathematical Synthesis and Conclusion
The static behavior of the coaxial dual-output cylindrical gear transmission can be synthesized into a governing equation derived from the compatibility condition. For a single input line with two parallel branches, ignoring transverse deflections for simplicity, the core relationship balancing the torques \( T_1 \) and \( T_2 \) in the two branches is:
$$
\frac{T_1}{r_{b1}^2 K_1(t)} + \frac{T_1}{K_{sh1}} + \frac{J_1}{2r_1} \approx \frac{T_2}{r_{b2}^2 K_2(t+\gamma T)} + \frac{T_2}{K_{sh2}} + \frac{J_2}{2r_2}
$$
given the total torque \( T_{total} = T_1 + T_2 \) is constant. Solving for \( T_1 \) yields:
$$
T_1(t) = T_{total} \cdot \frac{ \frac{1}{r_{b2}^2 K_2(t+\gamma T)} + \frac{1}{K_{sh2}} + C_2 }{ \frac{1}{r_{b1}^2 K_1(t)} + \frac{1}{K_{sh1}} + C_1 + \frac{1}{r_{b2}^2 K_2(t+\gamma T)} + \frac{1}{K_{sh2}} + C_2 }
$$
where \( C \) represents constant terms from backlash and errors. This equation explicitly shows that the instantaneous load share \( T_1/T_{total} \) is a function of the phase-shifted stiffness ratio \( K_1(t) / K_2(t+\gamma T) \). The load sharing coefficient fluctuation is therefore directly tied to the function:
$$
\Delta \Omega \propto f\left( \max_t \left[ \frac{1}{K_1(t)} – \frac{1}{K_2(t+\gamma T)} \right] \right)
$$
This model confirms that minimizing the phase difference \( \gamma \) minimizes the range of this inverse stiffness difference, leading to superior static load sharing.
In conclusion, this static analysis of a coaxial dual-output split-combined cylindrical gear transmission system has rigorously demonstrated that meshing phase difference is a critical design parameter influencing load equilibrium. The combining stage is more sensitive than the split-torque stage. Furthermore, the inherent asymmetry of the dual-output configuration leads to different sensitivity weights for the Left and Right input branches of the inner and outer paths. For optimal static load sharing performance, design and assembly processes should aim to minimize meshing phase differences, particularly in the combining stage cylindrical gear pairs. When zero phase difference is unattainable, the findings provide specific guidance on which branch’s phase orientation can be tuned to marginally improve performance. This work establishes a foundational static framework, which can be extended to include dynamic effects and multi-body system optimization for future research.