In this study, we analyze the static load sharing characteristics of coaxial dual-output spur gear split-combined-torsion transmission systems, which are widely used in aerospace and marine applications due to their compact structure, lightweight design, and high load-carrying capacity. Despite the inherent symmetry in the configuration of these spur gear systems, uneven load distribution among branches remains a critical issue. We focus on the impact of meshing phase differences between spur gear pairs on static load sharing performance, considering factors such as torsional deformation of shafts, support deformations, and time-varying meshing stiffness of spur gears. By establishing a static analysis model that incorporates equilibrium conditions and deformation compatibility, we derive equations to evaluate load sharing coefficients and investigate how phase differences influence system performance. Our findings provide insights into optimizing spur gear transmission designs for improved load distribution.
The static analysis model for the coaxial dual-output spur gear transmission system is developed by representing components as interconnected springs, accounting for deformations in spur gears, shafts, and supports. The coordinate systems are defined to analyze forces and displacements in the X and Y directions, with stiffness parameters including meshing stiffness for spur gear pairs and support stiffness for bearings. The model considers two input paths: one for the inner output shaft and another for the outer output shaft, each with left and right branches. The meshing stiffness of spur gears, denoted as \( K_{kijnps} \) and \( K_{kijnBh} \), varies with time due to phase differences, where indices represent specific branches and spur gear pairs. For instance, the meshing stiffness for spur gears in the split-level and combined-level stages is calculated using potential energy methods, incorporating Hertz contact, bending, shear, and axial compression effects. The general formula for time-varying meshing stiffness of a spur gear pair is given by:
$$ k = \sum_{i=1}^{2} \left[ \frac{1}{ \frac{1}{k_{h,i}} + \frac{1}{k_{b1,i}} + \frac{1}{k_{s1,i}} + \frac{1}{k_{f1,i}} + \frac{1}{k_{a1,i}} + \frac{1}{k_{b2,i}} + \frac{1}{k_{s2,i}} + \frac{1}{k_{f2,i}} + \frac{1}{k_{a2,i}} } \right] $$
where \( k_{h,i} \) is the Hertz contact stiffness, \( k_{b1,i} \) and \( k_{b2,i} \) are bending stiffnesses, \( k_{s1,i} \) and \( k_{s2,i} \) are shear stiffnesses, \( k_{f1,i} \) and \( k_{f2,i} \) are fillet foundation stiffnesses, and \( k_{a1,i} \) and \( k_{a2,i} \) are axial compression stiffnesses for the driving and driven spur gears, respectively. This approach allows us to model the dynamic behavior of spur gears under load, essential for understanding phase difference effects.
The static equilibrium equations for the spur gear transmission system are derived from torque and force balance conditions. For each branch, the torque equations ensure that input torques are distributed through the spur gear pairs, while force equations account for displacements due to deformations. The general torque balance for a branch is expressed as:
$$ \begin{cases}
T_{kif} – T_{kip} = 0 \\
T_{ki1s} + T_{ki2s} – i_1 T_{kip} = 0 \\
T_{kijs} – T_{kijh} = 0 \\
T_{kB} – i_2 \sum_{i} \sum_{j} \sum_{k} T_{kijh} = 0
\end{cases} $$
where \( T \) represents torques on spur gears, \( i_1 \) and \( i_2 \) are transmission ratios for split-level and combined-level spur gears, and indices denote specific branches and spur gear pairs. The relationship between meshing forces and torques for spur gears is given by:
$$ \begin{cases}
T_{kip} – (F_{ki1nps} + F_{ki2nps}) r_{bp} = 0 \\
T_{kijs} – F_{kijnps} r_{bs} = 0 \\
T_{kijh} – F_{kijnBh} r_{bh} = 0 \\
T_{kB} – r_{bB} \sum_{i} \sum_{j} \sum_{k} F_{kijnBh} = 0
\end{cases} $$
Here, \( F \) denotes meshing forces, and \( r_b \) represents base circle radii of the spur gears. The force equilibrium equations in the X and Y directions incorporate support stiffness and displacement components, as shown below:
$$ \begin{cases}
\frac{T_{ki1s}}{r_{bs}} \cos \alpha_{ki1p} + \frac{T_{ki2s}}{r_{bs}} \cos \alpha_{ki2p} – K_{kipx} x_{kip} = 0 \\
\frac{T_{ki1s}}{r_{bs}} \sin \alpha_{ki1p} + \frac{T_{ki2s}}{r_{bs}} \sin \alpha_{ki2p} – K_{kipy} y_{kip} = 0 \\
\frac{T_{kijs}}{r_{bs}} \cos \alpha_{kijs} + \frac{T_{kijh}}{r_{bh}} \cos \alpha_{kijh} – K_{kijsx} x_{kijs} = 0 \\
\frac{T_{kijs}}{r_{bs}} \sin \alpha_{kijs} + \frac{T_{kijh}}{r_{bh}} \sin \alpha_{kijh} – K_{kijsy} y_{kijs} = 0 \\
\frac{T_{kijs}}{r_{bs}} \cos \alpha_{kijs} + \frac{T_{kijh}}{r_{bh}} \cos \alpha_{kijh} – K_{kijhx} x_{kijh} = 0 \\
\frac{T_{kijs}}{r_{bs}} \sin \alpha_{kijs} + \frac{T_{kijh}}{r_{bh}} \sin \alpha_{kijh} – K_{kijhy} y_{kijh} = 0 \\
\sum_{k} \sum_{j} \sum_{i} \frac{T_{kijh}}{r_{bh}} \cos \alpha_{kijB} – K_{kBx} x_{kB} = 0 \\
\sum_{k} \sum_{j} \sum_{i} \frac{T_{kijh}}{r_{bh}} \sin \alpha_{kijB} – K_{kBy} y_{kB} = 0
\end{cases} $$
In these equations, \( x \) and \( y \) are displacements, \( K \) represents support stiffness, and \( \alpha \) angles define the direction of meshing forces for the spur gears. The deformation coordination equations account for relative angular displacements due to torsional deformations, meshing deformations, and backlash in spur gears. The total angular displacement for a branch is derived as:
$$ \phi_{kijp} = i_1 (\phi^1_{kij} + \phi^2_{kij} + \phi^3_{kij}) + \phi^4_{kij} + \phi^5_{kij} + \phi^6_{kij} $$
where \( \phi^1_{kij} \) and \( \phi^2_{kij} \) are angular displacements from meshing deformations of spur gears, \( \phi^3_{kij} \) is from shaft torsion, \( \phi^4_{kij} \) is from backlash, and \( \phi^5_{kij} \) and \( \phi^6_{kij} \) are from relative displacements of spur gear centers. The coordination condition requires that the total angular displacements for left and right branches are equal:
$$ \phi_{ki1p} – \phi_{ki2p} = 0 $$
To compute relative displacements, we consider the initial and loaded states of spur gears. For a spur gear pair under load, the pressure angle and center distance change, leading to a shift in the meshing position. The modified pressure angle \( \alpha^* \) and center distance \( O_1^*O_2 \) are calculated as:
$$ \alpha^* = \arccos \left( \frac{r_{b1} + r_{b2}}{O_1^*O_2} \right), \quad O_1^*O_2 = \sqrt{u^2 + (v + r_1 + r_2)^2} $$
where \( u \) and \( v \) are relative displacements, and \( r_1 \), \( r_2 \) are pitch radii of the spur gears. The angular shift \( \Delta \phi \) due to displacement is then determined based on gear rotation direction, which affects the load distribution in spur gear systems.
The meshing phase difference between spur gear pairs arises from designed backlash, causing nonsynchronous engagement and variations in meshing stiffness. For two spur gear pairs, the phase difference \( \gamma_{ba}(t_1) \) at time \( t_1 \) is defined as:
$$ \gamma_{ba}(t_1) = \frac{t_2 – t_1}{T_b} $$
where \( t_2 \) is the engagement time of the second spur gear pair, and \( T_b \) is its meshing period. This phase difference leads to disparities in meshing stiffness, as illustrated in the time-varying stiffness curves for split-level and combined-level spur gears. The load sharing coefficient \( \Omega_{ki} \) for each branch is derived from the torques on the left and right branches:
$$ \Omega_{ki1} = \frac{2 T_{ki1sh}}{T_{ki1sh} + T_{ki2sh}}, \quad \Omega_{ki2} = \frac{2 T_{ki2sh}}{T_{ki1sh} + T_{ki2sh}}, \quad \Omega_{ki} = \max(\Omega_{ki1}, \Omega_{ki2}) $$
This coefficient quantifies the uniformity of load distribution among spur gear branches, with higher values indicating poorer load sharing.
We analyze the impact of phase differences on load sharing coefficients for inner and outer output shaft branches of the spur gear transmission system. The basic parameters of the spur gears used in our analysis are summarized in the following table:
| Parameter | Split-Level Spur Gears | Combined-Level Spur Gears | Other Parameters |
|---|---|---|---|
| Module (mm) | 2.4 | 3.7 | Pressure Angle: 20° |
| Number of Teeth | 36 (driving), 100 (driven) | 31 (driving), 159 (driven) | Elastic Modulus: 2.1×10^5 MPa |
| Face Width (mm) | 25 | 35 | Poisson’s Ratio: 0.3 |
For the inner output shaft branch, phase differences in both split-level and combined-level spur gears cause significant fluctuations in the load sharing coefficient. Specifically, combined-level phase differences have a greater effect than split-level ones. In the inner branch, right input phase differences influence the load sharing coefficient more than left input phase differences, whereas in the outer branch, left input phase differences have a larger impact. The fluctuation amount of the load sharing coefficient, defined as the difference between maximum and minimum values over a phase difference range, increases with the magnitude of phase difference. For example, when phase difference is zero, minor fluctuations occur due to manufacturing errors in spur gears, but nonzero phase differences lead to abrupt changes, worsening load sharing performance.
The relationship between phase difference and load sharing coefficient fluctuation can be expressed mathematically. Let \( \Delta \Omega \) represent the fluctuation amount, and \( \gamma \) the phase difference angle. For spur gears in the split-level, the fluctuation is approximated by:
$$ \Delta \Omega \approx A |\gamma| + B $$
where \( A \) and \( B \) are constants derived from regression analysis of simulation data. Similarly, for combined-level spur gears, the fluctuation is more pronounced, as shown in the following table summarizing key results:
| Branch Type | Gear Stage | Phase Difference Range (rad) | Average Fluctuation \( \Delta \Omega \) | Remarks |
|---|---|---|---|---|
| Inner Output Shaft | Split-Level Spur Gears | -0.1 to 0.1 | 0.05 – 0.15 | Right input has higher effect |
| Inner Output Shaft | Combined-Level Spur Gears | -0.1 to 0.1 | 0.1 – 0.25 | Larger fluctuation than split-level |
| Outer Output Shaft | Split-Level Spur Gears | -0.1 to 0.1 | 0.04 – 0.12 | Left input has higher effect |
| Outer Output Shaft | Combined-Level Spur Gears | -0.1 to 0.1 | 0.08 – 0.20 | Phase difference sign matters |
The underlying mechanism for these fluctuations lies in the nonsynchronized meshing stiffness of spur gears. When phase difference is zero, meshing stiffness curves align, resulting in relatively stable load sharing. However, with nonzero phase differences, stiffness curves shift, creating instances where one branch of spur gears carries significantly more load than the other. This is modeled by the difference in meshing stiffness \( \Delta k = k_a – k_b \) over a meshing period, where \( k_a \) and \( k_b \) are stiffnesses of two spur gear pairs. The load imbalance \( \Delta F \) in a branch is proportional to this stiffness difference:
$$ \Delta F \propto \int_{0}^{T} |\Delta k(t)| \, dt $$
where \( T \) is the meshing period of the spur gears. This integral highlights how phase differences amplify load imbalances over time, leading to increased fluctuations in the load sharing coefficient.
In conclusion, our analysis of coaxial dual-output spur gear split-combined-torsion transmission systems demonstrates that meshing phase differences significantly degrade static load sharing performance. The combined-level spur gears exhibit a greater sensitivity to phase differences compared to split-level spur gears, and the influence varies between inner and outer output branches. To minimize load sharing fluctuations in spur gear systems, designers should aim to eliminate phase differences or optimize branch-specific parameters, such as setting negative phase differences for right input combined-level spur gears in inner branches and positive ones for left input in outer branches. This study underscores the importance of phase difference management in spur gear transmission design for achieving uniform load distribution and enhanced reliability.