In this work, I develop a flexible multibody dynamic model (FMDM) for a space closed-loop face gear split-torque transmission system (FGSTTS). The system combines face gear meshing, coaxial transmission, and split-torque paths. My model accounts for gyroscopic effects, flexible shaft bending and torsion, bearing compliance, and time-varying mesh stiffness. I solve the resulting equations using a fixed-step Newmark integration method and investigate the natural characteristics and vibration responses. The face gear meshing interface is represented with time-varying mesh stiffness, static transmission error, and piecewise mesh clearance. Gear rotors are modeled as rigid rotor elements with gyroscopic effects. Gear shafts are modeled as Timoshenko beam elements with high-speed gyroscopic effects. Bearings are modeled as six-degree-of-freedom support elements. I find that the system enters bifurcation or chaos only when all ten face gear pairs exhibit simultaneous jumps at the same rotational speed. This property reduces the risk of the closed-loop face gear system entering undesirable nonlinear vibrations.
1. Introduction
Face gear split-torque transmission systems are attractive for high-speed and heavy-load applications because they offer high power density, compactness, and insensitivity to axial mounting errors. In a face gear split-torque transmission system, multiple face gear pairs share the load, and the power path forms a spatial closed loop. The vibration signal transmission path in such a closed-loop face gear system is more complex than in a single face gear pair or an open gear chain. Therefore, understanding the vibration characteristics of the closed-loop face gear system is essential for reducing noise and ensuring reliable operation.
Previous studies on face gear dynamics have mainly focused on open-loop gear chains. They have examined the effects of force arm radius, transmission error, tooth surface accuracy, load, and modification on vibration. However, the influence of the closed-loop structure on the face gear system vibration has not been fully explored. The deformation coordination among branches in a closed-loop face gear system inevitably affects the transmission of vibration signals and changes the overall dynamic response. Some studies have investigated multi-branch face gear systems using lumped mass models, but they often neglect the bending and torsional deformations of the shafts and the gyroscopic effects at high speeds. My work aims to fill this gap by establishing a flexible multibody dynamic model for a space closed-loop face gear split-torque transmission system that includes gyroscopic effects and shaft flexibility.
I organize the remainder of this article as follows. Section 2 describes the system structure and finite element node distribution. Section 3 presents the face gear meshing element model. Section 4 details the gear rotor element. Section 5 describes the shaft beam element. Section 6 presents the bearing support element. Section 7 gives the system equations of motion. Section 8 analyzes the natural characteristics. Section 9 investigates the vibration responses. Section 10 validates the model. Section 11 concludes.
2. System Description and Finite Element Node Distribution
The FGSTTS consists of five pinions arranged around two face gears that are mounted face-to-face. The pinion tooth surface geometry is identical for all pinions, and the face gear tooth surface geometry is also identical. Power enters through the left and right input pinions and exits through the tail transmission gear and the upper gear. The remaining two pinions are idlers that increase the number of power branches and enhance load capacity. The circumferential positions of the pinions are determined by distribution angles \(\gamma_j\) for \(j=1,2,\dots,5\).
Any two pinions are connected through the upper and lower face gears either by meshing or by coaxial connection, forming a closed loop. The face gear has non-uniform dimensions in the tooth width, tooth height, and tooth thickness directions and is simultaneously loaded in multiple directions. Therefore, the system is a spatial closed-loop structure. The axes of the upper and lower face gears coincide, forming a coaxial system. This coaxial arrangement minimizes the overall size and allows each pinion to mesh with both face gears simultaneously, maximizing the number of power transmission paths. Thus, the FGSTTS integrates the advantages of spatial closed loop, face gear transmission, split-torque transmission, and coaxial transmission.
I discretize the system into finite element nodes. The left and right input pinions and the tail transmission gear are each divided into five elements and six nodes. The left and right idler pinions are each divided into four elements and five nodes. The upper face gear shaft is divided into two elements and three nodes. The lower face gear shaft is divided into two elements and three nodes. The meshing nodes of the five pinions are numbered 4, 10, 15, 20, and 26. The meshing nodes of the two face gears are numbered 31 and 33. The support nodes of the pinion bearings are located at the two ends of each pinion shaft. The support nodes of the upper and lower face gears coincide with their meshing nodes.
| Parameter | Pinion | Face gear |
|---|---|---|
| Module \(m\) (mm) | 3.75 | 3.75 |
| Pressure angle \(\alpha\) (°) | 25 | 25 |
| Shaft angle \(\eta\) (°) | 90 | 90 |
| Number of teeth \(Z_p / Z_g\) | 23 | 143 |
| Face width \(B\) (mm) | 40 | 35 |
| Distribution angles \(\gamma_1,\gamma_2,\gamma_3,\gamma_4,\gamma_5\) (°) | 245.455, 114.545, 303.375, 56.643, 0 | |
The material density is \(\rho = 7.84 \times 10^3 \,\mathrm{kg/m^3}\), Poisson’s ratio is \(\nu = 0.29\), and Young’s modulus is \(E = 207 \,\mathrm{GPa}\). The pinion force arm radius is \(r_p = 39.1 \,\mathrm{mm}\), and the face gear force arm radius is \(r_g = 243 \,\mathrm{mm}\). The upper face gear load torque is \(T_6 = 19\,530 \,\mathrm{N\cdot m}\), and the tail transmission load torque is \(T_5 = 1\,130 \,\mathrm{N\cdot m}\).
| Parameter | Value (mm) |
|---|---|
| Pinion shaft outer / inner diameter | 58 / 50 |
| Face gear shaft outer / inner diameter | 100 / 90 |
| Element length: 1-2-3; 7-8-9 | 60 |
| Element length: 3-4-5; 9-10-11; 14-15-16; 19-20-21; 25-26-27 | 20 |
| Element length: 5-6; 11-12; 13-14; 16-17; 18-19; 21-22; 27-28; 29-30 | 40 |
| Element length: 23-24-25; 30-31 | 30 |
| Element length: 32-33 | 50 |
3. Face Gear Meshing Element with Different Phases
There are ten face gear meshing elements in the FGSTTS, denoted by \(\Pi_{ij}\), where \(i=6,7\) represents the face gear and \(j=1,2,\dots,5\) represents the pinion. The large and small gears in each meshing element are treated as rigid rotors. The generalized coordinate vector of a meshing element \(\Pi_{ij}\) is
$$ q_{ij} = [x_i, y_i, z_i, \theta_{ix}, \theta_{iy}, \theta_{iz}, x_j, y_j, z_j, \theta_{jx}, \theta_{jy}, \theta_{jz}]^T. \tag{1} $$
The mesh deformation displacement along the pinion mesh line is
$$ \delta_{ij} = (x_i + z_j)\cos\alpha + (y_i – y_j)\sin\alpha + r_i\theta_{iz} – r_j\theta_{jz} – e_{ij}(t), \tag{2} $$
where \(\alpha\) is the pressure angle, \(r_i\) and \(r_j\) are the force arm radii, and \(e_{ij}(t)\) is the time-varying transmission error. To keep all ten meshing elements consistent with the reference coordinate system, I introduce transformation coordinate systems. The transformation between the intermediate coordinate system \(S_{6j}(x_{6j},y_{6j},z_{6j})\) or \(S_{7j}(x_{7j},y_{7j},z_{7j})\) and the meshing node coordinate system is
$$ [x_{ij}, y_{ij}, z_{ij}]^T = M_{ij} [x_s, y_s, z_s]^T, \tag{3} $$
$$ M_{ij} = \begin{bmatrix} -\sin\gamma_j & \mp\cos\gamma_j & 0 \\ \pm\cos\gamma_j & -\sin\gamma_j & 0 \\ 0 & 0 & 1 \end{bmatrix}. \tag{4} $$
When \(i=6\), the subscript \(s\) corresponds to node 31, and the upper signs in \(M_{ij}\) are used. When \(i=7\), the subscript \(s\) corresponds to node 33, and the lower signs are used. Thus, the mesh deformation becomes
$$ \delta_{mn} = (x_m + z_n)\cos\alpha + (y_m + \sin\gamma_j y_n \pm \cos\gamma_j x_n)\sin\alpha + r_m\theta_{mz}\cos\gamma_j – r_n\theta_{nz}\cos\gamma_j – e_{mn}(t), \tag{5} $$
where \(m\) is the pinion meshing node number (4, 10, 15, 20, 26) and \(n\) is the face gear meshing node number (31, 33). The plus sign corresponds to node 31, and the minus sign corresponds to node 33.
Because the face gear pair has a common clearance \(2b\), the mesh deformation is replaced by a nonlinear clearance function:
$$ f(\delta_{mn}) = \begin{cases} \delta_{mn} – b, & \delta_{mn} \ge b \\ 0, & -b < \delta_{mn} < b \\ \delta_{mn} + b, & \delta_{mn} \le -b \end{cases}. \tag{6} $$
The potential energy and kinetic energy of the meshing element are
$$ U = \frac{1}{2} k_{ij}(t) [f(\delta_{mn})]^2, \tag{7} $$
$$ T = \frac{1}{2}(J_m \dot{\theta}_{mz}^2 + J_n \dot{\theta}_{nz}^2) + \frac{1}{2}[m_m \dot{x}_m^2 + m_m \dot{y}_m^2 + m_m \dot{z}_m^2] + \frac{1}{2}[m_n \dot{x}_n^2 + m_n \dot{y}_n^2 + m_n \dot{z}_n^2]. \tag{8} $$
Using Lagrange’s equations, the equation of motion for the face gear meshing element is
$$ M_m \ddot{q}_{ij} + \lambda_0 C_m \dot{q}_{ij} + \lambda_0 K_m q_{ij} = H[\lambda_0 k_{ij}(t) e_{ij}(t) + \lambda_0 c_{ij} \dot{e}_{ij}(t) – b \lambda_1 k_{ij}(t)], \tag{9} $$
where the mass, stiffness, and damping matrices are
$$ M_m = \mathrm{diag}[m_i, m_i, m_i, J_i, J_i, J_i, m_j, m_j, m_j, J_j, J_j, J_j], \tag{10} $$
$$ K_m = \lambda_0 k_{ij}(t) H H^T, \tag{11} $$
$$ C_m = \lambda_0 c_{ij} H H^T, \tag{12} $$
and the coefficient vector is
$$ H = [\cos\alpha_n, \sin\alpha_n, 0, 0, 0, r_m \cos\gamma_i, \pm \cos\gamma_i \sin\alpha, \sin\gamma_i \sin\alpha, \cos\alpha, 0, 0, -r_n \cos\gamma_i]^T. \tag{13} $$
The switching parameters are
$$ \lambda_0 = \begin{cases} 1, & \delta_{ij} \ge b \\ 0, & \delta_{ij} < b \end{cases}, \quad \lambda_1 = \begin{cases} -1, & \delta_{ij} \ge b \\ 0, & \delta_{ij} < b \\ 1, & \delta_{ij} \le -b \end{cases}. \tag{14} $$
I compute the time-varying mesh stiffness of each face gear pair, which has a fluctuation period equal to the face gear meshing period. The mesh stiffness curves for different assembly positions have phase differences. The static transmission error excitation is periodic with the meshing period and has an amplitude of \(18 \,\mu\mathrm{m}\). The phase of the time-varying mesh stiffness and transmission error is the same for the same gear pair. I set the clearance \(2b = 0.06 \,\mathrm{mm}\) for all face gear pairs and the mesh damping coefficient \(c_{ij} = 1200 \,\mathrm{N\cdot s/m}\).
| Bearing number | Parameter | Value |
|---|---|---|
| 1 – 10 | Stiffness \(k_{xx}\) (N/m) | \(1.24 \times 10^8\) |
| Stiffness \(k_{yy}\) (N/m) | \(1.6 \times 10^8\) | |
| Stiffness \(k_{zz}\) (N/m) | \(4.37 \times 10^8\) | |
| Stiffness \(k_{\theta x\theta x}\) (N·m) | \(4.18 \times 10^8\) | |
| Stiffness \(k_{\theta y\theta y}\) (N·m) | \(1.06 \times 10^8\) | |
| Damping \(c_{xx},c_{yy},c_{zz}\) (N·s/m) | 1000 | |
| 11 – 12 | Stiffness \(k_{xx}\) (N/m) | \(1.59 \times 10^8\) |
| Stiffness \(k_{yy}\) (N/m) | \(2.02 \times 10^8\) | |
| Stiffness \(k_{zz}\) (N/m) | \(3.11 \times 10^6\) | |
| Stiffness \(k_{\theta x\theta x}\) (N·m) | \(7.72 \times 10^3\) | |
| Stiffness \(k_{\theta y\theta y}\) (N·m) | \(4.43 \times 10^3\) | |
| Damping \(c_{\theta x\theta x},c_{\theta y\theta y}\) (N·s/m) | 1000 |
4. Gear Rotor Elements
In the FMDM, I treat each gear body as a rigid rotor with high-speed gyroscopic effects. The global coordinate system is \(S(X,Y,Z)\), and the body-fixed coordinate system of a rotor node is \(S_0(X_0,Y_0,Z_0)\). The gear rotor has three translational and three rotational degrees of freedom. The generalized coordinate vector is
$$ q_{k,r} = [x_k, y_k, z_k, \theta_{xk}, \theta_{yk}, \theta_{zk}]^T, \tag{15} $$
where \(x_k\) and \(y_k\) are translations along the \(X\) and \(Y\) axes, \(z_k\) is the stretch along the \(Z\) axis, \(\theta_{xk}\) and \(\theta_{yk}\) are angular deformations about the \(X\) and \(Y\) axes, and \(\theta_{zk}\) is the torsional deformation about the \(Z\) axis. The undeformed shaft rotation speed is \(\Omega_k\). The kinetic energy of the gear rotor in generalized coordinates is
$$ T_{k,r} = \frac{1}{2} m_k (\dot{x}_k^2 + \dot{y}_k^2 + \dot{z}_k^2) + \frac{1}{2} J_{Dk} (\dot{\theta}_{xk}^2 + \dot{\theta}_{yk}^2) + \frac{1}{2} J_{Zk} (\Omega_k + \dot{\theta}_{zk})^2 – J_{Zk} \dot{\theta}_{yk} \theta_{xk} \Omega_k. \tag{16} $$
Using Lagrange’s equations without potential energy terms, the equation of motion for the gear rotor without external forces is
$$ M_{k,r} \ddot{q}_{k,r} + G_{k,r} \dot{q}_{k,r} = 0, \tag{17} $$
where \(M_{k,r}\) and \(G_{k,r}\) are the rotor mass matrix and gyroscopic matrix, respectively.
5. Shaft Beam Elements
I model each gear shaft as a two-node Timoshenko beam element. The element length is \(l\), the outer radius of the hollow shaft is \(r_o\), and the inner radius is \(r_k\). The distance from the initial node \(P_i\) to any point \(P\) on the flexible beam element is \(s\). The motion of point \(P\) is decomposed into three translational and three rotational degrees of freedom: \(x(t,s)\) and \(y(t,s)\) are bending deformations about the \(X\) and \(Y\) axes, \(z(t,s)\) is axial deformation along the \(Z\) axis, \(\theta_x(t,s)\) and \(\theta_y(t,s)\) are rotational deformations about the \(X\) and \(Y\) axes, and \(\theta_z(t,s)\) is torsional deformation about the \(Z\) axis. The 12-DOF generalized coordinate vector of the beam element is
$$ q_{k,s} = [x_k, y_k, z_k, \theta_{xk}, \theta_{yk}, \theta_{zk}, x_{k+1}, y_{k+1}, z_{k+1}, \theta_{x(k+1)}, \theta_{y(k+1)}, \theta_{z(k+1)}]^T. \tag{18} $$
The beam element is treated as a dynamic rotor, so its kinetic energy is
$$ T_{k,s} = \frac{1}{2} \int_0^l \rho A (\dot{x}^2 + \dot{y}^2 + \dot{z}^2) \, ds – \int_0^l J_{pk} \Omega_k \dot{\theta}_y \theta_x \, ds + \frac{1}{2} \int_0^l J_{dk} (\dot{\theta}_x^2 + \dot{\theta}_y^2) \, ds + \frac{1}{2} \int_0^l J_{pk} (\Omega_k + \dot{\theta}_z)^2 \, ds, \tag{19} $$
where \(\rho\) is the material density, \(A\) is the cross-sectional area of the hollow shaft, and \(J_{dk}\) and \(J_{pk}\) are the diametral and polar mass moments of inertia per unit length, respectively. The potential energy of the two-node beam element is
$$ U_{k,s} = \frac{1}{2} \int_0^l \Gamma I_s \theta_z’^2 \, ds + \frac{1}{2} \int_0^l E I (\theta_x’^2 + \theta_y’^2) \, ds + \frac{1}{2} \int_0^l \mu \Gamma A [(x’ – \theta_y)^2 + (y’ + \theta_x)^2] \, ds, \tag{20} $$
where \(E\) is the elastic modulus, \(\Gamma\) is the shear modulus, \(I\) is the cross-sectional diametral inertia, \(I_s\) is the polar inertia, and \(\mu\) is the shear correction factor. From Lagrange’s equations, the beam element equation of motion is
$$ M_{k,s} \ddot{q}_{k,s} + (C_{k,s} + G_{k,s}) \dot{q}_{k,s} + K_{k,s} q_{k,s} = 0. \tag{21} $$
Here, \(M_{k,s}\), \(G_{k,s}\), \(C_{k,s}\), and \(K_{k,s}\) are the mass, gyroscopic, damping, and stiffness matrices of the beam element, respectively.
6. Bearing Support Elements
I model each bearing as a six-degree-of-freedom support element. The stiffness and damping matrices are diagonal matrices with six rows and six columns in the local coordinate system. The bearing parameters are listed in Table 3. These support elements connect the shaft nodes to the ground and provide elastic constraints in three translational and three rotational directions.
7. System Equations of Motion
The FGSTTS is divided into 34 elements of different types, as shown in the node distribution. The elements are connected through nodes. The generalized coordinate vector of the system nodes is
$$ q = [q_1, q_2, q_3, \dots, q_{33}]^T = [x_1, y_1, z_1, \theta_{1x}, \theta_{1y}, \theta_{1z}, \dots, x_{33}, y_{33}, z_{33}, \theta_{33x}, \theta_{33y}, \theta_{33z}]^T. \tag{22} $$
The equation of motion of the FGSTTS can be written as
$$ M \ddot{q} + (G + C) \dot{q} + K f(q) = \Phi, \tag{23} $$
where \(M\) is the system mass matrix, \(C\) is the system damping matrix, \(G\) is the system gyroscopic matrix, \(K\) is the system stiffness matrix, \(f(q)\) is the nonlinear function containing the gear backlash, and \(\Phi\) is the external load vector.
For natural characteristic analysis, I replace the time-varying mesh stiffness with its mean value and treat the piecewise mesh clearance function as zero. Then the dynamic equation becomes an eigenvalue problem:
$$ Q = [q, \dot{q}]^T, \tag{24} $$
$$ \dot{Q} = A Q + B, \tag{25} $$
$$ A = \begin{bmatrix} 0 & I \\ -M^{-1} K & -M^{-1}(G + C) \end{bmatrix}, \tag{26} $$
$$ B = \begin{bmatrix} 0 \\ -M^{-1} \Phi \end{bmatrix}. \tag{27} $$
The eigenvalues are
$$ \lambda_i = -\omega_i (\xi_i \pm \mathrm{i} \sqrt{1 – \xi_i^2}), \quad i = 1,2,\dots,204, \tag{28} $$
where \(\xi_i\) is the damping ratio and \(\omega_i\) is the damped natural frequency.
8. Natural Characteristics
I compute the Campbell diagram for the FGSTTS in the speed range of \(0\) to \(8500 \,\mathrm{r/min}\). The intersections of the natural frequency curves with the synchronous whirling line give the critical speeds. The natural frequencies, critical speeds, and mode shapes are summarized in Table 4.
| Order | Natural frequency (Hz) | Critical speed (r/min) | Mode shape |
|---|---|---|---|
| 1 | 83.3 | 217.4 | 1-2: P & B & T |
| 2 | 88.7 | 231.3 | 1-2-3-4-5: T; 6: P & B |
| 3 | 118.0 | 307.9 | 5: B & T |
| 4 | 138.0 | 360.1 | 3-4: B & T |
| 5 | 192.7 | 502.8 | 7: B |
| 6 | 288.8 | 753.4 | 1-2: P |
| 7 | 308.8 | 805.6 | 5: P |
| 8 | 316.8 | 826.5 | 3-4: P |
| 9 | 571.6 | 1491.2 | 1-2: P & B & T |
| 10 | 577.0 | 1505.1 | 1-2: P & B & T; 3-4-5: T |
| 11 | 625.0 | 1630.4 | 1-2-3-4: T; 5: B & T |
| 12 | 641.0 | 1672.2 | 3-4: P & B & T |
| 13 | 643.7 | 1679.1 | 1-2-3-4-5: T |
| 14 | 675.7 | 1762.7 | 1-2-3-4-5: P & B & T; 6-7: P & T |
| 15 | 809.1 | 2110.7 | 1-2: P & B & T |
| 16 | 846.5 | 2208.2 | 1-2-3-4-5: P & B & T; 6-7: P & T |
| 17 | 883.8 | 2305.6 | 1-2-3-4-5: P & B & T; 6-7: P & T |
| 18 | 910.5 | 2375.2 | 3-4: P & B & T |
| 19 | 1385.5 | 3614.2 | 1-2-3-4-5: B & T; 6-7: P & B & T |
| 20 | 2121.9 | 5535.4 | 1-2: P & B & T; 3-4: T |
| 21 | 2714.2 | 7080.6 | 1-2-3-4-5: P & B & T; 6-7: P & T |
| 22 | 2965.1 | 7735.0 | 1-2: P & B & T |
| 23 | 3030.4 | 7905.5 | 1-2: P & B & T; 3-4-5: T |
| 24 | 3050.5 | 7957.7 | 1-2: P & B & T; 3-4: T; 6-7: P & T |
In Table 4, “P” denotes translation, “B” denotes bending, and “T” denotes torsion. The numbers 1 to 7 refer to the component numbers. The mode shapes include translational-bending-torsional coupled modes, translational-bending coupled modes, bending-torsional coupled modes, pure bending modes, pure translational modes, and pure torsional modes. Pure modes mainly appear in the low-frequency range below \(2000 \,\mathrm{r/min}\). For example, the 5th critical speed of \(502.8 \,\mathrm{r/min}\) corresponds to the bending mode of the lower face gear. The 6th, 7th, and 8th critical speeds correspond to the translational modes of the input pinions, tail transmission gear, and idler pinions, respectively. The 13th critical speed of \(1679.1 \,\mathrm{r/min}\) corresponds to the torsional mode of all five pinions. Thus, the system mainly exhibits unidirectional vibration below \(2000 \,\mathrm{r/min}\) and multi-directional coupled vibration above \(2000 \,\mathrm{r/min}\).
9. Vibration Responses
I solve the dynamic equations using the fixed-step Newmark numerical integration method. For each rotational speed, I set 500 meshing cycles, and each cycle contains 100 discrete points. I increase the rotational speed from \(200 \,\mathrm{r/min}\) to \(15\,000 \,\mathrm{r/min}\) in 400 steps to explore the sweep-frequency vibration characteristics. The root-mean-square (RMS) curves of the dynamic transmission error (DTE) of the face gear pairs are computed.
The RMS curves of the ten face gear pairs show similar trends. The DTE RMS curves of the pinion meshing with the upper and lower face gears fluctuate synchronously. All DTE RMS values increase significantly in the speed range of \(9000\) to \(12\,000 \,\mathrm{r/min}\). However, the RMS curves of different pinions differ. The DTE of \(\delta_{61}\), \(\delta_{71}\), \(\delta_{62}\), and \(\delta_{72}\) jump near \(10\,618 \,\mathrm{r/min}\), while \(\delta_{63}\), \(\delta_{73}\), \(\delta_{64}\), and \(\delta_{74}\) jump near \(8964 \,\mathrm{r/min}\). Near \(7889.4 \,\mathrm{r/min}\), the five face gear pairs with the upper face gear show peaks, while those with the lower face gear do not. Near \(5491.6 \,\mathrm{r/min}\), the lower face gear pairs show peaks, while the upper ones do not.
In the speed range of \(10\,000\) to \(12\,000 \,\mathrm{r/min}\), the vibration characteristics of different pinions also differ. The RMS curves of \(\delta_{61}\), \(\delta_{71}\), \(\delta_{62}\), and \(\delta_{72}\) peak near \(10\,618 \,\mathrm{r/min}\). The RMS curves of \(\delta_{63}\), \(\delta_{73}\), \(\delta_{64}\), and \(\delta_{74}\) peak near \(11\,279 \,\mathrm{r/min}\) and \(11\,445 \,\mathrm{r/min}\). The RMS curves of \(\delta_{65}\) and \(\delta_{75}\) peak near \(10\,949 \,\mathrm{r/min}\). At some special speeds, such as \(3507.3 \,\mathrm{r/min}\), \(5491.6 \,\mathrm{r/min}\), and \(7889.4 \,\mathrm{r/min}\), local peaks appear for both upper and lower face gear pairs. These special speeds are close to the critical speeds listed in Table 4. For example, \(3507.3 \,\mathrm{r/min} \approx n_{19th}\), \(5491.6 \,\mathrm{r/min} \approx n_{20th}\), and \(7889.4 \,\mathrm{r/min} \approx n_{23th}\). These phenomena are resonances. The difference between the resonance speed and the critical speed can be up to about 3% due to the discrete step size in the sweep calculation, the sensitivity of the numerical solution to initial conditions, and the use of mean mesh stiffness in the critical speed calculation versus time-varying mesh stiffness in the DTE RMS calculation.
Thus, I identify the first vibration characteristic of the FGSTTS: with the tail transmission axis as the symmetry axis, the left and right input pinions meshing with the same face gear vibrate almost identically, and the left and right idler pinions meshing with the same face gear also vibrate almost identically.
The jump speeds of \(8964 \,\mathrm{r/min}\), \(10\,618 \,\mathrm{r/min}\), and \(14\,091 \,\mathrm{r/min}\) can easily cause vibration bifurcation. At \(8964 \,\mathrm{r/min}\), the DTE phase diagrams of representative face gear pairs \(\Pi_{61}\), \(\Pi_{63}\), and \(\Pi_{65}\) show dispersed rings, and the Poincaré sections show ordered rings, indicating quasi-periodic vibration. At \(10\,618 \,\mathrm{r/min}\), the system also enters quasi-periodic vibration. At these two jump speeds, only some of the face gear pairs exhibit jumps in their DTE RMS curves. However, at \(14\,091 \,\mathrm{r/min}\), all ten face gear pairs exhibit simultaneous jumps in their DTE RMS curves. The DTE phase diagrams show disordered trajectories, and the Poincaré sections show irregular scattered points, indicating chaotic vibration.
Therefore, I identify the second vibration characteristic: the FGSTTS enters chaotic vibration only when all face gear pairs exhibit simultaneous jumps in their DTE curves at the same rotational speed. When only some face gear pairs jump at a certain speed, the system remains in quasi-periodic vibration.
Away from the jump speeds and critical speeds, I extract the time-domain DTE, frequency-domain DTE, and Poincaré sections of four representative face gear pairs \(\Pi_{61}\), \(\Pi_{71}\), \(\Pi_{63}\), and \(\Pi_{65}\) at \(7600 \,\mathrm{r/min}\). The DTE of \(\delta_{61}\) and \(\delta_{71}\) fluctuates periodically in the range of \(0\) to \(2.5 \times 10^{-4} \,\mathrm{m}\). The DTE of \(\delta_{63}\) fluctuates periodically in the range of \(-0.5 \times 10^{-4}\) to \(2.2 \times 10^{-4} \,\mathrm{m}\). The DTE of \(\delta_{65}\) fluctuates in the range of \(-1.0 \times 10^{-4}\) to \(2.2 \times 10^{-4} \,\mathrm{m}\). Both \(\delta_{63}\) and \(\delta_{65}\) show alternating positive and negative signs. In the frequency spectra, the dominant amplitudes are at \(0.043\) times the mesh frequency (i.e., the shaft frequency) and at the mesh frequency. The tail transmission gear pair shows a noticeable amplitude at twice the mesh frequency. For the tail transmission gear pair, the amplitude at the mesh frequency is greater than that at the shaft frequency, while for other gear pairs, the opposite is true. The DTE phase diagrams show concentrated closed loops, and the Poincaré sections show regular ring points, indicating quasi-periodic vibration.
Thus, I identify the third vibration characteristic: the upper and lower face gears have similar vibration severity. The input pinions vibrate more severely than the idler pinions and the tail transmission gear. The left idler, right idler, and tail transmission gear exhibit alternating positive and negative tooth contact when meshing with the upper and lower face gears. At non-jump speeds, the FGSTTS exhibits periodic vibration, mainly at the mesh frequency and shaft frequency.
| Characteristic | Description |
|---|---|
| Symmetry | With the tail transmission axis as the symmetry axis, left and right pinions vibrate symmetrically. |
| Closed-loop stability | The system enters bifurcation or chaos only when all face gear pairs jump simultaneously at the same speed. |
| Vibration severity | Input gears vibrate more severely than idler and tail transmission gears. |
10. Model Validation
I validate the proposed FMDM using the multibody dynamics software ADAMS. I import the geometric model of the FGSTTS into ADAMS and establish a simulation model. I create rotational joints for the seven gears about their own axes and define ten gear contact pairs between the pinions and the face gears. I set the time-varying mesh stiffness and mesh damping according to the parameters in Section 2. I set the mesh penetration depth to \(0.1 \,\mathrm{mm}\) and the force exponent to \(1.5\) based on the gear material. I apply input speeds to the two input pinions and load torques to the upper face gear and the tail transmission gear. I extract the four-cycle contact forces from the simulation results and compare them with the FMDM results.
The contact forces obtained from ADAMS and the FMDM show good agreement in magnitude and trend. The average contact force between the left input pinion and the upper face gear is \(3.47 \times 10^4 \,\mathrm{N}\) in ADAMS and \(3.21 \times 10^4 \,\mathrm{N}\) in the FMDM, a difference of \(8.1\%\). The differences for other gear pairs are all less than \(8.1\%\), which validates the effectiveness and accuracy of the FMDM. The contact force curves obtained from the FMDM are smoother than those from ADAMS because the imported geometric model in ADAMS has certain precision errors. Near \(0.0278 \,\mathrm{s}\), the FMDM predicts a sign change in the mesh force between the right input pinion and the upper face gear, indicating that the same single tooth experiences alternating positive and negative contact in a short time. The contact forces between the right idler pinion and the lower face gear, and between the tail transmission pinion and the lower face gear, are negative, indicating that the contact surface is the back surface, which is related to the power flow direction of the FGSTTS.
11. Conclusions
I have proposed and validated a flexible multibody dynamic model for a space closed-loop face gear split-torque transmission system based on the finite element method. The model considers the deformations of face gear meshing, shafts, bearings, and gear rotors. I have studied the natural characteristics and vibration responses and drawn the following conclusions.
(1) The mode shapes of the FGSTTS are mainly pure translational, bending, and torsional modes and their coupled modes. Pure modes are mainly distributed in the low-frequency range, while coupled modes dominate the high-frequency range.
(2) The quasi-periodic vibration of the FGSTTS is mainly mesh-frequency and shaft-frequency vibration. The input gear shafts are the most severely vibrating components. The left and right input pinions deform and vibrate synchronously, and the left and right idler pinions also deform and vibrate synchronously.
(3) The necessary condition for the closed-loop face gear system to enter vibration bifurcation or chaos is that all face gear pairs exhibit simultaneous jumps in their RMS curves at a certain rotational speed. When only some face gear pairs exhibit DTE jumps, the system does not enter vibration bifurcation.
The results show that the FMDM and its results can be directly used to analyze the vibration response at various positions of a coaxial face gear structure. The FMDM modeling method is universally applicable to all types of closed-loop gear systems. Compared with existing lumped mass models, the FMDM can extract more modes and more structural point vibration signals, making it more systematic and more accurate.
In future work, I plan to extend the model to include more complex nonlinear factors, such as friction and thermal effects, and to study the influence of parameter uncertainties on the vibration characteristics of the closed-loop face gear system. I also intend to apply the proposed modeling approach to other types of closed-loop face gear transmissions with different configurations.