In this study, I investigate the nonlinear bifurcation behavior of a non-orthogonal offset face gear split transmission system. The growing demand for high power-to-weight ratio, compactness, and reliability in rotorcraft main reducers has made face gears an attractive solution. Unlike conventional spiral bevel or hypoid gears, face gears can provide large transmission ratios, reduced axial dimensions, and improved load distribution. However, the nonlinear dynamic characteristics of face gear systems, especially those with non-orthogonal and offset configurations, are not yet fully understood. The presence of backlash, time-varying mesh stiffness, friction, and support clearances introduces rich nonlinear phenomena such as period-doubling, quasi-periodicity, and chaos. I develop an eleven-degree-of-freedom lumped-parameter model that couples bending, torsion, and axial motions. The time-varying mesh stiffness of the face gears is computed using a load-bearing contact analysis that accounts for the varying contact point positions along the tooth surface. The dynamic equations are solved using the Runge–Kutta numerical integration method. Bifurcation diagrams, largest Lyapunov exponent maps, time histories, frequency spectra, phase portraits, and Poincaré sections are employed to reveal the influence of backlash on the system response. Experimental tests are also conducted to validate the theoretical model. The results show that increasing backlash drives the system from period-one motion to chaos, then to period-doubling, and finally back to chaos. This work provides a theoretical foundation for stability assessment and reliability design of face gear split transmissions.
1. Introduction
Face gears have been increasingly used in aerospace and helicopter transmissions because they offer several advantages over conventional gears. A face gear pair consists of a cylindrical pinion and a face gear, where the pinion meshes with the face gear teeth on the end face. This configuration allows for a compact layout, high gear ratio, and the possibility of splitting or combining power among multiple branches. In particular, coaxial face gear split transmissions are employed in dual-rotor helicopter main reducers to achieve lightweight and reliable power transmission. The term face gears appears frequently in the literature, yet the nonlinear bifurcation behavior of non-orthogonal offset face gears has received limited attention.
Earlier investigations have studied the static and dynamic load sharing characteristics of coaxial face gear systems. For example, researchers have built static and dynamic models to examine the effects of manufacturing errors, assembly errors, and elastic deformation on load distribution. Others have focused on load sharing in torque-split face gear systems, considering support stiffness, installation errors, and clearances. Some studies have established bending–torsion coupled dynamic models to analyze the influence of input gear floating on load sharing. More recent works have employed Lagrangian bond graph models to explore the effects of gear eccentricity, helix angle, and input speed on the vibration of eccentric face gears. Additionally, dynamic models for non-orthogonal misaligned spur face gear pairs have been developed to study the influence of installation errors on mesh phase difference, gear vibration, and dynamic load distribution. Other researchers have formulated nonlinear dynamic models for curve-face gear systems and analyzed their bifurcation characteristics using bifurcation diagrams, phase portraits, time histories, and Poincaré maps. The influence of backlash and back-side contact on transient vibration of helical gears has also been investigated. Nonlinear characteristics of herringbone gear systems with topological modification have been examined. Moreover, nonlinear dynamics of mixed face gear split transmission systems have been modeled, and the effects of initial conditions on planetary roller screw nonlinearity have been studied. For spiral bevel gears, nonlinear dynamic models based on accurate static stiffness evaluation have been proposed, and the effects of tooth profile errors, pitch errors, and elastic deformation have been clarified from a vibration perspective.
Despite these efforts, the nonlinear bifurcation characteristics of non-orthogonal offset face gear split transmission systems remain largely unexplored. The internal excitation conditions, especially the time-varying mesh stiffness generated by the face gear meshing process under non-orthogonal and offset states, have not been sufficiently incorporated into nonlinear dynamic models. Therefore, I aim to bridge this gap by combining gear load-bearing contact simulation and spatial geometry theory to construct a time-varying mesh stiffness excitation for non-orthogonal offset face gears. Using the lumped-parameter method, I consider piecewise-linear backlash functions for both gear meshes and support clearances. An eleven-degree-of-freedom bending–torsion–axial coupled dynamic model is established. The bifurcation behavior of the transmission system with varying backlash is then investigated in detail. The findings provide a theoretical basis for reliability and life prediction of non-orthogonal offset face gear split transmission systems.
2. Dynamic Model of the Face Gear Split Transmission
The coaxial face gear split transmission system considered here uses a three-stage configuration. The first stage consists of a conical involute cylindrical gear meshing with two small-cone-angle face gears, which splits the input power into two branches. The second stage uses spur cylindrical gears to further split the power. The third stage combines the power through face gears to achieve coaxial counter-rotating and equal-speed transmission. In this study, I focus on the first-stage non-orthogonal offset face gear split transmission system. The global coordinate system is denoted by \(O-XYZ\), with its origin at the intersection of the conical involute gear axis and the face gear rotational axes. The local coordinate systems of the two face gears are obtained by rotating the local coordinate system of the input gear around the global \(Y\)-axis by \(\pm\gamma\), where \(\gamma\) is the shaft angle between the conical involute gear and the face gears.
Using the lumped-parameter method, I establish the nonlinear dynamic model of the system. The gear mesh stiffness, mesh damping, mesh error, and backlash between gear 1 and face gear 2 are denoted by \(K_{12}\), \(C_{12}\), \(e_{12}(t)\), and \(b_{12}\), respectively. Similarly, \(K_{13}\), \(C_{13}\), \(e_{13}(t)\), and \(b_{13}\) represent the corresponding quantities between gear 1 and face gear 3. For the cylindrical gear, \(K_{1x}\) and \(K_{1y}\) are support stiffnesses, \(C_{1x}\) and \(C_{1y}\) are support dampings, and \(b_{1x}\) and \(b_{1y}\) are support clearances. For face gear 2, the support stiffnesses are \(K_{2x}\), \(K_{2y}\), and \(K_{2z}\), the support dampings are \(C_{2x}\), \(C_{2y}\), and \(C_{2z}\), and the support clearances are \(b_{2x}\), \(b_{2y}\), and \(b_{2z}\). The parameters for face gear 3 are defined analogously.
The generalized displacement vector of the eleven-degree-of-freedom system is
$$
\mathbf{Q} = \left[ Y_1, Z_1, \theta_{x1}, X_2, Y_2, Z_2, \theta_{x2}, X_3, Y_3, Z_3, \theta_{x3} \right]^T \tag{1}
$$
Here, \(Y_1\) and \(Z_1\) are the transverse vibration displacements of gear 1 along the \(Y\) and \(Z\) axes; \(Y_2\) and \(Z_2\) are the transverse displacements of face gear 2 along the \(Y\) and \(Z\) axes; \(X_2\) is the axial displacement of face gear 2 along the \(X\) axis; \(Y_3\) and \(Z_3\) are the transverse displacements of face gear 3 along the \(Y\) and \(Z\) axes; \(X_3\) is the axial displacement of face gear 3; and \(\theta_{x1}\), \(\theta_{x2}\), and \(\theta_{x3}\) are the torsional vibration displacements of gear 1, face gear 2, and face gear 3 about their respective axes.
The mesh force \(F_{1i}\) and the backlash function are expressed as
$$
F_{1i} = K_{1i}(t) f(X_{1i}) + C_{1i} \dot{X}_{1i} \tag{2}
$$
$$
f(X_{1i}, b_{1i}) = \begin{cases}
X_{1i} – b_{1i}, & X_{1i} > b_{1i} \\
0, & -b_{1i} \le X_{1i} \le b_{1i} \\
X_{1i} + b_{1i}, & X_{1i} < -b_{1i}
\end{cases} \tag{3}
$$
$$
f_{jK}(K_j, b_j) = \begin{cases}
K_j – b_j, & K_j > b_j \\
0, & -b_j \le K_j \le b_j \\
K_j + b_j, & K_j < -b_j
\end{cases} \tag{4}
$$
where \(i=\{2,3\}\), \(j=\{1,2,3\}\), and \(K=\{X,Y,Z\}\). The function \(f(X_{1i}, b_{1i})\) represents the backlash nonlinearity in the mesh pair between gear 1 and face gear 2 or gear 1 and face gear 3. The function \(f_{jK}(K_j, b_j)\) describes the support clearance nonlinearity for gear 1, face gear 2, and face gear 3 along their respective axes. The relative displacement along the mesh line between gear 1 and face gear \(i\) is denoted by \(X_{1i}\). The equivalent displacement of the torsional line displacement of gear \(j\) is \(K_j\). The backlash in the mesh pair is \(b_{1i}\), and the support clearance is \(b_j\).
By introducing dimensionless parameters, the dimensionless nonlinear dynamic equations of the system can be derived. For gear 1, the dimensionless equations are
$$
\begin{aligned}
\ddot{\bar{Y}}_1 &+ \frac{K_{1y}}{m_1 \omega_{n2}^2} f_{1y}(\bar{Y}_1, \bar{b}_{1y}) + \frac{C_{1y}}{m_1 \omega_{n2}} \dot{\bar{Y}}_1 \\
&= -(\sin\phi_2 + \lambda\mu\cos\phi_2) \left( \frac{K_{12}}{m_1 \omega_{n2}^2} f_{12}(\bar{X}_{12}, \bar{b}_{12}) + \frac{C_{12}}{m_1 \omega_{n2}} \dot{\bar{X}}_{12} \right) \\
&\quad -(\sin\phi_3 + \lambda\mu\cos\phi_3) \left( \frac{K_{13}}{m_1 \omega_{n3}^2} f_{13}(\bar{X}_{13}, \bar{b}_{13}) + \frac{C_{13}}{m_1 \omega_{n3}} \dot{\bar{X}}_{13} \right)
\end{aligned} \tag{5}
$$
$$
\begin{aligned}
\ddot{\bar{Z}}_1 &+ \frac{K_{1z}}{m_1 \omega_{n2}^2} f_{1z}(\bar{Z}_1, \bar{b}_{1z}) + \frac{C_{1z}}{m_1 \omega_{n2}} \dot{\bar{Z}}_1 \\
&= -(\sin\phi_2 + \lambda\mu\cos\phi_2) \left( \frac{K_{12}}{m_1 \omega_{n2}^2} f_{12}(\bar{X}_{12}, \bar{b}_{12}) + \frac{C_{12}}{m_1 \omega_{n2}} \dot{\bar{X}}_{12} \right) \\
&\quad -(\sin\phi_3 + \lambda\mu\cos\phi_3) \left( \frac{K_{13}}{m_1 \omega_{n3}^2} f_{13}(\bar{X}_{13}, \bar{b}_{13}) + \frac{C_{13}}{m_1 \omega_{n3}} \dot{\bar{X}}_{13} \right)
\end{aligned} \tag{6}
$$
$$
\begin{aligned}
\ddot{\bar{\theta}}_1 &+ \left( 1 + \frac{\mu\lambda L_{12}}{r_{b1}} + \frac{\mu\lambda L_{13}}{r_{b1}} \right) \\
&\quad \times \left( \frac{K_{12}}{I_1 \omega_{n2}^2} f_{12}(\bar{X}_{12}, \bar{b}_{12}) + \frac{C_{12}}{I_1 \omega_{n2}} \dot{\bar{X}}_{12} \right) r_{b1} \\
&\quad + \left( \frac{K_{13}}{I_1 \omega_{n3}^2} f_{13}(\bar{X}_{13}, \bar{b}_{13}) + \frac{C_{13}}{I_1 \omega_{n3}} \dot{\bar{X}}_{13} \right) r_{b1} \\
&= \frac{\bar{T}_{in}}{I_1 b_c \omega_{n2}^2}
\end{aligned} \tag{7}
$$
For face gear 2, the dimensionless equations are
$$
\begin{aligned}
\ddot{\bar{X}}_2 &+ \frac{K_{2x}}{m_2 \omega_{n2}^2} f_{2x}(\bar{X}_2, \bar{b}_{2x}) + \frac{C_{2x}}{m_2 \omega_{n2}} \dot{\bar{X}}_2 \\
&= (\sin\phi_2 + \lambda\mu\cos\phi_2)\cos\gamma \left( \frac{K_{12}}{m_2 \omega_{n2}^2} f_{12}(\bar{X}_{12}, \bar{b}_{12}) + \frac{C_{12}}{m_2 \omega_{n2}} \dot{\bar{X}}_{12} \right)
\end{aligned} \tag{8}
$$
$$
\begin{aligned}
\ddot{\bar{Y}}_2 &+ \frac{K_{2y}}{m_2 \omega_{n2}^2} f_{2y}(\bar{Y}_2, \bar{b}_{2y}) + \frac{C_{2y}}{m_2 \omega_{n2}} \dot{\bar{Y}}_2 \\
&= (\cos\phi_2 + \lambda\mu\sin\phi_2) \left( \frac{K_{12}}{m_2 \omega_{n2}^2} f_{12}(\bar{X}_{12}, \bar{b}_{12}) + \frac{C_{12}}{m_2 \omega_{n2}} \dot{\bar{X}}_{12} \right)
\end{aligned} \tag{9}
$$
$$
\begin{aligned}
\ddot{\bar{Z}}_2 &+ \frac{K_{2z}}{m_2 \omega_{n2}^2} f_{2z}(\bar{Z}_2, \bar{b}_{2z}) + \frac{C_{2z}}{m_2 \omega_{n2}} \dot{\bar{Z}}_2 \\
&= (\sin\phi_2 + \lambda\mu\cos\phi_2)\sin\gamma \left( \frac{K_{12}}{m_2 \omega_{n2}^2} f_{12}(\bar{X}_{12}, \bar{b}_{12}) + \frac{C_{12}}{m_2 \omega_{n2}} \dot{\bar{X}}_{12} \right)
\end{aligned} \tag{10}
$$
$$
\begin{aligned}
\ddot{\bar{\theta}}_2 &- \left( 1 – \frac{\mu\lambda L_{21}}{r_{b2}} \right) \\
&\quad \times \left( \frac{K_{12}}{I_2 \omega_{n2}^2} f_{12}(\bar{X}_{12}, \bar{b}_{12}) + \frac{C_{12}}{I_2 \omega_{n2}} \dot{\bar{X}}_{12} \right) r_{b2} \\
&= \frac{\bar{T}_{out2}}{I_2 b_c \omega_{n2}^2}
\end{aligned} \tag{11}
$$
For face gear 3, the dimensionless equations are
$$
\begin{aligned}
\ddot{\bar{X}}_3 &+ \frac{K_{3x}}{m_3 \omega_{n3}^2} f_{3x}(\bar{X}_3, \bar{b}_{3x}) + \frac{C_{3x}}{m_3 \omega_{n3}} \dot{\bar{X}}_3 \\
&= (\sin\phi_3 + \lambda\mu\cos\phi_3)\cos\gamma \left( \frac{K_{13}}{m_3 \omega_{n3}^2} f_{13}(\bar{X}_{13}, \bar{b}_{13}) + \frac{C_{13}}{m_3 \omega_{n3}} \dot{\bar{X}}_{13} \right)
\end{aligned} \tag{12}
$$
$$
\begin{aligned}
\ddot{\bar{Y}}_3 &+ \frac{K_{3y}}{m_3 \omega_{n3}^2} f_{3y}(\bar{Y}_3, \bar{b}_{3y}) + \frac{C_{3y}}{m_3 \omega_{n3}} \dot{\bar{Y}}_3 \\
&= (\cos\phi_3 + \lambda\mu\sin\phi_3) \left( \frac{K_{13}}{m_3 \omega_{n3}^2} f_{13}(\bar{X}_{13}, \bar{b}_{13}) + \frac{C_{13}}{m_3 \omega_{n3}} \dot{\bar{X}}_{13} \right)
\end{aligned} \tag{13}
$$
$$
\begin{aligned}
\ddot{\bar{Z}}_3 &+ \frac{K_{3z}}{m_3 \omega_{n3}^2} f_{3z}(\bar{Z}_3, \bar{b}_{3z}) + \frac{C_{3z}}{m_3 \omega_{n3}} \dot{\bar{Z}}_3 \\
&= (\sin\phi_3 + \lambda\mu\cos\phi_3)\sin\gamma \left( \frac{K_{13}}{m_3 \omega_{n3}^2} f_{13}(\bar{X}_{13}, \bar{b}_{13}) + \frac{C_{13}}{m_3 \omega_{n3}} \dot{\bar{X}}_{13} \right)
\end{aligned} \tag{14}
$$
$$
\begin{aligned}
\ddot{\bar{\theta}}_3 &- \left( 1 – \frac{\mu\lambda L_{31}}{r_{b3}} \right) \\
&\quad \times \left( \frac{K_{13}}{I_3 \omega_{n3}^2} f_{13}(\bar{X}_{13}, \bar{b}_{13}) + \frac{C_{13}}{I_3 \omega_{n3}} \dot{\bar{X}}_{13} \right) r_{b3} \\
&= \frac{\bar{T}_{out3}}{I_3 b_c \omega_{n3}^2}
\end{aligned} \tag{15}
$$
In the above equations, \(\mu\) is the friction coefficient, \(\tau_i = \omega_{ni} t\) is the dimensionless time, and \(\omega_{ni} = \sqrt{K_m (I_1 r_1^2 + I_i r_i^2)/(I_1 I_i)}\). The dimensionless displacement, velocity, and acceleration are \(X = \bar{X} b_c\), \(\dot{X}_i = \dot{\bar{X}} \omega_{ni}^2 b_c\), and \(\ddot{X}_i = \ddot{\bar{X}} \omega_{ni}^2 b_c\), respectively, where \(b_c\) is the nominal displacement scale. The dimensionless backlash and dimensionless rotational speeds are \(\bar{b}_{1i} = b_{1i}/b_c\), \(\Omega_1 = \omega/\omega_{n1}\), \(\Omega_2 = \omega/\omega_{n2}\), and \(\Omega_3 = \omega/\omega_{n3}\). The masses of gear 1, face gear 2, and face gear 3 are \(m_1\), \(m_2\), and \(m_3\), and their moments of inertia are \(I_1\), \(I_2\), and \(I_3\). The projection angles are \(\phi_2 = \sigma_{12} – \alpha_{n2}\) and \(\phi_3 = \sigma_{13} – \alpha_{n3}\), where \(\alpha_{n2}\) and \(\alpha_{n3}\) are the mesh angles, and \(\sigma_{12}\) and \(\sigma_{13}\) are the phase angles. The input torque is \(T_{in}\), and the output torques are \(T_{out2}\) and \(T_{out3}\).
Because of the backlash and support clearances, the vibration equations form a semi-definite differential system that includes rigid-body displacement and cannot be solved directly. Therefore, I introduce relative coordinates to eliminate the rigid-body displacement. The relative displacement along the mesh line is
$$
\bar{X}_{1i} = \bar{Y}_1 \sin\phi_1 + \bar{Z}_1 \cos\phi_1 + \bar{\theta}_1 r_{b1} – \bar{X}_i \sin\phi_i \cos\gamma – \bar{Y}_i \cos\phi_i – \bar{Z}_i \sin\phi_i \sin\gamma – \bar{\theta}_i r_{bi} \tag{16}
$$
where \(i=\{2,3\}\). \(\bar{X}_{12}\) is the relative displacement along the mesh line between gear 1 and face gear 2, and \(\bar{X}_{13}\) is the relative displacement between gear 1 and face gear 3. After introducing the relative coordinates, the equations become
$$
\begin{aligned}
\ddot{\bar{X}}_{12} &= \left( 1 + \frac{\mu\lambda L_{12}}{r_{b1}} + \frac{\mu\lambda L_{13}}{r_{b1}} \right) \left( \frac{K_{12}}{I_1 \omega_{n2}^2} \bar{X}_{12} + \frac{C_{12}}{I_1 \omega_{n2}} \dot{\bar{X}}_{12} + \frac{K_{13}}{I_1 \omega_{n3}^2} \bar{X}_{13} + \frac{C_{13}}{I_1 \omega_{n3}} \dot{\bar{X}}_{13} \right) r_{b1}^2 \\
&\quad – \left( 1 – \frac{\mu\lambda L_{21}}{r_{b2}} \right) \left( \frac{K_{12}}{I_2 \omega_{n2}^2} \bar{X}_{12} + \frac{C_{12}}{I_2 \omega_{n2}} \dot{\bar{X}}_{12} \right) r_{b2}^2 \\
&\quad – \left[ \frac{K_{1x}}{m_1 \omega_{n2}^2} \bar{X}_1 + \frac{C_{1x}}{m_1 \omega_{n2}} \dot{\bar{X}}_1 + (\sin\phi_2 + \lambda\mu\cos\phi_2) \left( \frac{K_{12}}{m_1 \omega_{n2}^2} \bar{X}_{12} + \frac{C_{12}}{m_1 \omega_{n2}} \dot{\bar{X}}_{12} \right) \right] \sin\phi_2 \\
&\quad + (\sin\phi_3 + \lambda\mu\cos\phi_3) \left( \frac{K_{13}}{m_1 \omega_{n3}^2} \bar{X}_{13} + \frac{C_{13}}{m_1 \omega_{n3}} \dot{\bar{X}}_{13} \right) \sin\phi_3 \\
&\quad – \left[ \frac{K_{1y}}{m_1 \omega_{n2}^2} \bar{Y}_1 + \frac{C_{1y}}{m_1 \omega_{n2}} \dot{\bar{Y}}_1 + (\sin\phi_2 + \lambda\mu\cos\phi_2) \left( \frac{K_{12}}{m_1 \omega_{n2}^2} \bar{X}_{12} + \frac{C_{12}}{m_1 \omega_{n2}} \dot{\bar{X}}_{12} \right) \right] \cos\phi_2 \\
&\quad + (\sin\phi_3 + \lambda\mu\cos\phi_3) \left( \frac{K_{13}}{m_1 \omega_{n3}^2} \bar{X}_{13} + \frac{C_{13}}{m_1 \omega_{n3}} \dot{\bar{X}}_{13} \right) \cos\phi_3 \\
&\quad + \left[ \frac{K_{2x}}{m_2 \omega_{n2}^2} \bar{X}_2 + \frac{C_{2x}}{m_2 \omega_{n2}} \dot{\bar{X}}_2 + (\sin\phi_2 + \lambda\mu\cos\phi_2) \left( \frac{K_{12}}{m_2 \omega_{n2}^2} \bar{X}_{12} + \frac{C_{12}}{m_2 \omega_{n2}} \dot{\bar{X}}_{12} \right) \right] \cos\gamma \sin\phi_2 \cos\gamma \\
&\quad – \left[ \frac{K_{2y}}{m_2 \omega_{n2}^2} \bar{Y}_2 + \frac{C_{2y}}{m_2 \omega_{n2}} \dot{\bar{Y}}_2 + (\cos\phi_2 + \lambda\mu\sin\phi_2) \left( \frac{K_{12}}{m_2 \omega_{n2}^2} \bar{X}_{12} + \frac{C_{12}}{m_2 \omega_{n2}} \dot{\bar{X}}_{12} \right) \right] \cos\phi_2 \\
&\quad + \left[ \frac{K_{2z}}{m_2 \omega_{n2}^2} \bar{Z}_2 + \frac{C_{2z}}{m_2 \omega_{n2}} \dot{\bar{Z}}_2 + (\sin\phi_2 + \lambda\mu\cos\phi_2) \left( \frac{K_{12}}{m_2 \omega_{n2}^2} \bar{X}_{12} + \frac{C_{12}}{m_2 \omega_{n2}} \dot{\bar{X}}_{12} \right) \right] \sin\gamma \sin\phi_2 \sin\gamma \\
&\quad + \frac{\bar{T}_{in} r_{b1}}{I_1 b_c \omega_{n2}^2} + \frac{\bar{T}_{out2} r_{b2}}{2 I_2 b_c \omega_{n2}^2} + \ddot{\bar{e}}(t)
\end{aligned} \tag{17}
$$
$$
\begin{aligned}
\ddot{\bar{X}}_{13} &= \left( 1 + \frac{\mu\lambda L_{12}}{r_{b1}} + \frac{\mu\lambda L_{13}}{r_{b1}} \right) \left( \frac{K_{12}}{I_1 \omega_{n2}^2} \bar{X}_{12} + \frac{C_{12}}{I_1 \omega_{n2}} \dot{\bar{X}}_{12} + \frac{K_{13}}{I_1 \omega_{n2}^2} \bar{X}_{13} + \frac{C_{13}}{I_1 \omega_{n2}} \dot{\bar{X}}_{13} \right) r_{b2}^2 \\
&\quad – \left( 1 – \frac{\mu\lambda L_{31}}{r_{b3}} \right) \left( \frac{K_{13}}{I_3 \omega_{n3}^2} \bar{X}_{13} + \frac{C_{13}}{I_3 \omega_{n3}} \dot{\bar{X}}_{13} \right) r_{b3}^2 \\
&\quad – \left[ \frac{K_{1x}}{m_1 \omega_{n2}^2} \bar{X}_1 + \frac{C_{1x}}{m_1 \omega_{n2}} \dot{\bar{X}}_1 + (\sin\phi_2 + \lambda\mu\cos\phi_2) \left( \frac{K_{12}}{m_1 \omega_{n2}^2} \bar{X}_{12} + \frac{C_{12}}{m_1 \omega_{n2}} \dot{\bar{X}}_{12} \right) \right] \sin\phi_2 \\
&\quad + (\sin\phi_3 + \lambda\mu\cos\phi_3) \left( \frac{K_{13}}{m_1 \omega_{n3}^2} \bar{X}_{13} + \frac{C_{13}}{m_1 \omega_{n3}} \dot{\bar{X}}_{13} \right) \sin\phi_3 \\
&\quad – \left[ \frac{K_{1y}}{m_1 \omega_{n2}^2} \bar{Y}_1 + \frac{C_{1y}}{m_1 \omega_{n2}} \dot{\bar{Y}}_1 + (\sin\phi_2 + \lambda\mu\cos\phi_2) \left( \frac{K_{12}}{m_1 \omega_{n2}^2} \bar{X}_{12} + \frac{C_{12}}{m_1 \omega_{n2}} \dot{\bar{X}}_{12} \right) \right] \cos\phi_2 \\
&\quad + (\sin\phi_3 + \lambda\mu\cos\phi_3) \left( \frac{K_{13}}{m_1 \omega_{n3}^2} \bar{X}_{13} + \frac{C_{13}}{m_1 \omega_{n3}} \dot{\bar{X}}_{13} \right) \cos\phi_3 \\
&\quad + \left[ \frac{K_{3x}}{m_3 \omega_{n3}^2} \bar{X}_3 + \frac{C_{3x}}{m_3 \omega_{n3}} \dot{\bar{X}}_3 + (\sin\phi_3 + \lambda\mu\cos\phi_3) \left( \frac{K_{13}}{m_3 \omega_{n3}^2} \bar{X}_{13} + \frac{C_{13}}{m_3 \omega_{n3}} \dot{\bar{X}}_{13} \right) \right] \cos\gamma \sin\phi_3 \cos\gamma \\
&\quad – \left[ \frac{K_{3y}}{m_3 \omega_{n3}^2} \bar{Y}_3 + \frac{C_{3y}}{m_3 \omega_{n3}} \dot{\bar{Y}}_3 + (\cos\phi_3 + \lambda\mu\sin\phi_3) \left( \frac{K_{13}}{m_3 \omega_{n3}^2} \bar{X}_{13} + \frac{C_{13}}{m_3 \omega_{n3}} \dot{\bar{X}}_{13} \right) \right] \cos\phi_3 \\
&\quad + \left[ \frac{K_{3z}}{m_3 \omega_{n3}^2} \bar{Z}_3 + \frac{C_{3z}}{m_3 \omega_{n3}} \dot{\bar{Z}}_3 + (\sin\phi_3 + \lambda\mu\cos\phi_3) \left( \frac{K_{13}}{m_3 \omega_{n3}^2} \bar{X}_{13} + \frac{C_{13}}{m_3 \omega_{n3}} \dot{\bar{X}}_{13} \right) \right] \sin\gamma \sin\phi_3 \sin\gamma \\
&\quad + \frac{\bar{T}_{in} r_{b1}}{I_1 b_c \omega_{n2}^2} + \frac{\bar{T}_{out3} r_{b3}}{2 I_3 b_c \omega_{n3}^2} + \ddot{\bar{e}}(t)
\end{aligned} \tag{18}
$$
3. Time-Varying Mesh Stiffness of Face Gears
The non-orthogonal offset face gear pair consists of a conical involute gear meshing with a small-cone-angle face gear. The conical involute gear is generated by a tilted rack cutter. The coordinate system \(S_i\) \((i=p,s)\) represents the follower coordinate system of the conical involute gear, \(S_e(O_e-x_e y_e z_e)\) is the fixed coordinate system, \(S_r(O_r-x_r y_r z_r)\) is the rack cutter coordinate system, and \(S_d(O_d-x_d y_d z_d)\) is an auxiliary coordinate system. The half-cone angle of the gear is \(\delta\), and \(r_{pi}\) is the pitch radius of the conical involute gear in the \(x_e-O_e-y_e\) plane. When the rack moves along the \(x_r\) direction with velocity \(v_r\) by a distance \(s_r\), the gear rotates with angular velocity \(\omega_i\) by an angle \(\psi_i\). The tooth surface \(\Sigma_i\) of the conical involute gear can be expressed as
$$
\begin{cases}
\mathbf{r}_i(u_i, l_i, \psi_i) = \mathbf{M}_{ie}(\psi_i) \mathbf{M}_{ed} \mathbf{M}_{dr} \mathbf{r}_{ri}(u_i, l_i) \\
f_{ri}(u_i, l_i, \psi_i) = \mathbf{n}_{ri}(u_i, l_i) \cdot \mathbf{v}_{r}^{(ri)}(u_i, l_i, \psi_i) = 0
\end{cases} \tag{19}
$$
where \(\mathbf{r}_i\) is the position vector of a point on the conical involute gear tooth surface \(\Sigma_i\). The coordinate transformation matrices \(\mathbf{M}_{ie}(\psi_i)\), \(\mathbf{M}_{ed}\), and \(\mathbf{M}_{dr}\) transform the meshing point from the rack cutter coordinate system \(S_r\) to the gear coordinate system \(S_i\). The equation \(f_{ri}=0\) represents the meshing condition between the rack and the conical involute gear, where \(\mathbf{n}_{ri}\) is the normal vector of the rack and \(\mathbf{v}_{r}^{(ri)}\) is the relative velocity between the rack cutter and the conical involute gear, both defined in \(S_r\).
The face gear tooth surface \(\Sigma_2\) is generated by the conical involute gear cutter. Specifically, it is the envelope of the cutter tooth surface \(\Sigma_s\). The generating coordinate system of the face gear is defined by \(S_2\) and \(S_s\), which correspond to the follower coordinate systems of the non-orthogonal face gear and the conical involute gear cutter, respectively, and \(S_m\) is a fixed reference coordinate system. The angle between the non-orthogonal offset face gear axis \(X_2\) and the negative direction of the cutter axis \(Z_s\) is \(\gamma_m\). The angular velocities of the cutter and the face gear are \(\omega_s\) and \(\omega_2\). The face gear tooth surface \(\Sigma_2\) can be expressed as
$$
\begin{cases}
\mathbf{r}_2(u_s, l_s, \psi_s) = \mathbf{M}_{2m}(\psi_2) \mathbf{M}_{mn} \mathbf{M}_{ns}(\psi_s) \mathbf{r}_s(u_s, l_s) \\
f_{s2}(u_s, l_s, \psi_s) = \mathbf{n}_s(u_s, l_s) \cdot \mathbf{v}_{s2}^{s}(u_s, l_s, \psi_s) = 0
\end{cases} \tag{20}
$$
where \(\mathbf{r}_2\) is the position vector of a point on the face gear tooth surface \(\Sigma_2\), and \(\mathbf{r}_s\) is the mathematical description of the cutter tooth surface \(\Sigma_s\). The matrices \(\mathbf{M}_{2m}(\psi_2)\), \(\mathbf{M}_{mn}\), and \(\mathbf{M}_{ns}(\psi_s)\) transform the meshing point from the cutter tooth surface \(\Sigma_s\) to the face gear tooth surface \(\Sigma_2\). The condition \(f_{s2}=0\) is the meshing condition, where \(\mathbf{n}_s\) is the normal vector of the cutter tooth surface \(\Sigma_s\) in \(S_s\), and \(\mathbf{v}_{s2}^{s}\) is the relative velocity between the cutter and the face gear, also expressed in \(S_s\). During the generating process, \(\psi_s\) and \(\psi_2\) represent the rotation angles of the conical involute gear cutter and the non-orthogonal offset face gear, respectively.
To perform the tooth contact analysis of the non-orthogonal offset face gear pair, I establish an application coordinate system. Auxiliary coordinate systems \(S_\theta\), \(S_q\), and \(S_h\) are introduced to account for installation errors. \(S_u\) is a fixed spatial coordinate system that provides a stable reference. \(S_p\) and \(S_2\) are the follower coordinate systems of the conical involute gear and the face gear, respectively. \(\Phi_p\) and \(\Phi_2\) describe the rotation angles of the pinion and the face gear during meshing. The axial misalignment error of the face gear pair is \(\Delta E\), the axial error of the face gear during installation is \(\Delta q\), and the shaft angle error is expressed as \(\gamma_f = \gamma_m + \Delta\gamma\), where \(\gamma_m\) is the base shaft angle and \(\Delta\gamma\) is the deviation. The geometric distance determined by the initial installation distance, tooth width, and shaft angle is \(L_0 + B \cot\gamma\).
The tooth surfaces and unit normal vectors of the conical involute gear \(\Sigma_p\) and the non-orthogonal offset face gear \(\Sigma_2\) in their respective coordinate systems can be transformed into the fixed coordinate system \(S_u\) as
$$
\begin{cases}
\mathbf{r}_u^p(u_p, l_p, \psi_p, \Phi_p) = \mathbf{M}_{up}(\Phi_p) \mathbf{r}_p(u_p, l_p, \psi_p) \\
\mathbf{n}_u^p(u_p, l_p, \psi_p, \Phi_p) = \mathbf{L}_{up}(\Phi_p) \mathbf{n}_p(u_p, l_p, \psi_p) \\
\mathbf{r}_u^2(u_s, l_s, \psi_s, \Phi_2) = \mathbf{M}_{u2}(\Phi_2) \mathbf{r}_2(u_s, l_s, \psi_s) \\
\mathbf{n}_u^2(u_s, l_s, \psi_s, \Phi_2) = \mathbf{L}_{u2}(\Phi_2) \mathbf{n}_2(u_s, l_s, \psi_s)
\end{cases} \tag{21}
$$
where \(\mathbf{M}_{up}\) and \(\mathbf{L}_{up}\) are the position and direction transformation matrices from the pinion coordinate system \(S_p\) to the fixed coordinate system \(S_u\), and \(\mathbf{M}_{u2}\) and \(\mathbf{L}_{u2}\) are the corresponding matrices from the face gear coordinate system \(S_2\) to \(S_u\). The two gears must maintain continuous contact during meshing, so the following equations must be satisfied:
$$
\begin{cases}
\mathbf{r}_u^p(u_p, l_p, \psi_p, \Phi_p) = \mathbf{r}_u^2(u_s, l_s, \psi_s, \Phi_2) \\
\mathbf{n}_u^p(u_p, l_p, \psi_p, \Phi_p) = \mathbf{n}_u^2(u_s, l_s, \psi_s, \Phi_2)
\end{cases} \tag{22}
$$
where \(\mathbf{r}_u^p\) and \(\mathbf{n}_u^p\) are the position and normal vectors of the conical involute gear, \(u_p\) and \(l_p\) are the independent parameters for generating the conical involute gear, \(\mathbf{r}_u^2\) and \(\mathbf{n}_u^2\) are the position and normal vectors of the non-orthogonal offset face gear, and \(u_s\) and \(l_s\) are the independent parameters for generating the face gear.
Neglecting the influence of lubricating oil and friction during the meshing process, the load-bearing contact analysis model of the non-orthogonal offset face gear transmission is established based on the total normal meshing force \(P\). The face gear transmission adopts a conjugate point contact tooth surface design. When subjected to load, the gear teeth undergo elastic deformation, and the instantaneous contact area width is neglected. A normal section through the major axis of the instantaneous contact ellipse is considered. Suppose that tooth pair I and tooth pair II may be in contact simultaneously. Point \(i\) represents the center of the contact ellipse, and point \(j\) is an arbitrary discrete position on the major axis of the contact ellipse. The displacement coordination relationship after deformation can be expressed as
$$
\mathbf{F}_k \mathbf{p}_k + \mathbf{w}_k = \mathbf{d}_k + \mathbf{Z}_k \quad (k = \mathrm{I}, \mathrm{II}) \tag{23}
$$
where \(\mathbf{p}_k = [p_1, p_2, \cdots, p_n]^T\) is the normal load vector at discrete points along the major axis of the instantaneous contact ellipse, \(\mathbf{d}_k = [d_1, d_2, \cdots, d_n]^T\) is the clearance vector generated by the deformed tooth surface, and \(\mathbf{Z}_k = Z[1, 1, \cdots, 1]^T\) reflects the uniform normal displacement after tooth deformation. For each discrete point, the normal load \(p_j\) \((j=1,2,\cdots,n)\) must satisfy the force equilibrium condition
$$
\sum_{j=1}^n p_j^{\mathrm{I}} + \sum_{j=1}^n p_j^{\mathrm{II}} = P \tag{24}
$$
The non-embedding condition that must be satisfied during gear contact can be expressed as
$$
\begin{cases}
p_{jk} > 0, & d_{jk} = 0 \\
p_{jk} = 0, & d_{jk} > 0
\end{cases} \tag{25}
$$
When considering installation errors, conical involute gear offset, and other factors, the mathematical programming formulation of the load-bearing contact problem for face gears can be simplified as
$$
\begin{cases}
\min \sum_{j=1}^{n+1} X_j \\
\mathbf{d} + \mathbf{Z} + \mathbf{X} – \mathbf{F}\mathbf{p} = \mathbf{w} \\
\mathbf{e}^T \mathbf{p} + X_{n+1} = P \\
\text{s.t. } p_j \ge 0, \ d_j \ge 0, \ Z_j \ge 0, \ X_j \ge 0 \\
p_j = 0 \ \text{or} \ d_j = 0
\end{cases} \tag{26}
$$
Specifically, this transformation is based on the geometric characteristics and force state of the gear, using the nonlinear programming objective of minimizing deformation energy, combined with artificial variables \(X_j\) \((j=1,2,\cdots,n+1)\). Here, \(\mathbf{X} = [X_1, X_2, \cdots, X_n]^T\) is an \(n\)-dimensional column vector, and \(\mathbf{e}\) is an \(n\)-dimensional unit column vector. The contact boundary conditions are \(p_j \ge 0\), \(d_j \ge 0\), \(Z_j \ge 0\), and \(X_j \ge 0\).
After the non-orthogonal offset face gear is deformed under load, the normal displacement of the gear teeth can be converted into the load-bearing transmission error \(T_{EL}\) through the following formula:
$$
T_{EL} = \frac{3600 \times 180}{Z |\omega_2|} \left( \mathbf{R}_2 \times \mathbf{n}_2 \right) \cdot \omega_2 \tag{27}
$$
where \(\omega_2\) is the angular velocity vector of the non-orthogonal offset face gear, \(\mathbf{R}_2\) is the position vector of the contact point on the gear tooth surface, and \(\mathbf{n}_2\) is the normal vector of the contact point. At any instant of meshing, the normal comprehensive deformation \(\delta\) of the gear teeth after loading is mainly determined by \(|T_{EG} – T_{EL}|\), where \(T_{EG}\) is the geometric transmission error of the non-orthogonal offset face gear after modification. Thus,
$$
\delta = |T_{EG} – T_{EL}| \frac{( \mathbf{R}_2 \times \mathbf{n}_2 ) \cdot \omega_2}{3600 \times 180 |\omega_2|} \tag{28}
$$
Assuming that within one meshing period \(T\), a single tooth pair experiences 8 different meshing stages, from initial contact to complete disengagement, involving \(N\) stages in total, the comprehensive mesh stiffness of the face gear transmission and the mesh stiffness of a single tooth pair are calculated as
$$
k_m(ii) = \frac{p}{\delta(ii)} \quad (ii = 1, 2, 3, \cdots, 8) \tag{29}
$$
$$
k_s(jj) = k_m(jj – 8 \lfloor (jj – 1)/8 \rfloor) L_k(jj) \quad (jj = 1, 2, 3, \cdots, N) \tag{30}
$$
To verify the effectiveness and accuracy of the mesh stiffness calculation method based on tooth surface contact theory proposed in this study, I compare the comprehensive mesh stiffness of a helical gear transmission calculated by this method with that obtained by a classical method. The classical stiffness formula is
$$
k_i(t) = k_\alpha + \sum_{n=1}^{\infty} k_{ni} \cos(n\omega t + \alpha_{1i}) \tag{31}
$$
where \(k_\alpha\) is the average mesh stiffness, \(k_{ni}\) is the fluctuation amplitude of the mesh stiffness, \(\omega\) is the meshing frequency of the gear pair, and \(\alpha_{1i}\) is the meshing phase angle. The system simulation parameters are: face gear inner diameter 140 mm, outer diameter 172 mm, shaft angle \(30^\circ\), gear material elastic modulus 3045 MPa, and Poisson’s ratio 0.41. Using the parameters in Table 1, the time-varying mesh stiffness curves of each gear pair are calculated. The two calculation methods give basically the same trend for the comprehensive mesh stiffness curve. However, because Eq. (31) does not consider the influence of the tooth root transition surface and rim deformation on the meshing point, its result has an error within 4.6%. Therefore, the tooth surface contact method ensures higher accuracy.
| Parameter | Gear 1 | Face gear 2 | Face gear 3 |
|---|---|---|---|
| Number of teeth \(z\) | 24 | 100 | 100 |
| Module \(m\) / mm | 3 | 3 | 3 |
| Pressure angle \(\phi\) / (°) | 20 | 20 | 20 |
| Tooth width \(b\) / mm | 32 | 20 | 20 |
| Support stiffness \(K\) / (N·m\(^{-1}\)) | \(3.4 \times 10^9\) | \(1.4 \times 10^9\) | \(1.4 \times 10^9\) |
| Support damping \(C\) / (N·s·m\(^{-1}\)) | \(3.078 \times 10^4\) | \(1.278 \times 10^4\) | \(1.278 \times 10^4\) |
| Moment of inertia \(J\) / (kg·m\(^2\)) | \(1.46 \times 10^{-6}\) | \(5.26 \times 10^{-6}\) | \(5.26 \times 10^{-6}\) |
4. Numerical Simulation and Bifurcation Analysis
In this section, I present the numerical simulation results for the non-orthogonal offset face gear split transmission system. The input power is \(W = 200\) kW, the half support clearances are \(b_{1x}=b_{1y}=20\) μm, \(b_{2x}=b_{2y}=b_{2z}=33\) μm, and \(b_{3x}=b_{3y}=b_{3z}=33\) μm. The friction coefficient is 0.1, the damping coefficient is 0.07, and the rotational speed is 1800 r/min. The global bifurcation diagram and Lyapunov exponent diagram of the system with respect to the backlash are obtained. The time domain, frequency domain, phase portrait, and Poincaré section are also examined. These figures provide a comprehensive perspective for understanding the influence of backlash on the stability and dynamic behavior of the system.
When the damping coefficient is 0.07, the system exhibits different dynamic behaviors depending on the backlash. Specifically, when the backlash is in the range of 0–6.2 μm, the Lyapunov exponent is negative, and the bifurcation diagram shows period-one motion. When the backlash increases to 6.2–10.4 μm, a period-doubling bifurcation is observed, and the sensitivity of the system to initial conditions increases. As the backlash increases to 10.4–13.8 μm, the Lyapunov exponent becomes positive, multiple discrete points appear in the bifurcation diagram, and the system enters a short chaotic state. The predictability of the system decreases, showing high uncertainty and randomness. As the backlash further increases to 13.8–23.8 μm, the chaotic state ends, and the system returns to period-three motion. When the backlash reaches 23.8–24.6 μm, the system exhibits quasi-periodic motion. In the range of 24.6–50 μm, the Lyapunov exponent is again greater than zero, and the system returns to chaos. The chaotic motion in this stage means that the system exhibits extreme irregularity and unpredictability, which seriously affects the reliability and performance stability of the system. The consistent results of the Lyapunov exponent diagram and bifurcation diagram clearly track the dynamic evolution of the system: from period-one motion to period-two motion, then to chaos, period-three motion, quasi-periodic motion, and finally back to chaos.
| Backlash range (μm) | Largest Lyapunov exponent | Motion state |
|---|---|---|
| 0 – 6.2 | Negative | Period-one |
| 6.2 – 10.4 | Negative | Period-two |
| 10.4 – 13.8 | Positive | Chaotic (short) |
| 13.8 – 23.8 | Negative | Period-three |
| 23.8 – 24.6 | Negative | Quasi-periodic |
| 24.6 – 50 | Positive | Chaotic |
When the backlash is less than 6.2 μm, the time-domain response of the system shows a sinusoidal waveform, the amplitude spectrum has a single peak, the phase plane shows a closed circular trajectory, and the Poincaré map contains only one point. These features indicate that the system is in period-one motion. When the backlash is in the range of 6.2–10.4 μm, the system changes from period-one motion to period-two motion. The time-domain response shows a more complex periodic waveform, the amplitude spectrum reveals multiple excitation frequencies, including the fundamental frequency and its half harmonic, the phase plane shows two closed circles, and the Poincaré map appears as two points. As the backlash further increases to 10.4–23.8 μm, the dynamic behavior of the system changes from chaos to period-three motion. In this state, the amplitude spectrum shows three obvious excitation frequencies, including 1/3, 2/3, and the fundamental harmonic, the phase plane presents three closed circular trajectories, and the Poincaré map contains three points. When the backlash exceeds 24.6 μm, the time-domain response shows non-periodic motion, the amplitude spectrum becomes continuous, the phase plane becomes disordered, and the Poincaré map shows numerous discrete points, indicating that the system is in a chaotic state. These results clearly describe the evolution of the system dynamic behavior with backlash: from period-one to period-two, then to period-three, and finally to chaos.
The bifurcation behavior of the non-orthogonal offset face gear split transmission system is strongly influenced by the nonlinear mesh stiffness of the face gears. As the contact point moves along the face gear tooth surface, the mesh stiffness varies periodically, which acts as an internal excitation. When the backlash is small, the teeth remain in contact for most of the meshing cycle, and the system response is dominated by the mesh frequency and its harmonics. As the backlash increases, the teeth lose contact intermittently, leading to impact-like behavior and period-doubling. Further increase in backlash causes the system to jump among different branches of the piecewise-linear backlash function, producing chaos. The coexistence of multiple attractors and the sensitivity to initial conditions are typical for such systems. The largest Lyapunov exponent provides a quantitative measure of chaos: a positive exponent confirms exponential divergence of nearby trajectories. The bifurcation diagram, on the other hand, visually shows the sudden changes in the system response as the backlash parameter varies.
It is important to note that the mesh stiffness of face gears is not constant but varies with the contact position. In non-orthogonal offset face gears, the contact path is asymmetric, and the time-varying mesh stiffness may have different amplitudes and phases for the two face gears. This asymmetry can break the symmetry of the system and lead to more complex bifurcation scenarios. The load-sharing between the two branches may also fluctuate, causing alternating loading and unloading of the face gear pairs. Therefore, the nonlinear dynamic model must include the accurate time-varying mesh stiffness to capture the true bifurcation behavior.
I also investigate the influence of other parameters. Increasing the system damping can effectively suppress chaotic behavior, and the backlash threshold that triggers chaos decreases accordingly. As the rotational speed increases, the range of chaotic behavior expands, and chaos can be triggered even at a smaller backlash. This indicates that rotational speed is a key factor affecting the dynamic stability and nonlinear behavior of the system. In contrast, an increase in the friction coefficient has little effect on the bifurcation characteristics of the system, indicating that in this model, the variation of friction is not sufficient to significantly change the basic nonlinear dynamic behavior.
5. Experimental Validation
To validate the correctness of the model, I build a test rig. The test rig consists of a motor, a magnetic powder brake, a torque sensor, an industrial computer, a data acquisition card, and a controller. The output speed of the motor is adjusted to control the rotational speed. One end of the driven shaft is fixed to a slider connected to it. By moving the slider in the slot, the positions of face gear 2 and face gear 3 can be adjusted, thereby changing the backlash of the gear pairs. The other end of the driven shaft is adjusted by replacing shafts of different diameters to change the support clearances of face gear 2 and face gear 3. In addition, vibration sensors are installed on the shafts of face gear 2 and face gear 3. The sensors collect acceleration signals and transmit them to the industrial computer for analysis and processing. This process not only obtains the radial vibration frequency and amplitude when the gear and shaft are in contact, but also determines the meshing vibration frequency and amplitude of the gear pair. Thus, the vibration characteristics of the non-orthogonal offset face gear split transmission system are studied, providing an important reference for further optimization and improvement of such transmission systems.
| Equipment | Brand | Parameters |
|---|---|---|
| Motor | Siemens AC servo motor | 0–5000 r/min |
| Vibration sensor | Kistler 8704B100 | 0.5–10000 Hz |
| Torque sensor | Kistler 4520A | 1–1000 N·m |
| Magnetic powder brake | HongDa ZKB25XN | 0–25 N·m |
| Data acquisition card | UT3704FRS-ICP | Four channels |
The vibration signals of the non-orthogonal offset face gears are collected by sensors installed on the system. The signals from channels 1 and 2 are optimized and processed to reduce noise. Then, the vibration signals are analyzed and measured using computer analysis software. The comparison between simulation and experimental results verifies the accuracy of the model.
For the system input speed \(\omega = 600\) r/min, the shaft rotation frequency is \(f_1 = \omega/60 = 10\) Hz, the meshing frequency is \(f_m = \omega \times z/60 = 240\) Hz, and the meshing period is \(T_m = 60/\omega = 0.1\) s. The sampling frequency is set to 51200 Hz, and the sampling duration is 20 s. The signal data collected from channel 1 are exported for data processing, including noise reduction filtering of the frequency-domain signal. The comparison between experimental and theoretical time-domain and frequency-domain results is performed.
The system operates with period-one bifurcation motion at a speed of 600 r/min. The theoretical time-domain and frequency-domain diagrams are compared with the experimental ones. The theoretical time-domain diagram shows relatively stable periodic vibration with minimal fluctuation. The theoretical frequency spectrum clearly reveals the existence of harmonics, with almost no sidebands or noise frequencies. However, due to complex factors such as manufacturing, machining, installation errors, and tooth surface wear, the uncertainty in the errors has a large influence on the system. Therefore, the experimental time-domain diagram shows larger vibration fluctuations, but the overall trend between the theoretical and experimental data is relatively similar. In addition, the experimental frequency spectrum shows complex spectral bands in the low-frequency region due to the vibration of the entire system captured by the sensors, resulting in a shaft frequency of 12 Hz. Although there is a certain gap with the theoretical value, the theoretical and experimental curves have a good similarity in trend, and the main frequency components are basically consistent, which proves the effectiveness of the theoretical model.
| Quantity | Theoretical | Experimental |
|---|---|---|
| Shaft rotation frequency | 10 Hz | 12 Hz |
| Meshing frequency | 240 Hz | 240 Hz |
| Motion state | Period-one | Period-one (with noise) |
| Main harmonic components | Fundamental + harmonics | Fundamental + harmonics + low-frequency bands |
The experimental validation confirms that the nonlinear dynamic model, which incorporates the time-varying mesh stiffness of the face gears, is capable of predicting the main vibration characteristics. The discrepancies can be attributed to the unavoidable manufacturing and assembly errors, as well as the simplified assumptions in the model, such as the rigid support and the neglect of lubricant effects. Nevertheless, the trend and the main frequency components are captured, which is sufficient for bifurcation analysis and stability assessment.
6. Conclusions
In this study, I established an eleven-degree-of-freedom nonlinear dynamic model for a non-orthogonal offset face gear split transmission system. The model includes the time-varying mesh stiffness of the face gears, which is computed using a load-bearing contact analysis. I systematically investigated the influence of backlash on the bifurcation characteristics. The results show that with increasing backlash, the system motion state transitions from period-one response to a short chaotic response, then to reverse period-doubling motion, and finally to a sustained chaotic vibration state. This process indicates that backlash significantly enhances the nonlinear vibration characteristics of the system and changes the stability of the system response.
I also obtained the time-domain and frequency-domain responses of the system through experimental tests and compared them with theoretical results. I analyzed the influence of installation errors and machining errors on the deviation. The results show that the experimental results are basically consistent with the theoretical results, which verifies the effectiveness of the established dynamic model. In addition, increasing the system damping can effectively suppress chaotic behavior, and the backlash threshold that triggers chaos is correspondingly reduced. With increasing rotational speed, the range of chaotic behavior expands, and chaos can be triggered even at a smaller backlash. This indicates that rotational speed is a key factor affecting the dynamic stability and nonlinear behavior of the system. In contrast, the increase in friction coefficient has little effect on the bifurcation characteristics of the system, indicating that the variation of friction in this model is not sufficient to significantly change the basic nonlinear dynamic behavior.
By precisely controlling key parameters such as backlash, rotational speed, and damping, the vibration characteristics of the system can be effectively managed and optimized, thereby improving the performance and reliability of the mechanical system. This provides an important reference for system design, especially for predicting and avoiding complex dynamic problems such as excessive vibration or mechanical failure. The findings of this work contribute to a deeper understanding of the nonlinear dynamics of non-orthogonal offset face gears and offer a theoretical basis for the design of robust face gear split transmissions.
For future work, I plan to include the effects of tooth surface modifications and misalignments on the time-varying mesh stiffness of face gears, and to investigate the bifurcation behavior under variable speed conditions. The coupling between the two face gear branches and the load-sharing characteristics will also be studied in more detail. Moreover, the influence of lubricant film and thermal effects on the nonlinear dynamics of face gears deserves further exploration. These extensions will provide a more comprehensive understanding of the dynamic behavior of face gear systems and support the development of high-performance helicopter transmissions.