Introduction. I investigate the nonlinear bifurcation characteristics of a non-orthogonal offset face gear split transmission system. The face gear split transmission has become increasingly important in coaxial rotor helicopter main reducers because it offers a compact structure, a large transmission ratio, high reliability, and favorable power-to-weight ratio. My objective is to reveal how nonlinear factors such as gear error, tooth surface friction, and backlash influence the bifurcation behavior of the face gear system. I establish an eleven-degree-of-freedom bending-torsion-axial coupled nonlinear dynamic model by using the lumped mass parameter method. I precisely construct the time-varying meshing stiffness excitation condition of the non-orthogonal offset face gear by using a load-bearing contact analysis method that accounts for the variation of mechanical behavior at different meshing points. I solve the dynamic differential equations by using the Runge-Kutta numerical integration method. I obtain time-domain diagrams, frequency spectra, phase diagrams, Poincaré maps, bifurcation diagrams, and maximum Lyapunov exponent diagrams that characterize the nonlinear bifurcation behavior. The results show that as the backlash increases, the motion state of the system gradually transitions from single-period motion to chaotic motion, then further evolves into double-period motion, and finally exhibits pronounced nonlinear vibration. I also perform experiments to verify the correctness of the nonlinear dynamic model, which provides a theoretical basis for stability research of the face gear transmission system.
Keywords. nonlinear vibration; face gear; stiffness; backlash; bifurcation; non-orthogonal offset face gear; split transmission; dynamic model; Lyapunov exponent; Poincaré map
System configuration of the face gear split transmission. The coaxial face gear split-torque transmission system is composed of three stages. The first stage uses a small-cone-angle face gear and a conical involute cylindrical gear to split the power. The second stage uses spur cylindrical gears to form a secondary power split. The third stage uses a spur cylindrical gear and a face gear to combine the power, so that the upper and lower face gears rotate coaxially in opposite directions at equal speed. In this work, I focus on the first stage of the non-orthogonal offset face gear split transmission system and study its nonlinear bifurcation characteristics. The global coordinate system \(O-XYZ\) has its origin \(O\) at the intersection of the conical involute gear axis and the non-orthogonal offset face gear rotation axis. The local coordinate systems \(O-X_2Y_2Z_2\) and \(O-X_3Y_3Z_3\) of the small-cone-angle face gears 2 and 3 are obtained by rotating the local coordinate system \(O-X_1Y_1Z_1\) around the global \(Y\)-axis counterclockwise and clockwise, respectively. The angle \(\gamma\) is the shaft intersection angle between the conical involute cylindrical gear and the small-cone-angle face gear. This face gear configuration is critical because the non-orthogonal offset condition changes the meshing point trajectory and the resulting time-varying meshing stiffness of the face gear pair.
Dynamic model of the face gear system. I use the lumped mass parameter method to construct the nonlinear dynamic model of the non-orthogonal offset face gear split transmission system. The parameters \(K_{12}\), \(C_{12}\), \(e_{12}(t)\), and \(b_{12}\) represent the meshing stiffness, meshing damping, meshing error, and backlash between gear 1 and non-orthogonal offset face gear 2, respectively. Similarly, \(K_{13}\), \(C_{13}\), \(e_{13}(t)\), and \(b_{13}\) correspond to the meshing stiffness, meshing damping, meshing error, and backlash between gear 1 and non-orthogonal offset face gear 3. For the cylindrical gear, \(K_{1x}\) and \(K_{1y}\) are the support stiffnesses, \(C_{1x}\) and \(C_{1y}\) are the support dampings, and \(b_{1x}\) and \(b_{1y}\) are the support clearances. The support stiffnesses of non-orthogonal offset face gear 2 are denoted by \(K_{2x}\), \(K_{2y}\), and \(K_{2z}\), the support dampings by \(C_{2x}\), \(C_{2y}\), and \(C_{2z}\), and the support clearances by \(b_{2x}\), \(b_{2y}\), and \(b_{2z}\). The parameters of non-orthogonal offset face gear 3 are similar: \(K_{3x}\), \(K_{3y}\), and \(K_{3z}\) for support stiffness, \(C_{3x}\), \(C_{3y}\), and \(C_{3z}\) for support damping, and \(b_{3x}\), \(b_{3y}\), and \(b_{3z}\) for support clearance.
The eleven-degree-of-freedom generalized displacement vector \(Q\) of the face gear transmission system can be expressed as:
$$ Q = \left[ Y_1,\; Z_1,\; \theta_{x1},\; X_2,\; Y_2,\; Z_2,\; \theta_{x2},\; X_3,\; Y_3,\; Z_3,\; \theta_{x3} \right]^T $$
Here, \(Y_1\) and \(Z_1\) are the lateral vibration displacements of gear 1 along the \(Y\)-axis and \(Z\)-axis, respectively. \(Y_2\) and \(Z_2\) are the lateral vibration displacements of gear 2 along the \(Y\)-axis and \(Z\)-axis, respectively. \(X_2\) is the axial vibration displacement of gear 2 along the \(X\)-axis. \(Y_3\) and \(Z_3\) are the lateral vibration displacements of gear 3 along the \(Y\)-axis and \(Z\)-axis, respectively. \(X_3\) is the axial vibration displacement of gear 3 along the \(X\)-axis. \(\theta_{x1}\), \(\theta_{x2}\), and \(\theta_{x3}\) are the torsional vibration displacements of gear 1, gear 2, and gear 3 around their respective central axes.
The meshing force \(F_{1i}\) and the backlash function of the face gear pair can be written as:
$$ F_{1i} = K_{1i}(t) f(X_{1i}) + C_{1i} \dot{X}_{1i} $$
$$ f(X_{1i}, b_{1i}) = \begin{cases} X_{1i} – b_{1i}, & X_{1i} > b_{1i} \\ 0, & -b_{1i} \le X_{1i} \le b_{1i} \\ X_{1i} + b_{1i}, & X_{1i} < -b_{1i} \end{cases} $$
$$ f_{jK}(K_j, b_j) = \begin{cases} K_j – b_j, & K_j > b_j \\ 0, & -b_j \le K_j \le b_j \\ K_j + b_j, & K_j < -b_j \end{cases} $$
where \(i = \{2,3\}\), \(j = \{1,2,3\}\), and \(K = \{X,Y,Z\}\). The function \(f(X_{1i}, b_{1i})\) is the backlash nonlinear function on the meshing pair between gear 1 and gear 2 or gear 3. The function \(f_{jK}(K_j, b_j)\) is the support clearance nonlinear function of gear 1, gear 2, and gear 3 in their respective axial directions. \(X_{1i}\) is the relative displacement on the meshing pair between gear 1 and gear 2 or gear 3. \(K_j\) is the equivalent displacement of the torsional line displacement of gear 1, gear 2, and gear 3 in their respective axial directions. \(b_{1i}\) represents the backlash on the meshing pair between gear 1 and gear 2 or gear 1 and gear 3. \(b_j\) represents the support clearance of gear 1, gear 2, and gear 3.
Dimensionless dynamic equations of the face gear system. I introduce a dimensionless method to derive the dimensionless nonlinear dynamic differential equations of the system. The dimensionless time is defined as \(\tau_i = \omega_{ni} t\), where the natural frequency is:
$$ \omega_{ni} = \sqrt{\frac{K_m (I_1 r_1^2 + I_i r_i^2)}{I_1 I_i}} $$
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 speed are \(\bar{b}_{1i} = b_{1i}/b_c\), \(\Omega_1 = \omega/\omega_{n2}\), \(\Omega_2 = \omega/\omega_{n2}\), and \(\Omega_3 = \omega/\omega_{n3}\). The masses of gear 1, gear 2, and gear 3 are \(m_1\), \(m_2\), and \(m_3\). The 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 \(\alpha_{n2}\) is the meshing angle between gear 1 and gear 2, \(\alpha_{n3}\) is the meshing angle between gear 1 and gear 3, \(\sigma_{12}\) is the phase angle of gear 2 relative to gear 1, and \(\sigma_{13}\) is the phase angle of gear 3 relative to gear 1. \(T_{in}\) is the input torque of the system, and \(T_{out}\) is the output torque of the system.
Because the backlash and support clearances cause the gear system vibration equations to be a semi-definite differential equation system that contains rigid body displacement and cannot be solved directly, I introduce relative coordinates to eliminate the rigid body displacement. The relative coordinate expression for the face gear system is:
$$ \bar{X}_{1i} = \bar{Y}_1 \sin\phi_1 + \bar{Z}_1 \cos\phi_1 + \bar{\theta}_1 r_{b1} – \bar{X}_i \sin\phi_i \cos\gamma – \bar{Y}_i \cos\phi_i – \bar{Z}_i \sin\phi_i \sin\gamma – \bar{\theta}_i r_{bi} $$
where \(i = \{2,3\}\). \(X_{12}\) is the relative displacement on the meshing line between gear 1 and gear 2. \(X_{13}\) is the relative displacement on the meshing line between gear 1 and gear 3. After introducing the relative coordinates, the dimensionless dynamic equation for the relative displacement of the face gear pair can be expressed as:
$$ \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 – \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 – \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 + \left[ (\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) \right] \sin\phi_3 – \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 + \left[ (\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) \right] \cos\phi_3 + \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 – \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 + \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 + \frac{\bar{T}_{in} r_{b1}}{I_1 b_c \omega_{n2}^2} + \frac{\bar{T}_{out} r_{b2}}{2 I_2 b_c \omega_{n2}^2} + \ddot{\bar{e}}(t) $$
$$ \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 – \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 – \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 + \left[ (\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) \right] \sin\phi_3 – \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 + \left[ (\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) \right] \cos\phi_3 + \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 – \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 + \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 + \frac{\bar{T}_{in} r_{b1}}{I_1 b_c \omega_{n2}^2} + \frac{\bar{T}_{out} r_{b3}}{2 I_3 b_c \omega_{n3}^2} + \ddot{\bar{e}}(t) $$
Time-varying meshing stiffness of the non-orthogonal offset face gear. The non-orthogonal offset face gear pair is formed by a conical involute gear and a small-cone-angle face gear meshing with each other. The conical involute gear is generated by a tilted rack cutter. I establish the generating coordinate system of the conical involute gear. The gear tooth surface \(\Sigma_i\) can be expressed as:
$$ \begin{cases} r_i(u_i, l_i, \psi_i) = M_{ie}(\psi_i) M_{ed} M_{dr} r_{ri}(u_i, l_i) \\ f_{ri}(u_i, l_i, \psi_i) = n_{ri}(u_i, l_i) \cdot v_r^{(ri)}(u_i, l_i, \psi_i) = 0 \end{cases} $$
where \(r_i\) represents a point vector on the conical involute gear tooth surface \(\Sigma_i\). During the gear generating process, the meshing point is transformed from the rack cutter coordinate system \(S_r\) to the gear coordinate system \(S_i\) by the coordinate transformation matrices \(M_{ie}(\psi_i)\), \(M_{ed}\), and \(M_{dr}\). The condition \(f_{ri} = 0\) indicates that the meshing equation between the rack cutter and the conical involute gear is satisfied, where \(n_{ri}\) is the normal vector of the rack cutter, and \(v_r^{(ri)}\) represents the relative velocity of the rack cutter with respect to the conical involute gear, both defined in the rack cutter coordinate system \(S_r\).
The face gear tooth surface \(\Sigma_2\) is generated by the conical involute gear cutter. Specifically, it is the envelope surface of the gear cutter tooth surface \(\Sigma_s\). The face gear generating coordinate system is established. The coordinate systems \(S_2\) and \(S_s\) correspond to the non-orthogonal face gear and the conical involute gear cutter, respectively, and \(S_m\) is the fixed reference coordinate system. The angle \(\gamma_m\) is the angle between the non-orthogonal offset face gear axis \(X_2\) and the negative direction of the gear cutter axis \(Z_s\). The angular velocities \(\omega_s\) and \(\omega_2\) are the cutter speed and the non-orthogonal offset face gear speed. The face gear tooth surface \(\Sigma_2\) can be expressed as:
$$ \begin{cases} r_2(u_s, l_s, \psi_s) = M_{2m}(\psi_2) M_{mn} M_{ns}(\psi_s) r_s(u_s, l_s) \\ f_{s2}(u_s, l_s, \psi_s) = n_s(u_s, l_s) \cdot v_{s2}^s(u_s, l_s, \psi_s) = 0 \end{cases} $$
where \(r_2\) represents a point on the face gear tooth surface \(\Sigma_2\). \(r_s\) is the mathematical description of the conical involute gear cutter tooth surface \(\Sigma_s\). The meshing point is accurately transformed from the conical involute gear cutter tooth surface \(\Sigma_s\) to the face gear tooth surface \(\Sigma_2\) by the combined action of the matrices \(M_{2m}(\psi_2)\), \(M_{mn}\), and \(M_{ns}\). The condition \(f_{s2} = 0\) means that the meshing condition is reached, where \(n_s\) is the normal vector of the gear cutter tooth surface \(\Sigma_s\) in the coordinate system \(S_s\), and \(v_{s2}^s\) represents the relative velocity of the gear cutter with respect to the face gear, also given in the coordinate system \(S_s\).
I perform loaded tooth contact analysis for the non-orthogonal offset face gear pair and establish the application coordinate system. To comprehensively consider the influence of installation errors, I introduce auxiliary coordinate systems \(S_\theta\), \(S_q\), and \(S_h\). \(S_u\) serves as a fixed spatial coordinate system and provides a stable reference. \(S_p\) and \(S_2\) represent the conical involute gear and face gear follower coordinate systems, respectively. \(\Phi_p\) and \(\Phi_2\) describe the rotation angle states of the pinion and the face gear during meshing. The axis misalignment error of the conical face gear pair is \(\Delta E\), the axial error of the face gear during installation is \(\Delta q\), and the shaft intersection angle error is \(\gamma_f = \gamma_m + \Delta\gamma\), where \(\gamma_m\) is the base shaft intersection angle and \(\Delta\gamma\) is the deviation of the shaft intersection angle. The geometric distance determined by the initial installation distance, tooth width, and shaft intersection angle is \(L_0 + B \cot\gamma\).
The tooth surface equations and unit normal vectors of the conical involute gear tooth surface \(\Sigma_p\) and the non-orthogonal offset face gear tooth surface \(\Sigma_2\) in their respective coordinate systems are transformed into the fixed coordinate system \(S_u\):
$$ \begin{cases} r_u^p(u_p, l_p, \psi_p, \Phi_p) = M_{up}(\Phi_p) r_p(u_p, l_p, \psi_p) \\ n_u^p(u_p, l_p, \psi_p, \Phi_p) = L_{up}(\Phi_p) n_p(u_p, l_p, \psi_p) \\ r_u^2(u_s, l_s, \psi_s, \Phi_2) = M_{u2}(\Phi_2) r_2(u_s, l_s, \psi_s) \\ n_u^2(u_s, l_s, \psi_s, \Phi_2) = L_{u2}(\Phi_2) n_2(u_s, l_s, \psi_s) \end{cases} $$
where \(M_{up}\) and \(L_{up}\) correspond to the position and orientation transformation matrices from the pinion coordinate system \(S_p\) to the fixed coordinate system \(S_u\). Similarly, \(M_{u2}\) and \(L_{u2}\) represent the position and orientation transformation matrices from the face gear coordinate system \(S_2\) to the fixed coordinate system \(S_u\). The two gears must maintain continuous contact during meshing, so the following equations must be satisfied:
$$ \begin{cases} r_u^p(u_p, l_p, \psi_p, \Phi_p) = r_u^2(u_s, l_s, \psi_s, \Phi_2) \\ n_u^p(u_p, l_p, \psi_p, \Phi_p) = n_u^2(u_s, l_s, \psi_s, \Phi_2) \end{cases} $$
where \(r_u^p\) and \(n_u^p\) are the position vector and normal vector of the conical involute gear, \(u_p\) and \(l_p\) are independent parameters for generating the conical involute gear, \(r_u^2\) and \(n_u^2\) are the position vector and normal vector of the non-orthogonal offset face gear, and \(u_s\) and \(l_s\) are independent parameters for generating the non-orthogonal offset face gear.
Without considering the influence of lubricating oil and friction during the tooth meshing process, I establish the loaded contact analysis model of the non-orthogonal offset face gear transmission under the action of the total normal meshing force \(P\). The face gear transmission adopts a conjugate point contact tooth surface design. When a load acts, the gear teeth undergo elastic deformation, and the instantaneous contact area width is neglected. I consider a normal section through the long axis of the instantaneous contact ellipse. I assume that tooth pair I and tooth pair II may participate in contact simultaneously. Point \(i\) represents the center of the contact ellipse, and point \(j\) is any discrete position on the long axis of the contact ellipse. The displacement coordination relationship after deformation can be expressed as:
$$ F_k p_k + w_k = d_k + Z_k \quad (k = I, II) $$
where \(p_k = [p_1, p_2, \cdots, p_n]^T\) represents the normal load vector at each discrete point on the long axis of the instantaneous contact ellipse, \(d_k = [d_1, d_2, \cdots, d_n]^T\) represents the clearance vector generated on the tooth surface after deformation at these discrete points, and \(Z_k = Z[1, 1, \cdots, 1]^T\) reflects the uniform normal displacement of the tooth after deformation. For each discrete point, the normal load \(p_j\) must satisfy the basic principle of force balance:
$$ \sum_{j=1}^n p_j^I + \sum_{j=1}^n p_j^{II} = P $$
The non-embedding condition that must be satisfied during gear contact can be expressed as:
$$ \begin{cases} p_j^k > 0, & d_j^k = 0 \\ p_j^k = 0, & d_j^k > 0 \end{cases} $$
When considering installation errors, conical involute gear offset, and other factors in the loaded contact problem of the face gear transmission, the mathematical programming representation can be simplified as:
$$ \min \sum_{j=1}^{n+1} X_j $$
$$ \begin{cases} d + Z + X – F p = w \\ e^T p + X_{n+1} = P \\ p_j \ge 0, \; d_j \ge 0, \; Z_j \ge 0, \; X_j \ge 0 \\ p_j = 0 \text{ or } d_j = 0 \end{cases} $$
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 \(X = [X_1, X_2, \cdots, X_n]^T\) is an \(n\)-dimensional column vector, \(e\) is a unit \(n\)-dimensional column vector, and \(p_j \ge 0\), \(d_j \ge 0\), \(Z_j \ge 0\), \(X_j \ge 0\) are the contact boundary conditions.
After the non-orthogonal offset face gear is deformed under force, the normal displacement of the gear teeth can be converted into the loaded transmission error by a specific conversion formula:
$$ TE_L = 3600 \times 180 \frac{Z}{|\omega_2|} \frac{(R_2 \times n_2) \cdot \omega_2}{|\omega_2|} $$
where \(\omega_2\) is the angular velocity vector of the non-orthogonal offset face gear, \(R_2\) is the position vector of the contact point on the gear tooth surface, and \(n_2\) is the normal vector of the tooth surface at the contact point.
At any instantaneous meshing state, the normal comprehensive deformation \(\delta\) of the gear teeth after loading is mainly determined by \(|TE_G – TE_L|\), where \(TE_G\) is the geometric transmission error of the non-orthogonal offset face gear after typical modification. Thus:
$$ \delta = |TE_G – TE_L| \frac{(R_2 \times n_2) \cdot \omega_2}{3600 \times 180 |\omega_2|} $$
Assuming that within one meshing cycle \(T\), a single pair of gear teeth undergoes eight different meshing stages, from initial contact to complete disengagement, involving \(N\) stages. The comprehensive meshing stiffness of the face gear transmission and the meshing stiffness of a single tooth pair are calculated as:
$$ k_m(ii) = \frac{p}{\delta(ii)} \quad (ii = 1, 2, 3, \cdots, 8) $$
$$ k_s(jj) = k_m(jj – 8 \lfloor (jj-1)/8 \rfloor) L_k(jj) \quad (jj = 1, 2, 3, \cdots, N) $$
To verify the effectiveness and accuracy of my meshing stiffness calculation method based on tooth surface contact theory, I compare the comprehensive meshing stiffness of a helical gear transmission calculated by this method with that calculated by a classical method. The classical stiffness formula is:
$$ k_i(t) = k_\alpha + \sum_{n=1}^\infty k_{ni} \cos(n\omega t + \alpha_{1i}) $$
where \(K_\alpha\) is the average value of the meshing stiffness, \(K_{ni}\) is the fluctuation amplitude of the meshing stiffness, \(\omega\) is the meshing frequency of the gear pair, and \(\alpha_{1i}\) is the meshing phase angle. The comparison shows that the comprehensive meshing stiffness curves obtained by the two methods have basically the same variation trend. However, because the classical formula does not take into account the influence of the gear 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 precision for the face gear system.
System parameters and numerical solution. The simulation parameters of the face gear system are as follows: the inner diameter of the face gear is 140 mm, the outer diameter is 172 mm, the shaft intersection angle is \(30^\circ\), the elastic modulus of the gear material is 3045 MPa, and the Poisson’s ratio is 0.41. The main parameters are listed in Table 1.
| Parameter | Gear 1 | Gear 2 | 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\(^2\)) | \(1.46 \times 10^{-6}\) | \(5.26 \times 10^{-6}\) | \(5.26 \times 10^{-6}\) |
Using the parameters in Table 1, I calculate the time-varying meshing stiffness curves of each gear pair. 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. Under these conditions, I obtain the global bifurcation diagram and the Lyapunov exponent diagram of the system with respect to the backlash. These diagrams provide a comprehensive perspective for understanding the influence of backlash variation on the stability and dynamic behavior of the face gear transmission system.
Bifurcation behavior of the face gear system under varying backlash. From the bifurcation diagram and Lyapunov exponent diagram, I find that under the condition of damping coefficient 0.07, the system exhibits different dynamic behaviors that are directly related to the magnitude of the backlash. The detailed transition ranges are summarized in Table 2.
| Backlash range / μm | Lyapunov exponent | Motion state | Characteristics |
|---|---|---|---|
| 0–6.2 | Negative | Single-period motion | Sine waveform, single peak, one closed circle in phase diagram, one point in Poincaré map |
| 6.2–10.4 | Negative | Double-period motion | More complex periodic waveform, multiple frequencies including 1/2 harmonic, two closed circles, two points in Poincaré map |
| 10.4–13.8 | Positive | Short chaotic state | Multiple discrete points in bifurcation diagram, reduced predictability |
| 13.8–23.8 | Negative | Triple-period motion | Three excitation frequencies including 1/3 and 2/3 harmonics, three closed circles, three points in Poincaré map |
| 23.8–24.6 | Near zero | Quasi-periodic motion | Closed invariant curve in Poincaré map |
| 24.6–50 | Positive | Chaotic motion | Non-periodic time response, continuous spectrum, disordered phase diagram, many discrete points in Poincaré map |
When the backlash is less than 6.2 μm, the time-domain response of the system appears as a sinusoidal waveform, the amplitude spectrum shows a single peak, the phase plane diagram displays a closed circular trajectory, and the Poincaré map contains only one point. These features indicate that the face gear system is in a single-period motion state. When the backlash is in the range of 6.2–10.4 μm, the system changes from single-period motion to double-period motion. The time-domain response shows a more complex periodic waveform, the amplitude spectrum reveals multiple excitation frequencies including the fundamental frequency and its 1/2 harmonic, the phase plane diagram shows two closed circles, and the Poincaré map contains two points.
As the backlash further increases to 10.4–23.8 μm, the dynamic behavior of the system changes from chaos to triple-period motion. In this state, the amplitude spectrum shows three obvious excitation frequencies including 1/3, 2/3, and single harmonics. The phase plane diagram 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 a continuous spectrum, the phase plane diagram becomes disordered, and the Poincaré map shows many discrete points, indicating that the face gear system is in a chaotic state. This series of results clearly depicts the evolution of the dynamic behavior of the face gear system with increasing backlash: from single-period to double-period, then to triple-period, and finally to chaos.
Experimental validation of the face gear model. To verify the correctness of my model, I build an experimental test rig. The test rig consists of a motor, a magnetic powder brake, a torque sensor, an industrial computer, a data acquisition card, and an industrial control computer. The test rig adjusts the rotational speed by adjusting the output speed of the motor. One end of the driven shaft is fixed to a slider connected to it. By moving the slider in the slot, the positions of gear 2 and gear 3 can be adjusted, thereby changing the backlash of the gear pair. The other end of the driven shaft adjusts the support clearance of gear 2 and gear 3 by replacing shafts of different diameters. In addition, vibration sensors are installed on the shafts of gear 2 and gear 3. After the sensors collect acceleration signals, they transmit them to the industrial control computer for analysis and processing. This process not only obtains the radial vibration frequency and amplitude when the gear contacts the shaft, but also determines the meshing vibration frequency and amplitude of the gear pair, thereby completing the study of the vibration characteristics of the non-orthogonal offset face gear split transmission system. The experimental equipment models are listed in Table 3.
| 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 gear are collected by sensors installed on the system. I optimize and process the signals collected from channels 1 and 2 to reduce noise. Then I use computer analysis software to analyze and measure the vibration signals. The comparison between simulation results and experimental results verifies the accuracy of the model. The system input speed is \(\omega = 600\) r/min, the shaft transmission 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. I export the signal data collected from channel 1 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 summarized in Table 4.
| Quantity | Theoretical value | Experimental value | Deviation |
|---|---|---|---|
| Shaft frequency \(f_1\) | 10 Hz | 12 Hz | 20% |
| Meshing frequency \(f_m\) | 240 Hz | Approximately 240 Hz | Small |
| Motion state at 600 r/min | Period-1 | Period-1 with fluctuations | Trend consistent |
| Main harmonic components | Fundamental and harmonics | Fundamental and low-frequency bands | Good similarity |
The system operates at a speed of 600 r/min and performs period-1 bifurcation motion. The theoretical time-domain and frequency-domain diagrams at 600 r/min show that the overall system has relatively stable periodic vibration with minimal fluctuation. The 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 error has a great influence on the system. Therefore, the experimental time-domain diagram shows large vibration fluctuations, but the overall trend between the theoretical and experimental data is relatively similar. In addition, the experimental 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 good similarity in trend, and the main frequency components are basically consistent, which proves the effectiveness of the theoretical model of the face gear system.
Parameter sensitivity of the face gear bifurcation behavior. I further analyze the influence of key parameters on the nonlinear bifurcation characteristics of the non-orthogonal offset face gear system. The results are summarized in Table 5.
| Parameter | Trend | Effect on bifurcation |
|---|---|---|
| Damping coefficient | Increase | Suppresses chaotic behavior; the backlash threshold that triggers chaos decreases |
| Rotational speed | Increase | Expands the range of chaotic behavior; chaos can be triggered even at small backlash |
| Friction coefficient | Increase | Very small effect on the bifurcation characteristics; not enough to significantly change the basic nonlinear dynamic behavior |
| Backlash | Increase | Transitions from single-period to double-period, chaos, triple-period, quasi-periodic, and finally chaos |
| Support clearance | Increase | Enhances nonlinearity and can shift bifurcation boundaries |
The increase of system damping can effectively suppress chaotic behavior, and the backlash threshold that triggers chaos is correspondingly reduced. As the rotational speed increases, the range of chaotic behavior expands, and chaos is triggered even at a smaller backlash. This indicates that rotational speed is a key factor affecting the dynamic stability and nonlinear behavior of the face gear system. On the contrary, the increase of the 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 of the face gear transmission. 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 in the face gear system.
Conclusions. I established an eleven-degree-of-freedom nonlinear dynamic model of a non-orthogonal offset face gear split transmission system and systematically studied the influence of backlash on the bifurcation characteristics. The results show that with the increase of backlash, the motion state of the system transitions from a single-period response to a short chaotic response, then to a reverse double-period motion, and finally to a sustained chaotic vibration state. This process indicates that backlash significantly enhances the nonlinear vibration characteristics of the face gear system and changes the stability of the system response. The time-domain and frequency-domain responses obtained from experimental tests are basically consistent with the theoretical calculation 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. As the rotational speed increases, the range of chaotic behavior expands, and chaos is triggered even at a smaller backlash. This indicates that rotational speed is a key factor affecting the dynamic stability and nonlinear behavior of the face gear system. On the contrary, the increase of the 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 of the face gear transmission. By precisely controlling key parameters such as backlash, rotational speed, and damping, the vibration characteristics of the face gear system can be effectively managed and optimized, thereby improving the performance and reliability of the mechanical system. This provides an important reference for the design of non-orthogonal offset face gear split transmission systems, especially for predicting and avoiding complex dynamic problems such as excessive vibration or mechanical failure.