the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Study on dynamic loadsharing characteristics of face gear dualpower split transmission system with backlash, support and spline clearance
Hao Qin Zhang
Xiao Long Zhao
Ling Ling Duan
The dynamic loadsharing characteristics of aircraft face gear dualpower split transmission system (FGDPSTS) are taken as the research object. Considering the factors of timevarying meshing stiffness, comprehensive error, backlash, support clearance, spline clearance, torsional stiffness, and support stiffness, the dynamic loadsharing model was constructed based on the lumpedparameter method. The loaded tooth contact analysis (LTCA) simulation method was used to calculate the timevarying meshing stiffness. The dynamic loadsharing coefficient (DLSC) is obtained by using Runge–Kutta method. The influences of errors, backlash, support clearance, spline clearance, torsional stiffness and support stiffness on DLSC were analyzed, and the biggest factors affecting dynamic loadsharing performance were found out. The results show that the influence of the backlash of the twostage herringbone gear pair on the DLSC is more sensitive. The influence of support clearance on the DLSC is less. The loadsharing coefficient increases with the increase of the installation error and eccentricity error, and the influence of the error of the twostage gears on the system loadsharing performance is the most sensitive. The torsional stiffness has little effect on the loadsharing coefficient of one stage but has great effect on the twostage loadsharing coefficient. The influence of support stiffness on the DLSC of twostage is stronger. It provided a theoretical basis for the dynamic stability optimization design of the system.
The face gear transmission system (Heath and Bossler, 1993; Handschuh et al., 1996) replaces the original bevel gear transmission on the thirdgeneration Apache armed helicopter. Compared with the bevel gear transmission, the face gear is used in the onestage transmission, which has a good powersplitting effect, and can realize the reversing and splitting at the same time. The number of transmission stages is reduced from four to three stages, the weight of the transmission system is reduced by 40 %, and the bearing capacity is increased by 35 %. The support structure of spiral bevel gear is complex, and the pinion driven by face gear is not subject to axial force, which is conducive to reducing weight and noise, improving reliability and service life.
Many researchers at home and abroad have done a lot of research on gearrelated theory and loadsharing technology. Krantz (1996a, b) has done further research on the load sharing of the two branches of the cylindrical gear. By defining the synchronous angle, the influence on the loadsharing performance of the power split transmission is studied, and the mathematical relationship between the same step angle and loadsharing coefficient is tested and verified. Robert and James (2006) proposed applying the face gear to a configuration similar to the planetary gear transmission. Using two face gears with the same axis, the face gear on the input shaft and the small gear mesh, the small gear on the other end of the coaxial gear and a fixed rotating face gear mesh, the small gear shaft is installed on the output shaft. Stevens et al. (2009) put forward the variable speed configuration of the fixed axis star gear and face gear. The face gear is used to replace the fixed axis star gear on the transmission chain. The face gear is designed as an idler gear to achieve the same high and lowspeed output steering of the transmission system. Because the input and output shafts of the face gear are installed in the same axis, the structure is tight and the commutation is more stable. Peng et al. (2016) take the coaxial face gear transmission system as the research object, establish the geometric model and kinematic model of the face gear transmission, study the parameter instability of the face gear transmission, and analyze the load distribution characteristics of the coaxial face gear transmission system under different working speeds. Jin et al. (2019a, b) considered the backlash, the timevarying meshing stiffness between the gear pairs, the gear eccentricity error, the shaft torsion and the support stiffness and constructed the dynamic model of the face gear cylindrical gear two power split transmissions, focusing on the influence of the torsional stiffness of each transmission shaft on the loadsharing coefficient of the system. Mo et al. (2020) established the bending torsional coupling dynamic loadsharing model of multipower input face gear branch transmission system and studied the influence of meshing phase and error on the loadsharing coefficient. Dong et al. (2019) analyzed the assembly, power flow and load sharing of the coaxial face gear split torsional transmission system, deduced the assembly and installation conditions suitable for the coaxial face gear split torsional transmission system, and studied the power distribution direction of the system through the finite element method, and analyzed the influence of load conditions, distribution angle and installation angle. Wang et al. (2013) studied the influence of the dynamic loadsharing characteristics of the face gear split torsional transmission system. Bao et al. (2019) studied the loadsharing and dynamic characteristics of the variable speed helicopter transmission system and analyzed the influence of input power, input speed and friction factor on the loadsharing and dynamic characteristics of the system. Wang et al. (2019) studied the loadsharing performance of the twobranching cylindrical gear split torsional transmission system and analyzed the influence of working conditions, torsional stiffness, support stiffness, meshing phase difference and other factors on the loadsharing performance of the system. Zhang and Zhu (2018) studied the static loadsharing characteristics of a herringbone planetary gear system with floating combined inner gear ring and qualitatively analyzed the influence of the eccentricity error of each component, the floating mode and floating amount of the central component, the torsional rigidity of the flexible inner gear ring and other parameters on the static loadsharing characteristics of the system.
In previous studies, the influence of backlash and support clearance on the loadsharing characteristics of face gear drive is seldom considered, and the average mesh stiffness is mostly used in the establishment and calculation of the model, which cannot fully reflect the influence of the meshing process of tooth surface on the dynamic loadsharing characteristics of the system. According to the structure layout and transmission characteristics of the helicopter main reducer, and the advantages of two branch power split transmission of face gear, a face gear dualpower split transmission system configuration with dualpower split characteristics is proposed. Based on the lumpedparameter method, the dynamic model of the system is constructed, the dynamic differential equation is derived, the dynamic loadsharing coefficient (DLSC) is obtained, and the influence of parameters on the DLSC is analyzed. It provides the theoretical basis for the dynamic vibration stability optimization design of the face gear dualpower split transmission system.
Figure 1 is the configuration diagram of the face gear dualpower split transmission system (FGDPSTS), with the technical characteristics of onestage face gear pair power split, twostage pinion secondary power split, twostage idler gear convergence, twostage gear output. Compared with the face gear twobranch transmission system implemented by an art program of the United States (Handschuh et al., 1996), this configuration adopts a twostage split twist layout to meet the requirements of a large transmission ratio, high power density ratio and large thrust weight ratio. The transmission power of the gear is divided by multiple channels, and the torque transmitted by the gears in each branch is reduced, so that the bearing capacity of the gear is improved, and the service life and reliability of the system are further improved.
Figure 2 shows the torsional dynamic model of this system.
Here, T_{in} and T_{out}, k_{in} and k_{out}, and c_{in} and c_{out} are the input torque and output torque, torsional stiffness and torsional damping of input and output shafts, respectively. φ_{in} and φ_{out}, m_{in} and m_{out}, and I_{in} and I_{out} are the torsion angle, equivalent lumped mass and moment of inertia of prime mover and loaded, respectively. k_{ij}, c_{ij}, e_{(t)ij} and b_{ij} are the timevarying meshing stiffness, meshing damping, comprehensive error and backlash of each gear pair ij, respectively. φ_{iz} is the torsional vibration microdisplacement of each gear i. m_{i} is the equivalent concentrated lumpedmass gear i. I_{i} is the moment of inertia of component i. k_{24} and k_{35} and c_{24} and c_{35} are the torsional stiffness and torsional damping of connecting shaft 24 and connecting shaft 35.
The dynamic model of onestage face gear is established as shown in Fig. 3. k_{xi} and k_{yi} are the support stiffness in x direction and y direction of gear i, respectively. c_{xi} and c_{yi} are the support damping in x direction and y direction of gear i, respectively. b_{ij} is the average meshing backlash. b_{di} is the average elastic clearance. The input spur gear coordinate system is O_{1}x_{1}y_{1}z_{1}. The face gear coordinate system is O_{2}x_{2}y_{2}z_{2} and O_{3}x_{3}y_{3}z_{3}, respectively. The z axis is the vertical upward axial direction of the face gear. The input gear shaft is perpendicular to the x direction, y direction and xz plane.
The generalized vibration displacement vector δ with 32 degrees of freedom can be expressed as Eq. (1). Here, x_{i}, y_{i} and z_{i} are the transverse bending deformation of each gear in the x direction, y direction and z direction, respectively. φ_{iz}, φ_{in} and φ_{out} are the torsional angular displacement of gear i, prime mover and load components.
Force analysis of the onestage face gear pair is shown in Fig. 4.
Here, r_{1} is the pitch radius. r_{a1} is the top circle radius. The vibration displacement of pinion O_{1} is y_{1} and z_{1} along y direction and z direction, respectively. The vibration displacement of face gear O_{2} and O_{3} along y direction and z direction is y_{2}, y_{3} and z_{2}, z_{3}, respectively.
The meshing forces of F_{12} and F_{13} can be expressed as follows:
Based on the lumpedparameter method, the torsional vibration dynamic equation of the system is established as follows:
Here, ${\ddot{\mathit{\phi}}}_{iz}$, ${\ddot{\mathit{\phi}}}_{\mathrm{in}}$ and ${\ddot{\mathit{\phi}}}_{\mathrm{out}}$ are the angular acceleration of torsional vibration corresponding to gear i, prime mover and load component. ${\dot{\mathit{\phi}}}_{iz}$, ${\dot{\mathit{\phi}}}_{\mathrm{in}}$ and ${\dot{\mathit{\phi}}}_{\mathrm{out}}$ are the angular velocity of torsional vibration corresponding to gear i, prime mover and load component. r_{bi} represents the radius of base circle of gear i, where r_{b2} and r_{b3} represent the radius of base circle of face gear, ${r}_{b\mathrm{2}}={r}_{b\mathrm{3}}=$ (r_{L1}+r_{L2}), and r_{L1} is the inner radius of the face gear determined by the undercutting condition, and r_{L2} is the outer radius of the face gear determined by the tip sharpening condition. I_{i}(i=1, 2, …, 10) is the moment of inertia of each component. α_{n} is the normal meshing pressure angle of gear pair meshing.
The angular displacement form is transformed into the linear displacement form, which can be expressed as ${\ddot{u}}_{i}={r}_{bi}{\ddot{\mathit{\phi}}}_{iz},{\dot{u}}_{i}={r}_{bi}{\dot{\mathit{\phi}}}_{iz},{u}_{i}={r}_{bi}{\mathit{\phi}}_{iz}$. The mass m_{eq,i} can be expressed as ${m}_{\mathrm{eq},i}={I}_{i}/{r}_{bi}^{\mathrm{2}}$.
The expressions of torsional damping c_{24} and c_{35} are as follows:
Here, ζ_{24} and ζ_{35} are the damping coefficients.
Based on Newton's second law, the lateral vibration equation of the system is established as follows:
Here, ${\ddot{x}}_{i}$, ${\dot{x}}_{i}$ and x_{i} are the lateral xdirection vibration acceleration, velocity and displacement, respectively. ${\ddot{y}}_{i}$, ${\dot{y}}_{i}$ and y_{i} are the lateral ydirection vibration acceleration, velocity and displacement, respectively. ${\ddot{\mathrm{z}}}_{i}$, ${\dot{z}}_{i}$ and z_{i} are the lateral zdirection vibration acceleration, velocity and displacement, respectively.
F_{Wxi} and F_{Wyi} of gear i(i=4, 5) are the component of the supporting force of the spline shaft in x and y directions. Taking pinion O_{4} as an example, the dynamic mechanical analysis model is shown in Fig. 5.
k_{wxi} and k_{wyi} as well as c_{wxi} and c_{wyi} are bending stiffness and bending damping of spline shaft in x and y directions. F_{m}=τF_{N} is the friction force between the internal and external splines in the process of transmitting torque. F_{N} represents the normal positive pressure between the internal and external splines. τ is the coefficient of friction. $\mathit{\tau}={\mathit{\tau}}_{\mathrm{0}}\cdot sgn\left({v}_{\mathrm{s}}\right)$, τ_{0} is the amplitude of friction coefficient, v_{s} is the relative sliding speed, and sgn(v_{s}) is the symbol function of v_{s}. Here, τ_{0}=0.1, sgn(v_{s}>0)=1, τ=0.1.
Comprehensive displacement of gear i (i=4, 5) in x direction and y direction can be expressed as ${R}_{i}=\sqrt{{x}_{i}^{\mathrm{2}}+{y}_{i}^{\mathrm{2}}}$. R_{i} is the total displacement and also represents the value of spline.
The lateral bending supporting forces F_{Wxi} and F_{Wyi} can be described as follows:
Here, c_{m} is friction damping. S_{1} and S_{2} are the radial clearance of internal and external splines. ξ_{i} is direction angle of vector (x_{i}, y_{i}). By combining the differential equations of dynamic torsion and transverse vibration mentioned above, the bending–torsion coupled nonlinear dynamic differential equations can be obtained.
The backlash f_{ij}(x_{ij}, b_{ij}) and support clearance f_{xi}(x_{i}, b_{di}), f_{yi}(y_{i}, b_{di}) and fz_{i} (z_{i}, b_{di}) are expressed as follows:
where x_{ij} is the relative vibration displacement.
Dynamic meshing force F_{Lij} (L= I, II) can be expressed as follows:
The meshing damping c_{ij} is calculated according to the following empirical formula (Dong et al., 2018):
where k_{mij} is the average meshing stiffness. ζ_{gij} is the damping coefficient.
Elastic force P_{Iij} and P_{IIij} as well as meshing damping force D_{Iij} and D_{IIij} can be expressed as follows:
Here, α_{Iij} is the positive angle between the meshing line and the center line of the onestage face gear pair and the z axis, α_{Iij}(i=1, j=2, 3) $=\mathit{\pi}/\mathrm{2}+{\mathit{\alpha}}_{n}+{\mathit{\gamma}}_{\mathrm{I}ij}$, and γ_{Iij} is the positive angle between the center line of the onestage face gear pair and the z axis. α_{IIij} is the positive angle between the meshing line and the center line of the twostage cylindrical gear pair and the x axis, α_{IIij}(i=4, 5, j=6, 7, 8, 9) $=\mathit{\pi}/\mathrm{2}{\mathit{\alpha}}_{n}+{\mathit{\gamma}}_{\mathrm{I}ij}$, α_{IIij} (i=6, 7, 8, 9, j=10) $=\mathit{\pi}/\mathrm{2}+{\mathit{\alpha}}_{n}+{\mathit{\gamma}}_{ij}$, and γ_{IIij} is the positive angle between the center line of the twostage cylindrical gear pair and the x axis. r_{bi} and r_{bj} are the radius of base circle of gear i and j.
The eccentricity errors of onestage face gear and twostage cylindrical gear are shown in Fig. 6
Here, O_{i(j)} is theoretical center position. ^{′}O_{i(j)} is actual center position. α_{ij} is the angle between installation error and eccentricity error and x axis.
The equivalent meshing error e(t)_{ij} can be expressed as follows:
Here, E_{i} and E_{j} are eccentricity error. A_{i} and A_{j} are installation error. η_{i} and η_{j} are eccentricity error phase angle. δ_{i} and δ_{j} are installation error phase angle. ω_{i} and ω_{j} are excitation frequency.
The dimensionless time is defined as τ_{n}=ω_{n}t. Here, τ is the time variable of equations before dimensionless treatment. ${\mathit{\omega}}_{n}=\sqrt{{\stackrel{\mathrm{\u203e}}{k}}_{\mathrm{12}}\left({I}_{\mathrm{1}}{r}_{\mathrm{2}}^{\mathrm{2}}+{I}_{\mathrm{2}}{r}_{\mathrm{1}}^{\mathrm{2}}\right)/\left({I}_{\mathrm{1}}{I}_{\mathrm{2}}\right)}$. ${\stackrel{\mathrm{\u203e}}{k}}_{\mathrm{12}}$ is the average meshing stiffness of onestage gear pair. The displacement nominal scale b_{c} is given. The dimensionless displacement, dimensionless velocity and dimensionless acceleration are expressed as ${x}_{i}={\stackrel{\mathrm{\u203e}}{x}}_{i}{b}_{c}$, ${\dot{x}}_{i}={\dot{\stackrel{\mathrm{\u203e}}{x}}}_{i}{b}_{c}{\mathit{\omega}}_{n}$ and ${\ddot{x}}_{i}={\ddot{\stackrel{\mathrm{\u203e}}{x}}}_{i}{b}_{c}{\mathit{\omega}}_{n}^{\mathrm{2}}$, respectively. The backlash and support clearance are expressed as ${b}_{ij}={\stackrel{\mathrm{\u203e}}{b}}_{ij}{b}_{c}$ and ${b}_{di}={\stackrel{\mathrm{\u203e}}{b}}_{di}{b}_{c}$, respectively. Spline clearance is expressed as ${R}_{i}={\stackrel{\mathrm{\u203e}}{R}}_{i}{b}_{c}$. The dimensionless forms of excitation frequency are ${\mathrm{\Omega}}_{i}={\mathit{\omega}}_{i}/{\mathit{\omega}}_{n}$, ${\mathrm{\Omega}}_{j}={\mathit{\omega}}_{j}/{\mathit{\omega}}_{n}$, ${\mathrm{\Omega}}_{bi}={\mathit{\omega}}_{bi}/{\mathit{\omega}}_{n}$ and ${\mathrm{\Omega}}_{bj}={\mathit{\omega}}_{bj}/{\mathit{\omega}}_{n}$, respectively. The dimensionless damping variable is $\stackrel{\mathrm{\u203e}}{C}=c/\mathrm{2}m{w}_{n}$, the dimensionless stiffness variable is $\stackrel{\mathrm{\u203e}}{K}=k/m{w}_{n}^{\mathrm{2}}$, and the dimensionless excitation force is $\stackrel{\mathrm{\u203e}}{F}=F/m{b}_{c}{w}_{n}^{\mathrm{2}}$.
The abovementioned differential equations are normalized in dimension and solved by the Runge–Kutta method. The DLSCs of Ω_{I} and Ω_{II} are obtained as follows:
Here, Ω_{I} is the onestage DLSC. Ω_{II} is the twostage DLSC. Ω_{ij}(ij=12, 13, 610, …, 910) is the onestage and twostage DLSC of each branch. ${\stackrel{\mathrm{\u203e}}{P}}_{\mathrm{I}ij}$ and ${\stackrel{\mathrm{\u203e}}{P}}_{\mathrm{II}ij}$ are the average dynamic meshing force of each gear pair at onestage and twostage DLSC. I_{12} and I_{46} are the transmission ratio of gear pairs 12 and 46. The DLSC represents the load distribution of the system under the dynamic vibration response. The smaller the DLSC, the better the loadsharing performance of the system.
The loaded tooth contact analysis (LTCA) model of onestage face gear pair (Li and Zhu, 2010) and twostage gear pair (Wang et al., 2010) is shown in Fig. 7. j_{k} is a point along the relative principal direction, k= 1, 2 for onestage face gear, k= 1, 2, 3, 4 for twostage gear pair. Here, the tooth surface initial clearance w is $[w{]}_{k}=[\mathit{\delta}{]}_{k}+[b{]}_{k}$. $[w{]}_{k}=[{w}_{\mathrm{1}k},{w}_{\mathrm{2}k}$, …, w_{ik}, …, w_{jk},w_{nk}]^{T}, $[b{]}_{k}=[{b}_{\mathrm{1}},{b}_{\mathrm{2}}$, …, ${b}_{i},\mathrm{\dots},{b}_{n}{]}^{T}$, n is the number of discrete points. δ is the geometric transmission error, $[\mathit{\delta}{]}_{k}=[\mathrm{1},\mathrm{1},\mathrm{\dots},\mathrm{1},\mathrm{\dots},\mathrm{1}{]}^{T}$. b_{j} is tooth surface normal clearance ($j=\mathrm{1},\mathrm{\dots},n$). Z is the displacement direction.
Under the action of load P, the deformation coordination condition of the elastic deformation is as follows:
(F)_{k} is the normal flexibility coefficient matrix of the gear pair. p_{j}($j=\mathrm{1},\mathrm{2},\mathrm{\dots},nk$) is the contact loaded at the point j of the tooth k. Apparently, the contact force p_{j}($j=\mathrm{1},\mathrm{2},\mathrm{\dots},nk$) satisfies the following Eqs. (17)–(18):
If p_{jk}>0, [d]_{jk}=0. If [p]_{jk}=0, [d]_{jk}>0. f, P and w are known conditions. The contact forces [p]_{jk}, final backlash [d]_{jk} and tooth approach [Z] are unknown. The known parameters (f, P, w) and unknown parameters (p, d, Z) constitute a nonlinear program model. According to the tooth approach Z_{k}, the objective function is established as follows:
The objective function equation can be expressed as follows:
X_{j} (j=1, 2, …, 2n+1) is the artificial variables, (X) = (X_{1}, …, X_{2n})^{T}, (e) = 1.
The transmission error of the gear bearing in meshing is mainly composed of three parts of the geometric transmission error δ_{1}, the bending deformation δ_{2} and the contact deformation δ_{3}. The geometric transmission errors δ_{1} can be expressed as δ_{1}(T(k))=a. The bending deformation errors δ_{2} can be expressed as δ_{2}(T(k))=bT (k). The function relationship between the contact deformation δ_{3}(T(k)) and the loaded T(k) can be expressed as ${\mathit{\delta}}_{\mathrm{3}}\left(T\right(k\left)\right)=cT(k{)}^{\mathrm{2}/\mathrm{3}}$.
The relational expression of the integrated angular deformation Δφ_{ij}(T_{ij}(k)) and torque T_{ij}(k) of the gear pairs of the system can be expressed as follows:
Here, a, b and c are the constant terms in the formula. T_{ij}(k) is the torque of the kth meshing position.
By solving the LTCA equations, the load transmission error Δφ_{ij}[0.1T_{ij}[k]], Δφ_{ij}[0.5T_{ij}[k]] and Δφ_{ij}[0.9T_{ij}[k]] can be obtained. The coefficient a, b and c can be obtained by taking the obtained transmission error into Eq. (17).
The timevarying meshing stiffness k_{ij}(k) can be obtained as follows:
Input power P=2000 kW, input speed n=8780 r min^{−1}, installation error A_{i}=50 µm, eccentricity error E_{i}=50 µm, and onestage face gear backlash ${b}_{\mathrm{12}}={b}_{\mathrm{13}}=\mathrm{10}$ µm. Twostage gear pair backlash ${b}_{\mathrm{610}}={b}_{\mathrm{710}}={b}_{\mathrm{810}}={b}_{\mathrm{910}}=\mathrm{17}$ µm. Spline clearance S_{2}=50 µm. Support clearance b_{di}=10 µm. The gear parameters are shown in Table 1.
The parameters of equivalent support stiffness, torsional stiffness and meshing stiffness are shown in Table 2. The support stiffness is calculated according to GB/T 30731996.
The parameters of equivalent supporting damping, torsional damping, and meshing damping are shown in Table 3.
The calculated timevarying meshing stiffness curve is shown in Fig. 8. The meshing stiffness fluctuation range of onestage face gear pairs is (7.46 × 10^{5}–9.27 × 10^{5}) N mm^{−1}. The meshing stiffness fluctuation range of twostage idler gear pairs is (3.77 × 10^{6}–4.78 × 10^{6}) N mm^{−1}. The meshing stiffness fluctuation range of twostage big gear pair is (6.17 × 10^{6}–6.68 × 10^{6}) N mm^{−1}. Due to the different meshing stiffness of each meshing position of the tooth surface, the DLSC of different meshing positions will be changed slightly.
6.1 Effect of installation and eccentricity errors on the DLSC
Under the combined action of installation error and eccentricity error, the DLSC presents periodic changes with time, as shown in Fig. 9. The onestage DLSC is Ω_{12}=1.0383 and Ω_{13}=0.9617. Torque T_{12} and T_{13} are fluctuated around in 1129.33 N m and 1046.0 N m. Torque distribution is 51.92 % and 48.09 %. The twostage DLSC is Ω_{610}=0.914, Ω_{710}=1.086, Ω_{810}=0.948 and Ω_{910}=1.052. Torque T_{610}, T_{710}, T_{810} and T_{910} are fluctuated around in 6920.5, 8222.9, 7178.1 and 7965.4 N m, respectively. The influence of onestage gear pair 12 and twostage gear pair 710 on the loadsharing characteristics is maximum impact. The overall DLSC is Ω_{Z}=1.086. The DLSCs of the system are Ω_{I}=1.0383 and Ω_{II}=1.0860, respectively. The DLSC changed periodically with time.
Other parameters remain unchanged, when given ${b}_{\mathrm{12}}={b}_{\mathrm{13}}=\mathrm{0}$ µm. The DLSC under the combined action of errors is shown in Fig.10. The onestage DLSCs of Ω_{I} change from 1.0383 to 1.0921. The twostage DLSCs of Ω_{II} change from 1.0860 to 1.1220. The onestage face gear backlash has a great influence on the onestage DLSC and a little influence on the twostage DLSC.
Other parameters remain unchanged, when given onestage backlash ${b}_{\mathrm{12}}={b}_{\mathrm{13}}=\mathrm{0}$ µm, twostage backlash ${b}_{\mathrm{610}}={b}_{\mathrm{710}}={b}_{\mathrm{810}}={b}_{\mathrm{910}}=\mathrm{0}$ µm. The DLSC under the combined action of errors is shown in Fig. 11. The onestage DLSCs of Ω_{I} change from 1.0383 to 1.131. The twostage DLSCs of Ω_{II} change from 1.0860 to 1.3507. Under the combined action of the errors, the loadsharing performance becomes worse with the decrease of the backlash.
The DLSC changes with the errors as shown in Fig. 12. Figure 12a and b show the influence of installation errors on the DLSC, and Fig. 12c and d show the influence of eccentricity errors on the DLSC. When the given error range is from 0–50 µm, the influence range of installation error on onestage DLSC is 1.0003–1.0049, the influence range of installation error on twostage DLSC is 1.0001–1.1605, the influence range of manufacturing error on onestage DLSC is 1.0005–1.0125, and the influence range of manufacturing error on twostage DLSC is 1.0005–1.1160. The installation errors and eccentricity errors have less influence on the onestage DLSC and more influence on the twostage DLSC. The eccentricity errors are more sensitive than the installation errors on the onestage DLSC.
6.2 Effect of backlash, support clearance and spline clearance on the DLSC
The change of the DLSC Ω_{I} and Ω_{II} with the backlash is shown in Fig. 13. The onestage DLSC Ω_{I} changed strongly with the increase of onestage backlash, which changed from 1.121–1.016. The influence of onestage backlash on the twostage DLSC Ω_{II} changes little, which changed from 1.088–1.075. The influence of twostage backlash on twostage DLSC is more sensitive. With the increase of backlash, the DLSC gradually decreases from 1.268–1.021. The influence of twostage backlash on onestage DLSC is less, which changed from 1.045–1.035. In a certain range, the onestage DLSC and twostage DLSC decrease with the increase of the backlash. The backlash can compensate the deformation caused by the errors and improve the loadsharing performance to a certain extent.
When other parameters remain unchanged, the changes with the onestage and twostage DLSC support clearance and are shown in Fig. 14. With the increase of onestage support clearance, the DLSC decreases gradually. Ω_{I} changed from 1.0402–1.0341 and Ω_{II} changed from 1.0870–1.0842. With the increase of twostage support clearance, the DLSC Ω_{I} decreased from 1.0408–1.0297, and Ω_{II} decreased from 1.0879–1.0801. The influence of the support clearance on the DLSC is small.
Figure 15 shows the curves of onestage and twostage DLSC with the spline clearance from 0–450 µm. With the change of spline clearance, onestage DLSC does not change at 1.0383, while twostage DLSC changes from 1.0921–1.012. The influence of spline clearance on the onestage DLSC is almost unchanged and has a great impact on the twostage DLSC. When the spline clearance is 0 mm, it is equivalent to the bending stiffness of the spline shaft to provide support. The DLSC is Ω_{610}=0.9079, Ω_{710}=1.0921, Ω_{810}=0.9432 and Ω_{910}=1.0568. When the spline clearance is greater than the total displacement R_{5} (0.271 mm), the twostage pinion O_{5} is in a fully floating state. When the spline clearance is greater than the total displacement R_{4} (0.382 mm), the twostage pinion O_{4} and O_{5} enter a full floating state at the same time, and the loadsharing characteristic is improved, and the DLSC is stable at Ω_{II}=1.012. Among them, the DLSCs of twostage gear pairs are Ω_{610}=0.9861, Ω_{710}=1.0139, Ω_{810}=0.988 and Ω_{910}=1.012. Twostage pinion O_{4} and pinion O_{4} are independent in floating behavior.
6.3 Effect of torsional stiffness and support stiffness on the DLSC
The influence curve of DLSC of input shaft and output shaft is shown in Fig. 16 when the torsional stiffness changed from (0.5–10.0) × 10^{7} N mm rad^{−1}. Input shaft and output shaft have almost no influence on the dynamic loadsharing coefficient of the system. This conclusion is consistent with the conclusion of the influence of torsional stiffness on the loadsharing coefficient of power split transmission system analyzed in Jin et al. (2019a, b).
Other parameters remain unchanged, and the DLSC changes with the torsional stiffness of connecting shaft 24 and connecting shaft 35 at (0.5–10.0) × 10^{6} N mm rad^{−1}, as shown in Fig. 17. With the increase of the torsional stiffness of connecting shaft 24 and connecting shaft 35, the DLSC gradually becomes larger, the average load characteristics become worse, and the influence trend is the same. The influence of torsional stiffness on the twostage DLSC is more obvious than that of the onestage DLSC. Although the geometry of the system is symmetrical, but because of the different direction of meshing force and the geometry of the connecting axis, the force situation is not consistent, resulting in a certain difference in the dynamic loadsharing characteristics.
With other parameters unchanged, the overall support stiffness of each gear shown in Table 2 is (2–0.05) times that of the change, and the influence curve of the overall support stiffness on the DLSC of the system is shown in Fig. 18.
With the decrease of the support stiffness multiple, the DLSC becomes larger, and the dynamic performance becomes worse. Especially when the overall support stiffness multiple is (0.5–0.05), the influence of the dynamic loadsharing coefficient of twostage DLSC is stronger, and the change of Ω_{II} is from 1.0763–1.2714, but it has little influence on the dynamic loadsharing coefficient of onestage DLSC, and the change of Ω_{I} is from 1.0381–1.0397.
In this paper, a new configuration of face gear dualpower split transmission system is proposed, and the effects of backlash, support clearance, error, torsional stiffness and support stiffness on dynamic loadsharing characteristics are analyzed, which provides the theoretical basis for further optimization design of vibration stability. The conclusions are as follows:

With the increase of backlash, the DLSCs gradually decrease and tend to be stable. The twostage DLSC has a great influence with the change of backlash. Increasing the backlash of each twostage gear pair properly is conducive to improving the loadsharing characteristics. With the increase of the support clearance, the DLSC decreases, but the change value is small. The increase of the support clearance has little effect on the improvement of the dynamic loadsharing characteristics. The increase of spline clearance is beneficial to the improvement of even load characteristics.

With the increase of installation error and eccentricity error, the DLSC is larger and the system loadsharing performance is worse. The installation error and eccentricity error of each gear in level 2 have the greatest influence on the dynamic loadsharing coefficient. The installation and manufacture of each gear in level 2 should be considered in the design.

With the increase of torsional stiffness, the DLSC increases, and the loadsharing characteristics become worse. The change trend of the dynamic loadsharing coefficient is relatively consistent between the connecting shaft 24 and the connecting shaft 35. In the optimal design of the loadsharing structure, the flexible connecting shaft meeting the strength should be selected to improve the loadsharing performance. With the decrease of support stiffness, the change twostage DLSC is more intense. Choosing a reasonable rigid support in the optimal design is conducive to ensuring a better loadsharing performance.
All data, models, and code generated or used during the study appear in the article.
The raw/processed data required to reproduce these findings cannot be shared at this time as the data also form part of an ongoing study.
All authors contributed to the study conception and design. Material preparation, data collection and analysis were performed by HD, HQZ, XLZ and LLD. The first draft of the manuscript was written by HD, and all authors commented on previous versions of the manuscript. All authors read and approved the final manuscript.
The authors declare that they have no conflict of interest.
This research has been supported by the National Natural Science Foundation of China (grant no. 51705390), the Innovation Capability Support Program of Shaanxi (grant no. 2020KJXX016), the Scientific Research Program Funded by Shaanxi Provincial Education Department Program (grant no. 20JC015), the Principal Foundation Project of Xi'an Technological University (grant no. xgpy200201), and the Natural Science Foundation of Shaanxi Province (grant no. 2021JM428).
This paper was edited by Francisco Romero and reviewed by two anonymous referees.
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 Abstract
 Introduction
 Transmission system configuration and dynamic model establishment
 Establishment of differential equations of system dynamics
 DLSC calculation
 Timevarying meshing stiffness calculation based on LTCA
 Example analysis
 Conclusions
 Code and data availability
 Data availability
 Author contributions
 Competing interests
 Financial support
 Review statement
 References
 Abstract
 Introduction
 Transmission system configuration and dynamic model establishment
 Establishment of differential equations of system dynamics
 DLSC calculation
 Timevarying meshing stiffness calculation based on LTCA
 Example analysis
 Conclusions
 Code and data availability
 Data availability
 Author contributions
 Competing interests
 Financial support
 Review statement
 References