1 Introduction

Power split gear transmission system is suitable for high speed and overloaded occasions, and is widely used in power transmission of helicopters, ships, etc^[1]. But the branches of power split transmission system form a closed structure, contact each other through meshing with the input and output gear simultaneously, and inevitably produce dynamic characteristics in the operation which are different from the tandem transmission, so the dynamics study of power split gear transmission system is currently a hot issue^[2-16]. In order to obtain the power split gear transmission system with small vibration and low noise, many scholars devoted to the tooth profile modification effects on the gear dynamic characteristics. In 1994, Lin and Oswald^[17] considered the tooth profile modification as a time-varying gear backlash, established a single stage spur gear linear dynamic torsion model with tooth profile modification, and studied the tooth profile modification effect on spur gear dynamics characteristics. In 1999, Kahraman and Blankenship^[18] studied the spur gear nonlinear dynamics characteristics with addendum modification. In 2003, Wang^[19] established a single stage spur gear linear dynamic model with tooth profile modification and errors, the model considered the supporting function of bearing, and studied the tooth profile modification effect on dynamic transmission errors. In 2008, Liu and Parker^[20] established the nonlinear dynamic model of a two-stage spur gear transmission, and studied the effect of tooth profile modification to nonlinear dynamic system. In the same year, Bonori and Barbieri^[21] established a single stage spur gear nonlinear dynamic torsion model with tooth profile modification, and based on the model, the genetic algorithm was adopted to optimize the shape parameters. In 2011, Faggioni and Samani^[22] established a single stage spur gear nonlinear dynamic model with tooth profile modification, proposed a higher efficiency of random-simplex optimization algorithm, the algorithm was used to optimize the modification parameters in order to minimize the peak of dynamic transmission error. From the foregoing, the minimized dynamic response amplitude of modification parameters can be gotten effectively through dynamic model, and the studies provide powerful theoretical guidance to the gear transmission system damping noise reduction.

A bend-torsion-shaft coupling nonlinear dynamic model of the power split transmission system is established in this dissertation, the tooth profile modification is expressed as a time-varying gear backlash function varying according to the meshing process along the meshing line direction. The effects of gear modification parameters to the system dynamics characteristics are also analyzed through numerical method, and the dissertation provides a theoretical method and reference for the selection of the optimized modification parameters.

2 Dynamics Model and Equations of Power Split Transmission System

The power split gear transmission system is shown in Fig. 1,and the dynamic model is shown in Fig. 2.

Gear 1 meshes Gear 2 and 4 at the same time, which forms the power split. Gear 3 and 5 mesh Gear 6 at the same time, which forms the power interflow. The meaning of the parameters in Fig. 2: m_i(i=1, 2, …, 6) is the mass of each node. I_i(i=1, 2, …, 6) is the rotational inertia of each node. k_xi, k_yi, k_zi, c_xi, c_yi, c_zi(i=1, 2, …, 6) are the support stiffness and damp along the coordinate axises x, y, z. k_{ti, j}, k_{bi, j}, k_{tei, j}, c_{ti, j}, c_{bi, j}, c_{tei, j}(i=2, 4;j=3, 5) is the torsional, bending, axial stiffness and damp of the shaft between gear i, j respectively. k_ms(t), c_ms(t)(s=1, 2, 3, 4) is the time-varying mesh stiffness and damp of gear pair s. b_ms(s=1, 2, 3, 4) is the gear backlash along the contact line of gear pair s. e_ts(t)(s=1, 2, 3, 4) is the tooth profile error along the contact line of gear pair s. eⁱ_rs(t)(i=1, 2, …, 6, s=1, 2, 3, 4) is the projection of the radial run out error along the contact line of Gear i of gear pair s. e_ts(t)(s=1, 2, 3, 4) and eⁱ_rs(t)(i=1, 2, …, 6, s=1, 2, 3, 4) can be acquired by a method presented by Kubur and Kahraman^[23]. The input torque acts on Gear 1 with the same direction of Gear 1, and the output torque acts on Gear 6 with the opposite direction of Gear 6.

Fig. 2 shows the relationship of meshing force and displacement of spur gear pair s(s=1, 2, 3, 4). Gear i(i=1, 3, 5) is the driving gear, and Gear j(j=2, 4, 6) is the driven gear. Fig. 3(a) shows a local-coordinate x_sy_sz_s which the axis z overlaps the axle of driving gear. λ_s is the gear relative displacement along the meshing line of gear pair. Because of the gear backlash (b_ms), the meshing force (F_ms) should be written as a piecewise function:

${{F}_{ms}}=\left\{ \begin{align} & {{k}_{ms}}(t)({{\lambda }_{s}}-{{b}_{ms}}),{{\lambda }_{s}}>{{b}_{ms}} \\ & 0,-{{b}_{ms}}\le {{\lambda }_{s}}\le {{b}_{ms}},s=1,2,3,4 \\ & {{k}_{ms}}(t)({{\lambda }_{s}}+{{b}_{ms}}),{{\lambda }_{s}}-{{b}_{ms}} \\ \end{align} \right.$

(1)

The gear pair projection along the axis z_s is shown in Fig. 3(b), and the driving gear rotating direction is clockwise. F_mxs and F_mys are the component forces along the axises x_s, y_s. α_s is the gear pressure angle; ψ_s is the relative position angle between two gear, and Ψ_s is the angle between the meshing line and the axises x_s. x_i, y_i, z_i, x_j, y_j, z_j are the displacements of gear i, j along the axises x_s, y_s, z_s respectively, and θ_i, θ_j are the angular displacements around the axis z_s. The relative displacement along the meshing line is shown as follows:

$\eqalign{ & {\lambda _s} = ({x_i} - {x_j})\cos {\psi _s} + ({y_i} - {y_j})\sin {\psi _s} + {r_i}{\theta _i} - \cr & {r_j}{\theta _j} + e_{rs}^i(t) + e_{rs}^j(t) - {e_{ts}}(t) \cr} $

(2)

According to Newton′s second law, the dynamics equations of power split gear transmission system are deduced in the way of lumped parameter method^[24-26] as follows:

$\left\{ \begin{align} & {{m}_{1}}{{{\ddot{x}}}_{1}}+{{k}_{x1}}{{x}_{1}}+{{c}_{x1}}{{{\dot{x}}}_{1}}=-{{F}_{mx1}}-{{F}_{mx2}} \\ & {{m}_{1}}{{{\ddot{y}}}_{1}}+{{k}_{y1}}{{y}_{1}}+{{c}_{y1}}{{{\dot{y}}}_{1}}=-{{F}_{my1}}-{{F}_{my2}} \\ & {{m}_{1}}{{{\ddot{z}}}_{1}}+{{k}_{z1}}{{z}_{1}}+{{c}_{z1}}{{{\dot{z}}}_{1}}=0 \\ & {{I}_{1}}{{{\ddot{\theta }}}_{1}}=-{{F}_{m1}}{{r}_{1}}-{{F}_{m2}}{{r}_{1}}+{{T}_{1}} \\ & {{m}_{2}}{{{\ddot{x}}}_{2}}+{{k}_{x2}}{{x}_{2}}+{{c}_{x2}}{{{\dot{x}}}_{2}}+{{k}_{b2,3}}({{x}_{2}}-{{x}_{3}})+{{c}_{b2,3}}({{{\dot{x}}}_{2}}-{{{\dot{x}}}_{3}})={{F}_{mx1}} \\ & {{m}_{2}}{{{\ddot{y}}}_{2}}+{{k}_{y2}}{{y}_{2}}+{{c}_{y2}}{{{\dot{y}}}_{2}}+{{k}_{b2,3}}({{y}_{2}}-{{y}_{3}})+{{c}_{b2,3}}({{{\dot{y}}}_{2}}-{{{\dot{y}}}_{3}})={{F}_{my1}} \\ & {{m}_{2}}{{{\ddot{z}}}_{2}}+{{k}_{z2}}{{z}_{2}}+{{c}_{z2}}{{{\dot{z}}}_{2}}+{{k}_{te2,3}}({{z}_{2}}-{{z}_{3}})+{{c}_{te2,3}}({{{\dot{z}}}_{2}}-{{{\dot{z}}}_{3}})=0 \\ & {{I}_{2}}{{{\ddot{\theta }}}_{2}}+{{k}_{t2,3}}({{\theta }_{2}}-{{\theta }_{3}})+{{c}_{te2,3}}({{{\dot{\theta }}}_{2}}-{{{\dot{\theta }}}_{3}})={{F}_{m1}}{{r}_{2}} \\ & {{m}_{3}}{{{\ddot{x}}}_{3}}+{{k}_{x3}}{{x}_{3}}+{{c}_{x3}}{{{\dot{x}}}_{3}}+{{k}_{b2,3}}({{x}_{3}}-{{x}_{2}})+{{c}_{b2,3}}({{{\dot{x}}}_{3}}-{{{\dot{x}}}_{2}})={{F}_{mx3}} \\ & {{m}_{3}}{{{\ddot{y}}}_{3}}+{{k}_{y3}}{{y}_{3}}+{{c}_{y3}}{{{\dot{y}}}_{3}}+{{k}_{b2,3}}({{y}_{3}}-{{y}_{2}})+{{c}_{b2,3}}({{{\dot{y}}}_{3}}-{{{\dot{y}}}_{2}})={{F}_{my3}} \\ & {{m}_{3}}{{{\ddot{z}}}_{3}}+{{k}_{z3}}{{z}_{3}}+{{c}_{z3}}{{{\dot{z}}}_{3}}+{{k}_{te2,3}}({{z}_{3}}-{{z}_{2}})+{{c}_{te2,3}}({{{\dot{z}}}_{3}}-{{{\dot{z}}}_{2}})=0 \\ & {{I}_{3}}{{{\ddot{\theta }}}_{3}}+{{k}_{t2,3}}({{\theta }_{3}}-{{\theta }_{2}})+{{c}_{te2,3}}({{{\dot{\theta }}}_{3}}-{{{\dot{\theta }}}_{2}})={{F}_{m3}}{{r}_{3}} \\ & {{m}_{4}}{{{\ddot{x}}}_{4}}+{{k}_{x4}}{{x}_{4}}+{{c}_{x4}}{{{\dot{x}}}_{4}}+{{k}_{b4,5}}({{x}_{4}}-{{x}_{5}})+{{c}_{b4,5}}({{{\dot{x}}}_{4}}-{{{\dot{x}}}_{5}})={{F}_{mx2}} \\ & {{m}_{4}}{{{\ddot{y}}}_{4}}+{{k}_{y4}}{{y}_{4}}+{{c}_{y4}}{{{\dot{y}}}_{4}}+{{k}_{b4,5}}({{y}_{4}}-{{y}_{5}})+{{c}_{b4,5}}({{{\dot{y}}}_{4}}-{{{\dot{y}}}_{5}})={{F}_{my2}} \\ & {{m}_{4}}{{{\ddot{z}}}_{4}}+{{k}_{z4}}{{z}_{4}}+{{c}_{z4}}{{{\dot{z}}}_{4}}+{{k}_{te4,5}}({{z}_{4}}-{{z}_{5}})+{{c}_{te4,5}}({{{\dot{z}}}_{4}}-{{{\dot{z}}}_{5}})=0 \\ & {{I}_{4}}{{{\ddot{\theta }}}_{4}}+{{k}_{t4,5}}({{\theta }_{4}}-{{\theta }_{5}})+{{c}_{te4,5}}({{{\dot{\theta }}}_{4}}-{{{\dot{\theta }}}_{5}})={{F}_{m2}}{{r}_{4}} \\ & {{m}_{5}}{{{\ddot{x}}}_{5}}+{{k}_{x5}}{{x}_{5}}+{{c}_{x5}}{{{\dot{x}}}_{5}}+{{k}_{b4,5}}({{x}_{5}}-{{x}_{4}})+{{c}_{b4,5}}({{{\dot{x}}}_{5}}-{{{\dot{x}}}_{4}})={{F}_{mx4}} \\ & {{m}_{5}}{{{\ddot{y}}}_{5}}+{{k}_{y5}}{{y}_{5}}+{{c}_{y5}}{{{\dot{y}}}_{5}}+{{k}_{b4,5}}({{y}_{5}}-{{y}_{4}})+{{c}_{b4,5}}({{{\dot{y}}}_{5}}-{{{\dot{y}}}_{4}})={{F}_{my4}} \\ & {{m}_{5}}{{{\ddot{z}}}_{5}}+{{k}_{z5}}{{z}_{5}}+{{c}_{z5}}{{{\dot{z}}}_{5}}+{{k}_{te4,5}}({{z}_{5}}-{{z}_{4}})+{{c}_{te4,5}}({{{\dot{z}}}_{5}}-{{{\dot{z}}}_{4}})=0 \\ & {{I}_{5}}{{{\ddot{\theta }}}_{5}}+{{k}_{t4,5}}({{\theta }_{5}}-{{\theta }_{4}})+{{c}_{te4,5}}({{{\dot{\theta }}}_{5}}-{{{\dot{\theta }}}_{4}})={{F}_{m4}}{{r}_{5}} \\ & {{m}_{6}}{{{\ddot{x}}}_{6}}+{{k}_{x6}}{{x}_{6}}+{{c}_{x6}}{{{\dot{x}}}_{6}}=-{{F}_{mx3}}-{{F}_{mx4}} \\ & {{m}_{6}}{{{\ddot{y}}}_{6}}+{{k}_{y6}}{{y}_{6}}+{{c}_{y6}}{{{\dot{y}}}_{6}}=-{{F}_{my3}}-{{F}_{my4}} \\ & {{m}_{6}}{{{\ddot{z}}}_{6}}+{{k}_{z6}}{{z}_{6}}+{{c}_{z6}}{{{\dot{z}}}_{6}}=0 \\ & {{I}_{6}}{{{\ddot{\theta }}}_{6}}={{F}_{m3}}{{r}_{6}}+{{F}_{m4}}{{r}_{6}}-{{T}_{6}} \\ \end{align} \right.$

(3)

The nonlinear dynamics equation of power split transmission system is deduced from Eqs.(1),(2) and (3),and the matrix form is:

$M\ddot X + {K_{com}}X + {K_t}X + C\dot X = F + {F_n} + {F_e}$

(4)

Where M is the mass matrix; K_con is the constant part of stiffness matrix; K_t is the time varying part of stiffness matrix; C is the proportional damping matrix that is given as C=εK+ηM, here ε and η are proportionality constants, and the two proportionality constants could be acquired by a method presented by Clough and Penizen^[27]; X is the displacement vector; F is the external excitation vector; F_n is the nonlinear excitation vector caused by the gear backlash; F_e is the excitation vector caused by the tooth profile error.

3 Gear Backlash Equation with Tooth Profile Modification

Because the tooth profile modification is cutting a part from the original involute profile, which amounts to adding a backlash value in the process of tooth contact. However the backlash value is not constant, which is constantly changing in the meshing. And so the tooth profile modification is considered as a time-varying gear backlash along the meshing line. The gear backlash function with the tooth profile modification is:

${{g}_{s}}(t)=\left\{ \begin{align} & 0,0\le tt<\left( 2-{{c}_{s}} \right){{T}_{ms}}\text{or}\left( 2-{{c}_{s}}+{{\xi }_{ms}} \right){{T}_{ms}}<tt<{{T}_{ms}} \\ & -\frac{{{\rho }_{ms}}tt}{{{\xi }_{ms}}{{T}_{ms}}}+\frac{{{\rho }_{ms}}\left( 2-{{c}_{s}}+{{\xi }_{ms}} \right)}{{{\xi }_{ms}}},(2-{{c}_{s}}){{T}_{ms}}\le tt\le \left( 2-{{c}_{s}}+{{\xi }_{ms}} \right){{T}_{ms}} \\ \end{align} \right.$

(5)

Where tt={(t+p_sT_ms)/T_ms－floor[(t+p_sT_ms)/T_ms]}T_ms, here t is the time value, and T_ms is the meshing period, p_s is the meshing phase; c_s is the contact ratio; ρ_ms is the tooth profile modification quantity(TPMQ); ξ_ms is the proportion of modification area in the whole meshing period, which is the relative tooth profile modification length(TPML).

In consideration of the time-varying gear backlash caused by the profile modification, the nonlinear function of the gear backlash changes into the following form, and b_ms should replace b_ms in Eq.(1) during the calculation:

${{\bar{b}}_{ms}}=\left\{ \begin{align} & {{b}_{ms}}+{{g}_{s}}(t),{{\lambda }_{s}}>{{b}_{ms}}+{{g}_{s}}(t) \\ & {{\lambda }_{s}},{{b}_{ms}}\le {{\lambda }_{s}}\le {{b}_{ms}}+{{g}_{s}}(t) \\ & -{{b}_{ms}},{{\lambda }_{s}}-{{b}_{ms}} \\ \end{align} \right.$

(6)

4 Tooth Profile Modification Effect on Power Split Transmission System Dynamic Characteristics

The parameters of a power split gear transmission system are shown in Table 1.

After eliminating the rigid body displacement of the equations and nondimensionalization, the simultaneous Eqs.(4), (5), (6) become second order nonlinear differential equations. In this paper the equations are solved by a numerical method, Runge-Kutta method^[28], and the MATLAB ode45 function is used in the computational process. The input torque is T₁=600 N·m, and the input speed is 5 500 r/min. The TPMQ of each gear pair are assumed to be equal respectively, and the relative length of the modification(TPML) is ξ_ms=c_s－1. The system dynamic load factors when ρ_ms=0, ρ_ms=6 μm(s=1, 2, 3, 4) are shown in Fig. 4.

Fig. 4 shows that the maximum value of the first stage transmission dynamic load factor is 3.6 in the case of ρ_ms=0(s=1, 2, 3, 4), and a lot of zeros appear simultaneously, which shows that a phenomenon of the tooth surface complete separation arises in the first stage transmission. And the system is in a nonlinear motion state, and the maximum value of the second stage transmission dynamic load factor is 2.1. The dynamic load factors of all the gear pairs will reduce greatly in the case of ρ_ms=6 μm(s=1, 2, 3, 4), and the maximum value of the first stage transmission dynamic load factor reduce to 1.6. Zero and negative value no longer appears, and the maximum value of the second stage transmission dynamic load factor reduced to 1.5. As described in Refs.^[29-35], the tooth profile modification can reduce the gear meshing stiffness excitation and stabilize the gear meshing force. And so by means of tooth profile modification, the system vibration can be effectively alleviated and the nonlinear effects such as tooth surface separation and tooth back impact can even be inhibited.

The operating conditions and parameters are mentioned above. Then in the case of different TPMQs of the first and the second stages, the maximum dynamic load factor contour of the power split transmission system is shown in Fig. 5.

As shown in Figs. 5(a) and 5(b), the peak contours of the first stage transmission dynamic load factor K_v1 and K_v2 are almost parallel to the longitudinal axis, which indicates that the effect of the first stage transmission TPMQs to K_v1 and K_v2 is greater than to the second. In the case of ρ_m1=ρ_m2＜9 μm, the peaks of K_v1 and K_v2 decrease with the increase of ρ_m1=ρ_m2, and the contours are skewed slightly to the left which shows that the peaks of K_v1 and K_v2 decrease slowly with the increase of ρ_m3=ρ_m4. In the case of ρ_m1=ρ_m2>9 μm, the peaks of K_v1 and K_v2 increase with the increase of ρ_m1=ρ_m2, and the contours are skewed slightly to the right which shows that the peaks of K_v1 and K_v2 increase slowly with the increase of ρ_m3=ρ_m4. And so, if the second transmission TPMQs are ignored, the first transmission TPMQs minimized K_v1 and K_v2 is ρ_m1=ρ_m2=9 μm. As shown in Figs. 5(c) and 5(d), most of the peak contours of the second stage transmission dynamic load factor K_v3 and all of K_v4 are almost parallel to the horizontal axis, which indicates that the effect of the second stage transmission TPMQs to K_v3 and K_v4 is greater than to the first. The TPMQ that minimizes the peak of K_v3 locates in the area surrounded by the contour with value of 1.32, and the area is very narrow along the horizontal axis, which shows that the effect of the first stage transmission modification to K_v3 and K_v4 is lesser. And so, if the first transmission TPMQs are ignored, the second transmission TPMQs minimized K_v3 and K_v4 is ρ_m3=ρ_m4=9 μm. The tooth profile modification can ameliorate the gear vibration^[30-33], but overlarge TPMQ will lessen the contact ratio, even enlarge the transmission error^[30], which make against the stably operation of gear transmission on the contrary. ρ_m1=ρ_m2=ρ_m3=ρ_m4=9 μm are just the optimum TPMQs and in the range of the empirical value mentioned in Refs. ^[30-32].

In conclusion, the transmission TPMQs give a great effect on its own dynamic load factor, and give a small effect on the other transmission. So in the process of determining the TPMQs of the power split transmission system, if only for a rough estimate, the two stages of transmission can be analyzed separately, and the coupling effect between the two stages of transmission can be ignored. In fact, in the analysis of tooth profile modification of multistage gear transmission, the coupling effect of every stage is not taken into account usually, and the tooth profile modification parameters of every stage are calculated respectively^[34-35].

The TPMQs are assumed to be 8 μm, and the two stages of TPMLs are ξ_m1=ξ_m2＜c_1/2, ξ_m3=ξ_m4＜c_3/4 respectively. The operating conditions and other parameters are mentioned above. In the case of different TPMLs of the first and the second stage, the maximum dynamic load factor contour of the power split transmission system is shown in Fig. 6.

As shown in Figs. 6(a) and 6(b), in the case of ξ_m1=ξ_m2＜0.3, the peak contours of the first stage transmission dynamic load factor K_v1 and K_v2 are almost parallel to the longitudinal axis, which indicates that the effect of the first stage TPMLs to K_v1 and K_v2 far greater than to the second, and the peaks of K_v1 and K_v2 decrease with the increase of the TPMLs ξ_m1=ξ_m2. In the case of ξ_m1=ξ_m2＞0.3, both the first stage TPMLs ξ_m1=ξ_m2 and the second ξ_m3=ξ_m4 affect K_v1 and K_v2 greatly, and the relationship is relatively complicated. As shown in Figs. 6(c) and 6(d), in the case of ξ_m3=ξ_m4＜0.2, the peak contours of the second stage transmission dynamic load factor K_v3 and K_v4 are almost parallel to the horizontal axis, which indicates that the effect of the second stage TPMLs to K_v3 and K_v4 far greater than the first, and the peaks of K_v3 and K_v4 decrease with the increase of the TPMLs ξ_m3=ξ_m4. In the case of ξ_m3=ξ_m4＞0.2, both the first stage TPMLs ξ_m1=ξ_m2 and the second ξ_m3=ξ_m4 affect K_v3 and K_v4 greatly, and the relationship is also complicated. According to the principle of the tooth profile modification, there are two kinds of TPMLs: the short modification and the long modification^[29-35]. By comparison with the short modification, the long modification ameliorates the dynamic load factor more greatly. And that the contact ratio is almost invariant in the case of short modification, but can decrease in the case of long modification^[30-33]. The variation of the dynamic load factor and the contact ratio influence the running stability and the carrying capacity of the gear transmission, and then can magnify the coupling effect of the two stages of power split gear transmission system. And so, the relationship of the TPML and the dynamic load factor is summarized as above.

In addition, the variation of the dynamic load factor is smaller when the TPMLs vary than the TPMQs do. For example, the peaks of K_v1 and K_v2 reduce from 3.6 to 1.8, and the peaks of K_v3 and K_v4 reduce from 1.5 to 1.3 when the TPMQs vary. And the peaks of K_v1 and K_v2 reduce from 2.6 to 2.0, the peaks of K_v3 and K_v4 reduce from 1.4 to 1.35 when the TPMLs vary. That means the effect of the TPMQ to the dynamic load factor is greater than the TPML, so the TPMQ is the emphasis in the optimization of modification parameters. In accordance with the principle of the tooth profile modification, the TPMQ, the TPML and the tooth profile curve are the three elements of the tooth profile modification^[29-35]. As mentioned in Ref. ^[30], the TPMQ could optimize the single-tooth meshing force and meshing stiffness, and mitigate the violent fluctuation of the total meshing stiffness on the premise that the proportion of the single-tooth and double-teeth meshing area are kept invariant. On the other hand, the TPML would cause the variation of the proportion of the single-tooth and double-teeth meshing area, which have a certain negative effect on the dynamic meshing forces of the gear pair. Therefore the TPMQ has more positive effect on the dynamic load factor than the TPML. As a consequence, the TPMQ should be top-priority in the modification design.

5 Conclusions

1) The effect of the TPMQs of the first stage transmission to the vibration amplitude of the first stage transmission is larger than to the second stage. Similarly,the effect of the TPMQs of the second stage transmission to the vibration amplitude of the second stage is also larger than to the first stage. And the coupling effect between two stages of the power split transmission system can be ignored in the tooth profile modification design. When the TPMQs achieve optimal values,the vibration amplitude of the system would be the smallest.

2) In the case of the TPMLs of the first stage transmission ξ_m1=ξ_m2＜0.3, ξ_m1, ξ_m2 affect the vibration amplitude of the first stage more greatly than the TPMLs of the second stage transmission ξ_m3, ξ_m4. In the case of ξ_m1=ξ_m2＞0.3, ξ_m1, ξ_m2, ξ_m3, ξ_m4 all affect the vibration amplitude of the first stage transmission largely. In the case of ξ_m3=ξ_m4＜0.2, ξ_m3, ξ_m4 affect the vibration amplitude of the second stage more greatly than ξ_m1, ξ_m2. In the case of ξ_m3=ξ_m4＞0.2, ξ_m1, ξ_m2, ξ_m3, ξ_m4 all affect the vibration amplitude of the second stage transmission largely.

3) Reasonable tooth profile modification can conspicuously reduce the vibration amplitude of the power split transmission system, and even restrain the occurrence of the nonlinear characteristics caused by the gear backlash. The TPMQs affect the vibration amplitude of the power split gear transmission system more greatly than the TPMLs, and should be top-priority in the tooth profile modification design.