The continuous dynamic mesh process of a gear pair within one mesh cycle can be discretized into a series of engagement positions. The contact lines and contact points in one engagement position can be arranged on the sliced plane of action, as shown in Fig. a. The positional relationship of the contact points for a certain contact line before and after loading is shown in Fig. b, and the positional relationship of a single contact point pair before and after loading is shown in Fig. c. The deformation compatibility relationship in one engagement position can be written as 1λGF+uL+ε-ξI-d=0, where λG is the global flexibility matrix of the tooth surface, which can be determined using the potential energy and slice method ; uL is the nonlinear contact deformation vector, which can be calculated using the analytical contact mechanics formula ; ε is the initial clearance vector of contact point pairs; ξ is the transmission error of gear pairs; I is a vector of n×1 with all element values equal to 1; and d is the residual clearance vector of contact point pairs after loading. The load balance condition in one engagement position can be written as 2∑i=1nFi=HF=P, where F is the load vector of contact points, H is a vector of 1×n with all element values equal to 1, and P is the gear mesh force. The non-penetrating contact condition in one engagement position can be written as 3when Fi>0,di=0when Fi=0,di>0, where Fi is the load for contact point i, and di is the residual clearance of contact point i. By combining Eq. () with Eq. (), the LTCA equations can be obtained, and ξ and F can be solved by constructing an iterative solving algorithm. The mesh stiffness and composite mesh error are then derived from the LTCA results, with a detailed calculation procedure provided in the previous work .

The dynamic model of a gear pair with 2 nodes and 8 degrees of freedom is shown in Fig. . Op and Og are the rotational center of the driving and driven gears, rp and rg are the radii of base circles for driving and driven gears, βb is the helix angle of gear pairs, αt is the transverse mesh angle of gear pairs, ψ is the installation angle, φ is the angle between the radii of the base circle and the x axis, φ=α-ψ, Tp and Tg are the input torque and output torque, ki is the contact point mesh stiffness, and εi is the contact point mesh error introduced by tooth surface wear. The generalized coordinates of gear pair nodes can be defined as 4qm={xp,yp,zp,θzp,xg,yg,zg,θzg}T, where xp, yp, zp, xg, yg, and zg are the translational displacements of the driving and driven gears along the x, y, and z axes, and θp and θg are the rotational angles of the driving and driven gears. The relative displacement of gear pairs along the normal line of action (dynamic transmission error) can be written as 5δm=Vqm, where V is the projection vector of the displacements of the gear pair nodes along the normal line of action and can be written as 6V={cos⁡βbsin⁡φ,cos⁡βbcos⁡φ,-sin⁡βb,rpcos⁡βb-cos⁡βbsin⁡φ,-cos⁡βbcos⁡φ,sin⁡βb,rgcos⁡βb}. Considering the time-varying mesh stiffness and composite mesh error, according to Newton's second law, the motion differential equations for a gear pair considering tooth surface wear can be written as 7mpx¨p+{cmδ˙m+km(δm-em)}cos⁡βbsin⁡φ+cpxx˙p+kpxxp=0mpy¨p+{cmδ˙m+km(δm-em)}cos⁡βbcos⁡φ+cpyy˙p+kpyyp=0mpz¨p-{cmδ˙m+km(δm-em)}sin⁡βb+cpzz˙p+kpzzp=0Izpθ¨zp+{cmδ˙m+km(δm-em)}rpcos⁡βb=Tpmgx¨g-{cmδ˙m+km(δm-em)}cos⁡βbsin⁡φ+cgxx˙g+kgxxg=0mgy¨g-{cmδ˙m+km(δm-em)}cos⁡βbcos⁡φ+cgyy˙g+kgyyg=0mgz¨g+{cmδ˙m+km(δm-em)}sin⁡βb+cgzz˙g+kgzzg=0Izgθ¨zg+{cmδ˙m+km(δm-em)}rgcos⁡βb=Tg, where mi (i=p,g) is the mass of the pinion and wheel; Izi (i=p,g) is the moment of inertia around the z axis for the pinion and wheel; kix,kiy,kiz (i=p,g) are the supporting stiffnesses for the pinion and wheel; and cix,ciy,ciz (i=p,g) are the supporting damping for the pinion and wheel. By using the Newmark numerical integration method, the vibration displacement of the gear pair can be obtained, and the dynamic mesh force of the gear pair can then be solved.

Figure  shows the dynamic engagement process of a gear pair from end face view. N1N2 and B1B2 are the theoretical and actual meshing lines, ωp and ωg are the angular velocities of the driving and driven gears, αkp and αkg are the pressure angles for the two gears at the meshing point K, rkp and rkg are the distances from the meshing point K to the rotation centers of the two gears, rbp and rbg are the base circles of the two gears, vkp and vkg are the linear velocities of the two gears at meshing point K, and vp and vg are the tangential velocities of the two gears at meshing point K. According to the concept of “single point observation”, the relative sliding distance between tooth surfaces can be calculated using 9sp=2aλpsg=2aλg, where sp and sg are the relative sliding distances at the contact points of the driving and driven gears, respectively; a is the half width of the Hertz contact zone; and λp and λg are the relative sliding coefficients of the driving and driven gears, respectively, which can be calculated using 10λp=|Lp-Lg|Lp=|vp-vg|dtvpdt=1-vgvpλg=|Lp-Lg|Lg=|vp-vg|dtvgdt=1-vpvg, where Lp and Lg are the tangential displacement of the driving and driven gears at the contact point, and vp and vg are the tangential velocities of the driving and driven gears at the contact point, calculated using 11vp=vKpsin⁡αKp=ωprKpsin⁡αKp=ωpRpvg=vKgsin⁡αKg=ωgrKgsin⁡αKg=ωgRg, where ωp and ωg are the rotational angular velocities of the driving and driven gears. Rp and Rg are the curvature radii of the driving and driven gears, respectively, which can be determined using 12Rp=rp2+rbp2-epRg=rg2+rbg2-eg, where ri is the distance from the contact point to the rotational center of gears, rbi is the base circle radius of gears, and ei is the wear depth at the mesh point.

The wear coefficient k in the Archard wear model is an empirical parameter that depends on material properties and operating conditions. In practice, tooth surface errors, vibration displacements, and time-varying meshing stiffness lead to uneven load distribution, resulting in different lubrication regimes (boundary, mixed, or EHL) at different contact points. Therefore, a dynamic wear coefficient is required for accurate wear prediction under dynamic conditions. summarized the wear coefficients of piston ring contact surfaces under different lubrication conditions, and introduced them into the wear calculation process of gear pairs under dynamic working conditions. The dynamic wear coefficient can be calculated using 13k=k0,λ<1227k0(4-λ),12<λ<40,λ≥4, where λ is the film thickness ratio coefficient, and k0 is the wear coefficient under boundary lubrication. The film thickness ratio coefficient λ can be calculated using 14λ=hmin⁡Rqi2+Rqi2,i=(p,g), where Rqi is the root mean square roughness, and hmin⁡ is the minimum oil film thickness between tooth profiles, which can be determined using 15hmin⁡=2.65α00.54(ν0u)0.7Ee-0.03Re0.43W-0.13, where α0 is the pressure viscosity coefficient of the lubricant, v0 is the dynamic viscosity of the lubricant, Re is the equivalent curvature radius of the friction surface at the current contact point, Ee is the equivalent elastic modulus, W is the load per unit gear width, and u is the average tangential velocity of the gear pairs (where u=(vp+vg)/2). The wear coefficient k0 under boundary lubrication can be determined based on .

The vibration state of the gear system varies at different input speeds. The dynamic contact stress on tooth surface under different input speeds is shown in Fig. a–c. When the driving gear rotates 1 million revolutions, the wear depth of the tooth surface under different input speeds is given in Fig. d–f. It can be observed that there are significant differences in the contact stress distribution under different input speeds. Compared with the results obtained at the input speed of 1000 rmin-1, the dynamic contact stress fluctuations of the tooth surface are more pronounced at input speeds of 2000 and 4000 rmin-1, and the maximum contact stress gradually increases. This is because the main resonance speed of the system is around 4000 rmin-1, while the secondary resonance speed is around 2000 rmin-1, as shown in Fig. . At this time, the system has significant vibration displacement. The variation pattern of tooth surface wear distribution is similar to that of contact stress distribution, and there is a significant fluctuation in the tooth surface wear distribution when the input speed is 4000 rmin-1.

The dynamic contact stress distribution of the helical gear pair under different input torques is shown in Fig. a–c. When the driving gear rotates 1 million revolutions, the wear depth of the tooth surface is given in Fig. d–f. It can be observed that due to the small changes in dynamic mesh excitations of gear pairs under different input torques, the system vibration displacement changes are also weak. Therefore, there is no significant difference in the distribution of contact stress. However, the maximum contact stress on the tooth surface gradually increases significantly. The variation pattern of tooth surface wear distribution is similar to that of contact stress distribution. The difference in tooth surface wear distribution under different input torques is slight, but the maximum wear increases significantly.

Figures  and  show the dynamic evolution process of contact stress distribution and tooth surface wear of the unmodified helical gear pair when the input torque is 3200 Nm and the input speed is 1000 rmin-1. The contact stress distribution of the gear pair before wear is shown in Fig. a. Due to the higher contact ratio of helical gear pairs, there is no significant abrupt change in the contact stress distribution. The relative sliding between tooth profiles reaches its maximum at the tooth tip and root position, while there is almost no relative sliding near the pitch line. Therefore, gear wear is mainly concentrated in the tooth tip and root, as shown in Fig. a. In the early stage of gear wear, the contact stress at the tooth root of the driving gear is relatively high. As the wear cycle increases, the cumulative rate of wear depth at the tooth root of the driving gear is much larger than that at the tooth tip position. There is almost no wear near the pitch line of gears, so the contact stress distribution gradually presents a “U-shaped” distribution along the tooth profile direction. A clear contact stress concentration phenomenon can be observed. Specifically, the contact stress at the tooth tip and root is relatively small, while the contact stress near the pitch line increases significantly, as shown in Fig. d. Under the same working condition, the dynamic evolution process of contact stress distribution and tooth surface wear of the helical gear pair with tip relief is shown in Figs.  and . Both the driving and the driven gears are subjected to tip relief; the length of tip relief is 13 mm, and the maximum amount of tip relief is 20 µm. The tip relief curve is a quadratic parabola, and specific details about tip relief are shown in our previous work .

The tip relief introduces mesh errors on the tooth surface at the tooth tip and root positions of the driving and driven gears. Therefore, when the tooth surface is not yet worn, the contact stress between the tooth tip and root positions of the modified gear pair is significantly less than that of the unmodified gear pair. The wear depth of the driving gear from the tooth root to the pitch line first increases slightly and then decreases, and the trend of the wear amount from the tooth tip to the pitch line is also the same. As the wear cycle increases, the contact stress gradually concentrates towards the pitch line position, and the contact stress along the tooth profile direction shows an overall “U-shaped” distribution. Under the same wear cycle, compared with the unmodified gear pair, the maximum wear depth of the gear pair with tip relief is significantly reduced. Under the same working condition, the dynamic evolution process of the contact stress distribution and tooth surface wear of the helical gear pair with a lead crown is shown in Figs.  and . A lead crown of 20 µm is adopted for modifying the driving gear, and the curve of the lead crown is also a quadratic parabola. A detailed introduction to this modification method can be found in the published work of .

A lead crown introduces intentional errors along the gear width. Thus, the contact stress of the gear pair with lead crown shows a trend of first increasing and then decreasing along the gear width, and the maximum contact stress occurs near the middle of the tooth root of the driving gear. The wear amount along the gear width is also larger in the middle and smaller on both sides. Due to the fact that the load distribution is concentrated towards the middle of the gear width after the lead crown, the maximum wear amount of the gear pair with the lead crown is significantly larger than that of the unmodified gear pair under the same wear cycle. As the wear cycle increases, the wear amount at the tooth tip and root gradually increases, and the contact stress gradually decreases. Thus, the contact stress gradually exhibits an elliptical distribution, and the maximum contact stress is concentrated in the center of the tooth surface.

Under the same working condition, the dynamic evolution process of the contact stress distribution and tooth surface wear of the helical gear pair with bias modification is shown in Figs.  and . A bias modification of 35 µm is employed to modify the driving gear and is performed in the three-teeth engagement zone, and the modification curve along the normal direction of the contact line is also a quadratic parabola. A detailed introduction to this method can be found in our previous work .

Due to the meshing errors in mesh-in and mesh-out positions introduced by bias modification, the contact stress increases first and then decreases along the normal direction of the contact line after bias modification, indicating that the contact stress at the mesh-in and mesh-out positions is significantly less than in other areas. At the same time, the tooth surface wear at the mesh-in and mesh-out positions is also significantly smaller. The maximum contact stress is located on the left side of the tooth root of the driving gear before wear. At the tooth root of the driving gear, the contact stress gradually decreases along the gear width. The variation law of contact stress at the tooth tip along the gear width is the opposite. As the wear cycle increases, the maximum wear amount gradually increases, and a narrow area with significantly higher contact stress appears near the pitch line, while the wear amount in this area is relatively small.

The mesh stiffness curves are shown in Fig. a, the time-domain and frequency-domain results of the static transmission error of unmodified and modified helical gear pairs under different wear cycles are shown in Fig. b and c, and the variation law of the root mean squares of dynamic transmission errors with the input speed is shown in Fig. d, all at an input torque of 3200 Nm and input speed of 1000 rmin-1. It can be observed that the time-varying mesh stiffness curves of the unmodified and modified gear pairs remain basically unchanged before and after wear. This is because the load distributions of gear pairs with various tooth surface modifications under different wear cycles reach the full tooth surface, and the partial contact loss phenomenon that reduces the mesh stiffness value does not occur. For the unmodified helical gear pair, the fluctuation in static transmission error gradually increases as the wear cycle increases. Thus, under lower-input-speed conditions, the root mean square of dynamic transmission errors gradually increases. In addition, as the wear cycle increases, the amplitude of the first harmonic of the static transmission error first increases and then decreases, while the amplitudes of the second and third harmonics gradually increase. Therefore, the main resonance of the system first increases and then decreases, while the second and third harmonic resonances gradually increase.

For the helical gear pair with tip relief, the change in static transmission error fluctuation before and after wear is relatively small. Therefore, when the input speed is low, the root mean squares of the dynamic transmission error under different wear cycles do not change much. However, as the wear period increases, the first harmonic amplitude of the static transmission error of the gear pair gradually decreases, and the second and third harmonic amplitudes gradually increase. Therefore, as the wear period increases, the main resonance amplitude of the system gradually decreases, and the second and third harmonic resonance amplitudes increase significantly.

For the helical gear pair with lead crown, the fluctuation in the static transmission error of the gear pair gradually increases as the wear cycle increases. Therefore, when the input speed is low, the system vibration gradually increases with the increase in wear cycle. However, as the wear period increases, the amplitudes of the first, second, and third harmonics of the static transmission error gradually increase. Therefore, as the wear cycle increases, the amplitudes of the main resonance and second and third harmonic resonances of the system all increase significantly.

For the helical gear pair with bias modification, the fluctuation in static transmission error first decreases and then increases as the wear cycle increases. Therefore, in the lower-input-speed conditions, the root mean square values of the dynamic transmission error first decreases and then increases. However, as the wear cycle increases, the first harmonic amplitude of the static transmission error first decreases and then increases, and the second harmonic amplitude gradually increases. Therefore, the main resonance peak of the system near an input speed of 3800 rmin-1 first decreases and then increases, while the second harmonic resonance amplitude near an input speed of 1900 rmin-1 gradually increases.