Following the method presented by Xia et al. (2019), a misalignment model of the diaphragm coupling is established, with the left and right coupling halves used to define the coordinate system for parallel misalignment, as shown in Fig. 2. Coupling misalignment is defined as the relative offset between the left and right coupling halves. The right half of the coupling is used as the reference in the coordinate system, and it is assumed that the left half of the coupling undergoes parallel misalignment. Trigonometric functions are used to calculate the deformation at bolt holes A, B, and C in the left coupling half along each direction. These deformations are multiplied by the corresponding directional diaphragm stiffnesses to obtain the coupling force induced by parallel misalignment. The centers of the left- and right-hand coupling sections are located at points O and O^′, respectively, and the angle between the y axis and the OO^′ line is β. For the left coupling half, ϵx and ϵy represent the misalignments along the x and y axes, respectively. R denotes the distance from the rotor axis to the center of the bolt hole, and θ represents the angular position of bolt hole A about the coupling center O^′. The relationship between β and the parallel offsets ϵx and ϵy is tan⁡β=ϵx/ϵy, and the eccentricity is OO′=ϵx2+ϵy2.

According to the law of cosines, the deformations at bolt holes A, B, and C can be obtained as 1μ1=OA-O′A=(R2+ϵx2+ϵy2+2Rϵx2+ϵy2sin⁡(β+θ))-R,2μ2=OB-O′B=(R2+ϵx2+ϵy2+2Rϵx2+ϵy2cos⁡(π/6+β+θ))-R,3μ3=O′C-OC=R-(R2+ϵx2+ϵy2-2Rϵx2+ϵy2sin⁡(π/3+β+θ)). The deformation components along the x and y axes are 4μx=μ1cos⁡(θ)+μ2cos⁡(θ+2π/3)+μ3cos⁡(θ+4π/3),5μy=μ1sin⁡(θ)+μ2sin⁡(θ+2π/3)+μ3sin⁡(θ+4π/3). The x- and y-direction components of the coupling force induced by misalignment are then given by Eqs. (6) and (7), respectively, where kD is the diaphragm coupling stiffness, and μx and μy are the deformation components along the x and y axes. 6Fpx=kDμx7Fpy=kDμy

Parallel coupling misalignment is a planar mechanical problem, whereas angular coupling misalignment is a three-dimensional mechanical problem. Angular misalignment generates forces in the xy-plane, an axial force along the shaft axis, and bending moments about the x and y axes. As illustrated in the angular-misalignment schematic in Fig. 3, the left coupling half is fastened to the diaphragm by bolts passing through holes A, B, and C. γ is the offset angle. When one coupling half is angularly offset, the centers of the left and right coupling halves are separated by an axial distance ϵ_z, and the planes of the two halves intersect along a line l. The distance ΔR between this intersection line and the coupling center is calculated from ϵ_z and the offset angle γ using Eq. (8). 8ΔR=ϵztan⁡γ The deformations of the diaphragm at bolt holes A, B, and C are 9AA′=ΔR+Rsin⁡θ2-2cos⁡γ,10BB′=ΔR+Rsin⁡(θ+2π3)2-2cos⁡γ,11CC′=ΔR+Rsin⁡(θ+4π3)2-2cos⁡γ. The misalignment forces at each bolt hole are given by FA=ksAA′, FB=ksBB′, and FC=ksCC′, where ks is the diaphragm bending stiffness. The misalignment forces at each bolt hole are resolved along the axes of the three-dimensional coordinate system to obtain the components FAx, FBx, FCx, FAy, FBy, FCy FAz, FBz, and FCz. Summing the forces along each axis gives the resultant force induced by angular misalignment, as shown in Eq. (12). Since the torque T on the left coupling half acts along its axis, it must be resolved into components for the angularly offset right coupling half. The x- and y-direction moment components, Tx and Ty, are given in Eq. (13). 12Fx=FAx+FBx+FCxFy=FAy+FBy+FCyFz=FAz+FBz+FCz13Tx=Tsin⁡γcos⁡θTy=Tsin⁡γsin⁡θ

Within the framework of generalized finite-element theory, the system is represented by sliced mesh, bearing, shaft, and coupling elements, thereby forming the sliced static model illustrated in Fig. 4. At each meshing position, the mesh stiffness Km is obtained by iteratively solving the static equilibrium equations and the loaded tooth contact analysis model, with mesh misalignment calculated from the gear nodal displacements. The sliced coupling stiffness Kslice is simulated using Timoshenko beam stiffness. The method proposed by Gong et al. (2022b) is utilized to obtain the bearing element stiffness matrix Kb and the shaft and coupling element stiffness matrices Ks and Kc, respectively, followed by assembly of the element matrices. Section 3.1 shows that the coupling-misalignment forces and moments are equal in magnitude and opposite in direction for the left and right coupling halves. Therefore, the misalignment force/moment is applied to the respective nodes on either side of the coupling element in the static formulation. The equilibrium relation governing the sliced model is formulated in matrix form in Eq. (14), where P0 is the load column vector. The right-hand-side term Fc(ϵx,ϵy,ϵz,γ,θ) represents the misalignment-induced force/moment column vector of the coupling, which is related to the offset, offset angle, and rotation angle. The explicit expression for the misalignment force/moment column vector is shown in Eq. (15). For each paired nodal degree of freedom, the two coupling nodes carry oppositely directed forces and moments. 14Kmθ+Kslice+Kb+Ks+Kcqθ=P0+Fcϵx,ϵy,ϵz,γ,θ15FC(ϵx,ϵy,ϵz,γ,θ)=0,…,Fx,Fy,Fz,Tx,Ty,0,…,-Fx,-Fy,-Fz,-Tx,-Ty,0,…,0′ Each gear sub-slice is represented by a two-node element with 12 degrees of freedom. The generalized coordinates of the jth slice are defined as qmj=xpj,ypj,zpj,θxpj,θypj,θzpj,xgj,ygj,zgj,θxgj,θygj,θzgj. As shown in Fig. 5, O_p and O_g are the rotation centers of the driving and driven gears, βb is the base helix angle, γ=α∓θp denotes the coordinate system projection parameters, the symbol “∓” indicates the direction of rotation of the gear, α is the pressure angle, θp is the installation phase angle, and rp and rg are the base circle radii of the driving and driven gears. The driving and driven gears are connected through the slice mesh stiffness kmj, and j is the number of slices. The slices are connected through Timoshenko beam elements. The static equations of the jth gear pair slice are given in Eq. (16), where σmj and emj are the displacement and error along the line of action of the jth sliced mesh element. kp_i(i=x,y,z,θx,θy,θz) and kg_i(i=x,y,z,θx,θy,θz) denote the values of isotropic coupling stiffness between the slices. The matrix form of the static equilibrium equation of the sliced mesh element is shown in Eq. (17). The stiffness and displacement in the equations are related to the gear mesh position. When the relative displacement of a gear slice along the line of action is negative, the slice is out of contact. In the static equilibrium equation, the mesh stiffness of the corresponding slice should be set to zero. Based on the definition and calculation principles of mesh misalignment specified in clause 7.5.3.4 of ISO 6336-1:2019, the mesh misalignment formulation is further derived and extended according to the structural characteristics of the proposed model, as given in Eq. (18).

The deformation compatibility relationship before and after loading of the double-helical gear pair is shown in Fig. 6. The contact equations under load are established using the deformation compatibility relationship, as shown in Eq. (19), where δL_i and δR_i are the deformations at the ith contact point after loading, including macroscopic deformation and contact deformation. xL_i and xR_i are the rigid-body approaches, and dL_i and dR_i are the residual clearances after the contact point is loaded and deformed. ϵL_i and ϵR_i represent the initial gaps associated with the contact points, and the initial clearance is caused by the mesh misalignment of the double-helical gear pair due to the system deformation. 16kmj(σmj-emj)cos⁡βbsin⁡γ-kp_x(xpj-1-xpj)+kp_x(xpj-xpj+1)=0±kmj(σmj-emj)cos⁡βbcos⁡γ-kp_y(ypj-1-ypj)+kp_y(ypj-ypj+1)=0∓kmj(σmj-emj)sin⁡βb-kp_z(zpj-1-zpj)+kp_z(zpj-zpj+1)=0kmj(σmj-emj)rpsin⁡βbsin⁡γ-kp_θx(θxpj-1-θxpj)+kp_θx(θxpj-θxpj+1)=0±kmj(σmj-emj)rpsin⁡βbcos⁡γ-kp_θy(θypj-1-θypj)+kp_θy(θypj-θypj+1)=0±kmj(σmj-emj)rpcos⁡βb-kp_θz(θzpj-1-θzpj)+kp_θz(θzpj-θzpj+1)=0-kmj(σmj-emj)cos⁡βbsin⁡γ-kg_x(xgj-1-xgj)+kg_x(xgj-xgj+1)=0∓kmj(σmj-emj)cos⁡βbcos⁡γ-kg_y(ygj-1-ygj)+kg_y(ygj-ygj+1)=0±kmj(σmj-emj)sin⁡βb-kg_z(zgj-1-zgj)+kg_z(zgj-zgj+1)=0kmj(σmj-emj)rpsin⁡βbsin⁡γ-kg_θx(θxgj-1-θxgj)+kg_θx(θxgj-θxgj+1)=0±kmj(σmj-emj)rpsin⁡βbcos⁡γ-kg_θy(θygj-1-θygj)+kg_θy(θygj-θygj+1)=0±kmj(σmj-emj)rpcos⁡βb-kg_θz(θzgj-1-θzgj)+kg_θz(θzgj-θzgj+1)=017Kmj(t)+Ksliceqmj(t)=0Ifσmj≤0,Kmj=018ξmj=max⁡σmj-σmj19δL_iP+δL_iG+ϵL_i-xL_i-dL_i=0δR_iP+δR_iG+ϵR_i-xR_i-dR_i=0When pointiis in contact,dL_i=0,dR_i=0When pointiis not in contact,dL_i>0,dR_i>0 The total deformation is partitioned into macroscopic and contact components. Equation (20) gives the loaded contact relationship at each meshing position. The macroscopic deformation compliance matrix of the contact points along the meshing line is denoted by λ. ucontact denotes the contact deformation vector for points along the meshing line. xSTE represents the rigid-body approach vector, equivalent to the static transmission error. d is defined as the residual clearance vector at the contact points. ϵ refers to the initial clearance vector at the contact point. F represents the column vector of loads acting at the contact points, and P is the total load on the contact area. The slice mesh stiffness is obtained by summing the stiffnesses of the contact points within the corresponding gear slice. 20-λF-ucontact+xSTE+d=ϵ∑i=1nFi=PIfFi>0,di=0IfFi=0,di>0 In the loaded tooth contact analysis of a gear pair, the initial clearance at the contact point originates from the geometric errors of the gears and their assembly. Minor profile and lead deviations, installation tolerances of bearings and housings, and even bearing internal clearance can cause a discrepancy between the actual and theoretical meshing locations of the two gears, thereby forming an initial backlash when unloaded or lightly loaded. During operation, thermal expansion and shaft elastic deformation can further change the magnitude and distribution of this clearance. Because this study examines how coupling misalignment affects the contact and dynamic behavior of the gear system, the contact point clearance is attributed to the gear–mesh misalignment caused by shaft deformation induced by coupling misalignment. In Eq. (20), the contact point clearance is determined by the gear meshing misalignment through Eq. (21). Here, j denotes the number of contact points. 21ϵ=ξ1⋯ξj

To analyze the quasi-static contact characteristics of gear pairs considering coupling misalignment, an iterative procedure is established to solve the static equilibrium equation and LTCA model and to update the mesh misalignment, as shown in Fig. 8. The corresponding steps are outlined below:

Discretize the gear system. Obtain the misalignment force vector from the coupling-misalignment model. Use the LTCA model to calculate the initial mesh stiffness of the sliced mesh elements.

The model was experimentally verified with a test rig specifically designed to investigate the dynamic characteristics of a double-helical gearbox under various loading conditions. The setup, as detailed by Gong et al. (2022a), incorporates a purpose-built, isolation-mounted, power-circulating double-helical gear test rig, which allows for precise measurement of vibration acceleration and torque under controlled conditions. The experimental results, which focus on vibration response, torque effects, and resonance frequencies, closely match the model's predicted behavior. This agreement between the experimental and numerical results supports the capability of the adopted model to reproduce the overall dynamic behavior of the gear transmission system, providing a solid foundation for future gear system analysis and design. It should be noted that the previous experimental validation focused mainly on the overall dynamic response of the basic gear transmission model, while the coupling-misalignment-related effects introduced in the present study are investigated on the basis of this validated modeling framework, with corresponding experimental investigations to be conducted in subsequent work.

This section considers six coupling-misalignment cases. ϵx is 0.05, ϵy is 0.05, ϵz is 0.25 mm, and γ is 0.25° for case 1. The amount of misalignment for case 2 to case 6 is 2 to 6 times that of case 1, respectively. First, the effect of each misalignment parameter is analyzed using case 1 as the reference condition. While keeping all other parameters constant, one misalignment parameter at a time is increased by specified factors to investigate the independent effects of each parameter on the quasi-static contact characteristics of the gear pair. As shown in Fig. 7, the horizontal axis represents different coupling-misalignment parameters, and the vertical axis represents the magnification factor relative to the case-1 reference condition, where 1× denotes the reference condition, and 2× to 5× represent values that are 2 to 5 times the reference value, respectively. This single-factor control method effectively eliminates the coupled effects caused by simultaneous changes in different misalignment parameters, thereby allowing for a comparison of the independent effects of each parameter on meshing stiffness. The results show that the extent of the influence of different coupling-misalignment parameters on meshing stiffness varies significantly, with ϵx having the most significant effect, followed by ϵy, while γ has a relatively minor effect, and ϵz has no significant effect on meshing stiffness. As ϵx and ϵy increase, the meshing stiffness exhibits a generally pronounced upward trend, indicating a strong positive correlation between these parameters and meshing stiffness; in contrast, the effect of γ on meshing stiffness is weaker, and its variation does not follow a strictly monotonic pattern. The primary reason for this is that the meshing misalignment of a gear pair is more sensitive to torsional deformation of the shaft system, whereas deformation in the axial direction has a relatively limited effect on meshing misalignment. Furthermore, different misalignment parameters affect the form and extent of shaft system deformation in varying ways, which in turn leads to differing degrees of influence on meshing misalignment and meshing stiffness. It can thus be seen that the quasi-static contact characteristics of the gear pair are primarily sensitive to ϵx and ϵy, while the effects of γ and ϵz are relatively limited. It should be noted that this section primarily analyzes the impact of changes in coupling-misalignment parameters on the quasi-static contact characteristics of the gear pair, with a focus on the variation patterns of meshing stiffness. Since meshing stiffness is a critical internal excitation parameter in subsequent dynamic analyses, its variation will further influence the system's dynamic response. Therefore, the patterns of parameter sensitivity revealed by this single-factor quasi-static analysis can provide a basis for subsequently assessing the relative effects of different coupling-misalignment parameters on the system's dynamic response.

As shown in Fig. 9, the g2 gear pair mesh stiffness curve exhibits not only short-period fluctuations, that is, fluctuations at the mesh frequency, but also obvious long-period fluctuations. The fluctuation frequency is the same as the coupling-misalignment force frequency, which is twice the shaft frequency. The results indicate that the contact condition of the g2 gear pair is governed by the force/moment excitation induced by coupling misalignment. With increasing misalignment, the shaft undergoes greater bending deformation, which further intensifies the fluctuation of mesh stiffness over one rotational period. For the left helical gear, since complete tooth surface contact is established during meshing cycle intervals 42 to 86 and 142 to 186, the actual contact line length does not change, and so the mesh stiffness is the same. Outside these intervals, the reduction in mesh stiffness results from partial disengagement of the tooth surface, and the curve shape changes significantly. For the right helical gear, during mesh cycles 42 to 86 and 142 to 186, the actual contact line length changes due to partial disengagement of the tooth surface, which markedly alters both the level of mesh stiffness and its fluctuation characteristics.

Since the misalignment force/moment of the coupling changes periodically with variations in the rotational angle, the load-induced deformation of the system also varies over the period corresponding to twice the shaft frequency. Figure 10 shows the mesh misalignment of the g2 gear pair in the 13th and 66th mesh cycles. In the 13th mesh cycle, for the g2 gear pair, the left tooth exhibits markedly greater mesh misalignment than the right, and this left-side misalignment increases progressively with increasing coupling misalignment. Misalignment-induced forces and moments have an increasing influence on the bending deformation of the shaft carrying the g2 gear pair, causing the disengaged area on the left tooth surface to increase and the actual contact line length to decrease. Therefore, the mesh stiffness of the left tooth decreases significantly with the increase in misalignment in the 13th mesh cycle. Because the peak mesh misalignment on the right tooth is only 11.14 µm across the six prescribed misalignment levels and because the tooth surface remains in complete contact during the 13th meshing cycle, the right mesh stiffness remains almost unchanged as the misalignment level increases.

With increasing coupling misalignment, the force/moment excitation induced by misalignment exerts a progressively stronger effect on the bending deformation of the shaft carrying the g2 gear pair. In the 66th mesh cycle, as the coupling-misalignment level rises, the g2 left tooth exhibits a gradual reduction in mesh misalignment magnitude; in case 6, the peak misalignment is displaced toward the left-hand region of the tooth surface. The mesh misalignment of the right tooth surface gradually increases with the increase in coupling misalignment. Because the left tooth mesh misalignment is smaller in cases 2 to 6, the mesh stiffness of the left tooth of the g2 gear pair is the same in this mesh cycle; the curve shape is also the same, and the tooth surfaces are in complete contact. Figure 11 presents the tooth surface load pattern calculated for the g2 gear pair during the 66th mesh cycle under different coupling-misalignment amounts. In case 1, disengagement is observed only on the left tooth surface, and the mesh stiffness is significantly lower than that in the other misalignment cases. The shaft carrying the g2 gear pair bends under the combined action of the meshing force, and the force/moment excitation induced by misalignment. The bending deformation caused by its meshing force is symmetrically distributed along the axis. As the misalignment force/moment increases, the influence on shaft bending deformation gradually increases, and the differences in mesh misalignment and tooth surface load distribution between the left and right increase accordingly.

By comparing the tooth surface contact states shown in Fig. 11 with the face width mesh misalignment at the 66th meshing cycle in Fig. 10, it can be observed that the central relief groove of the herringbone gear separates the left- and right-hand helical tooth surfaces. Therefore, its meshing behavior can be approximately regarded to be the combined action of two single-helical gear pairs, allowing the mesh misalignment and contact states of the left and right tooth surfaces to be analyzed separately. It should be noted that the mesh misalignment of the gear pair is not governed by a single factor but results from the combined effects of the tooth meshing force under the actual loading condition and the additional forces and moments induced by coupling misalignment. Consequently, the contact states of the left and right tooth surfaces do not exhibit a simple positive correlation with the magnitude of coupling misalignment. In case 1, the coupling eccentricity is relatively small, and the additional forces and moments induced by the misalignment have only a limited influence on the shaft system attitude. Therefore, the mesh misalignment of the gear pair is mainly governed by the tooth meshing force under the actual loading condition. Although the mesh misalignment values of the left and right tooth surfaces are relatively close, the misalignment on the left tooth surface is slightly larger, resulting in local disengagement, whereas the right tooth surface maintains a relatively favorable contact state. This indicates that tooth disengagement in a herringbone gear cannot be determined solely by the magnitude of coupling eccentricity but depends primarily on the actual distribution of mesh misalignment between the left and right tooth surfaces under the combined effects of the tooth meshing force and coupling-misalignment excitation. Further examination of the results at the 66th meshing cycle in Fig. 10 shows that, as the coupling eccentricity increases, the additional forces and moments induced by the misalignment gradually increase, and their influence on the displacement and orientation of the shaft system becomes more pronounced. As a result, the original mesh misalignment distribution dominated by the tooth meshing force is progressively altered. During this process, the mesh misalignment of the left and right tooth surfaces exhibits opposite trends: the mesh misalignment on the left tooth surface gradually decreases with increasing misalignment excitation, whereas that on the right tooth surface gradually increases. Correspondingly, the tooth-surface contact states shown in Fig. 11 also change significantly, evolving from local disengagement on the left tooth surface with good contact on the right tooth surface in case 1 to good contact on the left tooth surface with disengagement on the right tooth surface in case 6. These results demonstrate that, for a herringbone gear transmission system, the magnitude of coupling misalignment itself does not directly determine the tooth surface contact state. Instead, coupling misalignment modifies the additional forces and moments acting on the shaft system, which interact with the tooth meshing force to redistribute the mesh misalignment between the left and right tooth surfaces, ultimately leading to corresponding changes in the tooth contact region and local load-carrying state. The above analysis shows that the system deformation and tooth surface contact state vary under coupling-misalignment forces and moments over the period corresponding to twice the shaft frequency. The mesh stiffness exhibits long-period fluctuations. With increasing misalignment magnitude, the mesh stiffness fluctuation of the gear pair, which is greatly affected by the misalignment force/moment, will also increase. Larger misalignment forces and moments lead to greater differences in mesh misalignment between the left and right tooth surfaces and a more asymmetric tooth surface load distribution. Under the combined effect of meshing force and misalignment force/moment, the mesh misalignment of the gear pair within certain mesh cycle ranges is small. When the tooth surface reaches a complete contact state, the actual contact line length is the same, and the mesh stiffness within this range is the same.

The load torque determines the meshing force of the gear pair. Shaft deformation at the gear pair location is governed jointly by meshing force excitation and the force/moment components generated by coupling misalignment. In this section, the coupling misalignment ϵx is 0.15, ϵy is 0.15, ϵz is 0.75 mm, and γ is 0.75°. The quasi-static characteristics of the gear system are analyzed at load torques of 22 705, 15 137, 11 353, 7568, 5676, and 3784 N m⁻¹. Figure 12 shows the mesh stiffness of the gear pair under different load torques. As the load torque gradually increases, the g2 and g3 gear pairs exhibit progressively reduced mesh stiffness fluctuations. Owing to the shorter distance between the g2 left helical gear and the coupling, it is more affected by coupling misalignment. Across the load torque levels considered, mesh stiffness fluctuation is more pronounced in the left helical gear than in the right. The tooth surfaces of the g2 left helical gear are in complete contact during certain mesh cycles, and so the mesh stiffness value does not change significantly. The g2 right helical gear has a disengaged area on the tooth surface during certain mesh cycles, and the mesh stiffness value changes significantly. Since the g3 gear pair has partial disengagement during the rotation period, the long-term fluctuation trend of the mesh stiffness curve under different load torques is sinusoidal. The greater distance between the g3 gear pair and the coupling weakens the influence of the misalignment-induced force. Therefore, the g3 gear pair has smaller long-term fluctuations compared with the g2 gear pair.

In general, higher load torque produces larger system deformation under load and correspondingly greater mesh misalignment. The formation of a disengagement region on the tooth surface reduces the actual contact line length; the mesh stiffness value will decrease, and the fluctuation amount will increase. However, when the coupling is misaligned, the misalignment force/moment and the gear pair meshing force jointly affect the system deformation. The meshing force is very small under a light load. The system deformation is mainly caused by the coupling-misalignment force/moment. Consequently, an extensive region of contact loss develops on the tooth surface. The actual contact line length changes with the mesh cycle, and the mesh stiffness value also changes accordingly, and so the long-period fluctuation increases under light loads. As the load torque gradually increases, the gear pair meshing force also increases. As the meshing force increases, the disengaged area on the tooth surface decreases until full contact is reached. Only limited variation occurs in the actual tooth surface contact line length, and so the mesh stiffness value changes little, and the long-period fluctuation is reduced.

Figure 13 shows the mesh misalignment of the gear pair under different load torques during the 66th mesh cycle. As the load torque increases, the influence of the g2 gear pair meshing force on the bending deformation of the shaft increases, and the mesh misalignment discrepancy between the left and right teeth is gradually reduced. For the g3 gear pair, the mesh misalignment is symmetrically distributed under the six load torques and is less affected by the misalignment force. As the torque increases, the meshing force increases, increasing the mesh misalignment. The proximity of the right tooth surface to the power output end causes the mesh misalignment associated with torsional deformation of the shaft to be slightly larger than that on the left tooth surface.

This section shows that coupling misalignment causes the mesh stiffness of the gear pair to fluctuate at twice the shaft frequency. Since mesh stiffness is an important source of excitation in the gear system, it can be inferred that the gear system dynamic model incorporating coupling misalignment presented below contains an excitation component at twice the shaft frequency.

According to the study by Gong et al. (2022b), the differential equation of motion of the gear system considering coupling misalignment is established, and its matrix form is given in Eq. (22). t is the time required for the slowest shaft in the system to rotate one revolution. The coupling-misalignment forces and moments are included as excitation terms in the gear system dynamics model, where Fc(t) represents the coupling-misalignment excitation vector. According to the above misalignment excitation equation, it is a low-frequency excitation associated with the rotational angle. Mglobal, C_global, and Kglobal are the system mass, damping, and stiffness matrices, respectively. x, x˙, and x¨ are displacement, velocity, and acceleration, respectively. e(t) is the meshing error. Since the research object of this paper includes multiple meshing excitations and coupling-misalignment excitations, to improve calculation efficiency, the parameters of Eq. (22) are normalized as shown in Eq. (23). Here, K‾global represents the mean stiffness, and Δ denotes its fluctuation. The system excitation force is calculated using the frequency domain solution method. The system vibration responses to time-varying mesh stiffness excitation, error excitation, and coupling-misalignment excitation in each gearbox are calculated separately using Fourier series expansions. The mesh stiffness, transmission error, and coupling-misalignment excitations are all expanded in Fourier series, with the gear meshing frequency taken as the fundamental frequency. The detailed formulation of the dynamic model and the corresponding solution procedure are provided in Gong et al. (2025). The system vibration response is then obtained by superposing the vibration responses associated with identical nodes and corresponding degrees of freedom. The dynamic meshing force of the gear pair is calculated using Eq. (24), where km and cm denote the mesh stiffness and mesh damping coefficient, respectively; em is the composite mesh error; and DTE is the dynamic transmission error. The vector qm represents the nodal vibration displacements of the gears, and V is the projection vector along the line of action. 22Mglobalx..t+Cglobalx.t+Kglobaltxt-et=P0+FCt23MglobalΔx..+CglobalΔx.+K‾globalΔx=K‾globalΔe-ΔKxSTEt-et+FCt+P024Fd=kmDTE-em+cmDT˙E-e˙m25DTE=V⋅qm

Figure 14 shows the root mean square (rms) of the dynamic meshing force of the g2 left helical gear. Coupling-misalignment excitation produces a new resonance peak for the g2 gear pair at 2350 r min⁻¹. In the resonance region induced by misalignment excitation, the linear model predicts a marked increase in dynamic meshing force fluctuations. The dynamic meshing force spectral map of the g2 gear pair is shown in Fig. 15. A distinct frequency band at twice the shaft frequency (2fn) is observed in the spectrum. fn denotes the rotational frequency of the shaft carrying the coupling. The maximum amplitude in the dynamic meshing force spectral map of the g2 gear pair appears at 2350 r min⁻¹ in the band at twice the shaft frequency. The mesh frequency band, fm1, of the adjacent gearbox GB1 appears in the dynamic meshing force spectrum of the g2 gear pair, which shows a coupling resonance between the GB1 and GB2 gearboxes at 2300 r min⁻¹.

The bearing reaction force under coupling misalignment is characterized in Fig. 16 using its rms value. The misalignment of the coupling significantly increases the root mean square reaction force of the b3 and b4 bearings in the GB1 gearbox that are closer to the coupling, and a larger resonance peak appears at 3100 r min⁻¹. The misalignment of the coupling significantly increases the root mean square reaction force of the b5 to b10 bearings in the GB2 gearbox, and large resonance peaks appear at 2350 and 3100 r min⁻¹. As the distance from the coupling gradually increases, the root mean square of the bearing reaction force gradually decreases, and the vibration weakens. The vibration response of the b9 and b10 bearings farthest from the coupling is also affected by coupling misalignment.

Figure 17 shows the root mean square of the dynamic meshing force under different coupling misalignments. For the g1 right helical gear, the RMS dynamic meshing force curves coincide under the aligned condition and cases 1 to 3. This is because the GB1 gearbox bearings have high stiffness and are unaffected by coupling misalignment. The mesh frequency causes the resonance peak at 1350 r min⁻¹. The dynamic meshing force of the g2 left helical gear fluctuates periodically at twice the shaft frequency. The amount of fluctuation increases with the amount of misalignment. Therefore, the RMS dynamic meshing force of the g2 left helical gear increases as the coupling misalignment becomes larger. Notably, when the coupling-misalignment excitation produces resonance at 2350 r min⁻¹, the resonance peaks for the aligned condition and cases 1 to 3 are 909.06, 30 579.46, 60 550.19, and 91 213.82 N, respectively. These results indicate a progressive enhancement of the vibration response of the g2 left helical gear. By contrast, the g3 left helical gear shows little variation in RMS dynamic meshing force within 0 to 850 r min⁻¹ as the coupling misalignment increases. This is because the vibration in this speed range is mainly caused by mesh frequency excitation and is less affected by coupling-misalignment excitation. Between 850 and 4000 r min⁻¹, resonance is caused by coupling-misalignment excitation. As misalignment increases, the resonance peak amplitude increases, and gear pair vibration intensifies. The change in coupling misalignment does not change the resonance speed of the gear pair.

Figure 18 shows the meshing force spectrum of the g2 left helical gear under different coupling misalignments. The corresponding amplitudes of the component at twice the shaft frequency, 2fn, in cases 1 to 3 are 1936.64, 3337.74, and 3735.16 N, respectively. This shows that, as the amount of misalignment increases, the impact of coupling-misalignment excitation on the vibration response of the gear pair increases. In the dynamic meshing force spectrum under cases 1∼3, the amplitude of the adjacent gearbox mesh frequency fm1 and the amplitude of its mesh frequency fm2 do not change significantly with the change in the misalignment amount. The component at the mesh frequency fm2 is modulated by the component at twice the shaft frequency, 2fn, producing a sideband near fm2.

The coupling misalignment ϵx is 0.15, ϵy is 0.15, ϵz is 0.75 mm, and γ is 0.75°. On this basis, the vibration response characteristics of the gear system under different load torques are analyzed in Fig. 19. Three load torques of 5676, 11 353, and 22 705 N m⁻¹ are selected. For the g1 right helical gear, the rms dynamic meshing force increases with load torque, and the increase is more obvious at the resonance speed. This is because the resonance peak of the g1 right helical gear in the range of 0 to 4000 r min⁻¹ is excited by its mesh frequency. Higher torque produces a larger excitation force in the gear pair, a higher rms dynamic meshing force, and more severe vibration. The resonance peak amplitude caused by the mesh frequency of the g2 and g3 gear pairs below 1000 r min⁻¹ increases with the increase in load torque. Since the misalignment excitation of the coupling is the same under different load torques, the resonance peak value of the g2 and g3 gear pairs caused by the misalignment excitation does not change much under the three load torques. The load torque has little effect on the resonance speed of the gear pair. It mainly affects the vibration response under mesh frequency excitation. The greater the torque, the more severe the vibration.