Influence of axis misalignments in stepped planetary gear stages on the excitation behavior—test rig development and simulative analysis

One challenge in the design of automotive gearboxes is the combination of high power density, high efficiency and low noise emission. With the electrification of the powertrain, the requirements in terms of noise emission and efficiency increase additionally. Stepped planetary gear stages are a potential topology to solve the current challenges in gearbox technology. Current research shows the pronounced misalignment behavior of planetary gear stages, especially with manufacturing or assembly deviations. However, the effects of dynamic misalignment behavior on tooth contacts in stepped planetary gear stages have not been adequately investigated. This paper presents a test rig that allows the investigation of the excitation and displacement behavior of stepped planetary gear stages taking into account adjustable axis misalignments. The axis misalignment of the stepped planetary shafts is introduced with eccentric bushings in this test rig concept. To evaluate the excitation and displacement behavior the transmission error and the displacements of different components can be measured. The test rig is modeled in the dynamic multibody simulation. The tooth contact is modeled using the force module GearForce6D. The axis misalignment is varied in the simulation model and the influence on the excitation and displacement behavior is evaluated. The simulation results show that planet pin position errors have the highest influence on the sun trajectory and the load sharing. The misalignments occurring in the tooth contacts due to inclination and skew of the stepped planetary shaft lead to higher tooth flank pressures and an increase in the total transmission error.


Motivation and introduction
Stepped planetary gear stages, as a further development of planetary gear stages, represent a potential alternative to solve the current challenges in drive technology.Advantages include in particular the short axial length in conjunction with a coaxial alignment of the input and output shafts as well as the comparatively high transmission ratio and power density.In stepped planetary gearboxes-in contrast to planetary gearboxes-the single planet is replaced by a stepped planet consisting of two joined gears.Due to the different number of teeth of the two gears of the stepped planet, a transmission ratio of up to itotal = 20 is possible between the sun gear (input) and the carrier (output) with a fixed ring gear.A major challenge in stepped planetary gearboxes is the design of the tooth flank modification due to the dynamic displacement behavior and the associated changing contact conditions in the gear meshes.
Stepped planetary gearboxes have a higher kinematic degree of freedom than simple cylindrical gears [1].If a stepped planetary gear stage is operated with only one stepped planet and a central element (sun or ring gear) or the planet carrier is fixed, the kinematics is completely determined.Each additional stepped planet results in a further load path, which increases the maximum transmittable torque for the same size.However, the static overdetermination, in conjunction with manufacturing deviations, has the effect of uneven load sharing between the different load paths [1].Load sharing to several stepped planets is thus in conflict between a high power density and the highest demands on manufacturing quality.In contrast to simple planetary gears, the radial force of the two meshes on the stepped planet acts at different positions, so that a torque acts orthogonally to the rotational axis of the stepped planet and can cause additional displacements.

Dynamic misalignment behavior
The influence of assembly and manufacturing deviations on the operational behavior of planetary gear stages has been the subject of intensive research for more than 20 years [2].Due to the kinematic similarity and the resulting transferability of the findings from planetary to stepped planetary gearboxes, various research results are considered in the following.
Sfar investigated the load distribution on the tooth flank of planetary gear stages with different tooth flank modifications [3].The load distribution was measured with strain gages and showed a dependence on the carrier position and the housing deformation.
Neubauer developed a method to calculate the static displacement and deformation behavior of planetary gear stages [4].The stiffnesses of the shafts, the bearings and the gear meshes were used in a closed linear-elastic stiffness system to calculate the system behavior.The gear mesh stiffness was approximated as a simplified average.The nonlinear stiffness characteristic was considered to investigate the excitation behavior.The misalignments and tooth flank modifications were converted to the ideal plane of contact, as in the work of Thoma [5].Neubauer showed that the floating of the sun gear shaft has a positive influ-K ence on the load sharing and a negative influence on the width load distribution [4].
Papies investigated the influence of the meshing sequence on the dynamic excitation behavior of planetary gear stages in the multi-body simulation (MBS) software Simpack [6].In the developed simulation model, with simplified calculation of tooth stiffness, it was shown that tangential axis misalignments of the planetary gear shaft have a significant influence on the displacement behavior of the sun gear shaft.
Various studies on the operational behavior of planetary gear stages with manufacturing deviations have been conducted at Ohio State University and calculation methods have been developed [7][8][9][10].Boguski et al. investigated the influence of manufacturing deviations on planet carriers on the load sharing and the displacement of the sun gear shaft [7].The measurements showed that the load on individual planets can increase by more than 100% due to tangential planet pin position errors.They also found that as the planet pin position error increases, the trajectory of the sun gear shaft becomes distorted and smaller in radial extent.
Hu et al. developed a method to calculate the load sharing in planetary gear stages taking into account the meshing sequence and manufacturing or assembly deviations [8].The position of the contact lines was calculated analytically, neglecting the misalignment of the gears.They compared the calculated load distribution at different deviations with the measurements of Boguski et al. and find partially good agreements [7,8].The validation of the sun gear displacement for different meshing sequences without manufacturing or assembly deviations showed that the qualitative characteristic of the sun gear displacement can be calculated.The different radial extent of the displacement for different meshing sequences agreed only partially.A validation of the sun gear displacement considering deviations was not performed.The authors justify the discrepancies between simulation and measurement with the missing consideration of the deformation of the planet carrier and the bearing stiffnesses of the sun shaft and the carrier [8].
Quasi-static investigations to validate the calculation method developed by Hu et al. were carried out by Ryali et al. [8,9].They presented a test concept for time-synchronous measurement of load sharing, total transmission error between the sun and the carrier, and sun gear displacement.The investigations included the phase position of the gear meshes, tangential planet pin position errors and tooth flank modifications.Their investigations showed that the periodicity of the sun gear displacement and the number of closed loops are directly influenced by the phase position of the gear meshes.In contrast, the sun gear displacement was only slightly influenced by the tooth flank modification.They concluded that the excitation and displacement behavior of planetary gear stages with manufacturing or assembly deviations can only be represented with complex simulation models [9].
In further work, Ryali et al. developed a simulation model for the dynamic excitation and displacement behavior of planetary gear stages [10].The calculation approach for the load distribution in the tooth contacts from the work of Hu et al. was used [8,10].The input and output shafts were modeled as Timoshenko beams and the bearings were modeled with 6 × 6 stiffness matrices.In this model, they investigated the influence of a speed variation on the total transmission error, tooth force and tooth flank pressure [10].Consideration of the exact planet carrier geometry and the associated modelling of the load-induced deformation was not included in this model.
Götz et al. investigated the dynamic load sharing on a double helical planetary gear stage with five planets [11].They showed that the load sharing factor is higher at low torques than at higher loads.They attributed this to the fact that manufacturing deviations have a greater effect at lower torques and the influence of deformations predominates at higher torques.In their investigations, no manufacturing or assembly deviations were specifically specified.
Matzke et al. investigated the influence of non-torque loads on the displacement behavior of a planetary gear stage of a wind turbine [12].They showed with measurements that the load distribution in the width direction is significantly influenced by external loads.With a simulation model in the MBS software Simpack, they were able to calculate the misalignments and validated their simulation method in a quasi-static tooth contact analysis with the measured tooth root strains considering the misalignments.
The various investigations show that planetary gear stages react sensitively to manufacturing or assembly deviations due to the load sharing.The simulation of the misalignment behavior is only possible with highly detailed models.The load distributions in the tooth contacts can only be predicted or optimized by taking the misalignments into account.Due to the kinematic overdetermination also present in stepped planetary gear stages, a similar sensitivity of the operational behavior to axis misalignment is expected.

Objective and approach
Increasing demands on the power density and the excitation behavior of gearboxes, particularly in the field of automotive engineering, result in rising requirements with regard to manufacturing quality.In power-split transmission systems such as stepped planetary gearboxes, kinematic overdetermination in combination with manufacturing deviations leads to different load sharing between the power paths.The influence of manufacturing-induced axis misalignments in stepped planetary gearboxes on the operational behavior has not been adequately investigated so far.
The objective of this paper is the development and simulative analysis of a test rig for the investigation of axis misalignments of planetary shafts in stepped planetary gear stages.The approach is divided into three steps and starts with a design extension of the planetary gear measurement cell, see Fig. 1.In the second step, the test rig is modeled in the dynamic multi-body simulation software Dassault Systèmes Simpack.The last step is the simulative analysis of the influence of axis misalignments on the operational behavior.

Test rig concept
The investigation of axis misalignments in stepped planetary gearboxes requires a test rig concept which allows a simple and repeatable adjustment of different misalignment conditions on the one hand and the integration of displacement and rotation angle measuring systems on the other hand.Two test rig concepts for planetary gear stages have already been developed at the WZL by Piel and Theling et al. [13,14].The planetary gear measuring cell developed by Piel has a standing planet carrier, whereas the measuring cell developed by Theling et al. has a standing ring gear [13,14].Due to the eccentric bushings at the bearing positions of the planetary shafts for the adjustment of axis misalignments in the measuring cell developed by Theling et al. this is now extended for the investigation of stepped planetary gear stages [14].The developed measuring cell for stepped planetary gearboxes is shown in Fig. 2 in both a sectional and detailed view.
The measuring cell is modular with three bearing blocks.First, the right side of Fig. 2 shows the input (1).The input shaft is rigidly supported in the input block by a doublerow angular contact ball bearing in O arrangement (type 3310, I in Fig. 2).Since the two existing measuring cells by Piel and Theling et al. contain a fixed sun shaft bearing, the dynamic displacement behavior of the sun could not be investigated completely [13,14].In the newly developed concept, the floating of the sun is ensured by a metal bellows coupling (manufacturer: R + W Antriebselemente GmbH, type: BKH/150/107, 2 in Fig. 2).When selecting the coupling, care was taken to ensure the lowest possible lateral and angular stiffness so that the floating of the sun is not impeded.To ensure angular adjustability, the sun shaft (4 in Fig. 2) is supported and axially secured by a spherical roller bearing (Type 22210, II in Fig. 2).
The planet carrier is supported by two spherical roller bearings (type 23022, III and VI in Fig. 2) in the input and output blocks.On the output side, a disk spring is provided for axial preloading of the planet carrier bearings.This allows the bearing clearance in the spherical roller bearings to be reduced.A detailed view of the planet carrier is shown in Fig. 2, top left.The assignment of the components of the planet carrier and the stepped planet is shown in Table 1.To ensure that the stepped planetary shafts (d in Fig. 2) can be mounted in the carrier, the carrier bolts (c and e in Fig. 2) must be designed in two parts, since the ring gear must already be positioned in alignment with planet 2 during the assembly process.The carrier bolts are screwed into the carrier side plates (b and f in Fig. 2) on both sides with covers (a and h in Fig. 2).The stepped planetary shafts are supported in the carrier side plates by spherical roller bearings (type 22205, IV and V in Fig. 2) as a fixed floating bearing arrangement.The planet carrier is connected to the output shaft (7 in Fig. 2) by a parallel key and an axial screw connection.A slip ring transmitter (8 in Fig. 2) is located on the output shaft, which transmits the measuring signals of the rotating planet carrier to the measuring system.In the following, the adjustability of the misalignment situations and the measuring equipment installed in the measuring cell will be discussed in detail.

Adjustability of misalignments
The reproducible adjustability of different misalignment situations in the measuring cell is of high importance for the investigations.The misalignments to be investigated can be divided into three groups-axis misalignment of the stepped planets, axis misalignment of the planet carrier and an angular misalignment between the two planets on one stepped planet shaft.First of all, axis misalignments of the stepped planet shaft relative to the planet carrier should be adjustable.These occur as a result of manufacturing tolerances of the positions of the planet carrier bores in series production transmissions.In the measuring cell, axis misalignments of the stepped planetary shafts are adjusted with eccentric bushings in the planet carrier, see Fig. 3 left.This principle has already been proved in the measuring cell developed by Theling et al. [14].The eccentric bushings have an eccentricity between the outer surface, which is positioned in the carrier side plates, and the inner surface, which is in contact with the spherical roller bearing of the stepped planetary shaft.By rotating the bushings, the phase position of the eccentricity can be adjusted, whereas the magnitude of the eccentricity is constant.In the measuring cell, with a phase position of the two eccentric bushings of a stepped planetary shaft set in the same direction, it is thus possible to set center distance changes and planet pin position errors.If the phase position is set in the opposite direction, an axis inclination and an axis skew can be investigated.The amount of eccentricity is selected based on a simulative investigation in the dynamic multi-body simulation.
The second group of axis misalignments relates to the planet carrier, see Fig. 3 center.The bearing of the planet carrier can be misaligned in series gearboxes due to manufacturing deviations in the housing.Of interest, similar to the axis position of the stepped planet, are co-axial and counter-axial eccentricities of the bearing positions.In terms of design, the adjustability is also implemented with eccentric bushings.In contrast to the eccentric bushings of the planetary shafts, the center points of the bearing bores in the bearing blocks are already manufactured offset to the ideal position by the amount of the eccentricity.This enables an investigation of the reference position (without misalignment of the carrier) as well as the misaligned situation without disassembly of the measuring cell.To set the reference position, the eccentric bushing is rotated to a phase position opposite to the bore in the bearing block so that the two eccentricities balance each other out.The adjustment of displacements is thus possible on a circular path that includes the reference case.This procedure is only possible because the resulting phase position of the misalignment is not relevant for the investigations due to the rotation of the carrier.Since the phase position of the eccentricity at the stepped planet results in different axis misalignments (e.g.center distance change vs. planet pin position error), the eccentric bushings must be exchanged at the stepped planetary shafts for the investigation of the reference variant (without misalignment).
Finally, the investigation of an angular offset between the two planets on a stepped planetary shaft is possible.The adjustability of the angle is realized with a conical locking element as shaft-hub connection of the larger planet, see Fig. 3 right.The alignment of the planet must be measured on a gear measuring center and adjusted if necessary.Only a difference in the mounting angle between the three stepped planets is relevant for the operational behavior.A rotation angle present at all stepped planets is compensated by the rotation of the central gears.

Measuring equipment
Different measurement systems are required to investigate the displacement and excitation behavior.The displacement behavior is measured with miniature inductive distance sensors.For this purpose, up to 14 sensors (manufacturer: Baumer GmbH, type: IR08.D02S-F46.UA1Z.7SL)can be attached to the carrier side plates.The position of the three planetary shafts is measured with two sensors per carrier side plate and the position of the sun shaft is measured with two sensors on the input side carrier side plate.Due to the spherical roller bearing of the sun shaft (II in Fig. 2), the pivot point of the shaft is assumed to be known, so that the translational measurement of the sun position can be converted into an angular position deviation.The measurement signals are transmitted from the rotating planet carrier to the measurement system by a slip ring transmitter.
The excitation behavior is evaluated in terms of the transmission error between the sun and the carrier and between the sun and the measurement planetary shaft.The transmission errors are calculated using signals from three magnetic incremental encoders by Baumer GmbH.The rotary encoder on the planet carrier is mounted on the input side, see (5) in Fig. 2 (type: MHGR 200 B5 G120 PN128).The rotation angle of the sun is measured as close as possible to the spherical roller bearing so that the radial displacement between rotor and stator remains as small as possible, see (3) in Fig. 2 (type: MHGR 100 B5 G80 PN64).One stepped planet is equipped with a rotary encoder as a measuring planetary shaft, see i in Fig. 2 top left (type: MHGR 100 B5 Z16 PN64).Since both the rotor and the stator of the encoder of the measuring planetary shaft rotate with the planet carrier, the signals are also transferred to the measuring system via the slip ring transmitter.To analyze the transfer path and the dynamic excitation resulting in the system, additional structure borne noise sensors are applied to the bearing blocks.

Gear geometry
The design of the gear geometry for the test rig is based on the algorithm-based method according to Westphal et al. [15].First, the quasi-static tooth contact analysis used in the algorithm-based design method was extended by the consideration of the power split in stepped planetary gearboxes.
The interactions of the meshing stiffnesses and the phase positions are considered quasi statically in the design.The aim of the design was to achieve an optimized macro geometry in terms of excitation behavior and efficiency, which ensures a balanced tooth flank and tooth root load carrying capacity.An axial force compensation at the stepped planetary shaft was specified as a boundary condition, so that the helix angles of the two stages are dependent on each other.The focus of this paper is on the analysis of the influence of axis misalignment on the operational behavior, which is why the design method of the macro geometry is not discussed further at this point.The resulting gear data are given in Table 2.The tooth flank modifications shown were selected to avoid edge wear on the basis of the maximum expected gear manufacturing deviations.Dynamic misalignments were not considered in the design of the tooth flank modifications.

Simulation of the dynamic operational behavior
The measuring cell described in the previous chapter will be used on the one hand to investigate axis misalignments in stepped planetary gearboxes and on the other hand to validate the simulation method developed.First, the measuring cell is modeled in the dynamic multi-body simulation software Dassault Systèmes Simpack 2022X [16].Subsequently, the evaluation variables for characterizing the operational behavior are described.Finally, an analysis of the influence of the preload of the disk spring on the dis-placement behavior of the planet carrier and the resulting operational behavior is carried out.

Setup of the simulation model
The system components are modeled in a high level of detail so that all influences relevant to the operational behavior can be represented as far as possible.The components are integrated into the model as modal reduced bodies (Linear Flexible Body) in order to realistically represent the stiffness and natural vibration behavior.The rolling bearings are integrated as maps provided by Schaeffler.A calculation model of the measuring cell in the FVA Workbench 7.1.1 is set up and evaluated to calculate the bearing clearance required for the generation of the map.The tooth contacts are modeled using the GearForce6D force element developed by Westphal et al. [17].A simplified overview of the force elements used and the connected bodies is shown in Fig. 4. First of all, the three bearing blocks are constrained to the inertial system at their screw connection points, each with a joint with zero degrees of freedom (DoF), see Fig. 4 top.The planet carrier is supported in the input and output blocks by two force elements of type 49 (Bearinx Map).The bearing in the output block is additionally combined with a force element of type 5 (Spring-Damper Parallel Cmp) for the disk spring.The illustration in Fig. 4 is simplified because the disk spring is also modeled as a rigid body but is not shown.The disk spring is fixed at the bearing point in the output block in five degrees of freedom by a joint which has one degree of freedom in the direction of the spring movement.The type 49 force element for modeling the bearing is connected to the disk spring and to the planet carrier.The forces and torques between the planet carrier and the output block in the five locked dimensions of the joint are thus applied directly.Whereas the force in the spring direction is passed on to the disk spring and via the force element of type 5 before it is also applied to the out- Stepped Planet Shaft (A) Fig. 4 Multi-body simulation model of the test rig put block.In the planet carrier, the three stepped planetary shafts are modeled in a similar form.The eccentric bushings are integrated as flexible bodies in the model of the planet carrier.To apply the position deviations, a rigid displacement of the markers is applied to position the bearing outer rings.The influence of the stiffness change of the planet carrier due to the eccentricity of the bores is estimated to be very small and is neglected.The gears are modeled on the one hand as flexible bodies in the form of cylinders with the diameter of the pitch circle to account for the modal vibration behavior and on the other hand as massless rigid bodies to specify the gear geometry.The force application from the tooth contacts into the flexible shafts takes place under the wheel body cylinders, since the stiffness of the wheel body is already included in the tooth contact calculation with the force element GearForce6D.The metal bellows coupling between the input and sun shafts is modeled with the force element of type 43 (Bushing Cmp).The stiffnesses for parameterizing the element were taken from the manufacturer's catalog.In addition, the mass of the coupling is divided between the two shaft ends and considered in the model.The control of the model is realized with a torque controller (PI controller).A torque defined as Excitation is specified at the input shaft with the force element of type 93 (Force/Torque by u(t)).In addition, the target time values of the output speed are defined with an Excitation.The difference between the actual speed and the target speed is used as error signal for the controller (control element type 129, P factor = 1000, I factor = 0.1).The actual output of the controller is the output torque, which is applied with the force element of type 110 (Proportional Actuator Cmp).

Evaluation values for characterizing the operational behavior
The simulation model enables a detailed analysis of the operational behavior because, in contrast to test rig investigations, state and load variables can be evaluated at any position in the model.The excitation behavior is characterized in terms of the transmission error.For this purpose, different system boundaries are considered in the calculation of the transmission error.In addition to the transmission error between the sun and the carrier and between the sun and the measurement planetary shaft, the transmission error can also be calculated between the sun and the other planetary shafts.Fast Fourier Transform (FFT) analyses are used to analyze the transmission error signals in the frequency domain.
For each body in the simulation model, the time histories of the positions, velocities and accelerations are available.In addition, the deformations of the flexible bodies and the forces or torques in the force elements can be evaluated.The displacement of the sun shaft is measured in the simulation model with a sensor at the position of the sun gear relative to the inertial system.
The GearForce6D force element enables a detailed evaluation of various parameters for each tooth contact.First, the relative positions of the gears are known, so that the rotational displacements (inclination and skew) can be converted into a resulting lead line deviation fHβ according to Wittke [18].The forces and torques acting on the gear teeth are output and used to evaluate the load sharing between the power paths in the form of the splitting factor Kγ. The tooth flank pressure is calculated during the simulation, taking into account the misalignments and the tooth flank modifi-K cations.The maximum value of tooth flank pressure is output for each time step and used for analysis.The tooth root stress is not calculated simultaneously to reduce the calculation time.The calculation can be performed subsequently in a stand-alone calculation kernel, taking into account the time history of the loads and displacements from the MBS simulation.

Influence of the preload force of the disk spring
The spherical roller bearings used in the test rig have a high operating clearance due to the angular adjustability.The operating clearance of the planet carrier bearings (III and VI in Fig. 2) is δeff = 128 µm.Gravity, which is also taken into account in the simulation model, causes the planet carrier to settle in the bearing clearance.For this reason, a disk spring is used to position the planet carrier, which reduces the resulting radial clearance during operation.The outer ring of the output-side carrier bearing is preloaded with the disk spring in the direction of the input.The bearing on the input side is designed as a locating bearing and supports the preload force.The non-linear stiffness behavior of the disk spring was taken into account in the form of a forcedisplacement function.
The influence of the preload force of the disk spring on the operational behavior is shown in Fig. 5.For this analysis, simulations with different preload distances were carried out and evaluated.As the preload distance increases and the axial force thus increases, the planet carrier is centered.The influence becomes clear in the trajectory of the sun.On the one hand, the radial expansion of the dynamic motion is reduced and, on the other hand, the center of the motion is shifted in the direction of the zero point, see Fig. 5 left.This is accompanied by a reduction in the fluctu- As can already be seen in the trajectory of the sun, the z the planet carrier, which vibrates in the axial direction.This axial excitation is converted into a rotational excitation in the planet 2-ring gear contact, since the ring gear cannot perform a compensatory movement like the sun shaft.This rotational excitation is shown in the first gear mesh order of the planet 2-ring gear contact in the transmission error between the sun and the planetary shaft, see Fig. 5 bottom right.Apart from this vibration, the transmission error in both meshes is reduced with increasing preload distance.Overall, the simulation results are plausible and show that the interaction between the rolling bearings and the gear meshes can be simulated.For the further investigations, a preload distance of z = 1.1 mm is used, since this has the lowest displacement of the sun shaft.

Simulative analysis of the influence of axis misalignments
The simulation model described is used for a variation of axis misalignments as they can be investigated on the test rig.First, a reference variant is simulated which has no misalignments.For the influence analysis, the eccentric bushings on the measuring planetary shaft are rotated in opposite radial directions (inclination, rotation of the planetary shaft about x-axis), in opposite tangential directions (skew, rotation of the planetary shaft about y-axis), in the same tangential direction (planet pin position error, displacement of the planetary shaft in x-axis) and in the same radial direction (change in center distance, displacement of the planetary shaft in y-axis).The displacement is simulated in each case in positive and negative direction and for two eccentricities.In addition, the sun shaft is constraint to the input shaft for another simulation series, so that the sun floating is restricted.The simulations are performed for three torques (MSun,1 = 30 Nm, MSun,2 = 90 Nm, MSun,3 = 150 Nm) at one constant speed (nSun = 500 rpm).The simulation time includes three revolutions of the planet carrier, whereby the last two revolutions are used for evaluation.The simulation results for different evaluation variables are analyzed below.

Trajectory of the sun gear
The trajectory of the sun is evaluated as a characteristic value for the dynamic displacements in stepped planetary gearboxes.In contrast to the simulations with different preload distances of the disk spring, the characteristic of the sun trajectory is circular in all simulated misalignment situations.A more or less pronounced additional motion is superimposed on the circular trajectory, depending on the type of misalignment.For analysis, the maximum radius of the sun trajectory around the center of the circular orbit is evaluated.The center of the sun trajectory is not significantly affected by the misalignments.In addition, the magnitude of the resulting lead line deviation in the sunplanet 1 mesh of the misaligned stepped planetary shaft is evaluated as described in the section before.
The results for an eccentricity of e = 100 µm and a floating sun are shown in Fig. 6, top left.For the reference case, a very small deflection of the sun is seen.The opposite orientations of the eccentric bushings (inclination and skew) show small influence on the radius of the sun trajectory, but result in higher lead line deviations in the tooth contact.The planet pin position error shows by far the largest influence on the radius of the sun trajectory.The resulting lead line deviations are comparable to those from inclination and skew.The center distance variation shows similar influences on the radius of the sun trajectory as inclination and skew.The influence on the lead line deviation is the least in this comparison.
The further simulations for an eccentricity of e = 50 µm show qualitatively comparable results, whereby the amounts are smaller by the same proportion as the eccentricity, see Fig. 6 bottom left.Constraining the sun shaft leads to a smaller radius of the sun movement, see Fig. 6 top right.However, the lead line deviations are comparable to the results with floating sun.This suggests that there is a bending of the sun shaft due to constraining forces, which is also shown in the bearing force of the sun shaft bearing.The lead line deviations of the non-misaligned stepped planetary shaft are smaller (fHβ < 10 µm) despite the trajectory of the sun.In all simulations, the lead line deviation shows the influence of the carrier torsion, which causes an increase or decrease in the amount of lead line deviation with increasing torque, depending on the sign of the given displacement.Overall, the trajectory of the sun is otherwise independent of the transmitted torque.

Load sharing and tooth flank pressure
The analysis of the trajectory of the sun has shown significant displacement influences.For this reason, the load sharing between the three stepped planetary shafts is analyzed in the next step.In addition, the tooth flank pressures K in the gear meshes of the misaligned stepped planet shaft are evaluated.
The load sharing is evaluated at the misaligned stepped planetary shaft (planet 0) and at the stepped planetary shaft offset by 120°in the positive direction of rotation (planet 120) in the sun-planet 1 mesh.The values shown describe the maximum torque occurring for the planetary shaft relative to the ideal sharing.With a floating sun shaft, the reference variant for the lowest torque shows a maximum load sharing of Kγ = 1.021.The load sharing approaches the ideal load sharing of Kγ = 1 as the torque increases.The signs of the displacements show an influence on the load sharing.With an eccentricity of e = 100 µm and a floating sun, the influences of a negative inclination and a positive skew are higher than with the opposite sign in each case, see Fig. 7, top left.In both cases, the contact distances in the sun-planet 1 mesh are reduced, which results in an increase of the transmitted torque.With a negative planet pin position error, this effect is also evident, resulting in a maximum load sharing of Kγ = 1.092 at a torque of MSun = 30 Nm.The more relevant load sharing value for the load carrying capacity at maximum torque of MSun = 150 Nm is Kγ = 1.039.Overall, it can be concluded that the load sharing increases only slightly for a system with floating sun even under misalignment conditions.
The load sharing with constrained sun shaft is shown in Fig. 7 top right.The constraining forces described in the previous section, which lead to bending of the sun shaft, result in a significantly higher load sharing factor.In particular, for the negative planet pin position error, the load sharing factor increases to Kγ = 2.250 at low torque and Kγ = 1.282 at maximum torque.In this case, the planet 120 transmits less load (Kγ = 0.510) at low torque.With a positive planet pin position error, the load sharing at the dis- placed stepped planetary shaft decreases accordingly and increases at planet 120.The higher transmitted torques usually also lead to higher tooth flank pressures.To evaluate the influence, the results for the sun-planet 1 and planet 2-ring gear meshes at the misaligned stepped planet shaft are shown in Fig. 8.The relative change in maximum tooth flank pressure in the sun-planet 1 mesh with floating sun is small with a maximum of σH,max = +3.3% and σH,min = -1.85%over all axis misalignments, see Fig. 8 left.In planet 2-ring gear mesh, the relative changes are larger with σH,max = +6.58%and σH,min = -2.34%,respectively.The maximum tooth flank pressure in the planet 2-ring gear mesh is achieved with a negative skew.For this misalignment, the torsion of the planet carrier and the skew of the planet shaft cause lead line deviations in the same direction, so that this is higher overall than in the case of positive skew.
Constraining the sun shaft results in a higher relative change of the maximum tooth flank pressure in both meshes.In the sun-planet 1 mesh, a tooth flank pressure higher by σH,max = +21.37% is reached for a negative planet pin position error at the lowest torque.This increase is reduced to σH,max = +8.86%at maximum torque.The relative changes in planet 2-ring gear mesh are similar with σH,max = +26.28%at low torque and σH,max = +7.54%at maximum torque.

Transmission error
The excitation behavior is analyzed based on the transmission error between the sun and the carrier.The simulation results show that especially the first gear mesh frequencies of the two stages are prominent in the frequency spectrum and are influenced by the axis misalignments.The results are shown in Fig. 9, where the transmission error is related to the carrier and given in the unit µrad.First, it can be noted that the maximum magnitude of the transmission error is very low over all simulations.A transmission error of TE = 1.5 µrad corresponds to a transmission error of TE = 0.169 µm with a radius for conversion of rreference = 112.5 mm.For the axis misalignments, it can be seen that the inclination and the skew of the stepped planetary shaft have the largest influence on the transmission error.The influence of the constrained sun shaft is more pronounced at low torque than at higher torque.

Summary
The investigated axis misalignments show a pronounced influence on the trajectory of the sun.A planet pin position error has a higher influence than other axis misalignments with the same amplitude.The constraint of the sun shaft leads to a smaller radius of the sun movement, but cannot prevent the displacement of the gears.One reason for this may be the long sun shaft used in the measuring cell and the resulting deformation.The forces acting on the bearing of the sun shaft are many times higher with a constrained sun shaft than with a floating sun shaft, even with the lowest torque of MSun = 30 Nm.
The influences continue in the load sharing between the stepped planets.Planet pin position errors lead to a higher difference in load sharing between the power paths than the other axis misalignments considered.The higher tooth flank pressures resulting from the higher load on the misaligned stepped planet shaft can be reduced with a floating sun shaft compared to a constrained sun shaft.The overall transmission error of the stepped planetary gearbox is mainly influenced by the rotational misalignments (inclination and skew).Overall, the first gear mesh frequencies of the two stages of the total transmission error are comparatively small.
In summary, the investigated axis misalignments can be compensated well with a floating sun shaft in the present test rig concept, so that the operational behavior is only slightly influenced.After validation of the results with test rig investigations, the simulation method can be transferred to series applications.

Conclusion and outlook
Increasing demands on the power density and the excitation behavior of gearboxes lead to higher quality requirements.In stepped planetary gearboxes, axis misalignments are of particular interest, since kinematic overdetermination can lead to constraining forces and higher load sharing.Thus, the objective of this paper is to analyze the influence of axis misalignments in stepped planetary gearboxes.First, a test rig concept is developed that allows investigation of various axis misalignments with eccentric bushings on the stepped planetary shafts and the planet carrier.Subsequently, the test rig is modeled in the multi-body dynamic simulation software Dassault Systèmes Simpack.The focus is on a high level of detail in the bearing and tooth contact calculations.The stiffness behavior of the components is modeled with flexible bodies.The investigation of the influence of the bearing preload force with a disc spring shows that the interactions between the structural elements, the bearings and the tooth contacts can be simulated.The built simulation model is used to analyze various axis misalignments of the stepped planetary shaft.The simulations include an inclination and skew of the stepped planetary shaft as well as a planet pin position error and a center distance deviation.The operational behavior analysis includes the sun trajectory, the load sharing, the tooth flank pressure, and the total transmission error.The results show that planet pin position errors have the highest influence on the sun trajectory and the load sharing.The misalignments occurring in the tooth contacts due to rotational misalignments (inclination and skew) of the stepped planetary shaft lead to high lead line deviations.These show their influence in particular in higher tooth flank pressures and an increase in the total transmission error.The amount of the transmission error is low overall, even under axis misalignments.Simulations with a constrained sun shaft show that the load sharing is significantly increased when the floating of the sun shaft is restricted.
Despite the high amount of eccentricity, the results are promising as they show, at least simulatively, that the axis misalignments only slightly affect the operational behavior for the designed stepped planetary gearbox.The planned test rig investigations will be used to validate the developed simulation model.The simulation model can then be used to further investigate the influence of the macro geometry and to check whether the macro geometry has a significant influence on the sensitivity of the operational behavior to axis misalignments.The next step is to design the tooth flank modifications for further optimization of the operational behavior, taking into account the dynamic misalignment and load behavior.

1 Fig. 5
Fig. 5 Influence of the disk spring preload force

Fig. 6
Fig.6 Influence of axis misalignments on the sun trajectory

Fig. 7
Fig. 7 Influence of axis misalignments on the load sharing

Fig. 8
Fig. 8 Influence of axis misalignments on the tooth flank pressure

Fig. 9
Fig.9 Influence of the axis misalignment on the transmission error

Table 1
Assignment of test rig components in Fig.2

Table 2
Test rig gear data