A simplified power-transfer system of gears and shafts connected through involute splines is used to study the influence of spline lead variations on gear-mesh behavior and load distribution on the gear teeth.

Involute splines are widely used in mechanical systems to connect power transmitting gears to their supporting shafts. These splines are as susceptible as gears to manufacturing variations, which change their loading pattern and may eventually lead to failures. The influence of manufacturing variations of spline teeth on performance and failure mechanisms of spline couplings is available in the literature. Similarly, the influence of manufacturing variations of gear teeth on gear tooth stresses and gear noise has been extensively studied. However, the effects of manufacturing variations of spline teeth on gear tooth contact, noise, and stresses remain unseen in publications. This study investigates how manufacturing variations of spline couplings affect gear performance. A parametric study was done on a spur gear set and a helical gear set to determine the amount of gear mesh misalignment caused by manufacturing variations of spline teeth. Spline parameters, such as tooth alignment and spline side fit were considered. The changes in gear contact and bending stress patterns were also investigated.

1 Introduction

Load distribution of meshing gear teeth plays a vital role on the performance of power transmitting gear pairs. Uneven load distribution along the tooth contact surfaces may result in localized contact pressure, high-root stresses, and noise issues. Gear tooth micro geometry, manufacturing tolerances, and key misalignment sources are meticulously designed by the gear engineer to optimize load distribution and tooth contact pattern under operating conditions. Misalignments coming from elastic deflections of elements such as gear teeth, gear bodies, shafts, shaft-gear connections, bearings, and housing [1] are load dependent, which may make the optimum load distribution solution valid only to a narrow range of operating loads [1].

Shaft-gear connections, used to transfer mechanical power between gears and shafts, are typically interference fits, where the gear is press fit onto the shaft, keyed or spline couplings [1-2]. In the design process, the couplings are usually overlooked as a potential source of gear-mesh misalignment. Involute side fit spline couplings are preferred over the other coupling methods due to various advantages, such as high load carrying capacity, and self-centering action under load [3]. They can also tolerate a certain amount of angular misalignment and relative sliding between their internal and external components. However, like gears, uneven load distribution of the contact surfaces affects the performance of the side fit splines [4-5]. Tolerances of spline-tooth thickness and space width are selected to suit design needs as well as manufacturing capabilities [6]. Manufacturing tolerances of spline teeth include profile variation, lead variation, and index variation [6]. These variations affect both the effective clearance or spline fit, and load distribution [6]. Studies show pitch errors have a major effect on reducing the number of active teeth sharing load [7], while profile and lead variations change the load distribution over contact interface [8]. Shaft torsional effects also result in a non-uniform load distribution in the axial direction along the engagement length of the tooth [9]. In certain cases, an intentional lead mismatch of the spline contact interface is introduced by a slight change of helix angle of the external spline to achieve an interference fit or zero backlash condition [10], which may also result in uneven load distribution on the spline teeth.

Involute splines of shaft-gear couplings are affected by gear loads apart from the torsional load component. Wink and Nakandakari [11] showed gear loads in the radial direction significantly change the load sharing among spline teeth of the supporting spline coupling. Hong et. al. [8] investigated the load distribution along the tooth interface of spline using combined finite element and surface integral contact analysis model under various gear-loading conditions and manufacturing errors. Under helical gear loads, a tilting movement of the gear about the supporting shaft is created causing misalignments and non-uniform load distribution on the splined interface [8]. The studies on load distribution and misalignments of spline teeth were primarily focused on spline strength and preventing common spline failure mechanisms such as tooth breakage, surface wear, and fretting. Although many studies have investigated load distribution changes of the spline contact interface, the influence of manufacturing variations of spline couplings on gear performance remains unseen in publications. The objective of this article is to investigate the effects of manufacturing variations of spline couplings on the performance of gears. The study was limited to spline teeth variations in the longitudinal direction, assuming that effects of spline tooth profile variations on gear tooth contact are negligible. An advanced commercial CAE tool for transmission analysis, named MASTA [12] was used for the analytical predictions under various input conditions and manufacturing variations through an extensive parametric study. MASTA’s Advanced Loaded Tooth Contact Analysis model was used for the gear calculations. Splines were modeled as a series of spline segments of variable stiffness, which was defined as a function of the initial gap between the spline teeth at each segment. The initial gap between spline teeth, under no load condition, was calculated from the spline geometry accounting for different manufacturing variations. A stiffness matrix of spline teeth in full contact was obtained from the SplineLDP program [13], which uses a custom finite element approach [14]. The base model results were verified using commercial finite element-based software, Transmission 3D [15].

The results show uneven load distribution due to spline coupling misalignment affects gear-mesh misalignment, which ultimately changes gear-load distribution, contact stress, and bending stress. The conclusions of this study point to the importance of accounting for misalignments of the spline coupling while designing and optimizing the macro and micro geometry of power transmitting gears.

2 Analytical Model

A simplified system of a helical gear pair along with the supporting shafts was used for the study. The driving gear was mounted on the shaft through a spline joint, and axial movement of the spline coupling was controlled on both sides. The driven gear was rigidly coupled to the supporting shaft. A schematic of the system considered is shown in Figure 1.

Figure 1: Gear system considered in the analytical model.

Parameters of the gear and the spline used are listed in Table 1 and Table 2 respectively. The system was supported by bearings at each end of the shafts, maintaining a perfect alignment between gears at no load.

Table 1: Parameters of example helical gear design.
Table 2: Parameters of example spline design.

Loaded tooth contact analysis was performed in MASTA, which uses a hybrid FE and Hertzian-based tooth contact model [16]. MASTA performs a system deflection analysis to predict misalignments at the gear mesh and couple it with the tooth contact model. Shafts and gear blanks are considered as classic Timoshenko beams [17] for the deflection analysis. Spline coupling between the driver gear and the supporting shaft was divided into multiple segments along the longitudinal direction. A node was defined at the center of each of these segments along with a corresponding stiffness matrix.

Twenty discrete points with specified stiffness matrix were used in the model to represent the spline coupling (Figure 2). Any misalignment in the spline coupling along the longitudinal direction was entered in the model as corresponding rotational clearances at each of these points as well as radial and tilt clearances along radial x and y directions, as shown in Equation 1 and Equation 2, respectively.

Figure 2: Spline coupling modeled as stiffness matrix at axial locations of gear-shaft interface.

Rotational clearance:

Equation 1

Radial clearance:

Equation 2


c is the nominal circumferential clearance from the spline design. That is half the difference between circular tooth thickness and space width of mating teeth.

ci    is the change in clearance due to lead error at  ith location.

d is the pitch circle diameter of spline.

α is the spline pressure angle.

i = 1, 2, 3, … n, where n is the number of segments in the spline longitudinal direction.

Change in clearance ∆ci are calculated as the variation of the tooth surface from its ideal position in the transverse plain at the specific location along the longitudinal direction. The effect of this is to either increase or reduce the nominal circumferential clearance c on the spline interface. For an angular lead error of θ, the change in clearance at a distance x is x tanθ. Figure 3 shows how misalignment due to linear lead variation, λ, is defined in the model as multiple connections along the spline contact length l with specific clearance matrix and stiffness matrix associated with it.

Figure 3: Clearance modeling for misaligned spline.

In the case of effective side clearance, that is cθ min  > 0, an assumption is made in the model that one member rotates relative to the other member making contact at the node with lowest clearance for the initial condition of the analysis. Therefore, at effective clearance conditions, cθi, cxi, and cyi are recalculated as:

Equation 3
Equation 4
Equation 5


cθ min  is the minimum value of rotational clearance along the spline length.

cx min  is the minimum value of radial clearance in x direction along the spline length.

cy min  is the minimum value of radial clearance in y direction along the spline length.

The SplineLDP program developed by the Gear Lab of The Ohio State University was used to calculate the stiffness matrix of the spline coupling. The stiffness values along with the clearance values were used on the model as shown in Figure 2 and Figure 3. Tooth contact analysis was performed on the model to study the impact of spline misalignments on the gear performance.

3 Impact of Spline Misalignment on Gear Flank Loading

An input torque of 2,500Nm was applied to the driver shaft, and contact stress distribution on the gear flank was calculated under different conditions of spline misalignment, λ = -50, -25, 0, 25, and 50 μm. Figure 4 shows the gear tooth contact patterns for the various spline misalignment values.

The contact stress distribution charts of Figure 4 represent gear tooth flank area with distance in the axial direction on the x axis and distance in profile direction on the y axis. The high contact area is located toward the center in both lead and profile direction due to the lead and profile crowning considered as part of gear micro-geometry. λ = 0 represents the condition with no misalignment on the spline teeth interface; a centralized load distribution pattern is observed on the gear flanks in this case. A positive λ value represents lead variation in the same direction of the helix hand of the gear, which is right hand in this model.

Figure 4: Variation of helical gear contact pattern under various spline misalignments: a) l =50, b) l =25, c) l = 0, d) l = -25, and e) l = -50.

This right-handed misalignment results in contact of spline teeth at the left edge (Side I) and gradually increasing clearance toward the right side (Side II) that leaves a clearance equal to λ at the right edge. A negative misalignment is left handed in this model, which is opposite to the helix hand of the gear. With the addition of spline misalignment in the positive direction, the high contact area moved from the center to the left side. The change in the contact pattern was due to the gear mesh misalignment caused by the tilting of the gear under load because of the spline teeth deflection. Effective gear mesh misalignment from the system deflection analysis compared in Figure 5 confirms that, under load, the misalignment on the spline influences the mesh misalignment of the gear, which is mounted on it. It is observed from Figure 4 and Figure 5 that, while the positive spline misalignment increased, the gear mesh misalignment and the negative spline misalignment hardly affected the gear-mesh misalignment and the contact pattern. In the case of right-hand spline misalignment, the tilting moment on the gear due to the helix angle was in the same direction as the tilting moment due to deflection, which resulted in a higher deflection. The opposite effect happened for left- handed misalignments of the spline teeth.

Figure 5: Variation of helical gear-mesh misalignment under various spline misalignments.

As the increase in gear-mesh misalignment causes the highest stressed area of the loaded flank to move away from the center, a corresponding change also happens in the root stress. When the contact stress pattern on the gear flank moves away from the center of face width, the highest bending stress location also moves toward the edges of the face width. This increases the probability of gear-bending fatigue failures, since the crack initiation points can develop toward tooth edges under high-stress conditions. The change in the distribution of tensile root stress along the axial direction from the analysis results are compared in Figure 6. In the example shown in Figure 6, the location along the gear face width of the maximum tensile root stress is shifted around 5mm to the left under a 50µm spline misalignment as compared to the original condition of no spline misalignment.

Figure 6: Variation of gear root bending stress due to spline misalignment.

The changes in gear mesh misalignment and load distribution on the tooth flanks might also affect the transmission error (TE) of the gear mesh, which is a main source of noise excitation in a geared system. TE varies as a function of the gear tooth elastic deflections, and the geometrical variations of the meshing tooth form from a true involute [18]. Since the variations on spline coupling affect the load distribution of the gear teeth, changes in the noise behavior of the system also are expected to happen.

4 Influence of Gear Helix Angle

To investigate the effect of spline variations on spur gears, the original gear geometry of Table 1 was modified to the helix angle, β = 0, by keeping the same transverse geometry parameters. Spur gear data is shown in Table 3. Analysis was repeated with the spur gear model. The results show the effect of spline-tooth misalignments on the gear surface loading pattern is similar in both directions — that is, positive and negative spline tooth misalignment values, unlike the helical gear model (Figure 7). As expected, maximum contact stress is high in the spur gear model compared to the helical gear under the same input torque due to the reduction of contact area with no axial contact ratio.

Table 3: Parameters of example spur gear design used.
Figure 7: Variation of spur gear contact pattern under various spline misalignments: a) l =50, b) l = 25, c) l = 0, d) l = -25, and e) l = -50.

The results confirmed the impact of spline misalignments on gear performance is affected by the helix angle and helix hand. Higher gear mesh misalignments are observed when the gear helix hand is in the same direction of the spline misalignment, and the spline misalignment effect is less when gear helix hand and spline misalignment are in opposite directions. The gear-mesh misalignment results of the helical gear set and the spur gear set are compared in Figure 8.

Figure 8: Impact of gear helix on variation of gear-mesh misalignment under various spline misalignments.

5 Influence of Load

Since the amount of deflection is load dependent, the impact of the spline misalignment on the gear-mesh misalignment also increases as the load transferred through the system increases. The graph in Figure 9 shows the influence of input torque on the spur gear-mesh misalignment under various spline misalignment conditions.

Figure 9: Impact of transmitted load on gear-mesh misalignment under various spline misalignments.

6 Influence of Effective Side Clearance

When there is a lead slope variation of the spline, the design clearances are consumed by the variation. This may result in a condition of effective side clearance or effective side interference between the external spline and the internal spline, depending on the amount of variation and the design fit. The effect of various fit conditions of the spline on the misalignment of the gear mounted on it were studied on the spur-gear model with λ = 50 μm, and the results are as shown in Figure 10. All conditions were analyzed, considering there was enough axial clearance on the model to allow gear blank to tilt under load. In Figure 10, the zero point on the horizontal axis represents the condition of no effective clearance or interference, when spline lead variation is equal to the design clearance. Moving toward the positive (left) side on the horizontal axis represents effective clearance conditions, and the negative (right) side represents effective interference conditions. As the amount of interference increases, the length in the longitudinal direction of the spline coupling under interference also increases depending on the amount of variation λ. The percentage of the spline coupling length in contact under interference is shown in the upper horizontal axis of the graph.

Figure 10: Impact of effective clearance on variation of spur gear mesh misalignment under 50µm spline misalignment.

It was observed that an increase in the effective side clearance of the spline does not contribute to the gear misalignment caused by the spline misalignment. As shown in Figure 10, gear-mesh misalignment remains the same under effective clearance condition, irrespective of the clearance value. That might be because of the modeling assumption of initial contact condition of the spline teeth, where the drive flanks are brought into contact, and any resultant clearance is moved to coast side. In case of helical gear loading or in the presence of other system misalignments, the effective side clearance may have an impact on the gear-mesh misalignment, which is beyond the scope of the analytical model considered. On the other hand, effective side interference condition drastically reduced the gear misalignments as the interference condition allows both right and left flanks of the spline to be in contact simultaneously, and more teeth area available for load transfer.

7 Reducing Spline Variation Effects on Gears

As manufacturing lead variations of spline teeth affect the performance of the geared systems, controlling these spline variation effects at the design stage might help the gear engineer to develop more robust gears. This can be achieved either by reducing the manufacturing variations of the splines or by designing gear micro geometries to accommodate mesh misalignments, including the contributions that come from the spline coupling. Selecting a tighter spline tolerance class reduces allowable manufacturing variations and consequently their impact on gear-mesh misalignment; however, it might increase manufacturing costs.

A solution in helical gear systems may be to use unidirectional tolerances of splines instead of bidirectional, so the hand of spline lead variation can be maintained relative to the gear hand to reduce the effect. Higher lead crowning on the gear micro geometry helps to accommodate more misalignments at the gear mesh. The amount of crowning is generally limited in the design by the increase in contact pressure and the TE levels, which need to be considered. A fair estimation of gear-mesh misalignment caused by the elastic deflections of the system, including the spline couplings at the design stage will help the gear engineer come up with an optimum design of micro geometry for the gear.
The axial clearances in the system, which results from the tolerance stack up of various assembly elements, might limit the amount of gear tilting over the spline coupling under load. Thus, lowering axial clearance value may help reduce the effects of spline variations on gear mesh.

8 Summary and Conclusions

A simplified power-transfer system of gears and shafts connected through involute splines was used to study the influence of spline lead variations on gear-mesh behavior and load distribution on the gear teeth. Spline misalignments caused by manufacturing variations were shown to increase misalignments in the gear mesh, which caused the gear flank contact pattern to change, which also changed the gear root bending stress distribution. The spline teeth with misaligned contact was observed to deflect under load and developed a tilting moment on the gear resulting in the misalignment of the gear mesh. Manufacturing variations of spline coupling act as a source of gear-mesh misalignment, causing changes in gear tooth load distribution, which might be detrimental to gear load capacity and gear noise.

The effect of spline manufacturing variations on helical gears was more pronounced when the spline lead variation and the gear helix hand were in the same direction. Conversely, the effect was reduced when spline lead variation and the gear helix hand were in the opposite direction. As the spline tooth deflections are load dependent, the effect also increased with the load transmitted through the system. The results of a parametric study performed on the effective clearances revealed that transitioning from interference to clearance condition drastically increases the gear-mesh misalignments.

A few suggestions were offered to reduce the impact of manufacturing variations of spline couplings on gear-mesh misalignment, which included selecting the right spline tolerance class, applying unidirectional tolerance instead of bidirectional in opposite direction to the gear helix hand, limiting the amount of gear axial clearance of the assembly, and designing gear tooth micro geometry to accommodate the mesh misalignment coming from the spline coupling.

Accounting for the effects of manufacturing variations of spline couplings on gear mesh might help gear engineers to develop more robust gears of high load capacity.


The authors thank Eaton Vehicle Group for their support in developing this article. 


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Printed with permission of the copyright holder, the American Gear Manufacturers Association, 1001 N. Fairfax Street, Suite 500, Alexandria, Virginia 22314. Statements presented in this paper are those of the authors and may not represent the position or opinion of the American Gear Manufacturers Association. (AGMA) This paper was presented October 2019 at the AGMA Fall Technical Meeting in Detroit, Michigan. 19FTM18