Load distribution analysis of clearance-fit spline joints using finite elements
Introduction
Splines are widely used in mechanical drive systems to transfer rotary motion and torsion from a shaft to a gear or from a gear to a shaft. The main advantage of splined shafts over keyed shafts is their higher load carrying capacity that often represents better durability performance. Moreover, spline couplings allow for a certain amount of angular misalignment and relative sliding between their internal and external components. They can transfer axial, rotational and torsional load effectively in case of helical gear loading.
The most common failure modes observed in spline joints include surface wear, fretting corrosion fatigue, and tooth breakage. Ku and Valtierra [1] studied wear of misaligned splines experimentally, demonstrating a significant effect of misalignments on the wear of a spline. Brown [2] reported accelerated wear of involute spline couplings in aircraft accessory drives primarily due to spline misalignment and undesirable lubrication conditions.
These experimental studies on splines were instrumental in defining and documenting failure modes in spline interfaces. Yet their contributions to the understanding of spline failure mechanisms were limited without knowing the load distributions along the spline contact interfaces. Review of the literature reveals only a few analytical models on splines, all of which were limited to simple loading conditions due to complexity of the contact in spline interfaces. Volfson [3] proposed a rough estimation of contact force distribution along the axial direction of splines under pure torsion or pure bending loading conditions. Tatur and Vygonnyi [4] developed an analytical model to estimate torque distribution along the face width direction of spline teeth for the case when the spline joint carries pure torsion. They proposed that the running torque m(z) along the axial direction of the spline be determined by the following differential equation:where Te(z) is the shaft torque, is the torsional stiffness of the spline joint which is assumed to be a constant along the axial direction, and φi(z) and φe(z) are twisting angles of the internal spline and external spline respectively. This simplified model requires a user-defined spline torsional stiffness. Barrot et al. [5], [6], [7] formulated the spline tooth torsional stiffness in the model of Tatur and Vygonnyi [4] by analyzing spline tooth deflections due to bending, shear, compression and base rotation. They calculated the load distribution along axial direction of splines under pure torsion loading conditions. These analytical models provide an estimate of load distribution along the face width direction of the splines under simple loading conditions, but they fail to predict the load distribution across the spline tooth profile direction. Furthermore, they fall short of handling load distribution of splines under combined loading conditions as is the case for gear–shaft spline joints. Other complicating effects such as spline tooth surface modifications and spline tooth manufacturing errors such as indexing or spacing errors are also not considered in these models.
Another group of more recent studies proposed computational models of splines using the finite element (FE) method or boundary element (BE) method. FE models by Limmer et al. [8], Kahn-Jetter and Wright [9] and Tjernberg [10], [11] used commercial FE packages to predict spline load distributions under pure torsion loading, while the last two accounted for effects of certain manufacturing errors as well. FE models for helical spline couplings were proposed by Leen et al. [12], [13], [14] and Ding et al. [15], [16], [17] for splines under combined torsional and axial loading. Adey et al. [18] developed a model using boundary element method for spline analysis. This model had the capability to analyze combined torsional and bending loading in the presence of certain manufacturing errors. Using Adey's model, Medina and Olver [19], [20] studied load distribution of misaligned splines, and the impact of spline pitch errors and lead crown modifications.
The above literature review indicates that there is no widely accepted general analysis tool for spline load distribution. Most of these did not include combined radial, torsional and bending loading conditions experienced by spur and helical gear splines. Under such complex loading conditions, effects of spline tooth modifications and manufacturing errors are not fully understood. Consequences of the common practice of applying an intentional mismatch of splines also remain unknown. Furthermore, extensive parameter studies of the effects of spline tooth modifications and manufacturing errors are also not available. Accordingly, this paper aims at developing FE based computation model of gear–shaft splines. The objectives of this paper are as follows. (i) Develop a computational model of a gear–shaft spline interface under combined torsion, radial forces and tilting moments. (ii) Establish nominal load distribution conditions under pure torsion, spur gear loading (torsion and radial force) and helical gear (torsion, radial force and tilting moment) loading conditions. (iii) Quantify the change to baseline load distributions caused by misalignments, spline tooth (lead and profile) modifications, spline helix angle and intentional helical mismatch. (iv) Investigate the influence of manufacturing errors on baseline spline load distributions.
Section snippets
Computational model
A commercial FE based contact mechanics model Helical 3D (Advanced Numerical Solutions, Inc.) designed specifically for loaded contact analysis of helical gears is modified here to analyze spline joints. An efficient and accurate finite quasi-prism (FQP) element [21] is used in this model to represent spline surfaces. The core contact solver of this software (CALYX) is based on a formulation by Vijayakar [22], which combines the finite element method and surface integral method to represent the
Finite element model of an example spline and analysis results
Fig. 1 shows the finite element model for an example spline joint that is designed according to ANSI B92.1-1996 standard [25]. The parameters of this clearance-fit spline are listed in Table 1.
The system model consists of a shaft, an external spline and an internal spline. Over the potential contact area, the model shown in Fig. 1 uses a contact grid with M number of elements in the face width direction and N number of elements along the profile direction. Within each contact element the user
Summary and conclusions
A finite element based computational model of a gear spline-shaft interface under combined torsional load, radial load and tilting moment was proposed. Load distributions of the baseline system of the spline coupling under pure torsion, spur gear loading and helical gear loading were characterized. Pure torsion loading results showed identical load distributions on all spline teeth, with each tooth exhibiting non-uniform load in axial direction. Spur gear loading was shown to cause increased
Acknowledgments
Appreciation is extended to the sponsors of the Gear and Power Transmission Research Laboratory of The Ohio State University whose financial support made this work possible. Authors also thank the Advanced Numerical Solutions, Inc. for making the spline analysis package CALYX available.
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