1. Introduction
In accord with prevalent statistics, rotor vibration occurrences contribute to approximately 80% of mechanical mishaps that occur during rotation processes. The resulting fallouts encompass fatigue, fractures, and additional malfunctions in, but not limited to, shafting components. Consequently, these failings precipitate substantial safety abnormalities [
1]. Consequently, an abundance of academic studies have pursued the vibrational traits of spline rotor systems in recent years. For instance, scholarly work carried out by Huang, Z.L. et al. [
2] acknowledged the nonlinear elements of spline meshing gaps and rolling bearing contact. Their research generated a nonlinear dynamic model of a rotor-bearing system exhibiting the properties of both flexural and torsional coupling. This breakthrough model now offers simplified ways to mitigate vibration and to design dynamically efficient rotor-bearing systems [
2]. In a separate study, Long, X. et al. [
3] embarked on the analysis of the impacts inflicted on the internal spline’s unusual wear due to determinants such as tooth surface hardness, vibration, and rotation precision. This exploration, which was premised on the features of involute spline connections, provides pivotal information for the conceptualization of accessory actuators [
3]. Zeng, G. et al. [
4] devised a coupling vibrational dynamics model for rotor–planetary gearing systems within the scope of the electromechanical compound transmission. Their research elucidated the vibration characteristics of the rotor–planetary gear system at various velocities. It extrapolated insights concerning the coupling vibration interaction between the gear system and the rotor of the propulsion motor. These enlightenments establish a theoretical scaffold for the expedited design of transmission systems [
4]. Du, W.L. et al. [
5] successfully derived the coupling dynamic equation applicable to a rotor system subjected to load excitation, utilizing the formidable computational prowess of Lagrange’s equation. The team conducted an array of numerical simulations reliant upon the Runge–Kutta computation methodology, permitting them to scrutinize the coupling vibration properties intrinsic to the motor rotor system under an assortment of load excitations. This seminal work yielded pivotal insights to inform and refine active vibration reduction stratagems pertinent to rotor systems [
5]. Wang, P.F. et al. [
6] embarked on a critical exploration centering on the repercussions of misalignment upon the dynamic contact attributes of bearings in addition to the vibration characteristics inherent in rotor systems. Pivotal among their findings is the revelation that bearing misalignment exponentially accelerates the resonance speed of the rotor while concurrently triggering amplitude hopping within the resonance domain. These seminal revelations provide a sturdy theoretical foundation as well as an invaluable cache of references for routine and advanced fault-identification tasks within rotor-bearing systems [
6]. Under the expert guidance of Wang, L.K. et al. [
7], dynamic models of intricate rotor-support systems, such as those found in aero-engines, were conceptualized. Experiments were thereafter conducted to authenticate the efficacy of these carefully constructed models. The team further dissected the nonlinear damping efficiency of parallel extruding oil film dampers (SFDs) in conjunction with elastic supports. Their research is a trove of quantitative references critical to the parallel design of aero-engine support stiffness and correlating SFDs [
7]. Ge, W.W. [
8] and his team established a comprehensive rotor-bearing system and pragmatically relied on the Runge–Kutta method for integral computations. The research team’s investigation hinged upon the discernible influences of support stiffness, damping, acceleration, and support position on rotor vibration traits [
8].
Scholars have conducted extensive investigations into the vibration characteristics of rotors featuring misaligned spline couplings. Xiao, L. et al. meticulously devised a nonlinear dynamic model tailored to misaligned spline couplings and scrutinized the dynamic meshing behavior exhibited by misaligned tooth pairs [
9]. Mura, A. et al. employed finite element analysis to delve into the ramifications of eccentricity angle and transfer torque, culminating in a theoretical model addressing inclination torque [
10,
11,
12]. Guo, Y. et al. put forth an innovative model to analyze gear couplings, taking into account variables such as load torque, misalignment, and friction [
13]. Zhao, G. et al. conducted simulations, unearthing the consequences of misaligned meshing force upon the dynamics of rotor–spline coupling systems, and subsequently derived a differential equation pertinent to the dynamics of toothed couplings [
14,
15,
16]. He, C.B. meticulously examined the misalignment concerns intrinsic to tooth couplings and their corresponding rotor systems [
17]. Fu, B. delved into the impact of parallel misalignment on fixed rigid couplings [
18]. Jing, J.P. and Gao, T. [
19,
20] computed the dynamic load coefficient emblematic of a secondary spline system, established a vibration model predicated on the finite element method, and analyzed the effects of dynamic engagement forces on the rotating subsystem’s stability. Through rigorous theoretical analysis and experimental research, they unveiled the influence of various dynamic parameters on the system’s characteristics and stability [
19,
20]. Furthermore, researchers have probed the effects of misalignment on the vibration characteristics intrinsic to spline coupling systems [
21,
22,
23].
The interpretation and analysis of load dispersion and spline principles have been acknowledged critical for the prognosis of fretting deterioration in spline couplings. Such matters have been earnestly explored by experts [
24,
25,
26]. Subsequently, Xue, X.Z. formulated a solely torsional dynamic model as well as a dynamic equation for involute spline combinations localized in aero-engines. They proceeded to compute the all-encompassing mesh rigidity, further assess the dynamic payload, and scrutinize the vibration displacement intrinsic to the spline coupling assembly [
27,
28,
29]. Surprisingly, it was found that the pragmatic logarithm of meshing teeth within involute spline combinations diverged from the theoretical estimation and varied in accordance with external payloads and lateral tooth flexibility, precipitating a temporal variation in meshing rigidity. In addition to this, installation-induced or operation-induced (originating from heat, payload, foundational deformation, etc.) misalignment of internal and external spline shafts, as well as manufacturing discrepancies resulting in mass eccentricity and tooth flexibility, engender bending and superior payloads on the spline shafts. This scenario culminates in the creation of nonlinear, flexural–torsional, amalgamated oscillations within the involute spline sub-categories. At the present moment, the majority of the research work pertaining to spline couplings is majorly concentrated on the coupling itself—ironically overlooking the reality that spline constructions are not solely employed within shaft couplings but also find widespread applications in various other transmission domains. Hence, the prevalence of studies evaluating vibration and its subsequent influence on fretting damage within this scenario is alarmingly scarce.
Thus, this investigation primarily pivots its attention towards precisely determining the dynamic parameters attributable to the secondary structure within involute spline pairs utilized in aero-engines. Subsequently, these findings are integrated with the execution of finite element simulation, which vigorously enables the analysis of various vibro-characteristics’ impact on fretting damage in involute spline pairs. Additionally, this study comprehensively examines how angulated misalignment influences nonlinear vibrations as well as fretting damage. The insights gleaned from this research hold instrumental value for the blueprinting of high-performance, reliable, and resilient involute spline pairs specifically tailored for utilization within aero-engines. Furthermore, aero-engine involute spline pairs inherently exhibit complex, nonlinear dynamic characteristics through the manifestation of transverse, torsional, and longitudinal vibrations within the scope of the rotor system. These vibrations, unfortunately, are potent enough to inflict destructive ramifications on the integrity both of spline pairs and the associated shafting [
30]. Misalignment aberrations represent a conventional fault within spline and rotor system frameworks. In instances where misaligned configurations morph into the predominant source of faults, one can expect detrimental outcomes such as pronounced structural damage to the spline mechanism, exacerbated system vibrations, and the emergence of substantial safety-related implications. Last but not least, the intricacies of the operational process tied to aero-engine involute spline pairs inevitably instigate micro-amplitude vibrations within the secondary system framework. These vibrations precipitate substantial fretting damage, drastically curtailing the operational lifespan of the implicated components [
31,
32].
In a concerted effort to efficaciously modulate the vibration and attenuate fretting wear associated with aero-engine involute spline couplings, the authors originated a bending–torsion unified nonlinear vibration model. This model was specifically constructed for involute spline couplings experiencing instances of misalignment. In tandem, a dynamic meshing stiffness function accommodating multi-tooth engagement was duly formulated [
33]. Nevertheless, reference [
33] did not account for the ramifications of spline vibration specifically on the extent of fretting damage, thereby leaving a gap in the study. Therefore, standing as a progression from the preceding investigation, this study delves into the mechanism underlying vibration-induced displacement, and extemporaneously investigates the influence of various contributory factors on fretting damage. These investigations specifically pertain to splines functioning under diversely angled misalignment conditions. A comprehensive depiction of the process is presented in
Figure 1.
3. Fretting Damage Prediction Model for Involute Spline Pairs of an Aero-Engine
Upon a careful investigation, existing studies [
34] have attested to the fact that although fretting fatigue pervades throughout the entire fretting cycle and exhibits competition and influence alongside fretting wear, the predominant manifestation of fretting damage failure within the involute spline pairs of an aerodynamic engine is ultimately and specifically characterized by the fretting wear failure mechanism. Thus, with a focus on highlighting the intricate interrelations between the observed phenomena, this current work opts to use the wear depth of the spline as a reference point. The aim here is to illuminate the underlying mechanisms through which nonlinear vibrations exert a potential impact on the ensuing fretting damage.
Based on the Archard abrasive wear and adhesive wear model, the fretting wear model suitable for involute spline pairs of an aero-engine was derived. In the abrasive wear model, i.e., the wear volume obtained by the normal load on a solid abrasive particle (Equation (11)), the wear volume obtained after the abrasive particle is pressed into the depth Z of the ground surface and slides
distance on the surface (Equation (12)), and the total wear rate of the abrasive particle under unit slip distance is obtained by dividing both ends of Equation (2) by the slip distance
(Equation (13)):
in which
is the normal load borne by the abrasive particles
;
is hardness;
is the abrasive wear volume (
);
is the abrasive slip distance;
is the total wear rate of abrasive particles;
is abrasive wear coefficient.
For adhesive wear, the microscopic cutting theory is also adopted. Different from abrasive wear, which assumes abrasive particles as cones, and adhesive wear, which assumes abrasive particles as spheres, the Archard adhesive wear model is obtained, as shown in Equation (14) [
29]:
in which
is the adhesive wear volume (
);
is the adhesion wear coefficient;
is the normal load borne by the abrasive particle (
);
is the yield stress (
).
For the calculation of fretting wear, domestic and foreign scholars have derived the Archard calculation model suitable for fretting wear by combining the above abrasive wear model and adhesive wear model, as shown in Equation (15):
in which
is the wear depth (mm);
is the wear coefficient;
is the relative slip distance (mm);
is the contact stress (
). However, in order to make the above formula more suitable for the special working condition of involute spline pairs of an aero-engine, the above formula is optimized as follows.
As the contact stress on the surface of spline teeth of involute spline pairs of an aero-engine is constantly changing during the working process, and the relative slip rate between spline couplings is also constantly changing within a certain period of time, the above equation is differentiated and integrated on both sides to obtain the optimized Archard calculation model, as shown in Equation (16):
in which
is the velocity change curve at a node of the spline tooth surface (mm/s);
is the stress change curve at a node of the spline tooth surface (
).
The methodology for discerning fretting damage experienced by secondary involute spline pairs in aero-engines under the exertion of nonlinear vibrations follows these outlined steps:
Solving the differential equation specified in
Section 1 utilizing MATLAB results in the derivation of the external spline’s vibration displacement curve, specifically oriented in the
x-axis direction.
Data relevant to the external spline’s vibration displacement in the x-axis direction are gleaned and solved through Fourier transform within MATLAB to yield a time-based function. This function delineates the vibrational displacement’s variability.
The resultant function is applied to the displacement boundary condition of the spline in the x direction using ABAQUS, an advanced computational platform, and ensuingly resolved.
Post resolution within ABAQUS, the CPRESS and rate FSLIPR in the resultant data file are scrutinized. Subsequently, MATLAB is employed to optimize the Archard fretting wear calculation model, facilitating the estimate of the wear depth distribution on the external spline’s tooth surface in conjunction with different vibrational displacements.
A more explicit illustration of this calculation process is demonstrated in
Figure 3.