Smooth Profile Research Articles

Accuracy improvement of surface measurement through phase correction in spectrally resolved interferometer

The spectrally resolved interferometer (SRI) is used to measure the surface profile by the linear fitting line of its phase distribution of spectral interference signal. However, the noise existed in the phase cannot be ignored if the nano-scale measurement accuracy is required. In this paper, a novel algorithm is proposed to improve the measurement accuracy by using the constant term oflinear fitting line of phase distribution in wavenumber domain. The simulations indicate that the algorithm achieves the similar measurement accuracy under the influence of three kinds of noise. Even though the proposed algorithm has the possibility of 4% to bring the additional minor error, it provides a better performance on measurement of entire surface in different SNRs compared with the other two common algorithms. In experiments, the variations of surface less than 10 nm and 30 nm and the repeatability of 2.2 nm and 4.9 nm are obtained in the measurements of a protected aluminium mirror with and without proposed algorithm. A smooth profile with a step shape of nearly 3 μm height is tested, where the maximum repeatability of step height and maximum magnitude of variations of the surfaces reaches to 3.8 nm and 10 nm from the 15.3 nm and 30 nm by using this algorithm. Both the simulations and experimental results demonstrate that the proposed algorithm is feasible for high-precision measurement.

Suppressing staircase-like electrochemical profile induced by P[sbnd]O transition by solid-solution reaction with continuous structural evolution in layered Na-ion battery cathode

Layered P2-type sodium manganese oxide materials (NaxMnO2) attract substantial interest due to their low cost and high security. However, the staircase-like electrochemical profiles in P2-type NaxMnO2 caused by unfavorable phase transitions and layer gliding upon sodium extraction/re-insertion, particularly from P-type stacking to O-type stacking (e. g. P2 to O2 or OP4), are difficult to be eliminated. In this work, the staircase-like electrochemical profiles induced by PO transition are effectively suppressed in P’2-type Na2/3Mn0.8Ti0.2O2 by a solid-solution reaction with continuous structural evolution during the charge/discharge processes. P’2-Na2/3Mn0.8Ti0.2O2 exhibits smooth voltage profiles and delivers a high specific capacity of 153 mA h g−1 within 2.0–4.0 V. Structural characterizations indicate that P’2-Na2/3Mn0.8Ti0.2O2 undergoes a solid-solution reaction with continuous phase transition from P’2 phase to so-called “Z” phase, considered as an intergrowth of P-type stacking and O-type stacking in the layered lattice. The structural evolution mechanism described in this work provides a solution to suppress the staircase-like electrochemical profiles induced by PO transition in P2-type oxide cathode materials and a versatile strategy for boosting the performance of P2-type cathode materials.

Kinematic-chemical Analysis and Time Tagging for the Diagonal Ridge Structure of the Galactic Outer Disk with LAMOST Red-giant Branch Stars

We investigate the kinematic-chemical distribution of red-giant branch stars from the LAMOST survey crossed matched with Gaia DR2 proper motions, and present time tagging for the well-known ridge structures (diagonal distributions for V R in the R, V ϕ plane) in the range of Galactocentric distance R = 8 to 15 kpc. We detect six ridge structures, including five ridges apparent in the radial velocity distribution and three ridges apparent in the vertical velocity, the sensitive time of which to the perturbations are from young population (0–3 Gyr) to old population (9–14 Gyr). Based on an analysis of the evolution of angular momentum distribution, we find that four ridges are relatively stationary, while another is evolving with time, which is confirmed by the difference analysis at different populations and supporting that there might be two kinds of dynamical origins. Furthermore, ridge features are also vividly present in the chemical properties ([Fe/H], [α/Fe]). The comparison between the north and south hemispheres of the Galaxy does show some differences and the ridge features are asymmetrical. Moreover, we find that diagonal ridge structures may affect the shape of the rotation curve, which is manifested as fluctuations and undulations on top of a smooth profile. Finally we speculate that the bar dynamics should be not enough to explain all ridge properties including the break feature in the V Z –L Z plane.

Mesoscale shock structure in particulate composites

Multiscale experiments in heterogeneous materials and the knowledge of their physics under shock compression are limited. This study examines the multiscale shock response of particulate composites comprised of soda-lime glass particles in a PMMA matrix using full-field high speed digital image correlation (DIC) for the first time. Normal plate impact experiments, and complementary numerical simulations, are conducted at stresses ranging from 1.1−3.1GPa to elucidate the mesoscale mechanisms responsible for the distinct shock structure observed in particulate composites. The particle velocity from the macroscopic measurement at continuum scale shows a relatively smooth velocity profile, with shock thickness decreasing with an increase in shock stress, and the composite exhibits strain rate scaling as the second power of the shock stress. In contrast, the mesoscopic response was highly heterogeneous, which led to a rough shock front and the formation of a train of weak shocks traveling at different velocities. Additionally, the normal shock was seen to diffuse the momentum in the transverse direction, affecting the shock rise and the rounding-off observed at the continuum scale measurements. The numerical simulations indicate that the reflections at the interfaces, wave scattering, and interference of these reflected waves are the primary mechanisms for the observed rough shock fronts.

Isogeometric vibration and material optimization of rotating in-plane functionally graded thin-shell blades with variable thickness

A combined formulation is proposed for modeling and material optimization of rotating in-plane functionally graded (IFG) thin-shell blades with variable thickness in this paper. The aim of this work is to study the vibration characteristics and optimize the IFG volume fraction of the rotating blade model. The isogeometric approach (IGA) and metaheuristic algorithms are employed to find the compositional profile of a two-constituent material blade with the goal of maximizing the fundamental frequency. By considering the rotating speed, the vibrational behaviors of the blades are predicted based on the thin-shell theory and IGA. The different effects due to the rotating speed, variable thickness and IFG materials are considered in the governing equations. For the optimization model, the high-order non-uniform rational B-spline (NURBS) can ensure a smooth material composition profile with fewer design variables. The changing of material composition at control points in plane provides more possibilities for blade material design. Several comparison studies are performed to verify the accuracy of current models for dealing with the vibration behaviors and optimizing the material distribution of the thin-shell blades. The correctness of the combined optimization approach is also proved by comparing those optimal results of two algorithms. Through the optimization process, the optimal material distribution of thin-shell blades with uniform and variable thickness is obtained.

Optimal Operation of Distributed Generations in Four-Wire Unbalanced Distribution Systems considering Different Models of Loads

The distribution network is generally unbalanced due to the distribution of consumers in different phases and the different status of their energy consumption. Also, in recent years, the utilization of distributed generations (DGs) has flourished in the distribution network, thanks to their favorable operational and environmental benefits. Therefore, it is predicted that the optimal energy management of DGs in the unbalanced distribution network can improve the operation state of the network. Moreover, the consumer load model is generally a combination of constant impedance, constant current, and constant power conditions. Therefore, the current study focuses on the analysis of the operation of unbalanced four-wire distribution systems containing DGs, in which load models are also involved. An optimization structure is established to make the voltage deviation, power loss, neutral wire loss, and voltage unbalance minimized. Limitations of the problem include those related to the network and those concerning operating indices of the system like voltage magnitude and phase angle of buses, power flow through the distribution lines, constraints of operation of DGs, load model, and unbalanced conditions. The problem formed in this study is nonlinear and is solved using sequential quadratic programming (SQP) by implementing it in the GAMS simulation environment. To test the benefits of the introduced approach, a 13-bus distribution system has been adopted so that the optimal operation of DGs in the distribution network has been able to reduce the unbalanced situation of the network compared to the power flow studies. Also, this scheme has improved the operation of network, where energy losses and voltage drop have been reduced compared to power flow studies, and a smooth voltage profile has been extracted.

On the effect of streamwise and spanwise spacing to height ratios of three-dimensional sinusoidal roughness on turbulent boundary layers

A developing zero pressure gradient (ZPG) turbulent boundary layer (TBL) over different three-dimensional (3D) sinewave roughnesses is investigated experimentally using single hot-wire anemometry. Seven different sinewave profiles are fabricated with the same amplitude and with different wavelengths in the streamwise (sx) and spanwise (sz) directions. The effects of varying sx and sz on turbulence statistics and the drag coefficient (Cf) are assessed. The wall-unit normalized streamwise mean velocity profile is shifted downward compared with the smooth wall profile for all roughnesses. The streamwise spacing to height ratio sx/k has a more significant effect on the roughness function ΔU+ and Cf compared with the spanwise spacing to height ratio sz/k. However, sz/k has a large impact on the streamwise turbulence intensities in the log and outer layer. An excellent collapse is observed among the mean streamwise velocity profiles plotted in defect form in the outer region. However, a lack of similarity between TBLs over different rough surfaces is observed in the outer region for the turbulence intensities profiles. For isotropic 3D sinusoidal roughness (equal streamwise and spanwise spacing to height ratios), the contours of premultiplied streamwise turbulent energy spectrograms show an increase in energy in the outer layer with increasing spacing to height ratios. For anisotropic 3D sinusoidal roughness (unequal streamwise and spanwise spacing to height ratios), the contours of premultiplied streamwise turbulent energy spectrograms show an increase in energy in the outer layer with increasing sz/sx from half to two in this study.

“Capillary” Structures in Transversely Trapped Nonlinear Optical Beams

A mathematical analogy between paraxial optics with two circular polarizations of light in a defocusing Kerr medium with positive dispersion, binary Bose–Einstein condensates of cold atoms in the phase separation regime, and hydrodynamics of two immiscible compressible liquids can help in theoretical search for unknown three-dimensional coherent optical structures. In this work, transversely trapped (by a smooth profile of the refractive index) light beams are considered and new numerical examples are presented, including a “floating drop,” a precessing longitudinal optical vortex with an inhomogeneous profile of filling with the second component, and the combination of a drop and a vortex filament. Filled vortices that are perpendicular to the beam axis and propagate at large distances have also been simulated.

Modeling and Characterization of Surface Discharges in Insulating Material for Spacers: Electrode Shape, Discharge Mode, and Revision of the Creepage Concept.

In the design of MV AC and DC spacers, the predominant factors are surface and interface conditions. Design is generally carried out on specifications and standards which are based on long-term experience and lab testing. However, the diffusion of power electronics with a trend to increase electric field, switching frequency, and rise time to achieve higher power density calls for an innovative, global approach to optimized insulation system design. A new methodology, based on field simulation, discharge modeling, and partial discharge inception measurements, called the three-leg approach, can form the basis to optimize insulation design for any type of supply voltage waveform. This paper focuses on the influence of the type of electrode on the inception and phenomenology of surface discharges and, as a consequence, on the interpretation of the results used for application of the three-leg approach. It is demonstrated that a typical electrode system used for insulating material testing can generate both gas and surface discharges at the triple point, when the electrodes have a smooth profile that is used to avoid corona or flashover. Hence, testing partial discharge may not provide a straightforward indication of the surface discharge inception and, thus, be partially misleading for insulation design. Another takeover is that such analysis must benefit from PD testing tools endowed with analytics able to provide automatic identification of the type of defect generating PD, i.e., internal, surface, and corona, since design and remedy actions can be taken, and adequate insulating materials developed, only knowing the type of source generating PD. Hence, testing partial discharge may not provide a straightforward indication of surface discharge inception and, thus, be partially misleading for insulation design. In addition to the importance of the three-leg approach to favor reliable and optimized design of insulation systems, there is a clear need to have a PD testing tool endowed with analytics. It should preferably be able to provide automatic identification of the type of defect generating PD, i.e., internal, surface, and corona.

Probabilistic load forecasting for the low voltage network: Forecast fusion and daily peaks

Short-term forecasts of energy consumption are invaluable for the operation of energy systems, including low voltage electricity networks. However, network loads are challenging to predict when highly desegregated to small numbers of customers, which may be dominated by individual behaviours rather than the smooth profiles associated with aggregate consumption. Furthermore, distribution networks are challenged almost entirely by peak loads, and tasks such as scheduling storage and/or demand flexibility maybe be driven by predicted peak demand, a feature that is often poorly characterised by general-purpose forecasting methods. Here we propose an approach to predict the timing and level of daily peak demand, and a data fusion procedure for combining conventional and peak forecasts to produce a general-purpose probabilistic forecast with improved performance during peaks. The proposed approach is demonstrated using real smart meter data and a hypothetical low voltage network hierarchy comprising feeders, secondary and primary substations. Fusing state-of-the-art probabilistic load forecasts with peak forecasts is found to improve performance overall, particularly at smart-meter and feeder levels and during peak hours, where improvement in terms of CRPS exceeds 10%.

Design and single-shot fabrication of lensless cameras with arbitrary point spread functions

Lensless cameras are a novel class of computational imaging devices, in which the lenses are replaced with a thin mask to achieve ultra-compact and low-cost hardware. In this paper, we propose a method for high-throughput fabrication of lensless cameras designed with arbitrary point spread functions (PSFs) for various imaging tasks. The workflow of our method includes designing the smooth phase mask profiles for a given PSF pattern and then fabricating the mask in a single shot via the gray-scale lithography technique. Compared to the existing approaches, our combined workflow allows an ultra-fast and cost-effective fabrication of phase masks and is suitable for mass production and commercialization of lensless cameras. We show that our method can be used for a flexible production of custom lensless cameras with various pre-designed PSFs and effectively obtain images of the scene via computational image reconstruction. Finally, we discuss and demonstrate the future directions and the potential applications of our custom lensless cameras, including the deployment of the learned reconstruction networks for fast imaging and fingerprint detection via optical template matching.

Real‐time estimation of the electron temperature profile in DIII‐Dby leveragingneural‐networksurrogate models

AbstractControl of both the magnitude and the shape of tokamak profiles will be necessary to achieve stable, high‐performance plasmas. In order to reject disturbances in real time, feedback‐control algorithms rely on accurate real‐time knowledge of the plasma state. When diagnostics alone are insufficient, because they are limited in number or their measurements are too noisy, observers can be used to combine diagnostic data with a response model to provide a better estimation of different plasma properties. An observer has been developed to estimate the electron temperature profile in real time using both diagnostic data from the Thomson scattering system and a model based on the electron heat transport equation describing the evolution of the electron temperature profile. Neural network surrogate models are leveraged to help improve the overall model prediction while staying within computation time constraints for real‐time use. The observer algorithm is shown in offline tests to produce smooth profiles that are consistent with both the diagnostic data and the electron heat transport equation. When implemented into the real‐time plasma control system, this observer will provide valuable information on the electron temperature profile to many potential feedback‐control applications.

Direct numerical simulations of a microswimmer in a viscoelastic fluid.

This study presents the application of the smoothed profile (SP) method to perform direct numerical simulations for the motion of both passive and active "squirming" particles in Newtonian and viscoelastic fluids. We found that fluid elasticity has a significant impact on both the transient behavior and the steady-state velocity of the particles. Specifically, we observe that the swirling flow generated by the squirmer's surface velocity significantly enhances their swimming speed as the Weissenberg number increases, regardless of the swimming type. Furthermore, we find that pushers outperform pullers in Oldroyd-B fluids, suggesting that the speed of a squirmer depends on the swimmer type. To understand the physical origin of the phenomenon of swirling flow enhancing the swimming speed, we investigate the velocity field and polymer conformation around non-swirling and swirling neutral squirmers in viscoelastic fluids. Our investigation reveals that the velocity field around the neutral swirling squirmers exhibits pusher-like extensional flow characteristics, as well as an asymmetric polymer conformation distribution, which gives rise to this increased propulsion. This is confirmed by the investigation of the force on a fixed squirmer, which revealed that the polymer stress, particularly its diagonal components, plays a critical role in enhancing the swimming speed of swirling squirmers in viscoelastic fluids. Additionally, our results demonstrate that the maximum swimming speeds of swirling squirmers occur at an intermediate value of the fluid viscosity ratio for all swimmer types. These findings have important implications for understanding the behavior of particles and micro-organisms in complex fluids.

Jerk-Continuous Feedrate Optimization Method for NURBS Interpolation

Feedrate scheduling is one of the most critical technologies in CNC machining, requiring a reasonable balance between efficiency and quality. This paper proposes a jerk-continuous feedrate smoothing (JCFS) method to generate a low-vibration and smooth feedrate profile for non-uniform rational B-spline (NURBS) interpolation. Firstly, the segmentation concept is introduced to subdivide the entire trajectory into segments to accommodate curvature changes of the NURBS curve, accelerating the acceleration/deceleration process. Secondly, a length threshold-based curve segment classification method is proposed to overcome the complexity of the traditional acceleration and deceleration algorithms. The curve segments are divided into long, medium, and short types, and the length threshold calculation model is derived. Next, to avoid computational complexity for engineering applications, a model is established for the first time to calculate the actual maximum feedrate for different types of segments. Finally, the horizontal-8-shaped and butterfly-shaped NURBS curves are simulated and analyzed. The simulation results indicate that the machining quality is steadily improved while several key indicators remain within the given tolerances. Compared with the traditional method, the proposed method reduces the computational and interpolation time by 17.2% and 22.8%, respectively, demonstrating the feasibility and effectiveness of the method.

Sensory Features of Partially Hydrogenated Oil and Their Connection with Melting and Crystalline Characteristics.

Although partially hydrogenated oil (PHO) provides foods with outstanding thick tastes and pronounced "creamy" flavor, the high level of artificial trans-fatty acids (TFA; about 30%) limits its usages around the world in the near future. It is necessary to produce trans-free alternatives with similar tastes to PHO. The relationship between sensory attributes and physicochemical characteristics of PHO and four typical specialty fats were therefore analyzed in the present study. PHO exhibited the highest greasiness score (8.19), accompanying by mild creaminess and aftertaste as well as a weak coolness during swallow, which were resulted from the close-packed arrangements of TFA, its cis-counterparts and other long chain fatty acids. None of artificial trans-fats, mainly anhydrous milk fat, cocoa butter, and coconut oil and its fully hydrogenated counterpart, were similar to PHO in terms of these sensory attributes. The unique fatty acid species of PHO and their arrangements contributed to the relatively smooth solid fat content profile and melting-crystallization curve, as well as forming uniform and dense β' crystal-structures (Db=1.80). The Pearson correlation analyses relevelled that long chain fatty acids, e.g., t-C18:1 and C18:1, and melting final temperatures were generally positive correlated with greasiness, creaminess and aftertaste; whereas these indices were negatively correlated with coolness. The melting enthalpy was highly connected with coolness, which reflected the endothermic effectiveness during the melting process of fats in the mouth. These indices screened by correlation analyses that were strongly correlated with sensory attributes could provide references for producing trans-free alternatives.

Robust Adaptive Autonomous Braking Control for Intelligent Electric Vehicles*

This paper presents a two-stage autonomous braking control for electric vehicles with preview information acquired by intelligent transportation systems (ITS) or connected and autonomous vehicle (CAV) technologies. The goal of the control method is firstly to plan a speed profile to comply with a target speed and a target residual distance to an upcoming deceleration event and then to track the planned speed profile by generating tracking-driven braking control torques, which maximally employ regenerative braking. Once deceleration circumstances are perceived, the intelligent reference speed planner computes a smooth speed profile by using a current vehicle speed, a target speed, and a target residual distance. For the braking control to robustly track the planned speed profile, a chattering-free sliding mode controller is designed on the interconnected vehicle and braking system model with braking system uncertainties and external disturbances, and a sufficient condition for the sliding mode gain is introduced. Also, adaptive update laws to compensate for the braking system uncertainties provide feed-forward control efforts for the tracking control inputs. To verify the effectiveness of the proposed control approach, the control methods are applied to a real commercial vehicle and the consecutive driving experiments with various deceleration circumstances illustrate a successful tracking-driven cooperative braking performance.

A Review of Microgrid Energy Management and Control Strategies

Several issues have been reported with the expansion of the electric power grid and the increasing use of intermittent power sources, such as the need for expensive transmission lines and the issue of cascading blackouts, which can adversely affect critical infrastructures. Microgrids (MG) has been widely accepted as a viable solution to improve grid reliability and resiliency, ensuring continuous power supply to loads. However, to ensure the effective operation of the Distributed Energy Resources (DER), Microgrids need to have Energy Management and Control Systems (EMCS). Therefore, considerable research has been conducted to achieve smooth profiles in grid parameters during operation at optimum running cost. This paper aims to provide a review of EMCS techniques that have evolved in recent years. Firstly, the fundamentals of microgrids are discussed for a general overview of the field. Then, a critical literature review is undertaken for the various methods applied for EM optimization in microgrid applications. Multiple factors have been explored in the objective functions throughout this review, including MG daily operational costs, energy storage degradation, revenue through trading with the grid or other parties, and Greenhouse Gas (GHG) emissions. A review of control systems has been conducted next by categorizing them based on the different applications in MGs for stable operation. This paper also focuses on IEEE standards related to MG operation and control to facilitate other researchers to build upon a standardized set of rules and to enhance the interoperability of the diverse EMCS techniques.

Molecular heterogeneity of pediatric choroid plexus carcinomas determines the distinctions in clinical course and prognosis.

Choroid plexus carcinomas (CPCs) are rare aggressive pediatric tumors of the brain with no treatment standards. Genetic profiling of CPCs is often confined to possible association with Li-Fraumeni syndrome, though only about a half of CPCs develop from syndromic predispositions. Whole-chromosome gains and losses typical of CPCs reflect genomic instability of these tumors, but only partially explain the aggressive clinical course. This retrospective study enrolled 25 pediatric patients with CPC, receiving treatment between January 2009 and June 2022. Molecular-genetic testing was performed for 20 cases with available tumor tissue and encompassed mutational status, chromosomal aberrations, and gene expression profiles. We analyzed several factors presumably influencing the outcomes, including molecular profiles and clinical parameters. The median follow-up constituted 5.2 years (absolute range 2.8-12.6 years). All studied CPCs had smooth mutational profiles with the only recurrent event being TP53 variants, either germline or somatic, encountered in 13 cases. Unbalanced whole-chromosome aberrations, notably multiple monosomies, were highly typical. In 7 tumors, chromosome losses were combined with complex genomic rearrangements: segmental gains and losses or signs of chromothripsis. This phenomenon was associated with extremely low 5-year survival: 20.0 ± 17.9% vs 85.7 ± 13.2%; P = .009. Transcriptomically, the cohort split into 2 polar clusters Ped_CPC1 and Ped_CPC2 differing by survival: 31.3 ± 17.8% vs 100%; P = .012. CPCs split into at least 2 molecular subtypes distinguished both genomically and transcriptomically. Clusterization of the tumors into Ped_CPC1 and Ped_CPC2 significantly correlates with survival. The distinction may prove relevant in clinical trials for dedicated and patient-oriented optimization of clinical protocols for these rare tumors.

Multi-objective topology optimization of thermal-mechanical coupling anisotropic structures using the isogeometric analysis approach

A multi-objective topology optimization model of thermal-mechanical coupling anisotropic structures based on the isogeometric analysis (IGA) approach is proposed, and the high order non-uniform rational B-spline (NURBS) is used as the basis functions. The correctness and superiority of the proposed model have been verified by numerical examples, and the present model is higher efficient than the FEM model. The results illustrate that the IGA optimal topological structures also have better performance and smoother profiles since the IGA approach can ensure the consistency of the discrete model with the geometry model in the curved boundaries problem. The effects of filter factors, thermal conductivity factors, Poisson's ratio factors and off-angles on IGA topological structures and their performance are explored, and the appropriate range of filter factors and above anisotropic parameters are recommended as 0.4∼0.8, 0.2∼0.5, 2∼10 and 15°∼45° for the heat dissipation gasket, respectively. The maximum temperature of the anisotropic IGA optimal topological structure under Ht = 0.2, Bt = 2 and θ = 45° is 2.59% lower than that of the isotropic structure.

Explicit Analytic Solutions for the Subsurface Stress Field in Single Plane Contacts of Elastically Similar Truncated Cylinders or Wedges

As has been pointed out recently, a possible solution strategy to the wear–fatigue dilemma in fretting, operating on the level of contact mechanics and profile geometries, can be the introduction of “soft” sharp edges to the contact profiles, for example, by truncating an originally smooth profile. In that regard, analysis of possible mechanical failure of a structure, due to the contact interaction, requires the knowledge of the full subsurface stress state resulting from the contact loading. In the present manuscript, a closed-form exact solution for the subsurface stress state is given for the frictional contact of elastically similar truncated cylinders or wedges, within the framework of the half-plane approximation and a local-global Amontons–Coulomb friction law. Moreover, a fast and robust semi-analytical method, based on the appropriate superposition of solutions for parabolic contact, is proposed for the determination of the subsurface stress fields in frictional plane contacts with more complex profile geometries, and compared with the exact solution. Based on the analytical solution, periodic tangential loading of a truncated cylinder is considered in detail, and important scalar characteristics of the stress state, like the von-Mises equivalent stress, maximum shear stress, and the largest principal stress, are determined. Positive (i.e., tensile) principal stresses only exist in the vicinity of the contact edge, away from the pressure singularity at the edge of the profile, and away from the maxima of the von-Mises equivalent stress, or the maximum shear stress. Therefore, the fretting contact should not be prone to fatigue crack initiation.