Hertzian Contact Stress Research Articles

A Vibration Model of Ball Bearings with a Localized Defect Based on the Hertzian Contact Stress Distribution

To study the vibration mechanism of ball bearings with localized defects, a vibration model of a ball bearing based on the Hertzian contact stress distribution is proposed to predict the contact force and vibration response caused by a localized defect. The calculation of the ball‐raceway contact force when the ball passes over the defect is key to establishing a defect vibration model. Hertzian contact theory indicates that the contact area between the ball and the raceway is an elliptical contact surface; therefore, a new approach is used to calculate the ball‐raceway contact force in the defect area based on the stress distribution and the contact area. The relative motion between the inner ring, the outer ring, and the balls is considered in the proposed model, and the Runge‐Kutta algorithm is used to solve the vibration equations. In addition, vibration experiments of a bearing with an outer ring defect under different loads are performed. The numerical signals and experimental signals are compared in the time and frequency domains, and good correspondence between the numerical and experimental results is observed. Comparisons between the traditional model and the proposed model reveal that the proposed model provides more reasonable results.

Hertz Contact Stress of Deep Groove Ball Bearing

Calculation of hertz contact stress between the deep groove ball bearing and inner race using analytical and MESYS tool analysis. The calculation procedure consists of calculation of maximum contact pressure at different loads using analytical formulas and comparison of results with MESYS tool to predict the contact pressure between inner race and ball of single row deep groove ball bearing and the contact between inner race and ball is considered as an elliptical contact.

A chromium(III) oxide-coated steel wire prepared by arc ion plating for use in solid-phase microextraction of aromatic hydrocarbons.

The authors introduce an arc ion plating method for the deposition of chromium oxide (Cr2O3) on a steel wire substrate, and its use as a coating for solid phase microextraction. The coating has a micro- and nano-scaled structure after annealing at 700°C. It is found that Cr2O3 exhibits a good extraction capability for the aromatic hydrocarbons naphthalene, anthracene, fluorene, fluoranthene, and biphenyl. Following desorption by high temperature at 300°C, the analytes were quantified by gas chromatography (GC). The limits of detection are in the range between 20 and 200ng·L-1, and calibration plots are linear within a wide range (0.2 to 400μg·L-1). The coating has excellent mechanical properties, with a hardness is as high as 31.7GPa, and the adhesion strength between coating and substrate reaches 20.1N (corresponding to the critical Hertzian contact stress of 10GPa). This, along with the chemical and thermal stability of the Cr2O3 coating, endows the wire with a long operational life. It was used for at least 100 times without any obvious decline of extraction capability. Graphical abstract An arc ion plating method was introduced for the deposition of chromium oxide (Cr2O3) on a steel wire substrate, and its use as a coating for solid phase microextraction with high mechanical strength, stability, and long operational lifetime.

Effect of off-sized balls on contact stresses in a touchdown bearing

Abstract Contact stresses of a ball bearing type touchdown bearing with off-sized balls are studied. The touchdown bearing model includes descriptions of the stiffness, damping and friction between bearing components. The model gives the contact deformations between the balls and bearing races, contact forces, and Hertzian contact stresses in each ball. The bearing model is used in simulation together with a model of a flexible rotor. The results show that off-sized ball or balls alters the contact stiffness between the balls and bearing race and the localized deformation of race. The contact force and stresses consequently change. The stress values are dependent not only on the dimension of the off-sized ball or balls but also their location.

Study of the effect of sigma phase precipitation on the sliding wear and corrosion behaviour of duplex stainless steel AISI 2205

The effects of σ-phase on the corrosion and unidirectional sliding wear behaviour of wrought duplex stainless steel (DSS) 2205 were investigated. The precipitation of σ-phase was induced by an isothermal aging at 850°C during 5, 15, 30 and 60min. Microstructural characterisation shows that the volume of δ-phase decreases as σ-phase increases. Dry sliding wear tests were carried out in a tribometer (model TRB, Anton Paar, Switzerland) with pin on disc technique at 7 and 15cm/s sliding rate with a load of 1N and an α-Al2O3 ball as counterpart. Results shown an important increase on wear resistance under dry sliding conditions after 5min of heat treatment due to the hardness provided by σ-phase on the DSS compared to base metal (BM). Nonetheless, cracks by Hertzian contact stress due to the difference of plastic deformation of matrix and σ-phase were observed at 15cm/s sliding rate in heat treated samples, induced by stress fatigue during the abrasion wear mechanism on the surface. Also there was observed that δ-phase is prone to dissolution in acidic media as σ-phase volume increases. Thus, results shown an improvement on wear resistance on samples aged for 5min in the starting critical intergranular corrosion susceptibility.

Extended Hertz Theory of Contact Mechanics for Case-Hardened Steels With Implications for Bearing Fatigue Life

The analytical expressions currently available for Hertzian contact stresses are applicable only for homogeneous materials and not for case-hardened bearing steels, which have inhomogeneous microstructure and graded elastic properties in the subsurface region. Therefore, this article attempts to determine subsurface stress fields in ball bearings for graded materials with different ball and raceway geometries in contact. Finite element models were developed to simulate ball-on-raceway elliptical contact and ball-on-plate axisymmetric contact, to study the effects of elastic modulus variation with depth due to case hardening. Ball bearings with low, moderate, and heavy load conditions are considered. The peak contact pressure for case-hardened steel is always more than that of through-hardened steel under identical geometry and loading conditions. Using equivalent contact pressure approach, effective elastic modulus is determined for case-carburized steels, which will enable the use of Hertz equations for different gradations in elastic modulus of raceway material. Nonlinear regression tools are used to predict effective elastic modulus as a weighted sum of surface and core elastic moduli of raceway material and design parameters of ball–raceway contact area. Mesh convergence study and validation of equivalent contact pressure approach are also provided. Implications of subsurface stress variation due to case hardening on bearing fatigue life are discussed.

Tribological study on a novel wear-resistant AlMgB14-Si composite

Abstract AlMgB 14 based composites are promising candidates for use as cutting tool etc. In this respect, tribological behavior is one of the most important performances among all the criteria. In this work, a novel AlMgB 14 -30 wt% Si composite (referred to as BMAS30) was prepared by Spark Plasma Sintering. and its tribological property was investigated under different counterface materials (316 L and Si 3 N 4 ) and loads in ambient environment. Besides, the fracture toughness and the hardness were measured. We found that BMAS30 has a good fracture toughness of 5.90 MPa m 1/2 and a super-high hardness of 27 GPa. In terms of the tribological properties, BMAS30 shows good tribological properties when sliding against 316 L, i.e., a lower friction coefficient of 0.19–0.28 and wear rate of 7.00–8.00 × 10 −6 mm 3 /N m, while it presents a relatively high friction coefficient of 0.35–0.46 and wear rate of 3.80–9.00 × 10 −5 mm 3 /N m when sliding against Si 3 N 4 . In the former case, large numbers of Si grains pull-out pits formed on the worn surface that can store wear debris, which is conducive to lower the friction and wear; the latter produces severe fatigue fracture under high Hertz contact stress and thereby aggravates the friction and wear.

Spall Propagation Characteristics of Refurbished VIM–VAR AISI M50 Angular Contact Bearings

During times of restricted supply, bearing refurbishment offers an attractive avenue to maintain operational readiness. However, bearing operation after fatigue spall initiation on refurbished bearings has not been extensively studied. In this study, spall propagation characteristics were compared between new and refurbished vacuum induction-melted, vacuum arc remelted AISI M50 208-size angular contact bearings. A control group of new AISI M50 bearings was evaluated for spall propagation characteristics as a baseline. Another group of AISI M50 bearings were subjected to an accumulated 11.5 billion stress cycles at a maximum Hertzian contact stress of 1.93 GPa and a temperature of 127 °C followed by Level II refurbishment. The refurbished bearings were evaluated for spall propagation characteristics and compared to the baseline bearings. Spalls were initiated via seeded Rockwell C hardness indents and propagated at a maximum Hertzian contact stress of 2.65 and 2.41 GPa, respectively, on both groups of bearings. The propagation rates of the bearings were measured in real time using an oil debris monitor. Pre- and post-tested bearings were examined for changes in microstructure, residual stress and retained austenite as a function of depth in the circumferential direction. No statistically significant difference in spall propagation characteristics was observed between new and refurbished bearings at the operating conditions and accumulated stress cycles studied here.

Mesostructure optimization in multi-material additive manufacturing: a theoretical perspective

As multi-material additive manufacturing technologies mature, a new opportunity for materials science and engineering emerges between the scale of the microstructure and the scale of an engineering component. Here we explore the problem of “mesostructure optimization,” the computational identification of preferred point-to-point distributions of material structure and properties. We illustrate the opportunity with two simple example problems for 1D and 2D mesostructure optimization, respectively, namely (1) a functionally graded cylinder that is computationally optimized to redistribute the Hertzian contact stress fields and (2) a thin plate made of digital materials computationally designed to simultaneously maximize bending resistance and minimize total weight. The mechanical performance of materials in these two problems is significantly improved as compared to any monolithic-material counterpart, including a topology-optimized monolith in case (2). These results point to new opportunities for multi-objective performance enhancement in materials.

Nonlinear vibration prediction of cylindrical roller bearing rotor system modeling for localized defect at inner race with finite element approach

In this paper, an analytical model is proposed to study the behavior of defective bearing and rotor system. An overhung rotor system and defective roller bearing are considered for the study. Rotor is considered with unbalanced mass and bearing is taken as the cylindrical roller bearing having localized surface defect. To analyze all the system components’ effect at one node, finite element method is used to predict exact system vibrations. Euler–Bernoulli’s beam element is used to discretize the shaft. Gyroscopic effect of overhung rotor is also taken into account and governing equations of motion have been modified according to our system. Hertz contact stress theory is used for every roller–race contact to calculate the overall nonlinear bearing force. Governing differential equation is solved by Newmark-β time integration method. Nonlinear matrix equation, which is generated at each time-step in Newmark’s method, is solved by Broyden’s method. Results for defective bearing are obtained and plotted in the time and frequency domain. Poincare map has been plotted to view the system’s minimum stability time. An experiment has been carried out to validate the proposed analysis work. In this paper, it has been shown how rotor dynamic analysis can be achieved numerically with minimum calculations.

Design Principles for Improved Fatigue Life of High-Strain Pneumatic Artificial Muscles

Abstract The fatigue life of pneumatic artificial muscles (PAMs) is a limitation to the development of reliability-intensive applications of soft robots in fields such as medical robotics, transportation, and industrial manufacturing. This article aims at improving the fatigue life of PAMs by (1) providing design principles for durable PAMs under high strains and (2) demonstrating these design principles by developing a representative optimal extensible pneumatic muscle (EPM) in the context of a soft surgical robot case study. Representative performance requirements are derived from an image-guided surgical robot taken as a case study. An experimental design study over relevant EPM geometries reveals three basic fatigue principles governing the failure of PAMs: fatigue limit, abrasion wear, and Hertz contact stress. Using these principles, a new extensible pneumatic muscle made of a silicone tube and a continuous orthotropic restraining sleeve is designed and characterized in terms of performance and fati...

Optimal design of the gerotor (2-ellipses) for reducing maximum contact stress

The oil pump, which is used as lubricator of engines and auto transmission, supplies working oil to the rotating elements to prevent wear. The gerotor pump is used widely in the automobile industry. When wear occurs due to contact between an inner rotor and an outer rotor, the efficiency of the gerotor pump decreases rapidly, and elastic deformation from the contacts also causes vibration and noise. This paper reports the optimal design of a gerotor with a 2-ellipses combined lobe shape that reduces the maximum contact stress. An automatic program was developed to calculate Hertzian contact stress of the gerotor using the Matlab and the effect of the design parameter on the maximum contact stress was analyzed. In addition, the method of theoretical analysis for obtaining the contact stress was verified by performing the fluid-structural coupled analysis using the commercial software, Ansys, considering both the driving force of the inner rotor and the fluid pressure, which is generated by working oil.

Evaluation of load capacity of gears with an asymmetric tooth profile

An ISO standard tooth profile has a symmetric pressure angle of 20°. However, the load capacity can be increased with respect to bending and contact pressure by increasing the pressure angle on the meshing side of an asymmetric tooth. Accordingly, we analyzed the torque transmission capacity of asymmetric gears with various pressure angles. We calculated the deflection and bending stress of teeth by the finite element method and found the root stress taking into account the load-sharing ratio. Hertzian contact stress was calculated with respect to contact pressure. Normal vector load was converted into a torque, and torque capacity was evaluated when the stress reached the allowable stress for each case. Reduced bending stress because of an increase in tooth thickness and decreased transmission torque because of a reduction in the base circle radius work together to maximize the load capacity for bending at a pressure angle of around 30°. Maximum load capacity with respect to contact pressure is achieved when the pressure angle is made 45° by increasing the radius of the contact surface. Both strength with respect to bending and contact pressure are found, and the torque transmission capacity of the gear is determined by the lower value of the two. For low-strength materials such as flame-hardened steel, damage due to contact pressure is expected for all forms of gears and the greatest torque capacity was at a pressure angle of 45°. In the case of assuming 800 Hv and an inclusion size $$ \sqrt{\boldsymbol{A}}=50 $$ μm for a high-strength material, the greatest torque transmission capacity is obtained at a pressure angle of 30°. In the case of assuming a moderate-strength material such as case-hardened steel, an optimal form exists at which strength with respect to bending and strength with respect to contact pressure are equal.

Stress analysis of a hollow sphere compressed between two flat platens

The theoretical basis for determination of tensile strength and elastic constants by compression of hollow spheres between two flat platens is provided in this paper. The method of solution follows the displacement function approach, and three displacement functions are introduced so that the fifteen governing equations can be uncoupled analytically. The Hertz contact between the out surface of the hollow sphere and the flat loading platens is considered, and Fourier-Legendre expansion technique is applied on the Hertz contact stress in order to determine the unknown constants in the general analytical expression for stress components. An analytical solution for the stress and displacement components within a hollow sphere compressed between two flat platens is obtained. Numerical results of the analytical solution show that the stress distributions within hollow spheres are not uniform, two tensile stress concentration zones are usually developed at the inner surface and near the loading area along the loading axis of the hollow spheres within both thick and thin hollow spheres, but the maximum tensile stress is usually developed at the inner surface, except few cases for relatively thick hollow sphere with a very small Poisson ratio. The maximum tensile stress developed along the axis of loading can be used to estimate the tensile strength of hollow spheres. The curves obtained by the analytical solution for the ratio between the vertical and horizontal strains at the equator of the out surface of the hollow sphere, together with those for nominal stress versus the normalized displacement between the two flat loading platens, may provide optional way to determine Poisson's ratio and Young's modulus of the hollow sphere, respectively.

Prediction of Tribological Limits in Sliding Contacts: Flash Temperature Calculations in Sliding Contacts and Material Behavior

In order to achieve greater efficiency or to meet light weight requirements, components are downsized. This, however, increases the load, e.g., Hertzian or nominal contact pressures and stresses of tribosystems. This load is expressed as pa·v-value, the product of nominal contact pressure and sliding velocity. pa·v-values are an effective tool for design engineers for predicting low wear/high wear transitions. Therefore, in the present work, topographical analysis has been combined with the plasticity of micro-asperities and the flash temperatures to estimate the limits of pa·v diagrams. The central piece of this set of models presented here is the calculations for flash temperatures and contact mechanics of micro-asperities. This central piece is used to predict the performance of materials in high velocity (turbines, machinery) and low velocity (human joint) applications. It is shown that the model combination suggested here is a useful tool for screening and preselecting a candidate and new materials with respect to tribological requirements before engaging in expensive testing.

3D statistical failure analysis of monolithic dental ceramic crowns

For adhesively retained ceramic crown of various types, it has been clinically observed that the most catastrophic failures initiate from the cement interface as a result of radial crack formation as opposed to Hertzian contact stresses originating on the occlusal surface. In this work, a 3D failure prognosis model is developed for interface initiated failures of monolithic ceramic crowns. The surface flaw distribution parameters determined by biaxial flexural tests on ceramic plates and point-to-point variations of multi-axial stress state at the intaglio surface are obtained by finite element stress analysis. They are combined on the basis of fracture mechanics based statistical failure probability model to predict failure probability of a monolithic crown subjected to single-cycle indentation load. The proposed method is verified by prior 2D axisymmetric model and experimental data.Under conditions where the crowns are completely bonded to the tooth substrate, both high flexural stress and high interfacial shear stress are shown to occur in the wall region where the crown thickness is relatively thin while high interfacial normal tensile stress distribution is observed at the margin region. Significant impact of reduced cement modulus on these stress states is shown. While the analyses are limited to single-cycle load-to-failure tests, high interfacial normal tensile stress or high interfacial shear stress may contribute to degradation of the cement bond between ceramic and dentin. In addition, the crown failure probability is shown to be controlled by high flexural stress concentrations over a small area, and the proposed method might be of some value to detect initial crown design errors.

Analyzing Hertzian contact stress developed in a double row spherical roller bearing and its effect on fatigue life

Purpose The purpose of this paper is to describe a method permitting the creation of a realistic model of spherical roller bearing with the aim of determining contact stress and fatigue life based on dynamic loading conditions. The paper also aims to recognize the effect of tolerance values on contact stress and fatigue life. Motion and load transmission in spherical roller bearing occurs within the assembly by elliptical curved contacting surfaces. The stress produced by the transmitted load would be very high because of least contacting area between these surfaces. Design/methodology/approach The paper describes a methodology to determine contact stress using analytically as well as finite element method for spherical roller bearing. The comparison for the both each approach for contact stress at different loading condition is carried out. Prediction of fatigue life based on dynamic loading conditions for bearing is also determined using finite element model. The effect on induced contact stress and fatigue life by varying tolerances on inner race dimensions have been found out. Findings The paper suggests that the maximum stress produces at the start or end of the contacting arc under static loading condition in spherical roller bearing. The analytical and finite element approach is in good agreement. The fatigue life prediction is useful for selecting loading conditions for various applications of double row spherical roller bearing. Tolerance level at inner ring raceway radius is kept high because of manufacturing constrain of complex curvature geometric shape. Research limitations/implications The present approach does not consider dynamic loading conditions for contact stress analysis. Therefore, researchers are encouraged to analyze the effect of wear, lubrication and other tribological aspects on bearing life. Originality/value The paper includes determination of contact stress and prediction of fatigue life for spherical roller bearing using analytical as well as finite element approach. The tolerance values at inner race are identified as per manufacturing constraint based on contact stress and fatigue life.

A Simulation of the Stress Intensity Factors KI and KII Variation in the Hertzian Stresses Field of Gear Teeth

The most outstanding parameter that governs the fatigue crack growth under tensile stresses field is the stress intensity factor, mode I, KI. This is a sufficient parameter to describe the whole stress field at the crack tip. An accurate stress intensity factor KI evolution was worked out taking into account the position of the crack centre depth, and also, the residual stresses that act on the surface of the tooth, tensions that are linearly decreasing with the depth in the contact zone. On the other hand, the parameter that governs the crack fatigue growth in the case of compression stresses field is the stress intensity factor mode II, KII. This paper also presents the KII variation along pitch line with respect to the Hertzian contact stresses, the residual stresses and the crack centre depth of an initial crack in the sub-surface of the pinion tooth, having different inclination angle α. As result of this study, some particular factors favorable to the propagation of the fatigue cracks towards the surface of the gear tooth were identified. The availability of a master curve for a particular material relating fatigue crack growth rate and range of stress intensity factor enables a designer to predict growth rates for any cracked body, and it is not limited to situations similar to those pertaining to the cracked stressed specimen used to generate the original data.

Theoretical and Experimental Stress Analysis of Cam With Simple Harmonic Motion

Cams are considered as one of the most important mechanical components that depends the contact action to do its job and suffer a lot of with drawbacks to be predicted and overcame in the design process. this work aims to investigate the induced cam contact and the maximum shear stress energy or (von misses) stresses during the course of action analytically using Hertz contact stress equation and the principal stress formulations to find the maximum stress value and its position beneath the contacting surfaces. The experimental investigation adopted two dimensions photoelastic technique to analyze cam stresses under a plane polarized light. The problem has been numerically simulated using Ansys software version 15 as FE solver and depending on Lagrange and Penalty contact algorithm. The effect of cam geometry, characterized by some parameters such as follower radius, face width, rise and return angles, and modulus of elasticity on the contact stress is investigated aiming to minimize the induced stresses.
 

Tribological Performance and Coating Characteristics of Sputter-Deposited Ti-Doped MoS2 in Rolling and Sliding Contact

In this work, the tribological behavior of Ti-containing MoS2 film is studied under sliding and rolling contacts. Ti-MoS2 films deposited on AISI 52100 and M50 specimens contain ∼16 at% Ti are slightly substoichiometric in sulfur and have an amorphous structure. Experiments show that the friction and wear performance of Ti-MoS2 films is influenced by humidity and temperature.For the first time, the rolling contact performance of Ti-MoS2 films was evaluated using a thrust bearing tribometer and ball-on-rod tribometers at ∼1 GPa Hertzian contact stress. Rolling contact measurements performed in humid air indicate that the useful life of Ti-MoS2 as a solid lubricant depends upon the amount of coating material retained in the contacts. Results of the rolling contact experiments performed in a vacuum indicate that Ti-MoS2 film on balls may function extremely well as a solid lubricant for ball bearings operating in vacuum.