APPLYING NUMERICAL METHODS FOR DETERMINING THE SERVICE LIFE OF GEARS

Abstract The service life of gears with a crack in a tooth root can only be determined by numerical methods. An initial fatigue crack is assumed on the tensile side of a tooth root at the site of maximum reference stress and is assumed to commence growth perpendicularly to the surface. An initial assessment can be made using a two dimensional analysis, which is relatively quick and cheap to perform. However, if we wish to take into account the influence of the contact area of load transfer, which can be distributed along the tooth width in different ways, the gear has to be treated by a three dimensional finite element analysis. Crack profile advance is made in stages, each using the strain energy release rate criteria and giving ultimately the stress intensity factor as a function of average crack depth. With known fracture mechanics material characteristics of different gear layers, through which the crack propagates, the service life of a gear is then determined by numerical integration of Paris' equation. A one-sided contact area causes the crack to propagate several times faster than the preferred load distribution across the middle of the tooth.     

Investigation of crack propagation scatter in a gear tooth’s root

This paper describes the problem of determining crack initiation location and its influence on crack propagation in a gear tooth’s root. Three different load positions on the gear tooth’s flank were considered for this investigation of crack initiation and propagation. A special test device was used for the single tooth test. It can be concluded from the measurements that a crack can be initiated at very different locations in a tooth’s root and then propagate along its own paths. A numerical investigation into a crack initiation’s position and its influences on its propagation were carried out within the framework of linear fracture mechanics. The influence of a tooth’s load position, the geometry of the tooth’s root, and the influence of non-parallel load distribution on the tooth’s flank were considered when investigating the crack initiation’s position. Results show that linear fracture mechanics can be used for determining crack propagation, if better initial conditions for crack initiation are considered.     

基于齿根动应力的船用人字齿轮疲劳分析与优化

为了合理预估人字齿轮齿根动应力疲劳寿命,利用考虑啮合刚度激励、啮合冲击激励和齿面摩擦激励的十二自由度人字齿轮弯-扭-轴耦合动力学模型,同时考虑了参与啮合齿对之间的齿根应力关联性,并由此合理有效地计算出了人字齿轮齿根受拉侧的动应力变化趋势。通过对轮齿疲劳寿命机理分析,选取三种常用载荷工况下小轮齿根中点受拉侧动应力作为疲劳寿命计算对象,采用考虑应力幅值和应力均值双参数的雨流计数法,结合Miner线性累积损伤理论对人字齿轮受拉侧齿根弯曲疲劳寿命进行了预估。同时比较了考察啮合轮齿在进入啮合前其它齿对产生的反向压缩应力对疲劳寿命的影响。通过三维修形技术,对人字齿轮齿面进行了以综合载荷工况下齿根动应力疲劳寿命为目标的优化设计,优化结果表明修形后齿根动应力变化趋势平缓,疲劳寿命增加25%。     

Interior fatigue fracture of gear teeth

A new type of gear tooth fatigue failure is presented and analysed. It is initiated in the interior of the tooth and is given the self-explanatory name tooth interior fatigue fracture (TIFF).
A fractographic examination of TIFF is presented. The fracture surface of TIFF has characteristic features that distinguish it from other gear failures.
A hypothesis for TIFF initiation is presented. The crack-initiating stresses in the interior of the gear tooth are: (i) a residual tensile stress due to case hardening; and (ii) alternating stresses due to idler usage of the gear.
A finite element analysis is used to compute the stress state history of engaging gear teeth throughout the load cycle. The critical plane fatigue criterion of McDiarmid is employed. The analysis shows that the risk of fatigue initiation is high in the root of the tooth and in a large region in the interior of the tooth. These findings agree with test results where both modes of failure occurred.     

Rim Thickness Effects on Gear Crack Propagation Life

Analytical and experimental studies were performed to investigate the effect of gear rim thickness on crack propagation life. The FRANC (FRacture ANalysis Code) computer program was used to simulate crack propagation. The FRANC program used principles of linear elastic fracture mechanics, finite element modeling, and a unique re-meshing scheme to determine crack tip stress distributions, estimate stress intensity factors, and model crack propagation. Various fatigue crack growth models were used to estimate crack propagation life based on the calculated stress intensity factors. Experimental tests were performed in a gear fatigue rig to validate predicted crack propagation results. Test gears were installed with special crack propagation gages in the tooth fillet region to measure bending fatigue crack growth. Good correlation between predicted and measured crack growth was achieved when the fatigue crack closure concept was introduced into the analysis. As the gear rim thickness decreased, the compressive cyclic stress in the gear tooth fillet region increased. The retarded crack growth and increased the number of crack propagation cycles to failure. This revised version was published online in August 2006 with corrections to the Cover Date.     

A NEW MODEL FOR THE NUMERICAL DETERMINATION OF PITTING RESISTANCE OF GEAR TEETH FLANKS

Abstract— A numerical model for determining the pitting resistance of gear teeth flanks is presented in this paper. The model considers the material fatigue process leading to pitting, i.e. the conditions required for crack initiation and then simulation of fatigue crack propagation. The theory of dislocation motion on persistent slip bands is used to describe the process of crack initiation, where the microstructure of a material plays a crucial role. The simulation of crack growth takes into account both short crack growth, where the modified Bilby, Cottrell and Swinden model is used for simulation of dislocation motion, and long crack growth, where the theory of linear elastic fracture mechanics is applied. The stress field in the contact area of meshing spur gear teeth and the functional relationship between the stress intensity factor and crack length are determined by the finite element method. For numerical simulations of crack initiation and crack propagation in the contact area of spur gear teeth, an equivalent model of two cylinders is used. On the basis of numerical results, and with consideration of some particular material parameters, the service life of gear teeth flanks is estimated. The developed model is applied to a real spur gear pair, which is also experimentally tested. The comparison of numerical and experimental results shows good agreement and it can be concluded that the developed model is appropriate for determining the pitting resistance of gear teeth flanks.     

Fatigue failure of a star–ratchet gear

We present an analysis of the fatigue failure of an 18 tooth star–ratchet gear (SRG). The subject gear was implemented in the freewheel assembly of a mountain bicycle. After 6 years of service, the gear failed unexpectedly during a typical off-road ride. The unique geometry of SRGs precluded a simple comparison to existing gear lifetimes. Scanning Electron Microscopy (SEM) analysis of the failed gears showed crack initiation at the root of the gear teeth, followed by fatigue crack propagation and eventual chip-out. A biomechanical analysis of pedaling forces, coupled with explicit power data obtained from instrumented rides over the same trails, in conjunction with a Finite Element Analysis (FEA) of the gear, were used to determine stress amplitudes for fatigue calculations. Energy dispersive spectroscopy (EDS) determined the alloy composition of the gear and thus set the strength and fatigue properties of the gear. Basquin’s law, Goodman’s mean stress correction, and Miner’s rule were used to estimate the lifetime, in bike rides, of the gear. Our analysis led to an estimate of 2288 rides, while failure was reported after roughly 312 rides. Given the uncertainties in fatigue life estimation and service use, we find this estimate acceptable.     

Three‐dimensional fatigue crack growth modelling in a helical gear using extended finite element method

In this paper, the fatigue crack growth in helical gear tooth root has been simulated using linear elastic fracture mechanics. The extended finite element method has been used to simulate 3D fatigue crack growth and obtain growth path. Paris equation has been used to calculate the fatigue life of the gear. The modelling time has reduced considerably compared to previous works carried out on 3D crack growth in gears. Some verifications have been carried out to ensure the reliability of the results.     

Analysis of crack propagation during tooth interior fatigue fracture

Tooth interior fatigue fracture is a failure mode that is initiated as a fatigue crack in the interior of the tooth of a gear. TIFF cracks have been observed in case hardened idler gears. A fracture mechanical analysis of a TIFF crack is performed utilising FEA. A 3D TIFF crack is modelled at a position in the tooth that corresponds with an observed crack surface. The different material properties in the case and the core, determined by mechanical testing, are considered, as well as the residual state of stress due to case hardening. Various crack lengths are analysed to estimate crack propagation both into the core and into the case. The following results have been found:

• A TIFF crack initiated slightly under the case layer will propagate into the case layer where it stops.

• The main crack propagation will take place in the core.

• The crack propagation is only a small portion of the total life (order of 10⁵ cycles).

• After reaching the case layer the TIFF crack eventually deflects toward the tooth root and the upper part of the tooth falls off. The crack deflection is due to redistribution of contact loading. Several gear teeth pairs are simultaneously in contact and the cracked tooth is loaded less than the uncracked during this stage of life.

AF1410与300M钢的腐蚀冲击疲劳行为

根据舰载飞机起落的服务条件提出了腐蚀冲击概念和试验方法，考察了两种起落架材料在盐水中的腐蚀冲击疲劳行为，包括冲击疲劳寿命，裂纹萌生与扩展速率。尽管两种材料在空气中的冲击疲劳寿命几乎相等。但300M钢在盐水中的冲击疲劳寿命下降幅度较大。在盐水介质中，氢脆加速300M钢冲击疲劳裂纹的萌生和扩展。局部塑性变形区优先腐蚀促使AF1410钢的裂纹萌生，盐水对AF1410钢的裂纹扩展速率没有影响。     

The Prediction of Fatigue Crack Initiation Life in Spot Welds

Abstract: In this paper, strain‐based fatigue life prediction method has been used to estimate the fatigue crack initiation life of spot‐welded joints of Mild Steel JSC270D and Ultra‐High Strength Steel JSC980Y. To do so, the joints were simulated using three‐dimensional finite‐element (FE) models, and then nonlinear FE analysis was performed to obtain the local stress and strain ranges and finally, the Morrow equation was applied to estimate the crack initiation lives. The results have been compared with those obtained from experimental crack growth morphology. In addition, the difference between fatigue limits for smooth specimens and spot‐welded joints for mentioned materials has been briefly discussed. It has been shown that mean stress values in the Ultra‐High Strength Steel can significantly decrease the fatigue limit of spot‐welded joint because even at very low load level the stresses exceed the yield point at the root of nugget of spot‐welded joint, while the amount of mean stress in the Mild Steel for the same load level is much less than that of Ultra‐High Strength Steel. The comparison between numerical results of fatigue crack initiation lives and experimental data provided good agreement between numerical predictions and crack growth morphology observations. The results also shows that in some cases, depending on the joint type, the life spent in the nucleation phase can be an important part of the final failure lifetime.     

Dynamic simulation of spur gear with tooth root crack propagating along tooth width and crack depth

Gear tooth crack will cause changes in vibration characteristics of gear system, based on which, operating condition of the gear system is always monitored to prevent a presence of serious damage. However, it is also a unsolved puzzle to establish the relationship between tooth crack propagation and vibration features during gear operating process. In this study, an analytical model is proposed to investigate the effect of gear tooth crack on the gear mesh stiffness. Both the tooth crack propagations along tooth width and crack depth are incorporated in this model to simulate gear tooth root crack, especially when it is at very early stage. With this analytical formulation, the mesh stiffness of a spur gear pair with different crack length and depth can be obtained. Afterwards, the effects of gear tooth root crack size on the gear dynamics are simulated and the corresponding changes in statistical indicators – RMS and kurtosis are investigated. The results show that both RMS and kurtosis increase with the growth of tooth crack size for propagation whatever along tooth width and crack length. Frequency spectrum analysis is also carried out to examine the effects of tooth crack. The results show that sidebands caused by the tooth crack are more sensitive than the mesh frequency and its harmonics. The developed analytical model can predict the change of gear mesh stiffness with presence of a gear tooth crack and the corresponding dynamic responses could supply some guidance to the gear condition monitoring and fault diagnosis, especially for the gear tooth crack at early stage.     

Molecular dynamics simulation of crack initiation and propagation in bcc iron under load within spur gear tooth root

Spur gears are widely used in practice, and one of their typical failures is tooth breakage. In general, the tooth breakage occurs at tooth root, and the amount of crack growth during a meshing cycle is in atomistic scale. This work aims at identifying the mechanisms of crack initiation and propagation at tooth root by using molecular dynamics simulation. The results prove that there are phase transition regions and edge dislocations at crack tips. According to the distribution characteristic of the atomic potential, its concentration can be observed obviously by visualization software. In these concentration regions, microvoids come into being and expand gradually, which results in the subcrack initiation. Additionally, the microvoids and subcracks propagate along the high potential direction and then come together to accelerate the crack growth. Through carrying out a comparative simulation, the effects of heavy load at single meshing area on crack initiation and propagation are addressed.     

Failure of an intermediate gearbox of a helicopter

An intermediate gearbox of a helicopter failed resulting in an accident. A systematic failure analysis was conducted to find out the cause of failure. Examination revealed that fatigue fracturing of the driving gear was responsible for the gearbox failure. The teeth of the gear were severely damaged by spalling. Fractographic study revealed multiple fatigue crack initiation at the tooth root regions. It was established that the failure was caused due to improper assembly of the gear. A detailed analysis of the failure is presented in this paper.     

Investigation of small fatigue crack initiation and growth behaviour of nickel base superalloy GH4169

Surface replication method was utilized to monitor the small fatigue crack initiation and growth process of single‐edge‐notch tension specimens fabricated by nickel base superalloy GH4169. Three different stress levels were selected. Results showed that small fatigue cracks of nickel base superalloy GH4169 initiated from grain boundaries or surface inclusions. The small fatigue crack initiation and growth stages took up about 80–90% of the total fatigue life. Multiple major cracks were observed in the notch root, and specimen with more major cracks seemed to have smaller fatigue life under the same test conditions. At the early growth stage, small crack behaviour might be strongly influenced by microstructures; thus, the crack growth rates had high fluctuations. However, the stress level effect on the small fatigue crack growth rates was not distinguishable for the three different stress levels. And no clear differences were found among the crack initiation lives by using replication technique.     

Influence of high speed on crack propagation path in thin rim gears

Thin rim gears need accurate design as cracks nucleated at the tooth root fillet may propagate in a safe way (through the tooth) or in catastrophic way (through the rim). Crack propagation direction is mainly influenced by both wheel geometry parameters and crack initiation point. For specific geometry configurations, crack propagation path may be influenced also by other parameters such as the centrifugal load. For this reason, in this work the effect of the centrifugal load (proportional to wheel speed), related to the bending one, has been investigated. The stress field at the tooth root fillet and near the crack has been considered to evaluate the crack initiation point and to explain the propagation direction. This research activity has been carried out by means of numerical models (traditional 2D and 3D finite elements and extended finite elements (XFEM)). Results show that both crack initiation point and crack propagation path are strongly influenced by the centrifugal load entity; this effect is mainly evident in the uncertainty zone of the backup ratio.     

SMALL CRACK GROWTH AND FATIGUE LIFE PREDICTIONS FOR HIGH-STRENGTH ALUMINIUM ALLOYS: PART I—EXPERIMENTAL AND FRACTURE MECHANICS ANALYSIS

The small crack effect was investigated in two high-strength aluminium alloys: 7075-T6 bare and LC9cs clad alloy. Both experimental and analytical investigations were conducted to study crack initiation and growth of small cracks. In the experimental program, fatigue tests, small crack and large crack tests were conducted under constant amplitude and Mini-TWIST spectrum loading conditions. A pronounced small crack effect was observed in both materials, especially for the negative stress ratios. For all loading conditions, most of the fatigue life of the SENT specimens was shown to be crack propagation from initial material defects or from the cladding layer. In the analysis program, three-dimensional finite element and weight function methods were used to determine stress intensity factors and to develop SIF equations for surface and corner cracks at the notch in the SENT specimens. A plasticity-induced crack-closure model was used to correlate small and large crack data, and to make fatigue life predictions. Predicted crack-growth rates and fatigue lives agreed well with experiments. A total fatigue life prediction method for the aluminium alloys was developed and demonstrated using the crack-closure model.     

Numerical calculation of bending fatigue life of thin-rim spur gears

Mechanical elements subjected to cyclic loading have to be designed against fatigue. The aim of this paper is to examine the bending fatigue life of thin-rim spur gears of truck gearboxes. The gear service life is divided into the initiation phase of the damage accumulation and the crack growth, respectively. The analysis of thin-rim gear fatigue life has been performed using the finite element method and the boundary element method. The continuum mechanics based approach is used for the prediction of the fatigue process initiation phase, where the basic fatigue parameters of the materials are taken into account. The remaining life of gear with an initial crack is evaluated using the linear-elastic fracture mechanics.     

Vergleichende Untersuchungen zur Anrisslebensdauervorhersage gekerbter Proben mit dem Örtlichen Konzept und dem Nennspannungskonzept am Beispiel des Stahls Cm15

Comparative Investigations on Service Life Assessment of Notched Specimens Based on the Local Strain and the Nominal Stress Approach to Fatigue for a Steel SAE 1017 It is still unclear whether the strain based approach to fatigue or the stress based approach to fatigue should be preferred for service life assessment of notched components. In order to clarify the similarities and differences between these concepts stress and strain controlled fatigue experiments have been performed with notched specimens. It has been found, that stress and strain controlled fatigue testing results in the same number of cycles until failure. Essential for this correlation is that the cyclic stable strain amplitude at the notch root is taken for the entry into the strain‐life diagram in both cases. Starting from an elastic‐plastic analysis of the material behaviour at the notch root it is shown, how the strain‐life curve can be converted into a stress‐life curve. Based on that result service‐life is calculated from both approaches mentioned above. The calculation gives nearly the same service‐lives for both cases, but overestimates the measured data. It becomes obvious, that a S‐N curve determined under one‐level loading doesn’t provide a proper basis for service life assessment. While strain or stress‐life curves always contain crack initiation phase as well as crack propagation phase, the fatigue process under irregular loads is mainly governed by crack propagation. As a consequence, the damage per cycle is underestimated for loads near the fatigue limit, if Miner’s rule is used.     

Fatigue crack growth and life prediction of a single interference fitted holed plate

To understand the different aspects of fatigue behaviour of complex structural joints it will be much helpful if the effects of different parameters are studied separately. In this article, to study the isolated effect of interference fit on fatigue life a pined hole specimen is investigated. This specimen is a single‐holed plate with an oversized pin which force fitted to the hole. The investigation was carried out both experimentally and numerically. In the experimental part, interference fitted specimens along with open hole specimens were fatigue tested to study the experimental effect of the interference fit. In the numerical part, three‐dimensional finite element (FE) simulations have been performed in order to obtain the created stresses due to interference fit and subsequent applied longitudinal load at the holed plate. The stress distribution obtained from FE simulation around the hole was used to predict crack initiation life using Smith–Watson–Topper method and fatigue crack growth life using the NASGRO equation with applying the AFGROW computer code. The predicted fatigue life obtained from the numerical methods show a good agreement with the experimental fatigue life.