Low Contact Ratio Research Articles

Analytical model of meshing stiffness, load sharing, and transmission error for internal spur gears with profile modification

The influence of the teeth geometry on the load distribution, transmission error, and meshing stiffness of internal spur gears is very similar to that of external gears. Therefore, the formulation of the models for strength and dynamic calculations for internal and external gears is also similar. However, the internal gears have usually greater values of the contact ratio, frequently above 1.8, even 1.9. In these cases, the lengthening of the contact interval due to the elastic deflections of the teeth may result in an effective contact ratio greater than 2, with the subsequent changes in the load sharing and transmission error. These changes are not properly described by the theoretical models for high contact ratio spur gears. In this paper, a model of load sharing, quasi-static transmission error, and time-varying meshing stiffness for internal spur gears with profile modification is presented, which has been applied to low contact ratio internal gears which, due to the transmitted load, become high contact ratio internal gears. The influence of the variations of the transmitted load on the load capacity is also studied.

Crack detection of a modified high contact ratio gear

For the purpose of vibration-based fault detection and to avoid gear failure consequences it is necessary to investigate the possibility of early fault detection. A reduction in the gear time-varying mesh stiffness (TVMS) due to the existence of a crack can be used to detect and evaluate tooth damage. The reduction in TVMS has an impact on the system’s dynamic response signal, and then the used fault detection indicators are affected. The current article sheds light on the fault detection possibility of high contact ratio (HCR) gears as compared to those standard ones of low contact ratio (LCR). The influence of modifying the gear tooth addendum dimension for obtaining a high contact ratio on the possibility of early fault detection is studied. Twenty different crack cases are examined by using both the LCR and HCR design cases. For every crack case, the mesh stiffness is evaluated and introduced to the used gear dynamic model for obtaining the system’s dynamic response. The TVMS evaluation is presented for both the LCR and HCR gear cases. Four fault detection indicators, namely the RMS, kurtosis, peak value and crest factor are applied as different measures to enhance the research outcome. The results show that the studied indicators will have less increase when the crack size increases in the HCR design than in the LCR one. Therefore, the HCR design will be less sensitive or more difficult to detect from the fault detection perspective. Lowering the amount of signal noise is applied to the HCR case to examine the impact on enhancing the obtained indicators. The four applied indicators show different behaviours with reducing noise.

Tribological Aspects Affecting Surface durability of Tooth-Sum Altered Spur Gears: A Load Sharing Approach

The performance of tooth-sum altered (ATS) gears is determined by the factors influenced by their profile geometry. This study aims to explore the influence of gear geometry modification on tribological aspects that affect surface wear in ATS spur gears. A computer code is developed to simulate surface wear numerically, using Archard's wear model, Greenwood-Williamson micro-asperity contact model, and Johnson’s load-sharing approach. The outcomes of the study indicate that the low contact ratio ATS gears promote the formation of thick oil film owing to reduced specific sliding and increased speed. However, high contact ratio ATS gears create unfavorable operating conditions resulting in extreme boundary lubrication. The effectiveness of lubricant oil film in reducing wear in ATS gears is associated with its modified profile, sliding velocities, load bearing, operating temperature, and oil viscosity.

Analytical study for low ground contact ratio of buildings due to the basemat uplift using a three-dimensional finite element model

The basemat uplift is a phenomenon that the bottom of the basemat of a building partially rises from the ground due to overturning moments and vertical motions during earthquake. The degree of the basemat uplift can be indicated using a ground contact ratio which is defined as a ratio of the contact area of the bottom of the basemat to its entire area. The problem is that, at a large earthquake, the basemat uplift becomes large, so-called low ground contact ratio state, and the basemat falls resultant to recontact between the basemat and the ground, and large acceleration response occur on the floor of the building. It is a crucial aspect in the seismic evaluation of a nuclear facility building. It affects not only structural integrity of the building but also the response of the equipment installed in the building. However, the building behavior under the low ground contact ratio state lacks sufficient study. In this study, we conducted seismic response analyses for the building using a three-dimensional finite element model and simulated shaking table experiments focused on the basemat uplift and confirmed the validity of this analysis method. Since the basemat uplift is a strong non-linear phenomenon, we conducted computer simulations under the same analysis conditions with three different analysis codes, namely E-FrontISTR, FINAS/STAR, and TDAPIII, and compared the results. We investigated the influence on the structural response caused by the difference of the adhesive force of the basemat and the bearing ground at the low ground contact ratio state. In addition, we studied the influence of numerical parameters to the structural response through sensitivity analyses. This paper reports the analysis results and the insights obtained from our investigations.

Effects of Flexural Rigidity on Soft Actuators via Adhering to Large Cylinders

This study proposes a soft pneumatic actuator with adhesion (SPAA) consisting of a top fluidic-driven elastic actuator and four bottom adhesive pads for adhering to large cylinders. Finite element models were developed to investigate the bending properties under positive air pressure and the effect of “rib” height on the flexural rigidity of the SPAA. A synchronous testing platform for the adhesive contact state and mechanics was developed, and the bending curvature and flexural rigidity of the SPAA were experimentally measured relative to the pressure and “rib” height, respectively, including the adhesion performance of the SPAA with different rigidities on large cylinders. The obtained results indicate that the SPAA can continuously bend with controllable curvature under positive air pressure and can actively envelop a wide range of cylinders of different curvatures. The increase in the “rib” height from 4 to 8 mm increases the flexural rigidity of the SPAA by approximately 230%, contributing to an average increase of 54% in the adhesion performance of the SPAA adhering to large cylinders. The adhesion performance increases more significantly with an increase in the flexural rigidity at a smaller peeling angle. SPAA has a better adhesion performance on large cylinders than most existing soft adhesive actuators, implying that is more stable and less affected by the curvature of cylinders. To address the low contact ratio of the SPAA during adhesion, the optimization designs of the rigid–flexible coupling hierarchical and differentiated AP structures were proposed to increase the contact ratio to more than 80% in the simulation. In conclusion, this study improved the adhesion performance of soft adhesive actuators on large cylinders and extended the application scope of adhesion technology. SPAA is a basic adhesive unit with a universal structure and large aspect ratio similar to that of the human finger. According to working conditions requirements, SPAAs can be assembled to a multi-finger flexible adhesive gripper with excellent maneuverability.

An investigation of nonlinear flow behaviour along rough-walled fractures considering the effects of fractal dimensions and contact areas

This study focuses on the investigation of the nonlinear flow behaviour along rough-walled fractures with contact areas in the framework of the lattice Boltzmann method. The numerical results indicate that the fracture surface roughness controls both the advective aperture and the eddy aperture in the analysis of pressure, velocity, and vorticity fields. The eddy volume fraction increases from 28% to 40% with the increase in the fractal dimension from D=2.0 to D=2.4. The parametric analysis based on Forchheimer’s equation further demonstrates that the enhancement in the vorticity field contributes to the strengthened nonlinear flow behaviour. The two competing roles of fractal dimension and contact area greatly impact the non-Darcy flow patterns. The fractal dimension is the main factor affecting flow nonlinearity at a low contact ratio, while the contact area dominates the nonlinear flow behaviour at a high contact ratio. The increase in the contact ratio strengthens the boundary effect, and thus greatly increases the volume of the eddy region and enhances the nonlinearity of the flow field. These observations provide a promising pathway to analyse the flow behaviour along the fracture based on vorticity analysis, and enlighten the understanding of the competing mechanism between fractal dimensions and contact areas for the influence of nonlinear flow behaviours.

A dynamic load distribution model of planetary gear sets

This study presents, a three-dimensional dynamic load distribution model of planetary gear sets, which can capture the influence of several component and system level design variations. A numerical integration scheme is used in conjunction with an iterative elastic contact algorithm to solve the multibody dynamics problem. This approach inherently captures the mesh stiffness fluctuation, without the need for an empirical mesh stiffness formulation or transmission error excitation of the system. The unique dynamic response of the parametrically excited system under various gear mesh phasing conditions is studied using the proposed model. Influence of gear tooth modifications and carrier pin position errors on the dynamics of the gearset is explored, also a brief study to demonstrate the model's capability to capture contact non-linearities in the case of low contact ratio gear sets is included. Discussed results not only illustrate the validity and potential of the proposed dynamic model but also reinforce the need for such computationally efficient models for design purposes.

Atomistic scale fracture behavior of the bcc iron with {1 1 0} crack under dynamic rectangular loading rate

Dynamic rectangular loading rate is a typical characteristic of the load on tooth root with gear transmission. Upload ratio and contact ratio are its key engineering factors. This work investigates the atomistic scale fracture behavior of bcc iron with {1 1 0} crack under dynamic rectangular loading rate through molecular dynamics simulation, where four crack fronts are considered to illustrate the orientation effect induced by the anisotropy of bcc iron at atomistic scale. The behavior of engineering stress-strain and atomistic scale behavior of crack propagation under various upload ratios and contact ratios are identified to clarify the effect of dynamic load. The {1 1 0} cracks with fronts of [1¯1¯2], [101¯], and [1¯1¯1¯] can propagate not only at model plastic deformation stage under low upload ratio or high contact ratio but also at model elastic deformation stage under high upload ratio or low contact ratio. However, the {1 1 0} crack with front of [01¯0] can propagate at model elastic deformation stage whatever the upload ratio and contact ratio are. According to the critical stress intensity factor for first partial dislocation nucleation and the variation of initiation stress with upload ratio and contact ratio, [01¯0] is proved to be the most dangerous crack front. Considering orientation and key engineering factor effects, this work can well satisfy engineering practice to analyze the atomistic scale mechanism of gear tooth breakage.

Analysis of Spur Gear with Tooth Circular Discontinuity for Weight Reduction

The design of gears and gear drives are becoming challenging in automobile applications in which low contact ratio gears are required to transmit higher loads and be lighter in weight. There are various methods available for optimizing the design. Some researches have been taken place in the areas of gear tooth profile modifications for providing internal stress relief and weight reduction. In most cases, the maximum loads are assumed to be acting on the tip of the tooth. In this study, investigations are carried out on the effect of circular discontinuity on the weight reduction without affecting the root stress. The maximum tangential load is applied on the highest point of single tooth contact. The results show that with the modified gear tooth having a combination of two circular discontinuities of size 0.4 times the module not only provides weight reduction but also gives stress relief at the root.

Comparison of High Contact Ratio And Low Contact Ratio Gears

Comparison of High Contact Ratio And Low Contact Ratio Gears

Low Friction and High Solid-Solid Contact Ratio—A Contradiction for Laser-Patterned Surfaces?

Recording of Stribeck-like curves is a common way to study the effect of laser-patterned surfaces on the frictional efficiency. However, solely relying on the coefficient of friction when identifying the lubrication regime and the underlying working principles can be misleading. Consequently, a ball-on-disc tribometer was combined with an electrical resistivity circuit to simultaneously measure Stribeck-like curves and solid-solid contact ratios for polished and laser-patterned samples. Line-like surface patterns with different periodicities were produced by direct laser interference patterning on steel substrates (AISI304). The reference shows a Stribeck-like behavior well correlating with the contact ratios. The behavior deviates for high sliding velocities (high contact ratios) due to a loss of lubricant induced by centrifugal forces pulling the lubricant out of the contact zone. In contrast, the solid–solid contact ratio of the laser-patterned samples is around 80% for all sliding velocities. Those values can be explained by higher contact pressures and the structural depth induced by the surface topography which make a full separation of the surfaces unlikely. Despite those high values for the contact ratio, laser-patterning significantly reduces friction, which can be traced back to a reduced real contact area and the ability to store oil in the contact zone.

Crack behavior in a high contact ratio spur gear tooth and its effect on mesh stiffness

The efficiency of high contact ratio (HCR) gearing can be achieved by proper selection of gear geometry for increased load capacity and smoother operation despite of their high sliding velocities. The prediction of variation in mesh stiffness of HCR gearing is critical as the average number of teeth being in contact is high at a given time as compared to conventional low contact ratio (LCR) gearing. In this paper, linear elastic fracture mechanics (LEFM) based finite element method is used to perform the crack propagation path studies of HCR spur gear having tooth root crack for two gear parameters viz. backup ratio and pressure angle. A total potential energy model has been adopted to analytically estimate the mesh stiffness variation. The results depict the mesh stiffness reduction in the presence of the crack. The percentage change in mesh stiffness with increasing crack length is an important parameter in fault diagnosis of geared transmission. Higher the percentage change in mesh stiffness, easier to detect the fault. Two gear parameters viz. back-up ratio and pressure angle has been studied and the effect of crack length on mesh stiffness have been outlined. With the increase of deterioration level gears having lower back-up ratio fault can be detected at an early stage, similarly, chances for early fault detection is more for gears having higher pressure angle.

Derivation of Optimum Profile Modifications in Narrow-Faced Spur and Helical Gears Using a Perturbation Method

A perturbation method is presented which makes it possible to obtain approximate closed-form expressions for profile relief that minimize the fluctuations of quasi-static transmission errors under load. A number of results are displayed which prove the theoretical effectiveness of the proposed solutions for low-contact ratio (LCR) and high-contact ratio (HCR) spur and helical gears. It is also shown that the corresponding relief performance is not significantly downgraded by center-distance (CD) variations. Finally, a number of practical considerations are brought up and commented.

Simulation of crack propagation in spur gear tooth for different gear parameter and its influence on mesh stiffness

Most of the gear dynamic model relies on the analytical measurement of time varying gear mesh stiffness in the presence of a tooth fault. The variation in gear mesh stiffness reflects the severity of tooth damage. This paper proposes a cumulative reduction index (CRI) which uses a variable crack intersection angle to study the effect of different gear parameters on total time varying mesh stiffness. A linear elastic fracture mechanics based two dimensional FRANC (FRacture ANalysis Code) finite element computer program is used to simulate the crack propagation in a single tooth of spur gear at root level. A total potential energy model and variable crack intersection angle approach is adopted to calculate the percentage change in total mesh stiffness using simulated straight line and predicted crack trajectory information. A low contact ratio spur gear pair has been simulated and the effect of crack path on mesh stiffness has been studied under different gear parameters like pressure angle, fillet radius and backup ratio. The percentage reduction of total mesh stiffness for the simulated straight line and predicted crack path is quantified by CRI. The CRI helps in comparing the percentage variation in mesh stiffness for consecutive crack. From the result obtained, it is observed that the proposed method is able to reflect the effect of different gear parameters with increased deterioration level on total gear mesh stiffness values.

Mesh stiffness calculation of a spur gear pair with tooth profile modification and tooth root crack

Gear tooth deviations are the main excitation sources for gear dynamic responses. However, it is seldom considered in the analytical mesh stiffness model except for some computational analysis like finite element method. To make up this gap, a general analytical mesh stiffness model is proposed in this paper to include the effect of the gear tooth errors. This proposed model establishes the relationship between the gear tooth errors and the total mesh stiffness, load sharing among different tooth pairs in mesh and loaded static transmission errors (LSTE). It is suitable for not only the gear pairs with low contact ratio (LCR), but also the gear pairs with high contact ratio (HCR). Two spur gear pair models, namely one with LCR between 1 and 2 and the other one with HCR between 2 and 3, are used to demonstrate the effectiveness of the proposed mesh stiffness model. Influences of the TPM, applied torque and gear tooth root crack on the mesh stiffness, load sharing and loaded static transmission errors are also investigated.

Some Analytical Results on Transmission Errors in Narrow-Faced Spur and Helical Gears: Influence of Profile Modifications

Some theoretical developments are presented, which lead to approximate analytical results on quasi-static transmission errors valid for low and high contact ratio spur and helical gears. Based on a multidegree-of-freedom gear model, a unique scalar equation for transmission error is established. The role of profile relief is analyzed by using Fourier series and it is shown that transmission error fluctuations depend on a very limited number of parameters representative of gear geometry and profile relief definition. An original direct solution to the optimum relief minimizing transmission error fluctuations is presented, which is believed to be helpful for designers. The analytical results compare well with the numerical results provided by a variety of models and it is demonstrated that some general laws of evolution for transmission error fluctuations versus profile modifications can be established for spur and helical gears.

Width modification for gears with low contact ratio

In this study, teeth width modification on spur gears with low contact ratios is introduced. The modification essentially relies on maintaining constant Hertz surface pressure along single and double teeth profiles. With this aim, adjustments were made in tooth width along the single mesh area. It is intended that, with this modification, negative effects caused by high Hertz pressures along the single mesh area will either be minimized or completely removed.

Some Guidelines on the Use of Profile Reliefs of High Contact Ratio Spur Gears

With reference to the previous articles of the author(s) based on the systematic approach to the profile relief design of both low contact ratio and high contact ratio spur gears, some guidelines are drawn for the use of short and double reliefs of the HCRG profile design under the effect of realistic parameters like variable tooth stiffness and manufacturing errors regarding the profile relief values. Root stress of the double relief can be minimized by optimizing the maximum tooth load value, hence the tooth load diagram with constant and variable tooth stiffness. Interaction of the manufacturing errors with the relief parameters is put into a form of analytical relation between manufacturing grades, hence the manufacturing tolerance and some gear parameters such as contact ratio, base pitch, and the load at which the smooth transmission error curve is required. In case of double relief, design load plays a major role in its selection. A relation and also a tabular form of guideline are given: for the required manufacturing grade if the contact ratio and other gear parameters are known and for the minimum contact ratio required if the manufacturing grade of the gears are known for either of the two cases short and double reliefs.

A systematic approach to profile relief design of low and high contact ratio spur gears

This is the first of a series of papers on the design of the tooth profile relief of both low and high contact ratio spur gears and its effects on transmission error (TE) and tooth loads. A systematic approach to profile relief design is introduced. The process of profile relief analysis is based on a number of simplifying assumptions to ease the understanding of the relief design. Useful relief cases determined during the simplified analysis are further investigated, with realistic parameter values such as variable stiffness, manufacturing tolerances and so on, in other papers of the series. Experimental validation of the cases proven to be good are also given in forthcoming papers. In the present paper, the systematic approach is applied to low contact ratio (LCR) spur gears first with some design regions and to high contact ratio (HCR) spur gears with some new and promising design regions and rules. Several smooth transmission error curves at different loads are shown to be possible for the relief designed, hence allowing a range of loads with uniform motion transfer. The advantages of HCRG over LCRG in terms of smooth TE curves and tooth load values are noted.

A new type of profile relief for high contact ratio spur gears

The second of a series of papers on the systematic approach to the profile relief design of both low contact ratio (LCR) and high contact ratio (HCR) spur gears and its effects on transmission error (TE) and tooth loads. Basics of the approach are given in the previous paper of the series. Application of the approach to HCR spur gears has produced two promising design cases, namely the short and long reliefs, with advantages and disadvantages. In this paper a new relief type, double relief, is introduced by making use of the two promising cases. Advantages of the two are brought together, with some compromise. The overall performance of the new relief type, however, is superior to the others in terms of peak-to-peak TE value, maximum tooth load value, rate of tooth loading and number of smooth TE curves, and hence in terms of the range of transmitted load with nearly uniform motion transfer. A procedure for compensation for adjacent pitch error (PE) with smooth TE curve at off-design load is also included.