Time-varying Contact Research Articles

Contact stiffness and dynamic behavior caused by surface defects of spiral bevel gear in mixed lubrication

Geometric defects may occur on the gear surfaces during manufacturing and service, such as pitting. However, the defect is rarely incorporated into gear mesh analysis. In this study, a transient mixed lubrication model and a two-dof-torsional dynamic model are proposed for spiral bevel gears, wherein the time-varying contact parameters and the defect on the meshing path are all considered. First, the time-varying contact parameters are calculated by Tooth-Contact-Analysis (TCA). After that, by simulating the gear meshing process, an engineering rough surface with a circular defect moves through the contact region. The transient mixed lubrication performances with the consideration of surface defect are obtained including the film thickness and pressure along the meshing path. Based on this, the normal stiffness of single tooth from engaged-in to engaged-out can be calculated through the normal contact deformation. Finally, the dynamic responses of gears are analyzed according to the normal stiffness through a two-dof-torsional model. Numerical results show that the surface defect has a significant effect on the meshing process of spiral bevel gears and the dynamic response also changes with the size, depth and location of the defect. This provides a theoretical reference for identifying early failures caused by defects based on dynamic signals.

Theoretical investigation of the improved nonlinear dynamic model for star gearing system in GTF gearbox based on dynamic meshing parameters

In the previous study, the gear meshing parameters are usually treated as constants, however, the translational motion, caused by bearing deformation, will lead to the variation of these parameters. In order to investigate the dynamic response of the GTF (Geared Turbofan engine) gearbox that more close to the actual working condition, the improved nonlinear dynamic model of GTF gearbox is proposed in this paper, in which the structure-based time-varying gear meshing parameters and multiple nonlinear clearances are fully considered, including time-varying mesh stiffness, pressure angles, contact ratios, position angles, and dynamic gear backlashes. Afterwards, the improved model is compared with the previous model, and their dynamic characteristics are demonstrated in time, and frequency domain. The results indicate that the nonlinear dynamic behaviors could be altered along with tooth impact state due to the time-varying meshing parameters, which could provide better understanding of the structural design and vibration control of the system.

Quantifying the dynamics of pig movements improves targeted disease surveillance and control plans.

Tracking animal movements over time may fundamentally determine the success of disease control interventions. In commercial pig production growth stages determine animal transportation schedule, thus it generates time-varying contact networks showed to influence the dynamics of disease spread. In this study, we reconstructed pig networks of one Brazilian state from 2017 to 2018, comprising 351,519 movements and 48 million transported pigs. The static networks view did not capture time-respecting movement pathways. For this reason, we propose a time-dependent network approach. A susceptible-infected model was used to spread an epidemic over the pig network globally through the temporal between-farm networks, and locally by a stochastic model to account for within-farm dynamics. We propagated disease to calculate the cumulative contacts as a proxy of epidemic sizes and evaluate the impact of network-based disease control strategies in the absence of other intervention alternatives. The results show that targeting 1,000 farms ranked by degree would be sufficient and feasible to diminish disease spread considerably. Our modelling results indicated that independently from where initial infections were seeded (i.e. independent, commercial farms), the epidemic sizes and the number of farms needed to be targeted to effectively control disease spread were quite similar; indeed, this finding can be explained by the presence of contact among all pig operation types The proposed strategy limited the transmission the total number of secondarily infected farms to 29, over two simulated years. The identified 1,000 farms would benefit from enhanced biosecurity plans and improved targeted surveillance. Overall, the modelling framework provides a parsimonious solution for targeted disease surveillance when temporal movement data are available.

Sparse HP filter: Finding kinks in the COVID-19 contact rate

In this paper, we estimate the time-varying COVID-19 contact rate of a Susceptible–Infected–Recovered (SIR) model. Our measurement of the contact rate is constructed using data on actively infected, recovered and deceased cases. We propose a new trend filtering method that is a variant of the Hodrick–Prescott (HP) filter, constrained by the number of possible kinks. We term it the sparse HP filter and apply it to daily data from five countries: Canada, China, South Korea, the UK and the US. Our new method yields the kinks that are well aligned with actual events in each country. We find that the sparse HP filter provides a fewer kinks than the ℓ1 trend filter, while both methods fitting data equally well. Theoretically, we establish risk consistency of both the sparse HP and ℓ1 trend filters. Ultimately, we propose to use time-varying contact growth rates to document and monitor outbreaks of COVID-19.

Visualizing the invisible: The effect of asymptomatic transmission on the outbreak dynamics of COVID-19

Understanding the outbreak dynamics of the COVID-19 pandemic has important implications for successful containment and mitigation strategies. Recent studies suggest that the population prevalence of SARS-CoV-2 antibodies, a proxy for the number of asymptomatic cases, could be an order of magnitude larger than expected from the number of reported symptomatic cases. Knowing the precise prevalence and contagiousness of asymptomatic transmission is critical to estimate the overall dimension and pandemic potential of COVID-19. However, at this stage, the effect of the asymptomatic population, its size, and its outbreak dynamics remain largely unknown. Here we use reported symptomatic case data in conjunction with antibody seroprevalence studies, a mathematical epidemiology model, and a Bayesian framework to infer the epidemiological characteristics of COVID-19. Our model computes, in real time, the time-varying contact rate of the outbreak, and projects the temporal evolution and credible intervals of the effective reproduction number and the symptomatic, asymptomatic, and recovered populations. Our study quantifies the sensitivity of the outbreak dynamics of COVID-19 to three parameters: the effective reproduction number, the ratio between the symptomatic and asymptomatic populations, and the infectious periods of both groups. For nine distinct locations, our model estimates the fraction of the population that has been infected and recovered by Jun 15, 2020 to 24.15% (95% CI: 20.48%-28.14%) for Heinsberg (NRW, Germany), 2.40% (95% CI: 2.09%-2.76%) for Ada County (ID, USA), 46.19% (95% CI: 45.81%-46.60%) for New York City (NY, USA), 11.26% (95% CI: 7.21%-16.03%) for Santa Clara County (CA, USA), 3.09% (95% CI: 2.27%-4.03%) for Denmark, 12.35% (95% CI: 10.03%-15.18%) for Geneva Canton (Switzerland), 5.24% (95% CI: 4.84%-5.70%) for the Netherlands, 1.53% (95% CI: 0.76%-2.62%) for Rio Grande do Sul (Brazil), and 5.32% (95% CI: 4.77%-5.93%) for Belgium. Our method traces the initial outbreak date in Santa Clara County back to January 20, 2020 (95% CI: December 29, 2019–February 13, 2020). Our results could significantly change our understanding and management of the COVID-19 pandemic: A large asymptomatic population will make isolation, containment, and tracing of individual cases challenging. Instead, managing community transmission through increasing population awareness, promoting physical distancing, and encouraging behavioral changes could become more relevant.

Nonlinear vibration response analysis of a double-row self-aligning ball bearing due to surface imperfections

This work attempts to study the vibration response of a double-row self-aligning ball bearing due to surface and localized imperfections. For the contact deformation at the ball–race interactions, the Hertzian load–deflection relation is used for the evaluation of time-varying contact stiffness. The elastohydrodynamic theory is applied to find out the central film thickness. For both the inner and outer race waviness cases, the system response is observed as periodic (with vibrations of high amplitude) at [Formula: see text], i.e. multiples of Nb and its vicinity, but gradually turns to quasi-periodic as the value of waviness order reach some intermediate value. In the case of a localized defect, the double impulse phenomenon marks the entry and exit events of the rolling element in and out of the rectangular spall. Hence, this analysis can be used as a diagnostic tool with system dynamic characteristics for distributed and localized defect identification.

Modeling behavioral change and COVID-19 containment in Mexico: A trade-off between lockdown and compliance

Sanitary Emergency Measures (SEM) were implemented in Mexico on March 30th, 2020 requiring the suspension of non-essential activities. This action followed a Healthy Distance Sanitary action on March 23rd, 2020. The aim of both measures was to reduce community transmission of COVID-19 in Mexico by lowering the effective contact rate. Using a modification of the Kermack–McKendrick SEIR model we explore the effect of behavioral changes required to lower community transmission by introducing a time-varying contact rate, and the consequences of disease spread in a population subject to suspension of non-essential activities. Our study shows that there exists a trade-off between the proportion of the population under SEM and the average time an individual is committed to all the behavioral changes needed to achieve an effective social distancing. This trade-off generates an optimum value for the proportion of the population under strict mitigation measures, significantly below 1 in some cases, that minimizes maximum COVID-19 incidence. We study the population-level impact of three key factors: the implementation of behavior change control measures, the time horizon necessary to reduce the effective contact rate and the proportion of people under SEM in combating COVID-19. Our model is fitted to the available data. The initial phase of the epidemic, from February 17th to March 23rd, 2020, is used to estimate the contact rates, infectious periods and mortality rate using both confirmed cases (by date of symptoms initiation), and daily mortality. Data on deaths after March 23rd, 2020 is used to estimate the mortality rate after the mitigation measures are implemented. Our simulations indicate that the most likely dates for maximum incidence are between late May and early June, 2020 under a scenario of high SEM compliance and low SEM abandonment rate.

The homotopy analysis for the nonlinear dynamics of planetary gear trains

In this paper, the nonlinear oscillation of planetary gear trains is investigated by the homotopy analysis method. The nonlinearity of planetary gear trains due to the periodically time-varying mesh stiffness and contact loss are included. In contrast to the perturbation analysis, the homotopy analysis method is independent of the contact loss ratio, and then can be applied to both small and large contact loss ratios. In this article, firstly the closed-form approximations for the primary resonance, sub-harmonic resonance, and super-harmonic resonance are obtained by homotopy analysis method. The accuracy of homotopy analysis method solutions is evaluated by numerical integration simulations. Results indicate that with relatively large contact loss ratios, the amplitude–frequency curves obtained by homotopy analysis method agree better with the results obtained by numerical integration than those obtained by the method of multiple scales. This study lays a higher accurate foundation for more complex nonlinear dynamic analysis of planetary gear trains.

Sparse HP Filter: Finding Kinks in the COVID-19 Contact Rate

In this paper, we estimate the time-varying COVID-19 contact rate of a Susceptible-Infected-Recovered (SIR) model. Our measurement of the contact rate is constructed using data on actively infected, recovered and deceased cases. We propose a new trend filtering method that is a variant of the Hodrick-Prescott (HP) filter, constrained by the number of possible kinks. We term it the sparse HP filter and apply it to daily data from five countries: Canada, China, South Korea, the UK and the US. Our new method yields the kinks that are well aligned with actual events in each country. We find that the sparse HP filter provides a fewer kinks than the l1 trend filter, while both methods fitting data equally well. Theoretically, we establish risk consistency of both the sparse HP and l1 trend filters. Ultimately, we propose to use time-varying contact growth rates to document and monitor outbreaks of COVID-19.

Estimating the Effectiveness of Non-Pharmaceutical Interventions on COVID-19 Control in Korea.

BackgroundThe coronavirus disease 2019 (COVID-19) pandemic has posed significant global public health challenges and created a substantial economic burden. Korea has experienced an extensive outbreak, which was linked to a religion-related super-spreading event. However, the implementation of various non-pharmaceutical interventions (NPIs), including social distancing, spring semester postponing, and extensive testing and contact tracing controlled the epidemic. Herein, we estimated the effectiveness of each NPI using a simulation model.MethodsA compartment model with a susceptible-exposed-infectious-quarantined-hospitalized structure was employed. Using the Monte-Carlo-Markov-Chain algorithm with Gibbs' sampling method, we estimated the time-varying effective contact rate to calibrate the model with the reported daily new confirmed cases from February 12th to March 31st (7 weeks). Moreover, we conducted scenario analyses by adjusting the parameters to estimate the effectiveness of NPI.ResultsRelaxed social distancing among adults would have increased the number of cases 27.4-fold until the end of March. Spring semester non-postponement would have increased the number of cases 1.7-fold among individuals aged 0–19, while lower quarantine and detection rates would have increased the number of cases 1.4-fold.ConclusionAmong the three NPI measures, social distancing in adults showed the highest effectiveness. The substantial effect of social distancing should be considered when preparing for the 2nd wave of COVID-19.

A fault dynamic model of high-speed angular contact ball bearings

Angular contact ball bearings (ACBBs) are widely used in rotary machines especially in high-speed rotating machines because of their excellent properties. However, the traditional dynamic model of a rolling bearing is mainly applied to simulate the dynamic response at low speed, and uses the simple displacement excitation function to characterize the fault, thus its accuracy will decrease under high speed condition. Aiming at the shortcomings of the traditional model, a new high-speed fault dynamic model of ACBB is proposed by considering the influences of centrifugal force, gyroscopic moment and time-varying contact angles on the rolling element under high-speed running and using a B-spline fitting displacement excitation method to represent the fault excitation. With this model, high-speed dynamic responses under different fault sizes and rotation speeds are calculated by the Runge–Kutta method and Newton– Raphson method. Then the corresponding time and frequency domain characteristics are analyzed. From the simulation results, it can be seen that all the simulated acceleration signals of ACBBs are more in line with the actual situation than the traditional model, especially for the shape of an impulse. Finally, the effectiveness and correctness of the proposed model are verified by the experiment on aero-engine spindle bearings.

Experimental and numerical investigations on dynamic characteristics of gear-roller-bearing system

In the present study, dynamic characteristics of gear-roller-bearing system is investigated by use of experimental test and theoretical modeling. In terms of experiment, a new test set-up dedicated to offset press is developed to measure the acceleration within a wide range of speed. After the completion of data acquisition, the vibration signals obtained from different conditions are analyzed both in time domain and frequency domain, thereby capturing the source of vibration excitation. For theoretical modeling, a novel dynamic model of gear-roller-bearing system including time-varying properties and contact nonlinearities is proposed based on the fundamental idea of finite element method, divided into the helical gear pair model, dynamic model of roller and shaft, bearing elements, shoulder iron and rubber contact elements. The unconditionally stable algorithm, known as Newmark-β method, is employed to solve the governing equations, and then the validity of mathematical model in predicting dynamic response is demonstrated by comparison with experimental results. Experimental and numerical results indicate that the proposed method can be utilized as a useful tool for parameter design and performance analysis of gear-roller-bearing system.

An investigation on the influence of modification parameters on transmission characteristics of planetary gear system

A new gear tooth modification model of planetary gear system was established by considering the characteristics of tooth longitudinal crowning modification and tooth profile modification. Using this model, the influences of modification parameters on the time-varying mesh stiffness, transmission error, contact spots and dynamic response of the planetary gear system were analyzed through numerical analysis. The results show that tooth profile modification can effectively restrain the mesh stiffness sudden variation in the single and double tooth alternation of both the internal and external mesh pairs, and significantly decrease the transmission error fluctuation. However, longitudinal crowning modification has no marked influence on mesh stiffness sudden variation or transmission error fluctuation. The tooth profile modification can significantly reduce the sudden variation of tooth surface load during gear mesh course. Longitudinal crowning modification can make the tooth surface load evenly distributed in tooth width direction. When the external meshing pairs are modified at the tooth profile individually, the magnitude values of main response resonance peaks of internal meshing pairs have a great reduction, but those values of external meshing pairs change slightly. In the case of tooth profile modification of the internal meshing pairs, the phenomenon is reversed. By contrast with tooth profile modification, longitudinal crowning modification has little influence on the resonance peaks of internal and external meshing pairs.

Analytical modeling of temperature distribution in lead-screw whirling milling considering the transient un-deformed chip geometry

High temperature often occurs at the local cutting area during lead-screw whirling milling, which significantly affects workpiece surface integrity as well as tool life. The cutting process of lead-screw whirling milling comprises several recurrent characteristics, such as the un-deformed chip thickness and un-deformed chip width. The combined effects of un-deformed chip thickness and width cause time-varying variation in the size of the heat source, thus further affect the temperature distribution in the cutting area. This paper proposes a transient thermal analytical model to analyze the temperature distribution in the cutting area for lead-screw whirling milling. According to the complex trajectory of the tool relative to the rotation workpiece, the un-deformed chip geometry has been studied in detail at the two cutting stages. The time-varying tool-chip contact area and boundary equations of frictional heat source are firstly formulated to describe the size of the time-varying heat source. Secondly, the transient thermal analytical model is constructed taking into account the heat liberation intensity of the time-varying heat source. A series of experiments of whirling milling for lead-screw shaft are carried out, and the theoretical results evaluated by the proposed analytical model agree well with the experimental results. Furthermore, the effects of cutting conditions and un-deformed chip geometry on the temperature distribution have been analyzed, which can be used for process designers to make decisions in choosing the cutting parameters with optimized surface integrity and tool life.

Maximum temperature of the disc during repeated braking applications

A computational finite element model of a brake disc for determining transient axisymmetric (two-dimensional) temperature field during repeated brake application has been proposed. The presented research is a subsequent stage of a previous study on the coupling of velocity and maximum temperature for a single braking in accordance with the system of equations of heat dynamics of friction and wear. In the analysed case, changes in the mean, flash, maximum and bulk temperature of the disc were determined and discussed. The calculations were carried out at the temperature-dependent coefficient of friction, the thermophysical properties of cast-iron disc combined with cermet brake pads and the time-varying contact pressure. The obtained results were compared with the reference values from the braking simulation at constant operating parameters and independent of temperature properties of materials. It was shown that the maximum values of the mean temperature for both cases differed slightly during the entire process. The flash temperature determined from the heat dynamics of friction and wear system of equations was the highest at the beginning and gradually decreased with the number of brake applications.

Thermal EHL Characteristics Investigation on Axle Box Bearings of Railway Vehicle Based on Slicing Method

Lubrication failures of axle box bearings can lead to accidents, such as bearing burnout and hot axle cutting. Presently, the modeling of the vehicle‐track system dynamics rarely considers the nonlinear contact load of axle box bearings, and this leads to imperfection in the vehicle‐track system dynamics calculation. And then, the load distribution and lubrication characteristics of axle box bearings are difficult to obtain. Therefore, in this paper, we fully consider the time‐varying nonlinear contact load of bearings and track irregularity in establishing the bearing‐wheel‐rail system coupling‐dynamics model. The dynamic response of axle box bearings is obtained by taking the vertical, strong impact‐time‐varying load on the carrying saddles as the external excitation. The load‐balance equation of dynamic pressure lubrication is then obtained, according to the slicing method of bearing rollers. Finally, the elastohydrodynamic lubrication (EHL) model of axle box bearings is established considering thermal and scale effects. The results show that the central film thickness under thermal EHL was decreased by 13.61% compared with that under isothermal EHL. As the velocity of the contact pair increases, the thickness difference between thermal and isothermal EHL became larger. Thermal effects should be considered in the EHL model, in order to truly reflect the characteristics of EHL under a high speed.

A model for evaluating the meshing characteristics and vibration characteristics of asymmetric spur gears with tooth modifications

Based on the potential energy method and slice assumption, an analytical model is proposed to evaluate the meshing characteristics and vibration characteristics of asymmetric spur gear pairs with tooth modifications. Considering the extended tooth contact, the nonlinear Hertzian contact, the revised foundation stiffness and the tooth modifications, the meshing characteristics of asymmetric spur gear pairs can be accurately evaluated. The time-varying mesh stiffness and contact stress obtained from the proposed method are compared with that obtained from the finite element method. The results show that the two methods are in good agreement but the computational efficiency of the proposed method is much higher than the finite element method. With the increase of the pressure angle on driving side, the contact ratio and the contact stress decrease and the mesh stiffness increases. On the basis of the meshing characteristics model, the dynamic model of asymmetric spur gear pairs is established to evaluate vibration characteristics. With the increase of the pressure angle on driving side, the amplitude of vibration increases and the first natural frequency decreases slightly. The proposed model can provide the theoretical basis for the parameter design of asymmetric spur gear pairs with tooth modifications.

Comparison of meshing characteristics of helical gears with spalling fault using analytical and finite-element methods

Abstract This paper investigates the effect of a gear tooth spalling in a helical gear. An analytical model was developed to study the effect of the damage to the time-varying mesh stiffness, contact stress and tooth root fillet stress. In order to validate the efficiency and the accuracy of the proposed method, a finite element method is presented for simulation, such to compare the mesh stiffness and stresses obtained from the analytical method under different spalling variables, including the spalling lengths, widths, axial positions (in the direction of face width) and tooth orientation positions (from dedendum to addendum). The results show that the proposed analytical method has higher computation efficiency than the finite element method, and the mesh stiffness obtained from two methods shows a good agreement. However, some errors exist between the time-varying mesh stiffness and stresses obtained by the two methods, but change laws of the tooth root stress and contact stress obtained from the two methods are similar. This is because the effects of flank-tip contact and spalling edge contact are considered in the finite element method, which makes the results more accurate than the results obtained by the proposed analytical method.

Mesh stiffness calculation of helical gears with profile modification

Time-varying contact line and mesh stiffness are important parameters for dynamic investigation of helical gear pair with profile modification; however, the calculation methods of the time-varying mesh with high accuracy and high efficiency are also very limited currently. A new mesh stiffness calculation method for helical gear pairs with profile modification is proposed. The new method is conducted by cutting the helical gear tooth into slices to obtain the change of the contact line and the time-varying mesh stiffness with considering the change of each slice's engagement performance introduced by the profile modification. Then, the mesh stiffness of the assumed three different modification cases is calculated by using the proposed method. The calculated results indicate that the proposed method can be used to calculate the mesh stiffness of helical gear pairs, especially of those with tooth profile modification.

A simple atomic force microscope-based method for quantifying wear of sliding probes.

Sliding wear is particularly problematic for micro- and nano-scale devices and applications, and is often studied at the small scale to develop practical and fundamental insights. While many methods exist to measure and quantify the wear of a sliding atomic force microscope (AFM) probe, many of these rely on specialized equipment and/or assumptions from continuum mechanics. Here we present a methodology that enables simple, purely AFM-based measurement of wear, in cases where the AFM probe wears to a flat plateau. The rate of volume removal is recast into a form that depends primarily on the time-varying contact area. This contact area is determined using images of sharp spikes, which are analyzed with a simple thresholding technique, rather than requiring sophisticated computer algorithms or continuum mechanics assumptions. This approach enables the rapid determination of volume lost, rate of material removal, normal stress, and interfacial shear stress at various points throughout the wear experiment. The method is demonstrated using silicon probes sliding on an aluminum oxide substrate. As a validation for the present method, direct imaging in the transmission electron microscope is used to verify the method's parameters and results. Overall, it is envisioned that this purely AFM-based methodology will enable higher-throughput wear experiments and direct hypothesis-based investigation into the science of wear and its dependence on different variables.