Contact Curves Research Articles

Study on the roller enveloping end face internal engagement worm gear

A novel worm gear, named roller enveloping end face internal engagement worm gear, with multiple worm-wheel meshing is proposed to provide multiple outputs with high transmission efficiency and good lubrication for furthering industrial advances. As such, an original worm transmission device with roller enveloping internal end-face meshing was proposed based on the innovation in worm structure and gear meshing theory, which can bear 1–4 working load(s) on a single worm. This paper presents mathematical models of meshing characteristics for roller enveloping end face internal engagement worm gear, including both the contact curve and the tooth profile function based on the gearing meshing theory. In doing so, the tooth profile was calculated by MATLAB and the 3D solid model of worm were produced by Creo to measure the meshing properties of roller enveloping end face internal engagement worm gear. The kinematic simulation was also conducted to analyze meshing position and the corresponding meshing teeth. Our study shows that the overall performance of the original worm transmission device with roller enveloping end face internal meshing is good without any interference, and the output curve is basically consistent with the engaging performance. This study is expected to provide a new multiple outputs worm drive for industry, and the results are beneficial for designers to choose the best parameters.

Optimization of the target profile for asymmetrical rail grinding in sharp-radius curves for high-speed railways

Asymmetrical rail grinding in sharp-radius curves could reduce the side wear of railheads and enhance curve capacity of rail vehicles. The wheel/rail contact performance and curve capacity could be further improved by the optimization of the asymmetrical rail grinding target profile. In order to modify the target profile smoothly, the nonuniform rational B-spline curve with adjustable weight factors is used to establish a parameterized model of railhead curves in the asymmetrical grinding region. The indices of contact performance and curve capacity for different weight factors are obtained using experiment design and service performance simulation. Two Kriging surrogate models are proposed, in which the design variables are the adjustable weight factors, and the response parameters are the indices of contact performance and curve capacity, respectively. The multi-objective optimization model of the target profile is established, in which the objective functions are the two Kriging surrogate models of contact performance and curve capacity. The optimized weight factors are sought using a nondominated sorting genetic algorithm II, and the corresponding optimal target profile is obtained. The wheel/rail service performance simulation before and after optimization indicates that the contact performance and curve capacity are improved significantly.

Geometry and parameter design of novel circular arc helical gears for parallel-axis transmission

Based on the space curve meshing equation, in this article, a geometry design of a novel circular arc helical gear mechanism with pure rolling for parallel transmission was presented. Different from conventional circular arc gears, the meshing points of circular arc helical gears were limited at the instantaneous centre of rotation. The parameter equations describing the contact curves for both the driving gear and the driven gear were deduced from the space curve meshing equation, and parameter equations of the concave–convex circular arc profiles were established both for internal meshing and external meshing. Furthermore, a formula for the contact ratio was presented, and the impact factors influencing the contact ratio were discussed. Then, the parameter design was presented for the geometry parameters of tooth profiles, such as normal pitch, tooth height and tooth thickness. Using the deduced equations, several numerical examples were then considered, and prototype samples were produced to experimentally validate the contact ratio equation and the theoretical kinematic performance. The circular arc helical gear mechanism investigated in this study showed a high gear transmission performance such as a pure rolling meshing, a high contact ratio and a large comprehensive strength, when considering engineering applications.

Experimental and theoretical analysis of the elastic-plastic normal repeated impacts of a sphere on a beam

The normal repeated impact of a sphere upon an elastic-plastic beam has been analyzed experimentally and theoretically. Hertz elastic contact model and seven well known elastic-plastic contact models were selected for theoretical analysis. A piecewise linearized method has been presented to linearize the nonlinear contact stiffness of eight selected contact models. Generalized closed-form solutions (theoretical solutions) of repeated impact response were derived for each piecewise linearization by vibration theory. Ahead of the theoretical study, to gain a deep insight into the eight selected contact models, an idealized repeated impact problem of elastic-plastic half space/foundation was solved by Runge–Kutta integration method. The numerical results show that the models are of large differences in predicting contact behavior, and reveal a strong correlation of the models with the two model parameters: the indentation of initial contact yield and the effective radius of contact curvature of unloading. Experiments were performed with regard to a repeated impact process of a sphere striking repeatedly at the center of the beam. The theoretical and experimental results show that selecting the appropriate contact model is very important on predicting contact behavior. No significant advantage of indentation type has been found over flattening type among these contact models. The divergences of contact behavior and the strong correlation for the eight contact models still exist. Compared with the experimental results, for the coefficient of restitution and the rebound velocity of the sphere, all of the eight contact models predict larger values and contrary trends on impact velocity compared to the experimental results. The reasons may come from wave effects and sub-impact phenomenon.

Sub-impacts of simply supported beam struck by steel sphere—part II: Numerical simulations

This part of the article describes numerical simulations of the problem investigated experimentally. A three-dimensional finite element model of elastic–plastic for sphere falling on beam has been implemented using the nonlinear dynamic finite element software LS-DYNA. From the numerical simulations, it was found that the LS-DYNA is suitable to study complex sub-impact phenomenon, and good agreement is in general obtained between the simulation and experimental results. The numerical simulations show that the initial impact velocity, equivalent elasticity modulus, contact curvature radius of the sphere, and equivalent mass have great influence on the contact–impact time of the sub-impact, and an applicable range of the theoretical expression of contact–impact time of the sub-impact was determined. In addition, the numerical simulations demonstrate the ratios of maximum amplitudes of the first-, second-, and third-order vibrations to the maximum amplitudes of the beam vibrations, and the phase angle of the first-order vibration will change suddenly when the sub-impacts occur. Furthermore, the occurrence conditions of the sub-impacts were clarified numerically. It was found that the occurrence conditions of the sub-impacts can be represented by a mass ratio threshold, and the thickness or length of the beam has also a great influence on the occurrence of the sub-impacts. Once the sub-impacts occur, which would result in an uncertain behavior of the apparent coefficient of restitution.

Design formulae for a concave convex arc line gear mechanism

Abstract. Line gear is a newly developed gear mechanism with point contact meshing according to space curve meshing theory. This paper proposes a new form of line gear with a couple of concave convex arc tooth profiles. It has four characteristics. First, contact curve of the driving line gear is a cylindrical spiral curve. Second, two axes of a pair of line gears are located in the same plane with an arbitrary angle. Third, at the mesh point, normal tooth profiles of a line gear pair are a couple of inscribed circles. Namely, they form a couple of concave convex tooth profiles. Fourth, the tooth profile of a driving line gear is a convex, that of a driven line gear is a concave, and they are interchangeable. If only consider that the arcs of teeth at meshing point are tangent, the actual tooth surfaces may interfere outside of the meshing point. In this paper, the geometric condition of the tooth surface for a concave convex arc line gear mechanism is derived, and the optimal formulae of the tooth profile parameters are derived on basis of interference proof conditions. Finally, the 3-D modeling and kinematic simulation of line gear pairs show that the proposed line gear pairs can perform transmission normally. The proposed method will extend the application of line gear in the conventional power drive.

Modulating Vesicle Adhesion by Electric Fields

We introduce an experimental setup for modulating adhesion of giant unilamellar vesicles to a planar substrate. Adhesion is induced by the application of an external potential to a transparent indium tin oxide-coated electrode (the substrate), which enables single-vesicle studies. We demonstrate tunable and reversible adhesion of negatively charged vesicles. The adhesion energy at different potentials is calculated from the vesicle shape assessed with confocal microscopy. Two approaches for these estimates are employed: one based on the whole contour of the vesicle and a second based on the contact curvature of the membrane in the vicinity of the substrate. Both approaches agree well with each other and show that the adhering vesicles are in the weak adhesion regime for the range of explored external potentials. Using fluorescence quenching assays, we detect that, in the adhering membrane segment, only the outer bilayer leaflet of the vesicle is depleted of negatively charged fluorescent lipids, while the inner leaflet remains unaffected. We show that depletion of negatively charged lipids is consistent Poisson-Boltzmann theory, taking into account charge regulation from lipid mobility. Finally, we also show that lipid diffusion is not significantly affected in the adhering membrane segment. We believe that the approaches introduced here for modulating and assessing vesicle adhesion have many potential applications in the field of single-vesicle studies and research on membrane adhesion.

Proposal of a face gear which generates virtual high mesh frequency by addition of grooves on the tooth flank, and the investigation via vibration simulator and actual samples

This report discusses the method of tooth flank modification of a face gear which improves the handle rotational feeling in a fishing reel. A vibration based on a gear pair engagement in fishing reel occurs when a handle of the reel rotates. When this vibration is large, an angler feels uncomfortable, and it is known that the rotational feeling strongly depends on the transmission error of a face gear which is mounted on the reel. In this report, a new method of tooth flank modification was proposed. Based on the new method, it is possible to increase the mesh frequency while keeping the module size and the outer diameter of the face gear unchanged. In this method, several grooves were added along the contact curve on the tooth flank of the face gear. The influence on vibration by these grooves was investigated using the vibration simulator which was introduced in the author’s previous reports. Moreover, sample face gears were manufactured and the rotational feeling was evaluated. As a result, it was confirmed that the rotational feeling was improved by applying a proper number of grooves and groove width.

Precision grinding of screw rotors using CNC method

A precision grinding method for screw rotors using a computer numerical control (CNC) grinding wheel dressing is presented in this study. A mathematical model for screw rotor grinding is developed for a prototype based on the new screw rotor grinding machine SK7032. A spatial contact curve equation for screw rotor grinding is proposed, which includes three parameters: center distance, swivel angle, initial phase angle. A parameterized program was designed by the contact curve equation, and the grinding wheel axial-section profiles were calculated for different parameters. With the calculation results, the grinding wheel profile errors were analyzed, and the maximum influence of the grinding wheel profile error was swivel angle. The minimum influence was the initial phase angle, but the position of grinding wheel profiles can be changed and the interference between the screw rotor and the grinding wheel can be avoided by its changing. A segmentation dressing method for the grinding wheel was designed based on a variable center distance, and the reasonable interval value was studied. A male rotor grinding experiment was conducted on SK7032, and the rotor profile error was measured on the gear tester P65. The results reveal that the screw rotor profile errors can be controlled within ±10 μm.

The first orbital parameters and period variation of the short-period eclipsing binary AQ Boo

We obtained the first VRI CCD light curves of the short-period contact eclipsing binary AQ Boo, which was observed on March 22 and April 19 in 2014 at Xinglong station of National Astronomical Observatories, and on January 20, 21 and February 28 in 2015 at Kunming station of Yunnan Observatories of Chinese Academy of Sciences, China. Using our six newly obtained minima and the minima that other authors obtained previously, we revised the ephemeris of AQ Boo. By fitting the O-C (observed minus calculated) values of the minima, the orbital period of AQ Boo shows a decreasing tendency P˙=−1.47(0.17)×10−7 days/year. We interpret the phenomenon by mass transfer from the secondary (more massive) component to the primary (less massive) one. By using the updated Wilson & Devinney program, we also derived the photometric orbital parameters of AQ Boo for the first time. We conclude that AQ Boo is a near contact binary with a low contact factor of 14.43%, and will become an over-contact system as the mass transfer continues.

Static Analysis of an Inverted Planetary Roller Screw Mechanism

The paper examines the static behavior of the inverted planetary roller screw (PRS) through numerical and experimental studies. The numerical analysis of the inverted PRS is first presented to capture the global and local deformations in different configurations. Using a three-dimensional finite element (3D FE) method, a sectorial model of the mechanism is built involving an entire roller. The model describes the static behavior of the system under a heavy load and shows the state of the contacts and the in-depth stress zones. The current work also investigates the axial stiffness (AS) and the load distribution (LD) under both compressive and tensile loadings. It is shown that the LDs are not the same at each contact interface of the roller and that they depend on the configuration of the system. Also, the nut is less stressed than the screw shaft because of their contact curvatures. In parallel, complementary experiments are carried out to measure the axial deflection of the screw shaft and the rollers in five cases with different numbers of rollers. In each situation, the mechanism is under the same equivalent axial and static load. The tests reveal that rollers do not have the same behavior, the difference certainly being due to manufacturing and positioning errors that directly affect the number of effective contacts in the device. This stresses the fact that the external load is unequally shared over rollers and contacting threads. By introducing the notion of an equivalent roller, the results are used to validate the previous numerical model of an inverted PRS. As they provide a better understanding of the inverted PRS, these investigations are useful to improve the existing analytical models of the device.

A note on splitting curves of plane quartics and multi-sections of rational elliptic surfaces

In this note, we study the relation between splitting curves of plane quartics and multi-sections of rational elliptic surfaces associated to the quartics. We give a criterion for when a simple contact curve splits and calculate the images of the components of the corresponding multi-sections in the Mordell–Weil group of the associated rational elliptic surface. We also introduce a notion called the splitting type for certain configurations of plane curves which allows us to distinguish the configurations topologically. To demonstrate its effectiveness we construct new examples of Zariski pairs and a Zariski triple.

Contact Angle Effects on Pore and Corner Arc Menisci in Polygonal Capillary Tubes Studied with the Pseudopotential Multiphase Lattice Boltzmann Model

In porous media, pore geometry and wettability are determinant factors for capillary flow in drainage or imbibition. Pores are often considered as cylindrical tubes in analytical or computational studies. Such simplification prevents the capture of phenomena occurring in pore corners. Considering the corners of pores is crucial to realistically study capillary flow and to accurately estimate liquid distribution, degree of saturation and dynamic liquid behavior in pores and in porous media. In this study, capillary flow in polygonal tubes is studied with the Shan-Chen pseudopotential multiphase lattice Boltzmann model (LBM). The LB model is first validated through a contact angle test and a capillary intrusion test. Then capillary rise in square and triangular tubes is simulated and the pore meniscus height is investigated as a function of contact angle θ. Also, the occurrence of fluid in the tube corners, referred to as corner arc menisci, is studied in terms of curvature versus degree of saturation. In polygonal capillary tubes, the number of sides leads to a critical contact angle θc which is known as a key parameter for the existence of the two configurations. LBM succeeds in simulating the formation of a pore meniscus at θ > θc or the occurrence of corner arc menisci at θ < θc. The curvature of corner arc menisci is known to decrease with increasing saturation and decreasing contact angle as described by the Mayer and Stoewe-Princen (MS-P) theory. We obtain simulation results that are in good qualitative and quantitative agreement with the analytical solutions in terms of height of pore meniscus versus contact angle and curvature of corner arc menisci versus saturation degree. LBM is a suitable and promising tool for a better understanding of the complicated phenomena of multiphase flow in porous media.

A variable-ratio line gear mechanism

Based on the previous studies of line gear (LG), which is also named as space curve meshing wheel (SCMW) in the published papers, a variable-ratio LG mechanism is proposed in this article. The formulae of contact curves of a variable-ratio LG pair were derived. Then prototypes of variable-ratio LG pairs were designed by these formulae and manufactured by 3D printing. Afterwards, kinematics experiments were conducted by measuring the transmission ratio. The result indicates that a variable transmission ratio is feasible to a variable-ratio LG mechanism designed by these derived formulae, and the change of the transmission ratio is steady according to a prescribed kinematic rule. It is capable of being employed to devices on demand of variable-ratio mechanical transmission mechanism in compact structure.

Analysis of the influence of the ramp and disk dynamics on the HDI response of 2.5-inch hard disk drive to a shock

Operational shock (op-shock) analysis has been developed to consider the dynamics of components of a hard disk drive (HDD). Op-shock simulation uses simplified models. However, the dynamics of the components affect the head–disk interface (HDI) response, especially the disk curvature and ramp–disk contact. This paper investigated the effect of the dynamics of HDD components on the HDI response. As the HDD is more sensitive to external shock and vibration, the effects of the dynamics of HDD components need to be considered for accurate op-shock analysis. The ramp–disk contact is an important factor in an op-shock simulation, because ramp–disk contact and disk deformation occur during the shock. Therefore, this op-shock simulation modeled the ramp–disk contact and considered the disk curvature. To obtain an accurate result, a two-way coupled method was used to investigate the HDD dynamic response during op-shock. Using the op-shock model, the fluid-structure coupling effect was investigated and compared with a decoupled model that used linear springs to replace the air bearing. Then, we investigated the effect of the ramp–disk contact and disk curvature on the HDI response during a shock.

An engineering model for yield inception in slip-stick elastic contacts

The failure of the mechanical contact due to plastic yielding is generally predicted employing stress analysis coupled with the von Mises yield criterion, which uses the maximum of the second deviatoric stress invariant as a threshold value. This paper aims to establish the relation between the frictional regime and the normal and tangential loading components which lead to yield inception in the slip-stick spherical contact between similarly elastic materials. The Boussinesq and Cerruti fundamental solutions for the elastic half-space are used in a robust semi-analytical method based on the superposition principle applicable in the frame of linear elasticity, and enhanced with an acceleration technique derived from the convolution theorem. A rapid algorithm for accurate computation of elastic stresses induced in subsurface by a known but arbitrary distribution of surface tractions, normal or shear, is advanced. The obtained data is normalized to allow model extension to any elastic constants or contact curvature, and curve fitting is employed to derive simple empirical formulas pertinent to practical engineering applications.

Finite element modeling and analysis of planetary gear transmission based on transient meshing properties

Planetary gear trains are widely applied in various transmission units. Whether strengths of all gears are accurately calculated or not can affect reliability of the entire system significantly. Strength calculation method for planetary gear trains usually follows the method for cylindrical gears, in which the worst meshing positions for both contact stress and bending stress cannot be determined precisely, and calculation results tend to be conservative. To overcome these shortcomings, a kinematics analysis for a planetary gear train is firstly performed, in which the influence of relative speed is investigated. Then the finite element strength analysis of a planetary gear train based on its transient meshing properties is carried out in ANSYS. Time–history curves of contact and bending stresses of sun gear, planetary gears and ring gear are respectively obtained. Also the accurate moment and its corresponding position of the maximum stress are precisely determined. Finally, calculation results of finite element method (FEM) and traditional method are compared in order to verify the effectiveness. Simulation and comparison show the stability of the proposed method in this paper. Researches in this paper establish the foundations for fatigue analysis and optimization for a planetary gear train.

Ricci flow on contact manifolds

This paper is devoted to Ricci flow on contact manifolds. We define the contact curvature flow and establish a short time existence. Meanwhile, we study a contact Ricci soliton and prove that every solution of the unnormalized contact curvature flow is a selfsimilar solution corresponding to a contact Ricci soliton which is a steady soliton. Finally we show that a time dependent family of contact Einstein, Sasakian, K-contact, or η-Einstein 1-forms ηt is a solution of the normalized contact curvature flow if it is a conformal variation of an initial 1-form η0.

Parametric analysis of the end face engagement worm gear

A novel specific type of worm drive, so-called end face engagement worm gear(EFEWD), is originally presented to minimize or overcome the gear backlash. Different factors, including the three different types, contact curves, tooth profile, lubrication angle and the induced normal curvature are taken into account to investigate the meshing characteristics and create the profile of a novel specific type of worm drive through mathematical models and theoretical analysis. The tooth of the worm wheel is very specific with the sine-shaped tooth which is located at the alveolus of the worm and the tooth profile of a worm is generated by the meshing movement of the worm wheel with the sine-shaped tooth, but just the end face of the worm(with three different typical meshing types) is adapted to meshing, and therefore an extraordinary manufacturing methods is used to generate the profile of the end face engagement worm. The research results indicates that the bearing contacts of the generated conjugate hourglass worm gear set are in line contacts, with certain advantages of no-backlash, high precision and high operating efficiency over other gears and gear systems besides the end face engagement worm gear drive may improve bearing contact, reduce the level of transmission errors and lessen the sensitivity to errors of alignment. Also, the end face engagement worm can be easily made with superior meshing and lubrication performance compared with the conventional techniques. In particular, the meshing and lubrication performance of the end face engagement worm gear by using the end face to meshing can be increased over 10% and 7%, respectively. This investigate is expect to provide a new insight on the design of the future no-backlash worm drive for industry.

Contact angle measurement on micropatterned surface using sessile drop shape fit profile detection

Micro-nano patterned surfaces have significant applications in various fields as they behave differently under the effect of catalysts, magnetic energy, electronic emission/absorption, optics and biological cells. Engineering these topologies demands a better understanding of the contact angle. The current contact angle measurement techniques assume the drop to be a perfect sphere, neglect gravitational and molecular dispersion effects; thereby leading to inaccuracies. This is because the micro-machined surfaces exhibit sub-micrometre scale porosity and pattern dimensions are comparable to the droplet size, resulting in composite interfaces at micro-nano scale. In this paper, the authors assessed the adaptability of conventional measurement techniques for textured surfaces and developed an algorithm that is based on curve fitting over sessile drop after edge detection. The algorithm performs edge detection, contact point identification and curve fitting and corrects uneven surfaces and was tested on micro-patterned surfaces fabricated over three different materials: polydimethylsiloxane, polystyrene and acrylic using laser.