Involute Curve Research Articles

Adaptive isogeometric gear contact analysis: Geometry generation, truncated hierarchical B-Spline refinement and validation

Gears are one of the most widely used transmission components. Their operation relies on the contact between mating gear teeth flanks for the transmission of power. Accurate prediction of the contact stresses at these regions, is crucial for the design and dimensioning of these systems. Gear design is centered around highly smooth involute curves that greatly influence their contact behaviour. In this paper, a fully adaptive isogeometric contact modelling scheme, based on hierarchical splines, is presented and applied to the simulation of gear contact problems. In particular, isogeometric simulation is performed for the modelling of mating pair of gear teeth, regarded as linearly elastic bodies. A boundary fitted B-Spline representation of the teeth is automatically generated from engineering design parameters and is used to define the initial discretisation basis. The numerical integration over the contact region is addressed using the so called, Gauss-Point to Surface formulation and a closest point projection procedure. Truncated hierarchical B-Splines are used to capture the highly localised nature of contact, while effectively reducing the number of degrees of freedom. The adaptivity is driven by the strain energy density gradient, which allows to automatically localise the mesh without a priori knowledge of the contact region between the teeth flanks. In our experiments we justify the choices made in different steps of our algorithm and we assess the performance of our adaptive solver with respect to classical tensor product B-Splines.

Analysis of involute-curve geometry using position gradients: Application to gear-tooth profile

This paper examines using non-rational polynomials to approximate the involute-curve geometry, which is fundamental in the design of gear systems. The parametric form of the involute curve in terms of an involute-angle parameter is first presented. To be able to compare the analytical geometry results and numerical results obtained using the finite-element (FE) method for describing the geometry, the relationship between the involute-angle parameter and the FE spatial coordinate is defined. A new relationship between the involute-curve arc length and the coordinate used in the FE parameterization is also developed. To this end, the involute-curve equation is used to determine the involute-angle parameter at the intersection with circles with arbitrary radii including the gear pitch, addendum, and base circles. The parameter relationship allows defining the nodal position-gradient vectors defined by differentiation with respect to the FE spatial coordinate from involute-curve tangent vector. Two procedures for determining the left involute curve from the right involute curve of the gear tooth are described. In the first procedure, a coordinate transformation based on Rodrigues transformation with axis of rotation defined by the tooth-center line is used; while in the second procedure, the analytical expression of the left involute curve is defined using the gear-tooth thickness at the base circle. A general procedure for determining the position gradients for a given geometry obtained from analytical curves or using imaging techniques is introduced. Such a procedure can be used in case of worn gear teeth that cannot be described by simple functions. A planar rectangular element based on the absolute nodal coordinate formulation (ANCF) is used to approximate the gear-tooth geometry. While the involute geometry cannot be described exactly using non-rational polynomials, the analysis and results of this investigation demonstrate that the FE solution leads to accurate geometry results, demonstrating the potential of using the position gradients in future strength-calculation studies.

Calculation of Time-Varying Mesh Stiffness of Internal Gears based on Precise Tooth Profile and Dynamic Analysis of Planetary Systems with Root Cracks

The internal meshing spur gear pair is the research object, and the potential energy method is applied to calculate the time-varying meshing stiffness of the internal gear. The internal gear tooth profile is divided into two parts: involute and transition curves, and the gear tooth stiffness is calculated by numerical integration based on accurate tooth profile, which improves the calculation accuracy. Analyzed the influence of different crack size parameters on the stiffness of internal meshing gears. A coupled dynamic model of a planetary system with internal gear crack faults was established, and the influence of cracks on the dynamic response of the planetary system was studied using Zoom-FFT spectrum and cepstrum analysis methods. The simulation results show that as the crack parameter size increases, the mesh stiffness of the internal gear pair gradually weakens, and the periodic vibration impact of the planetary-internal gear pair is also more severe. Cepstrum analysis can easily capture weak fault characteristic frequency information in the system and discover their variation patterns. The research results can provide a theoretical basis for the state monitoring and fault diagnosis of gear systems with crack defects.

B-Lift curves and involute curves in Lorentzian 3-space

The involute of a curve is often called the perpendicular trajectories of the tangent vectors of a unit speed curve. Furthermore, the B-Lift curve is the curve acquired by combining the endpoints of the binormal vectors of a unit speed curve. In this study, we investigate the correspondences between the Frenet vectors of a curve’s B-lift curve and its involute. We also give an illustration of a helix that resembles space in Lorentzian 3-space and show how to visualize these curves by deriving the B-Lift curve and its involute.

Experimental investigation and numerical simulation of the effect of type of bending die on the quality of tube forming in rotary draw bending process

This study introduces a new rotary draw bending method that utilizes a variable curvature bending die. Unlike traditional methods that bend tubes with a fixed radius, this method gradually deforms the tube from a large to a small radius. The curvature of the bending die is determined by using an involute curve as the equation for the geometric location of the variable curvature. Hydroforming technology, utilizing fluid under pressure, replaces the mandrel in the rotational tensile bending process. The research was conducted using a thin-walled AA6063 tube with a 13.88 diameter-to-thickness ratio. The bending process was examined at critical bending ratios of 1 and 1.6 times the diameter, with a 90° bending angle. The maximum pressure that can be applied in any bend radius ratio was predicted using the necking criterion. The simulation and experimental tests analyzed the effects of internal fluid pressure and bend die curvature on defects such as wall thinning in the outer curvature of the bend, thickening of the wall in the inner curvature of the bend, and cross-section non-roundness. The results indicate that, at constant pressure, the amount of thinning and thickening of the bent tube is significantly improved when using the variable radius bending die compared to the fixed radius die.

The Conchoidal Twisted Surfaces Constructed by Anti-Symmetric Rotation Matrix in Euclidean 3-Space

A twisted surface is a type of mathematical surface that has a nontrivial topology, meaning that it cannot be smoothly deformed into a flat surface without tearing or cutting. Twisted surfaces are often described as having a twisted or Möbius-like structure, which gives them their name. Twisted surfaces have many interesting mathematical properties and applications, and are studied in fields such as topology, geometry, and physics. In this study, a conchoidal twisted surface is formed by the synchronized anti-symmetric rotation matrix of a planar conchoidal curve in its support plane and this support plane is about an axis in Euclidean 3-space. In addition, some examples of the conchoidal twisted surface are given and the graphs of the surfaces are presented. The Gaussian and mean curvatures of this conchoidal twisted surface are calculated. Afterward, the conchoidal twisted surface formed by an involute curve and the conchoidal twisted surface formed by a Bertrand curve pair are given. Thanks to the results obtained in our study, we have added a new type of surface to the literature.

Mechatronic design of a modular artificial honeycomb for the automatic extraction of honey from Apis Melliferas bees in Huancayo, Peru

This research develops the mechatronic design of an artificial honeycomb for Apis Melliferas bees, which allows the automatic extraction of honey in the farms of the province of Huancayo, Peru, since Junin produces 208.26 tons of honey annually with traditional and obsolete techniques. This design was executed under the German VDI 2221 standard and contemplates the mechanical, electronic and control system, which are dimensioned according to the theories and laws of: distortion energy, polynomial 3 4 5, involute curve gears, Kirchhoff, Ohm, on-off control, among others. It was possible to design, in Inventor, the honeycomb structure whose cells have a 19° slope for the honey gravity flow, a gear-rack-gear system for the transmission of torque from the servomotor to the cam shaft to move the grid, which opens the flow duct; also, it was possible to design, in Proteus, the electronic system made up by the Arduino UNO, a servomotor of 2. 5 Kg-cm servomotor with rotation restricted to 110° and the 15mm force sensor; finally, the actuator control and the monitoring of the weight according to the on-off logic were programmed in Arduino IDE. According to the simulations performed, the design is technologically feasible.

An analytical approach for the determination of helical gear tooth geometry

Purpose This paper aims to present an analytical approach for the determination of helical gear tooth geometry and introduces the necessary parameters. Tooth geometry including tooth chamfer, involute curve, root fillet, helix as well as tooth microgeometry can be obtained using the presented approach. Design/methodology/approach The presented analytical approach involves deriving the equivalent equations at the transverse plane rather than the normal plane. Moreover, numerical evaluation of microgeometry modifications is presented for tooth profile, tooth lead and flank twist. Findings An analytical approach is presented and equations are derived and explained in detail for helical gear tooth geometry calculation, including tooth microgeometry. Method 1, which was presented by Lopez and Wheway (1986) for obtaining the root fillet, is examined and it is proven that it does not work accurately for helical gears, but rather it works perfectly in the case of spur gears. Changing the normal plane parameters in Method 1 to the transverse plane ones does not give correct results. Two alternative methods, namely, Methods 2 and 3, are developed in the current research for the calculation of the tooth root fillet of helical gears. The presented methods and also the numerical evaluation presented for microgeometry modification are examined against the geometry obtained from Windows LDP software. The results show very good agreement, and it is feasible to apply the approach using the presented equations. Originality/value In the gear design process, it is important to model the correct gear tooth geometry and deliver all related dimensions and calculations accurately. However, the determination of helical gear tooth geometry has not been presented adequately by equations to facilitate gear modelling. The detailed helical gear tooth root has been enveloped using software tools that can simulate the cutter motion. Deriving those equations, presented in this article, provides gear design engineers and researchers with the possibility to model helical gears and perform design calculations in a structured, applicable and accurate method.

Smarandache Curves of Involute-Evolute Curve According to Frenet Frame

In this paper, the invariants of the Smarandache curves, which consist of Frenet vectors of the involute curve, are calculated in terms of the evolute curve.

Experimental stress analysis of enhanced sliding contact spur gears using transmission photoelasticity and a numerical approach

Free-Form-Gear (FFG) is a recently published title for a gear that has a curved path of contact and offers contact between a convex addendum and a concave dedendum during the meshing cycle. Based on this method of designing a gear pair, the path of contact can be altered to improve the way the gears mesh and slide against each other by only adjusting the maximum pressure angle and the involute curve parameter. The analytical investigations into this gear pair indicated that, when compared to the standard involute gear pair, the sliding velocity, meshing efficiency, contact, and fillet strengths are improved while the contact ratio is decreased. In this study, experimental stress analysis is conducted on various configurations of the tooth profile and tooth fillet shape for the proposed FFG and the standard involute gear to validate the analytical results. Since it is difficult to measure the contact stress for the gear specimens, the contact stresses at critical points are estimated using an effective computational technique based on photoelastic data and a numerical approach utilizing the nonlinear least squares method. At the fillet zone, isochromatic fringe patterns are used to directly measure the maximum fillet stress. Using addendum modification coefficients, the reduction in contact ratio for the proposed FFG is eliminated, and the results are then verified experimentally. Based on the experimental estimations, the finite element analyses are conducted using ABAQUS to simulate the examined gear specimens. Under the same loading conditions, the experimental, analytical, and numerical results have all been in good agreement.

A new method for designing involute trajectory timelike ruled surfaces in Minkowski 3-space

In this study, we present the new concept of involute trajectory ruled surface in Minkowski 3-space. The involute trajectory timelike ruled surface is a surface generated by the motion of a timelike oriented line X along the spacelike involute curve γ(s) of a given timelike base curve r(s). The main purpose of this article is to present a new perspective on the generation of developable trajectory ruled surfaces in Minkowski 3-space. These surfaces are formed depending on the angle θ between the Darboux vector D and the binormal vector b of the evolute curve r(s). Also, some new results and theorems related to the developability of the involute trajectory timelike ruled surfaces are obtained. Finally, we illustrate these surfaces by presenting one example.

Eliptik Dişli Tasarımı, İmalatı, Analizi ve Eliptik Dişlilerin Endüstriyel Uygulamaları

Eliptik dişliler üzerine yapılan çalışmaların diş profillerinin tasarımı üzerine yoğunlaştığı ve prototip imalatlarına yönelik araştırmaların arttığı görülmektedir. Eliptik dişlilerin diş profillerinin silindirik dişliler gibi özdeş olmayacağından hareketle geliştirilen eliptik dişlinin diş profili; kramayer veya silindirik düz dişli formlu kesicinin bölüm elipsi üzerinde yuvarlanma yöntemine uygun olarak evolvent eğriye sahip diş yanaklarının şekillendirilmesiyle elde edilmektedir. Eliptik dişlilerin imalatında CNC tel erozyon tezgahlarından faydalanmanın yaygınlaştığı, bununla birlikte azdırma tezgâhları için elektronik dişli kutuları geliştirilerek eliptik dişlilerin standart modülde imalatının da mümkün olduğu tespit edilmiştir. Eliptik dişli çiftleri farklı makine ve teçhizatta kullanılmakta, özellikle eliptik dişli çiftine sahip debimetre ve pompaların yaygınlaştığı görülmektedir. Enerji verimliliğinin son derece önem kazandığı günümüzde, yüksek akışkan miktarlarının doğru ve hassas ölçülmesi ve işletmelerin akışkan transferindeki kayıplarının giderilmesi de gerekmektedir. Bu genel değerlendirme çalışmasında eliptik dişlilerin tasarımı, geleneksel ve geleneksel olmayan imal usulleri ile imalatı, dayanım ve titreşim analizleri ve literatürde yer alan muhtelif endüstriyel uygulamaları gözen geçirilmiş ve çalışmanın ortak bulguları elde edilmiştir. Eliptik dişlilerin yakın gelecekte bünyesinde mekanik kısım barındıran birçok akıllı ürünün vazgeçilmez bir organı olacağı öngörülmektedir.

Associated curves from a different point of view in $E^3$

In this paper, tangent, principal normal and binormal wise associated curves are defined such that each of these vectors of any given curve lies on the osculating, normal and rectifying plane of its partner, respectively. For each associated curve, a new moving frame and the corresponding curvatures are formulated in terms of Frenet frame vectors. In addition to this, the possible solutions for distance functions between the curve and its associated mate are discussed. In particular, it is seen that the involute curves belong to the family of tangent associated curves in general and the Bertrand and the Mannheim curves belong to the principal normal associated curves. Finally, as an application, we present some examples and map a given curve together with its partner and its corresponding moving frame.

Involute evolute curve family induced by the coupled dispersionless equations

In this paper, we derive the coupled dispersionless (CD) equation from the motion of the involute evolute curve pairs, which provides a favourable interpretation for the integrability conditions. This nonlinear differential equation allows us to introduce the Lax pair providing the integrability of this equation. The gauge equivalence between the CD equation and the Zhaidary-III equation is established. Integrable generalized Heisenberg ferromagnet type equations for the tangent, principal normal, and binormal unit vectors T,N,B are presented. Each of these three integrable nonlinear equations for the orthogonal frame is, separately, geometrical equivalent counterparts of the CD equation. Finally, we study the link between the involute evolute curves and the CD equation.

Study on a variable wall thickness profile of electric scroll compressor used for automobile air conditioner

Profile design plays an important role in improving the performance and reliability of scroll compressors. To expand design ideas of variable wall thickness profile (VWTP) of electric scroll compressor used for automobile air conditioner, a method to generate and modify VWTP is developed in this paper. The VWTP is generated according to the generalized profile theory and modified with symmetrical arc modification (SAM) and symmetrical arc plus line modification (SAPLM) at the central part. To make sure that the generated VWTP can form scroll profile, the generation conditions for coefficients of swing radius are discussed in detail. For the VWTP modified with SAM and SAPLM at the central part, a new way to classify scroll chambers and analyze the evolution of scroll chambers is proposed. Geometry analyses such as conjugate points on involute curves and arc segments, volume and its derivative with respect to orbiting angle are conducted as well. The generation and geometry analysis method of VWTP is validated through a design example of VWTP for electric scroll compressor with a displacement of 34 mL. The performance of the electric scroll compressor was tested through experiment and the results show that the cooling capacity can reach 7.5 kW and the COP can reach 3.0.

A novel design for enhancing the surface durability of the spur gear systems

In this study, a general mathematical model for defining the gear tooth profile and its geometrical aspects that meet a desirable sliding contact during the meshing cycle is built. This model is based on the path of contact shape predefinition. In the contact path, a straight-line segment through the pitch point and a universal transition curve for the other segments are combined to create a free-form tooth profile with desired radius of curvature in the designed gear. The maximum pressure angle during the meshing cycle and the involute curve length parameter are only introduced to characterize this model. The designed gear in this study is given the title “free-form-gear” since it provided a universal manner of establishing the path of contact that provides desirable sliding conditions during the meshing cycle. The proposed tooth form, interference condition, contact ratio, contact stress, sliding coefficient, and meshing efficiency are derived and investigated in this study. The fillet stress and torsional mesh stiffness of the proposed gear pair are obtained using the finite element method (FEM). The performance of the new gear is compared to that of the standard involute and non-involute gear types. The results indicated that the method proposed in this study can be utilized to construct a gear pair with higher surface durability and strength than typical involute gear of the same size.

Family of Surfaces with a Common Special Involute and Evolute Curves

In this paper, we define the necessary and sufficient conditions for a parametric surface on which both the involute and evolute of any given curve lie to be geodesic, asymptotic and curvature line. Then, the first and second fundamental forms of these surfaces are calculated. By using the Gaussian and mean curvatures, the developability and minimality assumptions are drawn, as well. Moreover we extended the idea to the ruled surfaces. Finally, we provide a set of examples to illustrate the corresponding surfaces.

High-contact ratio spur gears with conformal contact and reduced sliding

The high-contact ratio (HCR) spur gear is a possible choice for a quiet and smooth gearing system. However, HCR spur gear of the involute type has several constraints, such as high sliding contact under large loads and interference problems in high speed reduction systems, so its benefits may not be fully offered. As a result, the primary goal of this research is to design a non-involute HCR spur gear with reduced sliding contact, enhanced fillet strength, better lubrication conditions, and no interference issues compared to the traditional HCR involute pair. A generalized mathematical model for defining the gear tooth profile and its geometrical aspects is built based on the path of contact shape. The involute curve length parameter and maximum pressure angle during the meshing cycle are only introduced to characterize this model. This model is utilized to create a new non-involute HCR gear pair that provides a free-form tooth profile to meet the design requirements. To demonstrate this method, certain case studies are provided, and the analytical results are validated using the finite element method. The new gear's performance is compared to the traditional involute gear that is currently in use. The results showed that by employing the proposed method, the HCR spur gear can achieve better surface durability and fillet strength.

On Involute-Evolute Curve Pair in Semi-Euclidean Space

In this study, a kind of generalized involute and evolute curve pair is considered in 4 dimensional semi Euclidean space with 2 index. The curvatures and the associated Frenet Frame of this kind of generalized
 involute-evolute curve pair are presented.

Numerical Analysis of A Nonlinear Elastic Composite Leaf Spring

In this study, numerical analysis of a composite spring with a nonlinear elastic force-displacement relationship was performed. Due to the lightness and durability of composite materials, their use in the automotive industry is increasing and production methods allow for a high performance spring design. The behavior of a pair of composite leaf springs working in tandem with the cross-section profiles of the leaf springs to be obtained by using the involute curve of the spur gear pairs under load was examined in detail by numerical method, and the desired behavior was obtained. In the analyzes performed using Ls-Dyna software, samples were produced and many mechanical tests were performed for the determination of material parameters. The results obtained from these tests were used as the required material parameters in numerical analysis. The leaf spring thickness that will give the desired non-linear elastic behavior under 400 N load used in numerical analysis has been determined and successfully targeted force-elongation behavior has been created.