Rack Cutter Research Articles

Mathematical model and generation analysis for crossed helical gear system

Purpose Crossed helical gears have point contact and their surfaces are subject to high surface stress. Contact stress and root tooth stress are the most common sources of failure in crossed helical gear. This paper aims to study the load-carrying capacity and performance of crossed helical gear teeth with different gear tooth profiles. To overcome defects and reduce the sensitivity to small shaft angle changes. The combined tooth profile is designed to reduce the bending stresses, contact stresses and tooth deflection, and prevent pinion failure in the gearbox. Design/methodology/approach The principle of the method is a line contact is introduced instead of a point contact between two teeth in mesh with each other. The tooth surface of the helical gear is designed and cut by a modified tool. Higher normal pressure angles like 25° and 35° are used. The modified involute is accomplished to eliminate interference between the teeth. Engineering software packages have been applied to generate all crossed helical gears gear profiles. The modification is compounding curves consisting of an epicycloid-involute-hypocycloid gear teeth profile generated by the cutter modified. Findings The stresses in the crossed helical gear teeth profile were reduced by increasing the normal pressure angle values. Using a 35° pressure angle the enhancement percentage in contact stress and teeth fillet region will be about 29.345% and 15.421%, respectively. The best enhancement in a gear’s resistance is the epicycloid-involute-hypocycloid gear teeth profile. The enhancement was 32.610% and 18.588%. The skew in line of action in skewed helical gear will be sensitive when the crossing angles are small. Their teeth surface tends to be easily worn out; however, the wearing process will be reduced by using a proposed gear teeth profile. Practical implications The gear teeth to be modified are cut by a shaper process. The modifying rack cutter of this study can be used as a reference for creating a different helical gears sample. The helical teeth surface is modified to become an envelope of the other. This makes an original point contact change into a line contact. The epicycloid-involute-hypocycloid gear teeth profile is preferred for a higher contact ratio and a large load capacity. This work explicitly introduces a new method of kinematic consideration to improve the load capacity of crossed helical gear. Originality/value This paper showed some novel results by the unique shape of the rack-cutter designed to generate different helical angles and different gear positions. In the future, it will make valuable contributions to the further development of the dynamic performance of a crossed helical gear system through the study in the field of using asymmetric teeth profiles of helical gears with tip relief as the manner to enhance the crossed helical gear performance. Investigation of crossed helical gear by applying a predesigned parabolic function of transmission error enables the absorption of linear discontinuous functions caused by misalignment.

Manufacturing and contact characteristics analysis of a novel 2K-H internal meshing planetary reducer with involute-cycloid combined tooth profiles

Rotate Vector (RV) reducer involves multiple crankshafts to drive the cycloidal-pin transmission in parallel, leading to an over-positioning structure and high manufacturing and assembly requirements for the reducer, restricting the development of industrial robots. In view of the above problems, a novel configuration of Abnormal Cycloidal Gear (ACG) reducer with a 2K-H internal meshing planetary transmission is proposed. The design innovatively replaces the multiple crankshafts with a pin mechanism, thereby addressing the over-positioning and simplifying the manufacturing and assembly processes. To improve the performance of the reducer, the “epicycloid-involute-hypocycloid” combined tooth profiles are designed and adopted as the ACG reducer’s drive teeth. The configuration principle of the ACG reducer is conducted. Next, a manufacturing method utilizing a rack cutter to generate the profile of drive teeth is presented. Finally, the meshing performance and mechanical behaviors of the designed internal meshing pair for the ACG reducer are evaluated. Results show that ACG reducer has the advantages of a simple structure and large transmission ratio. The combined tooth profiles can be machined at one time using the designed cutter, simplifying the tooth machining process. The internal meshing pair is linear contact and exhibits an obviously multi-teeth meshing effect, which benefits the carrying capacity and transmission performance of the ACG reducer.

Analysis of arbitrary tooth profiles of cylindrical gears using normal polar coordinates (Application to the generation of a gear tooth profile by a given tooth profile of rack cutter and its interference problem)

Analysis of arbitrary tooth profiles of cylindrical gears using normal polar coordinates (Application to the generation of a gear tooth profile by a given tooth profile of rack cutter and its interference problem)

Balanced bending fatigue life for helical gear drives to enhance the power transmission capacity through novel rack cutters

This article proposes an idea for balancing the bending fatigue life between the gear and pinion of a novel helical gear drive. In this study, the helical gear is developed through analytical equations based on a novel rack cutter tooth thickness factor, and the estimation of principal stress and strain at the critical root fillet of the helical gear teeth is carried out through finite element analysis. Further, the fatigue module in COMSOL- Multiphysics software is employed to determine the bending fatigue life through strain life approach based on Smith-Watson-Topper with Neuber’s rule and fatigue usage factor based on Findley criteria. The effect of major gear parameters such as helical angle, pressure angle, teeth number, gear ratio, addendum height and addendum modification factor (S+, S− and S0) on bending fatigue life are computed based on novel helical gear with different rack cutter tooth thickness factor of gear and pinion. Finally, the above parametric study shows an optimum rack cutter tooth thickness factor between the gear and pinion to achieve a balanced bending fatigue life and fatigue usage factor, which aids in improving the power transmission capacity of the helical gear drive.

Proposed gear rack cutter for spur gear manufacturing by generating method. Part I: Theoretical and strength analysis arguments

The paper presents the cutting by generating of spur gears using a rack cutter with a modified geometry. In order to find the geometry of the new spur gears, a proper programme has been developed. The geometry of this spur gear is compared with one of a gear with the same parameters but obtained with a standard rack cutter. The differences between the two gears are insignificant but the new tool has a simpler geometry and a higher durability.

Proposed gear rack cutter for spur gear manufacturing by generating method. Part II: Comparison between the standard and modified profiles

The paper proposes a method of cutting by generating of spur gears using a rack cutter with a modified geometry. The analytical equations of the tooth profile of the modified rack-gear cutter are deduced, both for involute zone and for filleting region, based on the remark that the profile is the envelope of successive positions of the tool. The undercutting radius for the modified profile is obtained from a system of transcendental equations which requires and a numerical method for solving it. Compared to the standard profile, the differences are insignificant. A program was written and the profiles of the teeth are represented for two shifting coefficients, for the standard and the modified tooth profiles.

Fortification of diets with omega-3 long-chain polyunsaturated fatty acids enhances feedlot performance, intramuscular fat content, fat melting point, and carcass characteristics of Tattykeel Australian White MARGRA lambs.

Meat eating quality indices such as intramuscular fat content (IMF) and fat melting point (FMP) of the Longissimus thoracis et lumborum muscle and the feedlot performance, carcass traits, and commercial wholesale cuts of lot-fed Tattykeel Australian White (TAW) MARGRA lambs as a result of dietary fortification of the diet with omega-3 long-chain polyunsaturated fatty acids (n-3 LC-PUFA) were evaluated. A total of 75 TAW MARGRA lambs at 6 months of age with an average liveweight of 30 ± 1.2 kg were used. The lambs were randomly allocated to the following three dietary treatments of 25 lambs each in a 47-day feeding trial using a completely randomized experimental design: (1) control diet of hay plus pellets without omega-3 oil, (2) hay plus commercial whole grain pellets (MSM) without omega-3 oil, and (3) hay plus pellets fortified with omega-3 oil. It was hypothesized that dietary supplementation with omega-3 fortified pellets will improve feedlot performance, meat-eating quality indices of IMF, FMP, and carcass characteristics. Lot-fed lambs on the MSM whole grain had the highest feed intake of 1.69 kg/day, followed by the control at 1.57 kg/day and the lowest in the omega-3 diet at 1.01 kg/day (p = 0.0001). However, the omega-3 diet had the highest average daily gain of 230 g/head/day (p = 0.0001), indicating the greatest feed efficiency since it had the best growth response with minimal feed intake. Post-slaughter evaluation of the Longissimus thoracis et lumborum muscle revealed significant treatment variations in IMF (p = 0.0001), FMP (p = 0.0001), pH (p = 0.0380), and wholesale French rack primal cut (p = 0.0001). Strong correlations (p < 0.05) between liveweight, temperature, pH, FMP, and IMF were observed. Similarly, significant correlations between carcass characteristics of total saleable meat yield, lean trim, fat trims, bones, and leg shank were evident (p < 0.05). However, there were no treatment differences in the final liveweight, GR fat depth, hot standard carcass weight, or dressing percentage. The findings indicate that feedlot performance, meat-eating quality traits such as IMF and FMP, and commercial wholesale French rack cuts can be further improved during feedlot finishing of TAW MARGRA lambs through dietary supplementation with omega-3 oils, and hence the tested hypothesis of improved meat quality attributes is partially confirmed.

A Novel Curve for the Generation of the Non-circular Gear Tooth Profile

This paper presents a novel curve generated by a point attached to an ellipse as it rolls without slipping along a datum line of rack cutter. A mathematical model of the non-circular gear profile has been developed based on the theory of gearing. The effect of an axial ratio l (the ratio of the lengths of the major and minor axes of the ellipse) and the position of the point KR, at which the novel curve starts to generate on the tooth shape and the undercut of the non-circular gear pair is also taken into consideration. A numerical program developed from the mathematical model has been proposed for the calculation and design of the non-circular gears (NCGs). Case studies are presented to show the steps of tooth shape design and to examine the geometrical profile of the NCGs in relation to design parameters of the rack cutter etc. From that, the axial ration l and position of the point KR on the generating ellipse ƩE can be selected for each specific case in order to design the appropriate profiles of the NCGs. On that basis, an experiment to determine the gear ratio of the NCGs pair based on the meshing between gears has manufactured.

Double modification of a rack cutter using a variable pressure angle and modulus

This study modifies the straight edge of the rack cutter into an arc shape by changing the pressure angle. Using a variable modulus, the tooth surface is longitudinally modified along the axial direction of the rack cutter to create a bulge in the rack teeth in the center of the tooth width. This rack cutter with a double-crowned tooth is used to generate a pinion. An unmodified rack cutter is used to generate a gear. An undercutting analysis of the rack cutter is used to determine the undercutting condition for the double-crowned rack cutter and to determine the limits for the parameter. The transmission error for a gear pair is calculating using a tooth contact analysis. For this study, the assembly error for six cases is used to calculate the transmission error for the gear pair and the contact points for the gear pair are placed on a gear tooth. These contact points used to determine the contact for the gear pair. A transmission error analysis of the gear pair for a double-crowned gear gives a parabolic curve for the transmission error. The distribution of the contact points shows that, regardless of assembly errors, the contact points for the gear teeth are along the center of the tooth surface, starting from the top of the tooth, and gradually moving to the flank and away from the heel. The results show that the values for the transmission error for the modified gear pair for the proposed modification of the variable pressure angle and the variable modulus on a traditional rack cutter are small and the transmission error of the assembly errors are almost similar to those for an ideal assembly.

Using variable modulus to modify a rack cutter and generate a gear pair

In this paper, two normal imaginary helical rack cutters were first established. One of these cutters is a skewed-rack cutter with an asymmetrical straight edge. The other is a rack cutter with an asymmetric parabolic profile. Second, the gear’s tooth surface of the asymmetric parabolic rack cutter is modified to be barrel-shaped based on a variable modulus. The tooth thickness of the gear is gradually reduced along the face width of the tooth from the middle of the tooth surface. Then the coordinate relationship between the gears’ blanks and the imaginary helical rack cutters was established. Through the differential geometry, crowned and uncrowned helical gear pairs were generated. Because of human factors, when the gear pair is installed, it is easy to cause the gear pair edge contact. It is necessary to add artificial assembly error settings through the tooth contact analysis to investigate the kinematic errors and contact conditions of the crowned and uncrowned helical gear pair. The mathematical models and analysis methods proposed for the crowned imaginary rack cutter using variable modulus should be useful for the design and production of double crowned helical gears with asymmetric parabolic teeth.

Bone mineral concentration predicted by dual energy X-ray absorptiometry and its relationship with lamb eating quality

Lumbar bone mineral concentration, as predicted by dual energy x-ray absorptiometry (DEXA), may reflect changes in lamb maturity and eating quality. New season (n = 60) and old season (n = 60) lambs were slaughtered and DEXA scanned at a commercial abattoir across 2 kill groups. The second lumbar vertebra was isolated from the spine for determination of calcium, phosphorus, and magnesium concentration (mg/g). The loin and rack cuts were collected for consumer sensory grilling and roasting analyses. Mineral concentration was significantly higher in old season lambs within kill group 1 (P < 0.05). DEXA was a positive predictor of phosphorus and calcium concentration, but only when DEXA lean % (P < 0.05) was included in the model. Calcium and phosphorus were significant positive predictors of overall liking scores (P < 0.05), but only for the rack roast. These effects became insignificant when DEXA lean % was included. These results suggest that DEXA values likely reflect changes in both DEXA lean % and bone minerals, and that DEXA lean % was the driver of eating quality, rather than maturity.

Research and Manufacturing of Oval Gear Pair Applied in Rotor of a Roots Type Compressor

This study presents a method of shaping the tooth profile of an oval gear pair with the modified cycloid profile of the ellipse. The driving oval gear was shaped by a rack cutter with the profile of an improved cycloid, and the tooth profile of the driven oval gear was shaped by a shaper cutter to ensure that gear pair will mesh accurately with a function of transmission ratio. Mathematical model of the oval gear pair was established according to the theory of the non-circular gear with consideration to the conditions of the design parameters to avoid undercutting and tooth distribution of the oval gear. An improved pair of cycloid oval gears is designed and manufactured according to the results of the theoretical calculation. Furthermore, this gear pair is specifically applied to a Roots-type compressor to demonstrate its applicability in special cases where the traditional cylindrical gear pair cannot be replaced.

Issues related to an attempt to recreate the geometry of a non-standard spur gear

Modern machine manufacturers are making the design and technology of their products more and more complicated. This is to protect against a frequently used practice at customers, i.e. making extra parts on your own. This is because entrepreneurs often cannot afford to order expensive original spare parts and - using reverse engineering - prepare working drawings and commission the components to be made in their own machine park or externally from local suppliers. However, the matter is more complex in the case of gears, which so far have been designed on the basis of the selection of standard geometric parameters. A small modification of one or more of these parameters is enough and it becomes very difficult to recreate the geometry of such a gear. This paper presents the issues related to the reverse engineering of a spur involute gear with very non-standard parameters m = 4.98, α = 26.325 °, x = 0.0695, y = 0.795, c* = 0.383. Further metrological steps were proposed that should be taken to correctly identify at least the fact that the test object is not a part produced by standard modular tools (Fellows cutter, Maag rack cutter, worm cutter, etc.). The work also includes short graphical analyzes of the recreated geometry.

Effects of pressure angle on uneven wear of a tooth profile of an elliptical gear generated by ellipse involute

This paper presents a method to establish a mathematical model of an elliptical gears pair's tooth profile slip curve. The gear has an ellipse involute tooth profile generated by a rack cutter. The profile equation was developed based on the gearing theory with centrodes and center distance and consideration to conditions for avoiding undercutting. An elliptical involute curve and design parameters of the rack cutter were established corresponding to different points on the centrode of the elliptical gear. The profile equation was developed based on the theory of gearing with given centrodes and axis distance and consideration to conditions for avoiding undercutting. An elliptical involute curve and design parameters of the rack cutter determined corresponding to different points on the centrode of the elliptical gear. Wrote a number calculation program was in Matlab to examine and select appropriate pressure angles to limit the slippage between the mating teeth, to reduce wearing and enhance the lifespan of the elliptical gear pair. Case studies were also presented to confirm the influence of the pressure angle on the profile slippage phenomenon and select the rack cutter's pressure angle to reduce uneven wearing between the teeth of one elliptical gear. The survey results show that when the pressure angle of the rack cutter is large, the profile wear will occur faster since the ramp tooth rib, shorter length of the line of action, and pointed tooth tip. These research results are meaningful in selecting the rack cutter's pressure angle to gear the non-circular gear has tooth profile is an involute ellipse curve.

Shaping the tooth profile of elliptical gear with the involute ellipse curve

Non-circular gears (NCGs) with the advantage of compactness and simple mechanical structure in creating transmitters with variable gear ratios in the industry, so this is a topic that is being studied by many scientists. However, up to now the research usually focuses on NCGs with tooth profile that is an involute circle called involute tooth profile, while other curves have not been applied in the design of the non-circular gears. In this paper, the author proposes to apply the involute ellipse curve in shaping the tooth profile of the pair of elliptical gears with a rack cutter. A rack cutter with tooth profiles is trapezoidal weight is determined from the involute ellipse curve selected as the NCGs teeth. In order not to undercut and tip sharpening when shaping the gear, two algorithms for correcting the base ellipse according to the even distribution of teeth on the gear and undercutting avoidance condition are performed. A numerical computational software module is written according to the theory and the algorithm is implemented to explore the different cases in the design process. The results of this study show that the teeth are evenly distributed over the entire circumference of the ring elliptical gear, overcoming the phenomenon of uneven teeth in geometry and size when using the involute of a circle as other studies have shown. Also, with the tooth profile that is involute ellipse curve, the pressure angle is increased to improve efficiency the transmission capacity of a pair of gears. Therefore, this is the advantage of the new profile in creating variable automatic transmission for different script applications in industries such as the automotive industry or modern medical equipment.

Estimation of loss factor based on the load share model in improved bending strength spur gear drive system

This paper demonstrates the prediction of loss factor for non-standard gear drives with equal stresses in the root region of the pinion and wheel. This implementation of equal stresses in the root region is achieved in the described model of a spur gear drive with higher transmission ratio using non-standard rack cutter. The aim is to discover the optimal modification point in the tooth profile by varying values of tooth thickness with at most balanced stresses in the gear pair considered for the study. A loss factor is predicted along the path of contact for a complete meshing cycle using finite element method. This paper studies and compares the distinct models and thereby identifies the suitable and appropriate models. The predicted loss factor includes the distribution of load sharing effect during the meshing which is excluded in the compared models. The gear pair is analyzed, and the loss factor is predicted for influence over the major drive parameters. It was observed that the approach employed facilitate substantial enhancement in the efficiency which is confined by the detailed parametric study and numerical simulation of the balanced stresses in the drive.

Using a planar rack cutter with variable modulus to generate a spherical gear pair

In this study, an imaginary planar rack cutter with variable modulus and discrete conical teeth was used to generate a spherical gear pair with double degree of freedom. First, a geometric method was used to design the mathematical model of the imaginary planar rack cutter with variable modulus and discrete conical teeth. Next, the relationship of coordinate systems between the generating and generated surfaces was established. Then, the family of the imaginary planar rack cutter surfaces was obtained by homogeneous coordinate transformation matrix. Further, two equations of meshing between the generating and generated surfaces were determined by the two-parameter envelope theory. The mathematical model of spherical gear pair with variable modulus and discrete ring-involute teeth can be created by using the two equations of meshing and the family of the imaginary planar rack cutter surfaces. The mathematical models of the spherical gear pair with double degree of freedom and tooth contact analysis method were used to investigate the assembly errors of the gear pair in four cases.

Mathematical Modelling of the Profile Shifted Helical Gears

Bu çalışmada evolvent profilli helisel dişli çarkların kremayer takımla imalatının matematik modellenmesi ele alınmıştır. Litvin’in vektör yaklaşımından hareketle takım ve imal edilen dişli çark geometrisini tayin eden ifadeler verilmiştir. Modellemede profil kaydırma ve asimetrik diş profili de göz önüne alınmıştır. İmal edilen dişlide evolvent bölgenin üst sınırının analitik tayini araştırılmıştır. Bir bilgisayar programı geliştirilerek tasarım parametrelerinin imal edilen dişli geometrisindeki etkileri incelenmiştir.

A helical gear with discrete ring-involute teeth

This study produces a skewed-imaginary planar rack cutter with discrete conical teeth that is used to create a helical gear with discrete ring-involute teeth. A mathematical equation for the skewed-imaginary rack cutter with discrete conical teeth is firstly solved. The coordinate system for the rack cutter and gear pair is then established and a family of the rack-cutter surfaces is obtained using homogeneous coordinate transformation. The relative velocity method is used to produce the equation for meshing between the rack cutter and the gear pair. Substituting the equation of meshing into the family of the rack-cutter surfaces gives the mathematical models for the gear pair with discrete ring-involute teeth. The transmission error for the gear pair is calculated using the assembly error and a tooth contact analysis. A computer-aided design software package is used to establish solid model for the gear pair. A software interference function is used to simulate the contact condition for the gear pair for various assembly errors. Finite element analysis software is then used to determine the contact stress for the gear pair. The transmission error and the contact stress for the gear pair are insensitive to any horizontal misalignment in the assembly errors.

A novel non-involute gear designed based on control of relative curvature

The flank shapes of gears greatly affect tooth durability, and the relative curvature and sliding of tooth profiles are the main influences. This paper advances a design method of tooth profiles based on control of relative curvature of engaged tooth profiles. The mathematical model of the spur gear with constant relative curvature is built through employing the proposed method. Equations of conjugate tooth profiles, the virtual rack cutter and undercut condition are formulated according to principle of plane engagement. Operating Performance of the new gear drive is evaluated by comparing with that of the involute gear drive. The results show that this new type of gear drive has the following advantages:With constant relative curvature of engaged tooth profiles, it enhances the contact strength of tooth surfaces and abates maldistribution of contact stress on tooth surfaces.It decreases the wear of tooth surfaces, especially the dedendum of the pinion due to the smaller sliding coefficient.It improves the lubrication of tooth surfaces owning to the higher minimum film thickness.