Tooth Undercutting Research Articles

Nonlinear dynamic analysis of a cycloidal ball planetary transmission considering tooth undercutting

A cycloidal ball planetary transmission (CBPT) is a new type of transmission system with a large transmission ratio and a phenomenon of tooth undercutting. Nonlinear dynamic differential equations of the CBPT in the undercutting state are constructed. Firstly, the complete tooth profile equations of cycloidal grooves are established. With the increase of the radius of the deceleration balls, the tooth undercutting is firstly occurred in the inner part, and then in the outer part of the epicycloid grooves. Secondly, considering time-varying meshing stiffness, damping, meshing state, pre-tightening and external excitation, a strong nonlinear coupling dynamic model (70-DOF) of translation-torsion is established to further construct the nonlinear differential equations of the CBPT according to Lagrange equation. Thirdly, the response signal of the system is decomposed into low frequency and high frequency response signals. Fourthly, the nonlinear dynamic characteristics with three parameters (the damping, radius of rolling circle, short amplitude coefficient) are investigated. Finally, the effectiveness of the proposed model has been validated in part by testing vibration signals of CBPT. The stability of the system is affected by the damping, radius of rolling circle and short amplitude coefficient, which will provide useful support for the design of CBPT.

Wpływ modyfikacji asymetrycznej zarysu na odkształcenie zębów kół zębatych

The aim of the article was to determine the value of deformation of straight teeth with an asymmetrical outline with variation of the pressure angle and minimization of tooth undercutting. The geometry of the gear was developed as a parametric model in the NX 10 system. Simulation tests of tooth deflection were performed using the Pre/Pos module available in the NX10 program.

RETENTION AND FATIGUE RESISTANCE OF PEEK AND ACETAL THERMOPLASTIC RESIN CLASPS

Aim; The objective of this study was to evaluate the retentive force of clasps made from two thermoplastic resins by the insertion/removal test simulating two years. Methods: Over standardized premolar and molar metal crowns 36 clasps were fabricated, including 18 clasps (1.5 mm thick) from each of the following thermoplastic resins: polyetheretherketon (PEEK) and polyoxymethylene (POM). Each clasp assembly was subjected to an insertion/removal test on its abutment crown. Retention of each clasp at pre-test (Baseline) and after insertion/removal cycling intervals (360, 730, 1080, 1440, 2116 and 2880 cycles by chewing simulator) were measured by applying withdrawal force to it using universal testing machine. To analyze the retention over the course of insertion/removal test, retention was measured every 360 cycles. Data were statistically analyzed using 3-way ANOVA (α= 0.05). Results. PEEK Resin material group recorded statistically significant higher mean value than Acetal material group mean value (P=0.0013 0.05). Retention decreased significantly as indicated by three-way ANOVA (P= 0.007<0.05). Conclusion: Retention forces of PEEK clasps were clinically acceptable, and superior to retention forces of Acetal clasps. However, both the retention of adequately designed resin clasps might be enough for clinical use. Clasps made of PEEK showed the more promising retentive force with a non-fracture tendency at tooth undercut.

Numerical Algorithm of Tooth Profile of Noncircular Gear Pair

According to the generating method theory, noncircular gear with all convex pitch curve can be processed by generating, author designed noncircular gear tooth profile, described a method of precisely calculating noncircular gear tooth profile. And proposed automatically optimizes the noncircular gear tooth profile meshing curve part separately by the identical rack cutting tool different side envelope. Through rack cutting tool along two gear center of rotation segment deviation pitch curve change modified quantity to avoid tooth undercut, realization nonzero modified. The driving gear and the driven gear tooth profile meshing curve part separately by the identical rack cutting tool different side envelope, guarantees when meshing zero clearance. The noncircular gear machined by this method has been used in real machine, which verifies the feasibility of the method.

Tooth flank, undercutting and tooth pointing of spherical gears

Abstract By the developed mathematical model of spherical gears from literature [16], conditions of tooth undercutting of spherical gears with convex and concave teeth and tooth pointing conditions of the proposed concave spherical gear are derived. Besides, tooth flank profile deviations, limit curves of tooth undercutting and the beginning points of tooth undercutting and tooth pointing of the proposed spherical gear can be obtained by applying the spherical gear tooth mathematical model and the developed computer simulation programs, as illustrated by five numerical examples.

Computerized Tooth Profile Generation and Undercut Analysis of Gears Manufactured With Pre-Shaving Hobs

This paper investigated the feature of pre-shaving hob contour and the generated gear tooth profile. By tooth generation method, a complete geometry of the gear tooth can be mathematically derived in terms of the design parameters of the pre-shaving hob cutter. The mathematical model consisted of equations describing the generated fillet and involute profiles. The degree of undercutting and the radii of curvatures of a fillet were investigated by considering the model. Finally, a computer program for generating the profile of the gear teeth was developed by simulating the cutting methods. The methods proposed in this study were expected to be a valuable guidance for pre-shaving hob designers and manufacturers.

Tooth profile generation and analysis of oval gears with circular-arc teeth

Abstract An oval gear drive, which is used in an oil pump with larger pumping volume, is generated by using a rack cutter with circular-arc teeth. The generated oval gears have convex–concave tooth profile contact. The mathematical models of oval gears with circular-arc teeth are derived based on the theory of gearing and gear generation mechanism. Owing to its complex geometry, the pitch curves and tooth profiles of oval gears may exhibit convex–concave centrodes and tooth undercutting, respectively. A rack cutter with circular-arc teeth cannot be used to generate oval gears with convex–concave centrodes. Therefore, curvature analysis is utilized to design oval centrodes with convex curves. The equation of singular point is utilized to analyze tooth undercutting. In addition, a computer simulation program is developed to generate the tooth profiles of oval gears with circular-arc teeth having convex centrodes and without tooth undercutting. Various numerical examples illustrate the computerized design process.

1005 Webサービス技術を導入した歯車設計に関する研究 : 歯たけと圧力角が、歯先の尖りと切り下げに及ぼす影響(生産加工・工作機械II,生産システムI)

1005 Webサービス技術を導入した歯車設計に関する研究 : 歯たけと圧力角が、歯先の尖りと切り下げに及ぼす影響(生産加工・工作機械II,生産システムI)

Tooth profile design for the manufacture of helical gear sets with small numbers of teeth

Based on gear theory and generating mechanism, this investigation presents a complete mathematical model of a helical gear set with small number of teeth. The unavoidable tooth-profile undercutting of the gears with small number of teeth is examined by using the developed mathematical model and the conventional method of tooth-profile shifting. Furthermore, an alternative method for lessening the tooth-profile undercutting is also presented by considering a modification of the basic fillet geometry using a modified rack cutter. A third method, combining the aforementioned two methods for the design of helical gears with small number of teeth is also proposed to yield a gear set without tooth undercutting. The mating gear with profile shifting is generated using the pinion as a shaper. The tip fillet and root fillet are modified and a clearance between the pinion and the mating gear is also included in the design. Analysis results indicate that the change of distance between the centers of gear set depends only on gear shifting. Moreover, computer graphs are demonstrated the profile-shifted and the proposed modified gear tooth profiles.

Computerized tooth profile generation and analysis of characteristics of elliptical gears with circular-arc teeth

This study describes rack cutters with circular-arc profile teeth to generate elliptical gears which rotate about one of their foci. A mathematical model for elliptical gears with circular-arc teeth is developed according to gear theory. The influence of the design parameters, such as the number of teeth, gear module, pressure angle at the pitch point and the major-axis, are investigated. The effects of the circular-arc radius of the rack cutter on both the undercutting of teeth and on pointed teeth of the generated circular-arc elliptical gear are also studied. Three numerical examples are presented to elucidate the generation of gear tooth profiles using the proposed mathematical model, and to investigate the phenomena of tooth undercutting and pointed teeth.

Pitch Cone Design and Avoidance of Contact Envelope and Tooth Undercutting for Conical Worm Gear Drives

This paper presents a systematic approach for the design, generation and analysis of conical worm gear drives. The approach is based on the mathematical formulations on the mesh, tooth generation process and conditions for the appearance of contact envelope and tooth undercutting of this type of worm drives. The formulation is implemented by a computer model developed in MatLab environment. By model simulation on worm gear mesh or generation, the conditions for the formation of contact envelope and tooth undercutting are accurately determined for the proper design of worm gear blank and tooth geometry. The computer model can also be used to calculate worm gear tooth geometry and create solid models for the worm gear drives. Numerical examples on conical worm gear drive design, analysis on the problems of contact envelope, limit line and limit point are included in the paper.

Computer Aided Design of Elliptical Gears

An elliptical gear drive, whose rotation axis coincides with its geometric center, is simulated using generation with a rack cutter. The mathematical model of an elliptical gear based on the theory of gearing and gear generation mechanism is also developed. Owing to its complex geometry, the profile of elliptical gears may exhibit tooth undercutting, and pointed teeth. In addition, tooth undercutting of an elliptical gear based on the developed mathematical model of the elliptical gear and the theory of gearing is also investigated. Pointed teeth usually appear on the major axis of the gear elliptical pitch curve. Moreover, a geometric relation is developed and applied to prevent the pointed teeth appearing on elliptical gears. Furthermore, a computer simulation program is developed to generate the tooth profile of elliptical gears without tooth undercutting and pointed teeth. Various numerical examples illustrate the effectiveness of the computerized design process.

Computerized tooth profile generation of elliptical gears manufactured by shaper cutters

This work simulates an elliptical gear drive, the axis of rotation of which is coincident with its geometric center, manufactured by shaper cutters. The mathematical model of an elliptical gear is developed based on the theory of gearing and gear generation mechanisms. In addition, the tooth undercutting of the gear is also investigated based on the developed mathematical model of the elliptical gear, its unit normal vectors and a numerical method. A geometric relationship is developed and applied to prevent the occurrence of pointed teeth on elliptical gears. Further, this study also develops computer simulation programs to generate the tooth profile of elliptical gears without tooth undercutting and pointed teeth. Comparison of the angular velocity variations of the elliptical gear drives is also made. The results show that the developed elliptical gear drive can be utilized as an oil pump with a larger pumping volume and less angular velocity variation.

Mathematical model and undercutting of cylindrical gears with curvilinear shaped teeth

A rack cutter with a curved-tooth is considered the generating tool for the generation of the proposed gear, and a mathematical model of cylindrical gears with curvilinear shaped teeth is developed according to the gearing theory. The developed computer program can develop profiles of the curvilinear-tooth gear with point contacts. The developed gear tooth mathematical model is used to investigate the tooth undercutting of curvilinear-tooth gears via differential geometry and the numerical method.

Computer aided design of non-standard elliptical gear drives

An oil pump with a larger pumping volume and more compact size than conventional pumps is designed with elliptical gear drives. In this work, an elliptical gear drive with few teeth is used in the design of pumps with a large pumping volume. Also, a mathematical model of a profile-shifted elliptical gear is developed, based on the theory of gearing and the gear generation mechanism. This study also investigates the tooth undercutting of a non-standard elliptical gear, based on the proposed mathematical model of the profile-shifted elliptical gear and the theory of gearing. Pointed teeth generally appear on the major axis of the gear elliptical pitch curve. Moreover, a geometric relation is developed and applied to prevent pointed teeth on non-standard elliptical gears. Moreover, a computer simulation program is developed to generate the tooth profiles of non-standared elliptical gears with few teeth and a compact size, without tooth undercutting and pointed teeth. Numerical examples demonstrate the effectiveness of the computerized design process.

Tooth Undercutting of Beveloid Gears

A beveloid gear can be viewed as an involute gear of which the profile-shifted coefficient linearly decreases from the heel to the toe. Therefore, tooth undercutting occurs and singular points appear on the tooth surfaces near the toe. When undercutting occurs, the gear tooth is comparatively weak. In this study, the conditions of tooth undercutting of beveloid gears were derived and specific phenomena were also investigated by numerical illustrated examples. In addition, according to the characteristics of tooth undercutting on the beveloid gear tooth surface, two practicable methods were also proposed to avoid the tooth undercutting of beveloid gears.

Computerized Tooth Profile Generation and Undercut Analysis of Noncircular Gears Manufactured With Shaper Cutters

Pitch curves of a conjugate noncircular gear pair are derived based on kinematic considerations. A method for considering the inverse mechanism relationship and the equation of meshing, is proposed here to derive a complete mathematical model of noncircular gears manufactured with involute-shaped shaper-cutters. The proposed method is similar to the contact line method and the envelope method, but is easier to apply to the determination of tooth profiles. A computer program is developed for generation the tooth profile with backlashes. Undercutting analysis is also investigated by considering the relative velocity and equation of meshing. Finally, modified elliptical gears are presented to illustrate the tooth profile generation when the proposed mathematical model is applied, and to investigate the phenomenon of tooth undercutting.