Conjugate Surface Theory Research Articles

Tooth surface Hobbing analysis for unconventional gears with ultrasonic-assisted motion

As an atypical conventional gear pair, non-circular gear has a variable transmission ratio, which can be used to improve the structure and transmission characteristics in some applications, such as variable speed grinding mechanism, wool blending machine guide mechanism, continuously variable transmission structure and so on. According to the friction theory of conjugate surfaces, the transmission characteristics will be affected by the micro topography of tooth surface, but the shaped tooth surface of non-circular gear with normal hobbing method is not uniform, because of its variable radius and flank orientations of tooth surface. The research on the relationship between micro topography and parameters of non-circular gear is necessary, thus an improved manufacturing method with ultrasonic-assisted motion was presented in this paper, the mathematical equations for the theoretical tooth surface of non-circular gear with and without ultrasonic-assisted motion have been derived in this paper, also the equations for cutting parameters (step displacement, step rotating angle) have been proposed, shown that all the parameters (initial parameters of gear and gear hob, cutting parameters in manufacturing process) will affect the formed micro surfaces. In detail, the height Hm and width Sm of micro undulating tooth surface periodically increase and decrease, getting the maximum value at θn=π, where is the maximum radius. The height Hm and width Sm of micro tooth surface will increase with any control parameter increase, the height Hm of most micro peaks is near 6.5um in normal hobbing process. While, the tooth surface with ultrasonic-assisted motion is more regular and clearer with the maximum height Hm decrease from 6.5 um to 2.7 um, getting more compact cutting tracks and lower surface roughness. The experimental results shows that ultrasonic-assisted hobbing has obviously positive influence for micro tooth surface, provides the theoretical support for a further analysis and better application of unconventional gear pair.

Investigation on the semi-conjugate tooth model and disc milling process of logarithmic spiral bevel gears of pure-rolling contact

Conventional logarithmic spiral bevel gears (LSBGs) introduce full-conjugate surfaces (line contact) to generate teeth of the pinion and the gear. However, in general, these full-conjugate surfaces are constructed by spherical involute curves and it is difficult to manufacture these surfaces. Therefore, this article tries to use semi-conjugate surfaces (point contact) as the tooth profiles to make they can be manufactured by the disc milling process and of pure-rolling contact. The design method, manufacture kinematics and stress distribution situations of the LSBGs with semi-conjugate surfaces are investigated. Conjugate surface theory and spatial conjugate curve meshing theory are both introduced to complete the analytical arguments. Finite element analysis (FEA) is introduced to evaluate the contact mechanical characteristics of the LSBGs under loads. From the analytical and simulated results, it is concluded that, through the disc milling process, the LSBGs of continuous pure-rolling contact can be manufactured and mesh correctly. Besides, the manufactured LSBGs maintain pure-rolling contact approximately when they are under loads.

Calculation and Characteristic Analysis of Tooth Width of Eccentric Helical Curve-Face Gear

Eccentric helical curve-face gear pair is a new type of gear pair consisting of a helical non-circular gear and an eccentric helical curve-face gear, which can implement the rotation of intersected axes as well as the movement of output axes. According to the conjugate surface theory and the spatial coordinate transformation theory, the coordinate system, the meshing equation and the tooth surface equation of eccentric helical curve-face gear are obtained. Based on the spatial gear engagement theory, the undercutting and pointing phenomenon of eccentric helical curve-face gear are analyzed, and equations of tooth width of eccentric helical curve-face gear are proposed. By changing the different parameters of eccentric helical curve-face gear pair, the influencing factors and variation of undercutting radius, pointing radius and tooth width are derived. Furthermore, by building the experiment devices of the roll testing and the transmission, the correctness of the calculation method of tooth width of eccentric helical curve-face gear is verified by using the contact analysis and the experimental verification.

A Study on the Design and Performance of Epicycloid Bevels of Pure-Rolling Contact

To avoid the negative influence of sliding contact, this paper tries to investigate the spiral bevels of pure-rolling contact that can be manufactured by existing manufacture technology. In this process, spatial conjugate curve meshing theory and conjugate surface theory are both introduced to investigate the geometric principles and face hobbing process of the pure-rolling contact epicycloid bevel (PCEB for short in this paper). The tooth surface models of PCEBs by face hobbing process are obtained. Next, a sample is represented to show an application of this model. Then, finite element analysis (FEA) is applied to investigate the contact mechanics characteristics of these gears. Finally, the performance experiment of a prototype is completed to evaluate the deviations between theoretical expectations and practical results. From the FEA and experimental results, it is concluded that the PCEBs can mesh correctly and achieve a higher transmission efficiency.

DESIGN AND ANALYSIS OF CONSTANT-BREADTH CAM MECHANISM WITH OSCILLATING FLAT-FACED FOLLOWER

A simple yet comprehensive method is presented for the design and analysis of a constant-breadth cam mechanism with an oscillating flat-faced follower. In the proposed approach, the kinematic characteristics of the cam mechanism are first derived. The cam profile is then designed using homogenous coordinate transformation and conjugate surface theory. Moreover, the sliding velocity at the cam-follower contact points is determined. Finally, the pressure angle of the constant-breadth cam mechanism and the principal curvatures of the cam are analyzed. The validity of the proposed design methodology is verified by means of motion simulations performed using CAD software.

Mathematical model for design and analysis of power skiving tool for involute gear cutting

Power skiving is an efficient means of producing high accuracy gears; particularly internal gears. However, the literature lacks a systematic mathematical model for the design and analysis of such tools. Accordingly, the present study proposes a comprehensive yet straightforward methodology for the design of resharpening power skiving tools based on conjugate surface theory and full-field angle analysis. The proposed methodology has three important features. First, mathematical models of the virtual-conjugate-surface and rake surfaces are derived, and thus the cutting edges can be obtained. Second, the power skiving is modeled by transformation matrices, and the normal/tangent vectors of the relative surface can be explicitly obtained. As a result, the working rake and clearance angles can be investigated in accordance with recommended ISO-standards. Finally, a method is prescribed for determining the working rake angles, clearance angles and wedge angles of the cutter based on the ISO-defined reference planes. Utilizing this method, a simple technique is proposed for redesigning the power skiving tool so as to avoid negative working clearance angles. The results of the illustrative example are shown that the proposed approach provides a comprehensive, simple and versatile technique for modeling a wide range of power skiving tool design features.

DESIGN AND ANALYSIS OF CAM MECHANISM WITH NEGATIVE RADIUS ROLLER-FOLLOWER

This paper presents a simple method for the design and analysis of a cam mechanism with a negative radius roller-follower. In the proposed approach, conjugate surface theory is employed to derive a kinematic model of the cam mechanism. Analytical expressions for the pressure angle and principal curvatures of the cam profile are then derived. Finally, analytical expressions for the angular velocity and angular acceleration of the roller are derived. The validity of the proposed design methodology is demonstrated by machining a cam mechanism having a negative radius roller-follower with a radius of 100 mm.

Design of Cam Mechanism with Negative Radius Roller-Follower

Cam mechanisms with a negative radius roller-follower allow the design of simple machines capable of exerting remarkably high forces. Furthermore, the concave-convex nature of the contact surface between the cam and the follower results in a low contact stress and the ready formation of a lubricating layer. This paper presents a simple method for the design and analysis of a cam mechanism with a negative radius roller-follower. In the proposed approach, conjugate surface theory is employed to derive a kinematic model of the cam mechanism. Analytical expressions for the pressure angle and principal curvatures of the cam profile are then derived. The validity of the proposed design methodology is demonstrated by machining a cam mechanism having a negative radius rollerfollower with a radius of 100 mm.

DESIGN AND ANALYSIS OF CAM MECHANISM WITH TRANSLATING FOLLOWER HAVING DOUBLE ROLLERS

The pressure angle is one of the primary considerations in designing a cam mechanism since an inappropriate angle may cause excessive sliding loads on the follower. This paper presents a simple yet straightforward method for the design and analysis of a cam mechanism with a translating follower having double rollers. In the proposed approach, conjugate surface theory is employed to derive a kinematic model of the cam mechanism. Analytical expressions for the pressure angle and principal curvatures of the cam profile are then derived. The validity of the analytical expressions is confirmed by machining a designed cam using a 3-axis CNC machine tool.

Design of Cam Mechanisms with a Translating Follower Having Symmetrical Rollers

The pressure angle is one of the primary considerations in designing a cam mechanism since an inappropriate angle may cause excessive sliding loads on the follower. This paper presents a simple yet straightforward method for the design and analysis of a cam mechanism with a translating follower having double rollers. In the proposed approach, conjugate surface theory is employed to derive a kinematic model of the cam mechanism. Analytical expressions for the pressure angle is then derived. The validity of the analytical expressions is confirmed by machining a designed cam using a 3-axis CNC machine tool

DESIGN AND ANALYSIS OF GENEVA MECHANISM WITH CURVED SLOTS

A simple yet comprehensive method is proposed for the design of a Geneva indexing mechanism with curved slots. In the proposed approach, conjugate surface theory is used to derive an analytical description of the profile of the curved slots with and without an offset feature. Analytical formulae are then presented for the pressure angle of the Geneva mechanism and the principal curvatures of the curved slots. The effectiveness of an appropriate offset angle in eliminating the singular points and double-points on the curved slot profile is then demonstrated. Finally, a Geneva mechanism is fabricated in order to demonstrate the feasibility of the proposed approach.

Design and analysis of indexing cam mechanism with parallel axes

The indexing cam mechanism is one of the most crucial mechanisms in automated machinery. This study proposes a simple yet comprehensive method for the design and analysis of an indexing cam mechanism with parallel axes. In the proposed approach, an analytical description of the driving cam profile is obtained using coordinate transformation and conjugate surface theory. Formulae are then derived for the pressure angle of the indexing cam mechanism and the principal curvatures of the driving cam. Finally, the kinematic characteristics of the indexing cam mechanism are analyzed. The validity of the proposed design methodology is verified by machining an indexing cam on a 3-axis CNC machine. The results confirm that the proposed methodology enables the design and manufacturing tasks to be successfully integrated, thereby making possible a flexible, automatic, cost efficient and controllable production process.

Digital Conjugate of Tooth Surface of Logarithmic Spiral Bevel Gear

Based on the surface of the envelope principle and the digital conjugate surface theory, The taper logarithm helical gear meshing theory further perfect. According to the mathematical model of logarithmic spiral bevel gear tooth surface meshing, using MATLAB powerful scientific analysis and calculation visual features and functions for the conjugate tooth surface equation to get the enveloping surface discrete conjugate points and draw out the conjugate tooth surface. Again through the optimization of surface get rid of the noise, end up with conjugate tooth surface precision, and can B spline fitting. Said for further analysis of spiral bevel gear drive are very valuable.

Transmission Theory and Tooth Surface Solution of a New Type of Non-circular Bevel Gears

Non-circular bevel gear is a typical gear mechanism and often applied to the intersecting axes or interlacing axes transmissions. An orthogonal composite-motion bevel gear is put forward based on the combination of cam mechanism and non-circular bevel gear. This new gear pair can implement the rotation of intersecting axes as well as the movement of output axe. The mathematic modeling of conjugate tooth surface is derived based on the equations of spherical involute and the theory of conjugate surfaces. Base on the specific curves of the orthogonal composite-motion bevel gear, the influencing factors and variation is analyzed. According to the motion relationships of the gears and the method of generation, the design of tooth profile is completed according to the instantaneous contact micro planes and the establishment of solid model for the orthogonal composite-motion bevel gear is presented with the application of SolidWorks.

Transmission Theory and Tooth Surface Solution of a New Type of Non-circular Bevel Gears

Non-circular bevel gear is a typical gear mechanism and often applied to the intersecting axes or interlacing axes transmissions. An orthogonal composite-motion bevel gear is put forward based on the combination of cam mechanism and non-circular bevel gear. This new gear pair can implement the rotation of intersecting axes as well as the movement of output axe. The mathematic modeling of conjugate tooth surface is derived based on the equations of spherical involute and the theory of conjugate surfaces. Base on the specific curves of the orthogonal composite-motion bevel gear, the influencing factors and variation is analyzed. According to the motion relationships of the gears and the method of generation, the design of tooth profile is completed according to the instantaneous contact micro planes and the establishment of solid model for the orthogonal composite-motion bevel gear is presented with the application of SolidWorks.

Design of Geneva Mechanism with Curved Slots

A simple yet comprehensive method is proposed for the design of a Geneva indexing mechanism with curved slots. In the proposed approach, conjugate surface theory is employed to derive an analytical description of the profile of the curved slots with and without an offset feature. The use of an appropriate offset angle to eliminate the singular points and double-points on the profile of the curved slots is then demonstrated. Finally, a mock-up Geneva mechanism is constructed to demonstrate the feasibility of the proposed approach. The results confirm that the methodology presented in this study enables the integration of the design, analysis and machining tasks for a Geneva indexing mechanism, and therefore makes possible a flexible and automatic production process.

DESIGN AND ANALYSIS OF OFFSET SLIDER-CRANK WITH TRANSLATING ROLLER-FOLLOWER

This paper presents a systematic methodology for the design and analysis of an offset slider-crank mechanism with a translating roller-follower. In the proposed approach, a generic kinematic model of the offset slider-crank and roller-follower is constructed using a homogenous coordinate transformation method and the slider cam profile are then derived using conjugate surface theory. The pressure angle and principal curvatures of the designed slider cam are analyzed based on the analytical expression of the cam profile. Finally, the NC data required to machine the slider cam are produced by equating the ability matrix of the 3-axis CNC machine tool with the desired tool location matrix.

Method for Design of Integral Shaped Milling Cutter Profile

Nowadays the profile of integral shaped milling cutter is designed by diagram method. This method is low efficiency and bad precision. This paper deals with the application of conjugate surface theory to the profile design of integral shaped milling cutter. According to the structural characteristics of integral shaped milling cutter, the method for design of cutter profile to wood workpiece is investigated by use of conjugate surface theory. In the present study, the curve equations for integral shaped milling cutter profile are put forward, based on principles of coordinate transformation, mill cut mechanics and conjugate surface theory of differential geometry.

An Integrated Approach of CAD/CAM for Spatial Cam with Oscillating Cylindrical Rollers

In this paper, an integrated approach of CAD/CAM was presented for the spatial cam with oscillating cylindrical rollers. The relationship between the cam profiles and the meshing element’s surface is established by the conjugate surface theory. In the machining process, the cutter location for the rough and the finish machining using cylindrical-end mill is derived. To avoid interference, the principal curvatures and the principal induced normal curvatures between the cam surface and the cutting tool are analyzed. The geometric error was used as a basis for generating appropriate toolpaths. The postprocessor are developed for converting the cutter location file to the five-axis numerical control program. The generated NC program is verified before actual machining through solid cutting simulation. In order to demonstrate the applicability of the presented method, cam profile was cut using a five-axis machine tool with real material.

Surface design of roller gear cam mechanisms with a turret in both translation and rotation

Roller gear cam mechanisms (RGCMs) are used extensively as automatic tool changers of machining centres or as indexing mechanisms in various machinery. However, almost all the existing RGCMs have a turret in a rotational motion only. Thus for example, when they are used as a part of automatic tool changers, the translational motions of the tools have to be given by other mechanisms. An RGCM with a turret which can move with both translation and rotation simultaneously is proposed; such an RGCM may simplify the structure of some machines. By using D-H homogeneous coordinate transformation matrices, and applying the theory of conjugate surfaces, the model for the surface design of the RGCM is developed. Two examples are included to demonstrate the usages of the model and to verify its correctness.