Conjugate Surface Theory Research Articles

Surface Generation and Fabrication of Roller Gear Cam with Spherical Rollers

This paper derived the surface profile equations and the meshing equations of the spherical meshing elements from the homogeneous coordinate transformation matrix and the conjugate surface theory proved by the envelope theory. In the machining aspect, we demonstrated the position equation of cam profile using the coordinate transformation theory and obtained the perfected controlling position point of the tool path under the limitation of the chord errors. Finally, the design and the machining module was developed via Visual Basic that generated the cam profile points and the tool path points. The curves from UNIGRAPHICS to test and verify the cam profile equation. To verify the validation of the tool path, a cutting simulation software, VERICUT, is used to simulate the cutting geometry and kinematics of the roller gear cam, and demonstrate the practical application of the developed method.

Application of homogenous transformation matrix to the design and machining of a Geneva mechanism with curved slots

The current paper presents a simple yet comprehensive method for the design and machining of a Geneva indexing mechanism with curved slots. In the proposed approach, a kinematic model of the Geneva mechanism is developed using homogenous coordinate transformation and conjugate surface theory. The pressure angle of Geneva mechanism and the principal curvatures of the curved slots are analysed using an analytical expression derived for the slot profile. The NC data required to machine the driven with curved slots are derived from the CNC link variables by equating the machine tool ability function and the tool location matrix. The proposed design methodology is verified by machining the designed curved slots using a three-axis CNC machine. The results confirm that the approach presented in this study enables the Geneva mechanism design and manufacturing tasks to be successfully integrated, thus making possible a flexible, automatic, cost efficient, and controllable production process.

Geometric Design of Cycloidal Drives with Ceramic Ball Meshing Elements

Geometric design procedures for the cycloidal drive with ceramic ball meshing elements were introduced. Using the theory of conjugate surfaces, the equations of meshing of the cycloid drive and the cycloid raceway profiles were derived. Design examples were presented to demonstrate the design procedure and feasibility. The ceramic balls were used as the meshing elements. Prototype of this research was made and tested with satisfying results. The results of this work are suitable for computeraided design and manufacture of industrial applications.

Design and machining of cylindrical cam with meshing indexing disc

Cams are fundamentally important machine elements, which provide a simple means of generating virtually any follower motion. This paper presents a simple yet comprehensive method for the design and machining of a cylindrical cam with a meshing indexing disc. In the proposed methodology, homogenous coordinate transformation and conjugate surface theory are employed to develop a kinematic model for the cam profile design. The pressure angle and the principal curvature of the designed cam are analysed using the analytical expression derived for the profile. The required NC data equations are obtained by solving the CNC link variables by equating the machine tool ability function with the tool location matrix. The proposed methodology is verified by machining the designed cylindrical cam using a 4-axis CNC machine. The inspection results obtained from a 4-axis CMM show that the profile measurements of the machining cam are in excellent agreement with those of the designed profile. The results confirm that the methodology presented in this study enables the cam design and manufacturing tasks to be successfully integrated. As a result, the production process is flexible, automatic, cost efficient, and controllable.

Application of homogenous transformation matrix to measurement of cam profiles on coordinate measuring machines

A kinematic model is presented to aid the cam profile design task and to determine the NC data required to move an inspection probe to the necessary locations to inspect the surface profile of a spatial cam on a coordinate measuring machine (CMM). The cam profile is designed in accordance with the principles of conjugate surface theory. Homogenous transformation matrices are then employed to derive the CMM ability function matrix and the measuring probe location matrix. The required NC data are obtained by equating the elements in the ability function matrix with the corresponding elements in the probe location matrix in order to solve the CMM link variables. The validity of the proposed approach is verified by carrying out the automatic inspection of a machined cam on a CMM using the NC data computed by the developed algorithm. The methodology presented in this study combines the cam design, machining and inspection activities, and makes possible a flexible and automatic spatial cam design and manufacturing process.

Mathematical model and sensitivity analysis for helical groove machining

In any drill design, the helical groove shape plays a key role in ensuring an adequate flute space and an efficient chip removal capability. Moreover, the shape of the helical groove determines the principal drill angles. This study establishes a mathematical model of the helical groove and conducts a sensitivity analysis for helical groove machining performed on a 6-axis tool-grinding machine. Combining homogenous coordinate transformation and conjugate surface theory, a kinematic model is developed to facilitate the design of the helical groove shape (a direct problem). In determining the tool profile required to generate a desired helical groove (an inverse problem), this study exploits the condition that the common normal at the contact point between the tool and the helical groove surface must intersect the axis of the tool. The sensitivity of the helical groove profile with respect to the machining parameters is investigated. Finally, numerical examples are provided to demonstrate the validity of the developed models and algorithms. The numerical results reveal that the current design and sensitivity analysis methodology is comprehensive, simple and applicable to a wide range of helical groove machining applications.

Geometrical way for describing body motion and equations of relationships between geometrical and kinematical parameters

The geometrical/conjugate surfaces way for describing body motion by a pair of conjugate surfaces of point conjugation is detailed. The equations of relationships between the parameters of the geometrical/conjugate surfaces way (geometric parameters) and those of the traditional way (kinematical parameters) are derived by using concepts from the theory of conjugate surfaces. The conjugate surfaces of line conjugation are investigated as the degenerated case of the point conjugation cases, and are appropriated to describe body motion of one dof. The formula for determining the geodesic relative curvature of the instantaneous conjugation curve to guarantee line conjugation is derived and reported herein for first time. Both the line conjugation ruled surfaces and the axodes may be regarded as the special and singular case, respectively, of the general conjugate surfaces of line conjugation for describing body motion of one dof.

GEOMETRY DESIGN OF A ROLLER-DRIVE SPEED REDUCER

This work presents a mathematical model to design roller drives. It is based on coordinate transformation, triple vector product of differential geometry, and theory of conjugate surfaces and the equation of meshing is derived. Cycloid profiles are obtained and the centre of the cylindrical teeth of the pinion is determined by the equidistant curve principle, then apply the CAD (Pro/ENGINEER software package) to construct the solid modeling of the proposed roller drives. The mathematical model can design easily and the solid modeling is presented to demonstrate this method is feasibility.

GEOMETRIC DESIGN OF ROLLER DRIVES WITH CONICAL ROLLERS

This work presents a mathematical model to design the roller drives with conical rollers. It is based on coordinate transformation, theory of conjugate surfaces, from which the equation of meshing is derived. A program is developed to solve the conjugate surface equations and the centres of the teeth of the pinion is determined by equidistant offset methods, then apply the CAD to construct the solid modeling of the proposed roller drives. The mathematical approach can design the required speed ratios of the roller drives, which have cylindrical or conical rollers.

Geometric design of roller drives with cylindrical meshing elements

This paper presents geometric design procedures to design the roller drives which have cylindrical meshing elements. The equation of meshing of the cycloid drive is derived by theory of conjugate surfaces. The cycloid profile curvature is obtained by finite differences and the pinion-tooth curvature must be greater or equal to the cycloid profile curvature. The centers of the cylindrical teeth of the pinion are determined by equidistant offset a distance, pinion-tooth radius. Using CAD (Auto CAD and Pro/ENGINEER software packages) constructs the 2-D drawing and solid modeling of the proposed roller drives, respectively. The design procedure can design required speed ratios. Two design examples are presented to demonstrate the design procedure is feasibility. The results of this work are suitable for computer-aided design and manufacture about industrial applications.

Geometry analysis and optimal design of Geneva mechanisms with curved slots

A systematic procedure is proposed for the design of Geneva mechanisms with curved slots. Based on the theory of conjugate surfaces, mathematical expressions for the slot profile, pressure angle and cutter's location for manufacturing are presented. In addition, to evaluate the combined kinematics and structural performance of the mechanism, the maximum contact stress and degree of wear are established as the performance index. Effects of variations in various design parameters on the values of the performance indices are investigated. Using the indices as the objective function, the optimum design that takes into account the initial crank angle, offset and roller radius is performed.

Theory of digitized conjugate surface and solution to conjugate surface

In order to meet the needs of designing and processing digitized surfaces, the method to spreading digitized surface has been proposed. The key technique is to solve the problem of digitized conjugate surface. In the paper, the digitized conjugate surface was theoretically investigated, and the solution of conjugate surface based on digitized surface was also studied. The digitized conjugate surface theory was then proposed, and applied to build the model of solving conjugate surface based on digitized surface. A corresponding algorithm was developed. This paper applies the software Conjugater-1.0 that is developed by ourselves to compute the digitized conjugate surfaces of the drum-tooth surface. This study provides theoretical and technical bases for analyzing engagement of digitized surface, simulation and numerical processing technique.

Geometry design of an elementary planetary gear train with cylindrical tooth-profiles

This work presents a concept of elementary gear trains such that the tooth-profiles of the pinion are cylindrical. The pitch radius is replaced by the distance between the gear center and the tooth center as the radius of gears. Based on coordinate transformation, triple vector product of differential geometry, and theory of conjugate surfaces, equation of meshing is derived. A program is developed to solve the conjugate surface equations, and to show the solid modeling of the proposed elementary planetary gear trains.

Machine tool setting for the manufacturing of spherical cams

This paper employs the theory of conjugate surfaces and the notion of Denavit–Hartenberg (D–H) transformation to investigate the manufacturing requirements for generating the CNC machine setting for spherical cams. The general design model is first derived and presented to determine the equations for a spherical cam and to generate the associated NC data for a five-axis CNC machine system to produce the spherical cam-oscillating roller-follower system (SCORF) mechanism. A D–H transformation is employed to derive the ability-function matrix of the machine tool and to determine the desired cutter motion trajectory. The NC program data equation is obtained by equating the ability function of the production machine and the path planning matrices. The method presented here is general and systematic. It combines the functions of design and manufacture for general cam production. The resultant cutting velocity and material removal rate are also studied to allow the efficiency of the manufacturing process be properly controlled. A SCORF is employed to illustrate the effectiveness of the proposed approach. The direct measurement data for the final product indicates that the result is satisfactory. This result, in turns, indicates that the proposed theory and the systematic modeling procedure presented in this paper are both valid and effective.

Tool path generation and error control method for multi-axis NC machining of spatial cam

In this paper, based on the homogeneous coordinate transformation and conjugate surface theory, a tool path generation method is developed for generating spatial cam in order to establish the interface between the design and manufacture of this class of product. The mathematical error (chordal deviation) between the design and manufacture surface has been analysed and used as a basis for selecting the tool path control point. Moreover, the developed tool path generation method is verified through a cutting simulation software with solid model. It is also verified through the trial cut with model materials on a five-axis numerical controlled machine. The results show that the mathematical error of the cam surface can be controlled within given tolerance by the proposed method.

Kinetostatic analysis of variable lead screw mechanisms with 3° conic frustum meshing elements

The main concern of this paper is the theoretical kinetostatic analysis of a variable lead screw mechanism (VLSM) used in high speed shuttleless weaving looms. The mechanism consists of a slider-crank submechanism, with four conic frustum meshing elements (with a 3° conical angle) installed in the slider to drive the oscillatory motion of the variable lead screw. It is the spatial cam joint between the meshing elements and the screw. Then, d'Alembert's principle and theory of conjugate surfaces are applied to develop the models for the kinetostatic analysis; models for analyzing the frictional forces of various kinematic joints are also implemented. The result is a set of 27 nonlinear equations with 27 unknowns. The power equation is also applied to analyze the input and dissipated power of the VLSM, its results are also used to check the correctness of solutions to the nonlinear equation set. The VLSM is analyzed for a specified constant input speed or a measured state of motion of the crank. The results show that the peak input power and the peak input torque at constant input velocity are almost twice that of their values at nonconstant input velocities. There are obvious changes in the bearing loads when the slider is at both of its limit positions due to the change in the directions of the frictional forces.

Applications of D-H Notation in Machining and On-Line Measurement of Roller-Gear Cams on 5-Axis Machine Tools

A kinematic model is presented to aid in the design of cam-profiles and determine the NC data equations using Denavit-Hartenberg (D-H) notation to produce and to measure the dimension accuracy of roller-gear cams on 5-axis machine tools. First, cam profiles are determined based on the conjugate surface theory. Then, D-H notation is employed in the derivation of machine tool ability functions and in the generation of the desired cutter/measuring-probe location matrices. The desired NC data equations are obtained by solving the link variables of the machine tool by equating the machine tool ability function with the tool location matrices. The resultant cutting speed and material-removal rate during machining are also investigated. To verify the validity of this methodology, a designed roller-gear cam was machined and its surfaces were measured through the on-line measurement system equipped with a touch-trigger probe on a 5-axis machine tool. This methodology combines the cam design, machining, and measurement techniques, thus making the production process more flexible, automatic, and controllable.

Characteristics of the Variable Lead Screw Mechanism with 3.DEG. Conic Frustum Meshing Elements.

This paper outlines the theoretical and experimental analyses of the kinematic and kinetostatic characteristics of a variable lead screw mechanism(VLSM)used in high-speed shuttleless weaving looms. The mechanism consists of a slider-crank submechanism equipped with four conic frustum meshing elements with a 3° conical alngle to drive the variable lead screw in an oscillating manner. The screw and meshing elements act as a spatial cam pair. The conjugate surface theory and coordinate transformation are applied to develop the theoretical models used to analyze the mechanism. The kinematic curve is synthesized using optimization techniques that minimize the dissipated power due to the friction between the screw and meshing elements. The power equation is applied to analyze the input power of the VLSM, and the generalized Lanchester balancer technique is used in the design of the counterweight to reduce the shaking force. Experimental instruments are used for measuring the characteristics of the mechanism, including the angular velocity of the crank, the motion(including angular displacement, velocity and acceleration)of the screw, the input power, and the shaking forces before and after adding the counter-weight. Using a specified constant speed or the measured state of motion of the crank, the VLSM is analyzed theoretically, and results are compared with the measured results. It is shown that the input power of the prototype proposed is about one-quarter less than that of a commercial product, and the shaking force of the mechanism can be reduced by 46% once the proposed counterweight is added.

Geometry Design of Globoidal Cams With Generalized Meshing Turret-Rollers

This paper derives the generalized surface equation for cylindrical, conical, and hyperbolic meshing turret-rollers. Based on the generalized equation of meshing turret-rollers, generalized mathematical expressions of surface geometry for globoidal cams with cylindrical, conical, and hyperbolic meshing turret-rollers are derived using theory of conjugate surfaces, differential geometry, and coordinate transformation. A design example is presented for demonstrating procedures of surface generating of the globoidal cam. The result of this work is necessary for the computer-aided design and manufacturing of roller gear cams for industrial applications.

Geometry design of roller gear cams with hyperboloid rollers

This paper studies the surface geometry of roller gear cams with hyperboloid rollers as meshing elements. The surface equation of the hyperboloid roller is established. Coordinate systems and coordinate transformation matrices corresponding to the roller gear cam are defined. The mathematical expressions for the surface geometry of the roller gear cam, by applying the theory of conjugate surfaces, differential geometry, and coordinate transformation, are derived. Based on available surface geometry of the roller gear cam, the unit normal of the cam surface is obtained by differential geometry. The contact lines of the conjugate surfaces are defined according to the equation of meshing and the specified input-output relation. Once the contact line is transformed to the coordinate system which is rigidly connected to the globoidal cam, the surface profiles of the globoidal cam are obtained. Two design examples are given.