一种高性能精密凸轮CAD／CAM系统

Int J Adv Manuf Technol(1999)15:32–37©1999Springer-Verlag LondonLimitedA CAD/CAM System for High Performance Precision Drum CamsS.H.MasoodIndustrial Research Institute Swinburne,Swinburne University of Technology,Melbourne,AustraliaThis paper describes a computer-aided design and manufactur-ing system for the design and production of complex profiles for high-performance drum cams within the specified tolerance. The system graphically generates the cam profile on the cylin-drical drum after performing an analysis of the kinematic performance for the prescribed follower motion,using a B-spline representation of follower curves.Eight different types of follower motion for a translating follower are considered. The kinematic performance is based on the criteria of achieving the lowest levels of velocity and acceleration for each curve. The system is also able to simulate the motion of the designed cam graphically.The system also recommends the best cam profile on the basis of comparing the kinematic performance of all the B-spline representations of all the follower motions. Keywords:B-spline;Cam profiles;CNC machining;Cylindri-cal cams;Drum cams1.IntroductionA drum cam is a cylindrical cam which transmits motion to its follower in a groove cut on the surface of a cylinder.The cam rotates about its longitudinal axis and provides a trans-lational and/or oscillational displacement to the follower,simul-taneously.Applications of such cams are found in,for example, knitting machines,elevators,packing machines and indexing mechanisms.Requirements for high performance of such machinery demands efficient methods for the design and manu-facture of such cams.Conventional methods of design and machining complex drum cam profiles within a given accuracy are tedious and time consuming.Even programming them on a computer numerical control(CNC)machine can be a difficult job because of the complexity of the cam profiles.This paper describes a CAD/CAM system for the design and production of complex profiles for high performance drum cams within a specified tolerance.The system graphically generates the cam profile on the cylindrical drum,after per-Correspondence and offprint requests to:Dr S.H.Masood,IRIS, Swinburne University of Technology,Hawthorne,Melborne3122, Australia.forming an analysis of the kinematic performance of the pre-scribed follower motion,using a B-spline representation of follower curves.A number of different types of follower motions for a translating follower are considered.The kinematic performance is based on the criteria of achieving the lowest levels of velocity and acceleration for each curve.The system is also able to simulate graphically the motion of the designed drum cam.The system recommends the best cam profile on the basis of comparing the kinematic performance of all the B-spline representations of all the follower motions.The system also generates the tool cutter location data for the CNC machining of the drum cam profile.With the user specified tolerance,the system provides the minimum number of cutter location points required for machining the profile within the given accuracy.It is demonstrated that the adopted approach used in this CAD/CAM system provides a faster and more economical method of designing and manufacturing precision drum cams.2.Related WorkSeveral studies have been carried out on the design,production, and performance of cams,especially for disk or plate cams [1–3].Initially,the researchers concentrated on the method-ology for describing the surface of the profile.With the advent of powerful computers,work has been concentrated on the development of computer programs to assist in the design and manufacture of cams.Some of this work includes curvefitting for smoothing the motion curves of the cam when rotating at high speed.Most work is related to plate cams,but little work seems to have been done on cylindrical cams.Lin et al.[4]have developed a methodology by which cam drawings and NC codes can be automatically created after specifying the cam motion function.The procedure they have developed for cam design is based on a methodology called combined curvesfitting.The methodology is to establish the necessary equation for matching the velocities and accelerations at the curve junctions.From this equation,the cam profile coordinates can be ing the cam profile coordinates and the size and type of cam follower,the cutter centre coordinates can be calculated for a specified increment of cam rotation.Their work concentrated more on how to obtain anA CAD/CAM System for Precision Drum Cams33accurate profile,rather than on how to generate an accurate NC code.Yang and Kim[5]developed an interactive CAD/CAM system for the integrated design and manufacture of precision plate cams with three interpolation methods.The method using the cutting tool to assume the role of the follower was called R-⌰interpolation and generates the least number of NC curves compared to linear interpolation methods.Sadek and Daadbin[6]proposed a method of smoothing the profile curve once it had been specified.Polynomial curve fitting was used to replace the profile curve.A two-degree-of-freedom model for a cam mechanism is used as a test case. The conclusion is that with curvefitting using polynomials, the cam can cause less vibration than does the original cam and it has less tendency to bounce.However,their work does not deal with the development of diagrams,simulation and the manufacture of cams.Masood and Lau[7]developed a CAD/CAM system for the accurate machining of plate cams within a user specified tolerance.Their work includes the generation of a displacement diagram,the simulation of cam tool-path generation and the generation of actual CNC codes.They also proposed a new method for the interpolation of cam profiles called the half-angle algorithm,which produces more economical CNC part programs than other interpolation methods.Chan and Sim[8]developed a computer-aided design tool for optimum plate cam design.An exploratory search method called the Monte Carlo method was used to optimise the cam design.In this method,random points are generated over the range of all variables of the cam base circle,the width of the cam,the follower roller radius and its offset.The system is an integration of the design calculations and an optimisation algorithm.It provides an optimised solution,a graphical dia-gram,and a simulation of cam movement but does not provide the data for cam machining.Dhande et al.[9]developed a method of generating the profiles of planar and spatial cams.The method used is based on the concept that the common normal vector and the relative velocity vector are orthogonal to each other at the point of contact between the cam and follower.By this method,the contact line on the follower surface can be formed and then transformed in the coordinate system of the cam,giving the conjugate cam profile.This research focused only on how to obtain the coordinates of3D cams such as cylindrical or conical cams.None of this research concentrates on the full development of a CAD/CAM system for cylindrical cams. More recently,Tsay and Wei[10]developed a CAD system for cylindrical cams with a translating conical follower.Cams can be designed and graphics can be displayed once the follower motion program has been given.Based on the approach previously developed by in[9],the contact line of the follower at any angle of rotation can be obtained tofind analytical profile expressions.All these procedures were carried out without the assistance of a CAD/CAM system.Tsay and Lin[11]presented a procedure for the synthesis and analysis of the surface geometry of cylindrical cams with oscillating roller followers.B-splines are used to synthesise and refine the dwell-rise-dwell motion.To justify the method, the maximum acceleration and velocity of B-spline cams were compared with those using traditional curves of cycloidal and modified sine types.Work was under way for calculating the tool path for the machining of the cylindrical cam on a numerical control milling machine.3.Basic Follower Motion CurvesGenerally,thefirst task in the design of a cylindrical cam is to select a suitable follower motion curve that will satisfy the constraints of the application.Here,the constraints mean the relationship of the displacement of the follower to the angle of rotation.The follower motion curves play an important role in the performance of the cam mechanism.The desirable characteristics of the motion curves are that they should impart a continuous velocity,continuous acceleration,and low velocity and low acceleration to the follower.These may not be important when a cam is operating at low speed,but will be crucial at high speeds since they can result in high vibrations and may lead to a short life span for cam and other mechanisms in the machine.Cams with complex profiles are often required.This may involve a number of rise-dwell-return intervals in which the rise and return motions may be quite different to each other. To analyse the action of a cam,it is therefore necessary to study its displacement–time diagram and its associated velocity and acceleration curves.Some of the most common types of follower motion curve selected by cam designers are:1.Constant velocity.2.Parabolic motion.3.Simple harmonic motion.4.Cycloidal motion.5.The3–4–5polynomial motion.6.Modified sine motion.7.Trapezoidal motion.8.The4–5–6–7polynomial motion.In constant velocity motion,the velocity of the follower during the motion is constant,and the acceleration is zero,except at the beginning and the end of the motion,where the acceleration reaches infinity instantaneously.In parabolic motion,this curve has constant acceleration and retardation following a parabolic equation.Simple harmonic motion has a smooth,continuous acceleration but has a sudden change in acceleration at the ends.It is,therefore,suitable only for cams at medium or low speed.Cycloidal motion is obtained by rolling a circle on a straight line.The length of the line is equal to the circumference of the rolling circle.It has the smoothest motion among all of the basic curves,and therefore,it is suitable for high-speed applications.The3–4–5polynomial curve is worked out from a5th degree polynomial.It can generate a smooth and accept-able profile that will fulfil the design requirements.Its perform-ance at high speed is good.It is suitable for cams at medium speeds.The modified sine curve is a combination of cycloidal and harmonic quadrants occupying different parts of the work-34S.H.Masooding cycle.This curve has a low maximum velocity and good acceleration characteristics.Its performance at high speed is good or excellent.The trapezoidal motion curve,if machined accurately,will be good for high-speed cams.It has good acceleration characteristics.The4–5–6–7polynomial has been developed from a3–4–5polynomial.The smoothness of the curve is similar to that of a3–4–5polynomial.It has good acceleration characteristics and is used for high-speed cams.4.Synthesis of Follower Motion Curves Using B-SplinesWhen the kinematic and dynamic performances of the cam-follower mechanisms are to be considered,the follower motion must be carefully synthesised.For example,to reduce the amplitude of the inertial force,the maximum value of the acceleration curve of the follower motion must be as small as possible.One method of improving the follower motion curves would be to represent the curves by parametric B splines. 4.1B-spline CurvesThe term“spline”refers to a smooth curve drawn through a set of plotted control points.Such curves can be described mathematically as piecewise approximations of cubic poly-nomial functions having all three orders of continuity(zero-order,first-order,and second-order continuity).The smoothness of a curve is described in terms of curve continuity between sections.Zero-order continuity means that the curves intersect. First-order continuity means that the tangent lines(first derivatives)of two adjoining curve sections are the same at the intersection.Second-order continuity means that the curvatures (second derivatives)of the two curve sections are the same at the intersections.B-splines are a class of spline curves that cannot only interpolate a given set of control points,but can also allow localised modifications to be made easily without greatly affect-ing other parts of the curve[12].Another advantage of B-splines is that any number of control points can be specified by a designer without increasing the degree of the curve.A cubic curve could then be used for many different curve shapes,without the need to piece curve segments together. Any number of control points can be added or modified to manipulate the curve shape.Given an input set of n+1control points p i,with i varying from0to n,then the points on the approximating B-spline curve are given by,p(u)=⌺p i N i,k(u)where N i,k(u)is defined as a blending function,which is a polynomial of degree kϪ1.A method for setting up the polynomial form of the blending functions is to define them recursively over various subintervals of the range for para-meters u.This range now depends on the number of control points n and the choice for k,so that u varies from0to nϪk+2(instead of0to1).Setting up n+k subintervals,we define the blending functions recursively asN i,1(u)=1if t iϽuϽt i+1N i,1(u)=0otherwiseandN i,k(u)=[(uϪt i)N i,kϪ1(u)]/[t i+kϪ1Ϫt i]+[(t i+kϪu)N i+1,kϪ1(u)]/[t i+kϪt i+1]where k controls the degree(kϪ1)of the resulting polynomial in u and thus also controls the continuity of the curve.The values t i are called knot values.They relate the parametric variable u to p i control points.The knot values t i are given by t i=0if iϽkt i=iϪk+1if kՅiՅnt i=nϪk+2if iϾnwith0ՅiՅn+k4.2Applying a B-spline Approximation to a Basic CurveIn cam design,once the basic follower motion curve has been specified,it can then be approximated by B-splines.The following steps illustrate the procedure by a simple example.1.Create the basic curve of the selected follower motion(parabolic,simple harmonic,etc).2.Divide the angle interval of the curve into5parts.Eachpart has the same angle interval.This will create6points, p0,p1,p2,p3,p4,and p5lying on the curve.3.These6control points are interpolated by considering theconstraints of n=5,and k=3and using the parametric equations:p x,y=(1Ϫu)3p0+[u(1Ϫu)2+1/2(2Ϫu)(Ϫ3/2u2+2u)]p1+[u/2(Ϫ3/2u2+2u)+u2/6(3Ϫu)]p2+u3/6p3if0ՅuϽ1 p x,y=1/4(2Ϫu)3p1+[u/4(2Ϫu)2+(3Ϫu)/3(Ϫu2+3uϪ3/2)]p2+[u/3(Ϫu2+3uϪ3/2)+1/4(3Ϫu)(uϪ1)2]p3+1/4(uϪ1)3p4if1ՅuϽ2 p x,y=1/6(3Ϫu)3p2+[u/6(3Ϫu)2+(3Ϫu)/2(Ϫ3/2u2+7uϪ15/2)]p3[(uϪ1)/2(Ϫ3/2u2+7uϪ15/2)+(3Ϫu)(uϪ2)2]p4+(uϪ2)3p5if2ՅuϽ3 where p x is the x-coordinate point and p y is the y-coordinate point on the curve.Using the above procedure,each motion curve is converted into an equivalent B-spline curve.The new curve many look approximately the same as the former basic curve but it will have differences in the smoothness and maximum velocity and acceleration.These are the factors that affect the kinematicA CAD/CAM System for Precision Drum Cams35performance of the cam.The velocity and acceleration diagram of various cams can be examined using the CAD/CAM system which has been developed.For machining the cam on a CNC machine,the cutter location data required to machine the specified cam profile should be minimised.This will result in a shorter programming time and more economical CNC programs.The system calcu-lates the cutter location data required for the cam profile on the basis of the basic curve as well as the interpolated B-spline curve.5.The CAD/CAM SystemThe CAD/CAM system for a drum cam presented in this paper is an interactive computer program written in Pascal version 7.A computer with an80386or higher CPU with a DOS operating system is required.The system requires the user to specify the following input data from the main menu:1.Speed of cam rotation(r.p.m.).2.Tolerance to be achieved(mm).3.Type of basic curves for each segment of motion curve(selected from the list).4.Direction of the follower(lift/return).5.Interval angle for each segment of the motion curve(degrees).Then,the program computes the cam profile to provide the specified follower motion and generates the following infor-mation:1.Graphical display of displacement,velocity and accelerationdiagrams for the basic curve and the B-spline curve.2.Numerical values of maximum acceleration and maximumvelocity for the basic curve and the B-spline curve.3.Cutter location data points for CNC machining using thebasic curve,using the B-spline curve,and using the edited curve.4.2D animated graphical simulation of drum cam with itsprofile.5.3D animated graphical simulation of drum cam with itsprofile.Using the information from items1or2above,the user can decide whether the basic curve or the B-spline curve is to be used for profile generation.In general,it is expected that the B-spline curve will provide lower velocity and acceleration for a given set of combination of motion curves for the rise and return motions of the follower of a particular cam.The system also allows the user to generate CNC cutter location data for the cam profile on the basis of basic curves alone,or the B-spline curves alone,or the edited curves.The cam profile generated on the basis of the edited curve will generate the minimum number of CNC cutter location points. This is done by comparing the cutter location data for each segment of the basic curve with the cutter location data for the corresponding segment of the B-spline curve,and then selecting the segment with the lower number of cutter location data.In this way,an edited curve is the motion curve that contains a combination of portions of the basic curve as well as a B-spline curve.The system employs the half angle search algorithm.This provides a systematic searching method to determine the mini-mum number of NC points required for machining the cam profile to within a user specified tolerance[7].The limitation of the program is that for each of the curves defined,the value of the ratio of the angle interval to the height for rise,dwell or return,has to be more than10.For example,if we want to input the cam profile with a rise from 0°to120°with a cycloidal motion,a dwell from120°to 240°,and a return from240°to360°with a modified sine motion,then the height of the follower motion curve should be less than10to make the ratio of angle/height more than 10.This limitation is because the motion curve must have a certain horizontal spread in order to convert it to a B-spline, and also because of the requirements of the half angle search algorithm used for linear interpolation of the profile curve. 6.Analysis of Motion CurvesUsing the CAD/CAM system,an analysis was performed to demonstrate that B-spline curves of all the types of motion curves produce a lower number of cutter location data points. For this,a drum cam with a follower motion consisting of only rise and return(with no dwell),each with an angle interval of180°and a maximum height of100mm,was considered.The drum cam profile was created by considering each type of follower motion(parabolic,SHM,etc.)in turn. In each case,the motion curves for the rise and return motion were assumed to be identical.For example,in the case of parabolic motion,the rise and return motions were each assumed to be parabolic with an angle interval of180°each. The cam speed was assumed to be60r.p.m.The cam profiles were generated in each case for each of three different values of tolerance.Table1shows that the approximated B-spline curves for each type of follower motion produce fewer CNC cutter location points than those generated for the basic curve for the corresponding type of follower motion.An analysis was also performed to determine the maximum velocities achieved by a drum cam designed for different types of follower ing a follower displacement height of 100mm,with a tolerance of0.005,the CAD/CAM system was used to compute the maximum velocities for each drum cam in which the rise and return had equal angle intervals of 180°and with the rise and return having the same type of motion curve.This was done for all seven types of motion (parabolic,SHM,etc.).Three cam speeds were used.The results in Table2show that the approximated B-spline curve of each follower motion provides lower maximum velocities than the basic curve for the corresponding follower motion at different speeds of cam rotation.A similar analysis has been carried out for maximum acceleration,but the values of maximum acceleration are found to be lower for the basic motion curves than for the approximated B-spline curves in most cases.36S.H.MasoodTable1.Number of cutter location points required for basic curve and approximated B-spline curves for different follower motions.Type of Specified tolerance(mm)followermotion curve0.0010.0020.0051.Parabolic:Basic885835B-spline6940242.Simple harmonic:Basic714628B-spline7442263.Cycloidal:Basic745634B-spline6638224.The3–4–5polynomial:Basic745032B-spline7040245.Modified sine:Basic744831B-spline7040246.Trapezoidal:Basic805432B-spline6840227.The4–5–6–7polynomial:Basic986032B-spline643821Table2.Maximum velocity for basic and B-spline motion curves.Type of Cam speed(r.p.m.)followermotion curve1020601.Parabolic:Basic 6.6713.3340B-spline 5.7811.5634.672.Simple harmonic:Basic 5.2410.4731.42B-spline 5.5211.0433.133.Cycloidal:Basic 6.6713.3340B-spline 6.0012.1036.024.The3–4–5polynomial:Basic 6.2512.5037.5B-spline 5.8211.6434.925.Modified sine:Basic 5.8711.7335.19B-spline 5.6511.2933.886.Trapezoidal:Basic 6.6713.3340B-spline 5.9512.8335.727.The4–5–6–7polynomial:Basic7.2914.5843.75B-spline 6.2812.5637.69From the above analysis,it is clear that,from among thebasic curves,the curve that gives the least number of cutterlocation(CL)points is a simple harmonic motion,while thecurve with the most CL data is a4–5–6–7polynomial.How-ever,when comparing the B-splines of all the curves,the curvewith the least CL data is the4–5–6–7polynomial.Thus a B-spline curve of the4–5–6–7polynomial type is the preferablecurve when the quantity of CL data is the main consideration,and the cam does not operate at high speeds.Considering the maximum velocities,among the basiccurves,the one that gives the lowest velocity is the simpleharmonic motion,while the one with the highest maximumvelocity is the4–5–6–7curve.Therefore,the simple harmonicis preferable when the designer wants a cam running at a lowspeed.Considering the maximum accelerations,the parabolicmotion basic curve provides the lowest maximum acceleration Fig.1.Example of a drum cam designed by the CAD/CAM system.A CAD/CAM System for Precision Drum Cams37among the other basic curves,but other curves such as simple harmonic and trapezoidal motion also provide lower acceler-ations.When considering each curve and its B-spline form in turn,the results indicate that the cycloidal and the4–5–6–7 polynomial curves,with their B-spline forms will give better performance when low acceleration is required.Figure1shows an example of a drum cam design and a simulation created by the CAD/CAM system.7.ConclusionsThe CAD/CAM system for the design and machining of drum cams discussed in this paper is based upon the kinematic performance of the cam and includes a provision for improving the performance of the cam by considering an approximated B-spline representation of each motion curve.The system provides useful design information such as maximum velocities and maximum accelerations for each of the selected follower motion diagrams and for their approximated B-spline represen-tations,and the number of cutter location points required for the CNC machining of the cam profile.The system also allows the design of cams within user specified tolerance levels.The analysis of various follower motion curves for cutter location data,low maximum velocity and low maximum acceleration has been carried out to help the designer select the desirable type of motion curves for better cam performance.References1.F.Y.Chen,Mechanics and Design of Cam Mechanisms,JohnWiley,1982.2.P.W.Jensen,Cam Design and Manufacture,2nd edn,MarcelDekker,1987.3.D.Tesar and G.K.Matthew,The Dynamic Synthesis,Analysisand Design of Modelled Cam Systems,Lexington Books,1976.4.A.C.Lin,H.Chang and H.P.Wang,“Computerised design andmanufacturing of plate 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