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INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3, pp. 419-427 JUNE 2010 / 419DOI: 10.1007/s12541-010-0048-6 1. Introduction Plate cam mechanism is a widely used machine component with the continuous contact motion of cam and follower, and can easily produce any functional motion of follower due to the rotation of cam. Cam mechanism has the diverse types by the combination of different shape of cam and motion of follower; plate or cylindrical cam, roller or flat-faced follower, and reciprocating or oscillating motion. In spite of the advantages of a few number of links, simple structure, positive motion, and compact size, cam mechanisms require the accurate shape design and precise machining procedures for satisfying the mechanical requirements. Under the low leveled design and manufacturing, cam mechanisms give the heavy effects on vibration, noise, separation, and overloading to an overall system. To avoid these effects, cam mechanism must be well designed accurately and machined precisely. Actually, a hybrid CAD/CAM approach may be the best solution that the shape data from the design process are directly combined to the machining data for the manufacturing process. Line interpolation and circular interpolation are commonly used in construction of the machining data from the profile data of cam. Line interpolation has the low accuracy and circular interpolation can not keep the accuracy because of the disconnective radii of curvatures or the discontinuous slopes at the connected point by two circular arcs as presented in Shin et al.1-3 Recently, parameteric interpolation using B-spline and NURBS curve are suggested in Jung et al.5 and Yang et al.6 Also biarc interpolation is widely used and deeply dependent on the direction angle toward centers of biarc curves. Bolton7 described a biarc curve based on the tangential angles at two points, Parkinson and Moreton8 made a biarc curve based on a quadratic equation at three points, Meek and Walton9 used spline types for constuction of biarc curve. Schonherr10 introduced an approach to minimize the radii of biarc curves. Commonly these interpolation methods make the machining points increasing and then the excessive data for machining a curved shape make the machining errors increased. Thus, the precise machining process requires minimization of the machining points to keep the accuracy under a given machining tolerance. This paper introduces 3 steps of a hybrid CAD/CAM A Hybrid Approach for Cam Shape Design and Profile Machining of General Plate Cam Mechanisms Joong-Ho Shin1, Soon-Man Kwon1 and Hyoungchul Nam1,#1 Department of Mechanical Design & Manufacturing, Changwon National University, #9, Sarim-dong, Changwon, Kyungnam, South Korea, 641-773# Corresponding Author / E-mail: nhchulchangwon.ac.kr, TEL: +82-55-267-1106, FAX: +82-55-267-1106 KEYWORDS: Plate cam mechanism, Cam, Follower, Shape design, Profile machining, Contact point, NC data, Instant velocity center, Biarc curve fitting Plate cam mechanism can easily produce the positive and functional motions in contact of cam and follower. Generally cam mechanism is used in many fields of mechanical control, automation, and industrial machinery. To obtain the accurate motion of follower, the profile of cam must be designed and machined precisely. This paper proposes an instant velocity center method for the profile design and a biarc fitting method for the profile machining to 4 different types of plate cam mechanisms with reciprocating or oscillating motion and roller or flat-faced followers. The key of this paper is the introduction of a hybrid system combined the design procedures and the manufacturing procedures. The main idea is that the minimum machining data are built by the accurate biarc curves fitted directly from the design parameters. The radial direction angles toward biarc centers for the accurate biarc curve fitting can be defined directly by the contact angle of cam and follower given in the design procedures. An application of the proposed approach is verified the accurate profiles of a designed cam and a machining cam using the minimum NC data within a given machining tolerance. Manuscript received: July 16, 2009 / Accepted: February 18, 2010 KSPE and Springer 2010 420 / JUNE 2010 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3 approach11 for 4 different types of plate cam mechanisms. Firstly, the shape of cam is determined by the kinematic constraints at instant velocity centers and the contact angle at the contact point between cam and follower. The second step is to transform the contact angles into the center direction angles and then to calculate the radii of biarc curve. Finally, the machining data are minimized through expanding or contracting the biarc section whether the cam profile points are located inside or outside the range of a given machining tolerance. 2. Clarification of Accuracy on Biarc Fitting1-3 Only the profile data of cam shape are defined in common design process of a plate cam. Then, the machining data must be developed by any curve fitting for NC (Numerical control) process. The circular fitting, which is most widely used in machining, has unreliability as shown in Fig. 1. A circle developed by three points 1( ,P2,P3)P has a radius 1R and the other by points 2(,P3,P4)P has 2.R In the views of circular fitting two circles pass the profile points 1( ,P2,P3,P4),P but the discontinuous slopes are made at these points and also the disconnective radii at points in mid-span. These defects make the fitted curve in low accuracy and then higher vibration in high speed operation of cam mechanism. P4P3P2P1O2O1R1R2S1S2SSlope 1Slope 2 Fig. 1 Defects on circular fitting 3231R4R2P4P3O4O3S3S2R3S1P2O2O1P1R121112212 Fig. 2 Continuous fitting by biarc Fig. 2 shows a continuous curve fitted by biarcs, which passes the profile points 1( ,P2,P3,P4).P The biarc curve has 4 radii in this case. Radius 1R passes 1P to 1,S 2R for 1S to 2.S 3R for 2S to 3,S and 4R for 3S to4.P The slopes of the biarc curve are continuous and unique at every point. Also mid-points 1(,S2,S3)S are continuous without jump in radii. Thus, the biarc curve can keep the higher level of accuracy. As shown in Fig. 2, biarc fitting is highly dependent on radial direction angles ( ). The common design process of cam mechanism defines only the profile data and then machining process must use the angles from the circular fitting. This process gives the lower accuracy because of the incorrect angles. But the proposed approach in this paper can define the correct angles, which are given directly by design process of cam profile, and then keep the higher accuracy for the machining data. 3. Shape Design of Plate Cam 3.1 Displacement characteristics of cam mechanism For a plate cam mechanism with reciprocating roller follower shown in Fig. 3, the kinematic properties of follower motion can be defined as linear displacement ,Y first derivative ,Y and second derivative Y to the rotational angle c of cam. And the properties are given as angular displacement in case of oscillating follower. The instant velocity center method given in this paper uses the displacements and the 1st derivatives for determining the cam shape. CamFollowerContact point coordinateCam shape coordinatecCRSyx Fig. 3 Plate cam mechanism with reciprocating roller follower 3.2 Shape design based on instant velocity centers4 As shown in Fig. 4, Point Q is defined by a line through contact point C from roller center and a horizontal line and then it becomes instant velocity center. The velocity at point Q is proportional to a rotating speed of cam as in Eq.(1) and the velocity of roller at point R is defined in Eq.(2) as the linear velocity of follower. cQQdVLdt= (1) cRcdYdY dVdtddt= (2) INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3 JUNE 2010 / 421 By the kinematic characteristics of cam mechanism, the velocity at the instant velocity center ,QV is same as the velocity of follower .RV Thus, the velocity condition gives the location of the instant velocity center in Eq.(3). QcdYLYd= (3) QR(Rx, Ry)LQVQVRcRrYxyC(Cx, Cy) Fig. 4 Contact position of cam and follower The contact angle shown in Fig. 4 is defined in Eq.(4) by a angle between a sliding velocity line and a normal line at a contact point of follower roller. The coordinates of the contact point are given in Eq.(5) where the coordinates of a roller center (,xR)yR can be calculated from the displacement ( )Y and the geometric conditions (prime circle and eccentricity) of a given cam mechanism, and where rR is the radius of roller. Finally, the contact point (xC and )yC is given in Eq.(5) 1tanQxyLRR= (4) sinsinxxryyrCRRCRR=+= (5) QLQVQVfcYxC(Cx, Cy)F(Fx, Fy) Fig. 5 Plate cam with reciprocating flat-faced follower Fig. 5 shows a cam mechanism with reciprocating flat-faced follower. Instant velocity center Q is located on the horizontal line and defined in Eq.(6) based on the velocity conditions at instant velocity centers. Then, the contact point is defined in Eq.(7) QcdYLd= (6) xQyyCLCF= (7) For a mechanism with oscillating roller follower as in Fig. 6, the distance of instant velocity center from cam center becomes in Eq.(8). The contact angle between cam and roller is expressed in Eq.(9) and then the contact point is defined in Eq.(10). Here, Zxy is the distance to a pivot from cam center. 1fxycQfcdZdLdd=+ (8) 1tanQxyLRR= (9) sinsinxxryyrCRRCRR=+= (10) fQLQVQVRRZLZxycR(Rx, Ry)xyC(Cx, Cy)Rr Fig. 6 Plate cam with oscillating roller follower In a case of cam mechanism with oscillating flat-faced follower as shown in Fig. 7, the location of instant velocity center is formulated as in Eq.(11) and the contact point is given in Eq.(12) 1fxycQfcdZdLdd=+ (11) ()()2coscossinxxyxyQfyxyQffCZZLCZL= (12) 422 / JUNE 2010 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3 fQLQVQVfVfZLZxycxyC(Cx, Cy)F(Fx, Fy) Fig. 7 Plate cam with oscillating flat-faced follower Finally, the profile of cam shape can be determined by transforming the contact point with the reverse angle of cam rotation as in Eq.(13), where xS and yS are the coordinates of cam profile. cossinsincosxxcycyxcycSCCSCC=+= + (13) 3.3 Internal normal angle at contact point The normal line at each contact point is shown in Figs. 4-7 for 4 different cases of the plate cam mechanisms. In this paper an internal normal angle( ) is defined as an angle between lines connected to cam center and to instant velocity center from contact point as shown in Fig. 8. Because tool centers for machining and biarc centers for curve fitting are located on the normal direction line through contact point, the internal normal angle must be transferred to the machining data process in order to guarantee the precise shape of cam. The position angle ()c of contact point shown in Fig. 8 is easily defined as in Eq.(14). Also the normal line angle ()f at contact point for cam mechanism with roller follower in Fig. 8(a) and Fig. 8(c) is same as in Eq.(15). The normal line angles are defined in Eq.(16) for reciprocating flat-faced follower (Fig. 8(b) and in Eq.(17) for oscillating flat-faced follower (Fig. 8(d), respectively. 1tanycxCC= (14) 1tanyyfxxRCRC= (15) 90ffaceslopeangle= (16) 1tan90yyfxxCZCZ= (17) Finally, the internal normal angle at contact point on cam profile can be expressed in Eq.(18) for plate cam mechanisms as shown in Fig. 8. fc= (18) CamFollowerfcRollercoordinateContact pointcoordinateCRyx? (a) Reciprocating roller follower CamFollowercfface slope angleContact pointcoordinateCyxQ (b) Reciprocating flat-faced follower CamFollowerfcRollercoordinateContact pointcoordinateCRyxQ (c) Oscillating roller follower CamFollowerfcContact pointcoordinatePivotcoordinateCZx, ZyyxQ (d) Oscillating flat-faced follower Fig. 8 Internal normal angles of plate cam mechanism INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3 JUNE 2010 / 423 4. Biarc Interpolation7 4.1 Characteristics of biarc curve Biarc interpolation is to connect the two circular arcs inside of a span connected by 2 points, where the biarc must have one tangential component at each point. Thus, all of points on a biarc interpolated curve have the unique slope and also have the unique direction angle toward the biarc center (It is called by radial direction angle in this paper). The radial direction angles 1( and 2) in Fig. 9 can be defined angles between a radial line to a center and a connected line on a span. Biarc interpolation is deeply dependent on the radial direction angle in a view of accuracy. Biarc curve can be categorized into 4 different types by combination of radial direction angles in a span as shown in Fig. 9, i.e. Fig. 9(a) is case 1 1(0 Fig. 9(b) is case 2 1(0 and 20) Fig. 9(c) is case 3 1(0 and 20), and 20). Here, centers of a biarc curve in case 1 and case 2 are located in the same plane and biarc curve becomes continuous smoothly. Centers in case 3 and case 4 are positioned in the cross plane and a inflection point must be existed on biarc curve as shown in Fig. 9. 4.2 Definition of equation for biarc curve Fig. 10 shows biarc curves with same planar centers, where a radius 1R consists of an arc from point 1 to *S at a center 1O and a radius 2R makes an arc from *S to point 2 at a center 2.O The point *S is located on the common radial line. By connected two circular arcs continuously, all points on biarc curve have the continuous tangential components on the span with point 1 and 2. Biarc curve with the same planar centers in Fig. 10 have a convex or concave curve. By rearrangement of the radial direction angles 1( and 2), radii of arcs 1(R and 2)R and length of span ( ),L the equation for biarc curve with the same planar centers can be defined as in Eq.(19). ()()121212211222coscossinsin12coscos0R RL RRL+= (19) 12O1O2S*R1R221 12O1O2R1R2S*21(a) Case 1 (b) Case 2 Fig. 10 Biarc with same planar centers 12O1O2R1R2S*2112O1O2R1R2S*21(a) Case 3 (b) Case 4 Fig. 11 Biarc with cross planar centers Fig. 11 shows biarc curves with cross planar centers, where a center 1O makes a circular arc with radius 1R in one plane and the other center 2O builds a circular arc with radius 2R in the opposite plane. Thus, a inflection point *S must be satisfied the conditions 12t1t2n1n212 (a) Case 1 12t1t2n1n212 (b) Case 2 12t1t2n1n212 (c) Case 3 12t1t2n1n212 (d) Case 4 Fig. 9 Cases of biarc curves 424 / JUNE 2010 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3 of 3 points 1(,O2O and *)S located on the same line and the continuity of the tangential components of 2 circular arcs at *.S The equation for biarc curve with cross planar centers is defined in Eq.(20). ()()121212211222coscossinsin12coscos0R RL RRL+= (20) 5. Profile Machining Based on Biarc Curve 5.1 Radial direction angle for biarc curve On the profile of plate cam contoured by the contact points, any two consecutive points build a biarc span and can be connected by a biarc curve as shown in Fig. 12. Effectiveness of biarc interpolation depends on the accuracy of radial direction angle ( ) at each point, because the centers of biarc curve are located on the lines defined by the radial direction angle at span points. 1XYP2P1n1211n222 Fig. 12 Position angles, slope angle and length Arbitrary span on cam profile can be positioned at two points (1 and 2) by rotation of cam as shown in Fig. 12. Here, is the slope angle of a connected line between point 1 and point 2 and is the internal normal angle at each point defined in section 2.3. Position angles at span points 1(p and 2)p can be easily defined by the coordinates of the points from cam center. Thus, the radial direction angles at span points on biarc span are arranged as in Eq.(21). 111222180pp=+=+ (21) 5.2 Radius of curvature for biarc curve In the case of biarc curve with same planar centers shown in Fig. 13, the biarc equation of Eq.(19) can be rearranged as the following. 11212coscossinsin1Z=+ (22) ()2121112222coscos0R R ZL RRL+= (23) The optimization of biarc curve requires the minimum difference of radii 1(R and 2).R The minimum difference is defined as in Eq.(24) and the radius of 2R is reformed as in Eq.(25). The differentiation of to 1R 1(/0)ddR= from Eq.(24) and Eq.(25) gives a quadratic equation as in Eq.(26) and also the radii of arcs 1(R and 2)R can be calculated in Eq.(27) and Eq.(25). ()212RR = (24) ()21121122cos2cosLRLRZ RL= (25) ()22111212212124cos2coscoscos0Z RLZRLZ+ = (26) ()212111cos1cos2LRZ += (27) In the case of biarc curve with cross planar center as shown in Fig. 14, the biarc equation of Eq.(20) is arranged as in Eq.(28) and Eq.(29). 21212coscossinsin1Z=+ (28) ()2122112222coscos0R R ZL RRL+= (29) 12RC1(XC1, YC1)R1R2RC2(XC2, YC2)L12S* Fig. 14 Radii on cross planar centers 12R1R212RC1RC2L(XC2, YC2)(XC1, YC1)S* Fig. 13 Radii on same planar centers INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3 JUNE 2010 / 425 Here, it is assumed that the radii of arcs are proportional to the radial direction angles on biarc span as given in Eq.(31). By solving Eq.(29) and Eq.(30), the radius of 1R is defined as in Eq.(31) and 2R can be calculated in Eq.(30). 12221121;RRRR= (30) ()2111coscosLAABRB+= (31) where 122cos,A= 1222BZ= 5.3 Reduction of machining points Because the excessive machining points on cam profile can go down the accuracy and efficiency in the machining procedures, the reduction of machining points optimally is required. The concept of reduction process is defined as the elimination of cam profile points where the points have the allowable radii within a given machining tolerance. As shown in Fig. 15 the profile points 1( ,P2P and 3)P are located inside a biarc span. The points (1, 1,P2,P3,P 2) correspond to the cam profile designed by the contact points between cam and follower. R2O2R1O112P1P2P3 Fig. 15 Profile points on biarc fitted curve The biarc curve fitted by the designed radial direction angle at span points (1 and 2) has two regions: The points 1(P and 2)P are located in an arc consisted by radius 1R at center 1O and the point 3()P is in the other arc by radius 2R at center 2.O Radius of each point inside of biarc span can be defined as a distance from center of the corresponding region. Thus, the reduction process can be built by the expansion or contraction of the biarc span, whether all of the radial differences at the inside points are within the range of a given machining tolerance as shown in Fig. 15. All points on cam profile are located between two lines built by a tolerance shown in Fig. 16(a). Thus, end point for biarc span will be expanded as in Fig. 16(b) and vice versa. Generally profile of plate cam has a smooth shape and machining profile point can be reduced if biarc curves for cam shape are defined accurately. 12iji+1i+2j-1.S* (a) 12iji+1i+2j-1j+1.S* (b) Fig. 16 Diagram for data reduction procedure 5.4 Tool path and center direction angle NC data consists of radius of tool path, coordinates and direction angle of tool center at each machining point. Fig. 17 shows a machining point on cam profile and a tool center located on the normal line. Here, tR is a radius of tool and nR is a radius of biarc span. The tool center direction angle cut corresponds to the internal radial angle( ) defined in the shape design process of plate cam. Thus, tool center direction angle and radius of tool path are given in Eq.(32) and Eq.(33) respectively. The coordinates of tool center are defined by the geometric condition as in Eq.(34). cut= (32) tpntRRR=+ (33) ()()cossinxxtscutyytscutTSRTSR=+=+ (34) scutXYTSCenter of toolRtRn Fig. 17 Direction angle at tool center 6. Example for Shape Design and Profile Machining of Plate Cams11 The displacement conditions for a reciprocating roller follower are defined in Table 1 and Fig. 18. The design parameters for a cam mechanism are follows: Radius of base circle is 80mm, radius of roller is 10mm, and it has no eccentricity between cam and follower. The cam shape designed by the instant velocity center approach is shown in Fig. 19. 426 / JUNE 2010 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3 Table 1 Displacement conditions for plate cam with reciprocating roller follower section cam angle(o) motion curve type 1 0 60 dwell - 2 60 180 30mm rise Modified sinusoidal3 180 240 dwell - 4 240 360 30mm return Cycloid 090180270360Cam Rotating angle (deg.)0102030d i s p l a c e m e n t ( mm ) Fig. 18 Displacement curve of plate cam with reciprocating roller follower Fig. 19 Shape of cam with reciprocating roller follower In this case of example, 360 points are generated for profile of the designed cam shown in Fig. 20(a). Generally, linear interpolation method required the 25 times of the points for precise shape machining and circular interpolation method makes the additional machining points as the 1/2 of the design points because of a middle points inside of a fitted span. But, the proposed method reduces the machining points as 121 points under the condition of machining tolerance 0.001mm from 360 points for the designed profile of cam as shown in Fig. 20(b). The each span given in Fig. 20(b) has the unique radius by biarc curve interpolation. The accurate shapes of the design cam and the machining cam shown in Fig. 20 can give the verification of the proposed approach in accuracy and efficiency. In addition, the 3 cases of cam shapes for different cam mechanisms are shown in Figs. 19-20 for reciprocating roller follower, Fig. 21 for reciprocating flat-faced follower, Fig. 22 for oscillating roller follower, Fig. 23 for oscillating flat-faced follower. (a) Designed cam shape (b) Biarc curved cam shape Fig. 20 Profile of plate cam with reciprocating roller follower Table 2 Reduction for plate cam profile11 Number of initial design profile : 360 Machining tolerarce : 0.001 mm Follower type Number of initial design profile Reduction (%)Reciprocating roller 121(33.6%) 66.4 Reciprocating flat-faced 73(20.3%) 79.7 Oscillating roller 105(29.2%) 70.8 Oscillating flat-faced 133(36.9%) 63.1 (a) Designed cam shape (b) Biarc curved cam shape Fig. 21 Profile of plate cam with reciprocating flat-faced (a) Designed cam shape (b) Biarc curved cam shape Fig. 22 Profile of plate cam with oscillating roller follower INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 3 JUNE 2010 / 427 (a) Designed cam shape (b) Biarc curved cam shape Fig. 23 Profile of plate cam with oscillating flat-faced Application for 4 different types of plate cam mechanisms shows the results in Table 2. The proposed approach gives generally the well defined shapes of the design profile and machining profile with 6380% reduction of the profile data. 7. Conclusions This paper proposes a hybrid method for shape design and profile machining for general plate cam mechanisms. An instant velocity center method is used for shape design of plate cam and a biarc interpolation method is used for profile machining data processing. The main concept is that the design data generated in the design pr
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