Analysis And Simulation Of Cam Follower Mechanism Using .
International Journal of Multidisciplinary and Current ResearchResearch ArticleISSN: 2321-3124Available at: http://ijmcr.comAnalysis and Simulation of Cam Follower Mechanism Using Polynomial Cam Profilea*Tushar Kiran and S. K. SrivastavabaM. Tech. Student, Department of Mechanical Engineering, M.M.M.E.C., Gorakhpur- 273010, (UP), India.Associate Professor, Department of Mechanical Engineering, M.M.M.E.C., Gorakhpur- 273010, (UP), India.bAccepted 04 November 2013, Available online 01 December 2013, (Nov/Dec 2013 issue)AbstractThe cam follower mechanism is versatile and almost any arbitrarily-specified motion can be achieved. The use ofalgebraic polynomials to specify the follower motion is a new choice for cam profiles. This class of motion function ishighly versatile especially in high speed automobiles. In the present work, kinematic and dynamic analyses of camfollower mechanism with polynomial cam profiles are carried out. The kinematic analysis presents followerdisplacement, velocity, and acceleration driven by a cam rotating at a uniform angular velocity. Dynamic analysispresents static and inertial forces developed in the mechanism. A 2-3 polynomial cam profile shows discontinuousfollower acceleration at the ends of the stroke making it unsuitable at higher speeds. A 3-4-5 polynomial cam profile hasan extended control as it provides a zero acceleration at the end points and no control over the follower jerks at endpoints. The modelling and simulation of a cam follower mechanism is performed on SolidWorks and results arepresented for various cam speeds. The simulation results show substantially lower values of follower velocity andacceleration for 3-4-5 polynomial cam profile; hence, it is versatile and most suitable at higher speeds without muchmodifications.Keywords: Cam follower mechanism, kinematic analysis, dynamic analysis, polynomial cam profile, simulation.1. IntroductionIn any class of machinery where automatic control andaccurate timing are important, the cam is anindispensable part of mechanism. It is a curved outline ora curved groove, which, by its oscillation or rotationmotion, gives a predetermined specified motion to thefollower. Cam follower mechanisms find application in awide variety of devices and machines, such as printingpresses, shoe machinery, textile machinery, automobileengines and pumping devices. The application ofalgebraic polynomials for cams was developed by Dudley,in which the differential equations of motion are solvedusing polynomial follower motion equations. Stoddart [1]further showed an application of these polynomialequations to cam action. In polydyne cams, the profile isdesigned such that the follower lift curve matches adesired polynomial equation giving the cam followermechanism desired characteristics. Berzak andFreudenstein [2] stated about optimization criteria forpolydyne cam design.In polydyne cam it is possible to design a cam profilethat provides the features desired in kinematic behaviourat the start and end of the stroke. A 2-3 polynomial camprofile is cubic in nature and follower acceleration isdiscontinuous at the end points. 3-4-5 polynomial camprofile has six polynomial coefficient and a degree of five,provides added control over follower acceleration at theend points. Eight boundary conditions are needed to bespecified for finding all the polynomial coefficient in thecase of 4-5-6-7 polynomial cam profile. This polynomialcam profile extends the control feature producing zerojerks at the ends [3].In the present work, analyses of 2-3 polynomial camprofile, 3-4-5 polynomial cam profile and 4-5-6-7polynomial cam profile are presented. Kinematic anddynamic analyses are carried out using motion equations[3, 5]. The kinematic analysis presents followercharacteristics of displacement, velocity and acceleration.Dynamic analysis presents pressure angle, spring force,inertial force and resultant force. Combined plotsenlisting the follower characteristics of displacement,velocity and acceleration are presented for abovementioned polynomial cam profiles. Furthermore, plotsshowing the variation of pressure angle and forces fordifferent cam positions are presented. Next, a completeassembly of cam follower mechanism with 3-4-5polynomial cam profile is modelled and simulated inSolidWorks software. It is observed that a reduction incam speed from 650 rpm to 550 rpm, reduces maximum211 Int. J. of Multidisciplinary and Current research, Nov/Dec 2013
Follower Displacement, y(mm)Tushar Kiran et alAnalysis and Simulation of Cam Follower Mechanism Using Polynomial Cam Profile181614121086420-2 0Cam Speed 650 rpm2-3 Polynomial3-4-5 Polynomial4-5-6-7 Polynomial60120180240300360Cam Angle, θ (Degree)Figure 1: Follower displacement response for various polynomial cam profiles20Cam Speed 650 rpm2-3 PolynomialFollower Velocity, y'(mm/sec)153-4-5 polynomial4-5-6-7 Polynomial1050060120180240300360-5Cam Angle, θ (Degree)Follower Acceleration, y''(mm/sec2)Figure 2: Follower velocity response for various polynomial cam profiles50403020100-10 0-20-30-40-50Cam speed 650 rpm2-3 Polynomial3-4-5 Polynomial4-5-6-7 Polynomial60120180240300360Cam Angle, θ (Degree)Figure 3: Follower acceleration response for various polynomial cam profilesfollower velocity and maximum follower acceleration by15.5% and 21.2% respectively.2. Analysis2.1 Kinematic AnalysisAssuming Rise Dwell Return Dwell type cam profile, thegoverning equation of motion for follower displacementis expressed as [3]2-3 polynomial cam profile[ ( )( ) ](1)3-4-5 polynomial cam profile[( )( )( ) ]4-5-6-7 polynomial cam profile(2)[( )( )( )( ) ](3)0The follower characteristics are calculated for 130 rise,00040 dwell, 130 return and 60 dwell. Other parametersare assumed as: lift of follower 16 mm, speed ofocam 650 rpm and angle of rise 130 .Fig. 1 shows the follower displacements for 2-3, 3-4-5and 4-5-6-7 polynomial cam profiles at uniform camspeed of 650 rpm. As the degree of the polynomialincreases the slope of the displacement curve alsoincreases for rise stroke. Consequently, the valve openingduration becomes smaller; alternatively dwell periods atextreme positions of the follower increases.Fig. 2 shows the corresponding follower velocities forthree polynomial cam profiles. It is observed that thepeak velocity of the follower increases with an increase in212 Int. J. of Multidisciplinary and Current research, Nov/Dec 2013
Tushar Kiran et alAnalysis and Simulation of Cam Follower Mechanism Using Polynomial Cam ProfilePressure angle , α (Degree)30Cam Speed 650 rpm252-3 Polynomial3-4-5 Polynomial4-5-6-7 Polynomial201510500-560120180240300360Cam Angle, θ (Degree)Figure 4: Pressure angle variation with cam position for various polynomial cam profiles50Forces, Fi, Fs, F, P (Newton)Cam Speed 650 rpm40Inertial forceSpring forceStatic Force3020100060120180240300360-10Cam Angle, θ (Degree)Figure 5: Variations of inertial, spring and resultant forces for 3-4-5 polynomial cam profilethe degree of polynomial. Moreover, the velocity profileof follower is steeper during the middle of rise and returnstrokes. The peak values of follower velocity for 2-3polynomial, 3-4-5 polynomial and 4-5-6-7 polynomial camprofiles are 12.49 mm/sec, 15.52 mm/sec and 18.00mm/sec, respectively.Fig. 3 shows the corresponding follower accelerationsfor three polynomial cam profiles. The maximum value offollower acceleration/retardation for 2-3 polynomial , 3-45 polynomial and 4-5-6-7 polynomial cam profiles are22226.32 mm/sec , 31.26 mm/sec and 32.15 mm/sec ,respectively. It is observed that acceleration jumps at theends of rise and return strokes reduces when the degreeof polynomial increases. For, 2-3 polynomial the jerksbecomes infinite at the ends of strokes; however itreduces drastically (finite) for higher degree polynomials.Therefore, added control over the follower acceleration(alternatively, inertial forces) is achieved for higherdegree polynomial cam profiles.2.2 Dynamic AnalysisStatic and dynamic forces are used for the design ofdifferent elements of a mechanism. The pressure angleshould be between zero and about 30 for translatingfollowers to avoid jamming in the guide [6]. The pressureangle on the cam profile is calculated as [5](4)As a case study, the 3-4-5 polynomial cam profile isgenerated for the base circle radius ( ) 10 mm andfollower radius ( ) 5 mm. The values of inertial forces,0spring forces and resultant forces are calculated at 10 ofcam rotation using the empirical formulae [4]Inertial ForceSpring ForceResultant Force *() (5)These are derived with the assumptions that cam,follower, roller pin, guide and roller follower are perfectlyrigid; the coefficient of friction between cam and rollerfollower, roller and roller pin is zero; and roller followerrotates with the cam without slipping. The followingvalues are assumed for the dynamic analysis: follower mass (1.6 kg); spring constant (1.2 N/mm); force due to load (10 N); coefficient of friction (0.1); follower overhang (50 mm); follower bearinglength (10 mm).Fig. 4 shows the variation of pressure angle with camangle for three polynomial cam profiles at uniform camspeed of 650 rpm. As the degree of polynomial equationincreases successively from 2-3, 3-4-5, 4-5-6-7 there is an0increase in the peak value of pressure angle from 27.05 ,0027.78 and 28.38 , respectively. Moreover, the peakposition monotonically decreases during the rise strokeand increases during the return stroke. The values of213 Int. J. of Multidisciplinary and Current research, Nov/Dec 2013
Tushar Kiran et alAnalysis and Simulation of Cam Follower Mechanism Using Polynomial Cam ProfileFigure 6: Simulation model of cam follower mechanism with 3-4-5 polynomial cam profile40Follower Acceleration, y''(mm/sec2)30Cam Speed 650 rpm20AnalyticalSimulated100-10 060120180240300360-20-30-40-50Cam angle, θ (Degree)Follower Velocity, y'(mm/sec)Figure 7: Analytical and simulation responses for follower acceleration181614121086420650 RPM550 RPM450 RPM350 RPM060120180Cam angle, θ (Degree)240300360Figure 8: Velocity response of follower with speed as a parameteropressure angle is within reasonable limits ( 30 ) and thejamming of follower would not take place.Fig. 5 shows the variation of forces for 3-4-5polynomial cam profile. Spring force component hasmajor contribution on the resultant force whereas theinertial force has negligible effect. Thus, cam followermechanisms with a higher follower response rate can bedesigned using the polynomial profiles.3. Design and Simulationfollower mechanism modelling is based on the kinematicand dynamic results. The displacement of the follower isomarked at an interval of 10 cam rotation over the basecircle. The camshaft radius and cam width are assumed tobe 5 mm and 8 mm, respectively.The cam follower assembly is modelled in SolidWorkssoftware with following major dimensions: follower stemlength 50 mm; roller follower radius 5 mm; rollerfollower width 8 mm; roller pin radius 1 mm; roller pinwidth 8 mm; guide length 10 mm.3.1 Design3.2 SimulationThe design and simulation of cam follower mechanismwith 3-4-5 polynomial cam profile is carried out. The camComplete kinematic characteristics of the follower areobtained by simulation of the model shown in Fig. 6. The214 Int. J. of Multidisciplinary and Current research, Nov/Dec 2013
Tushar Kiran et alAnalysis and Simulation of Cam Follower Mechanism Using Polynomial Cam Profile50650 RPM550 RPM450 RPM350 RPMFollower Acceleration, y''(mm/sec2)403020100-10 060120180240300360-20-30-40-50Cam angle, θ (Degree)Figure 9: Acceleration response of follower with speed as a parametersimulation conditions for operating the mechanismassume static force 10 N, spring constant 1.2 N/mm andfollower overhang to guide length ratio 5:1. Themechanism is simulated at 650 rpm, 550 rpm, 450 rpmand 350 rpm, respectively.The simulation response for the followerdisplacement, velocity and acceleration are in affirmationwith the analytical response. Fig. 7 shows the analyticaland simulation responses for follower acceleration. Smallacceleration peaks are present at the start of the dwellooperiods (approximately at 130 and 300 ) due to theinertia of follower. The probability of the wear of thefollower is maximum for these two positions of camrotation.the velocity response of follower with speed as aparameter. When cam speed reduces from 650 rpm to550 rpm the maximum follower velocity reduces by15.5%. Subsequent reduction in cam speed by 100 rpmreduces the maximum follower velocities by 17.7% and22.8% respectively. It is observed that the duration of riseand return stroke is unaffected for entire speed range;thus, favours the use of polynomial cam profile at higherspeeds.Fig. 9 shows the follower acceleration response withspeed as a parameter. As cam speed reduces, a smootherfollower acceleration curve at lower values ofacceleration is observed. Moreover, the number of sharppeaks are less; consequently, the jerks are also reduced atlower speeds. Since, the magnitude of acceleration is less;hence, inertial forces are not significant. Hence, at highcam speed the use polynomial cam profiles is preferred.4. ConclusionsThe kinematic analysis of follower with higher degreepolynomial cam profile reveals shorter valve openingduration and larger dwell periods. A 2-3 polynomial camprofile shows discontinuous follower acceleration at theends of the rise and return stroke making it unsuitable athigher speeds. A 3-4-5 polynomial cam profile has anextended control as it provides a zero acceleration at theends of stokes but no control over the follower jerks. Thesimulation of cam follower mechanism with 3-4-5polynomial cam profile shows higher follower response athigh cam speeds mainly due to the low inertial forces;hence, should be preferred at higher speeds.References[1]. D. A. Stoddart (1953), Polydyne Cam Design, MachineDesign, pp. 121–135, 146–155, 159–162.[2]. N. Berzak and F. Freudenstein (1979), Optimization Criteriathin Polydyne Cam Design, Proceedings of 5 World Congresson Theory of Machine and Mechanisms, 1979, pp. 1303–1306.[3]. H. A. Rothbart (2004), Cam Design Handbook, McGraw Hill,New York.[4]. H. D. Desai and V. K. Patel (2010), Computer AidedKinematic and Dynamic Analysis of Cam and Follower,Proceedings of the World Congress on Engineering, Vol. II,ISBN: 978-988-18210-7-2.[5]. E. Söylemez (2011), METU Open Course Ware, .metu.edu.tr.[6]. R. L. Norton (2002), The Cam Design and ManufacturingHandbook, The Industrial Press, New York.[7]. J. Banks, J. Carson, B. Nelson and D. Nicol (2001), DiscreteEvent System Simulation, Prentice Hall, ISBN 0-13-0887021.[8]. R. S. Khurmi and J. K. Gupta (2008), Theory of Machines, S.Chand, New Delhi.[9]. F. Freudenstein (1960), On the Dynamics of High-SpeedCam Profiles, Int J. Mech. Sci. vol. 1, pp. 342–349.[10]. G. K. Matthew (1979), The Modified PolynomialSpecification for Cams, Proceedings of 5th World Congresson Theory of Machine and Mechanisms, pp. 1299–1302.E.E. Peisakh (1966), Improving the P[11]. olydyne Cam Design Method, Russian Engineering Journal,vol. XLVI, no. 12, pp. 25–27.215 Int. J. of Multidisciplinary and Current research, Nov/Dec 2013
Analysis and Simulation of Cam Follower Mechanism Using Polynomial Cam Profile . The modelling and simulation of a cam follower mechanism is performed on SolidWorks and results are presented for various cam speeds. . motion, gives
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