We first design and analyzethe contour surface of the globoidal indexing cam with the aid of computer, and then do optimum design according to the requirements of dynamics. Finally, we discuss the problem of the pressure angle of the globoidal indexing cam mechanism in detail and put forward a new concept of equivalent pressure angle.
Journal of Shanghai University (English Edition ) ISSN 1007-6417, Vol.4, No.l(Mar.2000), pp - - A n a l y s i s a n d D e s i g n o f the G l o b o i d a l I n d e x i n g C a m M e c h a n i s m FU Yan-ming School of Mechanical and Electronic Engineering and Automation, Shanghai University, Shanghai 200072, China Abstract We first design and analyze the contour surface of the globoidal indexing cam with the aid of computer, and then optimum design according to the requirements of dynamics Finally, we discuss the problem of the pressure angle of the globoidal indexing cam mechanism in detail and put forward a new concept of equivalent pressure angle Key words globoidal indexing cam, analysis of cam mechanism, design of cam mechanism, dynamics of mechanism, pressure angle Introduction The globoidal indexing cam mechanism (Fig ) i s Design of the Contour Surface of the Globoidal Indexing Cam globoidal cam and a frame It is a modern intermittent 2.1 Mathematical formulas of the contour surface of the cam composed mainly of a driven rotating disk 1, a driving indexing stepping device and is widely used in the high- The contour surface of globoidal indexing cam can be speed mechanical equipment of light industry, electronic designed according to the conjugate principle of the spa- industry and so on In order to design, manufacture and tial envelope surface In order to derive the mathematical formulas of contour surface of the cam conveniently, we adopt four sets of right-handed rectangular coordi- measure the globoidal indexing cam in a modern way, this paper attempts to put forward a method of analysis and design with the aid of computer aates (Fig 1) :two sets of moving coordinates $1( 01 X1, Y t , Z1 ) and $2 ( - X2, Y2, Z2) attached respectively to the rotating disk and cam 2, a set of fixed coordinates So ( Oo - Xo, Yo, Zo) attached to the frame, and a set of auxiliary fixed coordinates So" ( Oo.-Xo., Y,,., zo,) 2.1.1 Equation of the conjugate contact of the globoidal indexing cam mechanism Let P2 and P1 be the conjugate contact points on the cam and the rotating disk respectively, then the equation of the cylindrical surface of the roller in the coordinate $1 is (rpl)l = (Xl,Yl,Z1) T= xo~Xo' [x0' x~ (1) Its unit normal vector is = (O,cosO,sinO) r (Np)l Fig Globoidalindexing cam mechanism (r,pcosO,psinO) T (2) In order to find the relative velocity between the roller and the cam along their contact line, we can transform the radius vector of the contact point rpl from Received Apr 12, 1999 ; Revised Jun 12, 1999 the coordinates S~ to $2, that is (rpl)2 = M2o.Mo,oMol(rpl) = M21(rpl)l, (3) Vol No Mar 2000 FU Y M : Analysisand Design of the Globoidal Indexing Cam Mechanism where M2o,, Mo,o and Mol are respectively the matrixes of transformation of So" to $2, So to So" and S1 to So These matrixes can be got in terms of the rule of composition of coordinates transformation matrixes The derivation of the equation (3) is ( V p2l1) dM21 d ( r p l ) l - dM2t ~ (rp)l= td Y-(rPi)1 + M21 dt (4) Substituting the equation (4) to the equation ( 12 12 % )1 = M12(%,)2, dM21 (vp)l = MlZ d~(rp )1 (5) At the conjugate contact points, the relative velocity between the cylindrical working surface of the roller and the contour surface of the cam must be perpendicular to the common normal line, i.e (/~pt)l " (lt12) : Pl (6) Substituting Eqs (2) and (5) to Eq ( ) , we obtain the equation of conjugate contact of the globoidal indexing cam mechanism r tan0 C- ;COS~ If the globoidal indexing cam is right-handed, we may substitute - ~a to Eq (9) and obtain the equation of the contour surface of the cam Computer-aided design of globoidal indexing cam 2.2 The contour surface of the globoidal indexing cam can not be developed, so it is difficult for us to design by the usual engineering drawing In this paper, we design the cam with the aid of computer It can not only obtain different views of projective drawing, but also provide the data required for manufacturing by the numerical control machines and measurement The flowchart of the program is shown in Fig The main characters of we can get the formula 12 55 ' (7) where ~x is the turning angle of the disk, (COl/OJ2) is the ratio of the angular velocity of the disk and the cam the CAD software of the cam are as following ( ) The program is written by the Visual BASIC Language , and it is convenient to operate (2) The program adopts the form of person-computer dialogue By inputting the design parameters through keys, the range of use can thus be expanded ( ) The program is stored inside with the various laws of motion which are often used in the cam mechanisms It can display the diagrams of displacement, velocity, acceleration and jerk on the computer screen (4) The program can display and output the results of calculation of the cam contour surface and provide the data for the computer-aided manufacture and measureBent (5) The program can display the draft of the cam so that the designer can see the exterior contour of the Equation of the contour surface of the globoidal indexing cam cam Because the conjugate contact points of the roller and the cam are coincident, we can obtain Dynamic Design of Globoidal Indexing Cam (rp2) : 3.1 General dynamic equation of the mechanical system containing the globoidal indexing cam (rpl)2 Then Eq (3) can be expressed as below: (rp2)2 = M2l(rpl)l (8) Developing Eq ( ) , we can obtain the equations of the contour surface of the globoidal indexing cam ( lefthanded ) as shown in Fig X = XlCOS~lCOS~2 ylsin¢lcos~2 - z l s i n ~ - Ccos~2 Y2 = - xlcos~lsin~2 + ylsin~lsin~2 - ZlCOS~2 + Csin~ z2 = xlsin~ l + yleos~ (9) mechanism In the mechanical system containing the globoidal indexing cam mechanism, the electric motor usually used is of the ordinary type, the transmitting mechanism which we often adopt is V-belt or gear transmission, and the globoidal indexing cam mechanism is the executive mechanism As the system is single degree of freedom , we can take the cam as the equivalent link and establish a dynamic model in accordance with the dynamic equivalent principle The equivalent driving torque acting on the equivalent link is Journal of Shanghai University 56 can be got in terms of the given law of motion of the cam mechanism It is dependent to the turning angle of the cam ~e' therefore Mre is the function of ~e" The moment of inertia of the equivalent link is + Seleetthe law ofmotion / of the drivendisk Je = Jr + Jc + Jd + Ji i , (12) where J r , Jc and Ja are the moments of inertia of the turning disk, the cam and the electric motor respectively Ji and rol are the moment of the inertia and the an- parametem by keyboard + I Display the curve diagram of the ] displacement,velocity,accelerationandjerk gular velocity of i-th transmitting member, n is the number of the transmitting member In Eq ( ) , I + o3d//We and Calculate the rotatingangle of the driven disk from the rotating angle of the cam accordingto the law ofmotien + (13) i.e (14) Jre = Jr(cor/We )2, (15) The general dynamic equation of the mechanical system containing the globoidal indexing cam mechanism can be obtained through the Eqs ( ) , (11) and ( ) l Je = Jre(~e) + Jk = Je(~e), Yk = Jc + Ja(roa/%) z + ~ Y i ( r o i / W e )2 i=1 + + are definite values, % / % varies with ~e- Therefore Eq (12) can be expressed as below: I Calculatethe pressure angle, vectorand turning angle parameter of the roller contactpoint Calculate the coordinatesof the contact points and the cam contoursurface W i//rO e I I m ] Drawthe draft of the cam contour I /oisplay tho / + otho + [ d% Mde(coe)- Mr~(~) d~e (j, + J , ) % I oo.,o /" Fig Flowchart of the CAD program 3.2 (10) Where M& is the equivalent driving torque, Md is the driving torque of electric motor and it is the function of angular velocity cod The transmission ratio i = cod/% is a given value, therefore Mac is the function of the cam angular velocity The equivalent resistant torque acting on the equivalent link is = Mre(~e), - This differential equation can be solved by four-order Runge-Kutter method For reducing the number of iteration in solving the equation, the initial value (coe)0 is defined as the angular velocity of the earn corresponding to the normal velocity of the electric motor + Mr~ : Mr(co J % ) dcoe(~e'coe)'~ee (16) Print the output data / Md~ = Md(cod/%) = iMd = M d e ( % ) , d~e (11) where Mr, is the equivalent resistant torque, Mr is the resistant torque acting on the turning disk, and we usually adopt it as constant The ratio of the angular velocity of the disk cot to the angular velocity of the cam rOe T h e n o n - u n i f o r m c o e f f i c i e n t o f the a n g u l a r v e l o c i t y a n d the m o m e n t o f the i n e r t i a o f the cam The globoidal indexing cam mechanism bears mainly the inertia load As the inertia load has a considerable variation in a working circle , it will bring about a violent velocity fluctuation of the system and have influence on the regular function of the electric motor Therefore, we make the cam have a more suitable moment of inertia in the system to attain to the following two purposes (1) Choosing an electric motor of smaller power The law of acceleration motion of the cam has a greater periodic variation such as modified trapezoidal acceleration, modified sine acceleration and so on During the former Vol No Mar.2000 FU Y M : Analysis and Design of the Globoidal Indexing Cam Mechanism stage of the indexing period, the output disk is in the peak of the load and the system needs a greater driving torque During the latter stage of the indexing period, the acceleration is usually negative, the load of inertia changes its direction and it corresponds to the driving torque During dwell period, the system needs only the driving torque to overcome frictional torque and so on Therefore it is wasteful if we choose the power of the motor in terms of the peak of load Now if we can make the cam concurrently play the role of a flywheel, the electric motor of smaller power can be used to meet the requirements ( ) Moderating the velocity fluctuation of the mechanical system The velocity fluctuation not only has influence on the technology of the machine, but also is restricted by t h e p e r f o r m a n c e of the electric motor Three-phase alternating current electric motor has an allowable value of non-uniformity of the angular velocity [ s ] If the [ s ] is too big, the motor will be overloaded and thus cannot operate regularly If the cam has a more suitable moment of inertia , the fluctuation of the angular velocity in the me6t]anical system can be controlled in an allowable region through storage energy of the cam The fluctuation of the angular velocity in-the mechanical system can be controlled by restrictirrg the non-uniform coefficient of the angular velocity of the cam, i e 8= (O,e)m~CO m (%)~n 2((%)ma:,- (OOe)min) (( Oe)rnax _}_ (03e)min ~ [83, (17) where the allowable value [ ] is determined by the less allowable coefficient of the angular velocity between the requirements of technology of the machine and the electric motor The moment of inertia of the cam is determined as follows: (a) We estimate the size of the disk by the working condition and find its moment of inertia Jr (b) We initially choose a feasible region [Jkl ,Jk2 ] of Jk and use the method of the golden section search to determine Jk within the region (c) (%)m~x and ( O J e ) m i n c a n be found-in a working circle a@ording to the Eq (16) (d) 8min can be obtained within the initial feasible region [Jkl ,Jkz] according to the Eq (17) (e) If 8mi n is greater than [ ] ,we must change the region [Jkl ,Jk2] and return to the step (b) to repeat 57 (f) If 8mi n is equal to or smaller than [ ] ,we can obtain the corresponding Jk and find the moment of inertia of the cam according to Eq (15) If the obtained 8mi n is much smaller than [ ] , it means ,hat the chosen Jk is too big, we can change the region [J~l ,J~2] and return to (b) to repeat Optimum mathematical model of the globoidal indexing cam with m i n i m u m volume Designing variables The contour size of the cam (Fig 3) is determined by the maximum radius a , the minimum radius ao and the width of the cam L Therefore we specify the optimum designing variables of the mathematical model as X = [Xl,X2,X3,X4] T = [ a , a o , L , a ] T, (18) where a is the pressure angle of the cam mechanism I 280 28 I Fig.3 Contour size of the globoida[indexing cam Objective function In order to obtain the minimum volume of the cam when the moment of inertia of the cam is a given value, the objective function can be expressed as minV(X) X ~ R4 = {~r(C2 + R ) L - l ~ r r L XE R -~-+ R2arcsin2~ ) , (19) where C = a - a~ + L2/4 ( a - a0) + L 2 ( a - a0) , R = ( a - a0) Constrained condition The structural constrained equations are Sl(X) = ao > S ( X ) = a - a0 > I (20) S3(X) = L > The constrained equation of the moment of inertia is G I ( X ) = J - J c = O (21) 58 Journal, o f Shanghai University tact line can be obtained through Eq ( ) The constrained equation of the pressure angle is Through the derivative of Eq (25) with respect to r , we can obQI(X) = a - [ a 0] ~ 0, (22) where [ % ] is the allowed pressure angle of the cam tain the distributed values of the pressure angle on the contact line mechanism, it can generally be about 30", and Jc is a given value dr Therefore the mathematical model of the optimum design of the globoidal indexing cam should be - cos2a(C - rcos~l )2 " (26) The value of the right side of the Eq ( ) is always greater than zero, therefore the pressure angles on the rain V ( X ) , contact line increase with the increase of r Because the value of the pressure angle of the every point on a contact line is different, it is complicate to calculate precise- XE R4 and it is constrained by S~(X) > (i = 1,2,3), G,(X) = (i = 1), mechanism and discussing the influence of the pressure ( i = 1) angle on the dimensional parameters of the mechanism, O,(X)~O ly the resultant pushing force When designing the cam T h a t is the problem of the constrained nonlinear optim u m design of four dimensions It can be solved by the method of the mixed penalty function Pressure Angle of the Gioboidal In- dexing Cam M e c h a n i s m we often use the simplified method, that is , to substitute the pitch radius R of the disk to r in the Eq (25) and find the value of the nominal pressure angle: E a = arctan C - Rcos~ ~22 4.3 In the globoidal indexing cam mechanism , the pressure angles may be larger than the allowable value on some points and may be far below the allowable value on some other points Therefore the pressure angle Of the cam mechanism is not restricted by the usual way " (27) Equivalent pressure angle There sometimes occurs the condition in the practical mechanism, in which the pressure angles of the some points on a contact line are over the allowed value [ a ] , and the pressure angles of the other points are smaller than [ a ] , but the effective resultant pushing force may 4.1 Mathematical model of the pressure angle of the gioboidal indexing cam mechanism The pressure angle a of the cam mechanism is defined on the theoretical profile It is the acute angle between the direction of the acted force at a point on the axis of the roller and the direction of the velocity at the point (vpz) = to × r = (colZ 1) x (rX1) = o~lrrl, (23) still be enough If we use the nominal angle a o as a rule for checking, the result will be very inappropriate In order to reflect the condition of the acting force on each contact line, we present a new concept of equivalent pressure angle We think that the pressure angles of all points on a contact line can be equivalent to a value , that is, an equivalent pressure angle In designing, the equivalent pressure angle is taken as a nominal value If ( VPl)l cosa = ( N p ) l " the all equivalent pressure angles on the contact lines in I vpt)l I - (cosOYt +sinOZ1)" every turning angle of the cam are not over the allowed to r Y 601/" - cos0 (24) value , we think that the design of the cam mechanism is suitable Let Fr be the effective resultant pushing force Substituting Eq (7) to Eq ( ) , we obtain acting on the roller, Fi the pushing force acting on every conjugate contact point i along the contact line, a i a = = arctan C - 4.2 rcos~ l ~ o ! J " the pressure angle, Pl the factor of weight and n the division number of the roller width, so we can obtain Nominal pressure angle For every turning angle of the cam, we can find an instantaneous conjugate contact line through Eqs ( ) and ( ) T h e pressure angles of these points on the con- Fr = (piFicosai) Pi (i = 0,1, -,n) i=0 T h e equivalent pressure angle can be defined as (28) Vol.4 No t Mar 2000 FU Y M : picosot i a e =- a r c c o s i=0 (i Pi Analysis and Design of the Globoidal Indexing Cam Mechanism lent pressure angle are considered from the different O,l,.-.,n) i=0 (29) T h e factor of weight Pi in the Eqs (28) and (29) can be given from the point of the view of the probability in considering the condition of the acting force in the mechanism, the contact manner and elastic deformation of the roller , and the lubrication and friction of the contact surfaces of the roller and the cam, etc This~paper recommends that the factor of weight be given according to the normal distribution (/~ = n / , a = ) , and its analytical expression is [(i-n~2) Pi = C~-~exp - 4.4 2] • 59 (30) Conclusion Although the nominal pressure angle and the equiva- points of view, the former is to simplify the calculation and the latter is to reflect the practical condition of the force on the contact line , but the difference between the two calculations is very little in most cases Therefore it is suitable to use nominal pressure angle to simplify the calculation during the design and check of the cam mechanism References J.R.Jones, Cam and Cam Mechanisms, The Institution of Mechanical Engineers , London , 1978 : 51 - 62 [2 ] Yin Hongliang and Li Weiming, Profile of glohoidal indexing cam mechanism, Proceedings of the 1st Chinese National Symposium on Mechanisms, 1983:177 - 182 (in Chinese ) [3] Tang S k , jin D W , Dynamics of Machinery, China High Education Publishing House, 1984: - 135 (in Chinese ) [1] ... contour of the Equation of the contour surface of the globoidal indexing cam cam Because the conjugate contact points of the roller and the cam are coincident, we can obtain Dynamic Design of Globoidal. .. : Analysis and Design of the Globoidal Indexing Cam Mechanism stage of the indexing period, the output disk is in the peak of the load and the system needs a greater driving torque During the. .. gioboidal indexing cam mechanism The pressure angle a of the cam mechanism is defined on the theoretical profile It is the acute angle between the direction of the acted force at a point on the axis of