Analysis and Design of the Globoidal Indexing Cam Mechanism

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.

A n a l y s i s a n d D e s i g n o f t h e 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 do optimum de-

sign according to the requirements of dynamics Finally, we discuss the problem of the pressure angle of the globoidal indexing cam mecha- nism 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

1 Introduction

T h e globoidal indexing cam mechanism ( F i g 1 ) i s

composed mainly of a driven rotating disk 1, a driving

globoidal cam 2 and a frame It is a modern intermittent

indexing stepping device and is widely used in the high-

speed mechanical equipment of light industry, electronic

industry and so on In order to design, manufacture and

measure the globoidal indexing cam in a modern w a y ,

this paper attempts to put forward a method of analysis

and design with the aid of computer

Fig 1 Globoidal indexing cam mechanism

Received Apr 12, 1999 ; Revised Jun 12, 1999

2 Design of the Contour Surface of the Globoidal Indexing Cam

2.1 Mathematical formulas of the contour sur- face of the cam

T h e contour surface of globoidal indexing cam can be designed according to the conjugate principle of the spa- tial envelope surface In order to derive the m a t h e m a t i - cal formulas of contour surface of the cam conveniently,

we adopt four sets of right-handed rectangular coordi- aates (Fig 1) : t w o sets of moving coordinates $1( 01 - X1, Y t , Z1 ) and $2 ( 0 2 - X2, Y2, Z2) attached respec- tively to the rotating disk 1 and cam 2, a set of fixed co- ordinates 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 o f the conjugate contact o f the globoidal indexing c a m m e c h a n i s m

Let P2 and P1 be the conjugate contact points on the cam and the rotating disk respectively, then the equa- tion of the cylindrical surface of the roller in the coordi- nate $1 is

( r p l ) l = ( X l , Y l , Z 1 ) T = (r,pcosO,psinO) T ( 1 ) Its unit normal vector is

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 the coordinates S~ to $2, that is

(rpl)2 = M2o.Mo,oMol(rpl) 1 = M21(rpl)l, ( 3 )

Vol 4 No 1 Mar 2000 FU Y M : Analysis and Design of the Globoidal Indexing Cam Mechanism 55

where M2o,, Mo, o and Mol a r e 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 compo-

sition of coordinates transformation matrixes T h e

derivation of the equation (3) is

( V p l ) 2 1 2 dM21 d ( r p l ) l

= td -Y-(rPi )1 + M21 dt

dM2t -

(4) Substituting the equation (4) to the equation

% )1 = M12(%,)2,

we can get the formula

At the conjugate contact points, the relative velocity be-

tween the cylindrical working surface of the roller and

the contour surface of the cam must be perpendicular to

the common normal line, i.e

( / ~ p t ) l " ( l t 1 2 ) 1 : 0 ( 6 )

Pl

Substituting Eqs (2) and (5) to Eq ( 6 ) , we obtain the

equation of conjugate contact of the globoidal indexing

cam mechanism

r

where ~x is the turning angle of the disk, (COl/OJ 2) is

the ratio of the angular velocity of the disk and the cam

idal indexing cam

Because the conjugate contact points of the roller and

the cam are coincident, we can obtain

( r p 2 ) 2 : ( r p l ) 2

T h e n Eq (3) can be expressed as below:

Developing Eq ( 8 ) , we can obtain the equations of

the contour surface of the globoidal indexing cam ( left-

handed ) as shown in Fig 1

X 2 = XlCOS~lCOS~2 ylsin¢lcos~2 -

z l s i n ~ 2 - Ccos~2

Y2 = - xlcos~lsin~2 + ylsin~lsin~2 - (9)

ZlCOS~ 2 + Csin~ 2

z2 = x l s i n ~ l + yleos~ 1

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

2 2 Computer-aided design of globoidal index- ing cam

T h e 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 T h e flowchart of the program is shown in Fig 2 T h e main characters of the CAD software of the cam are as following

( 1 ) T h e 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

( 3 ) T h e program is stored inside with the various laws of motion which are often used in the cam mecha- nisms It can display the diagrams of displacement, ve- locity, 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 measure-

B e n t (5) T h e program can display the draft of the cam so that the designer can see the exterior contour of the

c a m

3 D y n a m i c Design o f Globoidal Index- ing Ca m

3.1 General dynamic equation of the mechani-

mechanism

In the mechanical system containing the globoidal in- dexing 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 execu- tive mechanism As the system is single degree of free- dom , we can take the cam as the equivalent link and es- tablish a dynamic model in accordance with the dynamic equivalent principle T h e equivalent driving torque act- ing on the equivalent link is

+

Seleetthe law ofmotion /

of the driven disk

parametem by keyboard

+

I Display the curve diagram of the ]

1

displacement,velocity, acceleration and jerk

I +

Calculate the rotating angle of the driven

disk from the rotating angle of the cam

according to the law ofmotien

+

I Calculate the pressure angle, vector and turning

angle parameter of the roller contact point

+

Calculate the coordinates of the contact I 1

points and the cam contour surface

I +

l

] Draw the draft of the cam contour [

m

+ /oisplay tho o tho oo.,o / "

+

/ Print the output data /

+

Fig 2 Flowchart of the CAD program

Md~ = M d ( c o d / % ) = iMd = M d e ( % ) , (10)

Where M& is the equivalent driving torque, Md is the

driving torque of electric motor and it is the function of

angular velocity co d 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

where Mr, is the equivalent resistant torque, Mr is the

resistant torque acting on the turning disk, and we usu-

ally adopt it as constant The ratio of the angular veloci-

ty of the disk cot to the angular velocity of the cam rOe

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" T h e moment of inertia of the equivalent link is

where J r , Jc and Ja are the moments of inertia of the

turning disk, the cam and the electric motor respective-

ly Ji and rol are the moment of the inertia and the an-

gular velocity of i-th transmitting member, n is the number of the transmitting member In Eq ( 1 2 ) , o3 d//W e and W i / / r O e are definite values, % / % varies with

~e- Therefore Eq (12) can be expressed as below:

Yk = Jc + J a ( r o a / % ) z + ~ Y i ( r o i / W e )2 (15)

i=1

T h e general dynamic equation of the mechanical sys- tem containing the globoidal indexing cam mechanism can be obtained through the Eqs ( 1 0 ) , (11) and ( 1 3 )

d% Mde(coe) - Mr~(~) 2 d~ e dcoe(~e'coe)'~ee

(16) This differential equation can be solved by four-order Runge-Kutter method For reducing the number of iter- ation 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

3 2 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 t h e a n g u l a r

v e l o c i t y a n d t h e m o m e n t o f t h e i n e r t i a o f t h e

c a m

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 vio- lent 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 T h e law of acceleration motion of the cam has a greater peri- odic variation such as modified trapezoidal acceleration, modified sine acceleration and so on During the former

Vol 4 No 1 Mar 2000 FU Y M : Analysis and Design of the Globoidal Indexing Cam Mechanism 57

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

( 2 ) Moderating the velocity fluctuation of the me-

chanical system T h e 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 al-

lowable 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 s y s t e m can be con-

trolled in an allowable region through storage energy of

the cam T h e 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

(O,e)m~- (%)~n

8 =

CO m

2((%)ma:,- ( O O e ) m i n )

(( Oe)rna x _}_ (03e)min ~ [83,

(17)

where the allowable value [ 8 ] is determined by the less

allowable coefficient of the angular velocity between the

requirements of technology of the machine and the elec-

tric motor The moment of inertia of the cam is deter-

mined as follows:

(a) We estimate the size of the disk by the working

condition and find its moment of inertia J r

(b) We initially choose a feasible region [Jkl ,Jk2 ] of

Jk a n d 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 ( 1 6 )

(d) 8mi n can be obtained within the initial feasible re-

gion [Jkl ,Jkz] according to the Eq ( 1 7 )

(e) If 8mi n is greater than [ 8 ] , w e must change the

region [Jkl ,Jk2] and return to the step (b) to repeat

(f) If 8mi n is equal to or smaller than [ 8 ] ,we can ob- tain the corresponding Jk and find the moment of inertia

of the cam according to Eq ( 1 5 ) If the obtained 8mi n is much smaller than [ 8 ] , it means ,hat the chosen Jk is too big, we can change the region [J~l ,J~2] and return

to (b) to repeat

3 3 O p t i m u m mathematical model of the globoidal indexing cam with m i n i m u m volume

3 3 1 Designing variables

T h e 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 = [ X l , X 2 , X 3 , X 4 ] T = [ a , a o , L , a ] T, (18) where a is the pressure angle of the cam mechanism

I

280

2 8

I

Fig.3 Contour size of the globoida[ indexing cam

3 3 2 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

m i n V ( X ) = min {~r(C2 + R 2 ) L - l ~ r r L 3 -

X ~ R 4 XE R 4

- ~ - + R 2 a r c s i n 2 ~ ) , (19)

where

a 2 - a~ + L2/4 4 ( a - a0) 2 + L 2

C = 2 ( a - a0) , R = 8 ( a - a0)

3 3 3 Constrained condition

The structural constrained equations are

S l ( X ) = ao > 0

S 2 ( X ) = a - a0 > 0 I (20)

S 3 ( X ) = L > 0 The constrained equation of the moment of inertia is

T h e constrained equation of the pressure angle is

where [ % ] is the allowed pressure angle of the cam

mechanism, it can generally be about 30", and Jc is a

given value

Therefore the mathematical model of the optimum de-

sign of the globoidal indexing cam should be

rain V ( X ) ,

XE R 4

and it is constrained by

S ~ ( X ) > 0 ( i = 1 , 2 , 3 ) ,

G , ( X ) = 0 ( i = 1 ) ,

O , ( X ) ~ O ( i = 1)

T h a t is the problem of the constrained nonlinear opti-

m u m design of four dimensions It can be solved by the

method of the mixed penalty function

dexing Cam M e c h a n i s m

In the globoidal indexing cam mechanism , the pres-

sure angles may be larger than the allowable value on

some points and m a y 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

of the gioboidal indexing cam mechanism

T h e 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) 1 = to 1 × r = ( c o l Z 1) x ( r X 1 ) = o~lrrl, (23)

( V P l ) l

cosa = ( N p ) l " I vpt)l I - (cosOYt +sinOZ1)"

to 1 r Y 1

601/"

Substituting Eq ( 7 ) to Eq ( 2 4 ) , we obtain

a = 0 = arctan C - rcos~ l ~ o 2 ! J "

4.2 Nominal pressure angle

For every turning angle of the cam, we can find an

instantaneous conjugate contact line through Eqs ( 1 )

and ( 7 ) T h e pressure angles of these points on the con-

tact line can be obtained through Eq ( 2 5 ) T h r o u g h the derivative of Eq (25) with respect to r , we can ob- tain the distributed values of the pressure angle on the contact line

dr - c o s 2 a ( C - r c o s ~ l )2 " (26)

T h e value of the right side of the Eq ( 2 6 ) is always greater than zero, therefore the pressure angles on the contact line increase with the increase of r Because the value of the pressure angle of the every point on a con- tact line is different, it is complicate to calculate precise-

ly the resultant pushing force W h e n designing the cam mechanism and discussing the influence of the pressure angle on the dimensional parameters of the mechanism,

we often use the simplified m e t h o d , that is , to substi- tute the pitch radius R of the disk to r in the Eq (25) and find the value of the nominal pressure angle:

E

a 0 = arctan C - Rcos~ 1 ~22 " (27)

4.3 Equivalent pressure angle

T h e r e 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 m a y 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 the all equivalent pressure angles on the contact lines in every turning angle of the cam are not over the allowed value , we think that the design of the cam mechanism is suitable Let Fr be the effective resultant pushing force acting on the roller, Fi the pushing force acting on ev- ery conjugate contact point i along the contact line, a i the pressure angle, Pl the factor of weight and n the di- vision number of the roller width, so we can obtain

Fr = (piFicosai) Pi ( i = 0 , 1 , - - - , n ) ( 2 8 )

i=0

Vol.4 No t Mar 2000 FU Y M : Analysis and Design of the Globoidal Indexing Cam Mechanism 59

(i O , l , - , n )

a e =- a r c c o s picosot i Pi

i=0 - - i = 0

( 2 9 )

T h e factor of w e i g h t Pi in the Eqs ( 2 8 ) and ( 2 9 ) can

be given from the point of the view of the probability in

considering the condition of the acting force in the

m e c h a n i s m , the contact m a n n e r and elastic deformation

of the roller , and the lubrication and friction of the con-

tact surfaces of the roller and the c a m , e t c This~paper

r e c o m m e n d s that the factor of weight be given according

to the normal distribution (/~ = n / 2 , a = 1 ) , and its

analytical expression is

1 [ ( i - n ~ 2 ) 2 ]

A l t h o u g h the nominal pressure angle and the equiva-

lent pressure angle are considered from the different 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

t w o 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

[1] 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 index- ing cam mechanism, Proceedings of the 1 st Chinese Nation-

al Symposium on Mechanisms, 1983:177 - 182 (in Chi- nese )

[3] Tang S k , jin D W , Dynamics of Machinery, China High Education Publishing House, 1984: 7 0 - 135 (in Chi- nese )