CAM GEAR DESIGN

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215 TECHNICAL UNIVERSITY OF CLUJ-NAPOCA

ACTA TECHNICA NAPOCENSIS

International Conference on Engineering Graphics and Design

12-13 June 2009

CAM GEAR DESIGN

Narcisa POPESCU, Relly Victoria PETRESCU, Florian Ion PETRESCU

Abstract: The paper presents an original method to determine the efficiency of a mechanism with camand follower. One analyzes four types of cam mechanisms: 1.The mechanism with rotary cam and platetranslated follower; 2.The mechanism with rotary cam and translated follower with roll; 3.Themechanism with rotary cam and rocking-follower with roll; 4.The mechanism with rotary cam and platerocking-follower. One takes into account the cams mechanism (distribution mechanism), which is the

second mechanism from the internal-combustion engines. The optimizing of this mechanism (thedistribution mechanism), can improve the functionality of the engine and may increase the comfort of thevehicle as well. Key words: Efficiency, force, power, cam, follower, roll, plate, force, speed.

1. INTRODUCTION

In this paper the authors present anoriginal method to calculate the efficiency ofthe cams mechanisms.One analyzes four kinds of cams and followersmechanisms: 1. A mechanism with rotary cam

and plate translated follower; 2. A mechanismwith rotary cam and translated follower withroll; 3. A mechanism with rotary cam androcking-follower with roll; 4. A mechanismwith rotary cam and plate rocking-follower. Forevery kind of cams and followers mechanism,one has utilizing a different method for thecams design with a better efficiency.

The dynamic velocities follow the samedirection like forces. The dynamic efficiencyuses the dynamic velocity. The normal(mechanical) velocities (known by kinematics)have the direction imposed by the linkage, andwith this velocity one calculates the mechanicalefficiency. When the kinematics (mechanical)velocity coincide with the dynamic velocity(the velocity has the same direction like theforce), one can say that the linkage (couple or

joint) is dynamically normal (natural). See thegeared transmissions, or the cam with platefollowers.

2. DETERMINING THE MOMENTARYDYNAMIC (MECHANICAL)EFFICIENCY OF THE ROTARY CAMAND PLATE TRANSLATED FOLLOWER

The consumed motor force, F c, perpendicular in A on the vector r A, is divided

in two components, [11-15]: a) F m, whichrepresents the useful force, or the motor forcereduced to the follower; b) F , which is thesliding force between the two profiles of camand follower, (see the picture 1). See thewritten relations (2.1-2.10):

sin= cm F F (2.1)

sin122 == vvv D (2.2)

212 sin=== v F v F P P cmu Du (2.3)

1v F P cc = (2.4)

22

1

21

cossin

sin

==

=

===v F

v F

P

P

c

c

c

ui

Di (2.5)

220

2

2

22

')(

''sin s sr

s

r

s

A ++== (2.6)


2/6

216

O

A

r 0

s

s

r A

1vr

2v

r

12vr

B

C

D

F

r m F r

c F r

F

E

2002 Florian PETRESCUThe Copyright-LawOf March, 01, 1989U.S. Copyright OfficeLibrary of CongressWashington, DC 20559-6000202-707-3000

Fig. 1. Forces and speeds to the cam with platetranslated follower. Determining the efficiency of the

cams mechanism

cos= c F F (2.7)

cos112 = vv (2.8)

2

112 cos== v F v F P c (2.9)

22

1

21 sincos

cos ==

==v F

v F P

P

c

c

ci (2.10)

3. DETERMINING THE MOMENTARYDYNAMIC AND MECHANICALEFFICIENCY OF THE ROTARY CAMAND TRANSLATED FOLLOWER WITHROLL

The pressure angle, , is determined by therelations (3.5-3.6), [2, 4, 6, 7]; The writtenrelations are the following:

20

22B s)(ser ++= (3.1)

2 B B r r = (3.2)

B B r

e= sincos (3.3)

B B r

s s += 0cossin (3.4)

220

0

)'()(cos

e s s s

s s

+++

= (3.5)

220 )'()(

'sin

e s s s

e s

++= (3.6)

sinsincoscos)cos( =+ (3.7)

)cos(2222 ++= Bbb B A r r r r r (3.8)

220

220

)'()(

)'()'()(cos

e s s sr

e sr e s s se

A

b A

++

+++= (3.9)

220

2200

)'()(

])'()([)(sin

e s s sr

r e s s s s s

A

b A

++

+++= (3.10)

cos'

)'()(

')()cos(

220

0 =++

+= A A

A r s

e s s sr

s s s (3.11)

2cos'

cos)cos( = A

A r s (3.12)

One can write the next forces, speeds and powers (3.13-3.20) (see the picture 2): F m, v m,are perpendicular on the vector r A in A. F m is

divided in F a (the sliding force) and F n (thenormal force). F n is divided too, in F i (the

bending force) and F u (the useful force). Themomentary dynamic efficiency can be obtainedfrom the relation (3.18), [11-15]:

==

)sin(

)sin(

Ama

Ama

F F

vv (3.13)

=

=

)cos(

)cos(

Amn

Amn

F F

vv (3.14)

==

sin

sin

ni

ni

F F

vv (3.15)

==

==

==

cos)cos(cos

coscos'

cos)cos(cos

22

2

2

Amnu

Am

Amn D

F F F

vr s

v

vvv

(3.16)


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217

0 A

A

B

A -

F n , v n

F m , v m

F a , v a

F i, v iF n , v n

F u , v 2

B

B 0

A 0

A

O

x

e

s 0

r 0

r A

r B

s

n

C

r b

2002 FlorianPETRESCUThe Copyright-LawOf March, 01, 1989U.S. Copyright OfficeLibrary of CongressWashington, DC 20559-6000202-707-3000

Fig. 2. Forces and speeds to the cam with translatedfollower with roll

==

==

mmc

Amm

Du

Du

v F P

v F

v F P

222

cos)(cos (3.17)

42

2222

22

cos'

]cos'

[]cos)[cos(

cos)(cos

==

=

==

A A A

mm

Amm

c

Du D

i

r s

r s

v F

v F P P

(3.18)

=

=

==

==

=

mmc

A

mm

A Amm

uu

Am

v F P r

sv F

r s

v F

v F P

r s

vv

22

2

2

2

cos'

'cos)cos(

'

(3.19)

22

2

22

2

cos'

cos'

=

=

==

A

mm

Amm

c

ui

r s

v F r

sv F

P P

(3.20)

4. DETERMINING THEMOMENTARY DYNAMIC ANDMECHANICAL EFFICIENCY OF THEROTARY CAM AND ROCKINGFOLLOWER WITH ROLL

The written relations are the following [11-15]:

d br r d b b

++=2

)(cos

20

22

0 (4.1)

02 += (4.2)

2222 cos)'1(2)'1( += bd bd RAD (4.3)

RADbbd += 'cossin 2 (4.4)

RADd 2sincos

= (4.5)

2222 cos2 += d bd br B (4.6)

B

B B

r d

br d

+=2

cos222

(4.7)

B B r

b 2sinsin

= (4.8)

cossincossin)sin( 222 +=+ (4.9)

sinsincoscos)cos( 222 =+ (4.10)

22

++= B B (4.11)

)sin(cos 2 B B ++= (4.12)

)cos(sin 2 B B ++= (4.13)

)cos(sincos)sin(cos 22 +++= B B B (4.14)

)cos(cossin)sin(sin 22 ++= B B B (4.15)

Br r r r r Bbb B A cos2222

+= (4.16)


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218

B A

b B Ar r

r r

+2

222r =cos (4.17)

Br r

A

b sinsin = (4.18)

+= B A (4.19)

sinsincoscoscos B B A = (4.20)

sincoscossinsin B B A += (4.21)

= 2 A (4.22)

A A

A

cos)cos(sin)sin(

)cos(cos

22

2

++==++=

(4.23)

cos'

cos = Ar

b (4.24)

2cos'

coscos = Ar

b (4.25)

The forces, velocities and powers are writtenin the relations (4.26-4.33) and the efficiency iswritten in the relations (4.31, 4.33), see thefigure 3 [11-15]:

==

sin

sin

ma

ma

vv

F F (4.26)

==

cos

cos

mn

mn

vv

F F (4.27)

==

sin

sin

nc

nc

vv

F F (4.28)

==

=====

22

2

2

coscos'

coscoscos

coscoscos

vr

bv

vvv

F F F

Am

mn D

mnu

(4.29)

=

==

mmc

mm D

u Du

v F P

v F v F P 222 coscos (4.30)

42

2222

222

cos'

)cos'

(

)cos(coscoscos

==

====

A A

c

Du D

i

r b

r b

P P

(4.31)

=

==

mmc

Ammuu

v F P

r b

v F v F P '

coscos2

(4.32)

22

222 cos

')cos

'(

'coscos

==

===

A A

Ac

ui

r b

r b

r b

P P

(4.33)

0

A

A

2

B

F n , v n

F m , v mF a , v a

F c , v c

F n , v n

F u , v 2B

B 0

A 0

x

r br 0

r Ar B

A

B

OD

0d

b

b

2002 FlorianPETRESCUThe Copyright-LawOf March, 01, 1989U.S. Copyright OfficeLibrary of CongressWashington, DC 20559-6000202-707-3000

Fig. 3. Forces and speeds at the rotary cam and rockingfollower with roll

Fm, v m, are perpendicular on the vector r A inA. F m is divided in F a (the sliding force) and F n (the normal force). F n is divided too in F c (thecompressed force) and F u (the useful force). Forthe mechanisms, with rotary cam and diversekind of followers, one must use differentmethods in creating the design with maximalefficiency for every type of follower [11-15].

5. DETERMINING THE MOMENTARYDYNAMIC (MECHANICAL)EFFICIENCY OF THE ROTARY CAMAND GENERAL PLATE ROCKINGFOLLOWER

The written relations are the following, (5.1-5.6), see the picture number four [11-15]:


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219

'1'

]sin)(cos

)([

0

20

2

=

br

br d AH (5.1)

sin)(

cos)(2

02

0

+

++=

br d

br bOH (5.2)

222 OH AH r += (5.3)

r 0

G

B

O D

d

A

A 0

B0

H

I

l

bG 0

l.

.

r

M m

x

1

2

F m;v m

F a ;v a

F n ;vn

2002 FlorianPETRESCUThe Copyright-LawOf March, 01, 1989U.S. Copyright OfficeLibrary of CongressWashington, DC 20559-6000

202-707-3000

Fig. 4. Forces and speeds at the rotary cam and general plate rocking follower

22

2

2

22sin

;sin

OH AH AH

r AH r

AH

+==

=

(5.4)

sincos

;sincos

====

mmn

mmn

vvv

F F F (5.5)

22

22

2

sin

sin

OH AH AH

v F v F

v F v F

P P

mm

mm

mm

nn

c

ni

Di

+==

=

=

=

===

(5.6)

6. CONCLUSION

The follower with roll determines the input-force, to be divided in more components. Thisis the reason for that, the dynamic and the

precisely-kinematics (the dynamic-kinematics[16]) of mechanism with rotary cam andfollower with roll, are more different anddifficult [17]. The presented dynamic efficiencyof followers with roll is not the same like theclassical-mechanical efficiency [11-15]. For the

plate followers the dynamic and the mechanicalefficiency are the same [14, 15]. This is thegreater advantage of the plate followers, but thedynamic efficiency in this conception results

just with the velocity which follows the forcedirection (not the linkage imposed direction),and in reality the dynamic velocity is moredifferent because the angular velocity isntconstant. The influence of the real variableangular velocity in dynamics is important, but ithasnt any influence to the efficiency (dynamicor mechanical) determination, because theangular velocity intervenes in the utile andconsumed power as well.

7. REFERENCES

[1] Petrescu, F., Petrescu, R., Contribu ii laoptimizarea legilor polynomiale, de mi carea tachetului de la mecanismele de distribu ieale motoarelor cu ardere intern ,Proceeding of 5 th Conference, ESFA,Bucure ti, Vol. I, p. 249-256, 1995.

[2] Petrescu, F., Petrescu, R., Contribu ii la sinteza mecanismelor de distribu ie ale

motoarelor cu ardere intern , Proceeding of5th Conference, ESFA, Bucure ti, Vol. I, p.257-264, 1995.

[3] Petrescu, F., Petrescu, R., Dinamicamecanismelor cu came (exemplificat pemecanismul clasic de distribu ie), Proceeding of 7 th International SymposiumSYROM, Bucharest, Vol. 3, p. 353-358,1997.

[4] Petrescu, F., Petrescu, R., Antonescu, O.,Contribu ii la maximizarea legilor

polinomiale pentru cursa activ amecanismului de distribu ie de la motoarele


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220cu ardere intern , Proceeding of 7 th

International Symposium SYROM,Bucharest, Vol. 3, p. 365-370, 1997.

[5] Petrescu, F., Petrescu, R., Antonescu, O.,Contribu ii la sinteza mecanismelor de

distribu ie, ale motoarelor cu ardereintern , cu metoda coordonatelorcarteziene, Proceeding of 7 th InternationalSymposium SYROM, Bucharest, Vol. 3, p.359-364, 1997.

[6] Petrescu, F., Petrescu, V., Sintezamecanismelor de distribu ie prin metodacoordonatelor rectangulare (carteziene),Proceeding of 7 th National Conference,GRAFICA, Craiova, p. 297-302, 2000.

[7] Petrescu, F., Petrescu, R., Legi de mi care pentru mecanismele cu came, Proceeding of7th National Symposium PRASIC, Bra ov,Vol. I, p. 321-326, 2002.

[8] Petrescu, F., Petrescu, R., Elemente dedinamica mecanismelor cu came Proceedingof 7 th National Symposium PRASIC,Bra ov, Vol. I, p. 327-332, 2002.

[9] Petrescu, I., Petrescu, V., Ocn rescu, C.,The cam synthesis with maximal efficiency, Proceeding of 7 th National SymposiumPRASIC, Bra ov, Vol. I, p. 339-344, 2002.

[10] Petrescu, F., Petrescu, R., The Cam Design for a Better Efficiency, Proceeding ofInternational Conference ICEGD, Bucharest,Vol. I, p. 245-248, 2005.

[11] Petrescu, F.I., Petrescu, R.V., Determiningthe dynamic efficiency of cams, Proceeding

of 9 th International Symposium SYROM,Bucharest, Vol. I, p. 129-134, 2005.

[12] Petrescu, F.I., Petrescu, R.V., Popescu, N.,The efficiency of cams, Proceedings of 2 th International Conference MME, Sofia, Vol.

II, p. 237-243, 2005.[13] Petrescu, F.I., Com nescu, A., Grecu, B.,

Ocn rescu, C., Petrescu, R.V., Determinarearandamentului la mecanismele cu cam ,Proceeding of 3 th National Seminar SNM,Craiova, p. 309-318, ISBN 978-973-746-910-6, 2008.

[14] Petrescu, F.I., Com nescu, A., Grecu, B.,Ocn rescu, C., Petrescu, R.V., Cams

Dynamic Efficiency Determination, In NEWTRENDS IN MECHANISMS, Ed.Academica Greifswald, ISBN 978-3-9402-37-10-1, 2008.

[15] Petrescu, R.V., Com nescu, A., Petrescu,F.I., Cam Gears Dynamics illustrated in theClassic Distribution, In NEW TRENDS INMECHANISMS, Ed. Academica Greifswald, ISBN 978-3-9402-37-10-1,2008.

[16] Petrescu, R.V., Com nescu, A., Petrescu,F.I., Antonescu, O., Cam Gears Dynamics inthe Module B (with Translated Followerwith Roll), In NEW TRENDS INMECHANISMS, Ed. Academica Greifswald, ISBN 978-3-9402-37-10-1,2008.

DESIGNUL MECANISMELOR CU CAM

Rezumat: Lucrarea prezint o metod original pentru determinarea randamentului unui mecanism cu cam i tachet.Se analizeaz 4 tipuri de mecanisme cu cam rotativ : 1. Cu tachet translant plat; 2. Cu tachet translant cu rol ; 3. Cu

tachet rotativ cu rol ; 4. Cu tachet rotativ plat. S-a considerat n mod special posibilitatea utiliz rii acestor mecanisme ladistribu ia motoarelor autovehiculelor, distribu ia reprezentnd al doilea mecanism al unui motor (ca importan ).Optimizarea acestui mecanism poate mbun t i func ionarea motorului, crescnd astfel i confortul vehiculului.

Narcisa Popescu, PhD. Eng., Associate Professor at Polytechnic University of Bucharest, GDGIDepartment (Department of Descriptive Geometry and Engineering Graphics),[email protected], 0214029136;

Relly Victoria Petrescu, PhD. Eng., Lecturer at Polytechnic University of Bucharest, GDGIDepartment (Department of Descriptive Geometry and Engineering Graphics),

[email protected], 0214029136;Florian Ion Petrescu , PhD. Eng., Assistant Professor at Polytechnic University of Bucharest, TMR

Department (Theory of Mechanisms and Robots Department), [email protected],0214029632.

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