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International Conference on Engineering Graphics and Design - Bucharest,Romania 2005

THE CAM DESIGN FOR A BETHER EFFICIENCY

Florian PETRESCU, Relly PETRESCU

Abstract: The paper presents an original method to determine the efficiency of a mechanism with cam

and follower. The originality of this method consists in eliminate of the friction modulus. In this paper on

analyze three types of cam mechanisms: 1.The mechanism with rotary cam and plate translated follower;

2.The mechanism with rotary cam and translated follower with roll; 3.The mechanism with rotary cam

and rocking-follower with roll. In every kind of cam and follower mechanism on utilize a different

method for the best efficiency design.

Key Words: efficiency, power, cam, follower, roll, force, speed.

1. INTRODUCTION

FIn this paper the authors present an original method

to calculate the efficiency of the cam mechanisms.

The originality consists in the eliminating of frictionforces and friction coefficients. On determine just the

mechanical efficiency of cam mechanism.

In every kind of cam and follower mechanism on are

utilizing a different method for the design with maximal

efficiency.

In this paper on analyze three kinds of cam and

follower mechanisms.

2. DETERMINING THE MOMENTARY

MECHANICAL EFFICIENCY OF THE ROTARY

CAM AND PLATE TRANSLATED FOLLOWER

The consumed motor force, Fc, perpendicular in A onthe vector rA, is dividing in two components, [1]:

1. Fm, which represents the useful force, or the motor

force reduced to the follower;

2. F, which is the sliding force between the two

profiles of cam and follower, (see the picture 1).

Pc is the consumed power and Pu represents the useful

power.

The written relations are the next:

sin= cm FF (2.1)

sin12 = vv (2.2)

212 sin== vFvFP cmu (2.3)

1vFP cc = (2.4)

22

1

21

cossin

sin

==

=

==

vF

vF

P

P

c

c

c

ui (2.5)

220

2

2

22

')(

''sin

ssr

s

r

s

A ++== (2.6)

cos= cFF (2.7)

cos112 = vv (2.8)

(2.9)2

112 cos== vFvFP c

Fig. 1. Forces and speeds to the cam with plate

translated follower. Determining the efficiency.

22

1

21

sincos

cos

==

=

==

vF

vF

P

P

c

c

ci (2.10)

In the relation number (2.11) on determine the

mechanical efficiency:

}]')[(

')(1{5.0

220

0

MM

MM

ssr

ssr

M

++

+= (2.11)



CAM AND TRANSLATED FOLLOWER WITH

ROLL


20

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

2BB rr = (3.2)

B

B

r

e= sincos (3.3)

B

Br

ss += 0cossin (3.4)

O

A

r0

s

s

rA

1vr

2v r

1v 2r

B

C

D

Fr

mFr

cFr

E

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0A

A

B

A-

Fn, vn

Fm

, vm

Fa, va

Fi, viFn, vn

Fu, v2

B

B0

A0

A

O

x

e

s0

r0

rA

rB

s

n

C

rb

Fig. 2. Forces and speeds to the cam with translated follower with roll. Determining the efficiency.

The pressure angle, , is determined by the relations(3.5-3.6):

220

0

)'()(

cos

esss

ss

++

+= (3.5)

220 )'()(

'sin

esss

es

++

= (3.6)

sinsincoscos)cos( =+ (3.7)

)cos(2222

++= BbbBA rrrrr (3.8)

220

220

)'()(

)'()'()(

cos

esssr

esressse

A

b

A

++

+++=

=

(3.9)

220

2200

)'()(

])'()([)(

sin

esssr

resssss

A

b

A

++

+++=

=

(3.10)

cos'

)'()(

')(

)cos(

220

0 =++

+=

=

AA

A

r

s

esssr

sss (3.11)

2cos'

cos)cos( =A

Ar

s(3.12)

On can write the next forces and speeds (see the

picture 2):

Fm, vm, are perpendicular on the vector rA in A.

Fm is dividing in Fa (the sliding force) and Fn (the

normal force).

Fn is dividing too in Fi (the bending force) and Fu (the

useful force).

=

=

)sin(

)sin(

Ama

Ama

FF

vv(3.13)

(3.14)

=

=

)cos(

)cos(

Amn

Amn

FF

vv

=

=

sin

sin

ni

ni

FF

vv(3.15)

==

==

cos)cos(cos

cos)cos(cos2

Amnu

Amn

FFF

vvv(3.16)

(3.17)22

2 cos)(cos == Ammuu vFvFP

mmc vFP = (3.18)

The momentary mechanical efficiency can be

obtained by the relation (3.19):

4

2

222

2

22

22

cos'

]cos'

[

]cos)[cos(

cos)(cos

cos)(cos

==

==

==

=

=

==

AA

A

A

mm

Amm

c

ui

r

s

r

s

vF

vF

P

P

(3.19)

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CAM AND ROCKING FOLLOWER WITH ROLL


db

rrdb b

++=

2

)(cos

2022

0 (4.1)

02 += (4.2)

2222 cos)'1(2)'1( +=

=

bdbd

RAD(4.3)

RAD

bbd +=

'cossin 2

(4.4)

RAD

d 2sincos

= (4.5)

2222

cos2 += dbdbrB (4.6)

B

BB

rd

brd

+=

2cos

222

(4.7)

B

Br

b 2sinsin

= (4.8)

cossincossin)sin( 222 +=+ (4.9)

sinsincoscos)cos( 222 =+ (4.10)

22

++= BB (4.11)

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

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

)cos(sin

cos)sin(cos

2

2

++++=

B

BB(4.14)

)cos(cos

sin)sin(sin

2

2

+

+=

B

BB(4.15)

Brrrrr BbbBA cos2222 += (4.16)

BA

bBA

rr

rrr

+=

2cos

222

(4.17)

Br

r

A

b sinsin = (4.18)

+= BA (4.19)

sinsincoscoscos BBA = (4.20)

sincoscossinsin BBA += (4.21)

= 2A (4.22)

A

A

A

cos)cos(

sin)sin(

)cos(cos

2

2

2

+

+=

=++=

(4.23)

cos'

cos

=Ar

b(4.24)

2cos

'coscos

=

Ar

b(4.25)

Forces and speeds are writhing in the relations (4.26)

and the efficiency is writhen in the relation (4.27):

=

==

==

==

==

=

=

=

=

mmc

mmuu

mn

mnu

nc

nc

mn

mn

ma

ma

vFP

vFvFP

vvv

FFF

vv

FF

vv

FF

vv

FF

222

2

coscos

coscoscos

coscoscos

sin

sin

cos

cos

sin

sin

(4.26)

4

2

22

222

22

cos'

)cos'

()cos(cos

coscos

=

=

==

===

A

A

c

ui

r

b

r

b

P

P

(4.27)

On demonstrate now the mode of deduction for the

relation (4.24). On can see now a very difficult algorithm

for the obtained of this relation (4.24):

RAD

bd )'1(cossincos

cossin)sin(

22

22

=+

+=+(4.28)

]cos)cos(sin)[sin(

sinsin

22 BB

A

b

A

br

rB

r

r

++

== (4.29)

]sin)cos(cos)[sin(

coscos

22 BB

A

b

A

B

A

bB

r

r

r

r

r

Brr

+++

=

=(4.30)

RAD

b )'1(sinsinsin

coscos)cos(

22

22

=

=+(4.31)

])'1(cos

cos[1

)]sin(cos[1

)]sin(cos[1

]cossin)cos(

sin)sin(

cossin)cos(

cos)[sin(cos

sinsincoscoscos

2

2

22

2

2

22

2

22

RAD

bdr

bd

r

rbdr

rrr

r

r

r

r

b

A

b

A

bBB

A

BB

B

BB

B

A

bB

A

B

BBA

=

=+=

=+=

=+

++

+++

+=

==

(4.32)

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Fig. 3. Forces and speeds to the cam with rockingfollower with roll. Determining the efficiency.

])'1(sin

sin[1

)cos(sin

)cos(sin

]cos)cos(

cossin)sin(

cossin)sin(

sin)[cos(sin

sincoscossinsin

22

22

2

22

2

2

22

RAD

brb

r

r

r

r

b

r

r

r

r

r

r

r

r

b

A

A

b

A

A

bB

A

B

B

BB

BB

B

A

bB

A

B

BBA

=

=+=

=+=

=++

++

++

+=

=+=

(4.33)

RAD

bd )'1(cos)sin( 22

=+ (4.34)

RAD

b )'1(sin)cos( 22

=+ (4.35)

])'1(sin

sin[1

sin 22RAD

brb

r

b

A

A

= (4.36)

])'1(cos

cos[1cos

2

2

RAD

drbr

bdr

bb

A

A

+

+=

(4.37)

In figure number three, on can see the forces and the

speeds of the mechanism with rotary cam and rocking

follower with roll.

The cam and the follower are represented in two

positions, successively.

The distance between the two rotary centers is noted

by d. The radius of follower is b.

The movement laws are known: , , , .On can write the next forces and speeds (see the

picture 3):

Fm, vm, are perpendicular

on the vector rA in A.

Fm is dividing in Fa (the

sliding force) and Fn (the

normal force).

Fn is dividing too in Fc

(the compressed force) and Fu(the useful force).

For the mechanisms, with

rotary cam and diverse kind

of followers, on must utilize

different methods for

realizing the design with

maximal efficiency to every

type of follower.

cos'sin'sin'

])'1(sin

)'1(cossin

)'1(cossin)'1(sin

)'1(cossin

)'1(sin

)'1(cossinsin[1

cos)cos(sin)sin(cos

22

2

222

2

222

2

222

2

2

222

2

22

=

=

=

=

+

+

+

+

+

=

=++=

AAA

b

b

b

b

A

AA

r

b

RAD

d

r

b

RADr

db

RAD

dbr

RAD

br

bdb

RAD

br

RAD

dbr

bdbRADr

(4.38)

0

A

A

2

B

Fn, vn

Fm, vmFa, va

Fc, vc

Fn, vn

Fu, v2

B

B0

A0

x

rbr0

rA

rB

A

B

OD

0d

b

b

5. CONCLUSION

The follower with roll, make input-force, to be

divided in more components. This is the motive for that,

the dynamic and the precisely-kinematics of mechanism

with rotary cam and follower with roll, are more

different and difficult.

6. REFERENCES

[1] Petrescu, R., Petrescu, F. The gear synthesis with the

best efficiency. ESFA 03, Bucharest, Romania, 2003,

Vol. 2, p. 63-70.

[2] Antonescu P., Oprean M., Petrescu, Fl.,La projection

de la came oscillante chez les mechanismes a

distribution variable. CONAT MATMA 85, Braov,

Romania, 1985.

Authors: Drd. Eng. Florian-Ion PETRESCU, proffesor-

asisstent, Univ. POLITEHNICA Bucureti, chair TMR,

phone: 021. 4029632;Dr. Eng. Relly-Victoria PETRESCU, lecturer,

Univ. POLITEHNICA Bucureti, chair GDGI, phone:

0722.529.840.

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