DECEMBER 2006 VOLUME 1 NUMBER 2 JIDEG

33

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

In this paper the authors present an original method to calculate the efficiency of the cam mechanisms.

The originality consists in the eliminating of friction forces 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 on the vector rA, is dividing in two components, [1]:

1. Fm, which represents the useful force, or the motor force reduced to the follower;

2. Fψ, 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: τsin⋅= cm FF (2.1)

τsin12 ⋅= vv (2.2)

τ212 sin⋅⋅=⋅= vFvFP cmu (2.3)

1vFP cc ⋅= (2.4)

δτ

τη

221

21

cossin

sin

==

=⋅⋅⋅

==vF

vFPP

c

c

c

ui (2.5)

220

2

2

22

’)(’’

sinssr

srs

A ++==τ (2.6)

τψ cos⋅= cFF (2.7)

τcos112 ⋅= vv (2.8)

τψψ2

112 cos⋅⋅=⋅= vFvFP c (2.9)

Fig. 1 Forces and speeds to the cam with plate translated follower. Determining the efficiency.

δτ

τψ ψ

221

21

sincos

cos

==

=⋅⋅⋅

==vF

vFP

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 ττ

ττ

τη

++⋅

⋅+−⋅= (2.11)

3. DETERMINING THE MOMENTARY MECHANICAL EFFICIENCY OF THE ROTARY CAM AND TRANSLATED FOLLOWER WITH ROLL The written relations are the next:

20

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

2BB rr = (3.2)

BB r

e=≡ τα sincos (3.3)

BB r

ss +=≡ 0cossin τα (3.4)

τ

O

A

r 0

s s’

r A

1 v &

2 v &

12 v & B

C

D τ

ω

δ

δ

δ

ψ F & m F

&

c F & F

E

Florian PETRESCU, Relly PETRESCU

THE CAM DESIGN FOR A BETTER EFFICIENCY

The CAM Design for a Better Efficiency

DECEMBER 2006 VOLUME 1 NUMBER 2 JIDEG 34

α0αA

ϕθA

θB

δ

µ

γ

αA-δ

Fn, vn

Fm, vm

Fa, va

Fi, viFn, vn

Fu, v2

B

B0

A0

A

O

x

e

s 0

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, δ, is determined by the relations (3.5-3.6):

220

0

)’()(cos

esss

ss

−++

+=δ (3.5)

220 )’()(

’sin

esss

es

−++

−=δ (3.6)

τδτδτδ sinsincoscos)cos( ⋅−⋅=+ (3.7)

)cos(2222 τδ +⋅⋅⋅−+= BbbBA rrrrr (3.8)

220

220

)’()(

)’()’()(

cos

esssr

esressse

A

b

A

−++⋅

−⋅+−++⋅=

=α

(3.9)

220

2200

)’()(

])’()([)(

sin

esssr

resssss

A

b

A

−++⋅

−−++⋅+=

=α

(3.10)

δ

δα

cos’

)’()(

’)(

)cos(

220

0 ⋅=−++⋅

⋅+=

=−

AA

A

rs

esssr

sss (3.11)

δδδα 2cos’

cos)cos( ⋅=⋅−A

A rs

(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(

δαδα

Ama

Ama

FF

vv (3.13)

−⋅=−⋅=

)cos(

)cos(

δαδα

Amn

Amn

FF

vv (3.14)

⋅=⋅=

δδ

sin

sin

ni

ni

FF

vv (3.15)

⋅−⋅=⋅=⋅−⋅=⋅=

δδαδδδαδ

cos)cos(cos

cos)cos(cos2

Amnu

Amn

FFF

vvv (3.16)

δδα 222 cos)(cos ⋅−⋅⋅=⋅= Ammuu vFvFP (3.17)

mmc vFP ⋅= (3.18)

The momentary mechanical efficiency can be obtained by the relation (3.19):

δδ

δδα

δδα

δδα

η

42

222

2

22

22

cos’

]cos’

[

]cos)[cos(

cos)(cos

cos)(cos

⋅=⋅=

=⋅−=

=⋅−=

=⋅

⋅−⋅⋅=

==

AA

A

A

mm

Amm

c

ui

r

srs

vF

vF

PP

(3.19)

4. DETERMINING THE MOMENTARY MECHANICAL EFFICIENCY OF THE ROTARY CAM AND ROCKING FOLLOWER WITH ROLL The written relations are the next:

The CAM Design for a Better Efficiency

DECEMBER 2006 VOLUME 1 NUMBER 2 JIDEG 35

dbrrdb b

⋅⋅+−+

=2

)(cos

20

22

0ψ (4.1)

02 ψψψ += (4.2)

2222 cos)’1(2)’1( ψψψ −−−+=

=

bdbd

RAD (4.3)

RADbbd −⋅+⋅

=’cos

sin 2 ψψδ (4.4)

RADd 2sin

cosψ

δ⋅

= (4.5)

2222 cos2 ψ⋅⋅⋅−+= dbdbrB (4.6)

B

BB rd

brd⋅⋅−+=

2cos

222

α (4.7)

BB r

b 2sinsin

ψα ⋅= (4.8)

δψψδψδ cossincossin)sin( 222 +=+ (4.9)

δψψδψδ sinsincoscos)cos( 222 −=+ (4.10)

22παψδ −++= BB (4.11)

)sin(cos 2 BB αψδ ++= (4.12)

)cos(sin 2 BB αψδ ++−= (4.13)

)cos(sin

cos)sin(cos

2

2

ψδααψδ

+⋅++⋅+=

B

BB (4.14)

)cos(cos

sin)sin(sin

2

2

ψδααψδ

+⋅−−⋅+=

B

BB (4.15)

Brrrrr BbbBA cos2222 ⋅⋅⋅−+= (4.16)

BA

bBA

rrrrr

⋅⋅−+

=2

cos222

µ (4.17)

Brr

A

b sinsin ⋅=µ (4.18)

µαα += BA (4.19)

µαµαα sinsincoscoscos BBA −= (4.20)

µαµαα sincoscossinsin BBA += (4.21)

δψαπα −−−= 2A (4.22)

A

A

A

αδψαδψ

αδψα

cos)cos(

sin)sin(

)cos(cos

2

2

2

⋅+−−⋅+=

=++−= (4.23)

δψα cos’

cos ⋅⋅=Arb

(4.24)

δψδα 2cos’

coscos ⋅⋅=⋅Arb

(4.25)

Forces and speeds are writhing in the relations (4.26)

and the efficiency is writhen in the relation (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):

⋅=⋅⋅⋅=⋅=

⋅⋅=⋅=⋅⋅=⋅=

⋅=⋅=⋅=⋅=

⋅=⋅=

mmc

mmuu

mn

mnu

nc

nc

mn

mn

ma

ma

vFP

vFvFP

vvv

FFF

vv

FF

vv

FF

vv

FF

δα

δαδδαδ

δδαα

αα

222

2

coscos

coscoscos

coscoscos

sin

sin

cos

cos

sin

sin

(4.26)

δψ

δψδα

δαη

42

22

222

22

cos’

)cos’

()cos(cos

coscos

⋅⋅=

=⋅⋅=⋅=

=⋅==

A

A

c

ui

r

b

rb

PP

(4.27)

RADbd )’1(cos

sincos

cossin)sin(

22

22

ψψψδ

ψδψδ−⋅⋅−

=⋅+

+⋅=+ (4.28)

]cos)cos(sin)[sin(

sinsin

22 BB

A

b

A

b

rr

Brr

αψδαψδ

µ

⋅+−⋅+⋅

⋅=⋅=(4.29)

]sin)cos(cos)[sin(

coscos

22 BB

A

b

A

B

A

bB

rr

rr

rBrr

αψδαψδ

µ

⋅++⋅+⋅

⋅−=⋅−=(4.30)

RADb )’1(sin

sinsin

coscos)cos(

22

22

ψψψδ

ψδψδ−⋅⋅

=⋅−

−⋅=+ (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

RADbd

r

bdr

rbdr

rrr

rr

rr

b

A

bA

bBBA

BB

B

BB

BA

bB

A

B

BBA

ψψ

ψ

ψδψ

ψδα

ααψδαψδ

ααψδ

αψδα

µαµαα

−⋅⋅−⋅−

−⋅−⋅=

=+⋅−⋅−⋅=

=+⋅−⋅⋅=

=⋅⋅+−−⋅++

+⋅⋅++

⋅+⋅−⋅=

=⋅−⋅=

(4.32)

The CAM Design for a Better Efficiency

DECEMBER 2006 VOLUME 1 NUMBER 2 JIDEG 36

Fig. 3 Forces and speeds to the cam with rocking follower 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

RADb

rbr

rr

rb

rr

rr

rr

rr

bA

A

b

A

A

bB

A

B

B

BB

BB

BA

bB

A

B

BBA

ψψψ

ψδψ

ψδα

αψδ

ααψδααψδ

αψδα

µαµαα

−⋅⋅⋅−⋅⋅=

=+⋅−⋅=

=+⋅−⋅=

=⋅++

+⋅⋅+−−⋅⋅++

⋅+⋅−⋅=

=⋅+⋅=

(4.33)

RADbd )’1(cos

)sin( 22

ψψδψ −⋅⋅−=+ (4.34)

RADb )’1(sin

)cos( 22

ψψδψ −⋅⋅=+ (4.35)

])’1(sin

sin[1

sin 22 RAD

brb

rb

AA

ψψψα −⋅⋅⋅−⋅⋅= (4.36)

])’1(cos

cos[1

cos

2

2

RADdrbr

bdr

bb

AA

⋅−−⋅⋅⋅+

+⋅−⋅=

ψψ

ψα (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: ψ, ψ’, ψ’’, ψ’’’. 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.

δψψψψψ

ψψ

ψψψ

ψψψψψ

ψψψ

ψψ

ψψψψ

αδψαδψα

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

222

22

222

2

2

222

2

22

⋅⋅=⋅

⋅⋅=⋅

⋅⋅⋅=

=−⋅⋅⋅⋅+

+−⋅⋅⋅⋅−

−−⋅⋅⋅+−⋅⋅⋅−

−−⋅⋅⋅⋅

+

+−⋅⋅⋅⋅

−

−−⋅⋅⋅−⋅⋅⋅⋅

=

=⋅+−⋅+=

AAA

b

b

b

b

A

AA

rb

RADd

rb

RADrdb

RADdbr

RADbr

bdbRAD

brRAD

dbr

bdbRADr

(4.38)

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, pp. 63-70. [2] Antonescu P., Oprean M., Petrescu, Fl., La projection de la came oscillante chez les mechanismes a distribution variable. CONAT MATMA’ 85, BUDúRY��Romania, 1985. Authors: Eng. Florian-Ion PETRESCU, associate professor, University POLITEHNICA of Bucureúti; Eng. Relly-Victoria PETRESCU, Ph.D., lecturer, University POLITEHNICA of Bucureúti.