Florian Ion PETRESCU, Barbu GRECU, Relly Victoria PETRESCU

Al III-lea Seminar National de Mecanisme, Craiova, 2008

1

DETERMINING THE DYNAMIC EFFICIENCY OF CAMS

Florian Ion PETRESCU, Barbu GRECU, Relly Victoria 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 of eliminating the friction modulus. In this paper one analyzes four 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; 4.The mechanism with rotary cam and plate rocking-follower. For every kind of cams and followers mechanism one uses a different method in determining the best efficiency design. One takes into account the cam’s mechanism (distribution mechanism), which is the second mechanism from the internal-combustion engines. The optimizing of this mechanism (the distribution mechanism), can improve the functionality of the engine and may increase the comfort of the vehicle too.

1 Introduction

In this paper the authors present an original method to calculate the efficiency of the cam’s mechanisms. One analyzes four kinds of cams and followers mechanisms: 1. A mechanism with rotary cam and plate translated follower; 2. A mechanism with rotary cam and translated follower with roll; 3. A mechanism with rotary cam and rocking-follower with roll; 4. A mechanism with rotary cam and plate rocking-follower. For every kind of cams and followers mechanism, one has utilizing a different method for the cam’s design with a better efficiency.

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 divided in two components, [1, 2]: a) Fm, which represents the useful force, or the motor force reduced to the follower; b) Fψ, which is the sliding force between the two profiles of cam and follower, (see the picture 1). See the written relations (2.1-2.10):

(2.1) τsin⋅= cm FF (2.2) τsin12 ⋅= vv


2

τ212 sin⋅⋅=⋅= vFvFP cmu (2.3)

(2.4) 1vFP cc ⋅=

δττη 22

1

21 cossinsin

==⋅⋅⋅

==vF

vFPP

c

c

c

ui (2.5)

220

2

2

22

')(''sin

ssrs

rs

A ++==τ

(2.6) (2.7) τψ cos⋅= cFF

(2.8) τcos112 ⋅= vv

τψψ2

112 cos⋅⋅=⋅= vFvFP c (2.9)

δττ

ψ ψ 22

1

21 sincos

cos==

⋅⋅⋅

==vF

vFPP

c

c

ci (2.10)

τ

O

A

r0

s

s’

rA

1vr2vr

12vrB

C

Dτ

ω

δ

δ

δ

ψFr mF

rcFr F

E

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

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


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3 Determining the momentary dynamic efficiency of the rotary cam and translated follower with roll

The pressure angle, δ, is determined by the relations (3.5-3.6), [1, 2]; One can write the next forces, speeds and powers (3.13-3.18) (see the picture 2): Fm, vm, are perpendicular on the vector rA in A. Fm is divided in F (the sliding force) and Fa n (the normal force). F is divided too, in F (the bending force) and Fn i u (the useful force). The momentary dynamic efficiency can be obtained from the relation (3.18):

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

s0

rb

r0

rA

rB

s

n

C

rb

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

The written relations are the following.

20

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

2BB rr = (3.2)

(3.3) B

B re

=≡ τα sincos


4

(3.4) B

B rss +

=≡ 0cossin τα

220

0

)'()(cos

esss

ss

−++

+=δ (3.5)

220 )'()(

'sinesss

es

−++

−=δ (3.6)

τδτδτδ sinsincoscos)cos( ⋅−⋅=+ (3.7) )cos(2222 τδ +⋅⋅⋅−+= BbbBA rrrrr (3.8)

220

220

)'()(

)'()'()(cos

esssr

esressse

A

bA

−++⋅

−⋅+−++⋅=α (3.9)

220

2200

)'()(

])'()([)(sin

esssr

resssss

A

bA

−++⋅

−−++⋅+=α (3.10)

δδα cos'

)'()(

')()cos(22

0

0 ⋅=−++⋅

⋅+=−

AAA r

s

esssr

sss (3.11)

δδδα 2cos'cos)cos( ⋅=⋅−A

A rs (3.12)

⎩⎨⎧

−⋅=−⋅=

)sin()sin(δαδα

Ama

AmaFFvv (3.13)

⎩⎨⎧

−⋅=−⋅=

)cos()cos(δαδα

Amn

AmnFFvv (3.14)

⎩⎨⎧

⋅=⋅=

δδ

sinsin

ni

niFFvv (3.15)

⎩⎨⎧

⋅−⋅=⋅=⋅−⋅=⋅=

δδαδδδαδ

cos)cos(coscos)cos(cos2

Amnu

AmnFFF

vvv (3.16)

⎪⎩

⎪⎨⎧

⋅=⋅−⋅⋅=⋅=

mmc

AmmuuvFP

vFvFP δδα 222 cos)(cos (3.17)

δδδδα

δδαη

42

2222

22

cos']cos'[]cos)[cos(

cos)(cos

⋅=⋅=⋅−=

=⋅

⋅−⋅⋅==

AAA

mm

Amm

c

ui

rs

rs

vFvF

PP

(3.18)


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4 Determining the momentary dynamic efficiency of the rotary cam and rocking follower with roll

Fm, vm, are perpendicular on the vector rA in A. Fm is divided in Fa (the sliding force) and F (the normal force). F is divided too in F (the compressed force) and Fn n c u (the useful force). The written relations are the following [1, 2] (4.1-4.31).

α0

αA

ϕθA

ψ2

μ

αB

Fn, vn

Fm, vmFa, va

Fc, vc

Fn, vn

Fu, vuB

B0

A0

x

rbr0

rA

rB

Aδ

α B

γ

OD

ψ

ψ0

d

b

b

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

dbrrdb b

⋅⋅+−+

=2

)(cos2

022

0ψ (4.1)

(4.2) 02 ψψψ +=

2222 cos)'1(2)'1( ψψψ −−−+= bdbdRAD (4.3)

RADbbd −⋅+⋅

='cossin 2 ψψδ (4.4)

RADd 2sincos ψδ ⋅

= (4.5)

2222 cos2 ψ⋅⋅⋅−+= dbdbrB (4.6)

B

BB rd

brd⋅⋅−+

=2

cos222

α (4.7)


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BB r

b 2sinsin ψα ⋅= (4.8)

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

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

22παψδ −++= BB (4.11)

(4.12) )sin(cos 2 BB αψδ ++=

(4.13) )cos(sin 2 BB αψδ ++−=

(4.14) )cos(sincos)sin(cos 22 ψδααψδ +⋅+⋅+= BBB (4.15) )cos(cossin)sin(sin 22 ψδααψδ +⋅−⋅+= BBB

Brrrrr BbbBA cos2222 ⋅⋅⋅−+= (4.16)

BA

bBArr

rrr⋅⋅−+

=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) AAA αδψαδψαδψα cos)cos(sin)sin()cos(cos 222 ⋅+−⋅+=++−= (4.23)

δψα cos'cos ⋅⋅

=Arb (4.24)

δψδα 2cos'coscos ⋅⋅

=⋅Arb (4.25)

⎩⎨⎧

⋅=⋅=

αα

sinsin

ma

mavvFF (4.26)

⎩⎨⎧

⋅=⋅=

αα

coscos

mn

mnvvFF (4.27)

⎩⎨⎧

⋅=⋅=

δδ

sinsin

nc

ncvvFF (4.28)

⎩⎨⎧

⋅⋅=⋅=⋅⋅=⋅=

δαδδαδ

coscoscoscoscoscos

2 mn

mnuvvvFFF (4.29)

⎪⎩

⎪⎨⎧

⋅=⋅⋅⋅=⋅=

mmc

mmuuvFP

vFvFP δα 222 coscos (4.30)

δψδψδαδαη 42

2222222 cos')cos'()cos(coscoscos ⋅

⋅=⋅

⋅=⋅=⋅==

AAc

ui

rb

rb

PP (4.31)


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5 Determining the momentary mechanical efficiency of the rotary cam and general plate rocking follower

The written relations are the following, (5.1-5.6), see the picture number four [1, 2]:

r0

G δ

B

O D

d

A

A0

B0

H

I

ρ

l

bG0

l.ψ’

ρ.ψ’r

τ

ψ

ψ

θ

β

αMαm

x

ϕ

γ ψ

1

2

τ

α

A

Fn; vn

Fm; vm

Fa; va

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

'1

']sin)(cos)([ 0

20

2

ψ

ψψψ

−⋅⋅−−⋅−−= brbrdAH (5.1)

ψψ sin)(cos)( 20

20 ⋅−−+⋅−+= brdbrbOH (5.2)

222 OHAHr += (5.3)

22

2

2

22sin;sin

OHAHAH

rAH

rAH

+=== ττ (5.4)

(5.5) τατα sincos;sincos ⋅=⋅=⋅=⋅= mmnmmn vvvFFF

22

22

2sinsin

OHAHAH

vFvF

vFvF

PP

mm

mm

mm

nn

c

ni

+==

⋅⋅⋅

=⋅⋅

== ττη (5.6)


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6 Conclusion

The follower with roll, makes input-force, to be divided in more components. This is the reason for that, the dynamic and the precisely-kinematics (the dynamic-kinematics) of mechanism with rotary cam and follower with roll, are more different and difficult. The presented dynamic efficiency of followers with roll is not the same like the classical-mechanical efficiency. For the plate followers the dynamic and the mechanical efficiency are the same. This is the greater advantage of the plate followers.

But the followers with roll can be calculated for a practical dynamics more good than the dynamics of the plate followers.

References

1- PETRESCU, F.I., PETRESCU, R.V. Determining the dynamic efficiency of cams. In the Ninth IFToMM International Symposium on Theory of Machines and Mechanisms, SYROM 2005, Bucharest, Romania, 2005, Vol. I, p. 129-134.

2- PETRESCU, F.I., PETRESCU, R.V., POPESCU N., The efficiency of cams. In the Second International Conference “Mechanics and Machine Elements”, Technical University of Sofia, November 4-6, 2005, Sofia, Bulgaria, Vol. II, p. 237-243.

3- PETRESCU, F.I., COMANESCU, A., GRECU, B., OCNARESCU, C., PETRESCU, R.V., Cams Dynamic Efficiency Determination. In New Trends in Mechanisms, Ed. Academica-Greifswald, 2008, ISBN 978-3-9402-37-10-1.

Florian Ion PETRESCU Universitatea Politehnica din Bucureşti, Departamentul TMR Splaiul Independenţei 313, Bucureşti, Sector 6, cod 060042

[email protected]

Barbu GRECU Universitatea Politehnica din Bucureşti, Departamentul TMR Splaiul Independenţei 313, Bucureşti, Sector 6, cod 060042

[email protected]

Relly Victoria PETRESCU Universitatea Politehnica din Bucureşti, Departamentul GDGI

Splaiul Independenţei 313, Bucureşti, Sector 6, cod 060042 [email protected]

Documents

Florian Ion PETRESCU, Barbu GRECU, Relly Victoria PETRESCU