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 cam
and follower. 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. 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 as well. Key words: Efficiency, force, power, cam, follower, roll, plate, force, speed.
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. The dynamic velocities follow the same direction like forces. The dynamic efficiency uses the dynamic velocity. The normal (mechanical) velocities (known by kinematics) have the direction imposed by the linkage, and with this velocity one calculates the mechanical efficiency. When the kinematics (mechanical) velocity coincide with the dynamic velocity (the velocity has the same direction like the force), one can say that the linkage (couple or joint) is dynamically normal (natural). See the geared transmissions, or the cam with plate followers.
2. DETERMINING THE MOMENTARY DYNAMIC (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, [11-15]: 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): Fm = Fc ⋅ sin τ
(2.1)
v 2D = v 2 = v1 ⋅ sin τ
(2.2)
PuD = Pu = Fm ⋅ v 2 = Fc ⋅ v1 ⋅ sin 2 τ
(2.3)
Pc = Fc ⋅ v1
(2.4)
Pu Fc ⋅ v1 ⋅ sin 2 τ = = Pc Fc ⋅ v1
(2.5)
η iD = η i =
= sin 2 τ = cos 2 δ
sin 2 τ =
s '2
=
rA2
s '2 2
(r0 + s) + s'
cos δ =
2
(2.6)
r Fc
© 2002 Florian PETRESCU The Copyright-Law Of March, 01, 1989 U.S. Copyright Office Library of Congress Washington, DC 20559-6000 202-707-3000
C
r v1 A
τ
s’
rA
δ D
δ
F
r Fm
δ
r v2
r Fψ
B
r v12
sin δ =
cos α A =
ω
Fig. 1. Forces and speeds to the cam with plate translated follower. Determining the efficiency of the cam’s mechanism.
Fψ = Fc ⋅ cosτ
(2.7)
v12 = v1 ⋅ cosτ
(2.8)
F ⋅ v ⋅ cos 2 τ ψi = = c 1 = cos 2 τ = sin 2 δ Pc Fc ⋅ v1
(2.9) (2.10)
3. DETERMINING THE MOMENTARY DYNAMIC AND MECHANICAL EFFICIENCY OF THE ROTARY CAM AND TRANSLATED FOLLOWER WITH ROLL The pressure angle, δ, is determined by the relations (3.5-3.6), [2, 4, 6, 7]; The written relations are the following: rB2 = e 2 + (s0 + s) 2 rB =
rB2
sin α B ≡ cos τ =
(3.1) (3.2)
e cos α B ≡ sin τ = rB s0 + s rB
(3.6)
( s0 + s ) 2 + ( s'−e) 2
cos(δ + τ ) = cos δ ⋅ cosτ − sin δ ⋅ sin τ
r0
Pψ
s'−e
rA2 = rB2 + rb2 − 2 ⋅ rb ⋅ rB ⋅ cos(δ + τ )
τ
Pψ = Fψ ⋅ v12 = Fc ⋅ v1 ⋅ cos2 τ
(3.5)
( s0 + s) 2 + ( s'−e) 2
(3.7)
E s
O
s0 + s
(3.3) (3.4)
sin α A =
cos(α A − δ ) =
e ⋅ ( s0 + s) 2 + ( s'−e) 2 + rb ⋅ ( s'−e) rA ⋅ ( s0 + s) 2 + ( s'−e) 2
( s0 + s) ⋅ [ ( s0 + s ) 2 + ( s'−e) 2 − rb ] rA ⋅ ( s0 + s) 2 + ( s '−e) 2 ( s0 + s) ⋅ s' 2
rA ⋅ ( s0 + s) + ( s'−e)
cos(α A − δ ) ⋅ cos δ =
2
=
s' ⋅ cos δ rA
s' ⋅ cos 2 δ rA
(3.8) (3.9)
(3.10)
(3.11)
(3.12)
One can write the next forces, speeds and powers (3.13-3.20) (see the picture 2): Fm, vm, are perpendicular on the vector rA in A. Fm is divided in Fa (the sliding force) and Fn (the normal force). Fn is divided too, in Fi (the bending force) and Fu (the useful force). The momentary dynamic efficiency can be obtained from the relation (3.18), [11-15]: ⎧va = vm ⋅ sin(α A − δ ) ⎨ ⎩ Fa = Fm ⋅ sin(α A − δ ) ⎧vn = vm ⋅ cos(α A − δ ) ⎨ ⎩ Fn = Fm ⋅ cos(α A − δ ) ⎧vi = vn ⋅ sin δ ⎨ ⎩ Fi = Fn ⋅ sin δ
⎧v 2D = v n ⋅ cos δ = v m ⋅ cos(α A − δ ) ⋅ cos δ ⎪ s' ⎪ 2 2 ⎨= v m ⋅ ⋅ cos δ = v 2 ⋅ cos δ r A ⎪ ⎪⎩ Fu = Fn ⋅ cos δ = Fm ⋅ cos(α A − δ ) ⋅ cos δ
(3.13)
(3.14)
(3.15)
(3.16)
© 2002 Florian PETRESCU The Copyright-Law Of March, 01, 1989 U.S. Copyright Office Library of Congress Washington, DC 20559-6000 202-707-3000
Fu, v2
4. DETERMINING THE MOMENTARY DYNAMIC AND MECHANICAL EFFICIENCY OF THE ROTARY CAM AND ROCKING FOLLOWER WITH ROLL
Fn, vn δ
Fn, vn
Fi, vi
B
Fm, vm
rb
αA-δ
s
A
Fa, va rB rA
s0
B0
A0
n α0 C
x
θB
μ
γ
θA ϕ
cosψ 0 =
αA e O
⎧ PuD = Fu ⋅ v 2D = ⎪ 2 2 ⎨= Fm ⋅ v m ⋅ cos (α A − δ ) ⋅ cos δ ⎪P = F ⋅ v m m ⎩ c PuD Fm ⋅ vm ⋅ cos 2 (α A − δ ) ⋅ cos 2 δ = = Pc Fm ⋅ vm
[cos(α A − δ ) ⋅ cos δ ]2 = [
s'2 Fm ⋅ vm ⋅ 2 ⋅ cos2 δ P rA ηi = u = = Pc Fm ⋅ vm =
s' ⋅ cos2 δ 2 rA
(4.1)
ψ 2 = ψ +ψ 0
(4.2)
RAD = d 2 + b 2 (1 − ψ ' ) 2 − 2bd (1 − ψ ' ) cosψ 2
(4.3)
sin δ =
(3.17)
d ⋅ cosψ 2 + b ⋅ψ '−b RAD cos δ =
(3.18)
s' s' 2 ⋅ cos 2 δ ]2 = 2 ⋅ cos 4 δ rA rA
⎧ s' ⎪v 2 = v m ⋅ rA ⎪ ⎪P = F ⋅ v = 2 u ⎪ u s' ⎪ ⎨= Fm ⋅ v m ⋅ cos(α A − δ ) ⋅ cos δ ⋅ = rA ⎪ ⎪ s'2 ⎪= Fm ⋅ v m ⋅ 2 ⋅ cos 2 δ rA ⎪ ⎪P = F ⋅ v m m ⎩ c
2
b 2 + d 2 − ( r0 + rb ) 2 2⋅b⋅d
r0
Fig. 2. Forces and speeds to the cam with translated follower with roll.
η iD =
The written relations are the following [1115]:
d ⋅ sinψ 2 RAD
rB2 = b 2 + d 2 − 2 ⋅ b ⋅ d ⋅ cosψ 2 cos α B =
d 2 + rB2 − b 2 2 ⋅ d ⋅ rB
sin α B =
(3.19)
b ⋅ sinψ 2 rB
(4.5) (4.6) (4.7)
(4.8)
sin(δ + ψ 2 ) = sin δ cosψ 2 + sinψ 2 cos δ
(4.9)
cos(δ + ψ 2 ) = cos δ cosψ 2 − sinψ 2 sin δ
(4.10)
B = δ +ψ 2 + α B −
(3.20)
(4.4)
π 2
(4.11)
cos B = sin(δ + ψ 2 + α B )
(4.12)
sin B = − cos(δ + ψ 2 + α B )
(4.13)
cos B = sin(δ + ψ 2 ) ⋅ cos α B + sin α B ⋅ cos(δ + ψ 2 )
(4.14)
sin B = sin(δ + ψ 2 ) ⋅ sin α B − cos α B ⋅ cos(δ + ψ 2 )
(4.15)
rA2 = rB2 + rb2 − 2 ⋅ rb ⋅ rB ⋅ cos B
(4.16)
r2 + r2 − r2 cos μ = A B b 2 ⋅ rA ⋅ rB
(4.17)
r sin μ = b ⋅ sin B rA
(4.18)
α A = αB + μ
(4.19)
cos α A = cos α B cos μ − sin α B sin μ
(4.20)
sin α A = sin α B cos μ + cos α B sin μ
(4.21)
α = π − α A −ψ 2 − δ
(4.22)
cos α = − cos(ψ 2 + δ + α A ) = = sin(ψ 2 + δ ) ⋅ sin α A − cos(ψ 2 + δ ) ⋅ cos α A cos α =
ψ '⋅b rA
⋅ cos δ
(4.23)
PuD = cos 2 α ⋅ cos 2 δ = (cos α ⋅ cos δ ) 2 = Pc (4.31) 2 2 ψ '⋅b ψ ' ⋅ b =( ⋅ cos 2 δ ) 2 = ⋅ cos 4 δ rA rA2
η iD =
ψ '⋅b ⎧ ⎪ Pu = Fu ⋅ v 2 = Fm ⋅ v m ⋅ cos α ⋅ cos δ ⋅ rA (4.32) ⎨ ⎪P = F ⋅ v m m ⎩ c
ηi =
Pu ψ '⋅b = cos α ⋅ cos δ ⋅ = Pc rA
ψ '⋅b
=(
rA
⋅ cos δ ) = 2
ψ ' 2 ⋅b 2 rA2
(4.24) A
cos α ⋅ cos δ =
rA
2
⋅ cos δ
μ
(4.25)
F n, v n B α
⎧ Fa = Fm ⋅ sin α ⎨ ⎩va = vm ⋅ sin α
(4.26)
⎧ Fn = Fm ⋅ cos α ⎨ ⎩vn = vm ⋅ cos α
(4.27)
⎧ Fc = Fn ⋅ sin δ ⎨ ⎩vc = vn ⋅ sin δ
(4.28)
⎧ PuD = Fu ⋅ v 2D = Fm ⋅ v m ⋅ cos 2 α ⋅ cos 2 δ ⎨ ⎩ Pc = Fm ⋅ v m
rB F a, v a
F u , v2 δ F n, v n
B Fc, vc
(4.29)
(4.30)
γ O
ϕ
A0 α0
r0 θA
rb
b
Fm, vm ψ2
αB B 0 α A
The forces, velocities and powers are written in the relations (4.26-4.33) and the efficiency is written in the relations (4.31, 4.33), see the figure 3 [11-15]:
⎧ ⎪ Fu = Fn ⋅ cos δ = Fm ⋅ cos α ⋅ cos δ ⎪⎪ D ⎨v 2 = v n ⋅ cos δ = v m ⋅ cos α ⋅ cos δ = ⎪ ψ '⋅b ⎪= v m ⋅ ⋅ cos 2 δ = v 2 ⋅ cos 2 δ ⎪⎩ rA
⋅ cos δ
© 2002 Florian PETRESCU The Copyright-Law Of March, 01, 1989 U.S. Copyright Office Library of Congress Washington, DC 20559-6000 202-707-3000
rA
ψ '⋅b
(4.33) 2
ψ
b d
ψ0 D
x
Fig. 3. Forces and speeds at 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 Fn (the normal force). Fn is divided too in Fc (the compressed force) and Fu (the useful force). For the mechanisms, with rotary cam and diverse kind of followers, one must use different methods in creating the design with maximal efficiency for every type of follower [11-15]. 5. DETERMINING THE MOMENTARY DYNAMIC (MECHANICAL) EFFICIENCY OF THE ROTARY CAM AND GENERAL PLATE ROCKING FOLLOWER
The written relations are the following, (5.15.6), see the picture number four [11-15]:
6. CONCLUSION
AH = [ d 2 − (r0 − b) 2 ⋅ ⋅ cosψ − (r0 − b) ⋅ sinψ ] ⋅
ψ ' (5.1) 1 −ψ '
OH = b + (r0 − b) ⋅ cosψ +
(5.2)
+ d 2 − (r0 − b) 2 ⋅ sinψ r 2 = AH 2 + OH 2
© 2002 Florian PETRESCU The Copyright-Law Of March, 01, 1989 U.S. Copyright Office Library of Congress Washington, DC 20559-6000 202-707-3000
(5.3)
Fn;vn
α Fm;vm
A
τ
l.ψ’ r
H τ
ρ.ψ’
Fa;va
I
G θ
A0 γ
ρ
ψ αM
G0
l
2
ψ ϕ
αm β
r0
δ
B0 x
b ψ
B
d
O
D 1
Fig. 4. Forces and speeds at the rotary cam and general plate rocking follower.
AH ; r AH 2 AH 2 sin 2 τ = 2 = r AH 2 + OH 2 sin τ =
Fn = Fm ⋅ cos α = Fm ⋅ sin τ ; v n = v m ⋅ cos α = v m ⋅ sin τ
η iD = η i = =
(5.5)
Pn F ⋅v = n n = Pc Fm ⋅ v m
Fm ⋅ v m ⋅ sin 2 τ = Fm ⋅ v m
AH 2 = sin τ = AH 2 + OH 2 2
(5.4)
(5.6)
The follower with roll determines the inputforce, to be divided in more components. This is the reason for that, the dynamic and the precisely-kinematics (the dynamic-kinematics [16]) of mechanism with rotary cam and follower with roll, are more different and difficult [17]. The presented dynamic efficiency of followers with roll is not the same like the classical-mechanical efficiency [11-15]. For the plate followers the dynamic and the mechanical efficiency are the same [14, 15]. This is the greater advantage of the plate followers, but the dynamic efficiency in this conception results just with the velocity which follows the force direction (not the linkage imposed direction), and in reality the dynamic velocity is more different because the angular velocity isn’t constant. The influence of the real variable angular velocity in dynamics is important, but it hasn’t any influence to the efficiency (dynamic or mechanical) determination, because the angular velocity intervenes in the utile and consumed power as well. 7. REFERENCES
[1] Petrescu, F., Petrescu, R., Contribuţii la optimizarea legilor polynomiale, de mişcare a tachetului de la mecanismele de distribuţie ale motoarelor cu ardere internă, Proceeding of 5th Conference, ESFA, Bucureşti, Vol. I, p. 249-256, 1995. [2] Petrescu, F., Petrescu, R., Contribuţii la sinteza mecanismelor de distribuţie ale motoarelor cu ardere internă, Proceeding of 5th Conference, ESFA, Bucureşti, Vol. I, p. 257-264, 1995. [3] Petrescu, F., Petrescu, R., Dinamica mecanismelor cu came (exemplificată pe mecanismul clasic de distribuţie), th Proceeding of 7 International Symposium SYROM, Bucharest, Vol. 3, p. 353-358, 1997. [4] Petrescu, F., Petrescu, R., Antonescu, O., Contribuţii la maximizarea legilor polinomiale pentru cursa activă a
mecanismului de distribuţie de la motoarele cu ardere internă, Proceeding of 7th International Symposium SYROM, Bucharest, Vol. 3, p. 365-370, 1997. [5] Petrescu, F., Petrescu, R., Antonescu, O., Contribuţii la sinteza mecanismelor de distribuţie, ale motoarelor cu ardere internă, cu metoda coordonatelor th carteziene, Proceeding of 7 International Symposium SYROM, Bucharest, Vol. 3, p. 359-364, 1997. [6] Petrescu, F., Petrescu, R., Designul (sinteza) mecanismelor cu came prin metoda coordonatelor polare (metoda triunghiurilor), Proceeding of 7th National Conference, GRAFICA, Craiova, p. 291296, 2000. [7] Petrescu, F., Petrescu, V., Sinteza mecanismelor de distribuţie prin metoda coordonatelor rectangulare (carteziene), Proceeding of 7th National Conference, GRAFICA, Craiova, p. 297-302, 2000. [8] Petrescu, F., Petrescu, R., Legi de mişcare pentru mecanismele cu came, Proceeding of 7th National Symposium PRASIC, Braşov, Vol. I, p. 321-326, 2002. [9] Petrescu, F., Petrescu, R., Elemente de dinamica mecanismelor cu came Proceeding of 7th National Symposium PRASIC, Braşov, Vol. I, p. 327-332, 2002. [10] Petrescu, I., Petrescu, V., Ocnărescu, C., The cam synthesis with maximal efficiency, Proceeding of 7th National Symposium PRASIC, Braşov, Vol. I, p. 339-344, 2002. [11] Petrescu, F., Petrescu, R., The Cam Design for a Better Efficiency, Proceeding of
International Conference ICEGD, Bucharest, Vol. I, p. 245-248, 2005. [12] Petrescu, F.I., Petrescu, R.V., Determining the dynamic efficiency of cams, Proceeding of 9th International Symposium SYROM, Bucharest, Vol. I, p. 129-134, 2005. [13] Petrescu, F.I., Petrescu, R.V., Popescu, N., The efficiency of cams, Proceedings of 2th International Conference MME, Sofia, Vol. II, p. 237-243, 2005. [14] Petrescu, F.I., Comănescu, A., Grecu, B., Ocnărescu, C., Petrescu, R.V., Determinarea randamentului la mecanismele cu camă, Proceeding of 3th National Seminar SNM, Craiova, p. 309-318, ISBN 978-973-746910-6, 2008. [15] Petrescu, F.I., Comănescu, A., Grecu, B., Ocnărescu, C., Petrescu, R.V., Cams Dynamic Efficiency Determination, In NEW TRENDS IN MECHANISMS, Ed. Academica – Greifswald, ISBN 978-3-940237-10-1, 2008. [16] Petrescu, R.V., Comănescu, A., Petrescu, F.I., Cam Gears Dynamics illustrated in the Classic Distribution, In NEW TRENDS IN MECHANISMS, Ed. Academica – Greifswald, ISBN 978-3-9402-37-10-1, 2008. [17] Petrescu, R.V., Comănescu, A., Petrescu, F.I., Antonescu, O., Cam Gears Dynamics in the Module B (with Translated Follower with Roll), In NEW TRENDS IN MECHANISMS, Ed. Academica – Greifswald, ISBN 978-3-9402-37-10-1, 2008.
DESIGNUL MECANISMELOR CU CAMĂ Rezumat: Lucrarea prezintă o metodă originală pentru determinarea randamentului unui mecanism cu camă şi tachet. Se analizează 4 tipuri de mecanisme cu camă rotativă: 1. Cu tachet translant plat; 2. Cu tachet translant cu rolă; 3. Cu tachet rotativ cu rolă; 4. Cu tachet rotativ plat. S-a considerat în mod special posibilitatea utilizării acestor mecanisme la distribuţia motoarelor autovehiculelor, distribuţia reprezentând al doilea mecanism al unui motor (ca importanţă). Optimizarea acestui mecanism poate îmbunătăţi funcţionarea motorului, crescând astfel şi confortul vehiculului. Popescu Narcisa, PhD. Eng., Associate Professor at Polytechnic University of Bucharest, GDGI Department (Department of Descriptive Geometry and Engineering Graphics), narcipop@yahoo.com, 0214029136; Petrescu Relly Victoria, PhD. Eng., Lecturer at Polytechnic University of Bucharest, GDGI Department (Department of Descriptive Geometry and Engineering Graphics), petrescurelly@yahoo.com, 0214029136; Petrescu Florian Ion, PhD. Eng., Assistant Professor at Polytechnic University of Bucharest, TMR Department (Theory of Mechanisms and Robots Department), petrescuflorian@yahoo.com, 0214029632.