Cams with High Efficiency

International Review of Mechanical Engineering (I.RE.M.E.), Vol. 7, n. 4

Cams with High Efficiency F.I. Petrescu1, R.V. Petrescu2 Abstract – The paper presents an original method to increase the efficiency of a mechanism with cam and follower. The distribution mechanisms work with small efficiency for about 150 years; this fact affects the total yield of the internal heat engines. Much of the mechanical energy of an engine is lost through the mechanism of distribution. Multi-years the yield of the distribution mechanisms was only 4-8%. In the past 20 years it has managed a lift up to the value of 14-18%; car pollution has decreased and people have better breathing again. Meanwhile the number of vehicles has tripled and the pollution increased again. Now, it’s the time when we must try again to rise the yield of the distribution mechanisms. This paper treats only two modules: the mechanism with rotary cam and plate translated follower and the mechanism with rotary cam and translated follower with roll.

Keywords: Cam dimensions, Efficiency, Distribution mechanism with high efficiency, Distribution settings, Distribution parameters

Nomenclature

i :

is the mechanical instantaneous efficiency of the

transmission with cam and pusher (tappet);  i : is the mechanical instantaneous coefficient of loss;

 : is the mechanical yield of the transmission with cam and pusher; : is the pressure angle of the transmission; : is the transmission angle; Pu: is the utile power, of the tappet movement; Pc: is the power consumption; v2: is the follower velocity; Fm: is the consumed or motor force; Fu: is the utile force; F: is the sliding force; k: is the elastic constant of the valve spring; x0: is the valve spring tension; r0: is the radius of the base circle of the cam; rb: is the radius of the roll of the tappet; h=smax: is the maximum displacement of the tappet; u: is the angle of lift; n: is the shaft speed. I.

Introduction

The mechanisms with cam and follower are utilized in many domains [1-14]. The principal utilization of cams gear is the mechanism of the distribution of a heat engine with internal combustion [1]. The paper presents an original method to increase the efficiency of a mechanism with cam and follower, used at the distribution mechanisms [2].

Manuscript received May 2013, revised May 2013

This paper treats only two modules: the mechanism with rotary cam and plate translated follower (the classic module C) and the mechanism with rotary cam and translated follower with roll (the modern module B). At the classical module C we can increase again the yield to about 30%. The growth is difficult. Dimensional parameters of the cam must be optimized; optimization and synthesis of the cam profile are made dynamic, and it must set the elastic (dynamic) parameters of the valve (tappet) spring: k and x0. The law used is not as important as the module used, sizes and settings used. We take the classical law cosine; dimensioning the radius cam, lift height, and angle of lift. To grow the cam yield again we must leave the classic module C and take the modern module B. In this way the efficiency can be as high as 60%. Yields went increased from 4% to 60%, and we can consider for the moment that we have gain importance, since we work with the cam and tappet mechanisms.

II. Determining of Momentary Mechanical Efficiency of the Rotary Cam and Plate Translated Follower The consumed motor force Fm, perpendicular in A to the vector rA, is divided into two components: a) Fu, which represents the useful force, or the motor force reduced to the follower; b) F, which is the sliding force between the two profiles of cam and follower (Fig. 1). See the written relations (2.1-2.10).

Fu  Fm  sin

(2.1)

Copyright © 2002 Florian Petrescu - All rights reserved

F.I. Petrescu, R.V. Petrescu

v2  v1  sin

(2.2)

Pu  Fu  v2  Fm  v1  sin 2 

(2.3)

Pc  Fm  v1 i 

(2.4)

Pu Fm  v1  sin 2    sin 2   cos 2  Pc Fm  v1

(2.5)

s '2 s '2 sin   2  rA (r0  s) 2  s'2 2

(2.6)

F  Fm  cos

(2.7)

v12  v1  cos

(2.8)

P  F  v12  Fm  v1  cos 2 

(2.9)

i 

P Pc



sc'  

    h  sin     2  c  c 

sc''     

(3.1)

    2 h  cos    2   c2  c 

sc''' 

   3 h  sin     2   c3  c 

 xc   s' cos   r0  s   sin    yc  r0  s   cos   s' sin   xc  s' cos   r0  s   sin    yc  r0  s   cos   s' sin 

(3.2)

(3.3)

The r0 (the radius of the base circle of the cam) is 0.013 [m]. The h (the maximum displacement of the tappet) is 0.006 [m]. The angle of lift, u is /2 [rad]. The cosine profile can be seen in the fig. 2.



s’

F

 D



r Fu



C

r v1 rA

   h h   cos    2 2  c 

(2.11)

r Fm

A

sc 

Fm  v1  cos 2   cos 2   sin 2  (2.10) Fm  v1

 i  i  sin 2   cos 2   1

© 2002 Florian PETRESCU The Copyright-Law Of March, 01, 1989 U.S. Copyright Office Library of Congress Washington, DC 20559-6000 202-707-3000

    h h s    cos    2 2  u      s '  vr    h  sin          2  u u    2    h   s ' '  a   cos    r  2   u2  u       3 h  sin    s ' ' '   r   2   u3   u

r v2

r F

B

r v12

E s

 r0

O

w

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

Fig. 2. The cosine profile at the cam with plate translated follower; r0=13[mm], h=6[mm], u=/2[rad]

The obtained mechanical yield (obtained by integrating the instantaneous efficiency throughout the climb and descent) is 0.06 or =6%. The dynamic diagram can be seen in the fig. 3 (the dynamic setting are normal) [3-4].

III. Increasing the Mechanical Efficiency at the Rotary Cam and Plate Translated Follower The used law is the classical law (3.1), cosine law. The synthesis of the cam profile can be made with the relationships (3.2) when the cam rotates clockwise and with the expressions from the system (3.3) when the cam rotates counterclockwise (trigonometric).


Fig. 3. The dynamic diagram at the cosine profile at the cam with plate translated follower; r0=13[mm], h=6[mm], u=/2[rad]; n=5000[rpm]; x0=9[cm]; k=40[kN/m]

International Review of Mechanical Engineering, Vol. 7, n. 4


It tries increase the yield; angle of climb is halved u=/4[rad] (see the profile of the Fig. 4) [5-6].

The obtained mechanical yield is 0.32 or =32%. The dynamic diagram can be seen in the fig. 7 (the dynamic setting are normal as well). As no longer climb using full profile, parameter h could be compromised.


Fig. 4. The cosine profile at the cam with plate translated follower; r0=13[mm], h=6[mm], u=/4[rad]

In this moment, we must leave the classical module C (Fig. 1), and take the module B (Fig. 8) which may increases further the yield of the distribution mechanism.

The obtained mechanical yield is 0.19 or =19%. The dynamic diagram can be seen in the fig. 5 (the dynamic setting are normal as well). As no longer climb using full profile, parameter h could be compromised.

IV. Determining of Momentary Dynamic Efficiency of the Rotary Cam and Translated Follower with Roll The pressure angle  (Fig. 8), is determined by relations (4.5-4.6). We can write the next forces, speeds and powers (4.13-4.18). Fm, vm, are perpendicular to the vector rA at A. Fm is divided into Fa (the sliding force) and Fn (the normal force). Fn is divided also, into Fi (the bending force) and Fu (the useful force). The momentary dynamic efficiency can be obtained from relation (4.18). Fu, v2

Fn, vn 


Fn, vn

Fi, vi

B

It tries increase again the yield; angle of climb is now u=/6[rad] (see the profile of the Fig. 6) [7-14].

Fm, vm

rb s

 A-

A

Fa, va rB rA

B0 rb n

s0

C

A0

x

B





A 

0 A e

O

r0

Fig. 8. Forces and speeds to the cam with translated follower with roll Fig. 6. The cosine profile at the cam with plate translated follower; r0=13[mm], h=6[mm], u=/6[rad]


The written relations are the following.



rB2  e 2  (s0  s) 2 rB  rB2

(4.1) (4.2)

i 

e cos B  sin   rB

(4.3)

s0  s rB

(4.4)

sin  B  cos 

cos  

sin  

V.

s0  s ( s0  s)2  ( s'e)2 s'e ( s0  s)2  ( s'e)2

(4.5)

(4.6)

(4.7)

rA2  rB2  rb2  2  rb  rB  cos(   )

(4.8)

sin  A 

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

(4.9)

(4.10)

( s0  s )  s '

s' cos( A   )   cos  (4.11) 2 2 rA rA  ( s0  s)  ( s'e)

cos( A   )  cos 

s'  cos2  rA

(4.12)

va  vm  sin( A   )  Fa  Fm  sin( A   )

(4.13)

vn  vm  cos( A   )  Fn  Fm  cos( A   )

(4.14)

vi  vn  sin   Fi  Fn  sin 

(4.15)

v2  vn  cos  vm  cos( A   )  cos  Fu  Fn  cos  Fm  cos( A   )  cos


Pu Fm  vm  cos2 ( A   )  cos2    Pc Fm  vm

 [cos( A   )  cos ]2  [

cos(   )  cos   cos  sin   sin 

cos  A 

2 2  Pu  Fu  v2  Fm  vm  cos ( A   )  cos  (4.17)   P  F  v c m m 

(4.16)

(4.18) 2

s' s'  cos2  ]2  2  cos4  rA rA

Increasing the Mechanical Efficiency at the Rotary Cam and Translated Follower with Roll

The used law is the classical law (3.1), cosine law. The synthesis of the cam profile can be made with the relationships (5.1) when the cam rotates clockwise and with the expressions from the system (5.2) when the cam rotates counterclockwise (trigonometric).  xC   e  rb  sin    cos   s0  s   rb  cos   sin  (5.1)    y  s  s   r  cos   cos    e  r  sin    sin  0 b b  C

 xc   e  rb  sin    cos   s0  s  rb  cos    sin   (5.2)   y  s  s  r  cos    cos    e  r  sin    sin  0 b b  c

The r0 (the radius of the base circle of the cam) is 0.013 [m]. The h (the maximum displacement of the tappet) is 0.020 [m]. The angle of lift, u is /3 [rad]. The radius of the tappet roll is rb=0.002 [m]. The misalignment is e=0 [m]. The cosine profile can be seen in the fig. 9.

Fig. 9. The cosine profile at the cam with translated follower with roll; r0=13[mm], h=20[mm], u=/3[rad], rb=2[mm], e=0[mm].



The obtained mechanical yield (obtained by integrating the instantaneous efficiency throughout the climb and descent) is 0.39 or =39%. The dynamic diagram can be seen in the fig. 10 (the dynamic setting are partial normal). Valve spring preload 9 cm no longer poses today. Instead, achieve a long arc very hard (k=500000[N/m]), require special technological knowledge.

normal). Valve spring preload 20 cm no longer poses today. Instead, achieve a long arc very-very hard (k=1500000[N/m]), require special technological knowledge.

Fig. 12. The dynamic diagram at the cosine profile at the cam with translated follower with roll; r0=15[mm]; h=10[mm]; u=/6[rad]; rb=2[mm]; e=0[mm]; n=5500[rpm]; x0=20[cm]; k=1500[kN/m]


It tries increase the yield; angle of climb is halved u=/6[rad] (see the profile in the Fig. 11). The r0 (the radius of the base circle of the cam) is 0.015 [m]. The h (the maximum displacement of the tappet) is 0.010 [m]. The angle of lift, u is /6 [rad]. The radius of the tappet roll is rb=0.002 [m]. The misalignment is e=0 [m]. The cosine profile can be seen in the fig. 11.

Camshaft runs at a shaft speed halved (nc=n/2). If we more reduce camshaft speed by three times (nc=n/6), we can reduce and the preload of the valve spring (x0=5[cm]); see the dynamic diagram in the Fig. 13. However, in this case, the cam profile should be tripled (see the Fig. 14). xpp [m/s2]

xpp [m/s2]

600

600

600

400

400

400

200

200

0

200

0 0

50

100

0 0

150

50 200

100 250

0 150 300

-200

-200

-200

-400

-400

-400

-600

-600

xpp [m/s2]

50 200 350

100 250 400

150 300

xpp [m/s2]

-600

-800

-800

-800

-1000

-1000

-1000

-1200

-1200

-1200

-1400

-1400

-1400

Fig. 13. The dynamic diagram at the cosine tripled profile at the cam with translated follower with roll; r0=15[mm]; h=10[mm]; u=/6[rad]; rb=2[mm]; e=0[mm]; n=5500[rpm]; x0=5[cm]; k=1500[kN/m]


The obtained mechanical yield (obtained by integrating the instantaneous efficiency throughout the climb and descent) is 0.428 or =43%. The dynamic diagram can be seen in the fig. 12 (the dynamic setting are not


Fig. 14. The cosine tripled profile at the cam with translated follower with roll; r0=15[mm], h=10[mm], u=/6[rad], rb=2[mm], e=0[mm].


200


It tries increase the yield again; angle of climb is reduced to the value u=/8[rad]. The r0 (the radius of the base circle of the cam) is 0.013 [m]. The h (the maximum displacement of the tappet) is 0.009 [m]. The angle of lift, u is /8 [rad]. The radius of the tappet roll is rb=0.002 [m]. The misalignment is e=0 [m]. The cosine profile can be seen in the fig. 15.

Camshaft runs at a shaft speed halved (nc=n/2). If we more reduce camshaft speed by four times (nc=n/8), we can reduce and the preload of the valve spring, x0=9[cm] and the elastic constant of the valve spring, k=15000[N/m]; see the dynamic diagram in the Fig. 17. However, in this case, the cam profile should be fourfold (see the Fig. 18).

xpp [m/s2] 4000

4000

4000

4000

3000

3000

3000

3000

2000

2000

2000

2000

1000

1000

1000

1000

0

0 0

50

xpp [m/s2]

100 0

150 50

0 100 2000

150 25050

0 0 200 300 100

-1000

-1000

-1000

-1000

-2000

-2000

-2000

-2000

-3000

-3000

-3000

-3000

-4000

-4000

-4000

-4000

50 250 350 150

xpp [m/s2] xpp [m/s2]

100 300 400 200

150 350 250

200 400 300

xpp [m/s2]

Fig. 17. The dynamic diagram at the cosine fourfold profile at the cam with translated follower with roll; r0=13[mm]; h=9[mm]; u=/8[rad]; rb=2[mm]; e=0[mm]; n=5000[rpm]; x0=9[cm]; k=15[kN/m]


The obtained mechanical yield (obtained by integrating the instantaneous efficiency throughout the climb and descent) is 0.538 or =54%. The dynamic diagram can be seen in the fig. 16 (the dynamic setting are not normal). Valve spring preload 30 cm no longer poses today. Instead, achieve a long arc very-very hard (k=1600000[N/m]), require special technological knowledge.

Fig. 18. The cosine fourfold profile at the cam with translated follower with roll; r0=13[mm], h=9[mm], u=/8[rad], rb=2[mm], e=0[mm].



With the same angle of climb u=/8[rad], can increase performance even further, if the size tappet race take a greater value (h=12[mm]). The r0 (the radius of the base circle of the cam) is 0.013 [m].



The h (the maximum displacement of the tappet) is 0.012 [m]. The angle of lift, u is /8 [rad]. The radius of the tappet roll is rb=0.002 [m]. The misalignment is e=0 [m]. The cosine profile can be seen in the fig. 19.

For now is necessary to stop here. If we increase h, or decrease the angle u, then is tapering cam profile very much. We must stop now at a yield value, =60%.

VI. Conclusion The distribution mechanisms work with small efficiency for about 150 years; this fact affects the total yield of the internal heat engines. Much of the mechanical energy of an engine is lost through the mechanism of distribution. Multi-years the yield of the distribution mechanisms was only 4-8%. In the past 20 years it has managed a lift up to the value of 14-18%; car pollution has decreased and people have better breathing again. Meanwhile the number of vehicles has tripled and the pollution increased again. Now, it’s the time when we must try again to grow the yield of the distribution mechanisms. The paper presents an original method to increase the efficiency of a mechanism with cam and follower, used at the distribution mechanisms. This paper treats only two modules: the mechanism with rotary cam and plate translated follower (the classic module C) and the mechanism with rotary cam and translated follower with roll (the modern module B). At the classical module C we can increase again the Fig. 19. The cosine profile at the cam with translated follower with roll; yield to about 30%. The growth is difficult. Dimensional r0=13[mm], h=12[mm], u=/8[rad], rb=2[mm], e=0[mm]. parameters of the cam must be optimized; optimization and synthesis of the cam profile are made dynamic, and it For correct operation it is necessary to decrease the must set the elastic (dynamic) parameters of the valve speed of the camshaft four times, and all four times (tappet) spring: k and x0. multiplication of the cam profile. Camshaft runs at a shaft The law used is not as important as the module used, speed halved (nc=n/2). If we more reduce camshaft speed sizes and settings used. We take the classical law cosine; by four times (nc=n/8), we can reduce and the preload of dimensioning the radius cam, lift height, and angle of lift. the valve spring, x0=9[cm]. The elastic constant of the To grow the cam yield again we must leave the classic valve spring is k=1500000[N/m]. See the dynamic module C and take the modern module B. In this way the diagram in the Fig. 20. However, in this case, the cam efficiency can be as high as 60%. profile should be fourfold. The obtained mechanical yield Yields went increased from 4% to 60%, and we can is 0.60 or =60%. consider for the moment that we have gain importance, since we work with the cam and tappet mechanisms. If we increase h, or decrease the angle u, then is xpp [m/s2] xpp [m/s2] xpp [m/s2] xpp [m/s2] tapering cam profile very much. We must stop now at a yield value, =60%. 500

500

0

0 0

-500

500

0100

50

-500

50 150

0 100 0 200

-500

500

0 150 50 250

0200 100 300

50 250 150 350

100 300 200 400

150 350 250

200 400 300

References

-500 xpp [m/s2]

-1000

-1000

-1000

-1000

-1500

-1500

-1500

-1500

-2000

-2000

-2000

-2000

[1]

[2]

[3] Fig. 20. The dynamic diagram at the cosine fourfold profile at the cam with translated follower with roll; r0=13[mm]; h=12[mm]; u=/8[rad]; rb=2[mm]; e=0[mm]; n=5000[rpm]; x0=9[cm]; k=1500[kN/m]


[4]

P. Antonescu, F. Petrescu, O. Antonescu, Contributions to the Synthesis of The Rotary Disc-Cam Profile, In VIII-th International Conference on the Theory of Machines and Mechanisms, Liberec, Czech Republic, p. 51-56, 2000. F.I. Petrescu, Theoretical and Applied Contributions about the Dynamic of Planar Mechanisms with Superior Joints, Doctoral Thesis, Bucharest Polytechnic University, 2008. F.I. Petrescu, R.V. Petrescu, Contributions at the dynamics of cams. In the Ninth IFToMM International Symposium on Theory of Machines and Mechanisms, SYROM 2005, Bucharest, Romania, 2005, Vol. I, p. 123-128. Z. Ge, a.o., Mechanism Design amd Dynamic Analysis of Hybrid Cam-linkage Mechanical Press, Key Engineering Materials



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Authors’ information 1

Dr. Eng. Florian Ion T. Petrescu, Senior Lecturer at UPB (Bucharest Polytechnic University), TMR (Theory of Mechanisms and Robots) department. 2 Dr. Eng. Relly Victoria V. Petrescu, Senior Lecturer at UPB (Bucharest Polytechnic University), TTL (Transport, Traffic and Logistics) department.


1. Ph.D. Eng. Florian Ion T. PETRESCU Senior Lecturer at UPB (Bucharest Polytechnic University), Theory of Mechanisms and Robots department, Date of birth: March.28.1958; Higher education: Polytechnic University of Bucharest, Faculty of Transport, Road Vehicles Department, graduated in 1982, with overall average 9.63; Doctoral Thesis: "Theoretical and Applied Contributions About the Dynamic of Planar Mechanisms with Superior Joints". Expert in: Industrial Design, Mechanical Design, Engines Design, Mechanical Transmissions, Dynamics, Vibrations, Mechanisms, Machines, Robots. Association: Member ARoTMM, IFToMM, SIAR, FISITA, SRR, AGIR. Member of Board of SRRB (Romanian Society of Robotics). 2. Ph.D. Eng. Relly Victoria V. PETRESCU Senior Lecturer at UPB (Bucharest Polytechnic University), Transport, Traffic and Logistics department, Citizenship: Romanian; Date of birth: March.13.1958; Higher education: Polytechnic University of Bucharest, Faculty of Transport, Road Vehicles Department, graduated in 1982, with overall average 9.50; Doctoral Thesis: "Contributions to analysis and synthesis of mechanisms with bars and sprocket". Expert in Industrial Design, Engineering Mechanical Design, Engines Design, Mechanical Transmissions, Projective and descriptive geometry, Technical drawing, CAD, Automotive engineering, Vehicles, Transportations. Association: Member ARoTMM, IFToMM, SIAR, FISITA, SRR, SORGING, AGIR.
