UNIVERSITATEA TRANSILVANIA DIN BRA)OV Catedra Design de Produs +i Robotic-
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P R A S I C ' 02 Vol. I – Mecanisme +i Tribologie 7-8 Noiembrie Bra6ov, România ISBN 973-635-064-9
THE CAM SYNTHESIS WITH MAXIMAL EFFICIENCY Ion PETRESCU, Victoria PETRESCU, Constantin OCN RESCU Bucharest Polytechnic University Abstract: The paper presents an original method to determinate the efficiency of cams mechanisms, which are used to the driving mechanism from OTTO engines. With the relations of cam efficiency, from this presentment, on can synthesize the cam and follower couple. On determine the constructive parameters of cam and follower couple, having in view some few criterions, such are: the achievement of the biggest efficiency and chronosection; the smallest values for the speed, the continuity of the movement, the projection of some best movement laws for the driving-mechanisms of the internal-combustion engines, and the optimal mechanism-couple, which are the F Module. Key words: Efficiency, F Module, constructive parameters, movement laws. 1.
Introduction
In this paper the authors presents an original method to synthesize the cam and follower mechanisms, with the biggest efficiency. On synthesize the cam mechanisms with the formulas of the great couple efficiency. The authors were analyzing more type of couples (more Modules), and they take the conclusion that, the best Module is the F Module, with rotation cam and with rotation follower with roll. This Module (F Module) allows the optimum regulation for the constructive parameters. The F Module is the best module, if he is use accordingly (see the picture number one). 2. The F Module In the picture number one (figure 1) on presents the mechanism with rotation cam and rotation follower with roll. The follower is in the right of the picture (the right of the cam), and the cam rotation is trigonometric. In this mode the pressure angle take the smallest
values (see the picture number one). Mechanism with rotation cam and rotation follower with roll; (Module F ). 2’ 1
rb
B
2 b
A rB
B0
A0
D
b
d
0
0
O r0
Fig. 1.
To be continued the presentation of the optimum constructive parameters for the F Module mechanism, and for varied movement laws of the follower: and ’. These constructive parameters are: r0 - the radius of the basic circle of the cam; rb - the radius of the follower roll; b – the length of the follower; d – the distance between the rotation centers of the cam and the follower;
– the maximum angle of follower rotation; - the maximum angle of cam rotation into the
M u
up run. 2.1.
The Efficiency to the F Module (for up run): 0.993772771 The efficiency of couple: = ( ) = f(x) i = f(
The LC1 law
1
u= /4[rad] r0 =13[mm] b=90[mm] rb =11[mm] d=109[mm] 0 M =30 The movement law, LC3: b = - 1.42 y=(b/2+1)x-b/2x2
0.9 0.8
The Efficiency to the F Module (for up run): The efficiency of couple: = 0.999961025 ( ) = f(x) i = f(
0.6 0.5
0.4 0.3 0.2
y=x+sin(2 x)/[(2 )2 a]-bx 2 /2+bx/2
0.1
x=
/
1
0.9
0.7
0.8
0.5
0.6
0.3
0.4
0.2
0
0.1
0
u
0.1
x=
/
u
1
0.9
0.8
0.7
0
0.6
0.5
0.2
0.5
0.6
0.3
0.4
0.7
0.4
0.3
0.8
0.2
0.9
0
= /4[rad] r0 =13[mm] b=99[mm] rb =8[mm] d=115[mm] 0 M =26 The movement law LC1: b = - 1.33 a=1000. u
0.1
1
0.7
Fig. 5.
2.5.
The
Power law
Fig. 2. The Efficiency to the F Module (for up run): 0.992648638 The efficiency of couple: = ( ) = f(x) i = f(
2.2. The LC2 law
1
u= /4[rad] r0 =8[mm] b=66[mm] rb =22[mm] d=91[mm] 0 M =27 The movement law,
0.9 0.8
The Efficiency to the F Module (for up run): 0.997442079 The efficiency of couple: = ( ) = f(x) i = f(
0.6
0.3 0.2
y=x+a[sin(2 x)/(2 )] 2 -bx 2 /2+bx/2
0.1
x=
/
u
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
0.1
0
Fig. 3.
2.3.
x=
2.6.
The
LC5 law
The Efficiency to the F Module (for up run): 0.882736114 The efficiency of couple: = i = f( ) = f(x)
The LC4 law
1
0.9 0.7
0.5 0.4 0.3 0.2 0.1
Fig. 4.
LC3 law
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
0.1
0
1
0.9
0.8
0.7
0 0.6
0.6
0.1 0.5
0.7
0.2
0.4
0.8
0.3
0.3
u
0.4
0.2
= /4[rad] r0 =13[mm] b=90[mm] rb =10[mm] d=108[mm] 0 M =29 The movement law, LC4: a = - 0.6 y=a(x-x2 )+x x= / u
0.9
0.5
0
1
0.6
0.1
The Efficiency to the F Module (for up run): 0.995919112 The efficiency of couple: = ( ) = f(x) i = f(
The
u
Fig. 6.
0.8
2.4.
/
1
0.9
0.8
0.7
0
0.6
0.4
0.1
0.5
0.5
0.2
0.4
0.6
Power: y=2 x -1
0.3
0.3
0.7
0.4
0.2
0.8
0.5
0
= /4[rad] r0 =13[mm] b=99[mm] rb =9[mm] d=116[mm] 0 M =22 The movement law, LC2: b = - 1.24 a= -0.1 u
0.9
0.1
1
0.7
u= /4[rad] r0 =11[mm] b=80[mm] rb =10[mm] d=96[mm] 0 M =27 The movement law, LC5: a = 1.76 y=2x-ax+ax 2 -x 3 x= / u
Fig. 7.
The present’s laws are synthesis by the authors of this paper, such as the dynamic response of these laws to be the best possible. The laws parameters, a or b, are settle for a maximum efficiency. The constructive parameters are settle too, for
the maximum efficiency of couple. 2.7.
The
The Efficiency to the F Module (for up run): 0.723109943 The efficiency of couple: = ( ) = f(x) i = f(
Radical law
1
u= /4[rad] r0 =8[mm] b=66[mm] rb =3[mm] d=72[mm] 0 M =29 The movement law,
0.9 0.8
The Efficiency to the F Module (for up run): 0.802063977 = The efficiency of couple: ( ) = f(x) i = f(
0.6 0.5
0.3
Radical:
0.2
1
0.9
0.7
0.8
0 0.5
0.4
0.1 0.6
0.5
LOGARITHM: y=ln(x+1)/ln2
0.2
0.3
0.6
0.3
0.4
0.7
0.4
0.2
0.8
0
= /4[rad] r0 =8[mm] b=66[mm] rb =10[mm] d=79[mm] 0 M =21 The movement law, u
0.9
0.1
1
0.7
x=
/
u
Fig. 11.
y=((1+x).5-1)/(2.5-1)
0.1
x=
/
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
0.1
0
u
2.11.
The COS law
Fig. 8.
2.8. The
The Efficiency to the F Module (for up run): The efficiency of couple: = 0.711000532 ( ) = f(x) i = f(
LC6 law
1
The Efficiency to the F Module (for up run): 0.734215978 The efficiency of couple: = ( ) = f(x) i = f(
0.8 0.7 0.6
= /4[rad] r0 =8[mm] b=66[mm] rb =5[mm] d=74[mm] 0 M =24 The movement law, LC6: a=0.32
0.5
u
0.2
x=
/
1
0.9
/
u
u
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
x=
Fig. 12.
y=(1-a)2 sin2 ( x/2)/[1-2asin( x/2)+a2 ]
0.1
0.8
0 0.7
0.3
0.1 0.5
0.4
0.2
0.6
0.5
COS : y=.5-.5cos( x)
0.3
0.4
0.6
0.4
0.2
0.7
0.3
0.8
0
0.9
0.1
1
u= /4[rad] r0 =8[mm] b=66[mm] rb =3[mm] d=72[mm] 0 M =29 The movement law,
0.9
2.12.
The
SIN law
Fig. 9. The Efficiency to the F Module (for up run): The efficiency of couple: = ( ) = f(x) i = f(
2.9. The Polynomial 2-3 law
0.61431855
1
u= /4[rad] r0 =8[mm] b=66[mm] rb =3[mm] d=72[mm] 0 M =29 The movement law,
0.9 0.7
0.72359524
0.6
0.3 0.2 0.1
x=
Fig. 10.
2.10.
The Logarithm law
/
1
0.9
Fig. 13.
u
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
0
y=x-sin(2 x):(2 )
0
0.8
0.4
SIN :
0.1
0.7
0.5
0.2
0.6
0.6
0.3
0.5
0.7
0.4
0.4
0.8
0.5
0.3
0.9
0.2
u= /4[rad] r0 =8[mm] b=66[mm] rb =3[mm] d=72[mm] 0 M =29 The movement law, Polynomial 2-3: y=3x 2 -2x 3
0
1
0.1
The Efficiency to the F Module (for up run): The efficiency of couple: = ( ) = f(x) i = f(
0.8
The last two laws are the classics. They are added purposeful for to see the advantages of the created laws (created for the authors), and the advantages of the method (the advantages of the F
Module) which are working with great efficiency even when the laws are the classic’s (COs and sin). The Module B realizes the smaller values for the efficiency of mechanism (of couple), (60-95%), but the C Module (the classic Module) realizes the smallest values for the couple efficiency (10-20%). 2.13.
The establish of the efficiency
y'
The relations to establish the efficiency for the F Module, are the next’s: i=
N/n
(1)
N = d2 . sin2
. V2
(2)
n = (rB2 +rb2 ). R2 - 2.rb .(rB2 +V.H).R
(3)
R2 = rB2 +V2 +2.V.H
(4)
2
rB2 = b2 +d2 - 2.b.d.cos
r0 b rb d =B23 psiM y
psi psi' psi0 psi2 rB2 V R2 R N n
(5)
2
i
2
=
0
= acos{[b2 +d2 - (r0 +rb )2 ]/(2.b.d)}
0
+
(6)
V = b. ’
(7) (8)
H = b - d . cos
(9)
2
u
= 1/
.
0
i
.d
(10)
On presents a program to calculate the efficiency for the F Module. The program is write in the Excel (see the table number one). The adimensional angular step ( x) is 0.05 , such as the integral precision through rectangles is roughly 5%. For that, the calculate efficiency can be greatest with 2-4 percent’s. For a greatest precision the program must be rewriting with a step x= 0.005, such as the approximate integral to be exactly the area under the i function. Table 1. A FIu
B =PI()/4
S b
=13 =99 =8 =B2+B4+B3-5 =0 =PI()/180*26 =B6+SIN(2*PI()*B6)/((2*PI())^2*1000) -B24*B6^2/2+B24/2*B6 =1+COS(2*PI()*B6)/(2*PI()*1000)B24*B6+B24/2 =B7*B8 =B7*B9/B1 =ACOS((B3^2+B5^2(B2+B4)^2)/(2*B3*B5)) =B12+B10 =B3^2+B5^2-2*B3*B5*COS(B13) =B3*B11 =B3-B5*COS(B13) =B14+B15^2+2*B15*B16 =SQRT(B17) =B5^2*B15^2*(SIN(B13))^2 =(B14+B4^2)*B172*B4*(B14+B15*B16)*B18 =B19/B20 =SUM(B21:V21) =B22*0.05 =-1.33
The program to calculate the efficiency is write by 24 rows and by more columns, for B to V, in accordance with the adopted step ( x= 0.05). On presents the B column relations. The others columns repeat the B column relations, with the variable x modified (in the 6 row). 3.
The
B
Module
The B Module is a kinematics couple too, with the rotation cam, and follower with roll, but the movement of this follower is the translation. The authors recommend (introduce) the F Module, but the eager for the translation movement can use the B Module, into the C Module’s detriment. On make the presentation of the B Module couple in picture number 14 (see figure 14.).
0
Module B couple: Rotation cam and translation follower with roll.
The movement laws for the B Module, are presents in the next pictures (see figures 15-23):
2 B 2’
1
O
rb
r0
3.2.
s
rB
B0 s0
The Power Law
The Efficiency to the B Module (for up run): The efficiency of couple: 0.952621077 = i = f( ) = f(x)
e
1
0
u= /4[rad] r0 =6[mm] e=14[mm] rb =9[mm] h=10[mm] The movement law,
0.9 0.8 0.7
Fig. 14.
0.6 0.5 0.4
Power: x y=2 -1
0.3
3.1. Calculate program
0.2 0.1
Table 2.
u
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
/
Fig. 15.
3.3. The Radical law
B The Efficiency to the B Module (for up run): 0.891572044 The efficiency of couple: = ( ) = f(x) i = f( 1
u= /4[rad] r0 =6[mm] e=14[mm] rb =9[mm] h=10[mm] The movement law,
0.9 0.8 0.7 0.6 0.5 0.4
RADICAL:
0.3
y=((1+x).5-1)/(2.51)
0.2 0.1 0.9
0.8
0.7
0.6
0.5
0.4
0.3
u
Fig. 16.
3.4.
The Logarithm law
The Efficiency to the B Module (for up run): = The efficiency of couple: 0.794681222 ( ) = f(x) i = f( 1
u= /4[rad] r0 =6[mm] e=14[mm] rb =9[mm] h=10[mm] The movement law,
0.9 0.8 0.7 0.6 0.5 0.4
LOGARITHM: y=ln(x+1)/ln(2)
0.3 0.2 0.1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
1
x=
0
0.1
The constructive parameters for the B Module are: e – the distance between the rotation center of cam and the follower guide; h = sMax - the maximum run of the follower; u – the maximum angle of cam rotation into the up run; r0 - the radius of the basic circle of the cam; rb - the radius of the follower roll.
/
1
x=
0 0.2
=PI()/4 =6 =14 =9 =SQRT((B2+B4)^2-B3^2) =0 =10 =2^B6-1 =LN(2)*2^B6 =B7*B8 =B7*B9/B1 =(B5+B10)^2 =B12+(B11-B3)^2 =B12+B3^2 =B14+B4^2-2*B4*(B14B3*B11)/(SQRT(B13)) =B11^2*B12/B13/B15 i s =SUM(B16:V16) rand ( ) =B17*0.05
0
FIu r0 e rb s0 =B18 h y y' s s' a b c rA2
0.1
A
x=
0 0.1
The calculate program writhed in Excel program language (see the table number two), for the power law (for example), is the next:
Fig. 17.
3.5.
The polynomial 2-3 law
/
u
The Efficiency to the B Module (for up run): 0.646367536 The efficiency of couple: = ( ) = f(x) i = f(
The Efficiency to the B Module (for up run): The efficiency of couple: = 0.715430581 ( ) = f(x) i = f(
1
0.9
0.4
= /4[rad] r0 =6.3[mm] e=14[mm] rb =9.6[mm] h=10[mm] The movement law,
0.3
Polynomial 2-3:
0.3
u
y=x+sin(2 x)/[(2 )2 a]-bx 2 /2+bx/2
0.1
x=
3.9.
The COS law
The Efficiency to the B Module (for up run): 0.697023351 The efficiency of couple: = ( ) = f(x) i = f( 1 u= /4[rad] 0.9 r0 =6[mm] 0.8 e=14[mm] 0.7 rb =14[mm] 0.6 h=10[mm] 0.5 The movement law,
COS : y=1/2 1/2.cos( .x) u
0.6 0.5 0.4 0.2
2
2
y=x+a[sin(2 x)/(2 )] -bx /2+bx/2
0.1
3.10.
0.4 0.3
LC6:
0.2
a = 0.35
0.1
0.9 0.8 0.7 0.6 0.5
0.2
1
0.9
0.7
0.8
0.6
0.5
0.3
1
0.9
u= /4[rad] r0 =6[mm] e=14[mm] rb =14[mm] h=10[mm] The movement law,
0.8 0.7 0.6 0.5 0.4
SIN :
0.3
y=x-sin(2 x)/(2 ) 0.9
0.8
0.7
0.6
0.5
0.4
0.3
u
0.2
/
0
x=
0.1
0
2
y=(1-a) sin ( x/2)/[1-2asin( x/2)+a ] 0.4
0.8
1
0.9
= /4[rad] r0 =6[mm] e=14[mm] rb =14[mm] h=10[mm] The movement law,
0.2
u
The Efficiency to the B Module (for up run): 0.595740012 The efficiency of couple: = ( ) = f(x) i = f(
u
0
/
The SIN law
x=
1
1
0.1
x=
Fig. 22.
The Efficiency to the B Module (for up run): The efficiency of couple: = 0.681008434 ( ) = f(x) i = f(
0
0.7
0.6
0.5
0
The LC6 law
2
LC2: a=-1,b=1.9
0.3
Fig. 19.
2
1
0.7
0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
/
1
x=
0
u= /4[rad] r0 =6[mm] e=14[mm] rb =14[mm] h=10[mm] The movement law,
0.8
0.4
0.1
1
0.9
0.3
0.2
The Efficiency to the B Module (for up run): The efficiency of couple: = 0.604058112 ( ) = f(x) i = f(
0.2
0.3
0.1
u
The LC2 law
0.1
0.4
3.7.
/
Fig. 21.
Fig. 18.
3.6.
0.9
0
u
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0
0.1
/
0
x=
0
0.2
0.8
y=3x -2x
0.1
3
0.4
0.7
2
0.5
0.6
0.2
0.6
0.5
0.5
0.7
0.4
0.6
0.3
0.7
u= /4[rad] r0 =6[mm] e=14[mm] rb =14[mm] h=10[mm] The movement law, LC1: a=1000,b=1.47
0.8
0.2
0.8
0.1
1
0.9
/
u
Fig. 23.
Fig. 20.
Bibliography
On can see easily the superiority of the Power movement law for the B and F Module, the superiority of the radical and logarithm movement laws, and the superiority of the LCn ( n= 1...6 ) laws. A good behaviour have too, the polynomial 2-3, the cos, and the sin movement laws. 3.8. The LC1 law
1. Petrescu, F., Petrescu, R. Designul (sinteza) mecanismelor cu came prin metoda coordonatelor polare (metoda triunghiurilor). ﾃ始 a VII-a Conferin01 Na0ional1 Grafica - 2000, Craiova, 2000, p. 291-296.