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Kinematic design of four-bar function-generators with optimum sensitivity
Mechanism and Machine Theory, 1975, Vol. 10, pp. 531-535.
Pergamon Press. Printed in Great Britain
Kinematic Design of Four-bar Function-Generators with Optimum Sensitivity A. C. Rao¢
Received on 27 November 1974 Abstract
In linkages used for instrumentation and controls sensitivity should be optimum so that the output link responds well, even for very small changes in the input. Also it is most desirable to incorporate good transmission characteristics in the linkage. A method to synthesise four-bar linkages for three separate precision points incorporating the above characteristics is explained in this paper. Introduction SENSITIVITYcan be defined as the ratio of the change in output to a corresponding small change in input. Linkages must be designed for optimum sensitivity to respond well, even for extremely small changes in the input. For a linkage which generated a function Y - - f ( x ) S = Theoretical sensitivity = ~Y = dy
8x
dx= f'(x)"
(1)
The designer must also keep in mind, at the design stage, the quality of motion transmission. Transmission angle is indicative of the quality of motion transmission. The best transmission angle is 90° and it should deviate as little as possible from 90° during a motion cycle. Linkages with minimum transmission angles of 40° and above are expected to give satisfactory performance.
Method The problem is to determine the link lengths of a four-bar linkage which is required to generate, at three precision points, Y = f(x), xs -< x -< xt with optimum sensitivity and good transmission characteristics. The solution involves writing down the displacement equation of the form of eqn (1) and replacing the angle a by sensitivity, crank angles, link lengths, etc. The resulting equation can be expressed, following Freudenstein[2], in terms of the ratios of link lengths, crank angles and sensitivity. In the usual course, three simultaneous equations are sufficient to solve the problem, i.e. to get three link ratios. However, from the viewpoint of good transmission characteristics, the transmission angle in the second phase is taken as 90°. This leads to a relation, eqn (12), which expresses one of the link ratios in terms of the crank angles, sensitivity and ranges of motion. The starting angles and the ranges of motion if chosen arbitrarily give directly one of the link ratios, eqn (12), and then the linkage can be designed to satisfy the given function only for any two values of the independent variable. Since the problem is concerned with three precision points, one of the angles, say, the angle of the input link corresponding to first precision point is not chosen arbitrarily. The above mentioned link ratio is then expressed in terms of this unknown angle, so necessitating three simultaneous equations for complete solution, as explained below. For a four-bar function generator, the theoretical or optimum sensitivity (eqn 1), can be related, in the following manner, to the ratio of angular velocities of output and input links: dy
dx
(2)
?Professorof MechanicalEngineering,Govt. EngineeringCollege,Ujjain, M.P., India.
531
532
Let x1, x2 and x3 be the three precision points and yl, y2 and y3be the corresponding values of y. Since the linkage can accept the input in angles only, let angles 0 and 4 represent the input x and output y respectively. Assuming a linear relationship between x and 0, and y and &, the following expressions can be written: x-x1 -=-
Ax
Y-Y1 -=
AY
e - 8,
(24
he ’
4-h 4
’
where Ax, by, A& AC#J are the ranges of x, y, 9 and 4. The ranges are defined by:
where the subscripts f and s refer to final and starting values. Angles 0 and C$correspond to any phase of the linkage within the range of motion. By differentiating eqns (2a) and substitution in (2) we obtain:
=- 04
s=
4,
(3)
o4 6 sin(a!-O)=S -=-. w2 d sin(cu-4) q’
(4)
02
.
where
For a four-bar linkage[l],
Expanding and rearranging,
bq sina cos$--cosO dS [
1
=cosa! sin+-asinbq
[
1
and
bq sin . 0 sin C#I - do > tan ff = bq ’ coscp--$OSe > (
(
(5)
Therefore,
1,2, where k, = i
533
d ×
4
b 2
Figure 1. Referring to Fig. 1, it can be seen that a =(6-~)...
(6)
w h e r e / z is the transmission angle. Also b cos 0 + c c o s a - d
cos 6 = a
or (7)
k, cos 0 + k2 cos a - k3 = cos 6, where, k , = ~ b;
k: = ~c
k 3 = ~a
and
Based on eqn (7), the following equations corresponding to three precision points can be written: k l c o s 0~+ k2. A - k3 = cos (b~
(8)
k, cos (0, + A0,) + k2. B - G = cos 62
(9)
k, cos (0, + A02) + k~. C - G = cos 63
(10)
where
klq
cos 6 , - - ~ - cos 0, A =
. kl2q 2 2k,q 1+ ~
] 112
S~- cos (0, - 6 0
]
cos 62 - ~ q cos (0, + a0,)
B =
1 _* - k,2q ~ f - - ~ -2klq 2 cos .~ to, + A01 _ (~2)11/'2 cos
C =
I
6~ --~q cos (0, +A02) 3
, k12q 2 2Gq
1 1- ~
- ~
COS (0, + A02
-
l~3)J 1/2
534
A01 = 05 - 01 A02 = 03 - 05 S, = i f ( x ) . . . . .
s2 = f'(x)
....
and $3 = f ' ( x ) . . . . •
For the output link, the angle 4), corresponding to the first precision point, and the range of motion A~b, may be chosen arbitrarily; ~b2 and 4~3 then follow from (2a). In case of the input link A0, the range of motion may be chosen arbitrarily. No such arbitrary choice is made for 0r, the angle corresponding to first precision point, which is left to be determined from eqn (15), to follow, for the reasons that will be evident from eqn (12). The angles 02 and 03 can then be expressed in terms of 01 with the help of relations (2a). The transmission angle for the phase of the linkage corresponding to second precision point may be assumed to be 90 ° and then by using the relation a = ( 4 ' - / z ) and eqn (5), tan a2 = tan (02 - 90) klq
sin 4'5 - ~ =
klq
sin (01 + A0D
cos 4'~--~- cos (01 +~01)
= p' say.
from which,
kr =
$2 (p cos 4'2 - sin 4') q [p cos (0, + A01) -'sin (0r + A01)]"
(12)
Subtracting eqn (8) from eqn (9) and eqn (10) from eqn (9) the following equations are obtained kl{cos (01 + A01)- cos 01}+ k z ( B - A ) = cos 4 ' : - cos 4',
(13)
kl{cos (0r + A0r) - cos (0r + A02)} + k~(B - C) = cos 4'2 - cos ~b~
(14)
and
Using eqns (13) and (14) the constant ks can be eliminated to give, k l [ ( B - C){cos (01 + A00 - cos 01} - (B - A){cos (0r + A0]) - cos (01 + A0:)}]
-~"[(B -- C) (COS (~2 -- COS ~bl) - (B - A) (cos 4'~ - cos $3)]
(15)
Replacing kl from eqn (12) it may be noted that eqn (15) contains only one unknown term 0,. The eqn (15) may be solved, say, graphically to give 01. Once 01 is obtained kl and hence k2 and k3 can be determined. Since k,, k2 and k3 are the ratios, one of the dimensions, e.g. frame or input link may be assumed to get the other dimensions. If the dimensions so obtained are not satisfactory calculations may be repeated with a different starting value of 4' or ranges A0 and A4'.
References [1] HINKLE R. T., Kinematics of Machines, 2nd Edn, p. 113, Prentice-Hall, New York. [2] FREUDENSTEINF., Approximate synthesis of four-bar linkages. ASME Transactions 77, 853 (1955). [3l HARTENBERG R. S. and DENAVITJ., Kinematic Synthesis of Linkages. McGraw-Hill,New York (1964).
535 Kurzfassung - Koppelgetriebe in Get,ten und f~r Steuerungen m~ssen optimal hinsichtlich der Empfindllchkeit ausgelegt werden, d. h., das Abtriebsglied mu@ auf kleinste Antriebsbewegungen reagieren. Gleichzeitig mu@ das Getriebe gute Ubertragungseigenschaften besitzen. Eine Methode f~r ~ie Synthese eines Viergelenkgetriebes mit dem genannten Eigenschaften fur drel Pr~zlsionpunkte wird erl~utert.
Pergamon Press. Printed in Great Britain
Kinematic Design of Four-bar Function-Generators with Optimum Sensitivity A. C. Rao¢
Received on 27 November 1974 Abstract
In linkages used for instrumentation and controls sensitivity should be optimum so that the output link responds well, even for very small changes in the input. Also it is most desirable to incorporate good transmission characteristics in the linkage. A method to synthesise four-bar linkages for three separate precision points incorporating the above characteristics is explained in this paper. Introduction SENSITIVITYcan be defined as the ratio of the change in output to a corresponding small change in input. Linkages must be designed for optimum sensitivity to respond well, even for extremely small changes in the input. For a linkage which generated a function Y - - f ( x ) S = Theoretical sensitivity = ~Y = dy
8x
dx= f'(x)"
(1)
The designer must also keep in mind, at the design stage, the quality of motion transmission. Transmission angle is indicative of the quality of motion transmission. The best transmission angle is 90° and it should deviate as little as possible from 90° during a motion cycle. Linkages with minimum transmission angles of 40° and above are expected to give satisfactory performance.
Method The problem is to determine the link lengths of a four-bar linkage which is required to generate, at three precision points, Y = f(x), xs -< x -< xt with optimum sensitivity and good transmission characteristics. The solution involves writing down the displacement equation of the form of eqn (1) and replacing the angle a by sensitivity, crank angles, link lengths, etc. The resulting equation can be expressed, following Freudenstein[2], in terms of the ratios of link lengths, crank angles and sensitivity. In the usual course, three simultaneous equations are sufficient to solve the problem, i.e. to get three link ratios. However, from the viewpoint of good transmission characteristics, the transmission angle in the second phase is taken as 90°. This leads to a relation, eqn (12), which expresses one of the link ratios in terms of the crank angles, sensitivity and ranges of motion. The starting angles and the ranges of motion if chosen arbitrarily give directly one of the link ratios, eqn (12), and then the linkage can be designed to satisfy the given function only for any two values of the independent variable. Since the problem is concerned with three precision points, one of the angles, say, the angle of the input link corresponding to first precision point is not chosen arbitrarily. The above mentioned link ratio is then expressed in terms of this unknown angle, so necessitating three simultaneous equations for complete solution, as explained below. For a four-bar function generator, the theoretical or optimum sensitivity (eqn 1), can be related, in the following manner, to the ratio of angular velocities of output and input links: dy
dx
(2)
?Professorof MechanicalEngineering,Govt. EngineeringCollege,Ujjain, M.P., India.
531
532
Let x1, x2 and x3 be the three precision points and yl, y2 and y3be the corresponding values of y. Since the linkage can accept the input in angles only, let angles 0 and 4 represent the input x and output y respectively. Assuming a linear relationship between x and 0, and y and &, the following expressions can be written: x-x1 -=-
Ax
Y-Y1 -=
AY
e - 8,
(24
he ’
4-h 4
’
where Ax, by, A& AC#J are the ranges of x, y, 9 and 4. The ranges are defined by:
where the subscripts f and s refer to final and starting values. Angles 0 and C$correspond to any phase of the linkage within the range of motion. By differentiating eqns (2a) and substitution in (2) we obtain:
=- 04
s=
4,
(3)
o4 6 sin(a!-O)=S -=-. w2 d sin(cu-4) q’
(4)
02
.
where
For a four-bar linkage[l],
Expanding and rearranging,
bq sina cos$--cosO dS [
1
=cosa! sin+-asinbq
[
1
and
bq sin . 0 sin C#I - do > tan ff = bq ’ coscp--$OSe > (
(
(5)
Therefore,
1,2, where k, = i
533
d ×
4
b 2
Figure 1. Referring to Fig. 1, it can be seen that a =(6-~)...
(6)
w h e r e / z is the transmission angle. Also b cos 0 + c c o s a - d
cos 6 = a
or (7)
k, cos 0 + k2 cos a - k3 = cos 6, where, k , = ~ b;
k: = ~c
k 3 = ~a
and
Based on eqn (7), the following equations corresponding to three precision points can be written: k l c o s 0~+ k2. A - k3 = cos (b~
(8)
k, cos (0, + A0,) + k2. B - G = cos 62
(9)
k, cos (0, + A02) + k~. C - G = cos 63
(10)
where
klq
cos 6 , - - ~ - cos 0, A =
. kl2q 2 2k,q 1+ ~
] 112
S~- cos (0, - 6 0
]
cos 62 - ~ q cos (0, + a0,)
B =
1 _* - k,2q ~ f - - ~ -2klq 2 cos .~ to, + A01 _ (~2)11/'2 cos
C =
I
6~ --~q cos (0, +A02) 3
, k12q 2 2Gq
1 1- ~
- ~
COS (0, + A02
-
l~3)J 1/2
534
A01 = 05 - 01 A02 = 03 - 05 S, = i f ( x ) . . . . .
s2 = f'(x)
....
and $3 = f ' ( x ) . . . . •
For the output link, the angle 4), corresponding to the first precision point, and the range of motion A~b, may be chosen arbitrarily; ~b2 and 4~3 then follow from (2a). In case of the input link A0, the range of motion may be chosen arbitrarily. No such arbitrary choice is made for 0r, the angle corresponding to first precision point, which is left to be determined from eqn (15), to follow, for the reasons that will be evident from eqn (12). The angles 02 and 03 can then be expressed in terms of 01 with the help of relations (2a). The transmission angle for the phase of the linkage corresponding to second precision point may be assumed to be 90 ° and then by using the relation a = ( 4 ' - / z ) and eqn (5), tan a2 = tan (02 - 90) klq
sin 4'5 - ~ =
klq
sin (01 + A0D
cos 4'~--~- cos (01 +~01)
= p' say.
from which,
kr =
$2 (p cos 4'2 - sin 4') q [p cos (0, + A01) -'sin (0r + A01)]"
(12)
Subtracting eqn (8) from eqn (9) and eqn (10) from eqn (9) the following equations are obtained kl{cos (01 + A01)- cos 01}+ k z ( B - A ) = cos 4 ' : - cos 4',
(13)
kl{cos (0r + A0r) - cos (0r + A02)} + k~(B - C) = cos 4'2 - cos ~b~
(14)
and
Using eqns (13) and (14) the constant ks can be eliminated to give, k l [ ( B - C){cos (01 + A00 - cos 01} - (B - A){cos (0r + A0]) - cos (01 + A0:)}]
-~"[(B -- C) (COS (~2 -- COS ~bl) - (B - A) (cos 4'~ - cos $3)]
(15)
Replacing kl from eqn (12) it may be noted that eqn (15) contains only one unknown term 0,. The eqn (15) may be solved, say, graphically to give 01. Once 01 is obtained kl and hence k2 and k3 can be determined. Since k,, k2 and k3 are the ratios, one of the dimensions, e.g. frame or input link may be assumed to get the other dimensions. If the dimensions so obtained are not satisfactory calculations may be repeated with a different starting value of 4' or ranges A0 and A4'.
References [1] HINKLE R. T., Kinematics of Machines, 2nd Edn, p. 113, Prentice-Hall, New York. [2] FREUDENSTEINF., Approximate synthesis of four-bar linkages. ASME Transactions 77, 853 (1955). [3l HARTENBERG R. S. and DENAVITJ., Kinematic Synthesis of Linkages. McGraw-Hill,New York (1964).
535 Kurzfassung - Koppelgetriebe in Get,ten und f~r Steuerungen m~ssen optimal hinsichtlich der Empfindllchkeit ausgelegt werden, d. h., das Abtriebsglied mu@ auf kleinste Antriebsbewegungen reagieren. Gleichzeitig mu@ das Getriebe gute Ubertragungseigenschaften besitzen. Eine Methode f~r ~ie Synthese eines Viergelenkgetriebes mit dem genannten Eigenschaften fur drel Pr~zlsionpunkte wird erl~utert.