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Geometric computing: Analogue-simulation of a linkage
Int. J. Mech. Sci.
Pergamon Press Ltd. 1965. Vol. 7, pp. 595-601. Printed in Great Britain
GEOMETRIC
COMPUTING: ANALOGUE-SIMULATION OF A LINKAGE
F . ERSK1NE CROSSLEY School of Mechanical Engineering, Georgia I n s t i t u t e of Technology, A t l a n t a , Georgia, U.S.A. (Received
11 A p r i l
1964)
S u m m a r y - - N e w circuits for t h e electronic analogue c o m p u t e r are described, for t h e s i m u l a t i o n of t h e m o t i o n s of t h e slider-crank a n d c r a n k - a n d - s w i n g i n g - a r m mechanisms. These m a k e use of t h e idea t h a t one or m o r e m e m b e r s m a y be considered to be elastic. T h e r e b y it is possible also to m e a s u r e t h e d y n a m i c forces occurring in these links. 1. I N T R O D U C T I O N
TrmEE different circuits have recently been described b y Lenk 1, Keller ~ and the author 8, b y which the path of any coupler-point or the angular motions of any member of either a slider-crank or a four-bar crank-and-swinging-arm mechanism can be simulated on the electronic analogue computer. Two new circuits are here described which achieve their stability from the geometrical concept that the connecting rods are slightly elastic, and that it is this stiff elasticity which constrains the mechanism. 2. T H E
CIRCLE
GENERATOR
A f u n d a m e n t a l idea for representing k i n e m a t i c a n d a n a l y t i c g e o m e t r y b y t h e oscilloscope or t h e x . y p l o t t i n g table, is t h e Lissajous figure, or phase portrait. A c r a n k of radins a r o t a t i n g w i t h c o n s t a n t a n g u l a r v e l o c i t y to projects on t h e x and y coordinate axes dimensions which v a r y h a r m o n i c a l l y : a z ---- a c o s t o t
a~ = a sin tot
(1)
T h e s e q u a n t i t i e s can be p r o d u c e d electronically b y a h a r m o n i c oscillator. I n its s i m p l e s t f o r m this appears as in Fig. 1. Two p o t e n t i o m e t e r s a d j u s t e d to e x a c t l y equal
l
~
Oy
l.C.- a
FIG. 1. The circle generator. v a l u e are necessary for t h e p r o d u c t i o n of a good circle. A c t u a l l y some d a m p i n g exists in all circuits of this simplified f o r m ; i m p r o v e d circuits t h a t m a y be used are discussed b y several a u t h o r s of t e x t s on analogue c o m p u t e r applications4, s. T h e y either consist of a d d i n g n e g a t i v e d a m p i n g a n d diode clipping, or t h e y are a r r a n g e d to follow t h e circular limit 595
596
F. ERSKINE CROSSLEY
cycle of a n o n - l i n e a r e q u a t i o n w h i c h is s o m e w h a t a k i n to a v a n d e r Pol equation2:
(2)
:i} - a ( ~ 2 b 2 - c o 2 y 2 - - i12 ) y + oo 2 y = 0
3. T H E
SLIDER-CRANK
MECHANISM
I n Fig. 2(a) c o n s i d e r t h a t t h e c r a n k O A r o t a t e s a t c o n s t a n t s p e e d a b o u t its fixed p i v o t O, w h i c h is also t h e origin of a c a r t e s i a n c o o r d i n a t e s y s t e m . T h e c o u p l e r A B (while a c t u a l l y rigid) m a y b e c o n c e i v e d to be l o n g i t u d i n a l l y elastic, so t h a t t h e p o i n t B is able to g e t o u t of its " p r o p e r " (i.e. r i g i d - b o d y ) position, a n d t h e i n s t a n t a n e o u s l e n g t h A B = b m a y n o t e x a c t l y agree w i t h t h e p r o p e r (rigid) l e n g t h b*. L e t t h e c o n s t a n t d i m e n s i o n s a, b*, e a n d t h e v a r i a b l e s xR, 0 = wt, be as s h o w n in Fig. 2(a).
I~
x.
"l
(o) K
e
(b)
!
l[
T
,o
a,)
FIG. 2. S i m u l a t i o n of t h e slider-crank. T h e n b y p r o j e c t i o n , t h e i n s t a n t a n e o u s l e n g t h of t h e c o u p l e r is defined b y t h e r e l a t i o n b = ~/[(x/~ -- a , ) 2 + (a~ - e) *]
(3)
I n b u i l d i n g u p a n electronic b l o c k d i a g r a m , see Fig. 2(b), for a n a n a l o g u e c o m p u t e r , t h e q u a n t i t i e s a , a n d a , in t h i s e x p r e s s i o n are d e r i v a b l e f r o m t h e h a r m o n i c g e n e r a t o r a l r e a d y
Geometric computing: analogue-simulation of a linkage
597
mentioned. Quantity e is a constant. Therefore it is possible to produce a voltage proportional to b, if the variable length xB could be obtained b y feedback. We conceive t h a t the difference between this instantaneous length b and the proper length b* of the coupler is the extension in length of the hypothetical spring. I f the stiffness of this spring is k, the spring force F b = It(b- b*)
(4)
and the component of this in the ( - x) direction is F_ x = F b ( x B b a z )
The denominator b is a variable, so t h a t this will entail a division in the computer; however, in view of the very high value of k t h a t will be used, we prefer to substitute the constant b* for b in the denominator. Thus we m a y write: k by (b - b*) (xB - a~) = - MS~B
(5)
where M is the mass of the slider and 5~ its acceleration. With two integrations the variable Xs m a y then be produced; it is the quantity desired for insertion in equation (3), and also in equation (5). One modification is however immediately necessary, for the mass M of the slider is attached only by a spring, and perpetual oscillation about its mean position is thus a possibility. To eliminate this, damping must be provided to remove this energy. The circuit evolved (see Fig. 2(b)) represents the relationship k
Mb* (b-b*) (x-ax)
= -&-yx
(6)
in which k / M b * is made as large as practicable, and y is a damping factor just big enough to eliminate any tendency to hunt. 4. C O U P L E R
CURVES
In this circuit there happen to be produced the quantities (x B - a ~ ) = b* cos~b -- b~ ( a ~ - e ) = b* sin~b -- b~
(7)
and these are useful in the production of coupler curves by the computer. Fig. 2 shows ma arbitrary point K of the coupler plane. This point can be defined by constant coordinates (~, 'q) with respect to a rectangular system attached to the coupler, with origin in the pivot A, and with ~-axis aligned with A B . The point K is then located in the (x, y) reference plane by the equations: x K = a ~ ÷ ~:cos¢+'q sin~b YK = a ~ - ~sin~b+'q cos~b
(8)
Let Pl = ~/b* and P2 = "q/b* define the proportional position of the chosen point K, with respect to the coupler length b*. Then equation (8) converts to XK = az + pl bz + pz b~ YK = a~-- pl by+P2 bx (9) and this allows coupler curves for any point K ( p l , P2) to be drawn by the circuit of Fig. 2, with the addition of a few potentiometers and summers.
5. S I M U L A T I O N OF THE CRANK-AND-SWINGING-ARM MECHANISM The idea of the elastic coupler m a y also be applied to build an analogue computer circuit for the crank-and-swinging-arm mechanism. Consider in Fig. 3(a) t h a t the coupler is subject to elongation, when angle ~bis slightly smaller than it should be in the rigid-body mechanism. The spring force due to this elongation will cause a m o m e n t about the pivot B 0, which will accelerate the arm B B o, to bring it into its correct angular position.
598
F. ERSKINE CROSSLEY
The e q u a t i o n s on which the circuit is built are as follows. I n Fig. 3(a), the p o i n t B m a y be located b y the projections along the x- and y-axes of the sides a, b, c and d of t h e quadrilateral: a~+c~
= b~+dx
(10}
au+c ~ = b~+d~
The i n s t a n t a n e o u s length c of the coupler is t h e n g i v e n by c = ~/(c~+c~)
(11)
where c x = b~+dx-a
x
c~ = b y -{- d~ - a~
B
-d
"llllll/
__~/11~1 I ~
FxG. 3. S i m u l a t i n g circuit for a four-bar mechanism. I t is p r e s u m e d a t this stage of building up t h e c o m p u t e r circuit, see Fig. 3(b), t h a t the two voltages p r o p o r t i o n a l to bx = b cos ~b and to b~ = b sin ~b are obtainable f r o m a served r i v e n resolver, a l t h o u g h it is not y e t shown where t h e driving voltage for the s e r v e - m o t o r is obtained.
Geometric computing: analogue-simulation of a linkage
599
The difference between the instantaneous length c and rigid-body length c* of t h e coupler represents the extension of the hypothetical spring, so t h a t the spring force is
k(c-c*) Resolving this into its x and y components, one obtains for its m o m e n t about B 0 the following expression : Accelerating torque -- I ~ ~b~ -- k(c - c*) (cx by - c~ bx) c*
(12)
in which the average value c* has been substituted for c in the denominator, in order to avoid a division. By two integrations the controlling voltage for the servo is then easily obtained. I n this part it is well to add damping by a feedback line around the first integrator. Note t h a t one desires to make the factor ]¢/IB c* extremely large in order t h a t the error ( c - c*) occurring shall be minimized: instead of the potentiometer shown in Fig. 3, set to an arbitrary value, an amplification of 10 or 100 is therefore to be preferred. This, however, has the tendency to overload these amplifiers initially, unless the servo-motor is carefully set to its initial equilibrium position ~b0(t = 0) by an initial condition on the second integrator. By connecting the additional elements in the circuit similar to equation (91 this circuit will produce excellent coupler curves. I t is worth noting also that the circuits of Fig. 3 ma y also be used for plotting the output angle ~b of the driven member against the input angle 4 ; t h a t is to say, plotting the function generated by the linkage. I n Fig. 3 some m a y prefer to substitute two diode function generators, preset to sine and cosine curves, for the servo and resolver. The operation of the whole circuit can then be much faster, and coupler curves m a y be displayed on an oscilloscope screen.
6. P R A C T I C A L
IMPROVEMENTS
IN CIRCUITRY
The circuit of Fig. 3 exhibits a strict adherence to the equations of the governing geometry. The electronic squaring and multiplying elements however, especially if t h e y work on the diode parabolic principle, involve approximations, and these in their t u r n allow simplifications of the circuit. First, if the circuit is to simulate sufficiently closely the action of the real rigld-linked mechanism, the value of c s must remain sensibly constant and equal to c *s. The squarerooting element is therefore scarcely used: if moreover this is the diode sort, the square root varies as a linear approximation. Elimination of this element therefore actually will improve the accuracy of the circuit. For similar reasons the two multipliers used to produce the factor (cx b y - c~ b®) m a y be removed, so long as this factor does not change sign. Bu t the sign cannot change in any mechanism, since it implies t h a t the two links c and b come into alignment and the transmission angle becomes 0 ° or 180 °. Thirdly, two integrators are not necessary to drive the servo to correspond with the dependent variable ~b. Only one integration of the error signal is better, causing less noise in the feedback line. The abbreviated and improved form of Fig. 3 then appears as shown in Fig. 4. One further improvement is to be noticed. I f the two squaring elements are of the diode form, it is better t h a t the output of these elements should be as nearly as possible to their full voltage. To this purpose two inverters are shown ahead of the squarers, to normalize their inputs, multiplying each input by the factor 1/c.
7. F U R T H E R
APPLICATIONS:
SIX-BAR
LINKAGES
I t is a simple m a t t e r to simulate a six-bar linkage, to produce either coupler curves of the fifth member, or the transfer function generated. Suppose the coupler point K of a four-bar is the point of a t t a c h m e n t of the fifth link; the fifth and sixth together form a binary dyad. On the computer circuit board the four-bar is first plugged in according to
600
F. ERSKINE CROSSLEY
(say) Fig. 4. Then a second similar unit is also connected, except that instead of the actuating circle of point A, the coordinates of point K obtained from the first circuit are the driving voltages for the second circuit.
~
ti]
___~'Cx -Qz
•
,
I.C.=%
F
-,
FIG. 4. I m p r o v e d circuit for simulating the crank-and-swinging-arm mechanism. 8. S P H E R I C A L
MECHANISMS
Meyer zur Capellen e has shown t h a t the motions of the spherical crank-and-swingingarm or slider-crank mechanisms are identical to the motions of the plane mechanism of similar proportion, provided t h a t the equivalent plane "crankpin" A moves in an elliptical path. I n Figs. 3 or 4 such a change requires only a change in the setting of the two potentiometers co in the driving oscillator.
Acknowledgements--This work was first described in lectures given to the Advanced Science Seminar held at Yale University in J u l y 1963, of which the author was director, and which was sponsored by the National Science Foundation under grant G-770. The author expresses his great appreciation to the Foundation for this generous support. The circuits themselves were first tested by Mr. R a y m o n d Carlson, to whom the author would like to express his thanks for his most helpful assistance. Furthermore, the author acknowledges the kind suggestions of Dr. R. E. Keller of Stanford, which have come through private correspondence. REFERENCES 1. E. W. LE~K, "Instrumentelle und elektrische Verfahren zur Erzeugung und Aufzeichnung von Koppelkurven". Konstruktion Nov. 1962. 2. R. E. KELLER, "Mechanisms Design by Electronic Analog Computer". Trans. 7th Conf. on Mechanisms, Purdue Univ., October 1962. Penton Publ. Co., Cleveland, Ohio. 3. F. E. CROSSLY.Y, "Die Nachbildung eines mechanischen Kurbelgetriebes mittels eines elektronischen Analogrechners". Feinwerktechnik 67, J u n e 1963.
Geometric computing: analogue-simulation of a linkage
601
4. W. GILOI and R. LAVBER, Analogrechnen. Springer Verlag, Berlin, Goettingen, Heidelberg (1963). 5. A. E. ROGERS and T. W. C o l o n Y , Analog Computation in Engineering Design, McGraw-Hill, New York (1960). 6. W. MEYER Z ~ CAPELLEN, "Ueber elliptische Kurbelschleifen". Werkstatt u. Betrieb, 91, 723 (1958).
Pergamon Press Ltd. 1965. Vol. 7, pp. 595-601. Printed in Great Britain
GEOMETRIC
COMPUTING: ANALOGUE-SIMULATION OF A LINKAGE
F . ERSK1NE CROSSLEY School of Mechanical Engineering, Georgia I n s t i t u t e of Technology, A t l a n t a , Georgia, U.S.A. (Received
11 A p r i l
1964)
S u m m a r y - - N e w circuits for t h e electronic analogue c o m p u t e r are described, for t h e s i m u l a t i o n of t h e m o t i o n s of t h e slider-crank a n d c r a n k - a n d - s w i n g i n g - a r m mechanisms. These m a k e use of t h e idea t h a t one or m o r e m e m b e r s m a y be considered to be elastic. T h e r e b y it is possible also to m e a s u r e t h e d y n a m i c forces occurring in these links. 1. I N T R O D U C T I O N
TrmEE different circuits have recently been described b y Lenk 1, Keller ~ and the author 8, b y which the path of any coupler-point or the angular motions of any member of either a slider-crank or a four-bar crank-and-swinging-arm mechanism can be simulated on the electronic analogue computer. Two new circuits are here described which achieve their stability from the geometrical concept that the connecting rods are slightly elastic, and that it is this stiff elasticity which constrains the mechanism. 2. T H E
CIRCLE
GENERATOR
A f u n d a m e n t a l idea for representing k i n e m a t i c a n d a n a l y t i c g e o m e t r y b y t h e oscilloscope or t h e x . y p l o t t i n g table, is t h e Lissajous figure, or phase portrait. A c r a n k of radins a r o t a t i n g w i t h c o n s t a n t a n g u l a r v e l o c i t y to projects on t h e x and y coordinate axes dimensions which v a r y h a r m o n i c a l l y : a z ---- a c o s t o t
a~ = a sin tot
(1)
T h e s e q u a n t i t i e s can be p r o d u c e d electronically b y a h a r m o n i c oscillator. I n its s i m p l e s t f o r m this appears as in Fig. 1. Two p o t e n t i o m e t e r s a d j u s t e d to e x a c t l y equal
l
~
Oy
l.C.- a
FIG. 1. The circle generator. v a l u e are necessary for t h e p r o d u c t i o n of a good circle. A c t u a l l y some d a m p i n g exists in all circuits of this simplified f o r m ; i m p r o v e d circuits t h a t m a y be used are discussed b y several a u t h o r s of t e x t s on analogue c o m p u t e r applications4, s. T h e y either consist of a d d i n g n e g a t i v e d a m p i n g a n d diode clipping, or t h e y are a r r a n g e d to follow t h e circular limit 595
596
F. ERSKINE CROSSLEY
cycle of a n o n - l i n e a r e q u a t i o n w h i c h is s o m e w h a t a k i n to a v a n d e r Pol equation2:
(2)
:i} - a ( ~ 2 b 2 - c o 2 y 2 - - i12 ) y + oo 2 y = 0
3. T H E
SLIDER-CRANK
MECHANISM
I n Fig. 2(a) c o n s i d e r t h a t t h e c r a n k O A r o t a t e s a t c o n s t a n t s p e e d a b o u t its fixed p i v o t O, w h i c h is also t h e origin of a c a r t e s i a n c o o r d i n a t e s y s t e m . T h e c o u p l e r A B (while a c t u a l l y rigid) m a y b e c o n c e i v e d to be l o n g i t u d i n a l l y elastic, so t h a t t h e p o i n t B is able to g e t o u t of its " p r o p e r " (i.e. r i g i d - b o d y ) position, a n d t h e i n s t a n t a n e o u s l e n g t h A B = b m a y n o t e x a c t l y agree w i t h t h e p r o p e r (rigid) l e n g t h b*. L e t t h e c o n s t a n t d i m e n s i o n s a, b*, e a n d t h e v a r i a b l e s xR, 0 = wt, be as s h o w n in Fig. 2(a).
I~
x.
"l
(o) K
e
(b)
!
l[
T
,o
a,)
FIG. 2. S i m u l a t i o n of t h e slider-crank. T h e n b y p r o j e c t i o n , t h e i n s t a n t a n e o u s l e n g t h of t h e c o u p l e r is defined b y t h e r e l a t i o n b = ~/[(x/~ -- a , ) 2 + (a~ - e) *]
(3)
I n b u i l d i n g u p a n electronic b l o c k d i a g r a m , see Fig. 2(b), for a n a n a l o g u e c o m p u t e r , t h e q u a n t i t i e s a , a n d a , in t h i s e x p r e s s i o n are d e r i v a b l e f r o m t h e h a r m o n i c g e n e r a t o r a l r e a d y
Geometric computing: analogue-simulation of a linkage
597
mentioned. Quantity e is a constant. Therefore it is possible to produce a voltage proportional to b, if the variable length xB could be obtained b y feedback. We conceive t h a t the difference between this instantaneous length b and the proper length b* of the coupler is the extension in length of the hypothetical spring. I f the stiffness of this spring is k, the spring force F b = It(b- b*)
(4)
and the component of this in the ( - x) direction is F_ x = F b ( x B b a z )
The denominator b is a variable, so t h a t this will entail a division in the computer; however, in view of the very high value of k t h a t will be used, we prefer to substitute the constant b* for b in the denominator. Thus we m a y write: k by (b - b*) (xB - a~) = - MS~B
(5)
where M is the mass of the slider and 5~ its acceleration. With two integrations the variable Xs m a y then be produced; it is the quantity desired for insertion in equation (3), and also in equation (5). One modification is however immediately necessary, for the mass M of the slider is attached only by a spring, and perpetual oscillation about its mean position is thus a possibility. To eliminate this, damping must be provided to remove this energy. The circuit evolved (see Fig. 2(b)) represents the relationship k
Mb* (b-b*) (x-ax)
= -&-yx
(6)
in which k / M b * is made as large as practicable, and y is a damping factor just big enough to eliminate any tendency to hunt. 4. C O U P L E R
CURVES
In this circuit there happen to be produced the quantities (x B - a ~ ) = b* cos~b -- b~ ( a ~ - e ) = b* sin~b -- b~
(7)
and these are useful in the production of coupler curves by the computer. Fig. 2 shows ma arbitrary point K of the coupler plane. This point can be defined by constant coordinates (~, 'q) with respect to a rectangular system attached to the coupler, with origin in the pivot A, and with ~-axis aligned with A B . The point K is then located in the (x, y) reference plane by the equations: x K = a ~ ÷ ~:cos¢+'q sin~b YK = a ~ - ~sin~b+'q cos~b
(8)
Let Pl = ~/b* and P2 = "q/b* define the proportional position of the chosen point K, with respect to the coupler length b*. Then equation (8) converts to XK = az + pl bz + pz b~ YK = a~-- pl by+P2 bx (9) and this allows coupler curves for any point K ( p l , P2) to be drawn by the circuit of Fig. 2, with the addition of a few potentiometers and summers.
5. S I M U L A T I O N OF THE CRANK-AND-SWINGING-ARM MECHANISM The idea of the elastic coupler m a y also be applied to build an analogue computer circuit for the crank-and-swinging-arm mechanism. Consider in Fig. 3(a) t h a t the coupler is subject to elongation, when angle ~bis slightly smaller than it should be in the rigid-body mechanism. The spring force due to this elongation will cause a m o m e n t about the pivot B 0, which will accelerate the arm B B o, to bring it into its correct angular position.
598
F. ERSKINE CROSSLEY
The e q u a t i o n s on which the circuit is built are as follows. I n Fig. 3(a), the p o i n t B m a y be located b y the projections along the x- and y-axes of the sides a, b, c and d of t h e quadrilateral: a~+c~
= b~+dx
(10}
au+c ~ = b~+d~
The i n s t a n t a n e o u s length c of the coupler is t h e n g i v e n by c = ~/(c~+c~)
(11)
where c x = b~+dx-a
x
c~ = b y -{- d~ - a~
B
-d
"llllll/
__~/11~1 I ~
FxG. 3. S i m u l a t i n g circuit for a four-bar mechanism. I t is p r e s u m e d a t this stage of building up t h e c o m p u t e r circuit, see Fig. 3(b), t h a t the two voltages p r o p o r t i o n a l to bx = b cos ~b and to b~ = b sin ~b are obtainable f r o m a served r i v e n resolver, a l t h o u g h it is not y e t shown where t h e driving voltage for the s e r v e - m o t o r is obtained.
Geometric computing: analogue-simulation of a linkage
599
The difference between the instantaneous length c and rigid-body length c* of t h e coupler represents the extension of the hypothetical spring, so t h a t the spring force is
k(c-c*) Resolving this into its x and y components, one obtains for its m o m e n t about B 0 the following expression : Accelerating torque -- I ~ ~b~ -- k(c - c*) (cx by - c~ bx) c*
(12)
in which the average value c* has been substituted for c in the denominator, in order to avoid a division. By two integrations the controlling voltage for the servo is then easily obtained. I n this part it is well to add damping by a feedback line around the first integrator. Note t h a t one desires to make the factor ]¢/IB c* extremely large in order t h a t the error ( c - c*) occurring shall be minimized: instead of the potentiometer shown in Fig. 3, set to an arbitrary value, an amplification of 10 or 100 is therefore to be preferred. This, however, has the tendency to overload these amplifiers initially, unless the servo-motor is carefully set to its initial equilibrium position ~b0(t = 0) by an initial condition on the second integrator. By connecting the additional elements in the circuit similar to equation (91 this circuit will produce excellent coupler curves. I t is worth noting also that the circuits of Fig. 3 ma y also be used for plotting the output angle ~b of the driven member against the input angle 4 ; t h a t is to say, plotting the function generated by the linkage. I n Fig. 3 some m a y prefer to substitute two diode function generators, preset to sine and cosine curves, for the servo and resolver. The operation of the whole circuit can then be much faster, and coupler curves m a y be displayed on an oscilloscope screen.
6. P R A C T I C A L
IMPROVEMENTS
IN CIRCUITRY
The circuit of Fig. 3 exhibits a strict adherence to the equations of the governing geometry. The electronic squaring and multiplying elements however, especially if t h e y work on the diode parabolic principle, involve approximations, and these in their t u r n allow simplifications of the circuit. First, if the circuit is to simulate sufficiently closely the action of the real rigld-linked mechanism, the value of c s must remain sensibly constant and equal to c *s. The squarerooting element is therefore scarcely used: if moreover this is the diode sort, the square root varies as a linear approximation. Elimination of this element therefore actually will improve the accuracy of the circuit. For similar reasons the two multipliers used to produce the factor (cx b y - c~ b®) m a y be removed, so long as this factor does not change sign. Bu t the sign cannot change in any mechanism, since it implies t h a t the two links c and b come into alignment and the transmission angle becomes 0 ° or 180 °. Thirdly, two integrators are not necessary to drive the servo to correspond with the dependent variable ~b. Only one integration of the error signal is better, causing less noise in the feedback line. The abbreviated and improved form of Fig. 3 then appears as shown in Fig. 4. One further improvement is to be noticed. I f the two squaring elements are of the diode form, it is better t h a t the output of these elements should be as nearly as possible to their full voltage. To this purpose two inverters are shown ahead of the squarers, to normalize their inputs, multiplying each input by the factor 1/c.
7. F U R T H E R
APPLICATIONS:
SIX-BAR
LINKAGES
I t is a simple m a t t e r to simulate a six-bar linkage, to produce either coupler curves of the fifth member, or the transfer function generated. Suppose the coupler point K of a four-bar is the point of a t t a c h m e n t of the fifth link; the fifth and sixth together form a binary dyad. On the computer circuit board the four-bar is first plugged in according to
600
F. ERSKINE CROSSLEY
(say) Fig. 4. Then a second similar unit is also connected, except that instead of the actuating circle of point A, the coordinates of point K obtained from the first circuit are the driving voltages for the second circuit.
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FIG. 4. I m p r o v e d circuit for simulating the crank-and-swinging-arm mechanism. 8. S P H E R I C A L
MECHANISMS
Meyer zur Capellen e has shown t h a t the motions of the spherical crank-and-swingingarm or slider-crank mechanisms are identical to the motions of the plane mechanism of similar proportion, provided t h a t the equivalent plane "crankpin" A moves in an elliptical path. I n Figs. 3 or 4 such a change requires only a change in the setting of the two potentiometers co in the driving oscillator.
Acknowledgements--This work was first described in lectures given to the Advanced Science Seminar held at Yale University in J u l y 1963, of which the author was director, and which was sponsored by the National Science Foundation under grant G-770. The author expresses his great appreciation to the Foundation for this generous support. The circuits themselves were first tested by Mr. R a y m o n d Carlson, to whom the author would like to express his thanks for his most helpful assistance. Furthermore, the author acknowledges the kind suggestions of Dr. R. E. Keller of Stanford, which have come through private correspondence. REFERENCES 1. E. W. LE~K, "Instrumentelle und elektrische Verfahren zur Erzeugung und Aufzeichnung von Koppelkurven". Konstruktion Nov. 1962. 2. R. E. KELLER, "Mechanisms Design by Electronic Analog Computer". Trans. 7th Conf. on Mechanisms, Purdue Univ., October 1962. Penton Publ. Co., Cleveland, Ohio. 3. F. E. CROSSLY.Y, "Die Nachbildung eines mechanischen Kurbelgetriebes mittels eines elektronischen Analogrechners". Feinwerktechnik 67, J u n e 1963.
Geometric computing: analogue-simulation of a linkage
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4. W. GILOI and R. LAVBER, Analogrechnen. Springer Verlag, Berlin, Goettingen, Heidelberg (1963). 5. A. E. ROGERS and T. W. C o l o n Y , Analog Computation in Engineering Design, McGraw-Hill, New York (1960). 6. W. MEYER Z ~ CAPELLEN, "Ueber elliptische Kurbelschleifen". Werkstatt u. Betrieb, 91, 723 (1958).