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A self-balancing crane
Mechanism and Machine Theory, 1974, Vol. 9, 359-366.
Pergamon Press. Printed in Great Britain
A Self-Balancing Crane To the memory of Malcolm Livingstone Urquhart.
C. F. Moppert* Abstract A simple mechanism is described which has the effect that an automatic displacement of a counterweight very nearly cancels the overturning moment caused by the load. This mechanism is then adapted to a hammerhead crane with movable trolley. An exact static analysis is given, variants are discussed and historical references made.
1. Basic model (Fig. 1) A HORIZONTALarm BD rests in its middle A on a tower. An arm AC which can swivel about A carries a pulley at its end C. The length r of the arm A C is equal to AD and AB. The cable starts at the winch W, goes o v e r a pulley at A to D, around the pulley at the hook E back to D then to B, to the pulley C supporting the counterweight K and back to a fixed point at B. At D there are two independent pulleys on the same axle, (for the details of the arrangement consult the figure).
Working action (1) Static. If there is no load L then the arm AC hangs vertically. If a load is picked up then the pulley C is lifted towards B. The load L starts being lifted at the m o m e n t the counterweight K is in the position to balance the load. F r o m then onwards the point C stays firm. If the load is put down, the point C first m o v e s down until AC is vertical. Only then can the load be r e m o v e d f r o m the hook. If the load L is small then the arm AC m o v e s only slightly f r o m the vertical position. The m a x i m u m load the crane can cope with is equal to the weight K of the counterweight, the latter being arbitrary. The bending m o m e n t , caused b y the load, acting on the arm BD about the point A is kept very small indeed. If the crane is built in such a way that the point A is the centre of gravity of this a r m (including pulleys and the weight of the sections of the rope along it), very little strutting will be necessary to keep it in a horizontal position. Furthermore, the t o w e r will be subjected almost entirely to compression. (2) Dynamic. The position of K reacts to forces of inertia as well as to static ones, i.e. C m o v e s out (in) if L is accelerated upwards (downwards). The balancing action works immediately: it is possible to cut the cable while the crane carries the m a x i m u m load and no overbalancing takes place (to m a k e this possible, the crane must be constructed in such a way that the arm AC can swing towards D). If the load is lowered by the crane at constant speed, onto a truck say, then at the m o m e n t of impact only the inertial force of the load together with part of the inertial force of the counterweight affect the platform of the truck. The weight of the load *Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia. 359
360 i
I
r-
X
c
-~
I
-W % I
Flgure I. The horizontal arms DA, AB of the crane have equal length r. The arm AC has length r also. It carries the c o u n t e r w e i g h t K at its end. This arm can swivel freely about A. The c o u n t e r w e i g h t will automatically assume a position such that the crane is in near balance.
proper only gradually builds up while the counterweight is lowered. We have investigated the corresponding mathematical problem but we feel it is not worth including here as complicating effects are likely to arise in a practical situation. (3) Practical. The action of the crane is not perfect due to friction in the pulleys and the cable. In lifting a load, the counterweight can slightly advance or stay behind the ideal position. It is a definite disadvantage that the load does not react immediately to the m o v e m e n t s of the winch. H o w e v e r , this "lagging behind" has an advantage: any load which is less than the m a x i m u m load can, while it is hanging on the crane, be moved up or down some distance using very little force (See Section 4).
2. Variants of the basic model 2.1 Counterweight The counterweight K can be placed anywhere along the arm AC or on an extension of it. If this is the case then the m a x i m u m load L is equal to a fraction or a multiple of K. 2.2 Ratio A D / A B The crane can be built for any given ratio AD/AB. The length of the movable arm AC must be equal to AB. If A D / A B = m / n with m, n integers, then the crane is always in near balance provided the load hangs on n cables and the connection of the points B and C is provided by m cables. It is always easy to lead the cable o v e r multiple pulleys to ensure that these conditions are satisfied. The counterweight K can again be placed anywhere along the arm AC. 2.3 H a m m e r h e a d crane We describe the h a m m e r h e a d version of the basic model, i.e. the model with A D / A B = 1. It is clear that this restriction is not necessary. The h a m m e r h e a d version makes use of the fact that the arm AC is under all
361 conditions under tension only and can therefore be replaced b y a cable. Suppose that the load is suspended f r o m a movable trolley D (Fig. 2). The crane is in balance provided the three distances AD, AB and AC are equal. It is shown in Fig. 2 that if the trolley D m o v e s in (out), so does the trolley B, both being connected to the endless loop P F G . The cable carrying the pulley C is connected to this same loop at P. If D m o v e s away f r o m A, P m o v e s towards A and as the distance A F is equal to the length PA + AC, the length AC is always equal to the length AB.
Tr°"e'D F I
_
_
.
.
.
.
--i
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I
Figure 2. Hammerhead version of the basic model. The trolleys B and D are always equidistant from A. The counterweight K is supported by the rope coming from the w h i n c h and by a cable P A C . This arrangement ensures that always DA ---AB = A C . The crane is in near balance independent of the load and of the position of the trolley. The h a m m e r h e a d crane is always in near balance, independent of the load and the position of the trolley. The luffing of the h a m m e r h e a d crane is horizontal only if the load is maximal (i.e. L = K ) . If the trolley D m o v e s then so does the trolley B and the length BC changes, h o w e v e r , the angle CAB remains constant. It follows that L m o v e s along a straight line while D m o v e s and that this line is roughly parallel to the line AC as long as the angle CAB is not too large.
3. Static equilibrium of the basic model The load L is suspended by two strands of the cable. Its tension t is therefore t = L [2. We denote b y k the c o m p o n e n t of the counterweight K along the suspending cables, i.e. t = k/2 and k = L. F r o m Fig. 3 we see that k / K = sin ( 9 0 - a ) / s i n (90 + a/2) = cos a / c o s a / 2 , thus K cos a = L cos a/2. We denote by d the m o m e n t about A, acting on the arm DB in the anticlockwise sense, thus d = L r - Kr cos a.
362 Moment
d
B A
k
I
90-a
Figure 3. The force K has component k acting on the rope. P u t t i n g )t = L / K , ~ = d / K , w e h a v e COS t~
x
(I)
cos a/2
= r(A - c o s or).
(2)
G i v e n )t, r e l a t i o n (1) d e t e r m i n e s a a n d t h u s ~ b y u s e o f r e l a t i o n (2). W e c a n e l i m i n a t e b y i n t r o d u c i n g u = c o s a/2. T h e n c o s a = 2u 2 - 1 a n d )t = (2u 2 - l)/u. It f o l l o w s t h a t u = (A + ~ ) / 4 . A s u is n o n - n e g a t i v e , w e c h o o s e t h e p o s i t i v e sign. S u b s t i t u t i n g this in ~ = r ( ( 2 u 2 - l)/u - (2u 2 - 1)) w e g e t 6 r
-
d Kr
-
x
;t 2 4
A~2+8 4
(3)
W e a r e i n t e r e s t e d in t h e b e h a v i o u r o f this f u n c t i o n if L v a r i e s b e t w e e n 0 a n d K, i.e. f o r 0 ~< X ~< 1. Its g r a p h is g i v e n in Fig. 4. T h e unit on t h e v e r t i c a l axis is t h e r e ten t i m e s t h e unit on t h e h o r i z o n t a l one. W e s e e t h a t 6/r = 0 f o r A = 0 a n d A = l a n d t h a t it r e a c h e s its m a x i m u m v a l u e o f 6/r = 0.0785 f o r A = 0.516. T h i s m e a n s t h a t o u r c r a n e is in p e r f e c t b a l a n c e f o r L = 0 a n d L = K o n l y a n d t h a t t h e i m b a l a n c e in t h e w o r s t c a s e o c c u r s r o u g h l y f o r L = K/2. It is t h e n less t h a n t h e i m b a l a n c e c a u s e d b y p u t t i n g 8 p e r c e n t o f t h e c o u n t e r w e i g h t K at D.
4. Vertical displacement of the load (basic model) L e t t h e w i n c h b e fixed, t h e l o a d h a n g i n g a d i s t a n c e x f r o m t h e p u l l e y D (Fig. 1). W e i n v e s t i g a t e t h e i n c r e m e n t a l v e r t i c a l f o r c e d K n e c e s s a r y to m o v e t h e l o a d b y an i n c r e m e n t dx. From Kcosct =Lcosa/2, it f o l l o w s t h a t - K s i n a da = c o s a / 2 dL1/2L sin a / 2 da; h e n c e K s i n a c o s a / 2 da = 1/2 K c o s a sin a / 2 da - c o s 2 a / 2 dL.
363
O-
o°d~ K-~ 0.06 0"040
-I'0
0"2
t_
0-516
I'O
K
Figure 4. Normalised momentum d / K r acting on the horizontal arm in the basic model as function of L / K . By shifting the support of the arm DB in the basic model by 2 per cent of r to the left of A (Fig. 6), the quantity d/Kr is the difference of the curve and the sloping straight line. In this way, the effectiveness of the mechanism is nearly doubled. A s t h e l e n g t h o f t h e r o p e r e m a i n s c o n s t a n t , w e h a v e 2x + 4 r sin a / 2 = c o n s t , a n d d x = - r c o s ct/2 d a . S t r a i g h t f o r w a r d c a l c u l a t i o n g i v e s t h e n dx
r
dL
2
K (tan a / 2 ) ( 3 + tan 2 c~/2)
(4)
w h e r e L, K a n d ot a r e l i n k e d t o g e t h e r b y (l). T h e f r a c t i o n on t h e r i g h t - h a n d side o f (4) b e c o m e s infinite f o r a = 0. T h i s c o r r e s p o n d s to t h e f a c t t h a t t h e s t r u c t u r e a l l o w s t h e n n o e x t e n s i o n o f t h e c a b l e . In o r d e r to i l l u s t r a t e t h e m e a n i n g o f t h e r e l a t i o n (4), w e c a l c u l a t e t h e c o r r e s p o n d i n g v a l u e s f o r a = 45 °. T h e n L / K = 0-765 a n d d x / r = 1.5 d L / K , i.e. t o d L = K / 1 0 0 c o r r e s p o n d s a d i s p l a c e m e n t d x o f a b o u t 1-5 p e r c e n t o f t h e length r. T h e r e l a t i o n (4) c o n n e c t i n g d x / r with d L / K h a s a n o t h e r i m p o r t a n t p r a c t i c a l applic a t i o n as it e n a b l e s us to s a y w i t h i n w h i c h limits x is d e t e r m i n e d b y v a r i a t i o n s o f L o r b y f r i c t i o n a l f o r c e s . W e h a v e p o i n t e d o u t e a r l i e r that it is a d i s a d v a n t a g e o f o u r c r a n e t h a t t h e c r a n e d r i v e r h a s n o t the i m m e d i a t e c o n t r o l o v e r x h e h a s with a c o n v e n t i o n a l c r a n e . T h e r e l a t i o n (4) is i l l u s t r a t e d in Fig. 5, u s i n g l o g a r i t h m i c scale f o r ( d x / r ) / ( d L / K ) .
5. Improvement of the basic model It is e v i d e n t f r o m Fig. 4 t h a t t h e m o m e n t d t a k e s o n l y p o s i t i v e v a l u e s . B y c h a n g i n g t h e d e s i g n in s u c h a w a y t h a t d is n e g a t i v e f o r L = 0 w e c a n k e e p Jdl b e t w e e n l o w e r b o u n d s . W e i n v e s t i g a t e t h e s i t u a t i o n w h e n t h e a r m A B is s u p p o r t e d at a p o i n t 0 at a d i s t a n c e (rr f r o m D w i t h tr a fixed p o s i t i v e n u m b e r < 1 (Fig. 6). T h e d e s i g n o f t h e b a s i c m o d e l is o t h e r w i s e u n c h a n g e d a p a r t f r o m t h e f a c t t h a t t h e p u l l e y l e a d i n g t h e r o p e f r o m W to D is n o w at 0. I n this c a s e w e h a v e d = o-rL - K((1 - t r ) r + r c o s ~ ) a n d this l e a d s e a s i l y to ~ / r = A - c o s a + (A + l)((r - 1) = A - A2/4 - A ~ 8 / 4 + (A + 1)(o" - 1). In o u r Fig. 4 w e h a v e t h e g r a p h o f t h e f u n c t i o n A - A2/4 - A ~ 8 / 4 . T h e f u n c t i o n (A + 1)(o~ - 1) is a s t r a i g h t line t h r o u g h t h e p o i n t s
364 20-15 I(~
5
I
0-5
I
b
0 o
pOo
90* I1
Figure 5. A change dL. of the load causes a change d x (Fig. 1). The relation of these quantities depends very much on the angle ~ (Fig. 3). D
r
,,I
_~
r
B
err
Figure 6. Improved version of the basic model. ( - 1, 0) and (0, o" - l). W e d r a w a line t h r o u g h ( - 1, 0) such that the deviation of o u r graph f r o m this line is minimal. As its intercept with the vertical axis is ~ 0.02 w e put 1 - tr = 0.02, i.e. tr = 0.98 (5) and we see that for this c h o i c e the value of I~/rl is surely < 0 . 0 5 . W e h a v e thus i m p r o v e d the p r e v i o u s value o f < 8 per cent to < 5 per cent. This i m p r o v e m e n t can clearly be carried o v e r to all other models.
6. History O u r c r a n e can be c o n s i d e r e d as a d e v e l o p m e n t of the ancient n o d d e r c r a n e w h i c h still can be seen in E g y p t (Fig. 7). In fact, the f a t h e r o f the n o d d e r and our c r a n e is the (historically later) letter b a l a n c e (Fig. 8). It is easily seen h o w the n o d d e r and o u r c r a n e strike t w o different c o m p r o m i s e s to a c h i e v e their end. F u r t h e r m o r e o n e sees that in our c r a n e s o m e imbalance is allowed in order to maintain simplicity of design and thus to keep d i s t u r b a n c e due to friction at a minimum.
365
Figure 7. The " n o d d e r crane", an ancient version of our crane. The pulley c o r r e s p o n d i n g to the pulley D in our crane moves along a circular arc.
I
I
Figure 8. A letter balan~,e. The wheel turns until the c o u n t e r w e i g h t compensates the load. Our crane as well as the n o d d e r crane can be considered as derivatives.
It would be surprising if our idea were new. I have been informed by Professor Hartenberg in Illinois that a similar construction is described in M. Ruehlmann's AUgemeine Maschinenlehre, 4. Band, p. 469 ff (Braunschweig 1875). In the crane described there, the counterweight is mounted on a trolley which moves on a rail forming a particular upward curve. This crane was designed by Jambille and exhibited at the international exhibition in Paris, 1867. Two final remarks. I have built several man-sized models of the basic crane with the variant AD/DB = 2 : l and a small model of the hammerhead crane. All these models work very well indeed. The idea as such is a very natural one: if a man lifts a weight with his right arm, he moves out the left one until he is in balance. It is reassuring to find our crane is imitating Nature.
Acknowledgement---I wish to express by thanks to Professor K. Hunt, Monash University, who enabled me to use the facilities of the School of Engineering. I am also greatly indebted to my friend Roger Hallett for encouraging me, building a model and trying to propagate the idea.
MMT Vol. 9, No. 3 - - F
366
Ein Kran der sich selbst im Gleichgewicht h~lt C. F. Moppert
Kurzfassung--Die Seilf(~hrung des Kran$ in der einfachsten Ausf~hrung is{ in Bild 1 ersichtlich. Der Arm AC ist um den Aufh~ngepunk{ A drehbar. Das Gegengewich{ K is{ an diesem Arm befestigt. Bevor der Kran beginn{, ein Gewicht zu heben, h~ngt der Arm AC senkrecht nach unten. Wird das Sell durch die Winde angezogen, so beweg{ sich der Arm nach aussen bis das Gegengewicht mit der Last im Gleichgewicht ist. Von diesem Moment an dreht sich der Arm AC nich{ mehr welter nach aussen, und die Last beginnt sich zu heben. Das Grundmodell kann mannigfach variien werden. Eine prinzipielle Welterentwicklung zeig{ Bild 2. Die vorher festen Leitrollen B und D sind hier dutch Trolleys ersetzt, die sich gegenl~ufig bewegen. Der Arm AC ist jetzt selbst ein Zugseil. Es is{ mit der Seilschleife, die die Trolleys bewegt, (gestrichelt gezeichnet) bel P verbunden. Dieser Kran behalt sein Gleichgewicht unabh~ngig v o n d e r Last und von der Auslegung. Bild 4 zeigt d / K r als Funktion yon L/K nach Gleichung (3). Hier ist d das Drehmoment um A (Bild 1). Man beachte die 0berh6hung der vertikalen Achse. Die praktischen Vor- und Nachteile werden kurz diskutiert. Der Kran hat verschiedene historische Vorl~ufer (Bild 7 und 8).
Pergamon Press. Printed in Great Britain
A Self-Balancing Crane To the memory of Malcolm Livingstone Urquhart.
C. F. Moppert* Abstract A simple mechanism is described which has the effect that an automatic displacement of a counterweight very nearly cancels the overturning moment caused by the load. This mechanism is then adapted to a hammerhead crane with movable trolley. An exact static analysis is given, variants are discussed and historical references made.
1. Basic model (Fig. 1) A HORIZONTALarm BD rests in its middle A on a tower. An arm AC which can swivel about A carries a pulley at its end C. The length r of the arm A C is equal to AD and AB. The cable starts at the winch W, goes o v e r a pulley at A to D, around the pulley at the hook E back to D then to B, to the pulley C supporting the counterweight K and back to a fixed point at B. At D there are two independent pulleys on the same axle, (for the details of the arrangement consult the figure).
Working action (1) Static. If there is no load L then the arm AC hangs vertically. If a load is picked up then the pulley C is lifted towards B. The load L starts being lifted at the m o m e n t the counterweight K is in the position to balance the load. F r o m then onwards the point C stays firm. If the load is put down, the point C first m o v e s down until AC is vertical. Only then can the load be r e m o v e d f r o m the hook. If the load L is small then the arm AC m o v e s only slightly f r o m the vertical position. The m a x i m u m load the crane can cope with is equal to the weight K of the counterweight, the latter being arbitrary. The bending m o m e n t , caused b y the load, acting on the arm BD about the point A is kept very small indeed. If the crane is built in such a way that the point A is the centre of gravity of this a r m (including pulleys and the weight of the sections of the rope along it), very little strutting will be necessary to keep it in a horizontal position. Furthermore, the t o w e r will be subjected almost entirely to compression. (2) Dynamic. The position of K reacts to forces of inertia as well as to static ones, i.e. C m o v e s out (in) if L is accelerated upwards (downwards). The balancing action works immediately: it is possible to cut the cable while the crane carries the m a x i m u m load and no overbalancing takes place (to m a k e this possible, the crane must be constructed in such a way that the arm AC can swing towards D). If the load is lowered by the crane at constant speed, onto a truck say, then at the m o m e n t of impact only the inertial force of the load together with part of the inertial force of the counterweight affect the platform of the truck. The weight of the load *Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia. 359
360 i
I
r-
X
c
-~
I
-W % I
Flgure I. The horizontal arms DA, AB of the crane have equal length r. The arm AC has length r also. It carries the c o u n t e r w e i g h t K at its end. This arm can swivel freely about A. The c o u n t e r w e i g h t will automatically assume a position such that the crane is in near balance.
proper only gradually builds up while the counterweight is lowered. We have investigated the corresponding mathematical problem but we feel it is not worth including here as complicating effects are likely to arise in a practical situation. (3) Practical. The action of the crane is not perfect due to friction in the pulleys and the cable. In lifting a load, the counterweight can slightly advance or stay behind the ideal position. It is a definite disadvantage that the load does not react immediately to the m o v e m e n t s of the winch. H o w e v e r , this "lagging behind" has an advantage: any load which is less than the m a x i m u m load can, while it is hanging on the crane, be moved up or down some distance using very little force (See Section 4).
2. Variants of the basic model 2.1 Counterweight The counterweight K can be placed anywhere along the arm AC or on an extension of it. If this is the case then the m a x i m u m load L is equal to a fraction or a multiple of K. 2.2 Ratio A D / A B The crane can be built for any given ratio AD/AB. The length of the movable arm AC must be equal to AB. If A D / A B = m / n with m, n integers, then the crane is always in near balance provided the load hangs on n cables and the connection of the points B and C is provided by m cables. It is always easy to lead the cable o v e r multiple pulleys to ensure that these conditions are satisfied. The counterweight K can again be placed anywhere along the arm AC. 2.3 H a m m e r h e a d crane We describe the h a m m e r h e a d version of the basic model, i.e. the model with A D / A B = 1. It is clear that this restriction is not necessary. The h a m m e r h e a d version makes use of the fact that the arm AC is under all
361 conditions under tension only and can therefore be replaced b y a cable. Suppose that the load is suspended f r o m a movable trolley D (Fig. 2). The crane is in balance provided the three distances AD, AB and AC are equal. It is shown in Fig. 2 that if the trolley D m o v e s in (out), so does the trolley B, both being connected to the endless loop P F G . The cable carrying the pulley C is connected to this same loop at P. If D m o v e s away f r o m A, P m o v e s towards A and as the distance A F is equal to the length PA + AC, the length AC is always equal to the length AB.
Tr°"e'D F I
_
_
.
.
.
.
--i
~W [
I
Figure 2. Hammerhead version of the basic model. The trolleys B and D are always equidistant from A. The counterweight K is supported by the rope coming from the w h i n c h and by a cable P A C . This arrangement ensures that always DA ---AB = A C . The crane is in near balance independent of the load and of the position of the trolley. The h a m m e r h e a d crane is always in near balance, independent of the load and the position of the trolley. The luffing of the h a m m e r h e a d crane is horizontal only if the load is maximal (i.e. L = K ) . If the trolley D m o v e s then so does the trolley B and the length BC changes, h o w e v e r , the angle CAB remains constant. It follows that L m o v e s along a straight line while D m o v e s and that this line is roughly parallel to the line AC as long as the angle CAB is not too large.
3. Static equilibrium of the basic model The load L is suspended by two strands of the cable. Its tension t is therefore t = L [2. We denote b y k the c o m p o n e n t of the counterweight K along the suspending cables, i.e. t = k/2 and k = L. F r o m Fig. 3 we see that k / K = sin ( 9 0 - a ) / s i n (90 + a/2) = cos a / c o s a / 2 , thus K cos a = L cos a/2. We denote by d the m o m e n t about A, acting on the arm DB in the anticlockwise sense, thus d = L r - Kr cos a.
362 Moment
d
B A
k
I
90-a
Figure 3. The force K has component k acting on the rope. P u t t i n g )t = L / K , ~ = d / K , w e h a v e COS t~
x
(I)
cos a/2
= r(A - c o s or).
(2)
G i v e n )t, r e l a t i o n (1) d e t e r m i n e s a a n d t h u s ~ b y u s e o f r e l a t i o n (2). W e c a n e l i m i n a t e b y i n t r o d u c i n g u = c o s a/2. T h e n c o s a = 2u 2 - 1 a n d )t = (2u 2 - l)/u. It f o l l o w s t h a t u = (A + ~ ) / 4 . A s u is n o n - n e g a t i v e , w e c h o o s e t h e p o s i t i v e sign. S u b s t i t u t i n g this in ~ = r ( ( 2 u 2 - l)/u - (2u 2 - 1)) w e g e t 6 r
-
d Kr
-
x
;t 2 4
A~2+8 4
(3)
W e a r e i n t e r e s t e d in t h e b e h a v i o u r o f this f u n c t i o n if L v a r i e s b e t w e e n 0 a n d K, i.e. f o r 0 ~< X ~< 1. Its g r a p h is g i v e n in Fig. 4. T h e unit on t h e v e r t i c a l axis is t h e r e ten t i m e s t h e unit on t h e h o r i z o n t a l one. W e s e e t h a t 6/r = 0 f o r A = 0 a n d A = l a n d t h a t it r e a c h e s its m a x i m u m v a l u e o f 6/r = 0.0785 f o r A = 0.516. T h i s m e a n s t h a t o u r c r a n e is in p e r f e c t b a l a n c e f o r L = 0 a n d L = K o n l y a n d t h a t t h e i m b a l a n c e in t h e w o r s t c a s e o c c u r s r o u g h l y f o r L = K/2. It is t h e n less t h a n t h e i m b a l a n c e c a u s e d b y p u t t i n g 8 p e r c e n t o f t h e c o u n t e r w e i g h t K at D.
4. Vertical displacement of the load (basic model) L e t t h e w i n c h b e fixed, t h e l o a d h a n g i n g a d i s t a n c e x f r o m t h e p u l l e y D (Fig. 1). W e i n v e s t i g a t e t h e i n c r e m e n t a l v e r t i c a l f o r c e d K n e c e s s a r y to m o v e t h e l o a d b y an i n c r e m e n t dx. From Kcosct =Lcosa/2, it f o l l o w s t h a t - K s i n a da = c o s a / 2 dL1/2L sin a / 2 da; h e n c e K s i n a c o s a / 2 da = 1/2 K c o s a sin a / 2 da - c o s 2 a / 2 dL.
363
O-
o°d~ K-~ 0.06 0"040
-I'0
0"2
t_
0-516
I'O
K
Figure 4. Normalised momentum d / K r acting on the horizontal arm in the basic model as function of L / K . By shifting the support of the arm DB in the basic model by 2 per cent of r to the left of A (Fig. 6), the quantity d/Kr is the difference of the curve and the sloping straight line. In this way, the effectiveness of the mechanism is nearly doubled. A s t h e l e n g t h o f t h e r o p e r e m a i n s c o n s t a n t , w e h a v e 2x + 4 r sin a / 2 = c o n s t , a n d d x = - r c o s ct/2 d a . S t r a i g h t f o r w a r d c a l c u l a t i o n g i v e s t h e n dx
r
dL
2
K (tan a / 2 ) ( 3 + tan 2 c~/2)
(4)
w h e r e L, K a n d ot a r e l i n k e d t o g e t h e r b y (l). T h e f r a c t i o n on t h e r i g h t - h a n d side o f (4) b e c o m e s infinite f o r a = 0. T h i s c o r r e s p o n d s to t h e f a c t t h a t t h e s t r u c t u r e a l l o w s t h e n n o e x t e n s i o n o f t h e c a b l e . In o r d e r to i l l u s t r a t e t h e m e a n i n g o f t h e r e l a t i o n (4), w e c a l c u l a t e t h e c o r r e s p o n d i n g v a l u e s f o r a = 45 °. T h e n L / K = 0-765 a n d d x / r = 1.5 d L / K , i.e. t o d L = K / 1 0 0 c o r r e s p o n d s a d i s p l a c e m e n t d x o f a b o u t 1-5 p e r c e n t o f t h e length r. T h e r e l a t i o n (4) c o n n e c t i n g d x / r with d L / K h a s a n o t h e r i m p o r t a n t p r a c t i c a l applic a t i o n as it e n a b l e s us to s a y w i t h i n w h i c h limits x is d e t e r m i n e d b y v a r i a t i o n s o f L o r b y f r i c t i o n a l f o r c e s . W e h a v e p o i n t e d o u t e a r l i e r that it is a d i s a d v a n t a g e o f o u r c r a n e t h a t t h e c r a n e d r i v e r h a s n o t the i m m e d i a t e c o n t r o l o v e r x h e h a s with a c o n v e n t i o n a l c r a n e . T h e r e l a t i o n (4) is i l l u s t r a t e d in Fig. 5, u s i n g l o g a r i t h m i c scale f o r ( d x / r ) / ( d L / K ) .
5. Improvement of the basic model It is e v i d e n t f r o m Fig. 4 t h a t t h e m o m e n t d t a k e s o n l y p o s i t i v e v a l u e s . B y c h a n g i n g t h e d e s i g n in s u c h a w a y t h a t d is n e g a t i v e f o r L = 0 w e c a n k e e p Jdl b e t w e e n l o w e r b o u n d s . W e i n v e s t i g a t e t h e s i t u a t i o n w h e n t h e a r m A B is s u p p o r t e d at a p o i n t 0 at a d i s t a n c e (rr f r o m D w i t h tr a fixed p o s i t i v e n u m b e r < 1 (Fig. 6). T h e d e s i g n o f t h e b a s i c m o d e l is o t h e r w i s e u n c h a n g e d a p a r t f r o m t h e f a c t t h a t t h e p u l l e y l e a d i n g t h e r o p e f r o m W to D is n o w at 0. I n this c a s e w e h a v e d = o-rL - K((1 - t r ) r + r c o s ~ ) a n d this l e a d s e a s i l y to ~ / r = A - c o s a + (A + l)((r - 1) = A - A2/4 - A ~ 8 / 4 + (A + 1)(o" - 1). In o u r Fig. 4 w e h a v e t h e g r a p h o f t h e f u n c t i o n A - A2/4 - A ~ 8 / 4 . T h e f u n c t i o n (A + 1)(o~ - 1) is a s t r a i g h t line t h r o u g h t h e p o i n t s
364 20-15 I(~
5
I
0-5
I
b
0 o
pOo
90* I1
Figure 5. A change dL. of the load causes a change d x (Fig. 1). The relation of these quantities depends very much on the angle ~ (Fig. 3). D
r
,,I
_~
r
B
err
Figure 6. Improved version of the basic model. ( - 1, 0) and (0, o" - l). W e d r a w a line t h r o u g h ( - 1, 0) such that the deviation of o u r graph f r o m this line is minimal. As its intercept with the vertical axis is ~ 0.02 w e put 1 - tr = 0.02, i.e. tr = 0.98 (5) and we see that for this c h o i c e the value of I~/rl is surely < 0 . 0 5 . W e h a v e thus i m p r o v e d the p r e v i o u s value o f < 8 per cent to < 5 per cent. This i m p r o v e m e n t can clearly be carried o v e r to all other models.
6. History O u r c r a n e can be c o n s i d e r e d as a d e v e l o p m e n t of the ancient n o d d e r c r a n e w h i c h still can be seen in E g y p t (Fig. 7). In fact, the f a t h e r o f the n o d d e r and our c r a n e is the (historically later) letter b a l a n c e (Fig. 8). It is easily seen h o w the n o d d e r and o u r c r a n e strike t w o different c o m p r o m i s e s to a c h i e v e their end. F u r t h e r m o r e o n e sees that in our c r a n e s o m e imbalance is allowed in order to maintain simplicity of design and thus to keep d i s t u r b a n c e due to friction at a minimum.
365
Figure 7. The " n o d d e r crane", an ancient version of our crane. The pulley c o r r e s p o n d i n g to the pulley D in our crane moves along a circular arc.
I
I
Figure 8. A letter balan~,e. The wheel turns until the c o u n t e r w e i g h t compensates the load. Our crane as well as the n o d d e r crane can be considered as derivatives.
It would be surprising if our idea were new. I have been informed by Professor Hartenberg in Illinois that a similar construction is described in M. Ruehlmann's AUgemeine Maschinenlehre, 4. Band, p. 469 ff (Braunschweig 1875). In the crane described there, the counterweight is mounted on a trolley which moves on a rail forming a particular upward curve. This crane was designed by Jambille and exhibited at the international exhibition in Paris, 1867. Two final remarks. I have built several man-sized models of the basic crane with the variant AD/DB = 2 : l and a small model of the hammerhead crane. All these models work very well indeed. The idea as such is a very natural one: if a man lifts a weight with his right arm, he moves out the left one until he is in balance. It is reassuring to find our crane is imitating Nature.
Acknowledgement---I wish to express by thanks to Professor K. Hunt, Monash University, who enabled me to use the facilities of the School of Engineering. I am also greatly indebted to my friend Roger Hallett for encouraging me, building a model and trying to propagate the idea.
MMT Vol. 9, No. 3 - - F
366
Ein Kran der sich selbst im Gleichgewicht h~lt C. F. Moppert
Kurzfassung--Die Seilf(~hrung des Kran$ in der einfachsten Ausf~hrung is{ in Bild 1 ersichtlich. Der Arm AC ist um den Aufh~ngepunk{ A drehbar. Das Gegengewich{ K is{ an diesem Arm befestigt. Bevor der Kran beginn{, ein Gewicht zu heben, h~ngt der Arm AC senkrecht nach unten. Wird das Sell durch die Winde angezogen, so beweg{ sich der Arm nach aussen bis das Gegengewicht mit der Last im Gleichgewicht ist. Von diesem Moment an dreht sich der Arm AC nich{ mehr welter nach aussen, und die Last beginnt sich zu heben. Das Grundmodell kann mannigfach variien werden. Eine prinzipielle Welterentwicklung zeig{ Bild 2. Die vorher festen Leitrollen B und D sind hier dutch Trolleys ersetzt, die sich gegenl~ufig bewegen. Der Arm AC ist jetzt selbst ein Zugseil. Es is{ mit der Seilschleife, die die Trolleys bewegt, (gestrichelt gezeichnet) bel P verbunden. Dieser Kran behalt sein Gleichgewicht unabh~ngig v o n d e r Last und von der Auslegung. Bild 4 zeigt d / K r als Funktion yon L/K nach Gleichung (3). Hier ist d das Drehmoment um A (Bild 1). Man beachte die 0berh6hung der vertikalen Achse. Die praktischen Vor- und Nachteile werden kurz diskutiert. Der Kran hat verschiedene historische Vorl~ufer (Bild 7 und 8).