Linkage

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The linkage is one of the six basic types of gearboxes . Of all types of gearboxes, they are the ones that have the greatest variety of structures and properties. They are used both as transmission gears and as guide gears. Together with the cam gears , they form the group of unevenly geared transmission gears.

Coupling gears "are gears with at least four solid-state links that are connected by sliding joints. All coupling gears contain at least one fixed coupling, i.e. a gear link that, in the simplest case, connects the two links mounted in the frame. ” As four-link and four-link coupling gears, they are also used as four- link gears or as movable squares, i. H. Articulated quadrilaterals .

Crank rocker with trajectory curve of a point of the coupling (coupling curve ) . The blue triangle represents the coupling. The connection drawn between the two fixed points makes it clear that the base ("the frame") is to be regarded as a link of the coupling mechanism.

If links of the linkage are mounted at "fixed points", the common base of the fixed points is referred to as "frame". The frame is synonymous with one of the links of the linkage. The four joints of the simplest basic form of the linkage each have only one degree of freedom . A swivel joint would be viewed as a combination of a swivel and a sliding joint, each with one degree of freedom. The connection between the two joints can then be viewed as a coupling (even if this connection is not recognizable as a separate link depending on the design of the joint).

According to the mutual position of the axes of rotation of the joints, they are divided into

  • flat gear with parallel axes of rotation of the joints and
  • spatial gear with intersecting axes of rotation of the joints.

The joints of planar gears can all have the degree of freedom f = 1. Spatial gears always contain some joints with a degree of freedom f> 1. or multiple degrees of freedom, such as ball joints. Two basic forms of the four-part planar coupling gear are:

Between the two links mounted on the frame is the transmission link called coupling , which explains the name coupling gear . Multi-link coupling gears have several intermediate links, all of which are referred to as coupling. In the case of transmission gears, the coupling functions as a transmission link between the input and output, corresponding to the other two movable links . In the case of guide gears, it is a guided or a leading (which leads further links connected to the coupling gear) gear link. The range of possible path shapes of points of the coupling or points connected to it is used here (see also the figure opposite). In the case of planar gears, these are algebraic curves of the 6th order, which are called coupling curves .

Designations of the transmission links

Paddock

"Gear links that are not directly connected to the frame are generally called coupling ."

Gear links connected to the frame

"Gear links connected to the frame are designated according to their mobility in relation to the frame" :

The following names exist:

crank

The crank is a gear member rotatably connected to the frame and to the coupling. It runs around the frame.

Swing arm

The rocker is a gear member rotatably connected to the frame and to the coupling, which only vibrates on the frame, but does not complete a full rotation. The term rocker arm is also used
for the automobile pendulum or oscillating axle and the motorcycle rear wheel swing arm without this swing arm being part of a coupling gear. The wheel carrier ( steering knuckle ) is firmly attached to the swinging end of the swing arm.

loop

The loop is a gear member that is rotatable on the frame and slidably connected to the coupling. It can revolve around the frame ( crank loop ) or just swing.

Slider

The slide is a gear element that is slidably mounted in the frame . Neighboring links are connected to it via a swivel joint .

Cross slide

The cross slide is a gear member that is slidably connected to the frame and to the coupling. It can only be pushed on the frame.

Plane, spherical and spatial coupling gears

Cardan joint with axes of rotation drawn in, all of which intersect at one point

Plane and spherical coupling gears

The axes of spherical gears meet at one point. An example of spherical gears is the combination of two shafts rotatably mounted on the frame and a universal joint connecting them (see adjacent figure). The axes of the shafts and the two swivel joints of the cardan joint cross at the point (from which the cardan joint owes its name to the universal joint ).

The flat gears are a special case of spherical gears in which the axes of the bearings are parallel to each other. With regard to the forces that occur, the rigidity of the transmission links and other physical factors, the construction and dimensioning of spherical transmissions are more complex.

A three-dimensional crank drive with a ball joint between the crank and the coupling link and a swivel joint between the coupling and the push rod. The green push rod is mounted in a swivel joint . Due to the three freedom of movement of the ball joint, the alignment of the push rod can be varied spatially as desired within a wide range. If the rotating red drive shaft is at an angle other than 90 degrees to the green push rod, the rotary motion of the drive shaft is transmitted in addition to the generated push movement (comparable to the universal joint).

Spatial coupling gear

With spatial coupling gears ( space gears) , the axes of rotation of the joints lie anywhere in space. In addition to simple swivel and sliding joints ( f = 1 ), there are also joints with f> 1 (f = 2: swivel and f = 3: ball joints ). The theoretical and practical mastery of spatial coupling gears is much more complex than that of plane and spherical gears. With the exception of a short section in the next chapter, the following illustrations relate exclusively to planar coupling gears.

An important field of application of spatial coupling gears are modern vehicle wheel suspensions .

Number of transmission links

Four links

A coupling gear has at least four links. With only three limbs, it would not be mobile unless one of the joints has two degrees of freedom of movement , e.g. B. a displaceable and rotatable pin in a slot (f = 2). The freedom of movement or the degree of running of a transmission as a whole should i. d. R. F = 1 so that all its links inevitably follow the drive link .

More than four links

If more than one link is to be driving ( F> 1 ), a minimum of five links are required in accordance with the Grübler equation (5-link with 2 drives; F = 2):

        Equation for flat gears that do not contain joints with more than one freedom of movement.
n = number of links
g = number of joints
c = number of joints with f = 1

The purpose of a multiple drive is to add up the movements of the driving elements in a special transfer function (output parameters ψ = ψ (φ 1 , φ 2 , ..); φ i = drive parameters).

Four links

 F = 2 ≠ 3 (4 - 1 - 4) + 4 >> F = 2 is not possible!
(F = 1 = 3 (4 - 1 - 4) + 4 >> F = 1 is possible, however.)

Five limbs

F = 2 = 3 (5 - 1 - 5) + 5 >> F = 2 is possible!

Six and more links

The Grübler equation shows that coupling gears with the desired degree of running F = 1, i.e. with only one driven link, always consist of an even number of links.

In contrast to four-link linkages, there is no comprehensive order (classification) for linkages with more than four links. In the case of the six-link linkage, the order extends to the division into two different swivel joint chains. These differ in the mutual position of the two three- hinged links and are called Stephenson's and Watt's chains .

Parallel crank drive on the Saxonia steam locomotive .
A second coupling rod is located on the opposite wheels.

Special dimensions: "locked gears"

In practice, functioning gears can also be found in which the forced running equations are not fulfilled. One example is the parallel crank mechanism for transmitting rotary motion from one shaft to a second, as in a steam locomotive . The second coupling rod is offset from the first by 90 °, as this is necessary to avoid turning a pair of wheels (should the locomotive not be on the tracks). Hence F = 0:

        F = 3 (5−1−6) + 6 = 0
This contradiction results from special dimensions: The two coupling rods are of the same length and all are mounted on the same radius. The equality must be guaranteed with high accuracy during production. In the event of inaccuracies, the gearbox jams or is only movable as far as the joint play allows.

Universal joint : right angles between adjacent axes of rotation

Four-part space transmission

An inevitable (F = 1) four-jointed space gear is required according to Grübler's equation

a total of seven freedom of movement of the joints, and can therefore have a maximum of two simple joints.
Control calculation: With F = 1 and n = g = 4 (4 limbs and 4 joints) is the sum of the freedom of movement of the joints     

A well-known example of a space transmission is the universal joint (cardan joint). As a result of special dimensions (right angles between adjacent joints that have to be manufactured with high accuracy) it works as a 4-part space drive with only 4 simple (f = 1) joints.
It does not represent the general case of space transmission.

transmission

The characteristic of coupling gears is that they convert a (uniform) rotary movement into a periodically variable movement. The output member rotates back and forth around a fixed axis (or revolves), goes back and forth on a straight path or is guided on a path of a higher order (coupling curve).

Transmission angle

The transmission angle (angle at the joint between the two limbs, which can be thought of as straight lines) is “a criterion for the quality of the force and movement transmission ”. It should stay within 90 ° ± 50 °. With the values ​​0 ° and 180 °, transmission is "not possible, the transmission gear cannot run."

Constant change in transmission

In most coupling gear applications, the transmission changes constantly.

Discontinuous transmission

Gearboxes in which an output member is temporarily in the rest position or executes a step movement are called indexing gears , switching mechanisms or locking gears . With coupling gears, a longer and exact rest position is more difficult to achieve than with cam gears and requires i. d. Usually more than 6 transmission links.

The order of the four-link linkage

"For a scientifically based work ... with the extraordinarily large variety of coupling gears, a clear order of this type of gear ... is essential."

Order of the four-link coupling gear according to three primary characteristics:

  • Structural features, d. H. Number of swivel and sliding joints and their mutual arrangement,
  • Length ratios of the transmission links and the resulting transfer functions and coupling curves and
  • Distribution of the member functions, d. H. Frame, drive or output link.

A distinction is made within the structural features:

  1. Coupling gear with four swivel joints ( four-joint chain) ,
  2. Coupling gear with three swivel joints and one sliding joint (sliding crank chain) ,
  3. Coupling gear with two adjacent swivel and sliding joints (cross- slide crank chain) ,
  4. Coupling gear with two opposing swivel and sliding joints each (sliding loop chain) .

Transmission of the four-link chain

Gear of the four-link chain
double crank - crank arm - double rocker arm - parallel crank (a penetrating gear )
Example from furniture construction: concealed hinge (animated)

Depending on whether the shortest link is the crank, the belt or the frame, it is:

  • a crank arm ,
  • a double swing arm (total swing arm) or
  • a double crank .

The lengths s , l , p and q of the members determine whether a link relative to its two adjacent links for circulation is. According to Grashof, the length condition for this is :
s + l <p + q.

In the event of a tie, there are sweeping gears . The shortest link can just about circulate, but there are positions in which the four swivel joints are in a straight line. In these there is no forced running: the gear can bottom out , but it can also hit back in the opposite direction (which can be prevented by additional design measures).

With the opposite condition
l + s> p + q.
all links are only able to oscillate relative to one another, as in the case of the double swing arm (or total swing arm ).

Gear of the crank chain

centric crank handle
a gear of the cross loop chain of
right-angled double slider (elliptical compass)

There are only the two links with the lengths l 1 and l 2 (see figure opposite; links 3 and 4 - the push stone and its path - are infinitely long in the kinematic sense).

The push- crank chain is centric when the push axis goes through a swivel joint on the push rod or push block. Depending on which link is the frame, a distinction is made between:

  • The push track is a frame - (centric) push crank .
  • Link l 1 (<l 2 ) or link l 2 (<l 1 ) is a frame - revolving crank loop (if l 1 = l 2 , then with isosceles turning ).
  • Link l 1 (> l 2 ) or link l 2 (> l 1 ) is a frame - swinging crank slider . (if l 1 = l 2 , then through isosceles ).
  • The pusher block is a frame - pusher arm , with a circumferential coupling l 1 (<l 2 ).

If the slider crank chain is eccentric (the slider axis does not go through the swivel joint; the distance from it is the eccentricity e), the conditions for circulability apply: rotatable
if             e <| l 1 - l 2 | ,
resounding if     e = | l 1 - l 2 | ,
not fit for circulation if   e> | l 1 - l 2 |        (The above list is supplemented by a non-rotating push rocker and an oscillating loop ).

Cross-loop chain gear

a gear of the cross loop chain
right-angled cross thrust crank l

For the design of cross-thrust crank chain gears, the information on the intersection angle of the thrust directions (90 ° is most favorable) and the length of the link between the two swivel joints are sufficient. In the case of the cross- thrust crank (picture left), the back and forth movement of the thrust element in the frame is exactly sinusoidal . With the usual crank handle , the sinus shape can only be approximated with a long push rod.

Relative to the thrust path cross, the paths of the points of the link with swivel joints are ellipses , which enables the gear unit to be used as an elliptical compass ( double slide , figure on the right).

Transmission of the thrust loop chain

It is characteristic of slider gears (two opposing swivel joints are replaced by sliding joints) that none of the links can rotate. Application in precision engineering in connection with switching and adjusting mechanisms.

Gear coupling

Combined gears are created by connecting gears of different types of gear in series or in parallel in order to combine the advantages of individual types of gear.

Gear coupling gears are combinations of coupling gears with gear wheels. They are mainly used to generate unevenly rotating or oscillating rotary movements.

Three-wheel linkage

Of the gear coupling gears , the three-wheel ( gear ) coupling gear combined with a crank arm is most frequently used. It is used to generate revolving, highly uneven rotary movements, also with a detent or pilgrim step (partial reverse rotation ) for applications in textile, packaging and other machines.

One example is the drive of a paper drum in a paper turning device in printing machines . Here, the drum, rotating at high speed, performs a momentary stop after each rotation, so that the gripper turning the printed sheet has the opportunity to grasp it precisely and safely.

Watt's planetary gearbox youtube.com ,
Behind the two gears there is a crank (barely visible) which, however, does not have to transmit any forces, but primarily serves to keep the gears at an even distance.

Watt's planetary gear

James Watt bypassed the patent on the latter, which had to be licensed at the time , when converting the lifting movements into rotary movements of a piston steam engine with an addition to the slider crank. He attached a gear (planetary gear) to the rotating end of the coupling , which meshed with a gear wheel mounted coaxially with the crank. The output was not the crank, but the gear wheel rotating twice as fast as it was.

analysis

The transmission analysis is a general task that has to be solved in a similar way for all transmissions. In the case of coupling gears, it is more extensive and complex than z. B. with constant transmission gear drives.

The kinematic and kinetic behavior of the parts of a given transmission is to be determined. The specified transmission can also be an approximate result of the transmission synthesis, which is to be analyzed in preparation for the next iterative development step.

  • Gear kinematics: Movement of the gear parts regardless of their masses and causes of movement. The movement of the gear parts essentially determines the function of the gear. Your knowledge is the basis of the
  • Gear kinetics (dynamics): Inclusion of the masses, causes of movement (including the drive torque ) and forces that are decisive for the stress on the parts. With the forces, their strength can be proven.

The physical basic parameters of the transmission analysis are time , distance and mass . The variables derived from this are speed , acceleration and inertial force . If the investigation is carried out at a certain moment at a certain point of a transmission link, a quasi-static task ( kineostatic analysis ) is present.

The transmission analysis is carried out graphically and mathematically, with the graphical methods having the advantage of clarity and quick implementation. When using CAD , the geometrical values ​​determined are already precise enough, they no longer have to be subjected to subsequent geometrical calculations. In the computational kinematic analysis, the determination of the transfer function (movement of the driven element depending on the driving element) is usually the first priority.

Particular questions in the analysis of linkages are those related to the

synthesis

A transmission should enable either a specified transfer function or a specified guideway. First, a suitable type of gearbox for the task at hand is determined with the help of the structural analysis. Possible transfer functions of four-link coupling gears result from the gear systematics (see section The order of the four-link coupling gears ). They are not always sufficient, e.g. B. a six-link coupling gear is absolutely necessary for a movement with detent. In contrast to that of the four-link coupling gear, their basic behavior is less well known. There are, however, numerous collections of examples ( transmission atlases) of tried and tested higher-link coupling transmissions on which the designer can rely.

Usually the synthesis is carried out using an iterative analysis. As with this one can proceed graphically and arithmetically. The clear drawing procedure can then - if necessary - computationally. Often the specified goal can only be achieved approximately. The transfer function to be realized is then either

  • only achieved in selected points, or
  • it is sufficient if the implemented function lies within a certain tolerance band.

The transfer function of a four-link linkage can only be determined precisely and explicitly for five function points.

See also

literature

Web links

Commons : Linkage  - collection of images, videos and audio files

Individual evidence

  1. Johannes Volmer (Ed.): Getriebetechnik - Koppelgetriebe Verlag Technik , 1979, p. 24
  2. a b Volmer: Linkage , p. 13
  3. Johannes Volmer (Ed.): Gear Technology - Basics Verlag Technik , 1995, ISBN 3-341-01137-4 , p. 27
  4. Volmer, coupling gear, page 25. At the beginning (19th century) the four-bar gear was still called the three-rod gear .
  5. guys uA: transmission technology . Springer-Verlag, 2015, p. 10 ( full text in Google Book Search).
  6. Volmer: Basics p. 240
  7. Volmer: Basics, p. 241
  8. cf. Volmer: Basics , pp. 184-185.
  9. a b c d e Volmer: Fundamentals , pp. 184-185
  10. Dankert / Dankert: Technical Mechanics. Examples of coupling gears including an animated display of coupling curves tm-aktuell.de
  11. Volmer: Basics , p. 195.
  12. Johannes Volmer: Transmission technology - textbook , Verlag Technik, 1969, p. 38, table 2.4.
  13. ^ Johannes Volmer, Basics, p. 185, Table 6.1.
  14. Johannes Volmer, coupling gear, p. 29, table 2.2.
  15. Volmer, Textbook ', p. 39.
  16. Volmer, coupling transmission, p. 56
  17. Volmer, coupling transmission, p. 34
  18. Volmer: Basics , p. 39.
  19. ^ Siegfried Hildebrand : Feinmechanische Bauelemente , Hanser, Munich 1968, p. 633.
  20. Volmer: Basics , p. 33.
  21. Volmer: Basics , p. 241.
  22. Hildebrand, p. 627.
  23. Volmer, coupling gear, p. 33/34
  24. Volmer: Basics , p. 327.
  25. Hildebrand, p. 751.
  26. Volmer: Basics , p. 194.
  27. Volmer: coupling transmission, p. 25 , p. 194.
  28. a b Volmer: Basics , p. 185.
  29. Kurt Luck, Karl-Heinz Modler: Transmission technology: Analysis, Synthesis, Optimization , Springer, 1990, page 31, table 2.2. Compilation of basic gears from the four-bar chain
  30. Volmer: Basics , pp. 185–188.
  31. Dankert / Dankert: Technical Mechanics. Double swing arm
  32. Dankert / Dankert: Technical Mechanics. Double crank
  33. Luck / Modler, page 36, table 2.4. Compilation of basic gears from the slider crank chain ( limited preview in Google book search)
  34. ^ Volmer: Basics , pp. 188-192.
  35. Dankert / Dankert: Technical Mechanics. swinging crank loop
  36. Volmer: Basics , p. 193.
  37. Luck / Modler, page 38, table 2.5. Compilation of basic gears from the cross loop chain
  38. a b Volmer: Basics , p. 192.
  39. Luck / Modler, page 39, table 2.6. Compilation of basic gears from the thrust loop chain ( limited preview in the Google book search)
  40. Hildebrandt, p. 639.
  41. Volmer: Basics , p. 323
  42. Volmer: Basics , p. 325
  43. Volmer, 1978, p. 336 ( limited preview in the Google book search)
  44. a b Volmer: Basics , p. 54.
  45. Volmer: Basics , p. 124.
  46. Volmer: Basics , p. 90.
  47. Volmer: Basics , p. 219.