A straight-line mechanism is a

mechanism Mechanism may refer to: *Mechanism (economics), a set of rules for a game designed to achieve a certain outcome **Mechanism design, the study of such mechanisms *Mechanism (engineering), rigid bodies connected by joints in order to accomplish a ...

that converts any type of rotary or angular motion to perfect or near-perfect straight-line motion, or ''vice versa''. Straight-line motion is

linear motion Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with ...

of definite length or "stroke", every forward stroke being followed by a return stroke, giving

reciprocating motion Reciprocating motion, also called reciprocation, is a repetitive up-and-down or back-and-forth linear motion. It is found in a wide range of mechanisms, including reciprocating engines and pumps. The two opposite motions that comprise a single ...

. The first such mechanism, patented in 1784 by

James Watt James Watt (; 30 January 1736 (19 January 1736 OS) – 25 August 1819) was a Scottish inventor, mechanical engineer, and chemist who improved on Thomas Newcomen's 1712 Newcomen steam engine with his Watt steam engine in 1776, which was f ...

, produced approximate straight-line motion, referred to by Watt as

parallel motion Parallel may refer to: Mathematics * Parallel (geometry), two lines in the Euclidean plane which never intersect * Parallel (operator), mathematical operation named after the composition of electrical resistance in parallel circuits Science a ...

. Straight-line mechanisms are used in a variety of applications, such as engines, vehicle suspensions, walking robots, and rover wheels.

History

In the late eighteenth century, before the development of the planer and the

milling machine Milling is the process of machining using rotary cutters to remove material by advancing a cutter into a workpiece. This may be done by varying directions on one or several axes, cutter head speed, and pressure. Milling covers a wide variety of ...

, it was extremely difficult to machine straight, flat surfaces. During that era, much thought was given to the problem of attaining a

straight-line motion Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with co ...

, as this would allow the flat surfaces to be machined. To find a solution to the problem, the first straight-line mechanism was developed by

, for guiding the pistons of early steam engines. Although it does not generate an exact straight line, a good approximation is achieved over a considerable distance of travel. Perfect straight-line linkages were later discovered in the nineteenth century, but they were not as needed, as by then other techniques for machining had been developed.

List of linkages

Approximate straight-line linkages

These mechanisms often use four-bar linkages as they require very few pieces. These four-bar linkages have coupler curves that have one or more regions of approximately perfect straight-line motion. The exception in this list is Watt's parallel motion, which combines Watt's linkage with another four-bar linkage – the

pantograph A pantograph (, from their original use for copying writing) is a Linkage (mechanical), mechanical linkage connected in a manner based on parallelograms so that the movement of one pen, in tracing an image, produces identical movements in a se ...

– to amplify the existing approximate straight-line movement. It is not possible to create perfect straight-line motion using a four-bar linkage, without using a

prismatic joint A prismatic joint is a one- degree-of-freedom kinematic pair which constrains the motion of two bodies to sliding along a common axis, without rotation; for this reason it is often called a slider (as in the slider-crank linkage) or a sliding ...

. *

Watt's linkage A Watt's linkage is a type of mechanical linkage invented by James Watt in which the central moving point of the linkage is constrained to travel a nearly straight path. Watt's described the linkage in his patent specification of 1784 for the ...

(1784) * Watt's parallel motion (1784) *

Evans "Grasshopper" linkage Evans or Evan's may refer to: People *Evans (surname) *List of people with surname Evans *Evans Welch, Trinidad and Tobago politician Places United States *Evans Island, an island of Alaska *Evans, Colorado *Evans, Georgia *Evans County, Georg ...

(1801) *

Chebyshev linkage In kinematics, Chebyshev's linkage is a four-bar linkage that converts rotational motion to approximate linear motion. It was invented by the 19th-century mathematician Pafnuty Chebyshev, who studied theoretical problems in kinematic Mechanism ...

Chebyshev lambda linkage In kinematics, the Chebyshev Lambda Linkage is a four-bar linkage that converts rotational motion to approximate straight-line motion with approximate constant velocity. It is so-named because it looks like a lowercase Greek letter lambda (λ). ...

(1878), a

cognate linkage In kinematics, cognate linkages are linkages that ensure the same coupler curve geometry or input-output relationship, while being dimensionally dissimilar. In case of four-bar linkage coupler cognates, the Roberts–Chebyshev Theorem, after ...

of the Chebyshev linkage *

Roberts linkage Roberts may refer to: People * Roberts (given name), a Latvian masculine given name * Roberts (surname), a surname Places * Roberts (crater), a lunar impact crater on the far side of the Moon ;United States * Roberts, Idaho * Roberts, ...

* Horse-head linkage *

Hoecken linkage In kinematics, the Hoecken linkage (named for Karl Hoecken) is a four-bar linkage that converts rotational motion to approximate straight-line motion. The Hoecken linkage is a cognate linkage of the Chebyshev linkage and Chebyshev's Lambda Mech ...

(1926) – requires a sliding joint

Perfect straight-line linkages

Eventually, perfect straight line motion was achieved. The Sarrus linkage was the first perfect linear linkage, made in 1853. However, it is a spatial linkage rather than a planar linkage. The first planar linkage would not be made until 1864. Currently, all planar linkages which produce perfect linear motion utilize the inversion around a circle to produce a hypothetical circle of infinite radius, which is a line. This is why they are called inversors or inversor cells. The simplest solutions are Hart's W-frame – which use 6-bars – and the quadruplanar inversors – Sylvester-Kempe and Kumara-Kampling, which also use 6-bars. *

Sarrus linkage The Sarrus linkage, invented in 1853 by Pierre Frédéric Sarrus, is a mechanical linkage to convert a limited circular motion to a linear motion or vice versa without reference guideways. It is a spatial six-bar linkage (6R) with two groups ...

(1853) * Peaucellier-Lipkin inversor (1864) * Hart's first inversor / Hart's antiparallelogram / Hart's W-frame (1874) * Hart's second inversor / Hart's A-frame (1875) * Perrolatz inversor * Kempe's double kite inversors (1875) * Bricard inversor *

Quadruplanar inversor The Quadruplanar inversor of Sylvester and Kempe is a generalization of Hart's inversor. Like Hart's inversor, is a mechanism that provides a perfect straight line motion without sliding guides. The mechanism was described in 1875 by James Josep ...

(1875) The

Scott Russell linkage A Scott Russell linkage is a linkage which translates linear motion through a right angle. The linkage is named after John Scott Russell (1808–1882), although watchmaker William Freemantle had already patented it in 1803. A different form of ...

(1803) translates linear motion through a right angle, but is not a straight-line mechanism in itself. The Grasshopper beam/Evans linkage, an approximate straight-line linkage, and the Bricard linkage, an exact straight-line linkage, share similarities with the Scott Russell linkage and the

Trammel of Archimedes An ellipsograph is a mechanism that generates the shape of an ellipse. One common form of ellipsograph is known as the trammel of Archimedes. () It consists of two shuttles which are confined to perpendicular channels or rails and a rod which is ...

Compound eccentric mechanisms with elliptical motion

These mechanisms use the principle of a rolling curve instead of a coupler curve and can convert continuous, rather than just limited, rotary motion to reciprocating motion and ''vice versa'' via elliptical motion. The straight-line sinusoidal motion produces no second-order inertial forces, which simplifies balancing in high-speed machines. * Cardan straight-line mechanism. Using the principle of the

Tusi couple The Tusi couple (also known as Tusi's mechanism) is a mathematical device in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller ...

(1247), a

spur gear Spur gears or straight-cut gears are the simplest type of gear. They consist of a cylinder or disk with teeth projecting radially. Viewing the gear at 90 degrees from the shaft length (side on) the tooth faces are straight and aligned parallel to ...

rolls inside an internally toothed

ring gear An epicyclic gear train (also known as a planetary gearset) is a gear reduction assembly consisting of two gears mounted so that the center of one gear (the "planet") revolves around the center of the other (the "sun"). A carrier connects the ...

of twice the diameter. The

hypocycloid In geometry, a hypocycloid is a special plane curve generated by the trace of a fixed point on a small circle that rolls within a larger circle. As the radius of the larger circle is increased, the hypocycloid becomes more like the cycloid creat ...

traced by any point on the pitch circle of the smaller gear is a diameter of the larger gear. The mechanism has been used in Murray's Hypocyclic Engine. *

. Originally an ellipsograph. Also known as the double-slider mechanism, it uses the fact that a circle and a straight line are special cases of an ellipse. It is based on much the same kinematic principle as Cardan's straight line mechanism (above) and could be considered as a spur gear with two teeth in a ring gear with four teeth. It has been used in the Baker-Cross engine. It has been used in inverted form in Parsons' steam engineParsons' epicyclic engine
/ref> and can still be found today in a further inversion as the

Oldham coupling A coupling is a device used to connect two shafts together at their ends for the purpose of transmitting power. The primary purpose of couplings is to join two pieces of rotating equipment while permitting some degree of misalignment or end mo ...

. The

scotch yoke The Scotch yoke (also known as slotted link mechanism) is a reciprocating motion mechanism, converting the linear motion of a slider into rotational motion, or vice versa. The piston or other reciprocating part is directly coupled to a sliding yo ...

is sometimes given as an inversion of the Archimedes trammel, but it has only one eccentric or crank with one slider and no parts with elliptical motion. * MultiFAZE is an acronym for Multiple Fixed Axis Shaft Compound Eccentric.

Gallery

Approximate straight-line linkages

Parts/links of the same color are the same dimensions. File:Watts Linkage.gif, Watt's linkage File:Watts Parallel Motion Linkage.gif, Watts parallel-motion linkage File:Evans "Grasshopper" Linkage.gif, Evans "Grasshopper" linkage File:Roberts Linkage.gif, Roberts linkage File:Chebyshev Linkage.gif, Chebyshev linkage File:Chebyshev Lambda Linkage.gif, Chebyshev lambda linkage File:Chebyshev Table Linkage.gif, Chebyshev table linkage File:Hoecken's Linkage.gif, Hoecken's linkage

Perfect straight-line linkages

Parts/links of the same color are the same dimensions. File:Sarrus Linkage - Links Ver.gif, Sarrus linkage (Bars variant) File:Sarrus Linkage.gif, Sarrus linkage (Plates variant) File:Peaucellier-Lipkin Inversor.gif, Peaucellier-Lipkin inversor File:Harts Inversor 1.gif, Harts inversor 1 File:Harts Inversor 2.gif, Harts inversor 2 File:Perrolatz Inversor.gif, Perrolatz inversor File:Kempe Kite Inversor 1.gif, Kempe kite inversor 1 File:Kempe Kite Inversor 2.gif, Kempe kite inversor 2 File:Kempe Kite Inversor 3.gif, Kempe kite inversor 3 File:Scott Russell Linkage.gif, Scott Russell linkage (slider connection) File:Scott Russell Demonstration.gif, Scott Russell linkage (connected to Peaucellier-Lipkin linkage) File:Bricard Inversor.gif, Bricard inversor File:Quadruplanar Inversor 1.gif, Sylvester-Kempe quadruplanar inversor 1 File:Quadruplanar Inversor 2.gif, Sylvester-Kempe quadruplanar inversor 2 File:Quadruplanar Inversor 3.gif, Sylvester-Kempe quadruplanar inversor 3 File:Quadruplanar Inversor 4.gif, Kumara-Kampling inversor

Tusi couple, elliptical motion: versions and inversions

File:T1 Tusi diag 90x20.gif,

(1247) according to the diagrams in the translation of the copy of Tusi's original description: Small circle rolls within large circle. File:T2 Tusi descr 90x20.gif, Tusi couple according to the translation of the copy of Tusi's original description: Circles rotate in same direction, speed ratio 1:2. File:Tusi by Copernicus 90x20.gif, Copernicus' (1473-1543) take on the Tusi couple: Direction of rotation and orbit of moving circle are equal and opposite. File:T4 Inv4 90x20.gif, Inversion No. 3 File:T9 Inv4 90x20.gif, Inversion No. 4 File:T5 180x20.gif, Inversion No. 5 - speed ratio 1:3 File:T6a 90x20.gif, Inversion No. 6 File:Tusi inside-out150x30.gif, Inside-out Tusi couple. The small circle is split into four fixed quadrants. Two 45° arcs of the large circle form the waist of the trammel. File:Parsons 190x20.gif, Parsons' mechanism (1877) combines Tusi Inversion No. 2 with an Archimedes trammel. Pistons at A and C balance each other. File:Oldham 120x30.gif,

(1821). Slotted ends of two misaligned shafts (black) are coupled by a cross piece (green). Compare with Tusi Inversion No. 4. File:MultiFAZE 150x25.gif, Kinematics of the Multiple Fixed Axis Shaft Compound Eccentric (MultiFAZE) mechanism (1982) characterised by parallelograms ABCD.

Compound eccentric mechanisms with elliptical motion

File:Tusicardmedes 120x40.gif, A spur gear with two teeth rolls inside a ring gear with four teeth: Archimedes, Tusi or Cardano? File:Cardano's hypocyclic gears.gif, Cardano's (1501-1576) hypocyclic gears: the red, green and blue pins reciprocate on diameters of the ring gear. File:Triple trammel of Archimedes.gif, Three Archimedes (287~212 BC) trammels on a triangular rotor showing the circular orbit of the trammel midpoints. File:MultiFAZE_60°_X4 100x40.gif, MultiFAZE mechanism (1982) in a 60° X4 engine with yokes, and rotary counterweights for full balance. File:MultiFAZE 90° X4 90x30.gif, MultiFAZE mechanism (1982) in a 90° X4 engine with crossheads, trammel gears, and reciprocating balance weights/sliders. File:Stiller-Smith 72x50 side view.gif, Stiller-Smith 90° X4 2T floating cantilever crank engine (1984) with MultiFAZE eccentric gear train. Wobble and overshoot exaggerated for effect.

Notes

References

* Theory of Machines and Mechanisms, Joseph Edward Shigley

External links

Cornell university (archived)
- Straight-line mechanism models * * {{cite web , url=http://pi.math.cornell.edu/~dwh/courses/M451-F02/PL-2.htm , title=How to Draw a Straight Line - a tutorial , author=Daina Taimina , author-link=Daina Taimina , website=

Cornell University Cornell University is a Private university, private Ivy League research university based in Ithaca, New York, United States. The university was co-founded by American philanthropist Ezra Cornell and historian and educator Andrew Dickson W ...

Simulations
using the Molecular Workbench software

- Hart's A-frame (draggable animation) 6-bar linkage Linkages (mechanical) Linear motion Straight line mechanisms

History

List of linkages

Approximate straight-line linkages

Perfect straight-line linkages

Compound eccentric mechanisms with elliptical motion

Gallery

Approximate straight-line linkages

Perfect straight-line linkages

Tusi couple, elliptical motion: versions and inversions

Compound eccentric mechanisms with elliptical motion

See also

Notes

References

External links