This paper presents the structural design and analysis of a novel compliant gripper based on the Scott-Russell (SR) mechanism. The SR mechanism in combination with a parallelogram mechanism enables the achievement of a pure translation of the gripper tips, which is attractive for practical micromanipulation and microassembly applications.
A Roberts linkage
A Sarrus linkage
Peaucellier–Lipkin linkage:
bars of identical colour are of equal length
bars of identical colour are of equal length
Cardan straight line mechanism
In the late seventeenth century, before the development of the planer and the milling machine, it was extremely difficult to machine straight, flat surfaces. For this reason, good prismatic pairs without backlash were not easy to make. During that era, much thought was given to the problem of attaining a straight-line motion as a part of the coupler curve of a linkage having only revolute connection. Probably the best-known result of this search is the straight line mechanism development by Watt for guiding the piston of early steam engines. Although it does not generate an exact straight line, a good approximation is achieved over a considerable distance of travel.
Nearly straight line linkages[edit]
- Watt's linkage (1784)
- Chebyshev's Lambda Mechanism (1878) (Can trade off straightness and near constant velocity)
- Hoeckens linkage (1926)
Perfect straight line linkages[edit]
Eventually, several linkages were discovered that produced perfect linear output:
- Sarrus linkage (1853)
- Peaucellier–Lipkin linkage (1864)
- Hart's inversor/Hart's A-frame (1874)
- Quadruplanar-Inversor (1875)
Rotary straight line mechanisms[edit]
- Cardan straight line mechanism[1]
- Tusi couple (1247) hypocycloid straight-line mechanism
See also[edit]
Sources[edit]
- ^https://web.archive.org/web/20180418124753/http://kmoddl.library.cornell.edu/model.php?m=139 Reuleaux Collection, Cornell university
- Theory of Machines and Mechanisms, Joseph Edward Shigley
External links[edit]
- Cornell university (archived) - Straight-line mechanism models
- Alfred Kempe (1877). How to Draw a Straight Line(PDF). Macmillan – via University of California at Irvine.
- Daina Taimina. 'How to Draw a Straight Line - a tutorial'. Cornell University.
- Simulations using the Molecular Workbench software
- bham.ac.uk - Hart's A-frame (draggable animation) 6-bar linkage
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Straight_line_mechanism&oldid=915288402'
The Chebyshev linkage is a mechanical linkage that converts rotational motion to approximate straight-line motion.
It was invented by the nineteenth-century mathematician Pafnuty Chebyshev, who studied theoretical problems in kinematic mechanisms. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight-line motion. This was also studied by James Watt in his improvements to the steam engine.[1]
The straight-line linkage confines the point P – the midpoint on the link L3 – on a straight line at the two extremes and at the center of travel. (L1, L2, L3, and L4 are as shown in the illustration.) Between those points, point P deviates slightly from a perfect straight line. The proportions between the links are
Point P is in the middle of L3. This relationship assures that the link L3 lies vertically when it is at one of the extremes of its travel.[2]
The lengths are related mathematically as follows:
It can be shown that if the base proportions described above are taken as lengths, then for all cases,
and this contributes to the perceived straight-line motion of point P.
- 1Equations of motion
Equations of motion[edit]
The motion of the linkage can be constrained to an input angle that may be changed through velocities, forces, etc. The input angles can be either link L2 with the horizontal or link L4 with the horizontal. Regardless of the input angle, it is possible to compute the motion of two end-points for link L3 that we will name A and B, and the middle point P.
while the motion of point B will be computed with the other angle,
And ultimately, we will write the output angle in terms of the input angle,
Consequently, we can write the motion of point P, using the two points defined above and the definition of the middle point.
Input angles[edit]
Illustration of the limits
The limits to the input angles, in both cases, are:
See also[edit]
Chebyshev's Lambda Mechanism (one blue and one green) shows an identical motion path
- Watt's linkage, a similar straight-line mechanism with the direction of one of the arms reversed.
- Hoeckens linkage (4-bar linkage that converts rotational motion to approximate straight-line motion)
- Peaucellier–Lipkin linkage ( an 8-bar linkage that gives perfect linear motion)
References[edit]
- ^Cornell university - Cross link straight-line mechanism
- ^Gezim BashaArchived August 19, 2014, at the Wayback Machine - Rotation to approximate translation using the Chebyshev Linkage
External links[edit]
Wikimedia Commons has media related to Chebyshev linkage. |
- A simulation using the Molecular Workbench software
- A Geogebra simulation of the linkage
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Chebyshev_linkage&oldid=793532957'