kinematics Kinematics is a subfield of physics, developed in classical mechanics, that describes the Motion (physics), motion of points, Physical object, bodies (objects), and systems of bodies (groups of objects) without considering the forces that cause ...

, Chebyshev's linkage is a

four-bar linkage In the study of mechanisms, a four-bar linkage, also called a four-bar, is the simplest closed- chain movable linkage. It consists of four bodies, called ''bars'' or ''links'', connected in a loop by four joints. Generally, the joints are configu ...

that converts

rotational motion Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...

to approximate

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 co ...

. It was invented by the 19th-century mathematician

Pafnuty Chebyshev Pafnuty Lvovich Chebyshev ( rus, Пафну́тий Льво́вич Чебышёв, p=pɐfˈnutʲɪj ˈlʲvovʲɪtɕ tɕɪbɨˈʂof) ( – ) was a Russian mathematician and considered to be the founding father of Russian mathematics. Chebyshe ...

, 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 (a

straight line mechanism A straight-line mechanism is a Mechanism (engineering), mechanism 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 of definite length ...

). This was also studied 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 fun ...

in his improvements to the

steam engine A steam engine is a heat engine that performs mechanical work using steam as its working fluid. The steam engine uses the force produced by steam pressure to push a piston back and forth inside a cylinder. This pushing force can be trans ...

, which resulted in

Watt's linkage In kinematics, Watt's linkage (also known as the parallel linkage) is a type of mechanical linkage invented by James Watt in which the central moving point of the linkage is constrained to travel on a nearly straight line. It was described in W ...

.Cornell university
– Cross link straight-line mechanism

Equations of motion

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 ''L''₂ with the horizontal or link ''L''₄ with the horizontal. Regardless of the input angle, it is possible to compute the motion of two end-points for link ''L''₃ that we will name A and B, and the middle point. :

x_A =  L_2\cos(\varphi_1) \,

y_A =  L_2\sin(\varphi_1) \,

while the motion of point B will be computed with the other angle, :

x_B =  L_1 - L_4\cos(\varphi_2) \,

y_B =  L_4\sin(\varphi_2) \,

And ultimately, we will write the output angle in terms of the input angle, :

- \arccos\left(\frac\right) \,

Consequently, we can write the motion of point P, using the two points defined above and the definition of the middle point. :

x_P = \frac \,

y_P = \frac \,

Input angles

The limits to the input angles, in both cases, are: :

\varphi_ =  \arccos\left( \frac\right) \approx 36.8699^\circ. \,

\varphi_ = \arccos\left( \frac\right) \approx 101.537^\circ. \,

Usage

Chebyshev linkages did not receive widespread usage in steam engines, but are commonly used as the 'Horse head' design of level luffing crane. In this application the approximate straight movement is translated away from the line's midpoint, but it is still essentially the same mechanism.

References

External links

{{Commons category, Chebyshev linkage
Cornell university, ''"How to draw a straight line, by A.B. Kempe, B.A."''
using the Molecular Workbench software
A Geogebra
simulation of the linkage
A 3D video
of the linkage Linkages (mechanical) Linear motion Straight line mechanisms

Equations of motion

Input angles

Usage

See also

References

External links