Cognate linkages

Three unique four-bar linkages can define the same coupler-curve. The linkages which describe the same coupler curve as a certain linkage are called 'cognate linkages'. This theorem in kinematics is called the Roberts-Chebychev theorem. The linkages can be constructed by using similar triangles and parallelgrams, and the Cayley diagram.

Roberts-Chebyschev Theorem

The theory states for a given coupler-curve there exists 3 four-bar linkages, three geared five-bar linkages, and more six-bar linkages which will generate the same path. The method for generating the additional two the four bar linkages from a single four-bar mechanism is described below, using the Cayley diagram.

How to Construct Cognate Linkages

Cayley Diagram

"From Original Triangle, ΔA₁,D,B₁"

*1. Sketch Cayley Diagram
*2. Using Parallelograms, Find A₂ & B₃
//O_A,A₁,D,A₂ & //O_B,B₁,D,B₃
*3. Using Similar Triangles, Find C₂ & C₃
ΔA₂,C₂,D & ΔD,C₃,B₃
*4. Using a Parallelogram, Find O_C
//O_C,C₂,D,C₃
*5. Check Similar Triangles ΔO_A,O_C,O_B
*6. Separate Left & Right Cognate
*7. Put Dimensions on Cayley Diagram

Dimensional Relationships

The lengths of the four members can be found through dimensional analysis. Using the Law of Sines both K_L and K_R are found as follows.

$K_L=frac{sin(alpha)}{sin(eta)}$ $K_R=frac{sin(gamma)}{sin(eta)}$

Conclusions

*If and only if the original is a Class I chain $(l+s)<(P+q)$ Both 4-Bar Cognates will be class I chains.
*If the original is a drag-link(double crank), both cognates will be drag links.
*If the original is a crank-rocker, one cognate will be a crank-rocker, and the second will be a double-rocker.
*If the original is a double-rocker, the cognates will be crank-rockers.

References

ee also

* Chebychev
* Mechanical Linkages
* Four bar linkages
* Kinematic pairs

Wikimedia Foundation. 2010.

Игры ⚽ Нужно сделать НИР?

Look at other dictionaries:

Cognate linkage — In Kinematics, cognate linkages are linkages that ensure the same input output relationship or coupler curve geometry, while being dimensionally dissimilar. In case of four bar linkage coupler cognates, the Roberts–Chebychev theorem states that… … Wikipedia
Four bar linkage — A four bar linkage or simply a 4 bar or four bar is the simplest movable linkage. It consists of 4 rigid bodies (called bars or links), each attached to two others by single joints or pivots to form a closed loop. Four bars are simple mechanisms… … Wikipedia
Dacian language — Dacian Spoken in Romania, northern Bulgaria, eastern Serbia; also (possibly): Moldova, SW Ukraine, eastern Hungary, southern Bulgaria, northern Greece, European Turkey, NW Anatolia (Turkey) Extinct probably by the 6th century AD … Wikipedia
Pafnuty Chebyshev — Chebyshev redirects here. For other uses, see Chebyshev (disambiguation). Pafnuty Chebyshev Pafnuty Lvovich Chebyshev Born May 16, 1821 … Wikipedia
History of molecular biology — The history of molecular biology begins in the 1930s with the convergence of various, previously distinct biological disciplines: biochemistry, genetics, microbiology, and virology. With the hope of understanding life at its most fundamental… … Wikipedia
St. Scholastica's College — Infobox University name = St. Scholastica s College, Manila native name = Kolehiyo ng Santa Eskolastika, Maynila motto = Ora et Labora (Prayer and Work) established = 1906 type = Private, Benedictine president = Sr. M. Angelica Leviste, OSB city … Wikipedia
In vitro compartmentalization — (IVC) is an emulsion based technology that generates cell like compartments in vitro . These compartments are designed such that each contains no more than one gene. When the gene is transcribed and/or translated, its products (RNAs and/or… … Wikipedia

Academic Dictionaries and Encyclopedias

Cognate linkages

Look at other dictionaries:

Share the article and excerpts

Academic Dictionaries and Encyclopedias

Wikipedia

Cognate linkages

Look at other dictionaries:

Share the article and excerpts

Direct link