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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
657:{\displaystyle \varphi _{2}=\arcsin \left-\arccos \left({\frac {L_{4}^{2}+{\overline {AO_{2}}}^{2}-L_{3}^{2}}{2\,L_{4}\,{\overline {AO_{2}}}}}\right)\,}
968:. In this application the approximate straight movement is translated away from the line's midpoint, but it is still essentially the same mechanism.
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106:. One of the problems was the construction of a linkage that converts a rotary motion into an approximate straight-line motion (a
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Consequently, we can write the motion of point P, using the two points defined above and the definition of the middle point.
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with the horizontal. Regardless of the input angle, it is possible to compute the motion of two end-points for link
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949:{\displaystyle \varphi _{\text{max}}=\arccos \left({\frac {-1}{5}}\right)\approx 101.537^{\circ }.\,}
872:{\displaystyle \varphi _{\text{min}}=\arccos \left({\frac {4}{5}}\right)\approx 36.8699^{\circ }.\,}
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Chebyshev linkages did not receive widespread usage in steam engines, but are commonly used as the
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Chebyshev's Lambda
Mechanism (one blue and one green) shows an identical motion path
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And ultimately, we will write the output angle in terms of the input angle,
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while the motion of point B will be computed with the other angle,
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356:{\displaystyle x_{B}=L_{1}-L_{4}\cos(\varphi _{2})\,}
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The limits to the input angles, in both cases, are:
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Link 1 (horizontal distance between ground joints):
1038:"How to draw a straight line, by A.B. Kempe, B.A."
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98:It was invented by the 19th-century mathematician
151:that we will name A and B, and the middle point.
790:{\displaystyle y_{P}={\frac {y_{A}+y_{B}}{2}}\,}
726:{\displaystyle x_{P}={\frac {x_{A}+x_{B}}{2}}\,}
102:, who studied theoretical problems in kinematic
420:{\displaystyle y_{B}=L_{4}\sin(\varphi _{2})\,}
274:{\displaystyle y_{A}=L_{2}\sin(\varphi _{1})\,}
210:{\displaystyle x_{A}=L_{2}\cos(\varphi _{1})\,}
8:
966:'Horse head' design of level luffing crane
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1053:using the Molecular Workbench software
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1022:– Cross link straight-line mechanism
23:Animation for the Chebyshev linkage.
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137:with the horizontal or link
16:Four-bar straight-line mechanism
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110:). This was also studied by
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71:Illustration of the limits
1059:simulation of the linkage
992:of the Chebyshev linkage.
1089:Straight line mechanisms
986:Chebyshev lambda linkage
25:Dimensions (unit length
1001:Straight line mechanism
108:straight line mechanism
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126:Equations of motion
81:Chebyshev's linkage
1049:2011-07-25 at the
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1084:Linear motion
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93:linear motion
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1044:A simulation
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801:Input angles
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116:steam engine
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1073:Categories
1063:A 3D video
1057:A Geogebra
1007:References
112:James Watt
104:mechanisms
77:kinematics
938:∘
930:≈
914:−
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889:φ
861:∘
853:≈
832:
817:φ
642:¯
590:−
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527:−
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488:φ
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457:
442:φ
405:φ
398:
341:φ
334:
318:−
259:φ
252:
195:φ
188:
1047:Archived
972:See also
990:cognate
934:101.537
857:36.8699
988:, the
901:arccos
829:arccos
530:arccos
454:arcsin
47:
32:
960:Usage
83:is a
893:max
821:min
478:sin
395:sin
331:cos
249:sin
185:cos
75:In
29:):
1075::
122:.
95:.
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39:+
943:.
926:)
921:5
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908:(
898:=
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844:5
841:4
836:(
826:=
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776:B
772:y
768:+
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759:y
752:=
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743:y
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708:x
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688:=
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632:O
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619:4
615:L
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537:(
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451:=
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236:=
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172:=
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146:L
142:4
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132:L
61:a
59:4
53:a
51:5
41:a
37:a
27:a
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