Hart's inversors
{{Short description|Planar straight-line mechanisms}}
[[File:Harts Inversor 1.gif|thumb|Animation of Hart's antiparallelogram, or first inversor.
Link dimensions:
{{legend-line|solid cyan|Crank and fixed: {{mvar|a}}}}
{{legend-line|solid #5ce336|Rocker: {{mvar|b}} (anchored at midpoint)}}
{{legend-line|solid #edc928|Coupler: {{mvar|c}} (joint at midpoint)}}
b &< c \\[4pt]
2a &< \tfrac{1}{2}b + \tfrac{1}{2}c \\[2pt]
\tfrac{1}{2}c &< \tfrac{1}{2}b + 2a
\end{align}
]]
Hart's inversors are two planar mechanisms that provide a perfect straight line motion using only rotary joints.{{cite web |url=http://alexandria.tue.nl/repository/freearticles/603890.pdf |title=True straight-line linkages having a rectlinear translating bar}} They were invented and published by Harry Hart in 1874–5.{{cite book |url=https://books.google.com/books?id=UG0RlFBqwrgC&pg=PA307 |title=International Symposium on History of Machines and Mechanisms|isbn=9781402022043|last1=Ceccarelli|first1=Marco|date=23 November 2007}}
Hart's first inversor
Hart's first inversor, also known as Hart's W-frame, is based on an antiparallelogram. The addition of fixed points and a driving arm make it a 6-bar linkage. It can be used to convert rotary motion to a perfect straight line by fixing a point on one short link and driving a point on another link in a circular arc.{{cite web |url=http://www.cut-the-knot.org/Curriculum/Geometry/HartInversor.shtml |title=Harts inversor (Has draggable animation) }}
=Rectilinear bar and quadruplanar inversors=
{{Main|Quadruplanar inversor}}
File:Quadruplanar Inversor Derivation Alt.gif from Hart's first inversor.]]
Hart's first inversor is demonstrated as a six-bar linkage with only a single point that travels in a straight line. This can be modified into an eight-bar linkage with a bar that travels in a rectilinear fashion, by taking the ground and input (shown as cyan in the animation), and appending it onto the original output.
A further generalization by James Joseph Sylvester and Alfred Kempe extends this such that the bars can instead be pairs of plates with similar dimensions.
Hart's second inversor
[[File:Harts Inversor 2.gif|thumb|Animation of Hart's A-frame, or second inversor.
{{legend-line|solid cyan|Double rocker: {{math|3a + a}} (distance between anchors: {{math|2b}})}}
{{legend-line|solid #5ce336|Coupler: {{mvar|b}}}}
{{legend-line|solid #edc928|Tip of the A: {{math|2a}}}}
]]
Hart's second inversor, also known as Hart's A-frame, is less flexible in its dimensions, but has the useful property that the motion perpendicularly bisects the fixed base points. It is shaped like a capital A – a stacked trapezium and triangle. It is also a 6-bar linkage.
=Geometric construction of the A-frame inversor=
Example dimensions
These are the example dimensions that you see in the animations on the right.
Mecanismo de Hart (2).png|{{ubl | Hart's first inversor: | AB {{=}} Bg {{=}} 2 | CE {{=}} FD {{=}} 6 | CA {{=}} AE {{=}} 3 | CD {{=}} EF {{=}} 12 | Cp {{=}} pD {{=}} Eg {{=}} gF {{=}} 6}}
Mecanismo de Hart.png|{{ubl | Hart's second inversor: | AB {{=}} AC {{=}} BD {{=}} 4 | CE {{=}} ED {{=}} 2 | Af {{=}} Bg {{=}} 3 | fC {{=}} gD {{=}} 1 | fg {{=}} 2}}
See also
- Linkage (mechanical)
- Quadruplanar inversor, a generalization of Hart's first inversor
- Straight line mechanism
Notes
References
{{reflist}}
External links
{{commons category|Hart's inversor}}
- [https://web.archive.org/web/20160712063400/http://web.mat.bham.ac.uk/C.J.Sangwin/howroundcom/straightline/Hart/Hart-A-frame.html bham.ac.uk] – Hart's A-frame (draggable animation) 6-bar linkage {{dead link|date=June 2017}}