Hart's inversors

Hart's inversors are two planar mechanisms that provide a perfect straight line motion using only rotary joints.[1] They were invented and published by Harry Hart in 1874–5.[1][2]

Animation of Hart's antiparallelogram, or first inversor.
Link dimensions:

Crank and fixed:

a

Rocker:

b

(anchored at midpoint)

Coupler:

c

(joint at midpoint)

{\begin{aligned}b&<c\\[4pt]2a&<{\tfrac {1}{2}}b+{\tfrac {1}{2}}c\\[2pt]{\tfrac {1}{2}}c&<{\tfrac {1}{2}}b+2a\end{aligned}}

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.[1][3]

Rectilinear bar and quadruplanar inversors

Animation to derive a Quadruplanar inversor 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

Animation of Hart's A-frame, or second inversor.
Link dimensions:[Note 1]

Double rocker:

3 a + a

(distance between anchors:

2 b

)

Coupler:

b

Tip of the A:

2 a

Hart's second inversor, also known as Hart's A-frame, is less flexible in its dimensions[Note 1], 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.

Example dimensions

These are the example dimensions that you see in the animations on the right.

- Hart's first inversor:
- AB = Bg = 2
- CE = FD = 6
- CA = AE = 3
- CD = EF = 12
- Cp = pD = Eg = gF = 6
- Hart's second inversor:
- AB = AC = BD = 4
- CE = ED = 2
- Af = Bg = 3
- fC = gD = 1
- fg = 2

Notes

The current documented relationship between the links' dimensions is still heavily incomplete. For a generalization, refer to the following GeoGebra Applet: [Open Applet]

References

"True straight-line linkages having a rectlinear translating bar" (PDF).
Ceccarelli, Marco (23 November 2007). International Symposium on History of Machines and Mechanisms. ISBN 9781402022043.
"Harts inversor (Has draggable animation)".

External links

bham.ac.uk – Hart's A-frame (draggable animation) 6-bar linkage

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[Note01-4] The current documented relationship between the links' dimensions is still heavily incomplete. For a generalization, refer to the following GeoGebra Applet: [Open Applet]

[Dijksman-1] "True straight-line linkages having a rectlinear translating bar" (PDF).

[2] Ceccarelli, Marco (23 November 2007). International Symposium on History of Machines and Mechanisms. ISBN 9781402022043.

[3] "Harts inversor (Has draggable animation)".