Animation of Hart's antiparallelogram, or first inversor. Link dimensions: Crank and fixed: a Rocker: b (anchored at midpoint) Coupler: c (joint at midpoint) b < c 2 a < 1 2 b + 1 2 c 1 2 c < 1 2 b + 2 a {\displaystyle {\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 inversors are two planar mechanisms that provide a perfect straight line motion using only rotary joints. They were invented and published by Harry Hart in 1874–5.

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.

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: Double rocker: 3a + a (distance between anchors: 2b) Coupler: b Tip of the A: 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.

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

See also

Notes

External links

  • – Hart's A-frame (draggable animation) 6-bar linkage [dead link]