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)

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

Hart's 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

Main article: Quadruplanar inversor

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: 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,^{[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.

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 = 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

See also

• Linkage (mechanical)
• Quadruplanar inversor, a generalization of Hart's first inversor
• Straight line mechanism

Notes

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