Physics:Hart's inversors

Short description: Planar straight-line mechanisms

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

  Crank and fixed: a

  Rocker: b (anchored at midpoint)

  Coupler: c (joint at midpoint)

[math]\displaystyle{ \begin{align} b &\lt c \\[4pt] 2a &\lt \tfrac{1}{2}b + \tfrac{1}{2}c \\[2pt] \tfrac{1}{2}c &\lt \tfrac{1}{2}b + 2a \end{align} }[/math]

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.

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

External links

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