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Transcript of Peaucellier–Lipkin linkage
Drawing a straight line looks very easy but making a mechanism that could synthesize one...well you gotta scratch your grey matter!!!
Before the development of the planar and the milling machine, it was extremely difficult to machine straight and flat surfaces.
Although we had approximate straight line generating mechanisms like Watt's, Hoeken's and Chebyshev's, a perfect straight line was still a dream.
The Peaucellier–Lipkin linkage was the first planar linkage capable of transforming rotary motion into perfect straight-line motion, and vice versa.
Goals of this investigation includes studying the current mechanism, creating an ADAMS simulation for it, and suggesting some improvement techniques that can enhance its abilities.
Geometry of the linkage
This linkage, in its rhomboidal eight-bar configuration shown in the figure, produces a
straight line as the input link rotates back and forth, reaching constant velocity as the input link passes the horizontal
Mathematics of the linkage
The mathematics of the Peaucellier–Lipkin linkage is directly related to the inversion of a circle.
Let x = BP = PD, y = OB,
and h = AP.
OB.OD = y(y + 2x) = y2 + 2xy
OA2 = (y + x)2 + h2 (by Pythagoras)
AD2 = x2 + h2 (by Pythagoras)
OA2 – AD2 = y2 + 2xy = OB.OD.
Since OA and AD are both fixed lengths, then the product of OB and OD is a constant:
OB.OD = k2 (say).
Since points O, B, D are collinear, then D is the inverse of B with respect to the circle with center O and radius k.
Well..you asked for it!!
No steam engine would have been possible without these.
No Automobiles!! :O Automobiles would have been a cripple without these.
Manufacturing with precision would not have been possible.
So our mechanism is.....
A slider rocker can be used as a driver for this linkage.
So basically this free point can be use as a joint where any mechanism can be fit in to just modify the present linkage to our need
Waldron-Kinzel : Kinematics,Dynamics and Design of Machinery
Another improvement can be using a follower with the linkage. This can enhance its properties
The follower can be either a simple 4 bar setup or a suitable structure to need of the client
We can also attach a gripper to point P in the mechanism.
Some more Improvements