Mechanical Linkages: SketchPad Files
This was a big interest of mine when I first bought Geometer's SketchPad. I think that it was a very good way for me to improve my understanding of the software and as such, several of the sketches below are not particularly mathematical but rather more experimental. However, there are several straight-line and approximate straight-line mechanisms that I hope will be of interest. Two of my particular favourites are "JCB" and "Delaunay's Ellipsograph Linkage 2": the JCB is just that! - a little extravagance on my behalf to design a Dumper-Truck Shovel Linkage! and Delaunay's Linkage was based on a design from a superb book: "MECHANISMS FOR THE GENERATION OF PLANE CURVES" by I. I. Artobolevskii, a Russian Professor of Applied Mechanics at the Academy of Sciences in the former USSR I got hold of an English Translation by R. D. Wills and W. Johnson from the British Lending Library (Search on the given Title) but as yet have not been able to locate a copy for myself. The book has the most exquisite type setting and diagrams - a beautiful thing to behold if you're a Maths book buff!
Notes:
(1). All of these filenames in this page are LONG! - Therefore, if you want
to preserve these (Because Version 3.* of SketchPad does not use long filenames)
you must either RIGHT-CLICK on the file links within the above
directories and then choose "Save Target As...". This allows you to
save the file directly to a disk/drive or after using a LEFT-CLICK
select the "Save this file to disk" Option Box rather
than selecting the "Open this file from its current location". If
you choose this last option then Geometer's SketchPad will automatically open
- if you have it! - but then after viewing the file if you then try to save
it you will not be able to use a long filename.
Sketches
D - Cardioid Linkage.gsp
Not of my making but a favourite of mine. See the host of superb
files by Chuan, et al. Geometric
Construction. A collection of advanced constructions, construction challenges
and exercises, with downloadable sketches and JavaSketchpad
illustrations.
D - Nephroid Linkage.gsp
Again one of the superb files by Chuan, et al. Geometric
Construction. A collection of advanced constructions, construction challenges
and exercises, with downloadable sketches and JavaSketchpad
illustrations.
D - Limacon of Pascal.gsp (added
11/04/02)
In trying to get an understanding of some of the superb linkages from Chuan,
et al. Geometric
Construction, I managed to get hold of a second-hand copy (Link1)
of "A Handbook on Curves and their Properties" by Robert C. Yates
(1947), as referenced by Chuan. I thoroughly recommend this wonderful little
book - especially if you can get hold of a cheap copy (around $40 or so!). To
help with the construction I found it useful to have a script that constructs
an isosceles trapezium from just three points
and which also draws in the cross-diagonals. I have also included an instruction
sheet written in Word or as a PDF
file which should be understandable if you have a little SketchPad experience.
There is a trick done part-way through the construction - a reflection - that
enables the full locus to be constructed. I hope it all helps!
D - Cardioid (Yates' Linkage).gsp
(added 12/04/02)
Another construction from "A Handbook on Curves and their
Properties" by Robert C. Yates (1947). In a manner similar to the
Limacon above, it is based upon two similar, isosceles trapeziums and for its
construction I again used my script that constructs an isosceles
trapezium from just three points and which also draws in the cross-diagonals
for use within the linkage.
D - Conics (added 12/04/02)
Another construction from "A Handbook on Curves and their
Properties" by Robert C. Yates (1947). This is made from a variable
trapeziodal 3-bar linkage onto which a Peaucellier Inverser linkage is attached.
Try varying the lengths of the bars to form either a hyperbolic, parabolic or
elliptical locus for the inverted point P'
D - Parallels to an Ellipse.gsp
(added 15/04/02)
Another construction from "A Handbook on Curves and their
Properties" by Robert C. Yates (1947). This mechanism is made from
two pairs of isosceles trapeziodal 3-bar linkages, a kite and a rhombus! It
draws a (blue) curve parallel to an ellipse that is itself the locus of a point
on the mechanism.
D - Nephroid
Linkage (Variable Links).gsp (added 19/04/02)
I've finally sorted out how to construct the Nephroid as done in the picture
above by Chuan.
It is based upon three pairs of isosceles trapeziodal 3-bar linkages and a parallelogram.
I have made many of the links of variable length and the relative positions
of some of the joints can also be varied. This means that many other, very surprising
locii can be displayed - click HERE
or on the smaller picture below to see some examples of these "Deformed
Nephroidal" locii (added 02/07/02)
. A button to return the linkage to that generating a Nephroid is included for
convenience (...to save you pulling your hair out!)
D - Delaunay's Ellipsograph Linkage 1.gsp
From "Mechanisms for the Generation of Plane curves"
D - Delaunay's Ellipsograph Linkage 2.gsp
From "Mechanisms for the Generation of Plane curves"
D - Ellipse Linkage (1).gsp
Not of my making but another favourite of mine.
D - Ellipse Linkage (2).gsp
D - Falling Ladder.gsp
A slight twist on the usual problem - this ladder slides through a hole as it
falls.
D - Garage Door Linkage.gsp
A little diversion but instructive in parts!
D - Gyershgorin's
General Three-Link Loci.gsp
From "Mechanisms for the Generation of Plane curves". I liked this
one so much that I have made it very 'customisable'. Some surprising locii result
- try to find a condition on the rod lengths such that the loci is an ellipse.
D
- Hoecken's Approximate Straight Line Linkage.gsp
This is a nice example of a rapid return system too.
D - JCB Piston.gsp
Yes - a piston with (green) oil in it! This construction is used in the full
JCB sketch below. In SketchPad select "Display --> Show All Hidden"
from the Toolbar to see the relatively painless construction!
D - JCB.gsp
This took me DAYS to make but I think it was worth it!
D - Mechanical Typewriter
Linkage.gsp
D -
Kempe's Line Straight Line Linkage (1).gsp
D -
Kempe's Line Straight Line Linkage (2).gsp
D - Kempe's Line Straight Line Linkage (3).gsp
As above but you can vary the lengths of some of the rods.
D - Lemniscate of Bernoulli.gsp
Not of my making but another nice one - though not as straightforward as it
may seem to plot the full loci without the trick that this sketch uses.
D - Corrected
Lemniscate of Bernoulli.gsp
The failings of the linkage above are easily corrected by using my script
that constructs an isosceles trapezium
D
- Loci of Two Fixed Length Rods with Fixed Angle between them.gsp
Another one of my early experiments!
D - My First Linkage.gsp
Enough said!
D - Peaucellier's Straight Line Linkage.gsp
D - Straight Line
5 Bar Linkage.gsp
From "Mechanisms for the Generation of Plane curves".
D - Sylvester's
Straight Line Linkage.gsp
D - Crosby's Staight
Line Linkage.gsp (added 09/04/02)
This was often used as the mechanism to draw a vertical line on a rotating,
vertical cylindrical drum. Applications include the recording of vibrations,
pressure variations, etc. In this sketch the aim is to adjust the lengths of
the bars so as to produce vertical motion.
D
- Tchebycheff's Approximate Straight Line Linkage.gsp
D - Tchebycheff's
Engineers' Table.gsp
Another bit of excess - but a valid, real-life application.
D - Double Bow.gsp
From "Mechanisms for the Generation of Plane curves".
Links