Planar Linkage Analysis using GeoGebra

Planar Linkage Analysis using GeoGebra
� Understand the Basic Types of Planar
Four-Bar Linkages
� Learn the Basic Geometric tools
available in GeoGebra
� Use GeoGebra to Construct Planar
Four-Bar Linkages
� Use GeoGebra to Create Animations of
Four-Bar Linkages
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Introduction to Four-Bar Linkages
A machine is a system or device consisting of fixed and moving parts that modifies
mechanical energy to do work. Assemblies within a machine that control movement are
often called mechanisms. A mechanism is a group of links connected together for the
purpose of transmitting forces or motions. A four-bar linkage or four-bar mechanism
is the simplest movable mechanism commonly used in machines. A four-bar linkage
consists of four rigid bodies (called bars or links) with one link fixed. The four links are
each attached to two others by single joints (pivots) to form a closed loop. The fixed link
is referred to as the frame; one of the rotating links is called the driver or crank; the
other rotating link is called the follower or rocker; and the floating link is called the
connecting rod or coupler. Some of the more commonly used four-bar linkages are
listed below.
� Depending upon the arrangement and lengths of the links, different types of
motions can be generated. Different mechanisms can also be formed by fixing
different links of the same chain.
Crank-Rocker
� The Crank-Rocker
mechanism is a four-bar
linkage in which the shorter
link makes a complete
revolution and the opposite
link rocks (oscillates) back
and forth.

Crank-Rocker
� The Double-Crank or Drag-
Link mechanism is a four-bar
linkage with two opposite links
making complete revolutions.
Double-Crank
Double-Rocker
Planar Linkage Analysis with GeoGebra 3
� Note that by fixing the
opposite link of the same
mechanism, another
Crank-Rocker
mechanism is formed.
� The Double-Rocker
mechanism is a four-bar
linkage in which the crank and
follower rock (oscillate) back
and forth; none of the links can
make full revolution.
� Note the four different
mechanisms above are formed
by fixing different links of the
same links.

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� The Slider-Crank mechanism is a special case of the four-bar linkage. As the
follower link of a crank-rocker linkage gets longer, the path of the pin joint
between the connecting rod and the follower approaches a straight line. Slider-
Crank is a mechanism for converting the linear motion of a slider into rotational
motion or vice-versa.
Slider-Crank Variation
Scotch Yoke
Slider-Crank
� The Scotch Yoke is
most commonly used
in reciprocating piston
pumps and in control
valve actuators in
high pressure oil and
gas pipelines.

Introduction to GeoGebra
Planar Linkage Analysis with GeoGebra 5
GeoGebra is an award winning interactive dynamic geometry software that joins
geometry, algebra and calculus. Interactive dynamic geometry software is a type of
computer program that allows the creation and then manipulation of geometric
constructions. In most interactive geometry software, constructions can be made with
points, vectors, segments, lines, polygons, conic sections, inequalities, implicit
polynomials and functions.
The development of GeoGebra was started by Prof. Markus Hohenwarter for
mathematic education. Prof. Markus Hohenwarter started the project in 2001 at the
University of Salzburg, continuing it at Florida Atlantic University (2006–2008), Florida
State University (2008–2009), and now at the University of Linz together with the help of
open-source developers and translators all over the world.
In GeoGebra, geometric entities can be entered and modified directly on screen, or
through the Input Bar. GeoGebra has the ability to use variables for numbers, vectors and
points; derivatives and integrals of functions can also be evaluated.
The main webpage of GeoGebra is at http://www.geogebra.org/cms/en
� GeoGebra is an open source free software, written in
Java and thus available for multiple platforms,
including Windows, Mac and Linux systems.

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� Two versions of GeoGebra are available for download. GeoGebra is the full
version and GeoGebraPrim has the same capabilities as the full version but uses a
more simplified user interface. Files created in GeoGebra can be loaded in
GeoGebraPrim and vice versa.
� Note the installation of the Java Platform is required as GeoGebra is written in
Java.
.
� For most versions of GeoGebra, the
installed program can be accessed through
the Start menu or the GeoGebra icon on
the desktop.

Planar Linkage Analysis with GeoGebra 7
1. Select the GeoGebra option on the Start menu or select the
GeoGebra icon on the desktop to start GeoGebra. The GeoGebra
main window will appear on the screen.
2. In the View pull-down menu, switch
on the Grid display by clicking on the
icon with the left-mouse-button as
shown.
� The GeoGebra screen layout contains the pull-down menus, the Standard toolbar,
the Algebra area, the graphics area, and the Input Bar at the bottom of the screen.
3. On your own, adjust the display of the GeoGebra main screen so that the Algebra
area and the Input Bar are available as shown.

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5. Click on the origin of the
coordinate system to place the
center of the circle. Note the
created center point is also
added in the Algebra area that is
toward the left.
7. Pick OK to accept the selected settings.
9. Use the mouse wheel to
Zoom/Pan and adjust the
current display.
4. Select the Circle with Center and
Radius icon in the Standard toolbar
area.
6. In the Radius input box, enter
4 to create a circle with a
radius of 4 units in size.
8. On your own, create another circle at
coordinates (12,0) and radius 9 units as
shown.

11. Select the X Axis as the first
entity to define the intersection
points.
� Note two intersection
points are created,
Point C and Point D.
Planar Linkage Analysis with GeoGebra 9
10. Activate the Intersect Two Objects
command by clicking on the icon as shown.
This will create points at the intersections of
two selected objects.
12. Select the smaller circle
as the second entity to
define the intersection
points.

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14. Select Point D as the
object to be rotated.
16. In the Angle input box,
enter 45 as the angle of
rotation.
17. Click OK to accept the
setting and create the
new point.
13. Activate the Rotate Object around
Point by Angle command by clicking
on the icon as shown. This will create a
new point by rotating an existing point.
15. Select Point A as the reference
axis of rotation.

19. Select Point DꞋ as the center
of the new circle. Note the
created center point is also
added in the Algebra area
that is toward the left.
21. Click OK to accept the selected settings.
Planar Linkage Analysis with GeoGebra 11
18. Select the Circle with Center and
Radius icon in the Standard toolbar
area.
20. In the Radius input box,
enter 10 to create a circle
with a radius of 10 units in
size.
22. Activate the Intersect Two Objects
command by clicking on the icon as shown.
This will create points at the intersections
of two selected objects.

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24. Select Circle d as the
second entity to define the
intersection points.
23. Select Circle e as the first
entity to define the
intersection points.
� Note two intersection
points are created, Point
E and Point F.

26. Click on Point A to place the
first point of the line.
28. On your own, create two
additional line segments, Line
DE and Line EB as shown.
Planar Linkage Analysis with GeoGebra 13
25. Activate the Segment between
Two Points command by
clicking on the icon as shown.
This will create a line by
selecting two endpoints.
27. Click on Point DꞋ to create a line as
shown.

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Turn off the Display of Objects
2. Select Circle d, the circle
on the right, as shown.
3. Inside the graphics area,
right-mouse-click once to
bring up the option menu.
4. Click the Show Object
icon to turn OFF the display
of the selected circle.
5. Inside the Algebra area,
click once with the rightmouse-button
on Circle
e to display the option
menu. Click on Show
Object to toggle OFF
the display of the circle.
1. In the Standard toolbar area, click on the Move icon to
activate the command.
� Note the corresponding circle,
Circle d, is also highlighted in the
Algebra area. The unfilled icon
indicates the display of the object
has been turned OFF.

8. In the Algebra area, click once with the
right-mouse-button on Point C to
display the option menu. Click on
Show Object to toggle OFF the
display of the selected item.
Planar Linkage Analysis with GeoGebra 15
6. Select Point F, the point below the
line segments, as shown.
7. On your own, turn OFF the display of
the object through the right-click option
menu as shown.

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Adding a Slider Control
2. In the graphics area, select a
location near the upper left corner
to place the Slider Control as
shown.
4. Set the Interval
settings to the three
values, 0, 360 and
1.0, as shown.
5. Click Apply to
accept the selected
settings
1. In the Standard toolbar select the
Slider command by left-clicking
once on the icon.
3. Set the Slider option to Angle as shown.
Note the angle name is automatically set
to α.

Planar Linkage Analysis with GeoGebra 17
6. In the Standard toolbar area, click on the Move
icon to activate the command.
7. Drag the handle of the Slider Control, and notice Angle α is
adjusted.
8. In the Algebra area, double click
with the left-mouse-button on
Point DꞋ to enter the Object
Redefine mode.
9. Modify the angle variable, the second variable in the edit box, to α as shown.
10. Click OK to accept the setting and exit the Redefine command.

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11. Drag the handle of the Slider Control, and notice the positions of the links of
the four-bar linkage are adjusted accordingly.

Using the Animate option
2. Inside the Algebra area, click once with the
right-mouse-button on Angle to display the
option menu. Click on Object Properties
to activate the command.
Planar Linkage Analysis with GeoGebra 19
1. Inside the graphics area, click once with the rightmouse-button
on Slider Control to display the
option menu. Click on Animation On to toggle
ON the animation of the Slider Control.
3. Set the animation repeat
option to Increasing as
shown.
4. Click Close to accept the setting and exit the object properties
command.
5. On your own, examine the animation of the four-bar
linkage; turn OFF the animation option before
proceeding to the next section.

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Tracking the Path of a point on the Coupler
2. Select Point DꞋ as the center of
the new circle.
5. On your own, create
another radius 7.5
circle centered at
Point E as shown.
1. Select the Circle with Center and
Radius icon in the Standard toolbar
area.
3. In the Radius input box,
enter 7.5 as the radius.
4. Click on the OK button to
accept the settings and
create the circle.

7. Select Circle g as
the first entity to
define the
intersection points.
Planar Linkage Analysis with GeoGebra 21
6. Activate the Intersect Two Objects command
by clicking on the icon as shown. This will create
points at the intersections of two selected objects.
8. Select Circle h as the
second entity to define
the intersection points.

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10. Select Point DꞋ as
the first corner of the
polygon.
11. Select Point G as
the second corner of
the polygon.
9. Activate the Polygon command by clicking on the
icon as shown. This will create a polygon by
defining the corner points of a polygon.
12. Select Point E as
the third corner of
the polygon.
13. Select Point DꞋ
again to form a
triangle.
14. On your own, turn
OFF the display of
Point H, Circle g
and Circle h.

16. In the graphics area, select
Point G as shown.
Planar Linkage Analysis with GeoGebra 23
15. In the Standard toolbar area, click on the Move
icon to activate the command.
17. Inside the graphics area,
click once with the rightmouse-button
to display
the option menu. Click on
Trace On to activate the
command.
18. Drag the handle of the Slider Control, and notice Angle α is
adjusted.

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� The locus of point G, also known as a coupler curve, is generated as Angle α is
adjusted. Note that by varying the lengths of the links in the mechanism, different
coupler curves can be generated. The interactive dynamic geometry software can
be used to aid linkage design and analysis.
� Note that the Animation On command can also
be used to generate the coupler curve.

Exercises:
1. Crank-Slider Mechanism
AB = 2.5, BC = 5
2. Watt Straight Line Mechanism
Planar Linkage Analysis with GeoGebra 25
ADꞋꞋ=BE, EDꞋꞋ=1/2 ADꞋꞋ, AB = 2 BE, G is at the midpoint of EDꞋꞋ.
A and B are fixed points.

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3. Hoekens Straight Line Mechanism
BE = ECꞋ = EH = 2.5 ACꞋ, AB = 2 ACꞋ
A and B are fixed points and Link CꞋ-E-H is a rigid link.