1. Graphical Analysis: Positions, Velocity and Acceleration (simple joints)
ME 3230
Dr. R. Lindeke
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11/20/2018

2. Topics For Review Positional Analysis, The Starting Point
The Velocity Relationship
The velocity polygon
Velocity Imaging
Acceleration Relationships
The acceleration polygon
Acceleration Imaging
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3. Positional Analysis Starts with drawing the links to scale
This is easily done in a graphical package like CATIA
This process creates the Linkage Skeleton
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4. From Here we need to address the “Trajectory” Models
This is the positional model – frame of ref. is fixed while the link containing both pts. A and B rotates at  an angular velocity which leads to this velocity model for the points – within the same link!
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5. Relative Velocity – as a vector
Magnitude is:
Direction is Normal to both  and the vector rB/A
We determine the relative velocity’s direction by rotating rB/A by 90 in the direct of  (CCW or CW)
Using Right Hand Rules!
We can determine a 3rd (direction or magnitude) from knowledge of either of the other two!
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6. Step 2:The Velocity Polygon
The lower right vertex is labeled o – this is called a velocity pole
Single subscripted (absolute) velocities originate from this velocity pole
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7. Step 3: “Graphical” Acceleration Analysis
This relationship can be derived by differentiating the absolute velocity model for B
Leading to an acceleration model:
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8. Looking deeper into Relative Acceleration we write:
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9. Graphically (the acceleration polynomial):
Note o’, it is the acceleration pole
Absolute (single subscripted) accelerations originate from this pole
Note: Tangential Acc is Normal to rB/A & Radial Component is Opposite in Direction to rB/A
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10. Lets Try an Example Lengths:
O2A: 30 mm
O4B: 56 mm
O2O4: 81 mm
AB: 63 mm
AC: 41 mm
BC: 31 mm
Link 2 is rotating CCW at 5 rad/sec (if we are given ‘speed’ in RPM we convert to rad/sec:  = N*(2 rad/rev60s/m))
At 120 wrt Base Axis (X0)
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11. After Drawing Linkage to Scale (CATIA)
Easily done – drew Base line to scale then O2A to scale at 120, 2 scaled construction circles to establish B and 2 more scaled construction circles to establish C and connected the lines
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12. Building Velocity Graph:
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13. As Seen here:
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14. Next We Compute 3 and 4:
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15. Computing Velocity of Point c:
We can solve these 2 equations using Graphical Simultaneous methods
Sketch both lines by starting a the “tip of the ‘oa’ or ‘ob’ vector” with a line normal to AC or BC respectively
Pt. c is their intersection
Draw and measure ‘oc’ is is vC
Alternatively we could do the vector math with either equation and the various vectors resolved to XYZ coordinates
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16. Graphically: Vc = 137.73 mm/s at 183.45 Vc/b & Vc/a as reported
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17. Now we build Acceleration “Graphs”
The steps for the 1st link:
I will scale acc. At 1/10 so the magnitude of aA is 75 units on the graph
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18. Means this: Note: accA line is opp. Direction to Link 2 ME 3230
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19. Getting Acc for Pt. B – first approach
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20. Getting Acc for Pt. B – first approach
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21. Getting Acc for Pt. B – second approach
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22. Getting Acc for Pt. B – second approach
AccB = *10 = mm/s2
Getting Acc for Pt. B – second approach
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23. Find b3’ & b4’ Where the 2 at Vectors Intersect
On Graph, measure the two at’s and solve for angular acceleration:
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24. Finding Acc. Of Pt. C
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25. Finding Acc. Of Pt. C o’c’ is at -35.648 or 324.352
And AccC = mm/s2
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26. Understanding “Imaging”: Velocity and Acceleration
If one knows velocity and/or acceleration for two points on a link any other point’s V & A is also know!
We speak of Physical Shape and the Velocity or Acc image of the Shapes
Ultimately we will develop analytical models of this relationship but we can use graphical information (simply in planer links) to see this means
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27. Velocity Imaging:
The Shape of the Velocity Polygon is determined by the physical dimensions of the linkage being studied
A Triangular link will produce a similar velocity triangle
For Planer Linkages, the similar polygon is rotated +90 in the direction of the link’s angular velocity (i)
The size of the Velocity Polygon is determined by the magnitude of the link’s angular velocity (i)
Rotated 90 CW!
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28. Acceleration Imaging:
The Acceleration Polygon is Similar to the Physical Linkage model
The ‘Magnification Factor’ is: (4 + 2)
Angle of Rotation is: =+Tan-1(/)
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29. For our Model:
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30. What it Means:
To get any additional point on a link (polygon shape) once two are know, Just make a similar shaped image on Velocity or Acceleration Graph.
Each added “Point of Interest” Velocity or Acceleration is then sketched in and relevant values can be determined for all needs
To make the similar shapes, use equal angles and ‘Law of Sines’ or sketching tricks
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