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Graphical Analysis: Positions, Velocity and Acceleration (simple joints)

Graphical Analysis: Positions, Velocity and Acceleration (simple joints). ME 3230 Dr. R. Lindeke. Topics For Review. Positional Analysis, The Starting Point The Velocity Relationship The velocity polygon Velocity Imaging Acceleration Relationships The acceleration polygon

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Graphical Analysis: Positions, Velocity and Acceleration (simple joints)

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

  2. Topics For Review • Positional Analysis, The Starting Point • The Velocity Relationship • The velocity polygon • Velocity Imaging • Acceleration Relationships • The acceleration polygon • Acceleration Imaging ME 3230

  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 ME 3230

  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! ME 3230

  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! ME 3230

  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 ME 3230

  7. Step 3: “Graphical” Acceleration Analysis • This relationship can be derived by differentiating the absolute velocity model for B • Leading to an acceleration model: ME 3230

  8. Looking deeper into Relative Acceleration we write: ME 3230

  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 ME 3230

  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) ME 3230

  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 ME 3230

  12. Building Velocity Graph: ME 3230

  13. As Seen here: ME 3230

  14. Next We Compute 3 and 4: ME 3230

  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 ME 3230

  16. Graphically: Vc = 137.73 mm/s at 183.45 Vc/b & Vc/a as reported ME 3230

  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 ME 3230

  18. Means this: Note: accA line is opp. Direction to Link 2 ME 3230

  19. Getting Acc for Pt. B – first approach ME 3230

  20. Getting Acc for Pt. B – first approach ME 3230

  21. Getting Acc for Pt. B – second approach ME 3230

  22. Getting Acc for Pt. B – second approach AccB = 77.399*10 = 773.99 mm/s2 ME 3230

  23. Find b3’ & b4’ • Where the 2 at Vectors Intersect • On Graph, measure the two at’s and solve for angular acceleration: ME 3230

  24. Finding Acc. Of Pt. C ME 3230

  25. Finding Acc. Of Pt. C • o’c’ is at -35.648 or 324.352 • And AccC = 589.81 mm/s2 ME 3230

  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 ME 3230

  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! ME 3230

  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(/) ME 3230

  29. For our Model: ME 3230

  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 ME 3230

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