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

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 11/20/2018

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 11/20/2018

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 11/20/2018

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 11/20/2018

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 11/20/2018

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 11/20/2018

Looking deeper into Relative Acceleration we write: ME 3230 11/20/2018

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 11/20/2018

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 11/20/2018

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 11/20/2018

Building Velocity Graph: ME 3230 11/20/2018

As Seen here: ME 3230 11/20/2018

Next We Compute 3 and 4: ME 3230 11/20/2018

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 11/20/2018

Graphically: Vc = 137.73 mm/s at 183.45 Vc/b & Vc/a as reported ME 3230 11/20/2018

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 11/20/2018

Means this: Note: accA line is opp. Direction to Link 2 ME 3230 11/20/2018

Getting Acc for Pt. B – first approach ME 3230 11/20/2018

Getting Acc for Pt. B – first approach ME 3230 11/20/2018

Getting Acc for Pt. B – second approach ME 3230 11/20/2018

Getting Acc for Pt. B – second approach AccB = 77.399*10 = 773.99 mm/s2 Getting Acc for Pt. B – second approach ME 3230 11/20/2018

Find b3’ & b4’ Where the 2 at Vectors Intersect On Graph, measure the two at’s and solve for angular acceleration: ME 3230 11/20/2018

Finding Acc. Of Pt. C ME 3230 11/20/2018

Finding Acc. Of Pt. C o’c’ is at -35.648 or 324.352 And AccC = 589.81 mm/s2 ME 3230 11/20/2018

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 11/20/2018

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 11/20/2018

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 11/20/2018

For our Model: ME 3230 11/20/2018

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 11/20/2018