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

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Graphical Analysis Positions, Velocity and
Acceleration (simple joints)

• ME 3230
• Dr. R. Lindeke

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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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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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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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Relative Velocity as a vector

• Magnitude is
• Direction is Normal to both ? and the vector
  rB/A
• We determine the relative velocitys 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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Step 2The 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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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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Looking deeper into Relative Acceleration we
write
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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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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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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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Building Velocity Graph
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As Seen here
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Next We Compute ?3 and ?4
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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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Graphically
Vc 137.73 mm/s at 183.45? Vc/b Vc/a as
reported
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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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Means this
Note accA line is opp. Direction to Link 2
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Getting Acc for Pt. B first approach
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Getting Acc for Pt. B first approach
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Getting Acc for Pt. B second approach
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Getting Acc for Pt. B second approach
AccB 77.39910 773.99 mm/s2
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Find b3 b4

• Where the 2 at Vectors Intersect
• On Graph, measure the two ats and solve for
  angular acceleration

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Finding Acc. Of Pt. C
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Finding Acc. Of Pt. C

• oc is at -35.648? or 324.352?
• And AccC 589.81 mm/s2

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Understanding ImagingVelocity and Acceleration

• If one knows velocity and/or acceleration for two
  points on a link any other points 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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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 links
  angular velocity (?i)
• The size of the Velocity Polygon is determined by
  the magnitude of the links angular velocity (?i)

Rotated 90? CW!
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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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For our Model
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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