Mechanism Analysis

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Mechanism Analysis
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Links and their kinematic representation
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Complex linkPin joint
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Slider jointCam joint
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Gear joint
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Kinematic diagram
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Kinematic diagram
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Grueblers Equation
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Grueblers Equation
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CompareGrueblers eqn degrees of freedom
3(links - 1) 2 jointsto degree of indet.
bars reactions 2 joints
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examples
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examples
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examples
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examples
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examples
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examples
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counts as two joints!
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examples
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Exceptions
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examples
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Four Bar Mechanism
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Grashofs Criterion s shortest, l longest,
p,q other two.Will have at least one revolving
link if s l p q
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Grashofs Criterion
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Circuits
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Path
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Slider-CrankAnother verycommonmechanismwith
four links.
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Slider-Crank
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Vector addition andsubtraction
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Vector Equations
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Position analysis, four bar mechanism
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Two link position synthesisdesign a mechanism
that moves this link between these two positions
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Two link position synthesisthe case of limited
space using two pivot points (four bar mechanism
synthesis)
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Four bar mechanism synthesis to move a link
through three positions
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Velocity Analysisvelocity of a point, v
dR/dtangular velocity of a link, ?
d?/dtvelocity of a point on a rotating element,
v r?, where r is distance from center of
rotation to the point.
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Relative Velocity
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Relative Velocity, what is vb/a?
va
vb
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Relative Velocity, what is vb/a?
va
va
vb/a
vb
vb
velocity polygon
Relative velocity equation vb va vb/a
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What is the angular velocity of the link?
va
va
vb/a
vb
vb
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What is angular velocity of the link?
va
va
vb/a
vb
vb
L
From perspective of a, b is rotating about it
with velocity vb/a, at a distance L.
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What is angular velocity of the link?
va
va
vb/a
vb
vb
L
Since v r?, then ? v/r, or in this case, ?
va/b/L.
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Also note that relative velocity of two points on
a single link is always perpendicular to the link.
va
va
vb/a
vb
vb
L
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Draw velocity polygonto find velocity ofeach
joint and angularvelocity of each link.Assume
the crank isrotating at 120 rpm.
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Draw velocity polygonto find velocity ofeach
joint and angularvelocity of each link.Assume
the crank isrotating at 120 rpm.
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Draw velocity polygonto find velocity ofeach
joint and angularvelocity of each link.Assume
the crank isrotating at 120 rpm.
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What hydraulic cylinder speed is needed for 5
rad/min rotation of bed?
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What hydraulic cylinder speed is needed for 5
rad/min rotation of bed?
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Velocity Image
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Instant Centers (of rotation)A link at any
instant intime appears to berotating about a
pointin space.Knowing instant centerlocations
will enable us todraw velocity diagramseasily.
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The instantaneous centeror rotation of link 3
w.r.t.the ground (link 1) islabeled 31. It is
identicalto 13 which is the apparent center
ofrotation of the groundfrom an observer
onlink 3.
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Where is instantcenter 23?34?How many
instantcenters are there?
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12, 13, 14, 23, 24, 34
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PrimaryInstantCenters
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Locations of primary instant centers
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Kennedys TheoremFor example, links 1,2,3
have instant centers 12, 23, 13 thatall fall
along a single straight line.
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Locking Brace Example
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Instant Center Diagram a graphical tool for
locating all centers
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Another Rock CrusherUse center diagram to
locateall instant centers
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Instant CenterDiagramProgression
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Final Instant CenterDrawing
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Drawing Velocity Images using Instant Centers
The velocities ofpoints on a rotatinglink are
proportionalto the distance fromthe pivot
point.Can be constructedwith just a ruler.
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Drawing Velocity Images using Instant Centers

• The instant center ab can be thought of as a
  point attached to either body a or body b
• The instant center serves as a transfer point
  for velocities between bodies

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Example
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Given velocity at B,find velocityat C
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Acceleration Analysisv dR/dta dv/dt
v1
v1
v2
a1
v2
R1
R2
R3
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Acceleration AnalysisAcceleration of a point is
usually broken into two componentsTangential
accel, at, producingchange in velocity
magnitudeNormal accel, an, producingchange in
velocity direction.
v
at
path
an
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Acceleration AnalysisComplete description of
the motion of a link requires a description of
the motion of a point on the link, and the
rotational motion of the link about that
point.Angular position, ?Angular velocity, ?
d?/dtAngular acceleration, ? d?/dt
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Acceleration of a point on a rotating linkGiven
a point located on a rotating link a distance rpt
from the center of rotation, and the angular
velocity of the link, ?link, then atpt rpt
?link anpt vpt ?link anpt rpt ?link2
(since vpt rpt ?link) anpt vpt2 / rpt
(since ?link vpt/rpt)
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Find acceleration at end of input link. ? 25
rad/s ? 500 rad/s2
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Relative Acceleration anB atB anA atA
anB/A atB/A
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anC/B atC/B anC atC - anB - atB Given VB
12 in/s (up and to the left), ?crank 37 rad/s2
CCW
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