ME321 - Kinematics and Dynamics of Machines Design Process
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Transcript ME321 - Kinematics and Dynamics of Machines Design Process
ME321
Kinematics and Dynamics of
Machines
Steve Lambert
Mechanical Engineering,
U of Waterloo
7/17/2015
Kinematics and Dynamics
Position Analysis
Velocity Analysis
Acceleration Analysis
Force Analysis
F ma
We will concentrate on four-bar linkages
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Four-Bar Linkages
What type of motion is possible?
q
l
s
p
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Grashof’s Criteria
Used to determine whether or not at least one of the links
can rotate 360o
the sum of the shortest and longest links of a planar
four-bar mechanism cannot be greater than the sum of
the remaining two links if there is to be continuous
relative rotation between the two links.
s + l< p + q
q
l
s
p
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Grashof’s Criteria
l
q
l
s
s
p
q
p
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s+l<p+q
s+l>p+q
Grashof Mechanism
Non-Grashof Mechanism
Grashof Mechanisms (s+l < p+q)
l
l
q
q
s
s
p
Crank-Rocker
p
Rocker-Crank
Shortest link pinned to ground and rotates 360o
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Grashof Mechanisms (s+l < p+q)
s
q
l
p
q
p
s
l
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Drag-Link
Double-Rocker
- Both input and output
links rotate 360o
- Coupler rotates 360o
Change-Point Mechanism
S+l = p+q
l
s
q
p
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Non-Grashof Mechanisms
•Four possible triplerockers
•Coupler does not
rotate 360o
s
q
p
l
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Transmission Angle
• One objective of position
analysis is to determine the
transmission angle,
• Desire transmission angle
to be in the range:
45o < < 135o
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coupler
input
link
output
link
Position Analysis
Given the length of all links, and the input angle,in, what
is the position of all other links?
Use vector position analysis or analytical geometry
coupler
output
link
input
link
in
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Vector Position Analysis
• ‘Close the loop’ of vectors
to get a vector equation with
two unknowns
• Three possible solution
techniques:
RB / A
RA s
O2
• Graphical Solution
• Vector Components
• Complex Arithmetic
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RB
RO2 / O4
O4
RA RB / A RO4 / O2 RB
Graphical Solution
• Draw ground and
input links to scale, and
at correct angle
• Draw arcs (circles)
corresponding to length
of coupler and output
links
•Intersection points
represent possible
solutions
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Vector Component Solution
y, i
R
3
x, j
4
2
O4
O2
‘Close the loop’ to get a vector equation:
R2 cos 2iˆ sin 2 ˆj R3 cos3iˆ sin3 ˆj R1iˆ R4 cos 4iˆ sin 4 ˆj
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Vector Component Solution (con’t)
Rewrite in terms of i and j component equations:
R2 cos 2 R3 cos 3 R1 R4 cos 4
R2 sin 2 R3 sin 3 R4 sin 4
• These represent two simultaneous transcendental
equations in two unknowns: 3 and 4
•Must use non-linear (iterative) solver
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Complex Arithmetic
• Represent (planar) vectors as
complex numbers
R Rei Rcos i sin
iy
R
x
• Write loop equations in terms of real and imaginary
components and solve as before
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Analytical Geometry
• Examine each mechanism
as a special case, and apply
analytical geometry rules
• For four-bar mechanisms,
draw a diagonal to form two
triangles
• Apply cosine law as
required to determine length
of diagonal, and remaining
angles
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B
3
A
2
O2
4
O4
l 2 a 2 b 2 2ab cos
Limiting Positions for Linkages
• What is the range of output motion for a crack-rocker
mechanism?
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