1. ME321 Kinematics and Dynamics of Machines
Steve Lambert
Mechanical Engineering,
U of Waterloo
9/17/2018

2. Kinematics and Dynamics
Position Analysis
Velocity Analysis
Acceleration Analysis
Force Analysis
We will concentrate on four-bar linkages
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3. Four-Bar Linkages
What type of motion is possible? s q l p
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4. 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 s q l p
9/17/2018

5. Grashof’s Criteria s + l > p + q s + l < p + q
Non-Grashof Mechanism
s + l < p + q
Grashof Mechanism
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6. Grashof Mechanisms (s+l < p+q)
Crank-Rocker
Rocker-Crank
Shortest link pinned to ground and rotates 360o
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7. Grashof Mechanisms (s+l < p+q)
Drag-Link
- Both input and output links rotate 360o
Double-Rocker
- Coupler rotates 360o
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8. Change-Point Mechanism
S+l = p+q s q l p
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9. Non-Grashof Mechanisms
Four possible triple- rockers
Coupler does not rotate 360o p s q l
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10. Transmission Angle
One objective of position analysis is to determine the transmission angle, 
Desire transmission angle to be in the range:
45o <  < 135o 
output
link
input
coupler
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11. 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
output
link
input
coupler
in
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12. Vector Position Analysis
‘Close the loop’ of vectors to get a vector equation with two unknowns
Three possible solution techniques:
Graphical Solution
Vector Components
Complex Arithmetic
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13. 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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14. Vector Component Solution R
x, j
y, i O2 O4 2 3 4
‘Close the loop’ to get a vector equation:
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15. Vector Component Solution (con’t)
Rewrite in terms of i and j component equations:
These represent two simultaneous transcendental equations in two unknowns:  3 and 4
Must use non-linear (iterative) solver
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16. Complex Arithmetic Represent (planar) vectors as complex numbers x iy R
Write loop equations in terms of real and imaginary components and solve as before
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17. 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 A B O2 O4 2 3 4   
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18. Limiting Positions for Linkages
What is the range of output motion for a crack-rocker mechanism?
9/17/2018