2. fundamentals
TRANSCRIPT
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THEORY OF MACHINEWith:
Sami Salama Hussen Hajjaj
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How are we going to design?
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In order to design Machine Components of a given shape
under a known loading, we need to follow this procedure:
1. Analyze and Simplify Applied loadingMechanics & TOM
2. Identify the MaterialssStress-Strain limits Materials & MOM
3. Check if workpiece can support applied load
4. If yes, Select part that fits this requirements
5. If no, redo (what?)
MD
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CHAPTER 2:
FUNDAMENTALS
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Contents
Degree of Freedom (DOF)
Links, Joints, and Kinematic Chains
Kinematic Diagrams Inversion
Our Three Best Friends
The Grashof Condition
Mobility (DOF of the whole system)
(Half Joints vs. Full Joints)
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DEGREE OF FREEDOM
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Degree of Freedom
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DOF is the number of INDEPENDENT free motionsthat can begenerated by a joint or a linkage
DOF is the number of INDEPENDENT free parametersneeded to
describe the motions generated by a joint or a linkage
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Degree of Freedom
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DOF is a very important concept is Mech Engineering
It is used in TOM, Machine Design, Robotics, and others
Links, Joints, Linkages, Machines, and Robots are classified
by their DOFs
Generally speaking, the higher the DOF, the more flexible
the design gets, but also the more complex it gets.
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LINKS, JOINTS, AND
KINEMATIC CHANES
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Links
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Link: a rigid body used to link to other components of the machine
Each link possesses at least 2 nodes (Points for attachment to other links)
Links are classified according to # of nodes
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Links
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Links can be any shape, but their classification still holds. Links are
classified according to # of nodes, not their shapes.
What are these links ?
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Joints
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A jointjoins two 2 or more links at their nodes which will a
motion between the links
Joints can be classified based on:
DOF (Degree Of Freedom)
Order of Joints
Joint
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DoF Classification
Name Diagram DoF
Pin 1
Slider 1
Link against
plane
2
Pin in slot 2
Rolling
cylinder
Pure roll: 1
Pure slide: 1
Roll & slide: 2
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Order Classification
Name Diagram DoF ?
Single Joint
(1storder joint)
Higher order Joint
(2ndorder joint)
Order of Joint = Number of links joint1
A 2ndOrder Joint = 2 single joints = 2 DOF
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Links & Joints
Kinematic Chain
An assembly of links and joints
MechanismA kinematic chain designed for a purpose, AND have at
least one link grounded (Fixed to the ground)
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A Machine
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AMachineis a system of multiple Mechanismsinteracting together toachieve a common purpose.
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KINEMATIC DIAGRAMS
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Kinematic Diagrams
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Actual Kinematic Diagram
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Kinematic Diagrams
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Actual Kinematic Diagram
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Kinematic Diagrams
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Kinematic Diagrams
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Kinematic Diagrams
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Kinematic Diagrams
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INVERSION
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Inversion
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Inversion is created by grounding a different link in the
kinemetic chain.
This is usually done to see the impact of the ground(from different places) on other links
Refer to section 2.12 for more
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The 4 BarCrank Rocker
This is the FourBar Crank Rocker Mechanism. In the mechanism,
the crank rotates a full rotatation, while the rocker rocks back and
forth. And the coupler couples them
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y
x
Crank
Coupler Rocker
Ground
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y
x
Slider
The 4 BarCrank Rocker
Crank
Ground
This is the FourBar Crank slider Mechanism. In the mechanism, the
rotaionl motion of the crank (full rotatation) is converted into a linear
horizontal motion, and the coupler couples them
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y
x
The 4 BarInverted Crank Slider
This is the FourBar Inverted Crank slider. It is very similar to the
crank slider, only that the slider is NOT grounded, therefore it is
alowed to slide along the coupler and rotate with the rocker.
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The Grashof Condition
The Grashof Condition is a test used to predict the
behavior of the fourbar linkage, it is based on length of
each link
Given:S: length of shortestlink
L: length of longestlink
P: length of intermediatelink
Q: length of another intermediatelink
Therefore ..
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The Grashof Condition
Case I: if S + L < P + Q, the fourbar behaves like a
Crank-Rocker (one link fully rotates, one rocks)
jj j
Grashof
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The Grashof Condition
Case II: if S + L > P + Q, then the fourbar behaves like a
Rocker-Rocker (Double Rocker), none of the links make a
full revolution.
jj j
Non-Grashof
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The Grashof Condition
Case III: if S + L = P + Q, then the fourbar behaves like
a Crank-Crank (Double Crank), both make full
revolution.
jj j
Grashof
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The Grashof Condition
Example:
Calculate the Grashof condition of the fourbar mechanism
defined below:
A. 2 4.5 7 9
B. 2 3.5 7 9
C. 2 4.0 6 8
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The Grashof Condition
Example:
Calculate the Grashof condition of the fourbar mechanism
defined below:
A. 2 4.5 7 9 => S+L< P+Q => Grashof (Crank-Rocker).
B. 2 3.5 7 9 => S+L > P+Q => non-Grashof (Double Rocker). C. 2 4.0 6 8 => S+L = P+Q => Grashof (Double Crank).
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MOBILITY
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Mobility
Mobility is the DOF of the whole mechanism (as a unit).
DOF => Joint
Mobility => Whole Mechanism
There are two kinds of Mobility:
Planar Mobility (2D Mechanisms)
Spatial Mobility (3D Mechanisms)
Our primary focus this semester will be on Planar Mobility
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Mobility
The meaning if Mobility:
Mobility = 0 ==> Rigid object
Mobility = 1 ==> all links and joints move in unison
(as one unit)
Mobility > 1 => 2 or more groups of lins/joints are able tomove independently from other groups
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Determining Mobility
Every component directly affects the mobility of the mechanism
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Determining Mobility
Solution for this example:
Number of Joints: 11.5
(11 Full Joints, 1 half Joint)
Number of Links: 9
Therefore:
M = 3(9-1)2(11.5) = 2423 = 1
Which means this system moves in unison (together)
Higher Order J
Half J
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Determining Mobility
In a 2D world, every link is able to make three motions.
Therefore: Every link adds 3 mobilities to the system
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Determining Mobility
If we add a joint to the link, then two motions are restricted.
Therefore: every joint removes 2 mobilities from the system
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Determining Mobility
If we add a joint to the link, then two motions are restricted.
Therefore: every joint removes 2 mobilities from the system
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Determining Mobility
And If we add a ground to the link (ground the link), then all
motions are restricted.
Therefore: every ground removes 3 mobilities from the system
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Determining Mobility
NO! .. some joints have 2 DOFs, they restrict only 1 motion.
Therefore:
- Joints with 1 DOF restrict only 2 motions
- Joints with 2 DOF restrict only 1 motion
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Determining Mobility
As such, Joints with 2 DOF restrict the amount of motions
restricted by 1 DOF joints, therefore
- Joints of 1 DOF (between 2 Links) are called: Full Joints
- Joints of 2 DOF (between 2 Links) are called: Half Joints
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Determining Mobility
Therefore, there are two approaches to calculating
mobility in a system:
One approach is to count each half joint as a (0.5 Joint)and simply add it to the numeber of joints in the system,
and then use the same equation as before
M = 3(L1)2J
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Determining Mobility
The second approach is to use the Kutzbachs/ Modified
Grueblersequation:
M= 3(L1)2J1J2
Where,
J1= Number of 1 DoF joints (Full Joints)
J2
= Number of 2 DoF joints (Half Joints)
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Determining Mobility
L (Gr)
L
L
L
L
L
JJ
J
JJ
1/2
J
J
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Higher order Joint vs. Joint
2 DOF, 2 Joints
2 DOF, Joint
To remove the confusion check the number of links
2 DOF & 2 Links = Joint
2 DOF & 3 Links = 2 Joints
3 Links
2 Links
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Determining Mobility
After counting links and Joints, we found:
Links = 6 links (inlcuding ground)
Joints = 7 Full Joints, and 1 half Joint
Lets calculate M using the (First Approach)
M = 3 (L- 1)2 (J)= 3 (6- 1)2 (7.5) = 1515 = 0
This system has 0 Mobility (Rigid)
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Determining Mobility
After counting links and Joints, we found:
Links = 6 links (inlcuding ground)
Joints = 7 Full Joints, and 1 half Joint
Now Lets calculate M using the (Second Approach)
M = 3 (L- 1)2 (J1
)(J2
)= 3 (6- 1)2 (7)(1) = 15141 = 0
This system still has 0 Mobility (Rigid)
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Determining Mobility
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Determining Mobility
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Determining Mobility
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Kinematic Diagrams
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Determining Mobility
Solution for this example:
Number of Joints: 11.5
(11 Full Joints, 1 half Joint)
Number of Links: 9
Therefore:
M = 3(9-1)2(11.5) = 2423 = 1
Which means this system moves in unison (together)
Higher Order J
Half J
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Determining Mobility
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Determining Mobility
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Determining Mobility
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Determining Mobility
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Determining Mobility
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Determining Mobility
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Determining Mobility
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Determining Mobility
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Determining Mobility