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Prof. Albert Espinoza
Mechanical Engineering Department
Polytechnic University of Puerto Rico
Spring 2015
Mobility (Degree of Freedom)
Mobility determines the “character” of the links assemblage
Degree of freedom (DOF) of a system can be defined as:
– The number of inputs which need to be provided in order to create a predictable output;
– The number of independent coordinates required to define its position
Kutzbach’s (modified Gruebler’s) Equation
𝑀 = 3 𝐿 − 1 − 2𝐽1 − 𝐽2
L= number of links
J1=number of full joints (e.g., pins, sliders)
J2=number of half-joints (e.g., roll-slide, cams, gears)
Mobility (Degree of Freedom)
Example
1 23 1 2M L J J
4L
1 4J
2 0J
Num. Links = 4
Num. full joint, (1 DOF) = 3 pins , 1 slide
Num. half joint, (2 DOF) = 0
3 4 1 2 4 0
1
M
M
Mobility (Degree of Freedom)
Example
1 23 1 2M L J J
8L
1 10J
2 0J
Num. Links = 8
Num. full joint, (1 DOF) = 7 (single pins)+ 2 (multiple joint) +1 (slide)
Num. half joint, (2 DOF) = 0
3 8 1 2 10 0
1
M
M
Multi-Nodes
Mobility (Degree of Freedom)
Example
1 23 1 2M L J J
6L
1 7J
2 1J
Num. Links = 6
Num. full joint, (1 DOF) = 5 (single pins)+ 2 (multiple joint)
Num. half joint, (2 DOF) = 1
3 6 1 2 7 1
0
M
M
Mobility (Degree of Freedom)
Example
1 23 1 2M L J J
Num. Links = 9
Num. full joint, (1 DOF) = 7(single pins)+ 2 (Multiple Joint) +2(slider)
Num. half joint, (2 DOF) = 0
L= 9
J1 = 11
J2 = 0
M = 3(9-1) - 2(11)
M = 2
Mobility (Degree of Freedom)
Example
1 23 1 2M L J J
Num. Links = 9
Num. full joint, (1 DOF) = 9 pins , 2 slide
Num. half joint, (2 DOF) = 1
L = 9
J1= 11
J2 = 1
M = 3(9-1) - 2(11) - 1
M = 1
Mobility (Degree of Freedom)
Example
1 23 1 2M L J J
Num. Links = 3
Num. full joint, (1 DOF) = 2 pins
Num. half joint, (2 DOF) = 1
L = 3
J1= 2
J2 = 1
M = 3(3-1) - 2(2) - 1
M = 1
Mechanisms vs. Structures
Mechanism: M>0
Paradoxes
Specific cases due to unique geometric considerations, a complete motion
analysis is needed instead
Kutzbach’s (or Gruebler’s) may lead to wrong answers (does not consider the
geometry of the linkage).
11
Linkage Transformation
Transforms basic links into other mechanisms
Rules
1. Pin joints can be replaced by sliders with no change in DOF (provided we keep at
least 2 pins).
2. A full joint can be replaced by a half-joint an increasing the DOF by 1.
3. Removal of a link reduces DOF by 1.
4. Combination of (2) and (3) keeps same DOF.
5. Any higher-order link (e.g., ternary) can be partially “shrunk” by coalescing
nodes in a multiple joint (with no change in DOF).
6. Complete shrinkage of higher-order link is equivalent to removing it. This
creates a multiple joint and reduces the DOF.
Linkage Transformation Examples
Behave as crank-rocker with infinite link 4
Slider-crank from Crank-Rocker (Rule #1)
Linkage Transformation Examples
Provides exact simple harmonic motion
Scotch Yoke from Slider-Crank (Rule #4)
Substitution of link by half-joint (2 DOF)
Inversion
Created by grounding a different link in the kinematic chain
Mechanism modifications
16
The Four-Bar Linkage
• The most fundamental linkage is the 4-bar
It consists of 3 moving links and 1 fixed link
The Links are:
• Ground/Base link (fixed)
• Input Link (“driver” mover and connected to ground)
• Output Link (“driven” mover and connected to ground)
• Coupler or Floating Link (moving and connects driver to driven link)
Grashof Condition
Predicts rotation behavior of four-bar linkages
Grashof Linkage if
𝑆 + 𝐿 ≤ 𝑃 + 𝑄
S = length of shortest link
L = length of longest link
P = length of one remaining link
Q = length of other remaining link
Driver
Frame
Coupler
Follower
Grashof Condition
Crank and/or rocking motion depending on grounded link
Class I: 𝑆 + 𝐿 < 𝑃 + 𝑄
• Grounding either link adjacent
to S yields a crank-rocker (1
and 2)
• Grounding S yields a double-
crank (3)
• Grounding the link opposite S
yields a Grashof-double
rocker
Grashof Condition
Non-Grashof, triple rockers
Class II: 𝑆 + 𝐿 > 𝑃 + 𝑄
• All inversions are triple rockers
(no link can fully rotate)
Grashof Condition
Special-case Grashof, will have “change points”
Class III: 𝑆 + 𝐿 = 𝑃 + 𝑄
• All are either double-
cranks or crank-
rockers, but will have
“change points”
Barker’s Classification
Further detailed classification for four-bar mechanism motion
Barker’s Classification
Grashof Condition and Barker’s Classification
Examples
Determine the Grashof Condition and Barker’s Classification
(a) (b)
(c)
Grashof Condition and Barker’s Classification
Examples
Determine the Grashof Condition and Barker’s Classification
(d) Oil Field Pump
Grashof Condition and Barker’s Classification
Examples
Determine the Grashof Condition and Barker’s Classification (Motion)
(e) Aircraft Overhead Bin Mechanism
