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Kinematics

Fundamentals

Chapter 2

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Definitions

• Mechanisms

 – A device which transform motion to some

desirable pattern and typically developvery low forces and transmits little power 

• Machine

 – Typically contains mechanism which are

design to provide significant forces and

transmit significant power 

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http://www.flyingmachines.org/davi.html

www.gizmag.com/pictures/hero/3533_01.jpg

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http://4dlab.info/images/engine_cutaway.jpg

www.50classicchevy.com/images/1950-chevrolet

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Kinematics

Fundamentals

• Degree of Freedom (DOF)

 – The system’s DOF equal to the number of

independent parameters(measurement)

which are needed to uniquely define its

position in space at any time

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Kinematics

Fundamentals

• Types of Motion

 –Pure translation

 –Pure rotation

 –Complex motion

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – A link is an rigid body which possesses at

least two nodes which are points for

attachment to other links

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – A joints (kinematic pairs) is a connection

between two or more links, which allows some

motion, or potential motion, between the

connected links

 – Classification

• Type of contact between the elements, line, point,or surface

• Number of DOF allowed at the joint

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – Classification

• Type of physical closure of the joint• Number of links joined

 – Type of Contact

• Lower pair (full joints)

 – Describe joints with surface contact

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – Type of Contact

• Lower pair 

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – Type of Contact

• Higher pair  – Describe joints with point or line contact

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – Number of DOF allowed Joint

• One DOF (full joint)

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – Number of DOF allowed Joint

• Two DOF (half joint/roll-slide)

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – Type of Physical Closure

• Form closed- closed by its geometry

• Force closed- closed by an external force

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – Number of links joined

• Order of the joint: the number of links minus one

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – Kinematic Chain

•  An assemblage of links and joints, interconnected

in a way to provide a controlled output motion in

response to a supplied input motion

 – Mechanism

•  A kinematic chain in which at least one link hasbeen “grounded,” or attached, to the frame of

reference

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – Ground

•  Any link or links that are fixed with respect to

the reference frame

 – Crank

•  A link which makes a complete revolution

and is pivoted to ground

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – Rocker 

•  A link which has oscillatory (back and forth)

rotation and pivoted to ground

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Kinematics

Fundamentals

• Links, Joints, and Kinematic Chains

 – Coupler 

•  A link which has complex motion and is

pivoted to ground

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Kinematics

Fundamentals

• Determining DOF

 – DOF or Mobility

• 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

 – Open or Closed

 – Dyads

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Kinematics

Fundamentals

• Determining DOF

 – DOF in Planar Mechanisms

• Gruebler’s Equation

where

 – M = degree of freedom or mobility – L = number of links

 – J = number of joints

 – G = number of grounded links

G J  L M    323  

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Kinematics

Fundamentals

• Determining DOF

 – DOF in Planar Mechanisms

• Gruebler’s Equation

• If more than one link is grounded, the net

effect will be to create one larger, higher-order ground link. G is always one, therefore

G J  L M    323  

  J  L M    213  

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Kinematics

Fundamentals

• Determining DOF

 – DOF in Planar Mechanisms

• Kutzbach Equation

 – Include full and half joints

where

 – M = degree of freedom or mobility

 – L = number of links

 – J1 = number of 1 DOF (full) joints

 – J2 = number of 2 DOF (half) joints

  21

213   J  J  L M   

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Kinematics

Fundamentals

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Kinematics

Fundamentals

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Kinematics

Fundamentals

• Mechanisms and Structures

 – The DOF of an assembly of links

completely predicts its character 

• If the DOF is positive→ mechanism

• If the DOF is zero→ structure

• If the DOF is negative→ preloaded structure

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Kinematics

Fundamentals

• Number Synthesis

 – The determination of the number and

order of links and joints necessary to

produce motion of a particular DOF

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Kinematics

Fundamentals

• Paradoxes

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Kinematics

Fundamentals

• Isomers

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Kinematics

Fundamentals

• Linkage Transformation – Revolute joints in any loop can be replaced by

prismatic joints with no change in DOF of the

mechanism, provided that at least two revolute joints remain in the loop

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Kinematics

Fundamentals

• Linkage Transformation

 – Any full joint can be replaced by a half

 joint, but this will increase the DOF by

one

 – Removal of a link will reduce the DOF by

one

 – The combination of rules 2 and 3 abovewill keep the original DOF unchanged

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Kinematics

Fundamentals

• Linkage Transformation

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Kinematics

Fundamentals

• Linkage Transformation –  Any ternary or higher  –order link can be partially

shrunk to a lower  –order link by coalescing

nodes. This will create a multiple but will notchange the DOF at the mechanism

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Kinematics

Fundamentals

• Linkage Transformation – Complete shrinkage of a higher-order link is

equivalent to its removal. A multiple joint will be

created, and the DOF will be reduced

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Kinematics

Fundamentals

• Intermittent Motion – Is a sequence of

motions and dwells

• Dwell; is a period in whichthe output link remains

stationary while the input

link continues to move

• Geneva Mechanism

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Kinematics

Fundamentals

• Intermittent Motion• Linear Geneva Mechanism

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Kinematics

Fundamentals

• Inversion

 – An inversion is created by grounding a

different link in the kinematic chain

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Kinematics

Fundamentals

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Kinematics

Fundamentals

• Grashof Condition

 – Is a simple relationship that predicts the

rotation behavior or rotatability of a four

linkage’s inversion based only on the link

lengths

• S = length of shorter link• L=length of longest link

• P=length of one remaining link

• Q=length of the other remaining link

Q P  LS   

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Kinematics

Fundamentals

• Grashof Condition

 – If the inequality is true, at least one link

will be capable of making a full revolution

with respect to the ground plane(Class I)

 – If not true, then the linkage is non-Grashof 

and no link will be capable of a complete

revolution relative to any other link (ClassII)

Q P  LS   

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Kinematics

Fundamentals

• Grashof Condition

 – For the class I case: S+L< P+Q

• Ground either adjacent to the shortest link and

you get a crank-rocker 

• Ground the shortest link and you will get a

double-crank

• Ground the link opposite the shortest and you

will get a Grashof double-rocker 

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Kinematics

Fundamentals

• Grashof Condition

 – For the Class II case: S+L> P+Q

•  All inversion will be triple-rockers in which no

link can fully rotate

 – For Class III: S+L=P+Q

•  All inversion will be either double-cranks, or

crank-rocker 

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Kinematics

Fundamentals

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Kinematics

Fundamentals

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Kinematics

Fundamentals

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Kinematics

Fundamentals

• Classification of the Four Linkage

 – C. Barker developed a classification

scheme that allows prediction of the type

of motion that can be expected from a

fourbar linkage based on the values of its

link lengths

 – Link ratio formation – Letter designation (C), (R) - GCRR

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Kinematics

Fundamentals

• Linkages of More Than Four Bars

 – Geared Fivebar Linkages

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Kinematics

Fundamentals

• Linkages of More Than Four Bars

 – Sixbar Linkages

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Kinematics

Fundamentals

• Spring as Links

• Compliant Mechanism

• Micro Electro-Mechanical Systems(MEMS)

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Kinematics

Fundamentals

 – Problems

Ki ti

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Kinematics

Fundamentals

 – Problems

Ki ti

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Kinematics

Fundamentals