MENG 371 Notes, Chapter 2

Dr. Keith Hekman

September 9, 2003

2.1 Degree of Freedom (DOF) or Mobility

• DOF – number of independent parameters (measurements) that are needed to uniquely define its position in space at any instant of time.

• Rigid body in a plane has 3 DOF. (x,y,θ)

• Rigid body in space has 6 DOF (3 translation, 3 rotation)

2.2 Types of Motion

• Pure Rotation – the body possesses on point (center of rotation) that has no motion with respect to the “stationary” frame of reference. All other points move in circular arcs

• Pure Translation – all points on the body describe parallel (curvilinear or rectilinear) paths.

• Complex motion – a simultaneous combination of rotation and translation

2.3 Links, joints, and kinematic chains

• Links – building blocks

• Node – attachment points– Binary link – two nodes

– Ternary link – three nodes

– Quaternary link – four nodes

• Joint – connection between two or more links (at their nodes) which allows motion– Classified by type of contact, number of DOF, type of

physical closure, or number of links joined

Joint Classification

• Type of contact - line, point, surface

• Number of DOF – full joint=1DOF, half joint=2DOF (removes half as many DOF)

• Form closed (closed by geometry) or Force closed (needs an external force to keep it closed)

• Joint order = number of links-1

Kinematic chains, mechanisms, machines, link classification

• Kinematic chains –links joined together for motion

• Mechanisms – grounded kinematic chain

• Machines – mechanism designed to do work

• Link classification– Ground – fixed w.r.t. reference frame

– Crank – pivoted to ground, makes complete revolution

– Rocker – pivoted to ground, has oscillatory motion

– Coupler - link has complex motion, not attached to ground

2.4 Determining Degree of Freedom

• For simple mechanisms calculating DOF is simple

Closed MechanismDOF=1

Open MechanismDOF=3

2.4 Determining Degree of Freedom

• Gruebler’s equation for planar mechanisms

M=3L-2J-3G

• Where– M=degree of freedom or mobility

– L=number of links

– J=number of joints (half joints count as 0.5)

– G=number of grounded links =1

M=3(L-1)-2J

2.4 Determining Degree of Freedom

• Kutzbach’s equation for planar mechanisms

M=3(L-1)-2J1-J2

• Where– M=degree of freedom or mobility

– L=number of links

– J1=number of full joints

– J2=number of half joints

• For Spatial Mechanisms

M=6(L-1) -5J1-4J2-3J3-2J4-J5

2.4 Determining DOF (Examples)M=3(L-1)-2J1-J2

Closed MechanismOpen Mechanism

2.4 Determining DOF (Examples)M=3(L-1)-2J1-J2

2.4 Determining DOF (Examples)M=3(L-1)-2J1-J2

2.5 Mechanisms and Structures

• Mechanism – DOF>0

• Structure – DOF=0

• Preloaded Structure –DOF<0, may require force to assemble

2.6 Number Synthesis• Number Synthesis – determination of the number

and order of links and joints necessary to produce motion of a particular DOF

• Book gives details

010168

002068

001258

000448

100078

001056

000246

000044

HexagonalPentagonalQuaternaryTernaryBinaryTotal Links

2.7 Paradoxes

• Greubler criterion does not include geometry, so it can give wrong prediction

• Usually when things are the same

E-quintetGears

2.8 Isomers

• Greek for having equal parts

• Refers to valid ways to assemble different types of links

• Only one valid fourbar isomer

• Two valid sixbar isomers

• Third one fails DOF test, as the DOF is not distributed over the linkage.

Fourbar Isomer

• Only way to construct a fourbar isomer is to have one binary link next to another binary link.

Watt’s Sixbar Isomer

• One way to construct a sixbar isomer is to have the two ternary links attached.

Stephenson’s Sixbar Isomer

• One way to construct a sixbar isomer is to have the two ternary links separated.

Invalid Sixbar Isomer

• This is an invalid isomer as the DOF is not distributed through the mechanism

This is a structureEffective link

2.9 Linkage Transformation• A slider can be replaced by a link of infinite length

2.10 Intermittent Motion

• Series of Motions and Dwells

• Dwell – no output motion with input motion

• Examples: Geneva Mechanism, Linear Geneva Mechanism, Ratchet and Pawl

Geneva Mechanism Linear Geneva Mechanism

Ratchet and Pawl 2.11 Inversion

• Created by grounding a different link in a kinematic chain

• Different behavior for different inversions

3 Stephenson 6-bar inversions 2 Watt’s 6-bar inversions

2.12 Grashof Condition

• Fourbar linkage is simplest linkage with 1DOF• Grashof condition predicts behavior of linkage

based only on links length– S=length of shortest link– L=length of longest link– P,Q=length of other two links

• If S+L�P+Q the linkage is Grashof with at least one link capable of making a complete rotation

• Otherwise the linkage is non-Grashof with no link capable of making a complete rotation relative to ground

For case of S+L<P+Q

• Ground link adjacent to shortest => crank-rocker

• Ground shortest link => double crank

• Ground link opposite shortest link – Grashof double rocker with shortest link capable of making a complete rotation

For the case of S+L>P+Q

• All inversions will be double rockers

For the case of S+L=P+Q

• Book says all inversions will be double cranks or crank rockers (true if S=P,L=Q)

• Indeterminate point when links are aligned (change points)

Parallelogram form

Anti parallelogram form

Deltoid form

Barker’s Complete Classification

SquareS3XTriple change pointIII-6L1=L2=L3=L4=14

Parallelogram or deltoid

S2XDouble change pointIII-5Two equal pairs=13

SC rocker-crankSRRCChange point rocker-rocker-crankIII-4L4=s=output=12

SC double-rockerSRCRChange point rocker-crank-rockerIII-3L3=s=coupler=11

SC crank-rockerSCRRChange point crank-rocker-rockerIII-2L2=s=input=10

SC double-crankSCCCChange point crank-crank-crankIII-1L1=s=ground=9

Triple-rockerRRR4Class 4 rocker-rocker-rockerII-4L4=l= output>8

Triple-rockerRRR3Class 3 rocker-rocker-rockerII-3L3=l= coupler>7

Triple-rockerRRR2Class 2 rocker-rocker-rockerII-2L2=l= input>6

Triple-rockerRRR1Class 1 rocker-rocker-rockerII-1L1=l=ground>5

rocker-crankGRRCGrashof rocker-rocker-crankI-4L4=s=output<4

double-rockerGRCRGrashof rocker-crank-rockerI-3L3=s=coupler<3

crank-rockerGCRRGrashof crank-rocker-rockerI-2L2=s=input<2

double-crankGCCCGrashof crank-crank-crankI-1L1=s=ground<1

Also Known asCodeBarker’s DesignationClassInversions+l vs p+qType

2.13 Linkages of more than 4 bars

5-bar 2DOFGeared 5-bar 1DOF

•Provides for more complex motion

•Watt’s sixbar – 2 fourbar linkages in series

•Stephenson’s sixbar – 2 fourbar linkages in parallel

2.14 Springs as links

• Springs remove a degree of freedom (1 more equation)

• Examples: desk arm lamp, garage door

2.15 Compliant Mechanisms

• Compliant “ link” capable of significant deflection acts like a joint

• Also called a “ living hinge”

• Advantage: simplicity, no assembly, little friction

2.16 Micro Electro-Mechanical Systems (MEMS)

• Micromachines range in size from few micrometers to a few millimeters

• Shape is made on large scale, then photographically reduced on wafer and etched.

• Can make compliant mechanisms in MEMS

2.17 Practical Considerations2.18 Motors and Drivers

• Read on your own