INTRODUCTION TO

MECHANISM SYNTHESIS

Mechanics Place in Science

Mechanisms and Structures

Number Synthesis

Paradoxes and Isomers

Transformations and Inversions

Grashof’s Law

MER312: Dynamics of Mechanisms 1

The Ultimate Goal is to

Synthesize Machine Elements

MER312: Dynamics of Mechanisms 2

Manufacturing

Materials

Mechanics

Thermo-Fluids

Controls

Statics

Dynamics

Strength of

Martials

Kinetics

KinematicsADV. DYN. &

KINEMATICS

Advanced

Strength Machine

Design

MECHANISM

DESIGN/SYNTHESIS

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Historical Perspective of

Kinematics

MER312: Dynamics of Mechanisms 3

4000-3000 BC

Mesopotamia

• Lever

• Inclined Plane

Wedge

28 BC

Marcus Vitruvius

• De Architecture

Earliest Writing

on the Subject

• Moving Heavy

Objects

1st Century AD

Hero of Alexandria

• Named Components

• Wedge, Lever, Screw,

Windlass, & Pulley

1588

Ramelli

• Arteficiose Machine

• Describes Each Machine

of the Time

Without recognition of

similarity of components

287-212 BC

Archimedes of Syracuse

• Archimede’s Screw

• Claw of Archimedes

• Heat Ray

• Equilibrium of Planes

Description of lever

1724-1739

Jacob Leupold

• First to Recognize

Machine Components

• 9 Volume Series of

Books Published

1736-1742

Euler

• Mechanica Sive Motus

Scienta Analytice Exposita

• Analytical Treatment of Mechanisms

• Planar Motion

1817

James Watt & Oliver Evans

• Straight-Line Linkage

for Steam Engine

• Coupler Link Motion in 4-Bar

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Historical Perspective of

Kinematics

MER312: Dynamics of Mechanisms 4

Early 1800’s

Gaspard Monge

• L’Ecole Polytechnic in Paris

• Inventor of Descriptive

Geometry

• Course in Elements

of Machines

Early 1811

Jean Nicolas Pierre Hachette

• L’Ecole Polytechnic in Paris

• First Mechanisms Book

1834

Andre-Marie Ampere

• L’Ecole Polytechnic

• Essai sur la Philosophie

de Science

• First to use term Cinématique

1841

Robert Willis

• University of Cambridge, England

• Mechanism Synthesis

1876

Alexander Kennedy

• Theoretische Kinematik

• Translated to English

1875

Franz Reuleaux

• Theoretische Kinematik

• Mechanism Synthesis

Europe and

Australia

1940’s

Interest Builds

In US

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Historical Perspective of

Kinematics

MER312: Dynamics of Mechanisms 5RBB

Machine de Marly (1684)

• Created to pump water to the

Gardens in the Palace of

Versailles

• Delivered water to aqueduct 533 ft

above river

• Pumps driven by parallelograph

linkages

Bachannan Paddle Wheel

(1813)

Machines/Kinetics and

Mechanisms/Kinematics

MER312: Dynamics of Mechanisms 6

• Machines: A combination of resistant bodies so arranged

that by their means the mechanical forces of the nature

can be compelled to do work accompanied by certain

determinate motions.

• MECHANISMS: An assemblage of resistant bodies,

connected by movable joints, to form a closed kinematic

chain with one link fixed and having the purpose of

transforming motion.

• Structures: An assemblage of resistant bodies

connected by joints (or not) that do no work, and do not

transfer motion. It is intended to be rigid.

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Synthesis of Several Mechanisms

will be Considered

MER312: Dynamics of Mechanisms 7

Linkages

Gears

CAMs

Belts, Pulleys,

And Chains

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Mechanisms are Synthesized to

Produce Various Types of Motion

MER312: Dynamics of Mechanisms 8

PLANAR MOTION: All motion contained to one

geometric Plane or Parallel Planes.

TRANSLATION

ROTATION

COMBINATION

• Rectilinear Translation: All Points of

the body move in parallel straight

line paths.

• Rotation: Each point the body

remains a constant distance from a

fixed axis that is perpendicular to

the plane of motion.

• Rotation and Translation:

Combination of the above two.

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Curvilinear Translation a

Special Case of Translation

MER312: Dynamics of Mechanisms 9

Curvilinear Translation: The paths of the points are

identical curves parallel to a fixed plane.

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Non-Planar Motion Can Also

Be Generated By Mechanisms

MER312: Dynamics of Mechanisms 10

• Helical Motion: each point of the body has motion

of rotation about a fixed axis and at the same time

has translation parallel to the axis.

• Spherical Motion: each point of the body has

motion about a fixed point while remaining at a

constant distance from it.

• Spatial Motion: the body moves with rotations

about three non-parallel axes and translates in

three independent directions.

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Cycle, Period,

and Phase of Motion

MER312: Dynamics of Mechanisms 11

• Cycle: When the parts of a

mechanism have passed through

all the possible positions they can

assume after starting from some

simultaneous set of relative

positions and have returned to

their original relative positions.

• Period: The time required for a

cycle of motion.

• Phase: The simultaneous relative

position of a mechanism at a

given instant during a cycle.

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A Link Is A Rigid Body Having

Two or More Nodes

12MER312: Dynamics of Mechanisms

Unary Binary Ternary Quaternary Pentagonal

Nodes/Pairing Elements: Points at which links can be attached. The

order of the link is determined by the attachments used.

Joints/Kinematic Pairs: Allows relative motion between links.

Joint Classes: a kinematic pair is of the jth class if it diminishes the

relative motion of linked bodies by j Degrees of Freedom (DoF)

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• Open Kinematic Chain: A chain with one or more

open loops.

• Closed Kinematic Chain: A chain that forms one or

more closed loops.

• Simple-Closed Chain: Chain consisting of entirely

binary links and is closed.

• Compound Closed Chain: Chain including other than

binary links that is closed.

Kinematic Chains are Formed

by Connecting Links with Pairs

MER312: Dynamics of Mechanisms 13

R

R

R

R

P

R

R

R

R

R

R

RR

B

B

R

R

R

R

R

R

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Joint Classification,

Kinematic Pairs

Type of contact between elements

Line

Point

Surface

Degrees of Freedom Allowed

Type of Physical Closure

Force

Form

Number of Links Joined (Order)MER312: Dynamics of Mechanisms 14

Higher Pairs

Lower Pairs

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Joint Closure Classified as

Lower Pairs and Higher Pairs

MER312: Dynamics of Mechanisms 15

Form Closed, Rotating FULL Pin Joint Form Closed, Translating FULL Slider Joint

Force Closed,

Link against a plane HALF JointForm Closed,

Pin in Slot HALF Joint

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Degrees of Freedom or Mobility

The number of inputs needed to

provide in order to create a

predictable output

The number of independent

coordinates required to define its

position

MER312: Dynamics of Mechanisms 16

RR

R

R

P

R

R

R R

R

R

R R

B

B

RR

R

R

R

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1 DOF, Class I Kinematic Pairs

as Defined by Reuleaux

MER312: Dynamics of Mechanisms 17

Revolute (R) Prismatic (P) Helical (H)

• Lower Pair Contact

• Plainer & 3D Joint

• 1 DOF -

• Lower Pair Contact

• Plainer & 3D Joint

• 1 DOF - s

• Lower Pair Contact

• Plainer & 3D Joint

• 1 DOF

• Input - , Output – s

• Input - s, Output -

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2 DOF, Class II Kinematic Pairs

as Defined by Reuleaux

MER312: Dynamics of Mechanisms 18

Cam (Ca) Cylinder (C) Slotted Spherical (Sl)

• Higher Pair Contact

• RP

• Plainer & 3D Joint

• 2 DOF - ,s

• Lower Pair Contact

• RP

• 3D Joint

• 2 DOF – s,

• Lower Pair Contact

• RR

• 3D Joint

• 2 DOF - ,

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3 DOF, Class III Kinematic Pairs

as Defined by Reuleaux

MER312: Dynamics of Mechanisms 19

Spherical (S) Spherical Slotted Cylinder (C) Plane Pair (Pl)

• Lower Pair Contact

• RRR

• 3D Joint

• 3 DOF: , ,

• Lower Pair Contact

• RRP

• 3D Joint

• 3 DOF: , , s

• Lower Pair Contact

• RRP

• 3D Joint

• 2 DOF - , , s

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4 DOF, Class IV Kinematic Pairs

as Defined by Reuleaux

MER312: Dynamics of Mechanisms 20

Spherical Grove (Sg) Cylindrical Plane Pair (Cp)

• Lower Pair Contact

• RRRP

• 3D Joint

• 4 DOF: , , , s

• Lower Pair Contact

• RRPP

• 3D Joint

• 4 DOF: , , s, t

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5 DOF, Class V Kinematic Pairs

as Defined by Reuleaux

MER312: Dynamics of Mechanisms 21

Spherical Plane (Sp)

• Lower Pair Contact

• RRRPP

• 3D Joint

• 4 DOF: , , , s, t

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Planar Mechanisms

Each link has 3 DoF when moving relative to a fixed link

n link planar mechanism (one link is considered FIXED) has 3(n-1) degrees of freedom before joints are connected

Connecting a revolute pair 1 DoF → 2 constraints

2 DoF → 1 constraint

Mobility of Mechanism Constraints of all joints

minus total DoF of unconnected links

MER312: Dynamics of Mechanisms 22RBB

Mobility Calculation

L- number of links

M- mobility of planar n-link mechanism

j1- number of 1 DoF pairs

j2- number of 2 DoF pairs

Kutzbach Criterion

M = 3·(L-1) - 2·j1 - j2

Grübler Criterion

M = 3·(L-1) - 2·j1

MER312: Dynamics of Mechanisms 23RBB

Mobility Criterion:

Kutzbach or Gruebler

M=1

Mechanism can be driven by a single input direction

M=2

Two separate input motions are necessary to produce constrained motion for the mechanism

Differential Mechanism

M=0

Motion is impossible and the mechanism is a structure

Exact Constraint

M=-1

Redundant constraint

Pre-Load

MER312: Dynamics of Mechanisms 24RBB

Slipping

MER312: Dynamics of Mechanisms 25

Example: Calculate the Mobility

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Example: Calculate the Mobility

MER312: Dynamics of Mechanisms 26RBB

MER312: Dynamics of Mechanisms 27

Example: Calculate the Mobility

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Order of a Joint is One Less

than the Number of Links Joined

MER312: Dynamics of Mechanisms 28

First order pin Joint Second order pin Joint

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MER312: Dynamics of Mechanisms 29

Example: Calculate the Mobility

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MER312: Dynamics of Mechanisms 30

Example: Calculate the Mobility

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MER312: Dynamics of Mechanisms 31

Example: Calculate the Mobility

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Kutzback Criterion for Half Joints

Particular attention should be paid to the

contact between the wheel and the fixed

link

Slipping

MER312: Dynamics of Mechanisms 32RBB

MER312: Dynamics of Mechanisms 33

Example: Calculate the Mobility

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Example: Calculate the Mobility

MER312: Dynamics of Mechanisms 34RBB

Spatial Mechanism Mobility

Kutzbach Criterion

Where

j3- 3 Dof joints

j4- 4 Dof joints

j5- 5 Dof joints

1 2 3 4 56 ( 1) 5 4 3 2M L j j j j j

MER312: Dynamics of Mechanisms 35RBB

Mobility Paradoxes

Over-constrained Linkage with Redundant

Constraint

MER312: Dynamics of Mechanisms 36RBB

E-quintet,

Delta Triplet

M=0

E-quintet

M=1

Mobility Paradoxes

Over-constrained Linkage with Redundant

Constraint

MER312: Dynamics of Mechanisms 37

Mobility Paradoxes

Passive or Idle Degree of Freedom

MER312: Dynamics of Mechanisms 38RBB

Example: Calculate the Mobility

MER312: Dynamics of Mechanisms 39RBB