Reconstruction Synthesis of the Lost Subsystem for the Planetary Motions of Antikythera Mechanism_presentation

Taipei Citys

Taipei City

Tainan City

Creative Machine Design E&R Lab, National Cheng Kung University, TAIWAN

ASME 2011 International Design Engineering Technical Conferences (IDETC) and Computers and Information in

R S h f h L

Conferences (IDETC) and Computers and Information in Engineering Conference (CIE)

Reconstruction Synthesis of the Lost Subsystem for the Planetary Motions Subsystem for the Planetary Motions

of Antikythera Mechanism

Authors: Hong-Sen Yan & Jian-Liang LinSpeaker: Jian-Liang Lin

Department of Mechanical EngineeringNational Cheng Kung University


Tainan, Taiwan

Outline

IntroductionOverall ContributionHistorical BackgroundHistorical BackgroundKinematic Analysis i i SFeasible Topological StructuresProcess of Reconstruction SynthesisConclusions


IntroductionDisplay the motions of heavenly bodiesDo the calculations of different calendarsDo the calculations of different calendarsPredict the time of eclipses


Systematic Approach of R t ti S th iReconstruction Synthesis

aHistorical Archives

Design specifications

Topological characteristicsNumber synthesisa

Generalized Kinematic Chains

Topological characteristicsNumber synthesis

Specialized Chains

Design constraintsSpecialization

Reconstruction Designs

Ancient science and technology a


a

Overall ContributionOverall Contribution Two examples of the lost planetary motions subsystem

are derived respectively with one and three design are derived respectively with one and three design concepts.

The inner displays the pointer of dateThe inner displays the pointer of date

41

35

The outer displays the pointer of the planetp y p

1

3

The outer displays the pointer of the planet

4 41

2

1

i b

2

4

b1

gear a1 axis baxis b

Creative Machine Design E&R Lab, National Cheng Kung University, TAIWAN6

Historical BackgroundgVenus indicator fixed wheel

80 teeth80 teeth

D 65 t th

L: 50 teeth

fixed wheel fixingD: ~65 teethrotates once per Earth year


Existing reconstruction designs from 2000 to date

Historical BackgroundgThe feasible astronomical theory

Apollonius developed the epicycles model to Apollonius developed the epicycles model to account for the motions of five planets. And he successfully produced the retrograde motion of successfully produced the retrograde motion of planets. Planet

Epicycle

Earth

Deferent

Earth


Deferent

Kinematic AnalysisApollonius’ epicycles model

The planet rotates on an epicycle whose center p p yrotates on the deferent with the same direction.

d de

cosed

r

cos

1

e ded

d ee

rr

1 cose

e dedr


Anomaly Function

Feasible Topological Structures

Q2 Th b f li k d j i ??Q1: Input Identification of Lost Subsystem…??

Q2: The numbers of links and joints………….??Q3: The position of pin-in-slot joint…………..??


Q1: Input Identificationp

Three feasible input conditions of lost planetary motions


Three feasible input conditions of lost planetary motions subsystem were supposed.

Q2: Numbers of Links and Joints QThe lost planetary subsystem with NL members, NJR revolute pairs, NJA cam pairs and NJG gear pairs has one degree of pairs, NJA cam pairs and NJG gear pairs has one degree of freedom.

Based on the mobility analysis：2 3 4 0.........(1)JR JG JA LN N N N

Based on the mobility analysis

For a gear train with N joints：

0.........(2)JR JG JA JN N N N

For a gear train with NJ joints：

Each link in a geared kinematic chain must have one revolute pair：

1 0.........(3)JR LN N

have one revolute pair：


Q2: Numbers of Links and Joints By solving above equations, the relations between th j i t d li k ld b d i d the joints and links could be derived as:

1N All Joint number2 3J LN N

1JGN

Four-bar mechanism Revolute Joint number

1JR LN N with five jointsRevolute Joint number

G i3JG LN N

2JGN

Five bar mechanism

Gear Joint number

1JAN

Five-bar mechanism with seven joints

Pin-in-slot Joint number


JA

Q3: Positions of Pin-in-slot Joint For a planetary gear train with n planet gears and one pin-in-slot joint:pin in slot joint:

The pin-in-slot joint is incident to the l d h li k

the rate of epicycles eT

planet gear and the output link.

N is even;the rate of epicyclethe rate of deferent

s e

pn dT

Feasible

The pin-in-slot joint is incident to two planet gears.

Feasible( 1)

( 1) the rate of deferentth t f i l

k k

ns p p o d

TT T T

TT


the rate of epicycleks p eTT

Design Specifications of Each Type

Type 1 Type 21. It is a planetary gear train.2. It is a four-bar mechanism

with five joints and one

1. It is a planetary gear train.2. It is a five-bar mechanism

with seven joints and one with five joints and one degree of freedom.

3. The joints include three

with seven joints and one degree of freedom.

3. The joints include four revolute joints, one external gear joint.

4. The types of mechanical

revolute joints, two external gear joints and one pin-in-slot joint.4. The types of mechanical

components are gears and links.

one pin in slot joint.4. The types of mechanical

components are gears and li klinks.


Specialization Process of Type 1p ypIt must have at least a ground link (member 1, KF), an input link (member 2, KI), an output link (member 3, KO), and a

G d li k

( , I), p ( , O),transmission link (member 4, KT1).

STEP 1

1 1

3step 1 step 2Ground link

Input link

STEP 1

STEP 2

4step 4

step 3Output linkSTEP 3

STEP 41

2 3

1

2 3p

Transmission link

Pin-in-slot joint

STEP 4

STEP 5

2 3

4JA

2 3

4JAJG

JRstep 6

j

Revolute joints &Gear joints

STEP 6


1 1JR JR

(a)

Gear joints

Feasible Design Concept of Type 1Feasible Design Concept of Type 1

One feasible design concept for the inferior and One feasible design concept for the inferior and superior planets


Specialization Process of Type 2

step 1step 1step 1 step 1step 1Ground link11

3

1

1

1

step 2 step 2step 2 step 2Not Feasible

step 2Input link1

3

1 3

1

3 1

3

step 3 step 3step 3 step 3Output link1

2

3

1 32

1

3

2

1

3

2

1

32

step 4 step 4step 4 step 4 step 4Not Feasible Not Feasible Not FeasibleTransmission link

1

2

45

33JA

1 32

4 5

JA

step 5step 5 step 5step 5Pin-in-slot joint1

2

45 JA1

2

45

JA

3 JJ

1 32

4 5

JA

J

1 32

4 5 JA

J

step 6 step 6step 6 step 6Not Feasible

j

Revolute joints & G j i t


1

2

45 JA

JR

JR

JR

JRJG

JG

1 32

4 5

JA

JR

JR JR

JRJG

JG

1 32

4 5 JA

JR

JR JR

JR JG

JG

(a) (b) (c)

Not FeasibleGear joints

Feasible Design Concepts of Type 2g p yp

The outer displays the pointer of the planetThe inner displays the pointer of date

41

3 5

41

2

s1

s2

b1


axis b

ConclusionsThis work synthesizes all feasible reconstruction

designs of the planetary subsystem through a designs of the planetary subsystem through a systematic design procedure.

aHistorical Archives

a

Design specifications

Topological characteristicsNumber synthesisa

Generalized Kinematic Chains

Design constraintsSpecialization

Specialized Chains

Ancient science and technology a


Reconstruction Designsa

ConclusionsThe kinematic analysis of epicycles model and

the discussions of feasible topological structures the discussions of feasible topological structures were presented.

Lost subsystemLost subsystem

Input linkInput link

s1

s2

a1b1

b2

a1

b2

b1Supposed compound

gear(a) Condition 1 (b) Condition 2

Lost subsystem

Input link

a1

b2

b1

(c) Condition 3


ContributionContribution Two examples of the lost planetary motions subsystem

are derived respectively with one and three design are derived respectively with one and three design concepts.

The inner displays the pointer of dateThe inner displays the pointer of date

41

35

The outer displays the pointer of the planetp y p

1

3

The outer displays the pointer of the planet

4 41

2

1

i b

2

4

b1

gear a1 axis baxis b


Acknowledgementg

The authors are grateful to the National Science Council (TAIWAN ROC) under Science Council (TAIWAN, ROC) under Grant NSC 99-2221-E-006-253 for the

fi i l t f thi kfinancial support of this work.



Documents

Reconstruction Synthesis of the Lost Subsystem for the Planetary Motions of Antikythera Mechanism_presentation