機構學
Mechanisms (機構學)
原名:
原名: 機動學,
機動學, 即機械運動學(
即機械運動學(kinematics)之簡稱
kinematics)之簡稱
Text:
K. J. Waldron and G. L. Kinzel, 2004, Kinematics,
Dynamics, and Design of Machinery,
Machinery 2nd ed., John Wiley
& Sons. (歐亞)
Reference:
1. 顏鴻森, 機構學, 東華書局.
2. G. Bögelsack, F. J. Gierse, V. Oravsky, J. M. Prentis,
and A. Rossi, 1983, Terminology for the Theory of
Machines and Mechanisms, Pergamon Press.
C. F. Chang, KUAS ME
1
Contents

Basic Concepts: (基本觀念)
– Chapter 1 Introduction

Linkages: (連桿組, 連桿機構)
– Chapter 2 Graphical Position, Velocity, and Acceleration Analysis
for Mechanisms with Revolute Joints of Fixed Slides
– Chapter 3 Linkages with Rolling and Sliding Contacts and Joints on
Moving Sliders
– Chapter 4 Instant Centers of Velocity
– Chapter 5 Analytical Linkage Analysis

Cam and Gears: (凸輪和齒輪)
–
–
–
–

Chapter 8 Profile Cam Design
Chapter 10 Spur Gears
Chapter 11 Helical, Bevel, and Worm Gears
Chapter 12 Gear Trains
We will focus on the so-called “
Planar Motion”
—Motion of links
whose points describe curves located in parallel planes. [桿件上
各點之運動皆在同一平面或其平行平面上]
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1
機構學
Chapter 1. Introduction
 Definition
of Mechanisms
–Mechanisms are assemblages of rigid
member connected together by joints. (p.3)
– (機構係由機件與接頭所構成之可動組合)
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Mechanism Vs Machine
Links
Joints


Constrained
Constrained
motion?
motion?
mechanism
Power
Mechanisms
Mechanismstransfer
transfermotion
motionto
to
one
or
more
output
members
one or more output members

 Machine
Machinetransfer
transfermotion
motionand
and
useful
usefulwork
workto
toone
oneor
ormore
more
output
outputmembers
members

(機器為可輸出有用之功的機構)
 (機器為可輸出有用之功的機構)
Controller
Output
Output
effective
effective
work?
work?




Constrained
Constrainedmotion:
motion:
(各機件皆產生確切且可預期之運動)
(各機件皆產生確切且可預期之運動)
machine
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機構學
Terminology For MMT

Kinematic chain [運動鏈]
– Assemblage of links and joints.

Mechanism [機構]
– Kinematic chain with one of its components (link
or joint) connected to the frame and with definite
motion
具有確切運動且至少有一桿固連於機架之運動鏈
– System of bodies designed to convert motions of
and forces on one or several bodies into
constrained motions of and forces on other bodies
(MMT)
一支或多支桿件之運動和受力轉換為其他桿件之拘
束運動和受力

Machine(機器
Machine(機器))
– Device performing mechanical motion to transform
and transfer energy, material and information
是一種執行機械運動的裝置, 用來變換和傳遞能量, 材料
與資訊
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Terminology For MMT

Link [連桿, 機件]
1. Mechanism element (component) carrying kinematic pairing
elements [機構元件, 用來帶動以運動對連接之元件]
2. Element of a linkage. [連桿組之元件]

Joint [接頭]
– The physical embodiment of kinematic pair.[運動對之具體化身]

Kinematic pair [運動對]
– Contacting elements of links permitting their constrained relative
motion. [桿件間之接觸部份, 它使桿件之間產生拘束的相對運動]
– Lower pair—Kinematic
pair which is formed by surface contact of
pair
its elements. [經由面接觸所構成之可動連接]
– High pair—Kinematic
pair which is formed by point or line contact
pair
of its elements [經由點或線接觸所構成之可動連接]
– Connectivity(Degree
Connectivity(Degree of freedom of a joint):
joint the number of
independent coordinates needed to describe the relative positions
of pairing elements [確定兩桿件之相對位置所需之獨立參數的數目]
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3
機構學
Lower Pair Joints [六種常見之低對接頭]
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1. Revolute Pair (R) [旋轉對]

Name:
Revolute hinge
2. turning pair
1.

Letter symbol:
–

R
Connectivity (DOF) :
1
Dof
Dofof
ofkinematic
kinematicpair
pair(connectivity)
(connectivity)
==the
thenumber
numberof
ofindependent
independent
coordinates
coordinatesneeded
neededto
todescribe
describethe
the
relative
relativepositions
positionsof
ofpairing
pairing
elements
elements
接頭之自由度
接頭之自由度==確定兩桿件之相對
確定兩桿件之相對
位置所需之獨立參數的數目
位置所需之獨立參數的數目
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4
機構學
2. Prismatic Pair (P) [滑行對]

Name:
1.
2.
3.

Prismatic joint
Slider
Sliding pair
Letter symbol:
P

Connectivity (DOF):
1
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3. Helical Pair (H) [螺旋對]

Name:
1.
2.
3.

Screw joint
Helical joint
Helical pair
Letter symbol:
H

Connectivity (DOF) :
1
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5
機構學
4. Cylindrical Pair (C) [圓柱對]

Name:
Cylindrical Joint
2. Cylindrical pair
1.

Letter symbol:
C

Connectivity (DOF) :
2
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5. Spherical Pair (S) [球面對]

Name:
1.
2.
3.

Spherical joint
Ball joint
Spherical pair
Letter symbol:
S

Connectivity (DOF) :
3
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6
機構學
6. Planar Pair (R) [平面對]

Name:
1.
2.

Planar joint
Planar pair
Letter symbol:
R

Connectivity (DOF) :
3
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Replacement of a Lower Pair Joint by a combination of Higher Pair
Pair Joints

In order to reduce the friction in lower pair joints, a simple joint
may be replaced by a kinematically equivalent compound joint.
 For instance,
C. F. Chang, KUAS ME
國立高雄應用科大機械系
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7
機構學
Antifriction Bearings
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Some Higher Pair Joints
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8
機構學
1. Cylindrical Roller [圓柱形滾子, 滾動對]

Name:
1. Cylindrical roller
2. Rolling Pair

Connectivity (DOF) :
1
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2. Cam Pair [凸輪對]

Name:
Cam Pair

Connectivity (DOF) :
2
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機構學
3. Rolling Ball

Name:
Rolling Ball

Connectivity (DOF) :
3
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4. Ball in Cylinder

Name:
Ball in Cylinder

Connectivity (DOF) :
3+1=4
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機構學
5. Spatial Point Contact

Name:
Spatial point contact

Connectivity (DOF) :
3+2=5
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Replacement of a Higher Pair Joint with Lower Pair Joints

In order to reduce the contact stress in higher pair joints, a joint
may be replaced by some kinematically equivalent lower pair
joints.
 For instance, a pin-in-a-slot joint may become a combination of
a revolute joint and a prismatic joint.
+
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11
機構學
Some Examples of Compound Joints
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Mechanism & Linkage (p.8)




A linkage is a closed kinematic chain with one link
selected as the frame.
A frame or base member is a link that is fixed.
The term mechanism is somewhat interchangeable
with linkage.
In normal usage,
– mechanism is somewhat more generic term encompassing
systems with higher pairs, or combinations of lower and
higher pair joints, whereas
– the term linkage tends to be restricted to systems that have
only lower pair joints.
C. F. Chang, KUAS ME
國立高雄應用科大機械系
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12
機構學
Planar Linkages


A planar linkage is one in which the velocities of all
points in all members are directed parallel to a plane,
called the plane of motion.
機構上各點之速度若皆與運動平面平行, 則稱其為平面
機構
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Representation of Links and frame

Binary links(
links(二接頭桿)
二接頭桿)
– those that have two joints mounted on them

Ternary links (三接頭桿)
三接頭桿)
– those that have three joints mounted on them
Slider-crank linkage

Quaternary links (四接頭桿)
四接頭桿)
– those that have four joints mounted on them
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國立高雄應用科大機械系
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13
機構學
Symbolic Designation of SingleSingle-Loop Linkages

RRRR Linkage (4R)

RRRP Linkage (3R-P)

RPRP Linkage (2R-2P)
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Visualization of the Motion of Linkages
 Modelling
with woods, paper cards, …
 Modelling
with computer graphics systems
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國立高雄應用科大機械系
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機構學
Constraint Analysis of Planar Linkages(pp. 1111-18)



Mobility (Degrees of freedom of a linkage)
– The minimum number of coordinates needed to specify the
positions of all members of the mechanism
– 確定機構各桿件之相對位置所需之獨立參數的數目
If the mobility is zero or negative, the assemblage is a structure.
structure.
– If the mobility is zero, the structure is statically determinate (靜定結
構)
– If the mobility is negative, the structure is statically indeterminate
(靜不定結構)
The mobility of planar linkages: (constraint criterion equation)
– n: the number of links
M 桿件之自由度 接頭所造成之拘束度
j
– j: the number of joints
3(n 1) (3 f i )
i
1
– fi : the connectivity of joint i (dof of joint i)
– the dof a link with planar motion = 3
j
M 3(n j 1) f i
i 1
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Degree of Freedom of a Body (Link)

The dof of a body is the number of independent
coordinates needed to specify its position
– A body moving freely in a plane has three degrees of
freedom. 2 translation + 1 rotation
– A body moving freely in space has six degrees of freedom. 3
translation + 3 rotation (pitch-yaw-roll)
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國立高雄應用科大機械系
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15
機構學
Examples
j
M 3(n j 1) f i
i 1

Mobility analysis of a planar four-bar linkage

Mobility analysis of a planar four-bar linkage
C. F. Chang, KUAS ME
Examples(cont.)
Examples(cont.) pp. 1414-15
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j
M 3(n j 1) f i
i 1

when more than two members come together at a single point
location (multiple joint 複接頭)
n=6, j=7, fi=7
M=3(6-7-1)+7=1
n=11, j=14, fi=15
M=3(11-14-1)+15=3
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機構學
Remark on those linkages
with all joints having connectivity one
j
M 3(n j 1) f i
i 1
Since all joints having connectivity one (fi=1), we have
–fi=j=number of joints
 Moreover, if the mobility of planar linkages is set to
one, the constraint criterion equation leads to
–1=3(n-j-1)+j
–3n=2j+4
–n must be a even number, say n=2, 4, 6, …


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Constraint Analysis of Spatial Linkages(pp. 1818-22)

The dof of a link with spatial motion = 6
M 桿件之自由度 接頭所造成之拘束度
j
6(n 1) (6 f i )
i
1
j
6(n j 1) f i
i 1

Where
– M = Mobility of spatial linkages
– n: the number of links
– j: the number of joints
– fi : the connectivity of joint i (dof of joint i)
 This equation is known as the Kutzbach criterion
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國立高雄應用科大機械系
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機構學
Example 1
j
M 6(n j 1) f i
i
1
n
= 4 (桿數)
 j = 4 (接頭數)

fi = 3+3+1+2 = 9 (接頭之總自由度)
 M = 6(4-4-1)+9 = -6+9=3
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Example 2

n=7
 j=6
 Five revolute joints: 1, 2, 4, 5, 6
 One prismatic joint: 3
1
link
joint


fi = 51+11 = 6 (接頭之總自由度)
M = 6(7-6-1)+6 = 6
C. F. Chang, KUAS ME
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機構學
Example 3






n=4
j = 4 (RSSR)
Two revolute joints (fi = 1)
Two spherical joint (fi = 3)
fi = 21+2 3 = 8 (接頭之總自由
度)
M = 6(4-4-1)+8 = -6+8 = 2

The result seems to conflict with our practical experience since there is a
unique value of for any given value of . i.e., the orientation of link 4 can be
determined when the orientation of link 2 is specified.
 Examining the mechanism carefully will reveal that we need an extra
parameter to identify the orientation of link 3. Because this parameter doesn't
affect the input-output relationship of the linkage, so we call it an idle degree
of freedom.
freedom
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Idle Degrees of Freedom (Redundant
(Redundant DOF 多餘自由度)
多餘自由度)

An idle dof is one that does not affect the input-output
relationship of the linkage.

Procedures for Locating the Idle dof are as following:
–Identify the input link and output link.
–Check to determine if a single link or a combination of
connected links can move without altering the relative
position of the input and output links. If the answer is
positive, there are some idle dof’
s.
C. F. Chang, KUAS ME
國立高雄應用科大機械系
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機構學
Idle Degrees of Freedom & Stewart Platform

For a Stewart platform, we have

n = 14
 (2
6 limbs+1 base link+1 output link )
j = 18

– Six prismatic joints (fi = 1)
– Twelve spherical joint (fi = 3)


fi = 61+12 3 = 42 (接頭之總自由度)
M = 6(14-18-1)+42 = -30+42 = 12


Indeed, this mechanism has six idle dof.
This is because each limb is free to spin about the line joining
the centers of its spherical joints.
C. F. Chang, KUAS ME
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Planar Mechanism with an Idle Degrees of Freedom

For the planar mechanism as shown in
the figure, we have

M = 1 if the kinematic pair at C is a rolling
pair (fi=1)

M = 2 if the kinematic pair at C is a cam
pair (fi=2)

However, the extra degree of
freedom does not affect the the inputoutput (link6 vs. link 2) relationship of
the linkage. So, the extra dof is an idle
dof.
C. F. Chang, KUAS ME
國立高雄應用科大機械系
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機構學
Paradoxical Mechanism (矛盾機構)
矛盾機構)
ref pp. 2525-29 overover-constrained linkage

A spatial 4R linkage is, in general, immovable because M=-2.
 However, it may have mobility one if special geometry are met.
 There are two well-know paradoxical mechanisms:
– Spherical four-bar mechanism (The axes of revolute joints all
pass through a single point)
– Bennett mechanism
a sin= b sin
C. F. Chang, KUAS ME
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Kinematic Inversion

Kinematic Inversion is the transformation of one mechanism to
another by choosing a different member to be the frame
 For example,
Toothbrush
mechanism
Walking
mechanism
Water
pump
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國立高雄應用科大機械系
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機構學
An Practical Application—Water Pump
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Classification of 4-bar Mechanisms &
Grashof’
s rule (pp. 32-37)



s: link length of the shortest link
l: link length of the longest link
p, q: link lengths of the other two links
Type
Grashof
condition
s+l<p+q
Shortest link
mechanism
Side link
Crank-rocker
Coupler
Double-rocker
Base, frame
Double-crank
ChangePoint
s+l=p+q
Any link
Change-point
Non-Grashof
s+l>p+q
Any link
Triple-rocker
Paper csme2001 csmmt2001
國立高雄應用科大機械系
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機構學
Example

AB=1.14 in, BC=2.26 in, AD=1.74 in
 AF=2.00 in, DE=2.68 in, c=1.09 in
 Determine the region for joint E that will allow full rotation of link
6, i.e., EF=?
Sol:
Link AB in loop ABC can make a full rotation
(BC-AB>c)
Link AF is not the shortest one (AF<DE)
Four-bar FEDA must be a crank-rocker
 s=EF  l=DE
1.74
2.68
2.0
E
s+l<p+q
EF+DE<AF+AD
EF+2.68<2.00+1.74
EF<1.06 in ANS
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Analysis of four-bar linkages-Centrodes
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機構學
Limit positions ( of Driven Link )
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Analysis of fourfour-bar linkageslinkages-Limit Positions
ref: csme2001.pdf
csme2001.pdf
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機構學
Classification of Spherical 44-bar Mechanisms
Ref: csmmt2001
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Interference
ref: csmeconf1995,1996,CSMMTconf2000
.pdf
csmeconf1995,1996,CSMMTconf2000.pdf
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國立高雄應用科大機械系
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25
機構學
Actuators
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Stable & Unstable Operation
load > driving torque
 angular velocity is decreased until state A is reached
 End
of Chapter 1
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