Chapter 3 planar linkage
3.1 Types of the four-bar linkages
3.2 Variation of revolute four-bar mechanism
3.3 Application of the linkages
3.4 Characteristics Analysis of four-bar linkages
3.5 Velocity analysis by the Method of instant centers
3.6 Kinematic synthesis of four-bar linkages
Chapter 3 planar linkage
Mechanisms are used in a great variety
of machines and devices,The simplest
closed-loop linkage is the
four-bar,which has three
moving links and
four pin joints,
4
3.1 Types of the four-bar linkages
Types,
1) Crank-rocker mechanism
2) Double-crank mechanism
3) Double-rocker mechanism
Crank-rocker
Double-crank
Double-rocker
Crane Blender
冲床
3.2 Variation of revolute four-bar
mechanism
Transformation techniques or rules that
can be applied
1,Replacing a revolute pair with a
sliding pair
2,Taking different links as the frame
3,Enlarge a revolute pair
4,Interchanging guide-bar and sliding
block
1,Replacing a revolute pair with a
sliding pair
e,represents eccentric or offset
For the same kinematic chain,different
kinds of mechanisms will be generated
by holding different links fixes as the
frame,Such kinds of variations are
called inversions,
2,Taking different links as the
frame
3,Enlarge a revolute pair
4,Interchanging guide-bar and
sliding block
3.3 Application of the linkages
Can you give some examples used in machinery?
Crank-rocker,Blender,
Crank-slider block,piston engine,punch
Others,bus door driving mechanism,
crank-shaper
hand-operated well pump
3.3 Application of the linkages(ct)
Please try to type the quite different
applications above,
They represent three different tasks,
1) Path generation,concern with the path of
a tracer point,For example?
2) function generator,the relative motion
between links connected to ground is of
interest,For example?
3) motion generation,the entire motion of
the coupler link is of concern,For example?
3.4 Characteristics Analysis of
four-bar linkages
1,The Grashof condition(曲柄存在条件 )
2,Fourbar Quick-Return （急回特性）
3,Transmission angle & Pressure angle
（传动角与压力角）
4,Toggle positions & Dead-points
（极限位置与死点）
5,Mechanical advantage（机械增益）
1.The Grashof condition(曲柄存在条件 )
It is used to predict the rotation
behavior or rotatability of a four-bar
linkage’s inversions based only on the
link lengths,
Two key positions
1) Let S=length of shortest link
L=length of longest link
P= length of one remaining link
Q= length of other remaining link
Then if S+L<=P+Q,the linkage is Grashof and at
least one link will capable of making a full revolution
with respect to the ground plane,Otherwise,the
linkage is non-Grashof and no link will be capable of
a complete revolution relative to any other link,
2) The Grashof condition are independent from the
order of the S,L,P,Q when the mechanism is
assembled,
Discussion(1)
Case 1 S+L<P+Q,
*The coupler makes a full revolution
Case 2 S+L>P+Q,
All inversions will be triple-rocker,no link can fully rotate,
Case 3 S+L=P+Q,
At the change points,the output behavior will
become indeterminate,The common used
mechanisms are parallelogram and antiparallelogram,
Mechanism Frame
Crank-rocker Ground either link adjacent to the
shortest
Double-crank Ground the shortest link
*Double-rocker Ground the link opposite the
shortest
Discussion(2)
Change points,and indeterminate
behavior,
How to solve this problem?
Table2-3 Type criteria for the revolute four-bar mechanisms
Link as the
frame
S+L<P+Q S+L>P+Q S+L=P+Q
Grashof Non-Grashof Change-point
The shortest
link
Double-crank Double-rocker
(without fully rotate)
or
Triple-rocker Opposite to the shortest link Double-rocker (with fully rotate)
Adjacent to the
shortest link
Crank-rocker
2,Fourbar Quick return （急回特性）
Toggle position
极限位置
The toggle
positions are
determined
by the
colinearity of
the two of the
moving links,
They also
define the
limits of
motion of the
driven rocker,
2,Fourbar Quick return （急回特性）
Definition of the Quick-Return(急回 )
It is used to describe the fact that the forward stroke
is faster than that of the return stroke,The time
ratio(TR) defines the degree of quick-return of the
linkage,
TR=?2 / ?1
K= ?f / ? s =?1 / ?2
=180o+? / 180 o -?
K,行程速比系数
3,Transmission angle ? & Pressure angle ?
（传动角与压力角）
What is the relation between the F and Fn or (Ft)
What is the function of the Fn & Ft
follower
Transmission angle ?,
1) Defined as the angle between the output link and coupler
2) Taken as the absolute value of the acute angle of the pair
of angles at the intersection of the two links
3) Varies continuously from min.to max,value
4) A measure of the quality of force transmission at the joint
Pressure angle ?,
The acute angle between the directions of the force and the
velocity of the point receiving the force on the follower,
3,Transmission angle ? & Pressure angle ?
（传动角与压力角）
Can you tell the position where the four-bar linkage
has the extreme transmission angle ??
3,Transmission angle ? & Pressure angle ?
（传动角与压力角）
One extreme value occurs
where Links 1 and 4 are
colinear and nonoverlapping;
The other extreme value
where occurs Links 1 and 4
are colinear and overlapping;
Please calculate them after
class,
For smooth running and good force transmission,keep the minimum
transmission angle above about 40o,
follower
3,Transmission angle ? & Pressure angle ?
（传动角与压力角）
4,Toggle positions & Dead-points
Toggle position
极限位置
4,Toggle positions & Dead-points
Dead-points
When the rocker is the driver,
the follower(crank) is
colinear with the coupler,the
transmission angle equals to
0o or 180o,The force applied
to the follower passes
through the fixed pivot of the
follower,In this situation,the
mechanism cannot move by a
torque on the driving rocker,
In this sense,these positions
are called dead-points,
Discussion,Pros/cons of the dead-points
Pros,in some circumstances,the dead-points
are used to lock the position,For example the
clamping device,landing mechanism in
airplane,
Cons,the dead-points are not good to the
transmission,But we can use either the
flywheel on the driven crank,or providing the
duplicate linkage 900 out of phase to
overcome the dead-points,
5,Mechanical advantage
Mechanical advantage (mA) can be
defined as,
mA =Fout / Fin
3.5 Velocity analysis by the Method
of instant centers
Velocity analysis
Analytical and Graphical method
3.5 Velocity analysis by the Method of
instant centers
Definition,
(a) A point in both bodies,(b) A point at which the
two bodies have no relative velocity,(c) a point
about which one body may considered to rotate
relative to the other body at a given instant,
Quantity of instant centers,
N = n(n -1)/2
How to find the instant centers?
Graphical method
Positions of the instant centers
By inspection,just use the definition of
the instant center;
By Kennedy’s rule,any three bodies in
plane motion will have exactly three
instant center,and they will lie on the
same line,
Example,
Graphical method
R joints
P joints
Higher pairs
Roll Roll-slide
Positions of the ICs for two links directly connected together
If υK2 equals υK3,,the point K must
be on the line of the P12P13 or its
extension,
Positions of the ICs for two links not connected together
Applications
The ICs of the four-bar linkage
Graphical method
Using Assur group to analyze the
velocity of the linkages,
Analytical method
3.6 Kinematic synthesis of four-bar
linkages
Synthesis,Qualitative synthesis and
Quantitative synthesis (定性综合和定量综合 ),
Type Synthesis is a form of qualitative,It refers to the
definition of the proper type of mechanism best suited
to the problem(型综合 ),
Dimensional Synthesis of a linkage is the determination
of the proportions(lengths) of the links necessary to
accomplish the desired motions (尺寸综合 ),
图解法、解析法
Dimensional Synthesis
Analytical Dimensional Synthesis
Graphical Dimensional Synthesis
Three types of synthesis tasks(1)
Example?(汽车转弯机构 )
Function Generation is defined as the
correlation of an input motion with an
output motion in a mechanism (函数发
生器 ),
Three types of synthesis tasks (2)
Path Generation is defined as the control of
a point in the plane such that it follows some
prescribed path (轨迹生成器 ),
Crane Blender
Three types of synthesis tasks (3)
Motion Generation is defined as the control of a line in
the plane such that it assumes some prescribed set of
sequential position (运动生成器 ),
Three types of synthesis tasks
Function Generation is defined as the correlation
of an input motion with an output motion in a
mechanism (函数发生器 ),
Path Generation is defined as the control of a
point in the plane such that it follows some
prescribed path (轨迹生成器 ),
Motion Generation is defined as the control of a
line in the plane such that it assumes some
prescribed set of sequential position (运动生成器 ),
Two-position synthesis
Two conditions:Rocker output and Coupler output
1,Rocker Output— Two position with angular displacement (Function
Generation)
Exp,Design a four-bar Grashof crank-rocker to give 450of rocker rotation
with equal time forward and back,from a constant speed motor
input;
2,Rocker Output— Two position with complex displacement (Motion
Generation)
Exp,Design a fourbar linkage to move link CD from position C1D1 to C2D2
3,Coupler Output— Two position with complex displacement (Motion
Generation)
Exp,Design a fourbar linkage to move link CD from position C1D1 to C2D2
(with moving pivots at C and D)
Exp,Design a four-bar Grashof crank-rocker to
give 450 of rocker rotation with equal time forward
and back,from a constant speed motor input;
1,Analysis the characteristics of the four-bar
linkage with equal time forward and back,from
a constant speed motor input,
2,Design the four-bar linkage(pp84 figure 3-4)
1,Rocker Output— Two position with angular displacement (Function Generation)
2,Rocker Output— Two position with complex displacement (Motion Generation)
Exp,Design a fourbar linkage to move link CD from
position C1D1 to C2D2
1,Use rocker to move the link CD;
2,Use coupler to move the link CD;
3,Output— Two position with complex displacement (Motion Generation)
Exp,Design a fourbar linkage to move link CD from
position C1D1 to C2D2 (with moving pivots at C and D)
Use coupler to move the link CD;
1,Three-position synthesis with specified
moving pivots(pp89 Exp.3-5)
2,Three-position synthesis with alternate
moving pivots(pp91 Exp.3-6)
3,Three-position synthesis known crank length
with specified fixed pivots
4,Three-position synthesis with specified fixed
pivots(pp93 Exp.3-7)
Three-position synthesis
Three-position synthesis known crank length with
specified fixed pivots
Fourbar Quick-Return
Fourbar Crank-Return Linkage for Specified Time Ratio
Exp,Design a four-bar Grashof crank-rocker to give 450of
rocker rotation with time ratio of 1:1.25,from a constant
speed motor input;
DOF Lower or higher pair Name & Kinematic diagram
Full
joint
Give two examples
Half
joint
Give one example
Draw a crank-rocker mechanism and find its toggle position
when the crank is the driver,then give the definition of the
time ratio,
Quiz
Homework (3/18)
1,Read book,pp40-49,pp55-60,pp76-
100,pp169-172,pp188-196,pp241,
pp249-261,
2,2-32,2-35,Sketch its kinematic
diagram,determine its DOF,the
Grashof condition,and its type,
3,3-15(a,b,c,),3-23
Homework (3/25)
1,Review
2,Pp 132 3-3(Add if the quick-return time ratio is
1:1.4,design the mechanism again),
3,Pp133 3-4,3-5
Discussion
1,Pp133 3-5,3-6
2,Do presentation
Homework and presentation
1,Draw a offset crank-slider mechanism in one
position,then locate all its instant centers,If the
crank is driver with ?,find the angular velocity
of the coupler,and velocity of the slider by
graphical method,
2,Discussion section:choose two
3,Brief summary of this chapter & your advice or
problems you meet,
Homework
Discussion section
1,Organized by your course master
2,Do presentation by groups(four students in
each group)
3,Requirement about your report,
* Finished your homework individually,
* Finished a Powerpoint file about 10 minutes by group for your
presentation,
* Upload your presentation report before 3:00pm,Apr.7
4,Time,IE21 4:00-5:30 Apr,7
IE22 4:00-5:30 Apr,8
Report Format
Cover page:Team members,Class (course #,university,
department,instructors),Date etc,
Content,analysis and design
Conclusion
References
Your questions & advices
3.1 Types of the four-bar linkages
3.2 Variation of revolute four-bar mechanism
3.3 Application of the linkages
3.4 Characteristics Analysis of four-bar linkages
3.5 Velocity analysis by the Method of instant centers
3.6 Kinematic synthesis of four-bar linkages
Chapter 3 planar linkage
Mechanisms are used in a great variety
of machines and devices,The simplest
closed-loop linkage is the
four-bar,which has three
moving links and
four pin joints,
4
3.1 Types of the four-bar linkages
Types,
1) Crank-rocker mechanism
2) Double-crank mechanism
3) Double-rocker mechanism
Crank-rocker
Double-crank
Double-rocker
Crane Blender
冲床
3.2 Variation of revolute four-bar
mechanism
Transformation techniques or rules that
can be applied
1,Replacing a revolute pair with a
sliding pair
2,Taking different links as the frame
3,Enlarge a revolute pair
4,Interchanging guide-bar and sliding
block
1,Replacing a revolute pair with a
sliding pair
e,represents eccentric or offset
For the same kinematic chain,different
kinds of mechanisms will be generated
by holding different links fixes as the
frame,Such kinds of variations are
called inversions,
2,Taking different links as the
frame
3,Enlarge a revolute pair
4,Interchanging guide-bar and
sliding block
3.3 Application of the linkages
Can you give some examples used in machinery?
Crank-rocker,Blender,
Crank-slider block,piston engine,punch
Others,bus door driving mechanism,
crank-shaper
hand-operated well pump
3.3 Application of the linkages(ct)
Please try to type the quite different
applications above,
They represent three different tasks,
1) Path generation,concern with the path of
a tracer point,For example?
2) function generator,the relative motion
between links connected to ground is of
interest,For example?
3) motion generation,the entire motion of
the coupler link is of concern,For example?
3.4 Characteristics Analysis of
four-bar linkages
1,The Grashof condition(曲柄存在条件 )
2,Fourbar Quick-Return （急回特性）
3,Transmission angle & Pressure angle
（传动角与压力角）
4,Toggle positions & Dead-points
（极限位置与死点）
5,Mechanical advantage（机械增益）
1.The Grashof condition(曲柄存在条件 )
It is used to predict the rotation
behavior or rotatability of a four-bar
linkage’s inversions based only on the
link lengths,
Two key positions
1) Let S=length of shortest link
L=length of longest link
P= length of one remaining link
Q= length of other remaining link
Then if S+L<=P+Q,the linkage is Grashof and at
least one link will capable of making a full revolution
with respect to the ground plane,Otherwise,the
linkage is non-Grashof and no link will be capable of
a complete revolution relative to any other link,
2) The Grashof condition are independent from the
order of the S,L,P,Q when the mechanism is
assembled,
Discussion(1)
Case 1 S+L<P+Q,
*The coupler makes a full revolution
Case 2 S+L>P+Q,
All inversions will be triple-rocker,no link can fully rotate,
Case 3 S+L=P+Q,
At the change points,the output behavior will
become indeterminate,The common used
mechanisms are parallelogram and antiparallelogram,
Mechanism Frame
Crank-rocker Ground either link adjacent to the
shortest
Double-crank Ground the shortest link
*Double-rocker Ground the link opposite the
shortest
Discussion(2)
Change points,and indeterminate
behavior,
How to solve this problem?
Table2-3 Type criteria for the revolute four-bar mechanisms
Link as the
frame
S+L<P+Q S+L>P+Q S+L=P+Q
Grashof Non-Grashof Change-point
The shortest
link
Double-crank Double-rocker
(without fully rotate)
or
Triple-rocker Opposite to the shortest link Double-rocker (with fully rotate)
Adjacent to the
shortest link
Crank-rocker
2,Fourbar Quick return （急回特性）
Toggle position
极限位置
The toggle
positions are
determined
by the
colinearity of
the two of the
moving links,
They also
define the
limits of
motion of the
driven rocker,
2,Fourbar Quick return （急回特性）
Definition of the Quick-Return(急回 )
It is used to describe the fact that the forward stroke
is faster than that of the return stroke,The time
ratio(TR) defines the degree of quick-return of the
linkage,
TR=?2 / ?1
K= ?f / ? s =?1 / ?2
=180o+? / 180 o -?
K,行程速比系数
3,Transmission angle ? & Pressure angle ?
（传动角与压力角）
What is the relation between the F and Fn or (Ft)
What is the function of the Fn & Ft
follower
Transmission angle ?,
1) Defined as the angle between the output link and coupler
2) Taken as the absolute value of the acute angle of the pair
of angles at the intersection of the two links
3) Varies continuously from min.to max,value
4) A measure of the quality of force transmission at the joint
Pressure angle ?,
The acute angle between the directions of the force and the
velocity of the point receiving the force on the follower,
3,Transmission angle ? & Pressure angle ?
（传动角与压力角）
Can you tell the position where the four-bar linkage
has the extreme transmission angle ??
3,Transmission angle ? & Pressure angle ?
（传动角与压力角）
One extreme value occurs
where Links 1 and 4 are
colinear and nonoverlapping;
The other extreme value
where occurs Links 1 and 4
are colinear and overlapping;
Please calculate them after
class,
For smooth running and good force transmission,keep the minimum
transmission angle above about 40o,
follower
3,Transmission angle ? & Pressure angle ?
（传动角与压力角）
4,Toggle positions & Dead-points
Toggle position
极限位置
4,Toggle positions & Dead-points
Dead-points
When the rocker is the driver,
the follower(crank) is
colinear with the coupler,the
transmission angle equals to
0o or 180o,The force applied
to the follower passes
through the fixed pivot of the
follower,In this situation,the
mechanism cannot move by a
torque on the driving rocker,
In this sense,these positions
are called dead-points,
Discussion,Pros/cons of the dead-points
Pros,in some circumstances,the dead-points
are used to lock the position,For example the
clamping device,landing mechanism in
airplane,
Cons,the dead-points are not good to the
transmission,But we can use either the
flywheel on the driven crank,or providing the
duplicate linkage 900 out of phase to
overcome the dead-points,
5,Mechanical advantage
Mechanical advantage (mA) can be
defined as,
mA =Fout / Fin
3.5 Velocity analysis by the Method
of instant centers
Velocity analysis
Analytical and Graphical method
3.5 Velocity analysis by the Method of
instant centers
Definition,
(a) A point in both bodies,(b) A point at which the
two bodies have no relative velocity,(c) a point
about which one body may considered to rotate
relative to the other body at a given instant,
Quantity of instant centers,
N = n(n -1)/2
How to find the instant centers?
Graphical method
Positions of the instant centers
By inspection,just use the definition of
the instant center;
By Kennedy’s rule,any three bodies in
plane motion will have exactly three
instant center,and they will lie on the
same line,
Example,
Graphical method
R joints
P joints
Higher pairs
Roll Roll-slide
Positions of the ICs for two links directly connected together
If υK2 equals υK3,,the point K must
be on the line of the P12P13 or its
extension,
Positions of the ICs for two links not connected together
Applications
The ICs of the four-bar linkage
Graphical method
Using Assur group to analyze the
velocity of the linkages,
Analytical method
3.6 Kinematic synthesis of four-bar
linkages
Synthesis,Qualitative synthesis and
Quantitative synthesis (定性综合和定量综合 ),
Type Synthesis is a form of qualitative,It refers to the
definition of the proper type of mechanism best suited
to the problem(型综合 ),
Dimensional Synthesis of a linkage is the determination
of the proportions(lengths) of the links necessary to
accomplish the desired motions (尺寸综合 ),
图解法、解析法
Dimensional Synthesis
Analytical Dimensional Synthesis
Graphical Dimensional Synthesis
Three types of synthesis tasks(1)
Example?(汽车转弯机构 )
Function Generation is defined as the
correlation of an input motion with an
output motion in a mechanism (函数发
生器 ),
Three types of synthesis tasks (2)
Path Generation is defined as the control of
a point in the plane such that it follows some
prescribed path (轨迹生成器 ),
Crane Blender
Three types of synthesis tasks (3)
Motion Generation is defined as the control of a line in
the plane such that it assumes some prescribed set of
sequential position (运动生成器 ),
Three types of synthesis tasks
Function Generation is defined as the correlation
of an input motion with an output motion in a
mechanism (函数发生器 ),
Path Generation is defined as the control of a
point in the plane such that it follows some
prescribed path (轨迹生成器 ),
Motion Generation is defined as the control of a
line in the plane such that it assumes some
prescribed set of sequential position (运动生成器 ),
Two-position synthesis
Two conditions:Rocker output and Coupler output
1,Rocker Output— Two position with angular displacement (Function
Generation)
Exp,Design a four-bar Grashof crank-rocker to give 450of rocker rotation
with equal time forward and back,from a constant speed motor
input;
2,Rocker Output— Two position with complex displacement (Motion
Generation)
Exp,Design a fourbar linkage to move link CD from position C1D1 to C2D2
3,Coupler Output— Two position with complex displacement (Motion
Generation)
Exp,Design a fourbar linkage to move link CD from position C1D1 to C2D2
(with moving pivots at C and D)
Exp,Design a four-bar Grashof crank-rocker to
give 450 of rocker rotation with equal time forward
and back,from a constant speed motor input;
1,Analysis the characteristics of the four-bar
linkage with equal time forward and back,from
a constant speed motor input,
2,Design the four-bar linkage(pp84 figure 3-4)
1,Rocker Output— Two position with angular displacement (Function Generation)
2,Rocker Output— Two position with complex displacement (Motion Generation)
Exp,Design a fourbar linkage to move link CD from
position C1D1 to C2D2
1,Use rocker to move the link CD;
2,Use coupler to move the link CD;
3,Output— Two position with complex displacement (Motion Generation)
Exp,Design a fourbar linkage to move link CD from
position C1D1 to C2D2 (with moving pivots at C and D)
Use coupler to move the link CD;
1,Three-position synthesis with specified
moving pivots(pp89 Exp.3-5)
2,Three-position synthesis with alternate
moving pivots(pp91 Exp.3-6)
3,Three-position synthesis known crank length
with specified fixed pivots
4,Three-position synthesis with specified fixed
pivots(pp93 Exp.3-7)
Three-position synthesis
Three-position synthesis known crank length with
specified fixed pivots
Fourbar Quick-Return
Fourbar Crank-Return Linkage for Specified Time Ratio
Exp,Design a four-bar Grashof crank-rocker to give 450of
rocker rotation with time ratio of 1:1.25,from a constant
speed motor input;
DOF Lower or higher pair Name & Kinematic diagram
Full
joint
Give two examples
Half
joint
Give one example
Draw a crank-rocker mechanism and find its toggle position
when the crank is the driver,then give the definition of the
time ratio,
Quiz
Homework (3/18)
1,Read book,pp40-49,pp55-60,pp76-
100,pp169-172,pp188-196,pp241,
pp249-261,
2,2-32,2-35,Sketch its kinematic
diagram,determine its DOF,the
Grashof condition,and its type,
3,3-15(a,b,c,),3-23
Homework (3/25)
1,Review
2,Pp 132 3-3(Add if the quick-return time ratio is
1:1.4,design the mechanism again),
3,Pp133 3-4,3-5
Discussion
1,Pp133 3-5,3-6
2,Do presentation
Homework and presentation
1,Draw a offset crank-slider mechanism in one
position,then locate all its instant centers,If the
crank is driver with ?,find the angular velocity
of the coupler,and velocity of the slider by
graphical method,
2,Discussion section:choose two
3,Brief summary of this chapter & your advice or
problems you meet,
Homework
Discussion section
1,Organized by your course master
2,Do presentation by groups(four students in
each group)
3,Requirement about your report,
* Finished your homework individually,
* Finished a Powerpoint file about 10 minutes by group for your
presentation,
* Upload your presentation report before 3:00pm,Apr.7
4,Time,IE21 4:00-5:30 Apr,7
IE22 4:00-5:30 Apr,8
Report Format
Cover page:Team members,Class (course #,university,
department,instructors),Date etc,
Content,analysis and design
Conclusion
References
Your questions & advices
