Short Notes: Dynamic Analysis of Linkages | Short Notes for Mechanical Engineering PDF Download

 Page 1


Dynamic Analysis of Linkages  
Linkages are the basic building blocks of all mechanisms. All common forms of 
mechanisms (cams, gears, belts, chains) are in fact variations on a common theme 
of linkages. Linkages are made up of links and joints.
• Links: rigid member having nodes.
• Node: attachment points.
• Binary link: 2 nodes
• Ternary link: 3 nodes
• Quaternary link: 4 nodes
• Joint: connection between two or more links (at their nodes) which allows 
motion; (Joints also called kinematic pairs)
Joints can be classified in several ways:
1. By the type of contact between the elements, line, point, or surface.
2. By the number of degrees of freedom allowed at the joint.
3. By the type of physical closure of the joint: either force or form closed.
4. By the number of links joined (order of the joint).
A more useful means to classify joints (pairs) is by the number of degrees of 
freedom that they allow between the two elements joined. A joint with more than 
one freedom may also be a higher pair.
• Joint order = number of links-1
Dynamic Force Analysis
Page 2


Dynamic Analysis of Linkages  
Linkages are the basic building blocks of all mechanisms. All common forms of 
mechanisms (cams, gears, belts, chains) are in fact variations on a common theme 
of linkages. Linkages are made up of links and joints.
• Links: rigid member having nodes.
• Node: attachment points.
• Binary link: 2 nodes
• Ternary link: 3 nodes
• Quaternary link: 4 nodes
• Joint: connection between two or more links (at their nodes) which allows 
motion; (Joints also called kinematic pairs)
Joints can be classified in several ways:
1. By the type of contact between the elements, line, point, or surface.
2. By the number of degrees of freedom allowed at the joint.
3. By the type of physical closure of the joint: either force or form closed.
4. By the number of links joined (order of the joint).
A more useful means to classify joints (pairs) is by the number of degrees of 
freedom that they allow between the two elements joined. A joint with more than 
one freedom may also be a higher pair.
• Joint order = number of links-1
Dynamic Force Analysis
• D'Alembert's Principle and Inertia Forces: An important principle, known as 
cTAIembert's principle, can be derived from Newton's second law. In words, 
d'Alembert's principle states that the reverse-effective forces and torques and 
the external forces and torques on a body together give statical equilibrium.
F +(-maG ) = 0 
Te g +(-Igo) = 0
• The terms in parentheses in above equations are called the reverse-effective 
force and the reverse-effective torque, respectively. These quantities are also 
referred to as inertia force and inertia torque. Thus, we define the inertia force 
F, as
F; = -maG
• This reflects the fact that a body resists any change in its velocity by an 
inertia force proportional to the mass of the body and its acceleration. The 
inertia force acts through the center of mass G of the body. The inertia torque 
or inertia couple C, is given by:
C; = -lG a
As indicated, the inertia torque is a pure torque or couple.
£ r G = £ r ,0 + c, = o
• Where £F refers here to the summation of external forces and, therefore, is 
the resultant external force, and UeG is the summation of external moments, 
or resultant external moment, about the center of mass G .
• Thus, the dynamic analysis problem is reduced in form to a static force and 
moment balance where inertia effects are treated in the same manner as 
external forces and torques.
• In particular for the case of assumed mechanism motion, the inertia forces 
and couples can be determined completely and thereafter treated as known 
mechanism loads.
• Furthermore, D'Alembert's principle facilitates moment summation about any 
arbitrary point P in the body, if we remember that the moment due to inertia 
force F, must be included in the summation. Hence,
U p = U~eP + C; + R pg x Ft = 0
• Where U p is the summation of moments, including inertia moments, about 
point P. UeP is the summation of external moments about P , C, is the inertia 
couple, is the inertia force, and RP G is a vector from point P to point C.
For a body in plane motion in the xy plane with all external forces in that plane.
£ ^ = z L A = ) = 0
£ F, = I X + F, = X X + )=o
£ T o = £ T t0 + c, = £ r , G+ (~IG a) = 0
• Where aG x and aG y are the x and y components of aG . These are three scalar 
equations, where the sign convention for torques and angular accelerations is
Page 3


Dynamic Analysis of Linkages  
Linkages are the basic building blocks of all mechanisms. All common forms of 
mechanisms (cams, gears, belts, chains) are in fact variations on a common theme 
of linkages. Linkages are made up of links and joints.
• Links: rigid member having nodes.
• Node: attachment points.
• Binary link: 2 nodes
• Ternary link: 3 nodes
• Quaternary link: 4 nodes
• Joint: connection between two or more links (at their nodes) which allows 
motion; (Joints also called kinematic pairs)
Joints can be classified in several ways:
1. By the type of contact between the elements, line, point, or surface.
2. By the number of degrees of freedom allowed at the joint.
3. By the type of physical closure of the joint: either force or form closed.
4. By the number of links joined (order of the joint).
A more useful means to classify joints (pairs) is by the number of degrees of 
freedom that they allow between the two elements joined. A joint with more than 
one freedom may also be a higher pair.
• Joint order = number of links-1
Dynamic Force Analysis
• D'Alembert's Principle and Inertia Forces: An important principle, known as 
cTAIembert's principle, can be derived from Newton's second law. In words, 
d'Alembert's principle states that the reverse-effective forces and torques and 
the external forces and torques on a body together give statical equilibrium.
F +(-maG ) = 0 
Te g +(-Igo) = 0
• The terms in parentheses in above equations are called the reverse-effective 
force and the reverse-effective torque, respectively. These quantities are also 
referred to as inertia force and inertia torque. Thus, we define the inertia force 
F, as
F; = -maG
• This reflects the fact that a body resists any change in its velocity by an 
inertia force proportional to the mass of the body and its acceleration. The 
inertia force acts through the center of mass G of the body. The inertia torque 
or inertia couple C, is given by:
C; = -lG a
As indicated, the inertia torque is a pure torque or couple.
£ r G = £ r ,0 + c, = o
• Where £F refers here to the summation of external forces and, therefore, is 
the resultant external force, and UeG is the summation of external moments, 
or resultant external moment, about the center of mass G .
• Thus, the dynamic analysis problem is reduced in form to a static force and 
moment balance where inertia effects are treated in the same manner as 
external forces and torques.
• In particular for the case of assumed mechanism motion, the inertia forces 
and couples can be determined completely and thereafter treated as known 
mechanism loads.
• Furthermore, D'Alembert's principle facilitates moment summation about any 
arbitrary point P in the body, if we remember that the moment due to inertia 
force F, must be included in the summation. Hence,
U p = U~eP + C; + R pg x Ft = 0
• Where U p is the summation of moments, including inertia moments, about 
point P. UeP is the summation of external moments about P , C, is the inertia 
couple, is the inertia force, and RP G is a vector from point P to point C.
For a body in plane motion in the xy plane with all external forces in that plane.
£ ^ = z L A = ) = 0
£ F, = I X + F, = X X + )=o
£ T o = £ T t0 + c, = £ r , G+ (~IG a) = 0
• Where aG x and aG y are the x and y components of aG . These are three scalar 
equations, where the sign convention for torques and angular accelerations is
based on a right-hand xyz coordinate system; that is. Counterclockwise is 
positive and clockwise is negative. The general moment summation about 
arbitrary point P,
= X X + ( - I 0a)-rRp a z (.-ma0 } ) -R p a > X-maG l)= 0
Where RpG x and RpG y are the x and y components of position vector RP G . This 
expression for dynamic moment equilibrium will be useful in the analyses to be 
presented in the following sections of this chapter.
• Equivalent Offset Inertia Force: For purposes of graphical plane force analysis, 
it is convenient to define what is known as the equivalent offset inertia force. 
This is a single force that accounts for both translational inertia and 
rotational inertia corresponding to the plane motion of a rigid body.
o Figure A shows a rigid body with planar motion represented by center of 
mass acceleration aG and angular acceleration a. The inertia force and 
inertia torque associated with this motion are also shown. The inertia 
torque -lG a can be expressed as a couple consisting of forces Q and (- Q) 
separated by perpendicular
(A)
(B)
(C)
(D)
Figure (A) Derivation of the equivalent offset inertia force associated with 
planer motion of a rigid body.
Figure(B) Replacement of the inertia torque by a couple.
Page 4


Dynamic Analysis of Linkages  
Linkages are the basic building blocks of all mechanisms. All common forms of 
mechanisms (cams, gears, belts, chains) are in fact variations on a common theme 
of linkages. Linkages are made up of links and joints.
• Links: rigid member having nodes.
• Node: attachment points.
• Binary link: 2 nodes
• Ternary link: 3 nodes
• Quaternary link: 4 nodes
• Joint: connection between two or more links (at their nodes) which allows 
motion; (Joints also called kinematic pairs)
Joints can be classified in several ways:
1. By the type of contact between the elements, line, point, or surface.
2. By the number of degrees of freedom allowed at the joint.
3. By the type of physical closure of the joint: either force or form closed.
4. By the number of links joined (order of the joint).
A more useful means to classify joints (pairs) is by the number of degrees of 
freedom that they allow between the two elements joined. A joint with more than 
one freedom may also be a higher pair.
• Joint order = number of links-1
Dynamic Force Analysis
• D'Alembert's Principle and Inertia Forces: An important principle, known as 
cTAIembert's principle, can be derived from Newton's second law. In words, 
d'Alembert's principle states that the reverse-effective forces and torques and 
the external forces and torques on a body together give statical equilibrium.
F +(-maG ) = 0 
Te g +(-Igo) = 0
• The terms in parentheses in above equations are called the reverse-effective 
force and the reverse-effective torque, respectively. These quantities are also 
referred to as inertia force and inertia torque. Thus, we define the inertia force 
F, as
F; = -maG
• This reflects the fact that a body resists any change in its velocity by an 
inertia force proportional to the mass of the body and its acceleration. The 
inertia force acts through the center of mass G of the body. The inertia torque 
or inertia couple C, is given by:
C; = -lG a
As indicated, the inertia torque is a pure torque or couple.
£ r G = £ r ,0 + c, = o
• Where £F refers here to the summation of external forces and, therefore, is 
the resultant external force, and UeG is the summation of external moments, 
or resultant external moment, about the center of mass G .
• Thus, the dynamic analysis problem is reduced in form to a static force and 
moment balance where inertia effects are treated in the same manner as 
external forces and torques.
• In particular for the case of assumed mechanism motion, the inertia forces 
and couples can be determined completely and thereafter treated as known 
mechanism loads.
• Furthermore, D'Alembert's principle facilitates moment summation about any 
arbitrary point P in the body, if we remember that the moment due to inertia 
force F, must be included in the summation. Hence,
U p = U~eP + C; + R pg x Ft = 0
• Where U p is the summation of moments, including inertia moments, about 
point P. UeP is the summation of external moments about P , C, is the inertia 
couple, is the inertia force, and RP G is a vector from point P to point C.
For a body in plane motion in the xy plane with all external forces in that plane.
£ ^ = z L A = ) = 0
£ F, = I X + F, = X X + )=o
£ T o = £ T t0 + c, = £ r , G+ (~IG a) = 0
• Where aG x and aG y are the x and y components of aG . These are three scalar 
equations, where the sign convention for torques and angular accelerations is
based on a right-hand xyz coordinate system; that is. Counterclockwise is 
positive and clockwise is negative. The general moment summation about 
arbitrary point P,
= X X + ( - I 0a)-rRp a z (.-ma0 } ) -R p a > X-maG l)= 0
Where RpG x and RpG y are the x and y components of position vector RP G . This 
expression for dynamic moment equilibrium will be useful in the analyses to be 
presented in the following sections of this chapter.
• Equivalent Offset Inertia Force: For purposes of graphical plane force analysis, 
it is convenient to define what is known as the equivalent offset inertia force. 
This is a single force that accounts for both translational inertia and 
rotational inertia corresponding to the plane motion of a rigid body.
o Figure A shows a rigid body with planar motion represented by center of 
mass acceleration aG and angular acceleration a. The inertia force and 
inertia torque associated with this motion are also shown. The inertia 
torque -lG a can be expressed as a couple consisting of forces Q and (- Q) 
separated by perpendicular
(A)
(B)
(C)
(D)
Figure (A) Derivation of the equivalent offset inertia force associated with 
planer motion of a rigid body.
Figure(B) Replacement of the inertia torque by a couple.
Figure(C) The strategic choice of a couple.
Figure(D) The single force is equivalent to the combination of a force and a 
torque in figure (A)
h= |/G ar| |»iaG |
Distance h, as shown in Figure B. The necessary conditions for the couple to be 
equivalent to the inertia torque are that the sense and magnitude be the same. 
Therefore, in this case, the sense of the couple must be clockwise and the 
magnitudes of Q and h must satisfy the relationship
I Q .h \= \ lG .a \
Otherwise, the couple is arbitrary and there are an infinite number of possibilities 
that will work. Furthermore, the couple can be placed anywhere in the plane.
Figure C shows a special case of the couple, where force vector Q is equal to ma6 
and acts through the center of mass. Force (- Q) must then be placed as shown to 
produce a clockwise sense and at a distance;
jv *) M
k ? r k i
Force Q will cancel with the inertia force F,-= - mag, leaving the single equivalent 
offset force, which has the following characteristics:
1. The magnitude of the force is | mac |.
2. The direction of the force is opposite to that of acceleration a.
3. The perpendicular offset distance from the center of mass to the line 
of action of the force .
4. The force is offset from the center of mass so as to produce a moment 
about the center of mass that is opposite in sense to acceleration a.
The usefulness of this approach for graphical force analysis will be demonstrated 
in the following section. It should be emphasized, however, that this approach is 
usually unnecessary in analytical solutions.. Including the original inertia force and 
inertia torque, can be applied directly.
• Dynamic Analysis of the Four-Bar Linkage:
o The analysis of a four-bar linkage will effectively illustrate most of the 
ideas that have been presented; furthermore, the extension to other 
mechanism types should become clear from the analysis of this 
mechanism.
• Dynamic Analysis of the Slider-Crank Mechanism:
o Dynamic forces are a very important consideration in the design of slider 
crank mechanisms for use in machines such as internal combustion 
engines and reciprocating compressors.
° Following such a process a kinematics analysis is first performed from 
which expressions are developed for the inertia force and inertia torque 
for each of the moving members, These quantities may then be 
converted to equivalent offset inertia forces for graphical analysis or 
they may be retained in the form of forces and torques for analytical 
solution.
° Following figure is a schematic diagram of a slider crank mechanism, 
showing the crank 1, the connecting rod 2, and the piston 3, all of which
Page 5


Dynamic Analysis of Linkages  
Linkages are the basic building blocks of all mechanisms. All common forms of 
mechanisms (cams, gears, belts, chains) are in fact variations on a common theme 
of linkages. Linkages are made up of links and joints.
• Links: rigid member having nodes.
• Node: attachment points.
• Binary link: 2 nodes
• Ternary link: 3 nodes
• Quaternary link: 4 nodes
• Joint: connection between two or more links (at their nodes) which allows 
motion; (Joints also called kinematic pairs)
Joints can be classified in several ways:
1. By the type of contact between the elements, line, point, or surface.
2. By the number of degrees of freedom allowed at the joint.
3. By the type of physical closure of the joint: either force or form closed.
4. By the number of links joined (order of the joint).
A more useful means to classify joints (pairs) is by the number of degrees of 
freedom that they allow between the two elements joined. A joint with more than 
one freedom may also be a higher pair.
• Joint order = number of links-1
Dynamic Force Analysis
• D'Alembert's Principle and Inertia Forces: An important principle, known as 
cTAIembert's principle, can be derived from Newton's second law. In words, 
d'Alembert's principle states that the reverse-effective forces and torques and 
the external forces and torques on a body together give statical equilibrium.
F +(-maG ) = 0 
Te g +(-Igo) = 0
• The terms in parentheses in above equations are called the reverse-effective 
force and the reverse-effective torque, respectively. These quantities are also 
referred to as inertia force and inertia torque. Thus, we define the inertia force 
F, as
F; = -maG
• This reflects the fact that a body resists any change in its velocity by an 
inertia force proportional to the mass of the body and its acceleration. The 
inertia force acts through the center of mass G of the body. The inertia torque 
or inertia couple C, is given by:
C; = -lG a
As indicated, the inertia torque is a pure torque or couple.
£ r G = £ r ,0 + c, = o
• Where £F refers here to the summation of external forces and, therefore, is 
the resultant external force, and UeG is the summation of external moments, 
or resultant external moment, about the center of mass G .
• Thus, the dynamic analysis problem is reduced in form to a static force and 
moment balance where inertia effects are treated in the same manner as 
external forces and torques.
• In particular for the case of assumed mechanism motion, the inertia forces 
and couples can be determined completely and thereafter treated as known 
mechanism loads.
• Furthermore, D'Alembert's principle facilitates moment summation about any 
arbitrary point P in the body, if we remember that the moment due to inertia 
force F, must be included in the summation. Hence,
U p = U~eP + C; + R pg x Ft = 0
• Where U p is the summation of moments, including inertia moments, about 
point P. UeP is the summation of external moments about P , C, is the inertia 
couple, is the inertia force, and RP G is a vector from point P to point C.
For a body in plane motion in the xy plane with all external forces in that plane.
£ ^ = z L A = ) = 0
£ F, = I X + F, = X X + )=o
£ T o = £ T t0 + c, = £ r , G+ (~IG a) = 0
• Where aG x and aG y are the x and y components of aG . These are three scalar 
equations, where the sign convention for torques and angular accelerations is
based on a right-hand xyz coordinate system; that is. Counterclockwise is 
positive and clockwise is negative. The general moment summation about 
arbitrary point P,
= X X + ( - I 0a)-rRp a z (.-ma0 } ) -R p a > X-maG l)= 0
Where RpG x and RpG y are the x and y components of position vector RP G . This 
expression for dynamic moment equilibrium will be useful in the analyses to be 
presented in the following sections of this chapter.
• Equivalent Offset Inertia Force: For purposes of graphical plane force analysis, 
it is convenient to define what is known as the equivalent offset inertia force. 
This is a single force that accounts for both translational inertia and 
rotational inertia corresponding to the plane motion of a rigid body.
o Figure A shows a rigid body with planar motion represented by center of 
mass acceleration aG and angular acceleration a. The inertia force and 
inertia torque associated with this motion are also shown. The inertia 
torque -lG a can be expressed as a couple consisting of forces Q and (- Q) 
separated by perpendicular
(A)
(B)
(C)
(D)
Figure (A) Derivation of the equivalent offset inertia force associated with 
planer motion of a rigid body.
Figure(B) Replacement of the inertia torque by a couple.
Figure(C) The strategic choice of a couple.
Figure(D) The single force is equivalent to the combination of a force and a 
torque in figure (A)
h= |/G ar| |»iaG |
Distance h, as shown in Figure B. The necessary conditions for the couple to be 
equivalent to the inertia torque are that the sense and magnitude be the same. 
Therefore, in this case, the sense of the couple must be clockwise and the 
magnitudes of Q and h must satisfy the relationship
I Q .h \= \ lG .a \
Otherwise, the couple is arbitrary and there are an infinite number of possibilities 
that will work. Furthermore, the couple can be placed anywhere in the plane.
Figure C shows a special case of the couple, where force vector Q is equal to ma6 
and acts through the center of mass. Force (- Q) must then be placed as shown to 
produce a clockwise sense and at a distance;
jv *) M
k ? r k i
Force Q will cancel with the inertia force F,-= - mag, leaving the single equivalent 
offset force, which has the following characteristics:
1. The magnitude of the force is | mac |.
2. The direction of the force is opposite to that of acceleration a.
3. The perpendicular offset distance from the center of mass to the line 
of action of the force .
4. The force is offset from the center of mass so as to produce a moment 
about the center of mass that is opposite in sense to acceleration a.
The usefulness of this approach for graphical force analysis will be demonstrated 
in the following section. It should be emphasized, however, that this approach is 
usually unnecessary in analytical solutions.. Including the original inertia force and 
inertia torque, can be applied directly.
• Dynamic Analysis of the Four-Bar Linkage:
o The analysis of a four-bar linkage will effectively illustrate most of the 
ideas that have been presented; furthermore, the extension to other 
mechanism types should become clear from the analysis of this 
mechanism.
• Dynamic Analysis of the Slider-Crank Mechanism:
o Dynamic forces are a very important consideration in the design of slider 
crank mechanisms for use in machines such as internal combustion 
engines and reciprocating compressors.
° Following such a process a kinematics analysis is first performed from 
which expressions are developed for the inertia force and inertia torque 
for each of the moving members, These quantities may then be 
converted to equivalent offset inertia forces for graphical analysis or 
they may be retained in the form of forces and torques for analytical 
solution.
° Following figure is a schematic diagram of a slider crank mechanism, 
showing the crank 1, the connecting rod 2, and the piston 3, all of which
are assumed to be rigid. The center of mass locations are designated b j 
letter G , and the members have masses m, and moments of inertia lei, i = 
1,2,3.
° The following analysis will consider the relationships of the inertia 
forces and torques to the bearing reactions and the drive torque on the 
crank, at an arbitrary mechanism position given by crank angle < p Friction 
will be neglected.
For the piston (moment equation not included):
+(-»¥*«) = 0
(Dynamic-force analysis of a slider crank mechanism)
(Free-body diagrams of the moving members of linkages)
• For the connecting rod (moments about point B):
FV., + F3 2 I + ( r m 2 aG2:)= 0 
F1 2 y + F32y+(-m2 aG2y)=°
F.2 y £cos0+ (— w;aG ,r)£G sin< ? 
+(-m ;< 3 G ,v K ccos<9-(-/c;ar:)= 0
• For the crank (moments about point 0 ? ):
FC lx ~ F2 1 x ~ ^l^Glx ) — ®
F C ly + F 2ly ~ m i a G ly ) = ®
T\ - Flb:rsm<f>~ F2lfrcos0+ ( - m ,^ ) ^ sin^ 
+(r mlaQ ly)r<icos^+(r I 6la l) = 0
Where T is the input torque on the crank. This set of equations embodies both 
of the dynamic-force analysis approaches described in Newton's Laws.