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Dynamic Simulation:
Lagrangian Multipliers
Objective
The objective of this module is to introduce Lagrangian multipliers
that are used with Lagranges equation to find the equations that
control the motion of mechanical systems having constraints.
The matrix form of the equations used by computer programs such asAutodesk Inventors Dynamic Simulation are also presented.
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Basic Problem in Multi-body Dynamics
In the previous module (Module 6)
we developed Lagranges equation
and showed how it could be used to
determine the equations of simple
motion systems.
Lagranges Equation
The examples we considered were for systems in which there
were no constraints between the generalized coordinates.
The basic problem of multi-body dynamics is to systematically findand solve the equations of motion when there are constraints that
bodies in the system must satisfy.
0
ii q
L
q
L
dt
d
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 2
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Non-conservative Forces
The derivation of Lagranges equation in the previous module (Module6) considered only processes that store and release potential energy.
These processes are called conservative because they conserve energy.
Lagranges equation must be modified to accommodate non-
conservative processes that dissipate energy (i.e. friction, damping, andexternal forces).
A non-conservative force or moment acting on generalized coordinate
qiis denoted as Qi.
The more general form of Lagranges equation is
i
ii
L
q
L
dt
d
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 3
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Simple Pendulum
Simple
Pendulum
c.g.
X
Y
The pendulum shown in thefigure will be used as an
example throughout this
module.
The position of the pendulumis known at any instance of
time if the coordinates of the
c.g.,Xcg,Ycg,and the angle q
are known.
Xcg,Ycgand qare the
generalized coordinates.
xy
Xcg
Ycg
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
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Kinetic and Potential Energies
The kinetic energy (T) and potentialenergy (V) of the pendulum are
These equations also give the
kinetic and potential energy of the
unconstrained body flying through
the air.
There needs to be a way to include
the constraints to differentiate
between the two systems.
cg
cgcg
mgYV
mYXmIT
2222
1
2
1
2
1 qc.g.
X
Y
xy
Xcg
Ycg
Unconstrained
Body
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
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Constraint Equations
In addition to satisfyingLagranges equations of motion,
the pendulum must satisfy the
constraints that the
displacements at X1and Y1are
zero.
The constraint equations are
c.g.
X
Y
xy
Xcg
Ycg
X1,Y1
0cos2
0sin2
1
1
YY
XX
cg
cg
q
q
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 6
The c.g. lies on they-
axis halfway along the
length .
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Lagrangian Multipliers
X1
Y1
X
Y
The kinetic energy is augmented by
adding the constraint equationsmultiplied by parameters called
Lagrangian Multipliers.
Note that since the constraint
equations are equal to zero, we have
not changed the magnitude of the
kinetic energy.
The Lagrangian multipliers are
treated like unknown generalized
coordinates.
What are the units of 1and 2?
12
11
222
cos2
sin2
2
1
2
1
2
1
yY
XX
YmXmIT
cg
cg
cgcg
ql
ql
q
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 7
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Governing Equations
Lagranges Equation
Lagrangian
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 8
i
ii
L
q
L
dt
d
i
n
i
i VTL 1
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
1
2
1
2
1qlqlq
In the following slides, Lagranges
equation will be used in a systematic
manner to determine the equations of
motion for the pendulum.
The governing equations that will be
used are shown here.
There are no non-conservative forces
acting on the system ( ).0iQ
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Equation for 1stGeneralized CoordinateSection 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 9
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
121
21 qlqlq
i
ii
Q
q
L
q
L
dt
d
Lagranges Equation
Generalized Coordinates
25
14
3
2
1
l
l
q
q
q
qYq
Xq
cg
cg1
1
1
1
l
q
L
Xmq
L
dt
d
Xm
q
L
cg
cg
1stEquation
01lcgXm
Mathematical Steps
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Equation for 2ndGeneralized CoordinateSection 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 10
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
121
21 qlqlq
i
ii
Q
q
L
q
L
dt
d
Lagranges Equation
Generalized Coordinates
25
14
3
2
1
l
l
q
q
q
q
Yq
Xq
cg
cg mgq
L
Ymq
L
dt
d
Ymq
L
cg
cg
2
2
2
2
l
2ndEquation
02
mgYm cg l
Mathematical Steps
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Equation for 3rdGeneralized Coordinate
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
121
21 qlqlq
i
ii
Q
q
L
q
L
dt
d
Lagranges Equation
Generalized Coordinates
25
14
3
2
1
l
l
q
q
q
q
Yq
Xq
cg
cg qlql
q
q
sin2
cos2
21
3
3
3
q
L
Iq
L
dt
d
Iq
L
3rdEquation
Mathematical Steps
0sin2
cos2
21 qlqlq I
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
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Equation for 4thGeneralized CoordinateSection 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 12
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
121
21 qlqlq
i
ii
Q
q
L
q
L
dt
d
Lagranges Equation
Generalized Coordinates
25
14
3
2
1
l
l
q
q
q
q
Yq
Xq
cg
cg1
4
4
4
sin2
0
0
XXq
L
q
L
dt
dq
L
cg
q
4thEquation
Mathematical Steps
0sin2
1 XXcg q
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Equation for 5thGeneralized Coordinate
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 13
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
121
21 qlqlq
i
ii
Q
q
L
q
L
dt
d
Lagranges Equation
Generalized Coordinates
25
14
3
2
1
l
l
q
q
q
q
Yq
Xq
cg
cg1
5
5
5
cos2
0
0
YYq
L
q
L
dt
d
q
L
cg
q
5thEquation
Mathematical Steps
0cos2
1 YYcg q
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Summary of EquationsSection 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 14
01lcgXm
02 mgYm cg l
0sin2
cos2
21 qlqlq I
0sin2
1 XXcg q
0cos2
1 YYcg q
There are five unknown generalizedcoordinates including the two
Lagrangian Multipliers. There are
also five equations.
Three of the equations aredifferential equations.
Two of the equations are algebraic
equations.
Combined, they are a system of
differential-algebraic equations
(DAE).
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Free Body Diagram Approach
1
2
cgmX
cgmY
cgI
The application of Lagrangesequation yields the same
equations obtained by drawing a
free-body diagram.
Free Body Diagram with
Inertial Forces
mg
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 15
Summation of Forces in the X-direction
Summation of Forces in the Y-direction
Summation of Moments about the c.g.
01lcgXm
02 mgYm cg l
0sin2
cos2
21 qlqlq I
q 2
qcos2
qsin2
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Physical Significance of Lagrangian Multipliers
Force required to impose
the constraint that X1is a
constant.
Newtons 2ndLaw in x-direction
Lagrangian Multipliers are simply the forces (moments) required to
enforce the constraints. In general, the Lagrangian Multipliers are a
function of time, because the forces (moments) required to enforce
the constraints vary with time (i.e. depend on the position of the
pendulum).
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 16
01lcgXm
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Matrix Format
i
i
ii q
VQ
q
T
q
T
dt
d
The computer implementation of Lagranges equation isfacilitated by writing the equations in matrix format.
Separating the Lagrangian into kinetic and potential
energy terms enables Lagranges equation to be written as
In this format, the conservative and non-conservative
forces are lumped together on the right hand side of the
equation.
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
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Matrix Format
The kinetic energy augmented with Lagrangian Multipliers can bewritten in matrix format as
tqqMqT TT ,2
1 l
Column array containing generalized coordinatevelocities.
Column array containing the constraint equations
(refer to Module 3 in this section).
Column array containing the Lagrangian multipliers.
Matrix containing the mass and mass moments of
inertia associated with each generalized coordinate.
q
tq,
l M
Inertia Matrix
000
000
000
000
A
cg
A
A
I
m
m
M
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
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Matrix Format
Another equation for accelerationwas obtained in Module 4 based
on kinematics and the constraint
equations.
Combining this equation with
Lagranges equation from the
previous slide yields:
qqj
i
l
q
qM
j
i
j
i
0
Matrix Form of Equations
This equation can be solved
to find the accelerations
and constraint forces at an
instant in time.
The accelerations must
then be integrated to find
the velocities and positions.
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
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Solution of Differential-Algebraic
Equations (DAE)
The solution of even the simplest system of DAE requirescomputer programs that employ predictor-corrector type
numerical integrators.
The Adams-Moulton method is an example of the type of
numerical method used.
Significant research has led to the development of efficient and
robust integrators that are found in commercial computer
programs that generate, assemble, and solve these equations.
Autodesk Inventors Dynamic Simulation environment is an
example of such software.
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
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Module Summary
This module showed how Lagrangian Multipliers are used inconjunction with Lagranges equation to obtain the equations that
control the motion of mechanical systems.
The method presented provides a systematic method that forms the
basis of mechanical simulation programs such as Autodesk InventorsDynamic Simulation environment.
The matrix format of the equations were presented to provide insight
into the computations performed by computer software.
The Jacobian and constraint kinematics developed in Module 4 of this
section are an important part of the matrix formulation.
Section 4Dynamic Simulation
Module 7Lagrangian Multipliers
Page 22