Dynamic Simulation:
Lagrangian Multipliers
Objective
 The objective of this module is to introduce Lagrangian multipliers that are used with Lagrange’s 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 as
Autodesk Inventor’s Dynamic Simulation are also presented.
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Basic Problem in Multi-body Dynamics
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 2
Lagrange’s Equation
In the previous module (Module 6) we developed Lagrange’s equation and showed how it could be used to determine the equations of simple motion systems.
d dt



L
 q  i




L
 q i

0
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 find and solve the equations of motion when there are constraints that bodies in the system must satisfy.
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Non-conservative Forces
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 3
 The derivation of Lagrange’s equation in the previous module (Module
6) considered only processes that store and release potential energy.
 These processes are called conservative because they conserve energy.
 Lagrange’s equation must be modified to accommodate nonconservative processes that dissipate energy (i.e. friction, damping, and external forces).


A non-conservative force or moment acting on generalized coordinate q i is denoted as Q i
.
The more general form of Lagrange’s equation is d dt



L
 q  i




L
 q i

Q i
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 The pendulum shown in the figure will be used as an example throughout this module.
 The position of the pendulum is known at any instance of time if the coordinates of the c.g., X cg
,Y cg
, and the angle q are known.

X cg
,Y cg and q are the generalized coordinates.
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 4
Simple
Pendulum
Y cg
Y c.g.
y x
θ
X cg
X
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Kinetic and Potential Energies
The kinetic energy (T) and potential energy (V) of the pendulum are
T
V

1
I q 
2
2
 mgY cg

1
2 m X

2 cg

1
2 mY cg
2
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.
Y cg
Y
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 5 c.g.
y x
θ
X
X cg
Unconstrained
Body
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Constraint Equations
 In addition to satisfying
Lagrange’s equations of motion, the pendulum must satisfy the constraints that the displacements at X
1 zero.
and Y
1 are
 The constraint equations are
X cg
Y cg



2 sin q

2 cos q

X
1

Y
1

0

0
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 6
Y cg
Y c.g.
y
X
1
,Y
1 x
θ
X cg
The c.g. lies on the y axis halfway along the
X
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 The kinetic energy is augmented by adding the constraint equations multiplied 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 l
1 and l
2
?
Y
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 7
Y
1
X
1
θ
T

 l
1
1
2
I q 
2 
1
2 m
2 cg

1
2 m Y
 cg
2
X cg


2 sin q 
X
1
 l
2
Y cg


2 cos q  y
1
X
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Governing Equations
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 8
 In the following slides, Lagrange’s 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
Lagrange’s Equation d dt



L
 q  i




L
 q i

Q i
Lagrangian
L
 i n 

1
T i

V i
L

1
2
I q 
2 
1
2 m X

2 cg

1
2 m Y

2 cg
 l
1
X cg


2 sin q 
X
1
 l
2
Y cg


2 cos q 
Y
1
 mgY cg
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Equation for 1 st Generalized Coordinate
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 9
L

1
2
I q 
2 
1
2 m X

2 cg

1
2 m Y

2 cg
 l
1
X cg


2 sin q 
X
1
 l
2
Y cg


2 cos q 
Y
1
 mgY cg
Lagrange’s Equation d dt



L
 q  i




L
 q i

Q i
Generalized Coordinates q
1 q
2 q
3 q
4 q
5

X cg

Y cg
 q

 l
1 l
2
Mathematical Steps

L
 q 
1
 m X
 cg d dt

L
 q
1



L
 q 
1
 l
1


 m X
 cg
1 st Equation m X
 cg
 l
1

0
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Equation for 2 nd Generalized Coordinate
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 10
L

1
2
I q 
2 
1
2 m X

2 cg

1
2 m Y

2 cg
 l
1
X cg


2 sin q 
X
1
 l
2
Y cg


2 cos q 
Y
1
 mgY cg
Lagrange’s Equation d dt



L
 q  i




L
 q i

Q i
Generalized Coordinates q
1 q
2 q
3 q
4 q
5

X cg

Y cg
 q

 l
1 l
2
Mathematical Steps

L
 q 
2
 m Y
 cg d dt

L
 q
2



L
 q 
2
 l
2


 m Y
  cg
 mg
2 nd Equation m
  cg
 l
2
 mg

0
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Equation for 3 rd Generalized Coordinate
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 11
L

1
2
I q 
2 
1
2 m X

2 cg

1
2 m Y

2 cg
 l
1
X cg


2 sin q 
X
1
 l
2
Y cg


2 cos q 
Y
1
 mgY cg
Lagrange’s Equation d dt



L
 q  i




L
 q i

Q i
Generalized Coordinates q
1 q
2 q
3 q
4 q
5

X cg

Y cg
 q

 l
1 l
2
Mathematical Steps

L
 q 
3

I q  d dt

L
 q
3



L
 q 
3
  l
1



I q  

2 cos q  l
2

2 sin q
3 rd Equation
I q    l
1

2 cos q  l
2

2 sin q 
0
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Equation for 4 th Generalized Coordinate
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 12
L

1
2
I q 
2 
1
2 m X

2 cg

1
2 m Y

2 cg
 l
1
X cg


2 sin q 
X
1
 l
2
Y cg


2 cos q 
Y
1
 mgY cg
Lagrange’s Equation d dt



L
 q  i




L
 q i

Q i
Generalized Coordinates q
1 q
2 q
3 q
4 q
5

X cg

Y cg
 q

 l
1 l
2
Mathematical Steps

L
 q 
4

0 d dt

L
 q
4



L
 q 
4

X cg




0

2 sin q 
X
1
4 th Equation
X cg


2 sin q 
X
1

0
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Equation for 5 th Generalized Coordinate
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 13
L

1
2
I q 
2 
1
2 m X

2 cg

1
2 m Y

2 cg
 l
1
X cg


2 sin q 
X
1
 l
2
Y cg


2 cos q 
Y
1
 mgY cg
Lagrange’s Equation d dt



L
 q  i




L
 q i

Q i
Generalized Coordinates q
1 q
2 q
3 q
4 q
5

X cg

Y cg
 q

 l
1 l
2
Mathematical Steps

L
 q 
5

0 d dt

L
 q
5



L
 q 
5

Y cg




0

2 cos q 
Y
1
5 th Equation
Y cg


2 cos q 
Y
1

0
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Summary of Equations
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 14
 There are five unknown generalized coordinates including the two
Lagrangian Multipliers. There are also five equations.
 Three of the equations are differential equations.
 Two of the equations are algebraic equations.
 Combined, they are a system of differential-algebraic equations
(DAE).
m X
 cg m
  cg
 l
1

0
 l
2
 mg

0
I q    l
1

2 cos q  l
2

2 sin q 
0
X cg


2 sin q 
X
1

0
Y cg


2 cos q 
Y
1

0
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Free Body Diagram Approach
Summation of Forces in the X-direction m X
 cg
 l
1

0
Summation of Forces in the Y-direction m
  cg
 l
2
 mg

0
Summation of Moments about the c.g.
I q    l
1

2 cos q  l
2

2 sin q 
0
The application of Lagrange’s equation yields the same equations obtained by drawing a free-body diagram.
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 15
2

2 cos q
 q 2
λ
1 mY cg

2 sin q
Iθ cg mX cg mg
Free Body Diagram with
Inertial Forces
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Physical Significance of Lagrangian Multipliers
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 16
Newton’s 2 nd Law in x-direction m X
 cg
 l
1

0
Force required to impose the constraint that X constant.
1 is a
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).
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Matrix Format
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 17
 The computer implementation of Lagrange’s equation is facilitated by writing the equations in matrix format.
 Separating the Lagrangian into kinetic and potential energy terms enables Lagrange’s equation to be written as d dt



T
  i




T
 q i

Q i


V
 q i
 In this format, the conservative and non-conservative forces are lumped together on the right hand side of the equation.
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Matrix Format
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 18
The kinetic energy augmented with Lagrangian Multipliers can be written in matrix format as
T

1
2
 
T
        
Column array containing generalized coordinate velocities.


  
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.
Inertia Matrix






 m
0
0
0
A
0 m
A
0
0
0
0
A
I cg
0
0
0
0





 
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Matrix Format
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 19
Lagrange’s equation for a mechanical system becomes
     
T



  i
 q j






  i
 q j


Is the constraint equation Jacobian matrix introduced in Module 4 in this section.
Column array containing both conservative and non-conservative forces.
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Matrix Format
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 20
 Another equation for acceleration was obtained in Module 4 based on kinematics and the constraint equations.



  i
 q j



   
Matrix Form of Equations





 

M q j i

  i
 q j
0






 l
 q 


 

Q



 This equation can be solved to find the accelerations and constraint forces at an instant in time.
 Combining this equation with
Lagrange’s equation from the previous slide yields:  The accelerations must then be integrated to find the velocities and positions.
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Solution of Differential-Algebraic
Equations (DAE)
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 21
 The solution of even the simplest system of DAE requires computer 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 Inventor’s Dynamic Simulation environment is an example of such software.
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Module Summary
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 22
 This module showed how Lagrangian Multipliers are used in conjunction with Lagrange’s 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 Inventor’s
Dynamic 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.
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