Upload
nerys
View
39
Download
0
Embed Size (px)
DESCRIPTION
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. - PowerPoint PPT Presentation
Citation preview
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
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.
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Basic Problem in Multi-body Dynamics
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.
Lagrange’s 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 find and solve the equations of motion when there are constraints that bodies in the system must satisfy.
0
ii qL
qL
dtd
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 2
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Non-conservative Forces
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 non-
conservative processes that dissipate energy (i.e. friction, damping, and external forces).
A non-conservative force or moment acting on generalized coordinate qi is denoted as Qi.
The more general form of Lagrange’s equation is
iii
QqL
qL
dtd
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 3
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Simple Pendulum
SimplePendulum
c.g.
θ
X
Y
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., Xcg,Ycg, and the angle q are known.
Xcg,Ycg and q are the generalized coordinates.
xy
Xcg
Ycg
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 4
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Kinetic and Potential Energies
The kinetic energy (T) and potential energy (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
222
21
21
21 q
c.g.
θ
X
Y
xy
Xcg
Ycg
Unconstrained Body
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 5
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Constraint Equations
In addition to satisfying Lagrange’s equations of motion, the pendulum must satisfy the constraints that the displacements at X1 and Y1 are zero.
The constraint equations are
c.g.
θ
X
Y
xy
Xcg
Ycg
X1,Y1
0cos2
0sin2
1
1
YY
XX
cg
cg
q
q
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 6
The c.g. lies on the y-axis halfway along the length .
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Lagrangian Multipliers
X1
Y1
θX
Y
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 l1 and l2?
12
11
222
cos2
sin2
21
21
21
yY
XX
YmXmIT
cg
cg
cgcg
ql
ql
q
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 7
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Governing Equations
Lagrange’s Equation
Lagrangian
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 8
iii
QqL
qL
dtd
i
n
ii VTL
1
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
121
21 qlqlq
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 acting on the system ( ).0iQ
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Equation for 1st Generalized CoordinateSection 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 9
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
121
21 qlqlq
iii
QqL
qL
dtd
Lagrange’s Equation
Generalized Coordinates
25
14
3
2
1
llq
qqq
Yq
Xq
cg
cg1
1
1
1
l
qL
XmqL
dtd
XmqL
cg
cg
1st Equation
01 lcgXm
Mathematical Steps
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Equation for 2nd Generalized CoordinateSection 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 10
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
121
21 qlqlq
iii
QqL
qL
dtd
Lagrange’s Equation
Generalized Coordinates
25
14
3
2
1
llq
qqq
Yq
Xq
cg
cg mgqL
YmqL
dtd
YmqL
cg
cg
22
2
2
l
2nd Equation
02 mgYm cg l
Mathematical Steps
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Equation for 3rd Generalized Coordinate
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
121
21 qlqlq
iii
QqL
qL
dtd
Lagrange’s Equation
Generalized Coordinates
25
14
3
2
1
llq
qqq
Yq
Xq
cg
cg qlql
q
q
sin2
cos2 21
3
3
3
qL
IqL
dtd
IqL
3rd Equation
Mathematical Steps
0sin2
cos2 21 qlqlq I
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 11
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Equation for 4th Generalized CoordinateSection 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 12
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
121
21 qlqlq
iii
QqL
qL
dtd
Lagrange’s Equation
Generalized Coordinates
25
14
3
2
1
llq
qqq
Yq
Xq
cg
cg1
4
4
4
sin2
0
0
XXqLqL
dtd
qL
cg
q
4th Equation
Mathematical Steps
0sin2 1 XX cg q
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Equation for 5th Generalized CoordinateSection 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 13
cgcgcgcgcg mgYYYXXYmXmIL
1211
222 cos2
sin22
121
21 qlqlq
iii
QqL
qL
dtd
Lagrange’s Equation
Generalized Coordinates
25
14
3
2
1
llq
qqq
Yq
Xq
cg
cg 15
5
5
cos2
0
0
YYqL
qL
dtd
qL
cg
q
5th Equation
Mathematical Steps
0cos2 1 YYcg q
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Summary of EquationsSection 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 14
01 lcgXm
02 mgYm cg l
0sin2
cos2 21 qlqlq I
0sin2 1 XX cg q
0cos2 1 YYcg q
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).
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Free Body Diagram Approach
1λ
2λ
cgmX
cgmY
cgIθ
The application of Lagrange’s equation yields the same equations obtained by drawing a free-body diagram.
Free Body Diagram with Inertial Forces
mg
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 15Summation of Forces in the X-direction
Summation of Forces in the Y-direction
Summation of Moments about the c.g.
01 lcgXm
02 mgYm cg l
0sin2
cos2 21 qlqlq I
q 2
qcos2
qsin2
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Physical Significance of Lagrangian Multipliers
Force required to impose the constraint that X1 is a constant.
Newton’s 2nd Law 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 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 16
01 lcgXm
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Matrix Format
ii
ii qVQ
qT
qT
dtd
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
In this format, the conservative and non-conservative forces are lumped together on the right hand side of the equation.
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 17
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Matrix Format
The kinetic energy augmented with Lagrangian Multipliers can be written in matrix format as
tqqMqT TT ,21
l
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.
q
tq,
l
M
Inertia Matrix
000000000000
Acg
A
A
Im
m
M
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 18
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Matrix Format
qMj
iT
l
Lagrange’s equation for a mechanical system becomes
j
i
q Is the constraint equation Jacobian matrix introduced in Module 4 in this section.
Q Column array containing both conservative and non-conservative forces.
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 19
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Matrix Format
Another equation for acceleration was obtained in Module 4 based on kinematics and the constraint equations.
Combining this equation with Lagrange’s equation from the previous slide yields:
qq ji
lQq
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 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 20
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Solution of Differential-Algebraic Equations (DAE)
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.
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 21
© 2011 Autodesk Freely licensed for use by educational institutions. Reuse and changes require a note indicating that content has been modified from the original, and must attribute source content to Autodesk. www.autodesk.com/edcommunity
Education Community
Module Summary
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.
Section 4 – Dynamic Simulation
Module 7 – Lagrangian Multipliers
Page 22