© 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:Lagrange’s Equation
Objective
The objective of this module is to derive Lagrange’s equation, which along with constraint equations provide a systematic method for solving multi-body dynamics problems.
© 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
Calculus of Variations
Problems in dynamics can be formulated in such a way that it is necessary to find the stationary value of a definite integral.
Lagrange (1736-1813) created the Calculus of Variations as a method for finding the stationary value of a definite integral. He was a self taught mathematician who did this when he was nineteen.
Euler (1707-1783) used a less rigorous but completely independent method to do the same thing at about the same time.
They were both trying to solve a problem with constraints in the field of dynamics.
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
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
Euler and Lagrange
1707-1783
Leonhard Euler
1736-1813
Joseph-Louis Lagrange
http://en.wikipedia.org/wiki/Leonhard_Euler http://en.wikipedia.org/wiki/Lagrange
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
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
Hamilton’s Principle
Hamilton’s Principle states that the path followed by a mechanical system during some time interval is the path that makes the integral of the difference between the kinetic and the potential energy stationary.
L=T-V is the Lagrangian of the system.
T and V are respectively the kinetic and potential energy of the system.
The integral, A, is called the action of the system.
2
1
t
t
LdtA
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
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
Principle of Least Action
Hamilton’s Principle is also called the “Principle of Least Action” since the paths taken by components in a mechanical system are those that make the Action stationary.
2
1
t
t
LdtA Action
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
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
Stationary Value of an Integral
The application of Hamilton’s Principle requires that we be able to find the stationary value of a definite integral.
We will see that finding the stationary value of an integral requires finding the solution to a differential equation known as the Lagrange equation.
We will begin our derivation by looking at the stationary value of a function, and then extend these concepts to finding the stationary value of an integral.
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 6
© 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
Stationary Value of a Function
A function is said to have a “stationary value” at a certain point if the rate of change of the function in every possible direction from that point vanishes.
In this example, the function has a stationary point at x=x1. At this point, its first derivative is equal to zero.
x
y
y=f(x)
x1
y1
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
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
3D Stationary Points
In 3D the rate of change of the function in any direction is zero at a stationary point. Note that the stationary point is not necessarily a maximum or a minimum.
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 8
© 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
Variation of a Function
x
y
y=f(x)dydx
dy
a bx x+dx
xgy
xxfxg
(x) is an arbitrary function that satisfies the boundary conditions at a and b.
g(x) can be made infinitely close to f(x) by making the parameter infinitesimally small.
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 9
Actual Path
Candidate Path
© 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
Meaning of dy
The Calculus of Variations considers a virtual infinitesimal change of function y = f(x).
The variation dy refers to an arbitrary infinitesimal change of the value of y at the point x.
The independent variable x does not participate in the process of variation. x
y
y=f(x)dydx
dy
a bx x+dx
xgy
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 10
© 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
Variation of a Derivative
dx
xdxdxd
xfxgdxd
dxyd
d
dx
xddx
xddx
xdfdx
xdgdxdy
d
In the calculus of variations, the derivative of the variation and the variation of the derivative are equal.
Derivative of the Variation Variation of the Derivative
dxdy
dxyd dd
The order of operation is interchangeable.
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
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
Variation of a Definite Integral
b
a
b
a
b
a
b
a
b
a
dxxdxxfxg
dxxfdxxgdxxf
d
Variation of an Integral Integral of a Variation
b
a
b
a
b
a
dxx
dxxfxgdxxf
d
dxxfdxxfb
a
b
a dd
In the calculus of variations, the variation of a definite integral is equal to the integral of the variation.
The order of operation is interchangeable.
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 12
© 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
Specific Definite Integral
The specific definite integral that we want to find the stationary value of is the Action from Hamilton’s Principle. It can be written in functional form as
The actual path that the system will follow will be the one that makes the definite integral stationary.
2
1
,,t
tii tqqLA tqVtqTL i
n
ii
1
qi are the generalized coordinates used to define the position and orientation of each component in the system.
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 13
© 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
Euler-Lagrange Equation Derivation
A first order Taylor’s Series was used in the last step.
ii
ii
iiiiiiii qL
qLtqqLtqqLtqqL d
,,,,,,
02
1
2
1
2
1
dtqL
qLLdtLdtA
t
ti
ii
i
t
t
t
t
ddd
The stationary value of an integral is found by setting its variation equal to zero.
02
1
dtqL
qLt
ti
ii
i
For an arbitrary value of ,
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 14
© 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
Euler-Lagrange Equation Derivation
Integration by Parts Substitutions0
2
1
dtqL
qLt
ti
ii
i
The second integral is integrated by parts.
dtqL
dtd
qLdt
qL
ii
t
t
t
ti
ii
t
t i
2
1
2
1
2
1
is equal to zero at t1 and t2.
02
1
dtqL
dtd
qL
i
t
t ii
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 15
vqLu
vduuvdudvvduudvuvd
i
© 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
Euler-Lagrange Equation Derivation
Lagrange’s equation
The only way that this definite integral can be zero for arbitrary values of i is for the partial differential equation in parentheses to be zero at all values of x in the interval t1 to t2.
02
1
dtqL
dtd
qL
i
t
t ii
0
ii qL
dtd
qL
or
0
ii qL
qL
dtd
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 16
© 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
Euler-Lagrange Summary
2
1
,,t
tii tqqLA
tqVtqTL i
n
ii
1
0
ii qL
qL
dtd
Finding the stationary value of the Action, A, for a mechanical system involves solving the set of differential equations known as Lagrange’s equation.
Solving these equations
Makes this integral stationary
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
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
Examples
Although the derivation of Lagrange’s equation that provides a solution to Hamilton’s Principle of Least Action, seems abstract, its application is straight forward.
Using Lagrange’s equation to derive the equations of motion for a couple of problems that you are familiar with will help to introduce their application.
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
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
Vibrating Spring Mass Example
Governing Equations
Equation of Motion
m
k
y
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 19
0
ii qL
qL
dtd
2
2
2121
kyV
ymT
VTL
y is measured from the static position.
0
21
21 22
kyym
kyyL
ymyL
dtd
ymyL
kyymL
Mathematical Operations
© 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
Falling Mass Example
Governing Equations
m
yg
x
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 20
Mathematical Operations
0
ii qL
qL
dtd
mgyV
ymT
VTL
2
21
0
21 2
mgym
mgyL
ymyL
dtd
ymyL
mgyymL
Equation of Motion
© 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
Lagrange’s equation has been derived from Hamilton’s Principle of Least Action.
Finding the stationary value of a definite integral requires the solution of a differential equation.
The differential equation is called “Lagrange’s equation” or the “Euler-Lagrange equation” or “Lagrange’s equation of motion.”
Lagrange’s equation will be used in the next module (Module 7) to establish a systematic method for finding the equations that control the motion of mechanical systems.
Section 4 – Dynamic Simulation
Module 6 – Lagrange’s Equation
Page 21
Top Related