Dynamic Simulation : Lagrange’s Equation

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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.


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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

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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



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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



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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



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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.



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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



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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.



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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.



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Actual Path

Candidate Path


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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



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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.



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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.



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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.



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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 ,



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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



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vqLu

vduuvdudvvduudvuvd

i


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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



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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



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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.



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Vibrating Spring Mass Example

Governing Equations

Equation of Motion

m

k

y



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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


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Falling Mass Example

Governing Equations

m

yg

x



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Mathematical Operations

0

ii qL

qL

dtd

mgyV

ymT

VTL

2

21

0

21 2

mgym

mgyL

ymyL

dtd

ymyL

mgyymL

Equation of Motion


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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.



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Dynamic Simulation : Lagrange’s Equation

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