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Euler-lagrange method

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PrefaceMotivationEL Method

Generalized coordinateLagrangianEuler-Lagrange equation

Illustrative ExamplesConclusion

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PrefaceWhat’re Models for systems and signals?

Basic conceptsTypes of models

How to build a model for a given system?Physical modelingExperimental modeling

How to simulate a system? Matlab/Simulink tools

Case studies Text-book: Modeling of dynamic systems Lennart Ljung and Torkel Glad

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Preface - Systems and models

Part one – Models, p13-78 System is defined as an object or a collection of objectswhose properties we want to study

A model of a system is a tool we use to answerquestions about the system without having to do an experiment

Mental model Verbal model Physical model

Mathematical model

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Preface - physical modeling

Conservation lawsMass balanceEnergy balance Electronics (Kirchhoff’s laws)

Constitutive relationships

m a = ∑ F

J dω/dt = ∑ τ

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DC motor with Permanent Magnet

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Euler-lagrange method

???Energy perspective…

Motivation (1)

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Motivation (2)

The mathematical model can be obtained from variationalprinciples applied to energy functions!

There exists a well established common terminology for all type ofsystems, whether electrical, mechanical, magnetic, etc., by defining energy functions in terms of the generalized coordinates

There are a number of different energy functions (e.g. theLagrangian, the total energy) which can be used as a energyfunction

The variational approach is quite formal analytically and insightinto physical processes may be lost. Nevertheless, if the method is properly understood, physical insight can be gained due to thegenerality of the method.

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PrefaceMotivationEL Method

Generalized coordinateLagrangianEuler-Lagrange equation

Illustrative ExamplesConclusion

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EL Method – Generalized coordinate(1)

For instance, for the specification of a rigid body, we need six coordinates, three for thereference point and three for the orientationA certain minimum number n of coordinates, called the degrees of freedom, is required to specify the configuration. Usually, these coordinates are denoted by qiand are called generalized coordinates. Thecoordinate vector

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EL Method – Generalized coordinate(2)

The choice of the generalized coordinates is usuallysomewhat arbitrary, but in general each individualenergy storage element of the system have a set ofgeneralized coordinates.For a dynamic system the generalized coordinates do not completely specify the system and an additional set of dynamic variables equal in number to thegeneralized coordinates must be used. These dynamic variables can be the first time derivatives of the generalized coordinates, thevelocities, or can be a second set variables (e.g. thegeneralized momenta).

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EL Method – Energy

The kinetic energy T in terms of Cartesiancoordinates is given by

The Potential energy V, e.g., V=mgh, 1/2kx^2

Lagrangian function: L=T-V

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EL Method – Euler Lagrange eq.

Q_j^e are generalized forces acting along the jth generalized axis

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EL Method – Summary

The general procedure : Select a suitable set of coordinates to represent theconfiguration of the system.

Obtain the kinetic energy T as a function of thesecoordinates and their time derivatives.

If the system is conservative, find the potential energyV as a function of the coordinates, or, if the system is not conservative, find the generalized forces Qje.

The differential equations of motion are then given by EL equations

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Example (1)

a set of generalized coordinates x_a and x_band their associated velocities va and vb.

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Example (2)

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Example (3)

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Conclusion

The application of the Lagrangian formulation is not restricted to mechanical systems.

The Lagrangian depends on the generalizedcoordinates q, the associated velocities , and thetime t.