Natural Response to Non-zero Initial Conditions Prof. Seungchul Lee - - PowerPoint PPT Presentation

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Natural Response to Non-zero Initial Conditions Prof. Seungchul Lee - - PowerPoint PPT Presentation

Aug 14, 2023 195 likes •644 views

Natural Response to Non-zero Initial Conditions Prof. Seungchul Lee Industrial AI Lab. The First Order ODE Solution will be exponential functions Unknown coefficient determined by initial conditions Stability unstable if

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Natural Response to Non-zero Initial Conditions

  • Prof. Seungchul Lee

Industrial AI Lab.

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The First Order ODE

  • Solution will be exponential functions

– Unknown coefficient determined by initial conditions

  • Stability

– unstable if 𝑙 > 0 – stable if 𝑙 < 0

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The First Order ODE

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The First Order ODE

  • 𝜐: time constant

– Large 𝜐 : slow response – Small 𝜐 : fast response

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Two First Order ODEs (Independent)

  • Suppose 𝑣1 and 𝑣2 are independent
  • In a matrix form

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ODE in Vector Form (Dependent)

  • Suppose 𝑣1 and 𝑣2 are dependent
  • In a matrix form

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Systems of Differential Equations

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Systems of Differential Equations

  • Given
  • Superposition

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Systems of Differential Equations

  • For a single ODE
  • Let us try
  • Linear ODE = Eigenvalue problem

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Eigenanalysis

  • Eigenanalysis
  • General solution
  • where

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Eigenanalysis

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Eigenanalysis

  • Linear Transformation
  • Solution

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Eigenanalysis

  • 𝑤 - frame is decoupled by Ԧ

𝑦1 and Ԧ 𝑦2

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Real Eigenvalues

  • Example 1

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Real Eigenvalues

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Phase Portrait

  • Geometric representation of the trajectories of a dynamical system in the phase plane

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Real Eigenvalues

  • Example 2

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Real Eigenvalues

  • Example 3

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Different Eigenvectors with the Same Eigenvalues

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De-coupling via Linear Transformation

  • Change variables

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De-coupling via Linear Transformation

  • Change variables

– Total amount of water – Difference in height

  • De-coupled

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Trajectory Comparison

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Systems of Differential Equations: Complex Eigenvalues

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Complex Eigenvalues (Starting Oscillation)

  • 𝜇 can be a complex number 𝜇 = 𝜏 + 𝑘𝜕

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Complex Eigenvalues (Starting Oscillation)

  • Example 1

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Complex Eigenvalues (Starting Oscillation)

  • Example 1

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What is the Corresponding Physical System?

  • Simple harmonic motion Revisited

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Pure Oscillation

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Complex Eigenvalues

  • Example 2

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Pure Oscillation

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Complex Eigenvalues

  • Example 3

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Pure Oscillation

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Complex Eigenvalues with Damping

  • Example 1

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Oscillation with Damping

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Mass-Spring-Damper System

  • Mass-spring-damper system

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Mass-Spring-Damper System

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Mass-Spring-Damper System

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State Space Representation

  • Define states
  • State space

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Eigenanalysis

  • Physical interpretation of 0 < 𝜂 < 1

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Eigenvalues in S-plane

  • Oscillating with damping (under damping)

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Eigenvalues in S-plane

Over damping Critical damping Pure oscillating

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The Second Order ODE

  • State space representation

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Stability

  • Scalar systems
  • Matrix systems

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Summary

  • Natural response with non-zero initial conditions
  • Systems of differential equations
  • Eigen-analysis
  • Complex eigenvalues

– Their locations in s-plane

  • The second order ODE

– Mass, spring, and damper system

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