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Natural Response to �Non-zero Initial Conditions

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The First Order ODE

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The First Order ODE

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The First Order ODE

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Two First Order ODEs (Independent)

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ODE in Vector Form (Dependent)

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Systems of Differential Equations

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Systems of Differential Equations

  • Given

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  • Superposition

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Systems of Differential Equations

  • For a single ODE

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  • Let us try

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  • Linear ODE = Eigenvalue problem

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Eigenanalysis

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  • Eigenanalysis

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  • General solution

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  • where

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Eigenanalysis

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Eigenanalysis

  • Linear Transformation

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  • Solution

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Eigenanalysis

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Real Eigenvalues

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  • 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

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  • Change variables

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De-coupling via Linear Transformation

  • Change variables
    • Total amount of water

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    • Difference in height

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  • 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)

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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

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  • Define states

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  • State space

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Eigenanalysis

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Eigenvalues in S-plane

  • Oscillating with damping (under damping)

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Eigenvalues in S-plane

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Over damping

Critical damping

Pure oscillating

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The Second Order ODE

  • State space representation

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Stability

  • Scalar systems

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  • Matrix systems

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Summary

  • Natural response with non-zero initial conditions

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  • Systems of differential equations

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  • Eigen-analysis

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  • Complex eigenvalues
    • Their locations in s-plane

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  • The second order ODE
    • Mass, spring, and damper system

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