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

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 𝑙 > 0 – stable if 𝑙 < 0 2

The First Order ODE 3

The First Order ODE • 𝜐 : time constant – Large 𝜐 : slow response – Small 𝜐 : fast response 4

Two First Order ODEs (Independent) • Suppose 𝑣 1 and 𝑣 2 are independent • In a matrix form 5

ODE in Vector Form (Dependent) • Suppose 𝑣 1 and 𝑣 2 are dependent • In a matrix form 6

Systems of Differential Equations 7

Systems of Differential Equations • Given • Superposition 8

Systems of Differential Equations • For a single ODE • Let us try • Linear ODE = Eigenvalue problem 9

Eigenanalysis • Eigenanalysis • General solution • where 10

Eigenanalysis 11

Eigenanalysis • Linear Transformation • Solution 12

Eigenanalysis • 𝑤 - frame is decoupled by Ԧ 𝑦 1 and Ԧ 𝑦 2 13

Real Eigenvalues • Example 1 14

Real Eigenvalues 15

Phase Portrait • Geometric representation of the trajectories of a dynamical system in the phase plane 16

Real Eigenvalues • Example 2 17

Real Eigenvalues • Example 3 18

Different Eigenvectors with the Same Eigenvalues 19

De-coupling via Linear Transformation • Change variables 20

De-coupling via Linear Transformation • Change variables – Total amount of water – Difference in height • De-coupled 21

Trajectory Comparison 22

Systems of Differential Equations: Complex Eigenvalues 23

Complex Eigenvalues (Starting Oscillation) • 𝜇 can be a complex number 𝜇 = 𝜏 + 𝑘𝜕 24

Complex Eigenvalues (Starting Oscillation) • Example 1 25

Complex Eigenvalues (Starting Oscillation) • Example 1 26

What is the Corresponding Physical System? • Simple harmonic motion Revisited 27

Pure Oscillation 28

Complex Eigenvalues • Example 2 29

Pure Oscillation 30

Complex Eigenvalues • Example 3 31

Pure Oscillation 32

Complex Eigenvalues with Damping • Example 1 33

Oscillation with Damping 34

Mass-Spring-Damper System • Mass-spring-damper system 35

Mass-Spring-Damper System 36

Mass-Spring-Damper System 37

State Space Representation • Define states • State space 38

Eigenanalysis • Physical interpretation of 0 < 𝜂 < 1 39

Eigenvalues in S-plane • Oscillating with damping (under damping) 40

Eigenvalues in S-plane Pure oscillating Critical damping Over damping 41

The Second Order ODE • State space representation 42

Stability • Scalar systems • Matrix systems 43

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 44