System Modeling: Complex Number and Harmonic Motion Prof. Seungchul - PowerPoint PPT Presentation

System Modeling: Complex Number and Harmonic Motion Prof. Seungchul Lee Industrial AI Lab.

Complex Number 2

Complex Number • Add 3

Euler's Formula • Complex number in complex exponential 4

Complex Number • Multiply 5

Geometrical Meaning of 𝒇 𝒋𝜾 • 𝑓 𝑗𝜄 : point on the unit circle with angle of 𝜄 • 𝜄 = 𝜕𝑢 • 𝑓 𝑗𝜕𝑢 : rotating on an unit circle with angular velocity of 𝜕 • Question: what is the physical meaning of 𝑓 −𝑗𝜕𝑢 ? 6

Sinusoidal Functions from Circular Motions • Real part (cos term) is the projection onto the Re{} axis. • Imaginary part (sin term) is the projection onto the Im{} axis. 7

Sinusoidal Functions from Circular Motions 8

Sinusoidal Functions from Circular Motions 9

𝒋 Multiplying • 𝑗 multiplication ⟹ 90 𝑝 rotation forward 10

n-th Power of the Complex Exponential • Example – Find the solutions of 𝑨 12 = 1 11

Circular Motion 12

Circular Motion • Particle rotates on the circle with angular velocity of 𝜕 • Velocity in Circular Motion 13

Circular Motion • Acceleration in Circular Motion 14

Harmonic Motion 15

Harmonic Motion • Spring and Mass System • Equations of motion 16

Harmonic Motion • Differential Equation – 2 nd order ODE (ordinary differential Equation) – No additional external force (suppose our system contains 𝑛, 𝑙 ) – spring force ( −𝑙𝑦 ) is internal force – No input (= external) force – Two initial conditions determine the future motion • Solutions – Assume (or educated guess from Physics 1) that the solution is – Unknowns 𝑆 and ∅ are determined by 𝑦 0 , 𝑤 0 17

Seen as a Projection of a Circular Motion • Sinusoidal can be seen as a projection of a circular motion 18

Seen as a Projection of a Circular Motion • We know that two initial conditions ( 𝑦 0 , 𝑤 0 at 𝑢 = 0 ) will determine every motions. 19

Determine Unknown Coefficients • How to obtain 𝐵, ∅ from 𝑦 0 , 𝑤 0 • Determine Unknown Coefficients from circle 20

Pendulum • Equations of motion • From Nonlinear to Linear – Nonlinear system approximation possible? – Period is independent of mass (non-intuitive) 21

Period is Independent of Mass (non-intuitive) From Physics (MIT 8.01) by Prof. Walter Lewin 22

Simulation of Free Vibration • 𝜕 : angular velocity, [rad/sec] • 𝑔 : frequency, [rev/sec = Hz] • One revolution per sec 23

Simulation of Free Vibration 24

Damped Free Vibration 25

Experiment First PHY245: Damped Mass On A Spring, https://www.youtube.com/watch?v=ZqedDWEAUN4 26

Damped Oscillating • In a mathematical form (again from the educated guess) • Exponentially decaying while oscillating 27

Damped Oscillating • Assume damping causes exponential decay while oscillating • Show 𝑨 𝑢 = 𝑓 −𝛿𝑢 𝑓 𝑘𝜕𝑢 also satisfies 28

Damped Oscillating • Given the differential equation • Solution is a linear combination of • 𝐵, 𝐶 are determined by initial conditions 29

Mass, Spring, and Damper System • Parameters • Solution 30

Simulation of Damped Vibration 31

Example: Door Closer 32

Example: Torsional Pendulum From Physics (MIT 8.01) by Prof. Walter Lewin 33

Example 34