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

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

Aug 13, 2023 18 likes β€’359 views

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

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System Modeling: Complex Number and Harmonic Motion

  • Prof. Seungchul Lee

Industrial AI Lab.

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Complex Number

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Complex Number

  • Add

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Euler's Formula

  • Complex number in complex exponential

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Complex Number

  • Multiply

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Geometrical Meaning of π’‡π’‹πœΎ

  • π‘“π‘—πœ„: point on the unit circle with angle of πœ„
  • πœ„ = πœ•π‘’
  • π‘“π‘—πœ•π‘’: rotating on an unit circle with angular velocity of πœ•
  • Question: what is the physical meaning of π‘“βˆ’π‘—πœ•π‘’ ?

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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.

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Sinusoidal Functions from Circular Motions

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Sinusoidal Functions from Circular Motions

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𝒋 Multiplying

  • 𝑗 multiplication ⟹ 90𝑝 rotation forward

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n-th Power of the Complex Exponential

  • Example

– Find the solutions of 𝑨12 = 1

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Circular Motion

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Circular Motion

  • Particle rotates on the circle with angular velocity of πœ•
  • Velocity in Circular Motion

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Circular Motion

  • Acceleration in Circular Motion

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Harmonic Motion

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Harmonic Motion

  • Spring and Mass System
  • Equations of motion

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Harmonic Motion

  • Differential Equation

– 2nd order ODE (ordinary differential Equation) – No additional external force (suppose our system contains 𝑛, 𝑙) – spring force (βˆ’π‘™π‘¦) 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 βˆ… are determined by 𝑦0, 𝑀0

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Seen as a Projection of a Circular Motion

  • Sinusoidal can be seen as a projection of a circular motion

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Seen as a Projection of a Circular Motion

  • We know that two initial conditions (𝑦0, 𝑀0 at 𝑒 = 0) will determine every motions.

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Determine Unknown Coefficients

  • How to obtain 𝐡, βˆ… from 𝑦0, 𝑀0
  • Determine Unknown Coefficients from circle

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Pendulum

  • Equations of motion
  • From Nonlinear to Linear

– Nonlinear system approximation possible? – Period is independent of mass (non-intuitive)

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Period is Independent of Mass (non-intuitive)

22 From Physics (MIT 8.01) by Prof. Walter Lewin

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Simulation of Free Vibration

  • πœ•: angular velocity, [rad/sec]
  • 𝑔: frequency, [rev/sec = Hz]
  • One revolution per sec

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Simulation of Free Vibration

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Damped Free Vibration

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Experiment First

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

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Damped Oscillating

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

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Damped Oscillating

  • Assume damping causes exponential decay while oscillating
  • Show 𝑨 𝑒 = π‘“βˆ’π›Ώπ‘’π‘“π‘˜πœ•π‘’ also satisfies

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Damped Oscillating

  • Given the differential equation
  • Solution is a linear combination of
  • 𝐡, 𝐢 are determined by initial conditions

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Mass, Spring, and Damper System

  • Parameters
  • Solution

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Simulation of Damped Vibration

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Example: Door Closer

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Example: Torsional Pendulum

33 From Physics (MIT 8.01) by Prof. Walter Lewin

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Example

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