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Modeling and Simulation of Physical Systems for Hobbyists Manuel Aiple

35C3 — 29 December 2018

CC-BY 4.0 version f59c89b

Modeling and Simulation of Physical Systems for Hobbyists

Mathematical description

• f a system

Running a model Hardware With commonly available tools

Motivation

Why use Simulation?

Motivation

Why use Simulation?

• Placeholder for hardware components
• Virtual test bench

Modeling

Picture: Newton portrait with apple tree by Mariana Ruiz Villarreal (LadyofHats) CC0-1.0 (Recomposed)

Simple Detailed Very Detailed Apple Moves Down Apple Accelerates Down Apple Accelerates Down Until Saturation

t x t x t x

As simple as possible, as detailed as necessary

Differentiation & Integration

Differentiation & Integration

Differentiation & Integration

Differentiation & Integration Position x(t) Velocity v(t) Acceleration a(t)

Differentiation & Integration

Differentiation & Integration Position x(t) Velocity v(t) Acceleration a(t) Differentiate

Differentiation & Integration

Differentiation & Integration Position x(t) Velocity v(t) Acceleration a(t) Differentiate Integrate

Differentiation & Integration

Differentiation & Integration Position x(t) Velocity v(t) Acceleration a(t) Differentiate Integrate

v(t) = lim

h→0

x(t + h) − x(t) h

Differentiation & Integration

Differentiation & Integration Position x(t) Velocity v(t) Acceleration a(t) Differentiate Integrate

v(t) = lim

h→0

x(t + h) − x(t) h lim

h→0 x(t + h) = x(t) + lim h→0 v(t) h

Differentiation & Integration

Differentiation & Integration Position x(t) Velocity v(t) Acceleration a(t) Differentiate Integrate

v(t) = lim

h→0

x(t + h) − x(t) h lim

h→0 x(t + h) = x(t) + lim h→0 v(t) h

Looking into the past Looking into the future Always integrate for simulation

Euler Method

Euler Method lim

h→0 x(t + h) = x(t) + lim h→0 v(t) h

→ Not usable for computation

Euler Method

Euler Method lim

h→0 x(t + h) = x(t) + lim h→0 v(t) h

→ Not usable for computation Replace infinitesimal limh→0 h with finite Ts and only calculate for integer multiples k of Ts: t = k Ts

Euler Method

Euler Method lim

h→0 x(t + h) = x(t) + lim h→0 v(t) h

→ Not usable for computation Replace infinitesimal limh→0 h with finite Ts and only calculate for integer multiples k of Ts: t = k Ts x t + Ts Ts

• = x

t Ts

• + v

t Ts

• Ts

Euler Method

Euler Method lim

h→0 x(t + h) = x(t) + lim h→0 v(t) h

→ Not usable for computation Replace infinitesimal limh→0 h with finite Ts and only calculate for integer multiples k of Ts: t = k Ts x t + Ts Ts

• = x

t Ts

• + v

t Ts

• Ts

x(k + 1) = x(k) + v(k) Ts

Euler Method

Euler Method lim

h→0 x(t + h) = x(t) + lim h→0 v(t) h

→ Not usable for computation Replace infinitesimal limh→0 h with finite Ts and only calculate for integer multiples k of Ts: t = k Ts x t + Ts Ts

• = x

t Ts

• + v

t Ts

• Ts

x(k + 1) = x(k) + v(k) Ts Keep Ts small

Building Blocks Mechanics

Building Blocks Mechanics

Building Blocks Mechanics

Building Blocks Mechanics · M v

Building Blocks Mechanics

Building Blocks Mechanics · M v F F = M dv dt Second law of motion

Building Blocks Mechanics

Building Blocks Mechanics · M v F F = M dv dt Second law of motion I ω

Building Blocks Mechanics

Building Blocks Mechanics · M v F F = M dv dt Second law of motion I ω T T = I dω dt

Building Blocks Mechanics

Building Blocks Mechanics · M v F F = M dv dt Second law of motion I ω T T = I dω dt F = M g Weight

Building Blocks Mechanics

Building Blocks Mechanics · M v F F = M dv dt Second law of motion I ω T T = I dω dt F = M g Weight F x κ F = −κ (x − x0) Spring force

Building Blocks Mechanics

Building Blocks Mechanics · M v F F = M dv dt Second law of motion I ω T T = I dω dt F = M g Weight F x κ F = −κ (x − x0) Spring force F = −b v Viscous damping

Building Blocks Electric

Building Blocks Electric

Building Blocks Electric

Building Blocks Electric R V = R i Resistor

Building Blocks Electric

Building Blocks Electric R V = R i Resistor L V = L di dt Inductance

Building Blocks Electric

Building Blocks Electric R V = R i Resistor L V = L di dt Inductance C i = C dV dt Capacitor

Building Blocks Electromechanics

Building Blocks Electromechanics

M T ω

Building Blocks Electromechanics

Building Blocks Electromechanics

M T ω

T = Kt i Motor

Building Blocks Electromechanics

Building Blocks Electromechanics

M T ω

T = Kt i Motor V = Kv ω Generator

Building Blocks Electromechanics

Building Blocks Electromechanics

M T ω

T = Kt i Motor V = Kv ω Generator V = R i + L di dt + Kv ω I dω dt = Kt i − b ω Electric motor

Tips & Tricks

Tips & Tricks

• 1. Sampling Period (Ts): min. 100x faster than system time constant
• 2. Block Diagram: helps to keep overview
• 3. Adapt the model to your needs: different questions might need different models
• 4. Specialized Tools (SciPy, OpenModelica/OMEdit, Scilab/XCos):
• for complex models or as reference
• better differential equation solving (BDF, Runge-Kutta, etc.)
• efficient through variable time-step
• nice data logging and visualization tools

Motor Model Block Diagram 1/I B

• −

+ + Kv

Text θ dω dt ω

1/L R

• +

+ − Kt

Vsup i di dt i

Background & Further Reading

Background & Further Reading (Wikipedia)

• Scientific modeling
• Ordinary differential equation
• Numerical methods for ordinary differential equations

– Euler Method – Runge-Kutta – Backward differentiation formula (BDF)

• Discrete time and continuous time
• State-space representation