Simulation of Physical Systems

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Simulation of Physical Systems
Applying physical principles to engineering
problems results in a model described by
differential equations Ordinary differential
equations (ODEs) with initial conditions are the
most common type of model Closed form solutions
derived from mathematical techniques cannot
normally be found for nonlinear
systems Simulation involves the solution of
models (including nonlinear) using computer
algorithms
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Eulers Method
A simple algorithm for solving an ODE can be
developed using a Taylor series approximation of
the solution Eulers method uses the first
term of the Taylor series to approximate the
ODE The step size approximation error is
proportional to
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Illustration of Eulers Method

actual solution
x(t)
approximation
t
0
Eulers method is based on extrapolation of the
tangent slope
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A Simple Example
or alternatively
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Systems of ODEs
A system of 1st order ODEs can be expressed in
the form
The system can also be expressed using vector
notation
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Eulers Method for Systems of ODEs
Taylor series approximation
Eulers method
Using vector notation
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High Order ODEs
High order ODEs can be converted to a system of
ODEs, e.g.
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Example A Mass-Spring-Damper System

x
k
m
b