ENGINEERING COMPUTING ENG1602

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ENGINEERING COMPUTING ENG1602
BACHELOR of ENGINEERING
Week 3 Lecture 1 Microsoft EXCEL CASE
STUDY REGRESSION FOURIER SERIES
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CASE STUDY LEAST SQUARES

• Often the Engineer has to fit a (Mathematical)
  model to some experimental data - the data will
  not generally fit perfectly as there may be
  errors ( in measurement, or un-modelled small
  effects, etc.)
• Take the case of a simple linear model - one has
  to find a line that best fits What does that
  mean?

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CURVE FITTING

• If f(x) is an unknown function defined by pairs
  of x values and corresponding y values, so that y
  f(x) over the given range.
• It is desired to select another known function
  g(x, C1, C2,...... Cn)
  and then determine the constants
  C1, C2, ...... Cn such that g(x) fits
  the given values of f(x) with minimum Error.

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Best Fit - Minimum Error - 1.

• Simple Strategy - If there are m pairs of data
  and n unknown constants in the chosen function
  g(x), it being assumed that n lt m
• Then Select n pairs of values from the m
  available.
• Substitute Solve for the values of the n
  constants from the n resulting simultaneous
  equations.

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Best Fit - Minimum Error - 2.

• Then the function g(x) will fit the data exactly
  at the n chosen points.
• but there may be serious errors at the (m-n)
  unused points.
• Note that the chosen function g(x) must be such
  that n lt m, otherwise there are not enough
  equations to solve for the unknown constants.
• However if m gt n then which n points should be
  selected ?

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Example Fit y g(x) A Bx ( n 2 ) to 3
Data Points ( m 3 )
Point 2
y
Error
Point 3 Exact
B
Simple Solution using Data Points 1 3 to
solve for A B
1
Point 1 Exact
A
x
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Analysis of the Solution

• Whilst the function fits exactly at two points,
  there are serious errors elsewhere
• Also totally different results would result
  depending which two of the three available points
  were chosen to determine the constants
• It is possible to do better !

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Intuitive Line of Best Fit
Point 2 Error
y
Point 3 Error
Intuitive Solution using Data Points 1, 2 3
Point 1 Error
x
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Using All the Specified Points in the Analysis -
1.

• If the Error at a given point i is defines
  as
• Then Best Fit could be defined by the values of
  the constants that minimise
  

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Using All the Specified Points in the Analysis -
2.

• This approach is not so easy to analyse
• A better approach is to minimise the Sum of
  (errors)2
• the so called LEAST SQUARES Criteria
• Where S is called the sum of the residuals

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For our Straight Line Application
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Solving for Constants a b
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Least Squares Fit - Conclusions

• These relationships involve all the data points
  specified
• One could write a Java Program to read the data
  and work our the constants or
• EXCEL has a wide range of REGRESSION facilities (
  which also return some diagnostics on how well
  the data fits the model - since you always get an
  answer even if the data doesnt fit the model by
  any reasonable intuitive meaning - you should
  always check how good the answer is).

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Using EXCEL - TRENDLINEs -1.

• First you must create a chart or graph of the
  given data
• Use Excels Graph Wizard icon after
  highlighting the cells containing the given x and
  y data
• Then pull open an area on your work sheet for the
  graph or select INSERT CHART for a separate sheet

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Using EXCEL - TRENDLINEs -2.

• For technical applications, always use the
  XYScatter Graph option - follow the other screen
  prompts to create a graph with straight line
  segments between each point
• To insert a TRENDLINE select the graph by Left
  clicking on it with the mouse
• Select INSERT TRENDLINE from the pull down Menu
  and follow the instructions
• When finished you can check how well the
  generated Trendline fits the given data

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CASE STUDY Fourier Series

• A famous French Engineer, Fourier, proposed that
  reasonable functions could be expressed as the
  sum of sinusoids of different frequencies
• f(x) -gt Sum of sinusoids - called Fourier
  Analysis
• Sum of sinusoids -gt f(x) - called Fourier
  Synthesis

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Typical Sinusoids
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Fourier Series - 1.

• If a function is periodic ( and meets a few other
  criteria for reasonable), then it may be
  represented by a Fourier Series and there are
  formulae for calculating the coefficients
  (amplitudes and phases of the sinusoids)
• Important applications of this theory involve
  certain Electrical and Mechanical systems that
  are linear and Time/Shift Invariant).

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Fourier Series Example
Show how a square wave can be created from its
Fourier Sine Series Components
A square wave comprises only odd harmonics
(1,3,5,7) (see Lab 2)
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Fourier Series - 2.

• Without developing the theory further - we can
  say that if we excite the system with a sinusoid
  of a certain frequency - then the response is
  also sinusoidal of the same frequency but with
  different amplitude and phase.
• Moreover if the excitation consists of a sum of
  sinusoids, then the system responds to each
  sinusoid independently.
• To determine what a system does to input f(x) we
  simply decompose f(x) into a sum of sinusoids

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Fourier Series - 3.

• Then work out the new amplitude and phase of each
  component from the equations describing the
  systems performance
• Finally Add together the various sinusoids (
  using the new amplitudes and phases) to get the
  output

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Damped Vibration Analysis
Example Shock Absorber in a car Assuming that
platform vibration is driven by the a square
wave, what how will the mass vibrate?
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Mass Vibration Waveform
Plot of the mass vibration waveform using
1,3,5,7,9 harmonics of the platform driving
force waveform (See Lab 2)