<!DOCTYPE OMNotebook>
<Notebook>
 <GroupCell closed="false" >
  <GroupCell closed="false" >
   <TextCell style="Title" >
    <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Arial; font-size:38pt; font-weight:600; color:#000000;">Frequency Analysis of Simulation Data&lt;/p>&lt;/body>&lt;/html></Text>
   </TextCell>
   <GroupCell closed="false" >
    <TextCell style="Section" >
     <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:18pt; font-weight:600; color:#000000;">Fourier Analysis&lt;/p>&lt;/body>&lt;/html></Text>
    </TextCell>
    <GroupCell closed="true" >
     <TextCell style="Subsection" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:14pt; font-weight:600; color:#000000;">ModelDiagramCode&lt;/p>&lt;/body>&lt;/html></Text>
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      <Input>Model[WeakAxis,
  Modelica.Mechanics.Rotational.Torque Torque1;
  Modelica.Mechanics.Rotational.Inertia Inertia1;
  Modelica.Mechanics.Rotational.Spring Spring1[{c == 0.7`};
  Modelica.Mechanics.Rotational.Inertia Inertia2;
  Modelica.Mechanics.Rotational.Spring Spring2[{c == 1}];
  Modelica.Mechanics.Rotational.Inertia Inertia3;
  Modelica.Blocks.Sources.Pulse Pulse1[{width == {1}, period == {100}}];
  Equation[
    Connect[Pulse1.outPort, Torque1.inPort];
    Connect[Torque1.flange_b, Inertia1.flange_a];
    Connect[Inertia1.flange_b, Spring1.flange_a];
    Connect[Spring1.flange_b, Inertia2.flange_a];
    Connect[Inertia2.flange_b, Spring2.flange_a];
    Connect[Spring2.flange_b, Inertia3.flange_a];
    ]
  ]</Input>
      <Output></Output>
     </InputCell>
    </GroupCell>
    <GroupCell closed="false" >
     <TextCell style="Subsection" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:14pt; font-weight:600; color:#000000;">The Model&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">Consider a weak axis excited by a torque pulse train. The axis is modeled by three segments joined by two torsion springs. The following diagram is imported from the &lt;span style=" font-style:italic;">MathModelica&lt;/span> &lt;span style=" font-style:italic;">Model Editor&lt;/span> where the model was defined.&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;img src="file:///C:/OpenModelica132/bin/OMNoteboook_tempfiles/3.png" width="516" height="69" />&lt;/p>&lt;/body>&lt;/html></Text>
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     </TextCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p align='center' style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">Diagram: WeakAxis&lt;/p>&lt;p align='center' style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;/p>&lt;/body>&lt;/html></Text>
      <Rule name="GeneratedCell" >True</Rule>
      <Rule name="TextAlignment" >Center</Rule>
     </TextCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">The corresponding Modelica code ...&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <InputCell style="Input" closed="true" >
      <Input>model WeakAxis
   Modelica.Mechanics.Rotational.Torque Torque1;
   Modelica.Mechanics.Rotational.Inertia Inertia1;
   Modelica.Mechanics.Rotational.Spring Spring1(c=0.7);
   Modelica.Mechanics.Rotational.Inertia Inertia2;
   Modelica.Mechanics.Rotational.Spring Spring2(c=1);
   Modelica.Mechanics.Rotational.Inertia Inertia3;
   Modelica.Blocks.Sources.Pulse Pulse1(width={1},period={200});
equation
   connect(Pulse1.outPort, Torque1.inPort);
   connect(Torque1.flange_b, Inertia1.flange_a);
   connect(Inertia1.flange_b, Spring1.flange_a);
   connect(Spring1.flange_b, Inertia2.flange_a);
   connect(Inertia2.flange_b, Spring2.flange_a);
   connect(Spring2.flange_b, Inertia3.flange_a);
end WeakAxis;
</Input>
      <Output></Output>
     </InputCell>
    </GroupCell>
    <GroupCell closed="false" >
     <TextCell style="Subsection" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times; font-size:14pt; font-weight:600; color:#000000;">Simulation of WeakAxis&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">We simulate the model in 100 seconds ...&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <InputCell style="Input" closed="true" >
      <Input>simulate( WeakAxis, stopTime=200 );</Input>
      <Output></Output>
     </InputCell>
     <TextCell style="Output" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Courier New; font-size:11pt; color:#000000;">&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;br />... and plot the angular velocity of the rightmost axis-segment.&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <InputCell style="Input" closed="true" >
      <Input>plot( { Inertia3.w, Torque1.tau } );</Input>
      <Output>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style="-qt-paragraph-type:empty; margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Courier New; font-size:11pt; color:#000000;">&lt;/p>&lt;/body>&lt;/html></Output>
     </InputCell>
     <TextCell style="Output" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Courier New; font-size:11pt; color:#000000;">&lt;img src="file:///C:/OpenModelica132/bin/OMNoteboook_tempfiles/4.png" width="372" height="276" />&lt;/p>&lt;/body>&lt;/html></Text>
      <Image name="file:///C:/OpenModelica132/bin/OMNoteboook_tempfiles/4.png" >[encoded data, 4336 characters]</Image>
     </TextCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;br />Now, lets sample Inertia3.w using a sample frequency 4Hz ...&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <InputCell style="Input" closed="true" >
      <Input>data1=Table[Inertia3.w[t],{t,0,200,.25}];</Input>
      <Output></Output>
     </InputCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;br />... remove the mean ...&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <InputCell style="Input" closed="true" >
      <Input>mean=Apply[Plus, data1]/Length[data1];</Input>
      <Output></Output>
     </InputCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;"> &lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <InputCell style="Input" closed="true" >
      <Input>data2=data1-mean;</Input>
      <Output></Output>
     </InputCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;br />Compute the absolute values of the discrete Fourier transform ...&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <InputCell style="Input" closed="true" >
      <Input>fdata1=Abs[Fourier[data2]];</Input>
      <Output></Output>
     </InputCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;br />... and plot the 80 first points of data.&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
     </TextCell>
     <InputCell style="Input" closed="true" >
      <Input>ListPlot[fdata1[[Range[80]]],PlotRange->All,PlotStyle->{Red,PointSize[0.015]}];</Input>
      <Output></Output>
     </InputCell>
     <TextCell style="Text" >
      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">&lt;img src="file:///C:/OpenModelica132/bin/OMNoteboook_tempfiles/5.png" width="398" height="245" />&lt;/p>&lt;/body>&lt;/html></Text>
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      <Text>&lt;html>&lt;head>&lt;meta name="qrichtext" content="1" />&lt;/head>&lt;body style=" white-space: pre-wrap; font-family:MS Shell Dlg; font-size:8.25pt; font-weight:400; font-style:normal; text-decoration:none;">&lt;p style=" margin-top:0px; margin-bottom:0px; margin-left:0px; margin-right:0px; -qt-block-indent:0; text-indent:0px; font-family:Times New Roman; font-size:12pt; color:#000000;">It is easy to write a function that repeat the above operations. FourierPlot also scales the axes such that amplitude of trigonometric components are plotted against frequency (Hz).&lt;br />&lt;/p>&lt;/body>&lt;/html></Text>
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:=Module[{data1,n,mean,data2,fdata1,fdata2,f
}
,[IndentingNewLine]data1=Table[signal,{t,tmin,tmax,T
}

]

;[IndentingNewLine]n=Length[data1]

;[IndentingNewLine]mean=Apply[Plus, data1
]
/n

;[IndentingNewLine]data2=data1-mean

;[IndentingNewLine]fdata1=2/n

  Abs[Fourier[data2]
]


;[IndentingNewLine]fdata2=Drop[fdata1,-Round[n/2.
]


]

;[IndentingNewLine]f=Range[0,1/(2*T
)

,(1/(2 T
)

-0
)
/Round[n/2.
]


]

;[IndentingNewLine]ListPlot[Transpose[{f,fdata2
}
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,options
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;
[IndentingNewLine]
]

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+5Sin[2 [Pi] 1.55t
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,PlotJoined[Rule]True

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