hw.lyx

#LyX 2.2 created this file. For more info see http://www.lyx.org/
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\usepackage{fancyhdr}
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\lhead{ME 585 Special Topics in Mech. Eng., Fall 2015}
\chead{C. Ça\u{g}r\i{} ÖZKAN}
\rhead{Assignment 2}
\rfoot{Yeditepe University}
\cfoot{\thepage}
\lfoot{Mechanical Engineering}
\usepackage{tikz}
\usepackage{pgfplots}
    % recommended as of Pgfplots 1.3:
\pgfplotsset{compat=newest}
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\begin_body

\begin_layout Paragraph*

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Question Modification:
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\begin_layout Standard
Since ID=XXXXXXX9X03 the coefficients are 
\begin_inset Formula $a=3;\:b=0;\:c=9$
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Therefore;
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\begin_layout Standard
\begin_inset Formula $y=f(t)=3t^{2}+9$
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 in interval of 
\begin_inset Formula $0,1$
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Solution:
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\begin_inset Formula $T=1\rightarrow\omega=\nicefrac{2\pi}{T}=2\pi$
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\begin_layout Standard
\begin_inset Formula $a_{0}=\frac{2}{1}\int_{0}^{1}f(t)dt=2\int_{0}^{1}(3t^{2}+9)dt=2\left[t^{3}+9t\right]_{0}^{1}=20$
\end_inset


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\begin_layout Standard
\begin_inset Formula $a_{n}=\frac{2}{1}\int_{0}^{1}(3t^{2}+9)cos(n\omega t)$
\end_inset


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\begin_layout Standard
\begin_inset Formula $b_{n}=\frac{2}{1}\int_{0}^{1}(3t^{2}+9)sin(n\omega t)$
\end_inset


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\begin_layout Standard
Maxima is used to integrate the 
\begin_inset Formula $f(t)*cos(2\pi n)$
\end_inset

 and 
\begin_inset Formula $f(t)*sin(2\pi n)$
\end_inset

 to get cosine and sine compliance.
 The commands are listed below.
 
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\begin_layout Standard
\begin_inset listings
lstparams "basicstyle={\small\ttfamily}"
inline false
status open

\begin_layout Plain Layout

ratsimp(integrate((3*t^2+9)*cos(w*t*n),t,0,1));
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\begin_layout Plain Layout

ratsimp(integrate((3*t^2+9)*sin(w*t*n),t,0,1));
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\begin_layout Plain Layout

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\end_inset


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\begin_layout Standard
\begin_inset Formula 
\[
a_{n}=2\times\frac{6\cdot n\cdot w\cdot\mathrm{cos}\left(n\cdot w\right)+\left(12\cdot{{n}^{2}}\cdot{{w}^{2}}-6\right)\cdot\mathrm{sin}\left(n\cdot w\right)}{{{n}^{3}}\cdot{{w}^{3}}}
\]

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\begin_layout Standard
\begin_inset Formula 
\[
b_{n}=2\times\frac{-6+9\cdot{{n}^{2}}\cdot{{w}^{2}}+\left(6-12\cdot{{n}^{2}}\cdot{{w}^{2}}\right)\cdot\mathrm{cos}\left(n\cdot w\right)+6\cdot n\cdot w\cdot\mathrm{sin}\left(n\cdot w\right)}{{{n}^{3}}\cdot{{w}^{3}}}
\]

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\begin_layout Standard
Therefore;
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\begin_layout Standard
\begin_inset Formula 
\[
y(t)=\frac{a_{0}}{2}+\sum_{n=1}^{100}a_{n}cos(\omega nt)+b_{n}sin(\omega nt)
\]

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\begin_layout Standard
To simplify the output; 
\begin_inset Formula $\pi$
\end_inset

 is set to 3.14 therefore 
\begin_inset Formula $\omega=6.28$
\end_inset

 since 
\begin_inset Formula $\omega=2\pi$
\end_inset


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\begin_layout Standard
\begin_inset Formula 
\[
y(t)=\frac{20}{2}+\sum_{n=1}^{100}a_{n}cos(6.28nt)+b_{n}sin(6.28nt)
\]

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\begin_inset listings
lstparams "numbers=left,numberstyle={\scriptsize},basicstyle={\small\ttfamily},breaklines=true"
inline false
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sum((2*6*n*(2*3.14)*cos(n*(2*3.14))+(12*n^2*(2*3.14)^2-6)*sin(n*(2*3.14)))/(n^3*(2*3.
14)^3)*cos((2*3.14)*n*t)+2*(-6+9*n^2*(2*3.14)^2+(6-12*n^2*(2*3.14)^2)*cos(n*(2*3.14)
)+6*n*(2*3.14)*sin(n*(2*3.14)))/(n^3*(2*3.14)^3)*sin((2*3.14)*t*n), n, 1, 100),
 simpsum;
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\end_inset


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placement H
wide false
sideways false
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\noindent
\align center
\begin_inset Graphics
	filename ../YuksekLisans/muzik/grafik1aa
	lyxscale 50
	width 50text%

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\begin_layout Plain Layout
\begin_inset Caption Standard

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f(t) vs.
 Fourier Series
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\begin_layout Standard
Following Octave code is used to determine intensity of each sine/cosine
 component for the frequency from 
\begin_inset Formula $\pi$
\end_inset

 to 
\begin_inset Formula $10\pi$
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function ycos=an(n)
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\begin_layout Plain Layout

  ycos=2*(6*n*2*pi*cos(n*2*pi)+(12*n^2*(2*pi)^2-6)*sin(n*2*pi))/(n^3*(2*pi)^3);
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endfunction
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\begin_layout Plain Layout

function ysin=bn(n)
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\begin_layout Plain Layout

  ysin=2*(-6+9*n^2*(2*pi)^2+(6-12*n^2*(2*pi)^2)*cos(n*(2*pi))+6*n*(2*pi)*...
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\begin_layout Plain Layout

  sin(n*(2*pi)))/(n^3*(2*pi)^3);
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endfunction
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for i=1:1:10
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  m(i,1)=sqrt(an(i)^2+bn(i)^2);
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endfor
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f1 = fopen('~/YuksekLisans/muzik/hw3_m.dat','w');
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for i = 1:length(m)
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    fprintf(f1,'%d
\backslash
n',m(i));
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end
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fclose(f1);
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wide false
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\noindent
\align center
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	filename ../YuksekLisans/muzik/22
	lyxscale 50
	width 50text%

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\begin_layout Plain Layout
\begin_inset Caption Standard

\begin_layout Plain Layout
Intensity vs.
 frequency
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\end_body
\end_document