% From - Fri Aug 24 10:16:57 2001
% From: Jorge Angeles <angeles@athena.mcrcim.mcgill.edu>
% To: svetlana@cim.mcgill.ca
% Subject: Latex job

% Svetlana:

% I am enclosing the Latex file of Project 1 of 305-577A, that I want to
% hand out on Tuesday Sept. 4 to the students. Please include therein
% the figure I asked you to edit for me, and give me a printout for
% checking. Thx.

% JA

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\begin{document}
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\centerline{\bf\Large 305-577A Optimum Design}
\vskip .7cm

\begin{center}
{\bf Project \# 1:  Single-Variable Optimization\\
The Optimum Design of a Vehicle Suspension}
\end{center}

\vskip .5cm

\noindent Assigned: September 4, 2001\hfill Due: October 3, 2001\medskip
\vskip .3cm

Shown in Fig.~\ref{f:susp} is the iconic model of the suspension of
a terrestrial vehicle. The suspension has been modelled with three
identical masses $m$ coupled by different springs of stiffness $k$,
$\alpha k$ and $\alpha^2 k$, where $\alpha$ is a real number, as yet
to be determined, as explained below.

\begin{figure}[h]
\centerline{
\epsfig{file=/home/diestro/svetlana/help/305-577/3dof_suspen.plt,width=50mm}}
         \caption{Three-degree-of-freedom mode} 
\label{f:susp}
\end{figure}
The natural frequencies of the system are found as the square roots of
the eigenvalues of what is called the {\em dynamic matrix} \negr D,
namely,

\bed
\negr D = \omega^2\smatrix{1+\alpha & -\alpha          & 0     \cr
                            -\alpha  & (1+\alpha)\alpha & -\alpha^2\cr
                                0   & -\alpha^2        & \alpha^2 \cr}
\equiv \omega^2\negr A
\eed
where $\omega$ is a scalar with units of frequency, and defined as
\bed
\omega \equiv \sqrt{\frac{k}{m}}
\eed

It is important to specify $\alpha$ judiciously because the natural
frequencies of the suspension are proportional to the square roots of
the eigenvalues of \negr A, which depends on $\alpha$.

The physical significance of the natural frequencies lies in the
effect of {\em resonance}: When the system is excited with a harmonic
perturbation of frequency equal to one of the natural frequencies of
the system, the system responds with oscillations of an amplitude that
grows unboundedly. For this reason, it is important to choose $\alpha$
in such a way that the {\em bandwidth} $b$ of the system, given by
$b\equiv 20 \log (\omega_M/\omega_m)$, in db, is made as small as
possible. In this definition, $\omega_M$ and $\omega_m$ denote the
largest and the smallest natural frequencies of the system,
respectively.

Ideally, we would like to have $b = 0$, but this is, in
general, not possible. Whenever this is not possible, we attempt to
make $b$ a minimum. An expert designer tells us that the bandwidth
attains its minimum when $\alpha$ lies ``somewhere between 0 and 2.'' 

Using a) dichotomous search, b) Fibonacci numbers and c) golden
search, find an estimate of the optimum value of $\alpha$ within an
interval of uncertainty of 0.01. Comment on your results. What are the
pros and the cons of each method? What is the most likely optimum
value of $\alpha$? How about the most likely optimum value of $b$?

\vskip 3mm
\noindent{\bf Remarks:} Note that, if $\alpha = 0$, \negr A becomes
singular, and hence, at least one of its eigenvalues vanishes. In
fact, when $\alpha = 0$, two eigenvalues of \negr A vanish, which
means that two of the natural frequencies of the system vanish. Give
a physical explanation to this limit case. Now, if one natural
frequency of the system vanishes, its bandwidth obviously tends to
$\infty$. This means that $\alpha = 0$ is to be avoided.

\end{document}











