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\begin{document}
\title{List of course projects (Preliminary)}
\author{Take one (First come -- first served)}\maketitle

{\it
These are drafts of the course projects. Please feel free to modify
them replacing electrical network with mechanical one, adding new
goals, etc. A group of two can take one project or modify two close
projects; the grade will be shared.  
}


\section{Project. Vibrating chains}
\paragraph{Sequential connections}
Consider a chain of $N=10$ large masses $M_+$ alternating with the small
masses $M_-= {1 \over 2}M_+$. The masses are separated by the linear spring with
stiffness $C$. A periodic force $F(t)=  \cos(\omega t) $ 
is applied to the left end of the chain, the right end is attached to
the mass $M_0$.

Find the motion of the right end of the chain. Graph the dependence of
the frequency of
oscillations $\omega$. 
 Choose the frequency  $\omega$ to minimize and to maximize 
the norm $ N(u_0)$ of deflection $ u_0$ of the
mass  $M_0$, 
$$ N(u_0)= \frac {1}{2 \pi \omega  } \int_0^{2 \pi \omega} u_0^2 dt
$$
\paragraph{Parallel connections}
Consider a chain of $N=10$ sequential  large masses $M_+$ with 
 the linear spring with
stiffness $C$ between them. To each mass $M_+$ in parallel
is attached a spring $C_-$ with  the small
masses $M_-$ at the end. A periodic force $F(t)=  \cos(\omega t) $ 
is applied to the left end of the chain, the right end is attached to
the mass $M_0$.

Find the motion of the right end of the chain. Graph the dependence of
the frequency of oscillations $\omega$.  Choose the frequency
$\omega$  to minimize and to maximize 
the norm $ N(u_0)$. 


\section{Network design}

Consider an electrical network of $5 \times 5$ nodes joined vertically
and horizontally by resistances $ R_{nm}$ such that $$ R_- \leq R_{nm}
\leq R_+.$$ The left lower node is grounded (has zero potential).

\paragraph{Excitation}
Two type of excitations are considered:

 (1) One can apply the voltage
sources on any boundary node or any combination of them.

or

 (2)  One can apply the current
sources on any boundary node or any combination of them provided that
the sum of currents is zero.


\subsection{Project. Sources optimization: Minimization of the whole
  resistivity} Assume that the resistances $ R_{nm}$ are equal.
The resistance of the whole network
is defined as the energy dissipated in it.  The sum of absolute values
of the applied currents is equal to one. 
\begin{enumerate}
\item
How to distribute the current between the nodes to{\em minimize} the
resistance of the whole network.
\item
How to distribute the current between the nodes to {\em maximize} the
resistance of the whole network. 
\end{enumerate}



\subsection{Project. Sources optimization: Maximization of the current
in a link}
Assume that the resistances $ R_{nm}$ are equal. Any potential source
can be applied to any boundary node but the grounded one, the sum of
squares of these potentials is equal to one. 

What potentials should be applied to maximize the current through the
vertical link in the center of the network.

Assume in addition that the potentials must be applied only on the lower
boundary of the network.  What potentials should be applied to maximize the current through the
vertical and horizontal links in the center of the upper side of the network.


\subsection{Project. Detection of the broken link}
 
Assume that that the resistances $ R_{nm}$ are equal but one link is
broken. Apply any boundary sources as many times as you wish to detect
the position of the broken link. 

How to minimize the number of measurements?


\subsection{Project. Network design}

The network transports the currents from the left to the right side.
The total current to be transported through the network equals
one. The equal positive current is applied  to all nodes at the left side and
the equal negative current -- to all nodes at the right side. 

Choose the resistivity of the links in a network (subject to
 constraints  $ R_- \leq R_{nm}
\leq R_+$)
 that

\begin{enumerate}
\item
Maximizes the current through a vertical link at the center of the
network.
\item 
Maximizes the potential difference between two inner nodes

\end{enumerate}







\subsection{Project. Network design}

The network transports the currents from the left to the right side.
The total current to be transported through the network equals
one. The equal positive current is applied  to all nodes at the left side and
the equal negative current -- to all nodes at the right side. 

Assume that $N$ links have the resistivity $ R_-$ and the rest have 
 the resistivity $ R_+$. 

\begin{enumerate}
\item
Maximize the resistance of the whole 
network.
\item 
Minimize the resistance of the whole 
network.
\end{enumerate}
\subsection{Complex-valued conductivity}
Replace the resistors around the main diagonal with the capacitors of
equal capacity $C$. Assume that the harmonic current source $ I(t)=
I_0 \sin (\omega t) $ is applied
to a node in the right side and current $ - I(t)$ 
is applied to one node at the left side. 

 Derive the real-valued system for
the network and compute the currents in the network. Derive the
quadratic function which minimization leads to the solution of the
network.


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