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{\bf EML 5060 \hfill Analysis in Mechanical Engineering \hfill 10/18/94 }\\
{\it Closed book \hfill Van Dommelen \hfill 9:00-9:50 EST}
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Show all reasoning and intermediate results leading to your answer.
One book of mathematical tables, such as Schaum's Mathematical Handbook,
may be used.
Jack Tsai may use one English or English-Chinese dictionary.

\begin{enumerate}

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\item
For the truss shown below, the tension forces in the bars satisfy the
equations
$$-T_1 + T_2 = 0$$
$$(T_1 + T_2) \cos (45) + T_5 = 0$$
$$- T_3 +T_4 = 0$$
$$(T_3 + T_4) \cos(60) - T_5 = 0$$
If the vector of unknowns is token to be $(T_1,T_2,T_3,T_4,T_5)^T$,
give the matrix representing the above equations.
Reduce to echelon form.  From the echelon form,
show that the bars may be under tension or
compression, even though there are no external forces.
The only thing we can be sure of are the ratios of the tension
forces in the bars.  Determine these.

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\item
Consider the two masses suspended by springs below.
Newton's equations of motion are:
$$m {\ddot x}_1 + 2 k x_1 - k x_2 = 0 \qquad
m {\ddot x}_2 + 2 k x_2 - k x_1 = 0$$
We will assume that $k=m=1$.
If we look for solutions of the form
$$x_1 = X_1 \sin(\sqrt{\lambda}t) \qquad
x_2 = X_2 \sin(\sqrt{\lambda}t)$$
we find that
$$
\pmatrix{2 & -1\cr
-1 & 2}
\pmatrix{X_1\cr
X_2}
= \lambda
\pmatrix{X_1\cr
X_2}$$
Find the two values of $\lambda$ for which such motion is possible.
Also find the  corresponding ratio of $X_1/X_2$, i.e. of $x_1/x_2$.
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\item
For the 2 by 2 matrix shown in question 2,
evaluate the matrix
$$I + A + A^2 + A^3 + \cdots + A^{14}$$
Do not use laborious procedures that are error prone.

\end{enumerate}

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