Fall 2016 Final Exam

EML 5060 Analysis in Mechanical Engineering I 12/9/16
Closed book Van Dommelen 5:30-7:30 pm

Solutions should be fully derived showing all intermediate results, using class procedures. Show all reasoning. Bare answers are absolutely not acceptable, because I will assume they come from your calculator (or the math handbook, sometimes,) instead of from you. You must state what result answers what part of the question. Answer exactly what is asked; you do not get any credit for making up your own questions and answering those. Use the stated procedures. Give exact, fully simplified, answers.

One book of mathematical tables, such as Schaum's Mathematical Handbook, may be used, as well as a calculator, and a handwritten letter-size formula sheet.

  1. Background: Differential equations of second order are common in engineering because of dynamics. Below is an equation for the forced dynamics of an unstable system.

    Question: Solve the ordinary differential equation

    \begin{displaymath} y'' - 2 y' + y = e^x \end{displaymath}

    using variation of parameters.

    Solution.

  2. Background: The Laplace transform is a primary way to study the stability and evolution of linearized dynamical systems, because it turns them into algebraic systems.

    Question: Use the Laplace transform to solve the following damped vibrating system that experiences a constant force:

    \begin{displaymath} y'' + 2 y' + 5 y = 5 \qquad y(0)=0 \quad y'(0)=2 \end{displaymath}

    A table of Laplace transforms is attached. Everything not in this table must be fully derived showing all reasoning. The convolution theorem may only be used where it is absolutely unavoidable. Do not use any complex numbers in your analysis (besides $s$.) You can only use one Laplace transform table entry at each step (except P2), and its table number must be listed. No funny (discontinuous) functions in your answers.

    Solution.

  3. Background: Since any system of differential equations can be reduced to a first order system, all you really need to know is how to solve these systems.

    Question: Solve using the class procedures for systems of ODE, including variation of parameters:

    \begin{displaymath} \dot x = 4 x - y \qquad \dot y = x + 2 y + e^{3t} \end{displaymath}

    In particular, find the solution for the initial condition

    \begin{displaymath} x(0)=1 \quad y(0)=1 \end{displaymath}

    Solution.


Table 1: Properties of the Laplace Transform. ($k,\tau,\omega>0$, $n=1,2,\ldots$)

\begin{picture}(402,593)(1,0) % put(1,0)\{ framebox(402,593)\{\}\} \put(0,295){\... ...u)$\ & $\displaystyle e^{-\tau s}$\ \\ \hline \end{tabular*}}} \end{picture}