AME 302 Final Exam Checklist

1. Compute the eigenvalues and eigenvectors for a given matrix.
2. Compute the eigenvalues and eigenvectors and generalized eigenvectors for matrices with repeated eigenvalues.
3. Determine whether a set of vectors is linearly independent or linearly dependent and explain the importance of linear (in)dependence.
4. Classify a system of differential equations as homogeneous or inhomogeneous.
5. Compute, utilize and explain basic matrix computational methods including inverses, transposes, inner products and conjugates and identify and explain the consequences of a matrix being singular, symmetric or Hermitian.
6. Explain the importance and use of the eigenvalue (and eigenvector) problem.
7. Derive the and solve equations of motion for multi-degree-of-freedom mechanical vibration systems (both forced and unforced).
8. Determine the fundamental modes of vibration for a multi-degree-of-freedom mechanical vibration system.
9. Convert a differential equation of order greater than one to a system of first order differential equations in matrix form.
10. Explain the consequences of and prove the principle of superposition for systems of first order differential equations.
11. For a set of solutions for an nth order system of first order differential equations, compute the Wronskian and explain the consequences of both a zero and nonzero Wronskian.
12. Solve systems of first order, constant coefficient, ordinary, linear differential equations including:
    1. computing a fundamental set of solutions, computing the fundamental matrix, computing a general solution and sketching the phase portrait when
       1. all eigenvalues are real and distinct;
       2. some eigenvalues occur in complex conjugate pairs; and,
       3. some eigenvalues are repeated
    2. Solve an initial value problem.
13. Solve nonhomogeneous systems of linear first order ordinary differential equations with constant coefficients using the methods of
    1. undetermined coefficients;
    2. undetermined coefficients where the naturally assumed form of the particular solution is identical to the homogeneous solution;
    3. diagonalization; and
    4. variation of parameters.
14. Derive the equations of motion for mechanical systems using Largange's equations.
15. Sketch the root locus of a (simple) transfer function K G(s) as K varies from zero to plus infinity, including
    1. computing asymptote angles;
    2. computing the intersection of the asymptote with the real axis;
    3. compute the departure angles of the locus from the poles of G(s);
    4. compute the arrive angles of the locus to the zeros of G(s);
    5. compute the break away point(s) from the real axis between two poles; and,
    6. compute the break in point(s) to the real axis between two zeros.
16. Explain each of the rules or formulas for sketching root locus plots in terms of the phase of G(s).
17. Explain the relationship between the phase of G(s) and the location of s relative to the poles and zeros of G(s).
18. Explain the relationship between the magnitude of G(s) and the location of s relative to the poles and zeros of G(s).
19. Determine the gain value, K corresponding to a particular point on a root locus plot.
20. For a non-unity feedback system, compute the characteristic equation and manipulate it to be in a form so that the root locus can be plotted to determine transient response characteristics and stability of the system.
21. Explain why the phase of G(s) must always be -180^o on the root locus plot of G(s).
22. Explain in terms an Arts & Letters major can understand, what is represented on a Bode plot.
23. Sketch the Bode plot for a (simple) transfer function G(s), including sketching the contribution to the Bode plot of each of the individual factos of G(s).
24. Determine the stability of a system, if possible, from its Bode plot.
25. Determine the gain and phase margin of a system from a Bode plot.
26. Write a computer program to solve an initial value problem for linear, first order differential equations including
    1. using Euler's method;
    2. using improved Euler's method;
    3. using the fourth order Runge-Kutta method;
    4. using the above methods to solve systems of first order equations; and,
    5. converting second order (and higher order) differential equations into a system of first order equations.
27. Identify when separation of variables may be used to solve a partial differential equation.
28. Solve boundary value problems, or show there is no solution, for second order ordinary differential equations.
29. For a given boundary value problem, determine the corresponding eigenvalues and eigenfunctions.
30. Explain the relationship between the eigenvalue problem for matrices and the eigenvalue and eigenfunction problem for boundary value problems.
31. Determine the Fourier series for a periodic function.
32. Use the method of separation of variables to solve the heat and wave equations given appropriate boundary conditions.