GUIDELINES FOR THE PHD WRITTEN EXAMINATION
AEROSPACE ENGINEERING GRADUATE STUDENTS
The written qualifying examinations ascertain that students intending to pursue the PhD degree in Aerospace
Engineering have a general knowledge of the principles and techniques in engineering analysis or system theory
and two of the following disciplines: aerothermodynamics, control theory, dynamics, fluid mechanics, orbital
mechanics, and solid/structural mechanics. For each discipline, questions are formulated to test the student's
knowledge of the assumptions and limitations of the engineering theory, the derivation of the governing
equations, and the solution of typical problems and interpretation of the solution.
The written examinations are composed of three-hour examinations in each of the three subjects. The
examination is offered once a year, usually during the second week of June. If a student does not pass the
written examinations, the ASE Graduate Studies Committee decides if the student may take the examinations
again. If a student is given another opportunity, all three examinations must be retaken.
The exam combinations which are automatically approved are shown below for students in the various areas of
study. If you are interested in another combination, you should discuss it with the Graduate Advisor.
AERODYNAMICS
Engineering Analysis
Aerothermodynamics
Fluid Mechanics
ORBITAL MECHANICS
System Theory
Dynamics
Orbital Mechanics
GUIDANCE AND CONTROL
System Theory
Dynamics
Control Theory
STRUCTURES
Engineering Analysis
Dynamics
Solid/Structural Mechanics
Listed below are courses, topics and references which give some idea of the extent and depth of the material
examined in each area. The graduate courses which are listed are offered on a regular basis, but they do not
review all the material of the undergraduate courses which are listed nor do they contain all of the material for
which you are responsible.
Finally, copies of the three previous examinations are available at
http://www.ae.utexas.edu/graduate-programs/ase/degree-info/302-ase-qual-guidelines-samples.
ENGINEERING ANALYSIS
COURSES:
Undergraduate mathematics courses relevant to engineering analysis:
ASE 380P.1
Mathematical Analysis - Analytical Methods I
ASE 380P.2
Mathematical Analysis - Analytical Methods II
TOPICS COVERED IN FIRST COURSE:
1. Real Analysis - limits and continuity, sequences and series in general, power series, Fourier series,
differentiation and integration of the real line, ditto for R (n=2 in particular), functions of several
variables.
2. Vector Calculus - differential operators (e.g., gradient, divergence, curl), basic integral identities (e.g.,
divergence theorem, Stoke's theorem, etc.), applications in geometry.
3. Complex Analysis - sequences and series in general, differentiation and integration, power series,
analytic function theory, residue theory.
4. Transform Calculus - basis theorems and identities, convolution and inversion of Fourier and Laplace
transforms.
TOPICS COVERED IN SECOND COURSE:
5. Linear Algebra and Analysis - matrix algebra, elementary properties of abstract linear spaces, inner
product spaces, eigenvalues and eigenvectors, eigenfunction expansions.
6. Variational Methods - minimization of a functional, basic calculus of variations.
7. Ordinary and Partial Differential Equations - classification and properties, elementary methods of
solution, characteristics, Green's function, special functions, Strum-Liouville theory, etc.
REFERENCES:
1. Greenberg, Foundations of Applied Mathematics, Prentice-Hall, 1977.
ADDITIONAL REFERENCES:
1. Butkou, Mathematical Physics, Addision Wesley.
2. Hildebrand, Advanced Calculus for Applications, McGraw-Hill, 1960.
3. Kreyszig, Advanced Calculus for Applications, McGraw-Hill, 1960.
4. Pipes/Harvill, Applied Mathematics for Engineers and Physicists, McGraw-Hill.
5. Sokolnikoff/Redheffer, The Mathematics of Modern Engineering, McGraw-Hill.
Rev. 6/02
SYSTEM THEORY
BACKGROUND COURSES:
ASE 330M
Linear System Analysis
EE 351K
Probability and Random Processes
REQUIRED COURSES:
ASE 380P.1
ASE 381P.3
ASE 381P.6
Analytical Methods 1
Optimal Control Theory
Statistical Estimation Theory
TOPICS:
1. Math – Topics covered in Analytical Methods 1, linearity, periodicity, superposition, complex variable
transforms (Fourier and Laplace Transforms), frequency and time domain representations of signals,
state transition matrix, solution of homogeneous and nonhomogeneous differential equations, stability,
least squares method, matrix manipulations.
2. Optimization – Parameter Optimization: first differential and second differential conditions for the
minimum of a function of n variables subject to equality constraints and inequality constraints. Optimal
Control Theory: Euler-Lagrange equations, natural boundary conditions, Legendre-Clebsch condition,
Jacobi (conjugate point) condition, Weierstrass condition, natural corner conditions, control inequality
constraints, state inequality constraints.
3. Estimation - Nonlinear estimation (Newton-Raphson), linearization, batch estimation, sequential
estimation, random variables, density functions, mean and variance, conditional probability, covariance
matrix, probability ellipse, Gauss-Markov theorem, process and measurement noise statistics, linear
unbiased minimum variance estimation, extended sequential estimation, continuous Kalman filter, Least
Squares, Weighted Least Squares, geometric derivation of normal equations, apriori estimate, shifting an
estimate to a different nominal trajectory, Shur Identity, Minimum Norm estimate, observability,
Maximum Likelihood estimate, time derivative of apriori covariance and apriori estimate, process noise,
Cholesky decomposition, Consider Covariance analysis, Orthogonal decomposition based on Givens
transformations.
REFERENCES:
1. Bryson/Ho, Applied Optimal Control, Hemisphere, 1975.
2. Chen, Linear System Theory and Design, Holt, Rinehart, and Winston, 1975.
3. Gelb, Applied Optimal Estimation, MIT Press, 1974.
4. Greenberg, Foundations of Applied Mathematics, Prentice-Hall, 1978.
5. Hull, Optimal Control Theory for Applications, Springer Verlag, 2003.
6. Tapley, Schutz, Born, Statistical Orbit Determination, Elsevier Academic Press, 2004.
7. Maybeck, Stochastic Models, Estimation and Control, Vol. I, Academic Press, 1979.
8. Oppenheim/Willsky, Signals and Systems, Prentice-Hall, 1997.
Rev. 9/13
AEROTHERMODYNAMICS
COURSES:
ASE 362K
ASE 346
ASE 347
(or)
EM 393N
ASE 382Q
Compressible Fluid Mechanics
Viscous Fluid Flow
Introductory Computational Fluid Dynamics
Numerical Methods for Flow Problems
Advanced Problems in Compressible Flow
TOPICS:
1. Viscous flows - laminar and turbulent, compressible and incompressible boundary layers.
2. Compressible flows, including shock waves, Prandtl-Meyer expansions, methods of characteristics, and
convective heat transfer.
3. Computational aerodynamics - panel methods, finite difference and finite element methods for compressible and
boundary layer flows; aerodynamic flow modeling.
REFERENCES:
1. Anderson, Modern Compressible Flow, McGraw-Hill, 1984.
2. Anderson/Tannehill/Fletcher, Computational Fluid Mechanics and Heat Transfer, Hemisphere Publishing Co.,
McGraw-Hill, 1984.
3. Bertin/Smith, Aerodynamics for Engineers, Prentice-Hall, 1989.
4. Saad, Compressible Fluid Flow, Prentice Hall, 1992
5. Schlichting, Boundary Layer Theory, Pergamon Press, 1955.
Rev. 6/02
CONTROL THEORY
BACKGROUND COURSES:
ASE 330M
Linear System Analysis
ASE 370L
Flight Control Systems
REQUIRED COURSES:
ASE 381P.1
ASE 381P.2
Linear Systems Analysis
Multivariable Control Systems
SUBJECT TOPICS:
1. Linear algebra and matrix analysis fundamentals; vector and matrix norms; signal and operator norms.
2. Input-output mathematical models for
finite-dimensional LTI and LTV systems; linearization of
nonlinear equations about nominal motions; existence and uniqueness results; state-transition operator
and its properties.
3 Convolution integral and s-plane analysis for LTI systems; the Routh stability test; frequency response;
Bode plots; root-locus methods; Nyquist stability criterion; phase and gain margin; closed-loop system
performance measures.
4. Zero-input and zero-state stability analysis for LTI and LTV systems; controllability and observability;
pole placement and state-feedback controllers; observer design and reduced-order observers.
5. Smith form and the Smith-McMillan form; MIMO poles and zeros; matrix fraction descriptions; Bezout
identity; co-prime factorizations; linear fractional transformations; realization theory.
6. Controller synthesis methods; Youla parameterization; small gain theorem; structured singular values;
robust stability and performance analysis.
REFERENCE MATERIAL:
1. Antsaklis and Michel, Linear Systems, Mc-Graw Hill, 1997.
2. Chen, Linear System Theory and Design, Oxford Press, 3rd ed, 2004.
3. Dorf and Bishop, Modern Control Systems, Prentice-Hall, 10th ed, 2005.
4. Goodwin, Graebe, and Salgado, Control System Design, Prentice-Hall, 2001.
5. Rugh, Linear System Theory, Prentice-Hall, 2nd ed, 1996.
6. Maciejowski, Multivariable Feedback Design, Addison-Wesley, 1989.
Rev. 9/07
DYNAMICS
COURSES:
ASE 311
ASE 372.K
EM 381
Dynamics
Advanced Space Craft Dynamics
Advanced Dynamics
TOPICS:
1. Newtonian Mechanics - particle and rigid body kinematics and dynamics, vibrations of lumped-mass
systems, dynamical stability.
2. Lagrangian Mechanics - LaGrange’s equations, conservative and non - conservative systems, impulsive
considerations, applications to particle and rigid body systems. Lagrangian Density Functions for
distributed parameter systems and approximation theory.
3. Introductory Hamiltonian Mechanics - Hamilton's equations, phase space, canonical transformations.
REFERENCES:
1. Beer/Johnston, Vector Mechanics for Engineers, McGraw-Hill, 1977.
2. Goldstein, Classical Mechanics, Addision-Wesley, 1980.
3. Greenwood, Classical Dynamics, Prentice-Hall, 1977.
4. Greenwood, Principles of Dynamics, Prentice-Hall, 1965.
5. Meriam, Dynamics, Wiley, 1975.
Rev. 6/02
FLUID MECHANICS
COURSES:
ASE 320
ASE 340
ASE 346
ASE 382Q.1/EM 387
ASE 347
(or)
EM 393N
Introduction to Incompressible Fluid Mechanics
Boundary Layer Theory & Heat Transfer (or equivalent undergraduate courses in
general fluid mechanics and convective heat transfer in lieu of the above courses)
Viscous Fluid Flow
Foundations of Fluid Mechanics
Introductory Computational Fluid Dynamics
Numerical Methods for Flow Problems
TOPICS:
1. Convective heat transfer.
2. The governing equations of fluid motion: continuity, momentum, energy, state, including variable
density; integral and differential forms.
3. Supplementary physical laws and concepts: Kelvin's theorem, vorticity, stream functions, potential
functions, etc.
4. Applications: inviscid and viscous incompressible flow, as typified by material in the above four
courses.
5. Approximations of flow and heat and mass transport equations using finite difference and finite element
methods at introductory level. Basic solution techniques.
REFERENCES:
1. Chapman, Heat Transfer, MacMillan, 1974.
2. Craig/Oden, Finite Elements, Fluid Mechanics, (Vol. VI) Prentice Hall, 1986.
3. Currie, Fundamental Mechanics of Fluids, McGraw-Hill, 1993.
4. Fox & McDonald, Introduction to Fluid Mechanics, 4th ed, Wiley.
5. Panton, Incompressible Flow, 2nd ed., Wiley, 1996.
6. Schlichting, Boundary Layer Theory, Pergamon Press, 1979.
7. Tannehill, Anderson and Pletcher, Comp. Fluid Mechanics and Heat Transfer, 2nd ed., 1997.
8. White, Fluid Mechanics, 1999.
9. White, Viscous Flow, McGraw-Hill. 1991.
Rev. 6/02
ORBITAL MECHANICS
BACKGROUND COURSES:
ASE 366K
Spacecraft Dynamics
ASE 366L
Applied Orbital Mechanics
ASE 372K
Advanced Spacecraft Dynamics
REQUIRED COURSES:
ASE 387P.2
ASE 388P.2
ASE 3890.1
Mission Analysis and Design
Celestial Mechanics I
Satellite Geodesy (includes topics from #4 below: periodic and secular
perturbations from a non-spherical central body, spherical harmonic
representation of planetary gravitational fields, atmospheric resistance, solar
radiation pressure.)
TOPICS:
1. Gravitational Two-Body Problem - equations of motion, coordinates, orbital elements, Kepler’s equation,
Lambert’s problem, satellite navigation; applications such as orbit transfer, ground tracks, mission
constraints, planning, rendezvous, orbit determination, patched conics, and effect of spacecraft
maneuvers.
2. Rigid Body Dynamics - reference frames, rotational dynamics, mass properties, torque free motion,
equations of motion, gravity gradient stability, attitude representations.
3. Celestial Mechanics- gravitational n-body problem, integrals of motion, the general three body problem,
the restricted three body problem, zero velocity curves, libration points, libration point orbits, stability
analysis, periodic gravity fields, restricted 3-body problem approximations to Sun-Planet-Spacecraft or
Earth-Moon-Spacecraft problems, escape trajectories, planetary flyby trajectories, lunar assisted
injection/capture.
4. General and Special Perturbations – Lagrange Planetary Equations, Gauss’s form of the Lagrange
Planetary equations, periodic and secular perturbations, resonance, perturbations from a non-spherical
central body, spherical harmonic representation, of planetary gravitational fields, perturbation due to
third bodies, atmospheric resistance, solar radiation pressure, numerical approximation and integration of
solutions, Linearization along nominal trajectories, the state transition matrix and its applications.
REFERENCES:
1. Bate/Mueller/White, Fundamentals of Astrodynamics, Dover, 1971.
2. Battin, An Introduction to the Mathematics and Method of Astrodynamics, AIAA, Revised Edition,
1999.
3. Kaplan, Modern Spacecraft Dynamics and Control, Wiley, 1976.
4. Kaula, Theory of Satellite Geodesy, Blaisdell Publishing, 1966 (also Dover).
5. Prussing and Conway, Orbital Mechanics, Oxford, 1993.
6. Roy, Foundations of Astrodynamics, Adam-Hilger, 1988.
7. Tapley, Schutz, Born, Statistical Orbit Determination, (Chapter 2), Elsevier, 2004.
8. Vallado, Fundamentals of Astrodynamics and applications, Microcosm Press, 2001.
9. Wiesel, Spaceflight Dynamics, 2nd Edition, McGraw Hill, 1999.
Rev. 9/07
SOLID/STRUCTURAL MECHANICS
COURSES:
ASE 365
ASE 384P.1/EM 388
ASE/EM 339
Structural Dynamics
Solid Mechanics I
Advanced Strength of Materials
TOPICS:
1. Beams - symmetric and unsymmetric bending; curved beams, beams on elastic foundation, inelastic
bending.
2. Torsion - circular and noncircular sections, approximate theory of thin-walled sections.
3. Stability - column buckling, beam columns.
4. Energy Methods - formulation and solution of problems, methods of minimum potential energy and
virtual work.
5. Structural Analysis - stiffness method of analysis of elastic trusses and frames.
6. Structural Dynamics - free and forced response of single degree of freedom systems, modeling of linear
multi-degree of freedom systems, modal analysis, vibration of continuous structural elements.
7. Elasticity - analysis of stress and strain; material response: linear elastic, elastic-plastic (elementary);
formulation of boundary value problems; solution of plane problems, St-Venant problems, axisymmetric
problems.
REFERENCES:
1. Craig, Structural Dynamics, Wiley, 1981.
2. Fung, Foundations of Solid Mechanics, Prentice-Hall, 1965.
3. Gere & Timoshenko, Mechanics of Materials, PWS Publishers, 1984 (or equivalent).
4. Gould, Introduction to Linear Elasticity, Springer-Verlag, 1983.
5. Oden & Ripperger, Mechanics of Elastic Structures, McGraw-Hill, 1981.
6. Timoshenko & Goodier, Theory of Elasticity, McGraw-Hill, 1975.
Rev. 6/02