• Normally, the doctoral student is expected to complete at least 48 credits of course work (beyond the bachelor's degree) and at least 24 credits of research work.
  
  A student who started in the department with a B.S. degree must take a minimum of 10 MAE courses. A student who started in the department with an M.S. degree must take a minimum of six (6) MAE courses.
  
  A student in the Ph.D. program may register for one Independent Study course (in addition to the one he/she may have taken at the M.S. level).
  
  Students in the Ph.D. program may transfer up to 24 course credits from another institution Click here for more details.
  
  All students must take the two Applied Mathematics Courses (642:527 and 642:528) and at least one (1) graduate level course from each of the following four (4) areas:
  
  

  Design and Control Fluid Mechanics Solid Mechanics, Materials and Structures Thermal Processes

  
  
  Courses are classified according to areas in the section Descriptions of Graduate Courses. Transferred course credits can be counted for these minimum MAE course requirements.
• At least one year of full-time residence is required for the Ph.D. While the student may be a full-time student throughout his/her studies, the one-year residence requirement is normally satisfied after the student has passed the qualifying examination and is mainly devoted to research activities. This residency permits the student to benefit from interactions with the faculty and other students of the Department, College and University.
• When the full-time doctoral student has completed (or is near completion) two academic years of full-time graduate work past the B.S. degree, he/she is generally prepared to take the qualifying examination.
• Prior to taking the qualifying examination, a doctoral student should take a number of required courses that depend on the area of specialization. At present, the program offers four areas: (l) Design and Control, (2) Fluid Mechanics, (3) Solid Mechanics, Materials and Structures, and (4) Thermal Processes. The required number of credits and courses in these areas are listed below.
  
  

  Design and Control      Methods of Applied Mathematics, 642:527, 528    Design of Mechanisms, 650:514    Analytical Dynamics, 650:522    Mechanics of Materials, 650:550    Optimal Design in Mechanical Engineering, 650:614    Total Credits: 18

  Fluid Mechanics      Methods of Applied Mathematics, 642:527, 528    Fluid Mechanics I and II, 650:530, 630    Conduction Heat Transfer, 650:570    Thermodynamic Theory, 650:574    Total Credits: 18

  Solid Mechanics, Materials and Structures      Methods of Applied Mathematics, 642:527, 528    Analytical Dynamics, 650:522    Mechanics of Materials, 650:550    Mechanics of Continua (Solid Mechanics I), 650:554    Theory of Elasticity (Solid Mechanics II), 650:556    Total Credits: 18

  Thermal Processes      Methods of Applied Mathematics, 642:527, 528    Fluid Mechanics I, 650:530    Conduction Heat Transfer, 650:570    Thermodynamic Theory, 650:574    Convection Heat Transfer, 650:578    Total Credits: 18

B. Qualifying Examination

Every doctoral student must take the Qualifying Examination in order to be admitted into cnadidacy. The following general rules apply:

• Qualifying examinations are usually held in May, after the final examinations and before Commencement Day.
• A doctoral student is not permitted to take the qualifying examination if his/her cumulative average of all courses taken at Rutgers since matriculation into the Graduate School is lower than B, or if the student has not completed all requirements for the M.S. degree. A student wishing to take the qualifying examination should apply in writing to the Graduate Director for permission to take the examination. Early in the spring semester, an announcement is set to all graduate students regarding the qualifying examination. Students wishing to take the qualifying exam should respond to that announcement with a letter that includes a detailed statement showing how the course requirements have been met and how all M.S. degree requirements are or are about to be met.
• A student whose application to take the examination is granted should complete Part I of the form Application for Admission to Candidacy for the Degree of Doctor of Philosophy, and submit it to the chairperson of the examination committee prior to the examination. The Ph.D. Qualifying Examination Committee consists of four members, one for each of the four major areas of concentration. A fifth member, usually from the Mathematics Department, may be added to the committee.
• The type, content and duration of the Qualifying Examination are at the discretion of the Examination Committee and the Graduate Director. The normal format, however, consists of a written and sometimes an oral part, as described below.

1. The type of written examination, e.g., open book, equations sheets, etc., is at the discretion of the Examination Committee.
   
   
2. The examinee will be notified of the outcome of the examination in writing by the graduate director.
   
   
3. If a student fails the examination or any part of it, the question of reinstatement and make-up of the exam is at the discretion of the Examination Committee. A student who fails the examination may be allowed to repeat the examination at the most, one more time.
   
   
4. Ph.D. candidates, i.e., those who pass the qualifying examination, are required to register for credit in the one-credit research seminar (650:691,692) for two (2) semesters after the examination. They should subsequently register in it for no credit.

Oral Part

At the discretion of the Ph.D. Qualifying Examination Committee, an oral examination may be scheduled.

Guidelines for Ph.D. Qualifying Examination

The purpose of studying for the Ph.D. Qualifying Exams is to motivate the students, now as more mature individuals than when they started their graduate studies, to review and study their course material in an intensive environment with the intent that such a review will enable them to better understand the original material and to begin to see the larger connections between subjects and disciplines.

The purpose for taking the Ph.D. Qualifying Exams is to enable the faculty to assess whether each student has mastered the material necessary to pursue more advanced study and research. This includes mastery of fundamental concepts, physical understanding needed for modeling phenomena and processes relevant to the subjects that comprise the discipline, working knowledge of the mathematical tools needed for quantification and for solving the relevant equations, and the ability to extract practical conclusions from the results.

Even though students are required (or strongly recommended) to take a certain number of courses before the qualifying exam, there really is not a one-to-one correlation between recommended courses and contents of the exam. In each exam students are required to know certain basic topics, as described below. Many Ph.D. students join our program after they receive their M.S. degrees from other institutions and the course material covered in courses they took before coming to Rutgers may be different than what we cover here. Students are required to know the basic undergraduate material in the subjects they are examined.

Click here for a more detailed description of the rules and procedures for the qualifying exam. Following is a description of the subject material for the qualifying examination:

Design and Control
The exam in Design and Control tests the basic knowledge of students in the areas of study listed below:

Kinematics: Students should have a basic understanding of mechanism analysis and design. They should be familiar with basic approaches to mechanism kinematic analysis as well as synthesis. Students should be familiar with measures of merit such as mechanical advantage, transmission angle, and structural error. In analysis, students should know about two and three-dimensional kinematics, including Denavit-Hartenberg parameters and screw algebra. In synthesis, students should know analytical and numerical precision point approaches as well as numerical optimization-based methods.

Dynamics (including basic vibration): The student should be able to apply the basic concepts of dynamics to the modeling and analysis of particles and rigid bodies. The ability to model and analyze the following are expected: (1) Kinematics - Particles, 3D rotation of rigid bodies, interconnected bodies, and rolling; (2) Kinetics - Derivation of equations of motion using Newtonian and Lagrangian approaches; (3) Qualitative Analysis - Energy and momentum methods, as well as other motion integrals; (4) Response - Eigenvalue analysis of single and multi-degree of freedom linear vibrating systems, as well as continuous systems.

Design: Students should have developed a basic ability of converting real-life design problems into design models, applying various analytical, computational and optimization methods, generating realistic solutions and interpreting the results. A basic understanding of function-based performance criteria and constraints, of strength of materials, of the role played by material properties, and of failure and safety factors is necessary. Basic understanding of underlying concepts of different design representations, manufacturing methods, finite element methods and optimization algorithms is needed.

Mechanics of Materials: Students should know plane stress and plane strain formulations, principal stresses in two and three dimensions, beams, membranes, plates and shells, as well as basic constitutive equations.

Solid Mechanics
The student should have a basic understanding of the foundations of solid mechanics, analytical dynamics and applied mathematics. The analytical dynamics and mechanics of materials requirements are described in the previous subsection. The other two main areas of study in solid mechanics are Continuum Mechanics and Elasticity, as described below:

Continuum Mechanics: Elementary tensor analysis, Cartesian components, traction and Cauchy stress, finite and infinitesimal strain, kinematics and material rates of change, balance laws, material symmetry. Elasticity, plasticity, fracture and fluid mechanics as special branches of continuum mechanics.

Elasticity: Finite strain, elastic energy, linear elastic constitutive equations, material symmetry. Elementary solutions for extension and bending. Torsion of cylindrical rods including non-circular cross-sections, Prandtl's stress function, complex variable methods for torsion. Plane stress and strain, Airy's stress function, 2D problems in rectangular and cylindrical coordinates. Betti's reciprocal theorem, the principle of virtual work, minimum energy theorems. 3D solutions in cylindrical and spherical coordinates, point load solutions: Kelvin's and Boussinesq's problems.

Suggested References:
Y.C. Fung, Foundations of Solid Mechanics, Prentice Hall, 1965.
Y.C. Fung, A First Course in Continuum Mechanics, Prentice Hall, 1977.
L.E. Malvern, Introduction to the Mechanics of a Continuous Medium, Prentice Hall, 1969.
I.S. Sokolnikoff, Mathematical Theory of Elasticity, Krieger, 1983.
S.P. Timoshenko and J.N. Goodier, Theory of Elasticity, McGraw-Hill, 1969.

Fluid Mechanics
The specific objectives are to test the student's understanding of fundamental principles in fluid mechanics, as well as his/her ability to synthesize the various topics into a single global perspective on the field. In this context, understanding implies being able to derive the important governing equations, and to apply well-developed physical insight coupled with strong theoretical skills to solve complex problems.

The exam is typically given at two distinct levels. The first exam (specialist) is targeted at students intending to specialize in some aspect of fluid mechanics. The second exam (cognate) is for students whose chosen field of expertise requires some degree of understanding of fluid mechanics, but is not the central focus, such as for students specializing in the thermal sciences.

Both exams test for the student's mastery of introductory fluid mechanics concepts at the undergraduate and graduate levels. Students are asked to demonstrate their ability to derive key governing equations (mass, momentum, energy and vorticity) in both differential and integral form (where appropriate) as well as to provide exact solutions of these equations for different conditions.

The level of the specialist exam transcends the cognate exam by testing the student's detailed knowledge in specific topical areas along with his/her ability to synthesize material from different topical areas. Questions for the specialist exam, in addition to those described in the previous paragraph, are drawn from the full spectrum of fluid mechanics courses offered by the Department. In the context of testing synthesis skills and probing the student's depth of knowledge, open-ended questions combining a number of the above topics are also appropriate. The exam is structured so that the student need only demonstrate proficiency in two to three specialized topics in addition to the base knowledge covered in the two semester graduate fluid mechanics series.

Thermal Processes
This exam, like the fluid mechanics exam, is offered at two levels, one for students specializing in thermal sciences and the other for students is for students whose chosen field of expertise requires some degree of understanding of thermal sciences, but is not the central focus, such as for students specializing in fluid mechanics. The areas of study are:

Thermodynamics: Laws of classical thermodynamics, fundamental relations, energy analysis of processes, prediction of material properties, stability and phase transitions, reactive processes, equilibrium. Introduction to statistical thermodynamics, micro-and macro-canonical approach, the Boltzman distribution, modeling ideal gases, generation of property data.

Heat Transfer: Basic processes involving heat transfer by conduction, convection and radiation. Formulation and modeling different single and multi-dimensional processes with temperature gradients in solid and fluid media. Conjugate heat transfer with stationary and also moving boundaries due to phase change. Effect of radiation interaction at the boundaries. Effects of material properties, and scaling.

Exact, approximate, and finite difference methods of solution for basic steady and unsteady conduction heat transfer problems. Exact and approximate solutions for basic internal and external fluid flow problems with laminar and turbulent flow. Effect of body forces in free and combined free and forced convection. Boundaries with injection or blowing, and phase change. Effect of temperature dependent properties. Compressible flow. Analysis using empirical correlations.

Radiation: Radiation analysis in enclosures without a participating medium. Directional and spectral emission and absorption characteristics of surfaces.

C. Dissertation

Dissertation Committee

For each student that passes the Ph.D. Qualifying Exams, the Graduate Director in consultation with the student's research advisor forms a committee of three faculty, one of who is the student's research advisor, from the ranks of the MAE Department, which will eventually comprise the student's Dissertation Committee. The external examiner may, but need not be, appointed at this time. Within a year of the forming of this committee, a dissertation proposal will be presented to the committee, in a public meeting, for review. This is important: The student must defend his/her proposal within a year of passing the Qualifying Exam. The prevailing opinion on this proposal will be that of the student's advisor. The other two faculty members will have an advisory role where suggestions are provided. These two faculty members will also follow the work of the student to the extent that is reasonable and warranted.

There are a number of reasons for following this procedure. The student and faculty advisor benefit by having more formal access to their colleagues. The two committee faculty members can follow the research and can make constructive suggestions long before the actual defense. This makes the committee input more meaningful and removes some of the negative aspects of providing the committee too short a time to review a substantial amount of research at the very end of the process when there is little likelihood of change. The committee would be seriously encouraged to bring in the external examiner at six to twelve months prior to the estimated completion of the student's research for similar reasons. This process in no way reduces the authority of the faculty research advisor as the lead in the research effort, its methods and goals, and in determining issues such as publications, presentations, and milestones.

Dissertation Proposal

What is a Dissertation Proposal? In principle, research begins with a study of the literature. We examine what work has been done on a topic before we decide how we are to contribute to a discipline. This process: (i) a review of the state-of-the-art, (ii) an examination where there is a lack in understanding, and (iii) a focusing on the resolution of some part of that lack, is what the Dissertation Proposal is all about.

Your advisor, with the Graduate Director, will form your Committee, minus the external examiner. You are given a maximum of one year from passing the Quals to defend your research proposal, i.e., the Dissertation Proposal, before this committee. The defense will be very similar to an MS thesis defense. This proposal is very similar, except there is no budget, to those written by your advisor when external funding is sought.

You should use the Dissertation Style format. The remaining details are up to your advisor and your committee. Your proposal should be of sufficient detail to convince your committee that you understand the problem area, are aware of the state-of-the-art, and have a well-defined problem to focus on.

Some of you have already done much of this in an informal way and therefore, need only write this as a report and then present it to your committee.

Completion of a Dissertation

Two semesters before the anticipated completion date of the Ph.D. degree the student should meet with the Graduate Director and confirm that the student has met (or is going to meet) all graduation requirements. A checklist for this purpose is provided here. Following is the procedure to be followed for the completion of a dissertation:

• The Ph.D. dissertation advisor supervises the research progress and provides guidance as necessary. In addition to consulting with his/her advisor, the candidate is encouraged to discuss his/her research with his/her Dissertation Committee and other members of the faculty. The advisor will assign a grade of Satisfactory or Unsatisfactory for each semester in which the candidate is registered for research credit.
• When the research is completed, the candidate will write a Ph.D. dissertation in accordance with the guidelines set forth in the booklet Style Guide for Thesis and Dissertation Preparation.
• After the advisor approves the dissertation, he/she will notify the Graduate Director, who in consultation with the advisor, select the outside member of the Dissertation Defense Committee (note: this selection can be done before the dissertation is written). The advisor, the two previously selected faculty from the graduate program and the outside member will form the student's Ph.D. Dissertation Committee.
• The Graduate Director will send the dissertation to each reader, requesting a review in approximately two weeks.
• After the readers indicate their approval, the Graduate Director will schedule the public defense of the dissertation, and a public notice shall be distributed to all relevant departments, faculty and students. This public notice has to be posted at least five days before the disertation defense.
• The candidate provides a copy of the dissertation for the department's thesis collection at least five days prior to the final examination date. This copy will be available in the MAE office for public inspection until the day of the examination.

• The candidate obtains from the Graduate School the previously completed candidacy form and brings it to the defense.
• The examination is presided over by the candidate's dissertation advisor and consists of an open portion and a closed portion. The open portion consists of an approximately 45 minute presentation by the candidate on his/her dissertation. This is followed by approximately 30-45 minutes of questions from the committee and from the public. The closed portion is attended by the examining committee only, who may elect to question the student further on his/her dissertation, or in areas related to it.
• The candidate is excused and the student's Dissertation Committee, and the Graduate Director, if so interested, meets to reflect on the examination and to complete the candidacy form.
• The candidate returns the candidacy form to the graduate secretary. The form will be kept in the student's folder until recommended thesis revisions are made. A bound copy of the revised thesis must be presented to the Graduate Director along with the candidacy form. The Graduate Director's name will be the final signature on the form. This bound copy is used as a reference within and outside the department.

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