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Mechanical Engineering and Mechanics
Subhrajit Bhattacharya

Teaching >>
Analytical Methods in Dynamics and Vibrations (Mech 425, Spring 2017, Lehigh U.)
Feb 03, 2017


Topics to be covered include coordinate systems, conservation laws, equilibrium and stability, systems of particles, variable-mass systems, transport equation; basic concepts from variational calculus; generalized coordinates, holonomic & nonholonomic constraints, generalized forces, D'Alembert's principle, Hamilton's principle, Lagrange's equations, generalized momenta; 3D rigid-body motion, inertia tensors, Euler angles, axis-angle representation, Hamilton's and Lagrange's equations for rigid bodies; oscillations, free and forced response of linear systems, linearization of nonlinear systems, discrete eigenvalue problem; chaotic systems, perturbation theory; additional topics: configuration spaces, forward and inverse kinematics, Jacobian, singularities, position control, nonholonomic systems.

Course Site:

Course Information and Policies:

Catalog Description:
Topics to be covered include coordinate systems, conservation laws, equilibrium and stability, systems of particles, variable-mass systems, transport equation; basic concepts from variational calculus; generalized coordinates, holonomic & nonholonomic constraints, generalized forces, D'Alembert's principle, Hamilton's principle, Lagrange's equations, generalized momenta; 3D rigid-body motion, inertia tensors, Euler angles, axis-angle representation, Hamilton's and Lagrange's equations for rigid bodies; oscillations, free and forced response of linear systems, linearization of nonlinear systems, discrete eigenvalue problem; chaotic systems, perturbation theory; additional topics: configuration spaces, forward and inverse kinematics, Jacobian, singularities, position control, nonholonomic systems.

Books:
We will primarily use the following books (any edition should be fine) for this class:
  1. "Analytical Dynamics" by Haim Baruh, WCB/McGraw-Hill.
  2. "Advanced Engineering Dynamics" by J.H. Ginsberg, Cambridge, UK: Cambridge University Press.
  3. "Classical Mechanics" by Herbert Goldstein, Charles P. Poole Jr., John L. Safko, Pearson.
Out of the above books, #1 (book by Baruh) is the preferred text. However, if you are unable to get a copy of that book, #2 (book by Ginsberg) is a reasonable alternative. #3 (book by Goldstein) is an excellent reference text that you should definitely try to have a copy of.

Software and Programming Languages:
In this course we'll use MATLAB (alternatively, Octave) and Mathematica. Both are available for your use through Lehigh University's LTS: https://software.lehigh.edu/install/ . While I will run a few simple in-class tutorials on these software, if you have never used either of these before, it is expected that you'll self-teach yourself the basics of these tools.

Grading Policy:
  • There will be weekly homeworks assigned, which you will not need to turn in and on which you'll not be graded. It is however important that you practice the problems assigned in the homeworks in order to properly learn the class material and be able to do well in the projects and the exams.
  • There will be a project assigned every two to three weeks, which you'll need to turn in. The projects will contain one or two problems that you will have about two weeks to work on. The projects will be worth 30% of the total course credit.
  • There will be one midterm exam, which will be worth 30% of the total course credit.
  • The final exam will be worth 35% of total course credit.
  • The remaining 5% of the total course credit will be on class participation.

More details

Course Schedule:

The following is a tentative / partial schedule, which will be updated with details as the class progresses.
 
The "additional reading" section numbers are from the text book by Hiam Baruh (HB) and the book by Herbert Goldstein (HG). Note that the material from the books are meant to be purely supplementary. Your notes from the class should be your primary reference. There may be topics that we did in class which you may not find in the books, and likewise there may be topics in the books that we skipped. Homeworks, projects and exams will be based on what we do in class, and not what's there in the section numbers mentioned from the books.
 
Day Tentative List of Topics Comments/Remarks



Mon, Jan 23 Coordinate systems (cartesian, cylindrical, spherical), velocity & acceleration, Jacobian, degrees of freedom, constraints (holonomic, non-holonomic); [additional reading: 1.5, 4.2, 4.3 of HB]

Wed, Jan 25  Holonomic & non-holonomic constraints, examples; Configuration spaces; Kinetics (Newton's laws of motion), review problems, [additional reading: 1.4, 1.6, 3.2, 3.3, 4.2, 4.3 of HB]

Sun, Jan, 29 Last Day for Web Registration, Last Day to Add without instructor permission
Mon, Jan 30 Conservation laws (momentum, energy, angular momentum) for single and multi- particle systems; Equilibrium & stability; [additional reading: 1.7, 1.8, 3.3-3.5, 3.12 of HB]    

Wed, Feb 01 review problem, variable-mass system; Vibration (under-damped, over-damped, critically-damped oscillations) [additional reading: 1.7, 3.6, 1.9 of HB]

Fri, Feb 3 Last Day to add/drop without a "W"
Mon, Feb 06 Forced Vibration, impulse response; [additional reading: 1.11, 1.10 of HB]

Wed, Feb 08 Forced vibration example; vibration of multi-d.o.f. undamped systems, modes of vibration [additional reading: 1.11, 5.5 of HB]

Fri, Feb 10 Last Day to select OR cancel Pass/Fail
Mon, Feb 13 vibration of multi-d.o.f. undamped systems, modes of vibration; [additional reading: 5.5 of HB]

Wed, Feb 15 vibration of multi-d.o.f. undamped systems with damping; forced vibration of multi-d.o.f. systems [additional reading: 5.5, 5.6, 5.7 of HB]

Mon, Feb 20 Introduction to Calculus of Variations; Euler-Lagrange Equations, examples [additional reading: 2.1, 2.2 of HG]
Project 1 due
Wed, Feb 22 Euler-Lagrange Equations, examples; Derivation of Lagrangian for Newtonian mechanics (Hamilton's principle, virtual displacement) [additional reading: 2.1, 2.2 of HG; 4.8 of HB]

Mon, Feb 27 Lagrange's equation of motion; Generalized forces [additional reading: 4.8, 4.9 of HB]

Wed, Mar 01 Constrained Lagrange's equation of motion; Lagrange multipliers [additional reading: 4.10 of HB]

Mon, Mar 06 Insights into constrained Lagrange's equation of motion; Solving a set of differential-algebraic equations [additional reading: 4.10 of HB]

Wed, Mar 08 In-class midterm exam (open book, open notes)
Mon, Mar 13 Spring Break
Wed, Mar 15
Mon, Mar 20 Example of a constrained system, application of constrained Lagrange's Equations; General method for solving a set of differential-algebraic equations [additional reading: 4.10 of HB]

Wed, Mar 22 Computation of constrained Lagrange's equations using Mathematica; Numerically solving the set of differential-algebraic equations arising from constrained Lagrange's equation [additional reading: 4.10 of HB]
Project 2 due
Mon, Mar 27 Generalized momenta; Jacobi Integral; Hamiltonian Mechanics [additional reading: 5.8, 5.11 of HB]

Wed, Mar 29 Hamiltonian Mechanics; spatial rotions [additional reading: 5.8, 5.11 of HB]

Mon, Apr 03 Spatial rotations, Direction cosines, rotation matrices, Euler angles, axis-angle representation of rotation [additional reading: 2.4, 7.5 of HB]

Wed, Apr 05 axis-angle representation of rotation, Rotation matrix corresponding to an axis-angle reprsented rotation, angular velocity, Relationship between angular velocity and rotation; velocity & acceleration of points on a rotating & translating rigid body; center of mass [additional reading: 4.7, 4.7 of HG; 2.7, 7.2, 6.2, 7.7.5 of HB]

Mon, Apr 10 Linear and Angular momentum of rigid bodies; Moment of inertia matrix, transformation under rotation of coordinate frame -- moment of inertia tensor; principal moments of inertia [additional reading: 8.2, 8.3, 6.2, 6.3, 6.4, 6.5 of HB]

Wed, Apr 12 Interconnected rigid bodies -- additivity of angular velocity; body-fixed fame; velocity and acceleration for points on interconnected rigid bodies (velocity/acceleration relative to body-fixed frames, identification of Coriolis and centripetal terms); Example problem; Force equation for rigid bodies [additional reading: 2.6, 2.7, 2.8, 7.8, 7.9 of HB]

Mon, Apr 17 Angular momentum of rigid bodies, rate of change of angular momentum (in body-fixed frame), Moments of forces and couples, Moment equation in body-fixed frame -- Rigid body dynamics [additional reading: 8.5 of HB]
Project 3 due
Tue, Apr 18 Last Day to withdraw with a "W"
Wed, Apr 19 Rigid body dynamics example problem [additional reading: 8.5 of HB]

Mon, Apr 24 Rigid body dynamics example problem, work done by moments and kinetic energy of rigid bodies [additional reading: 8.5, 8.9 of HB]

Wed, Apr 26 work done by moments and kinetic energy of rigid bodies; Lagrangian and Lagrange's equations for rigid bodies, example problem [additional reading: 8.9, 8.10 of HB]

Mon, May 01 Dynamics of a spinning top -- derivation using Lagrange's equations, analysis of the motion and stability of a spinning top; constrained system example (rolling disk) [additional reading: 8.10 of HB]

Wed, May 03 Review
Project 4 due
Fri, May 05 Last day for May masters candidates to electronically upload thesis and deliver final paperwork to the Registrar's Office
Wed, May 10 -- Mon, May 15
Final exam Take-home exam due at noon on Mon, May 15.


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