Analytical Methods in Dynamics and Vibrations (Mech 425, Spring 2019, Lehigh U.)
Jul 03, 2019
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:
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. |
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| Day | Tentative List of Topics | Comments/Remarks |
| Mon, Jan 21 | Coordinate systems (cartesian, cylindrical, spherical), velocity &
acceleration, Jacobian [additional reading: 1.5, 4.2, 4.3 of HB] |
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| Wed, Jan 23 | Degrees of freedom, constraints (holonomic, non-holonomic); Holonomic & non-holonomic constraints, examples; Configuration spaces; Kinetics (Newton's laws of motion) [additional reading: 1.4, 1.6, 3.2, 3.3, 4.2, 4.3 of HB] |
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| Sun, Jan, 27 | Last Day for Web Registration, Last Day to Add without instructor permission | |
| Mon, Jan 28 | Holonomic & non-holonomic constraints review; Kinetics (Newton's laws of motion), review problem 1.4, 1.6, 3.2, 3.3, 4.2, 4.3 of HB] |
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| Wed, Jan 30 |
Kinetics review problem; Conservation laws (momentum, energy, angular momentum) for single and multi- particle systems; [additional reading: 1.7, 1.8, 3.3-3.5, 3.12 of HB] |
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| Fri, Feb 1 |
Last Day to add/drop without a "W" | |
| Mon, Feb 04 | Review of work-energy principle, example problem; Equilibrium & stability; variable-mass systems [additional reading: 1.7, 1.8, 3.3-3.5, 3.12 of HB] |
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| Wed, Feb 06 | variable-mass system; Vibration (under-damped, over-damped, critically-damped oscillations) [additional reading: 1.7, 3.6, 1.9 of HB] |
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| Fri, Feb 8 |
Last Day to select OR cancel Pass/Fail | |
| Mon, Feb 11 | Forced Vibration, impulse response; [additional reading: 1.11, 1.10 of HB] |
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| Wed, Feb 13 | Forced vibration example; vibration of multi-d.o.f. undamped systems, modes of vibration [additional reading: 1.11, 5.5 of HB] |
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| Mon, Feb 18 | vibration of multi-d.o.f. undamped systems, modes of vibration; modal coordinates [additional reading: 5.5 of HB] |
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| Wed, Feb 20 | Introduction to Calculus of Variations; Euler-Lagrange Equations, examples [additional reading: 2.1, 2.2 of HG] |
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| Mon, Feb 25 | 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] |
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| Wed, Feb 27 |
Lagrange's equation of motion; Generalized forces [additional reading: 4.8, 4.9 of HB] |
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| Mon, Mar 04 | Constrained Lagrange's equation of motion; Lagrange multipliers [additional reading: 4.10 of HB] |
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| Wed, Mar 06 | Insights into constrained Lagrange's equation of motion; Solving a set of differential-algebraic equations [additional reading: 4.10 of HB] |
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| Mon, Mar 11 | Spring Break | |
| Wed, Mar 13 | ||
| Mon, Mar 18 | 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] |
Tentative week for midterm exam |
| Wed, Mar 20 | 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] |
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| Mon, Mar 25 | Generalized momenta; Hamiltonian Mechanics [additional reading: 5.8, 5.11 of HB] |
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| Wed, Mar 27 | Hamiltonian Mechanics; spatial rotions [additional reading: 5.8, 5.11 of HB] |
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| Mon, Apr 01 | Spatial rotations, Direction cosines, rotation matrices, Euler angles, axis-angle representation of rotation [additional reading: 2.4, 7.5 of HB] |
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| Wed, Apr 03 | 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] |
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| Mon, Apr 08 | 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] |
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| Wed, Apr 10 | 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] |
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| Tue, Apr 12 | Last Day to withdraw with a "W" |
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| Mon, Apr 15 | 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] |
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| Wed, Apr 17 | Rigid body dynamics example problem [additional reading: 8.5 of HB] |
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| Mon, Apr 22 | Rigid body dynamics example problem [additional reading: 8.5, 8.9 of HB] |
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| Wed, Apr 24 | ||
| Mon, Apr 29 | ||
| Wed, May 01 | ||
| Fri, May 03 | Last day for May master's candidates to electronically upload thesis & deliver final paperwork to Registration & Academic Services | |
| May 7-10 |
Final exam (take-home) |
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