Homework and syllabus, evolving
MAE 6700, Spring 2013
Homework due at the lecture occuring at least 6 days after
the posting date. If this is ambiguous, use the later date if
It's OK to hand in HW just once per week. Save all returned and improved homeworks to hand in at the end of the semester.
Date of lecture listed. Solutions should be self-complete. That is, they should include a brief restatement of the question.
Jan 22, Tuesday: Euler's theorem (rigid motion is a translation plus a rotation about an axis).
HW 1) Euler's thm.
Jan 24, Thursday: Geometric addition of rotations. Vector formula for rotation using vectors.
2) Successive rotations examples
3) Race between a wheel and a block (Q type), v1.
Jan 29, Tuesday: Rotation is linear, rotation is independent of reference point. Matrix formula for rotation.
4) Matlab code for rotations
5) Make a 3D graphic of rotation about an axis.
Jan 31, Thursday: Euler angles, Matrix formula for rotation (cont'd), dyads. (Posted Feb 1.)
6) Vector vs matrix formula for rotations
7) Find axis from rotation matrix.
Feb 5, Tuesday: Various forms for rotation: R. Commutivity of small rotations. Introduction to angular velocity.
8) Wheel and block again (Q type), v2
Feb 7, Thursday: More rotations
Feb 12, Tuesday: More rotations, Intro to 3D dynamics
Feb 14, Thursday: 3D dynamics of rigid objects.
9) Find center of mass ang mom eqn from general AMB
10) Block vs wheel again, again. This time more theoretically (Q type). v3.
11) Exercise on rotation notation.
12) Escape velocity (Q type).
Feb 19, Tuesday: Euler equations for motion of one rigid object in space. Evolution of angular velocity.
13) Integration of Euler eqs to discover stability, numerically.
14) General motion of a rigid object in space. Use equations in the fixed basis.
Feb 21, Thursday: More on rigid object. Stability of rotation about various principle axes. Also, axisymmetric object. Discussion of degradation of $R$ matrix and need for Graham-Schmidt or Euler angles or something.
Feb 26, Tuesday: Calculus of variations and intro to Lagrange Equations. Guest lecture by Todd Murphey.
Feb 28, Thursday:Moment of inertia tensor (and its symmetries). Also calculation of I for a complicated object.
March 5 Tuesday: Harmonic Oscillator eqn using Murphy method. Guest lecture by student Spyros (Andy in Zurich)
March 12, Tuesday: Special motions of axisymmetric objects.
March 14, Euler Angles revisited, relation between rates of change and angular velocity.
15) Steady motion of disk on a plane. Find all solutions with no slip, no friction, and both (no slip and no friction), use algebraic equations, not ODE solutions.
March 26, Tuesday: Analytical dynamic. D'Alembert's principle, constraints, workless constraints.
16) Steady precession of a top. What is the slowest speed where such motion is possible near upright.
March 28, Thursday: Analytical Dynamics and Lagrange Equations. Principle of Least Action.
April 2, Tuesday: Derive Lagrange eqs from F=ma. Including non-conservative forces.
April 4, Thursday: Lagrange equations derivation continued.
17) Q-type question. Disk on table cloth.
April 9, Tuesday: Axioms of mechanics. The big thing is the independence, or not of Angular Momentum Balance. Principle of the lever implies all of mechanics. Problems with some classical assumptions.
April 11, Thursday: Deriving mechanics from the principle of the lever. Intro to rolling disk (with bad figure).
April 16, Tuesday: Rolling disk, again more carefully. Setting up governing equations.
18) Find, solve and animate solutions for a rolling disk. 3D.
April 18, Thursday: Class puzzle problem. The ice skater and the pole. Everyone worked at the board the whole time.
19) Find, solve and animate solutions for a spinning top. 3D.
April 23, Tuesday: Intro to the Lagrange multipliers; discussed the idea of constraints w/r/t the sleigh. Chatterjee notes are good.
April 25, Thursday: Lagrange multipliers for non-holonomic systems. determined "Qi" for the sleigh and generated Q1...Q3.
April 30, Tuesday: Beam vibrations with lagrange equations.
19) Find approximate solutions for first few modes of a vibrating beam. Length is L, mass m, stiffness EI. Half way down is point mass m and hanging from that with spring k another point mass m. Use a value of 1 for all parameters so you can compare solutions with other students.
May 2, Thursday: 3D double pendulum (2 DOF, hinges at arbitrary angles)