Classical Mechanics Homework Solutions

Classical Mechanics Homework Solutions-9
Lecture 13: Fri Sept 27 Topics: Root finding for finding periodic motions.

Lecture 13: Fri Sept 27 Topics: Root finding for finding periodic motions.Reading: Matlab help and doc for root finding and minimization using , etc.Lecture 31: Mon Nov 11 (Guest lecturer: Matt Kelly) Notes corrected on 11/29/2013 Topics: Double pendulum Lecture 32: Wed Nov 13 (Guest lecture: Hod Lipson) Topics: Design automation of kinematic and dynamical systems Lecture 33: Fri Nov 15 (Guest lecture: Mark Psiaki) Topics: Inverted-pendulum tight-rope walker with a balance beam.

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3) Mechanisms (linked rigid objects) 4) DAE f (Differential Algebraic Equations) formulation of equations of motion (Guest lectures on Aug 30, Oct 2, 25, Nov 11,13,15) If highlighted, notes are linked, courtesy of a student in the class. Reading: Taylor Ch 2.1-4, RP Ch 11.1-2 Associated homeworks: (due Wed Sept 11): 1) 2.12 2) 2.13 3) 2.21 4) 2.36 *** Labor Day, no class on Monday Sept 2*** Lecture 3: Wed Sept 4 Topics: and solution using Matlab with FEVAL (ODE solver in a separatefunction fom right-hand-side function).Lecture 5: Mon Sept 9 Topics: Change and conservation of Linear and Angular Momentum Reading: Taylor Ch 3 Lecture 6: Wed Sept 11 Topics: Energy Reading: By now you should know all of Taylor through Chapter 4 and RP Chapter 1,2,3,11& 14. Challenge bonus: using numerical root finding find another periodic motion of this system.(lecture and homework will cover the multi-particle aspects of these chapters in coming days and weeks, so you can skim those now). This is a redo from last week, but few seem to have done it completely.. 4) Handout #8 5) Taylor 3.20 (easy, soln is in RP section 3.2) 6) Bonus: Any problems from Taylor that interest and challenge you.Reading: Same as above look at Matlab samples from TAM 2030 (linked from Ruina's home page) and last year's ME 4735 (linked from this course home page).Associated homeworks: (due Wed Sept 11): 5) 2.22, Also, on your final plot (most any solution assumes a plot or two to go with it) show the analytic solution with linear drag.2) For some fairly complicated example compare your solution with ODE45 and make any observations about, say, time of computation. Prob 22 from handout, but instad of )) 4) Simple pendulum. a) When G is 45 degrees from straight down what is the direction of the force at O on the hoop from the hinge.3) Assume damping is a linear combination of the mass and stiffness matrices and solve the problem above using a superposition of normal modes and compare with the solution (for some example problem of your choosing) with solution by one of the two means above. Lecture 26: Wed Oct 30 Topics: Structural Vibrations (computer demo) Lecture 27: Friday Nov 1 Topics: Structural Vibes 2 Associated homeworks: (due Fri ): No new homework this week. Lecture 28: Monday Nov 4 Topic: Introduction to kinematic constraints and DAEs Lecture 29: Wed Nov 6 Topics: Constrained particles: bead on curved wire, a rigid triangle. b) can you prove that this force is more than, less than or equal to 45 degrees from vertical?Chapters 1 Newton laws (40 pages) 2 Projectiles, but not too much with magnetic fields (60 pages) 3 Momentum and Angular Momentum (20 pages) 4 Energy (55 pages) 5 Oscillations (70 pages) 7.1, 2, 3, 5 Elementary use of Lagrange Equations (20 pages) 8 Two body central force, but not too much on analytic solutions (30 pages) 9 Non-inertial reference frames, 2D only (40 pages) 10 Rotation of rigid objects, 2D only (50 pages) 11 Couple oscillations and normal modes (30 pages) 12 Nonlinear mechanics and chaos, just the elements (45 pages) [13 Collision theory, but not following this book (which emphasizes atomic collisions)] 16.1-3 String vibrations and 1D waves (15 pages) A.1-2 Diagonalizing matrices (5 pages) Tongue, Vibrations: Most of the book, overlaps some with vibrations concepts above.Not all will be in lecture, but you should master most of the book. 1) Using Matlab for simulations, plotting and animation. There are many interesting and useful problems at the end of Taylor Ch1.c) [tarray xarray] = Springmass Sqrt M(tspan, x0,v0, K, M) This should use a superposition of normal mode solutions based on the methods of lecture on 10/18 (using two changes of coordinates) d) For some fairly complex problem show that your three methods agree as well as they should.e) Animate the solution (using moving dots, circles or squares, your choice). a) Make the functions above work even if K is singular (has some modes with zero frequency).

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