*b) Instead of using cosine and sine for the normal modes, use exponentials and do complex math to solve for the initial conditions. 6) A uniform hoop (a circular line) with mass m and radius R swings in the plane from a stationary frictionless pivot on the hoop at O.*Lecture 22 (with audio): Mon Oct 21 Topics: State Space, Matrix exponential ***Prelim 1: Oct 22. Prelim1, Solns1, Matlab for prob 1.**** Lecture 23: Wed Oct 23 Topics: Matrix exponential Lecture 24: Fri Oct 25 (Guest lecturer Ephrahim Garcia) Topics: Normal modes with damping Associated homeworks: (due Friday ): 1) Use the matrix exponential to solve the initial value problem for the general MDOF damped oscillator with given initial position and velocity. The center of mass G (the center of the circle) swings back and forth 90 degrees (from to the right of the hinge, to the left, and back and so on).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.

*b) Instead of using cosine and sine for the normal modes, use exponentials and do complex math to solve for the initial conditions. 6) A uniform hoop (a circular line) with mass m and radius R swings in the plane from a stationary frictionless pivot on the hoop at O.*

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.

Associated homeworks: (due Wed ): Lecture 30: Fri Nov 8 Topics: omega. c)When G is directly below the hinge what is the force on the hoop from the hinge (in terms of)?

Dynamics of one or more rigid objects in 2D (Lin Mom, Ang Mom, Kin Energy). Associated homeworks: (due Fri Nov 15): 1) Forced 3 masses. Parts (b) and (c) can be done without numerical integration.

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).

Check your result by using the same numbers and initial conditions as used in the sample from lecture.

If you need to look at the sample to do this, then do it again and again until you can do it from start to stop, free body diagrams, variable definitions and all, without looking at lecture notes or a sample.

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.

Associated homeworks: (due Fri ): Lecture 34: Mon Nov 18 Topic: Double pendulum on the computer ***Prelim 2: Nov 19.

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