Classical Mechanics Homework Solutions

Classical Mechanics Homework Solutions-48
You don't know if you can do this unless you actually do it.So don't say, "now I know" unless you have done it from start to stop on your own. Your solution should include the problem setup as well as enough printed graphics you need to show that you have a working animation.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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2) Use of Angular Momentum balance to get equations of motion for complex systems. Position, velocity and acceleration in cartesian and polar coordinates. Taylor Ch 1 Associated homeworks: (due Wed Sept 4, see HW policy): 1) Taylor 1.17b (use the definition of derivative) 2) Taylor 1.23 3) Taylor 1.45 4) 1D, no friction, no gravity. In principle you should be able to do almost all of them. Lecture 2: Fri Aug 30 (Guest lecturer: Anoop Grewal) Topics: Newtons laws and ballistics.

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).

Associated homeworks: (due Friday Oct 25 in class): 1) Write three matlab functions that solve the general spring-mass IVP (Initial Value Problem) a) [tarray xarray] = Springmass NUM(tspan, x0,v0, K, M) This can use ODE45 or your own ODE integrator, your choice.

It should work with arbitrary positive definite symmetric M and K matrices of any size.

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).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 14: Mon Sept 31 Topics: Sinusoidal forcing and resonance Reading: RP Ch 10.2, Tongue 2.1-8, Taylor 5.5-6 Lecture 15: Wed Oct 2 (Guest lecturer: Ephrahim Garcia) Topics: Logarithmic decrement, friction, measurement Reading: Tongue 2.10-12 Lecture 16: Fri Oct 4 Topics: Superposition, Fourier Series, Impulse response Reading: Taylor 5.7-8, Tongue 2.4-5, 3.1-4 (esp 3.2-3) Associated homeworks: (due Fri Oct 11): 1) Handout problem 9: two masses connected by a spring (not really a vibrations problem) 2) RP 10.2.4 3) RP 10.2.5 CANCELLED. 4) RP 10.2.11 5) Tongue 2.79 Lecture 17: Mon Oct 7 Topics: Intro to multi-DOF vibrations Readings: RP Ch 10.3, Tongue Ch 4.1-3, Taylor 11.1-3 Lecture 18: Wed Oct 9 Topics: Multi Do F cont'd Readings: RP Ch 10.3, Tongue Ch 4.1-3, Taylor 11.1-3 Lecture 19: Fri Oct 11 Topics: Forcing and vibration absorption Readings: Tongue 4.3-6 Associated homeworks: (due Friday Oct 18, Noon): New Handout.1) Handaout 23 2) Handout 25 3) Handout 40 4) Tongue 4.18 5) Tongue 4.60 6) Tongue 4.68 (easy) 7) Tongue 4.70 (570 students only) *** Fall break *** Lecture 20: Wed Oct 16 Topics: Normal modes using Inv(M) Lecture 21 (with audio): Fri Oct 18 Topics: Normal modes using sqrt(M).b) [tarray xarray] = Springmass Minv(tspan, x0,v0, K, M) This should use a superposition of normal mode solutions based in either (your choice) eig(K, M) or eig(M^-1*K) .Hint: to turn a diagonal matrix into a column vector use the DIAG command.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.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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