Readings and Homework, Fall 2013
ME 4730/5730, Dynamics and Vibrations

----    This page was updated on 12/24/2013    ----

Please read the Homework policy a few times, once before doing each of the first few homework assignments.

Homework Solutions: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14

Planned coverage (what you should know by the end of the semester)

Ruina & Pratap (RP): The whole book (mostly review, but you should master all of it).

Taylor, Classical Mechanics:

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.

Other topics: Going a bit beyond the books above.

1) Using Matlab for simulations, plotting and animation.
2) Use of Angular Momentum balance to get equations of motion for complex systems.
3) Mechanisms (linked rigid objects)
4) DAE f (Differential Algebraic Equations) formulation of equations of motion

Lecture schedule & homework assignments

(Guest lectures on Aug 30, Oct 2, 25, Nov 11,13,15)

If highlighted, notes are linked, courtesy of a student in the class. If you find errors in the notes, write to Andy.

Lecture 1 : Wed Aug 28
Topics: Intro. Position, velocity and acceleration in cartesian and polar coordinates.
Reading: RP Ch 2 and 11. Taylor Ch 1
Associated homeworks: (due Wed Sept 4, see HW policy):
Taylor 1.17b (use the definition of derivative)
2) Taylor 1.23
3) Taylor 1.45
4) 1D, no friction, no gravity. Consider a mass (m=1) connected to one end of spring (k=1) the other end of which is anchored and stationary. The mass is released from rest at t=0 with a displacement of 1. Find the position at t=2 pi two different ways and compare.
Analytic solution (see, e.g., RP Ch 9.3, Tongue Ch 1.2, Taylor Ch 5.2).
b) Numerical solution using self-coded Euler method with step size h=10^(-n) with n = 1,2,3,4,5,6, ... (as big as your computer can handle). Plot the log (base 10) of the error (difference between numerical and analytic) as a function of n. If you push your computer enough, you should find a local minimum in the error for some relatively large n. Why doesn't the accuracy simply get better and better as n gets bigger?
5) Optional. There are many interesting and useful problems at the end of Taylor Ch1. In principle you should be able to do almost all of them. Do any that please you.

Lecture 2: Fri Aug 30 (Guest lecturer: Anoop Grewal)
Topics: Newtons laws and ballistics.
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: F=ma and solution using Matlab with FEVAL (ODE solver in a separatefunction fom right-hand-side function).
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.
2.44 using your own Matlab code and Euler's method. For the last of the 3 plots compare the step size needed for 7 digit accuracy using Euler's method and (optionally) midpoint methods.
7) Optional. There are many interesting problems in the book. Do one or more of them that interest and appropriately challenge you.

Lecture 4: Fri Sept 6
Topics: Continuation of Matlab solution of ODEs using Euler's method and FEVAL
Readings: See Matlab code posted on Piazza.

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.
(lecture and homework will cover the multi-particle aspects of these chapters in coming days and weeks, so you can skim those now).
Associated homeworks: (due Wed Sept 18):
Some of these are from Fall 2012 homeworks (save the pdf for future homework assignments).
Problem 3 from fall 2012
2) Problem 4 from fall 2012
3) Problem 5 from fall 2012
4) 3.27 Show how well angular momentum is conserved in a nuerical solution (that you make) of the governing ODEs.
5) 4.4
6) 4.23

Lecture 7: Fri Sept 13
Topics: Conservative Forces

Lecture 8: Mon Sep 16
Topics: Multi-particle systems and the axioms of mechanics

Lecture 9: Wed Sept 18
Topics: Multiparticle systems and Angular Momentum
Multiparticle parts of Taylor through Chapter 4. RP Chapter 12 and appendix D.
Associated homeworks: (due Wed ): New Handout.
Handout #5. For advanced students or as a challenge. This is a redo from last week, but few seem to have done it completely..
2) Handout #6.
3) Handout #7 (a beautiful problem). Challenge bonus: using numerical root finding find another periodic motion of this system.
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.

Lecture 10: Fri Sept 20
Topics: Multiparticles and Konig's Theorem

Lecture 11: Mon Sept 23
Topics:      Intro. to Vibrations
Reading: RP Chapter 10.1-2, Taylor 5.1-2, Tongue 1.1-5.


Lecture 12: Wed Sept 25
Topics : More Single DOF vibrations
Associated homeworks: (due Wed Oct 2):
Taylor 5.13
Taylor 5.30
RP 10.1.4 (In c assume springs are relaxed when mass is in the middle)
4) RP 10.1.13
5) Tongue 1.4
Tongue 1.31
7) For those taking 5730: Use root finding to find periodic motions from Handout problem #5.
If you can, do likewise with the three body problem.

Lecture 13: Fri Sept 27
Topics: Root finding for finding periodic motions.
Reading: Matlab help and doc for root finding and minimization using

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):
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. Don't do it.
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 DoF 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

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).
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] = SpringmassNUM(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) [tarray xarray] = SpringmassMinv(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.
c) [tarray xarray] = SpringmassSqrtM(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).
2) Extra things for 5730 students.
a) Make the functions above work even if K is singular (has some modes with zero frequency).
b) Instead of using cosine and sine for the normal modes, use exponentials and do complex math to solve for the initial conditions.

Lecture 22 (with audio): Mon Oct 21
Topics: State Space, Matrix exponential

***Prelim 1: Oct 22. Covers through HW turned in on Oct 18. 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.
2) For some fairly complicated example compare your solution with ODE45 and make any observations about, say, time of computation.
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 25: Mon Oct 28
Topics: Modal damping.

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. Catch up on re-dos.

Lecture 28: Monday Nov 4
Introduction to kinematic constraints and DAEs

Lecture 29: Wed Nov 6
Topics: Constrained particles: bead on curved wire, a rigid triangle.
Associated homeworks: (due Wed ):

Lecture 30: Fri Nov 8
Topics: omega. Dynamics of one or more rigid objects in 2D (Lin Mom, Ang Mom, Kin Energy). Pendulum.
Associated homeworks: (due Fri Nov 15):
Forced 3 masses. Prob 37 from handout.
2) Dumbell in space. Prob 14 from handout.
3) Bead on wire. Prob 22 from handout, but instad of y=cx^2 use y=c(1-cos (b x))
4) Simple pendulum. Prob 11 from handout.
DAEs and simple pendulum. Problem 12 from handout.
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. 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). 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) can you prove that this force is more than, less than or equal to 45 degrees from vertical? c)When G is directly below the hinge what is the force on the hoop from the hinge (in terms of R, m and g)? Parts (b) and (c) can be done without numerical integration.

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. Covers through HW turned in on Nov 15.****
Prelim with Solutions.

Lecture 35: Wed Nov 20
Topic: Double pendulum on the computer
Associated homework: (due Wed Nov 27)
1) Simulate and animate a double pendulum. 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. 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.
Looking up all manner of online Matlab help is ok. Your solution should include the problem setup as
well as enough printed graphics you need to show that you have a working animation.
Write on the top of your HW and any graphics you hand in: "I did this
version of this write-up without looking at any double-pendulum examples."

Lecture 36: Fri Nov 22
Topics: Vibrating inverted pendulum.

Lecture 37: Mon Nov 25
Topics: Equivalence between gravity and accelerating reference frames.
Chaplygin Sleigh (grocery cart, braked car directional stability)

Lecture 38: Wed Nov 27 (We have class, University rules)
Topics: 5-term acceleration formula (moving and rotating frames)
Reading: RP Chapter 17 especially 17.3, read earlier if needed.
Homework: (Due Friday December 6 with final project on Dec 10-11)
1) Handout #16: braking stability
) Don't hand in. But do this early so you have practice.
Handout #19b: double pendulum with Lagrange Eqs (you may learn from the lecture example,
but what you hand in should be written by you without looking at it when you write).
Handout #20: Mass in slot in rotating turntable
Handout #27: Cart and pendulum by two methods.

*** Thanksgiving ***

Lecture 39: Mon Dec 2
Topics: Lagrange Eqs using Matlab symbolics, DAEs with multiple rigid objects

Lecture 40: Wed Dec 4
Topics: DAEs for many bodies, Rolling.

Lecture 41: Fri Dec 6
Topics: Generalized forces and Lagrange eqs.
Associated homeworks: (not handed in):

***Final Exam: Dec 19, 7-10 PM. Comprehensive. **** Exam, Solutions