MASSACHUSETTS INSTITUTE OF TECHNOLOGY
Department of Mechanical Engineering
2.010 Modeling Dynamics and Control III
Fall 2000

Notes for PS #4

• Here's the link for Tutorial Web-notes for HW #4.
  (These notes address the individual problems a bit more directly than the last notes, but they'll still only help you get started...)
  
  - Hopefully, some stuff on this page is somewhat useful (conceptually, for odd vs even mode shapes), too.
  
  
• Office hours will be held on Wednesday, Oct. 11, 3-6pm, Rm 1-390 [just upstairs from our usual classroom...] (The switch is being made to accommodate hours missed hours on Monday, Oct. 9, which is an institute holiday).
  
  
• To see some examples of odd and even mode shapes, copy the following two lines into a MATLAB window (at athena!):
  
  addpath /mit/2.010/www/psets/hw4_dir/tutor4_dir
  mode_movie
  
  If you are patient enough to wait for this matlab program to finish creating all the movie frames, it's relatively quick to replay the resulting movies, 'Modd' and 'Meven'. (Type 'help movie' in MATLAB for more information on playing movies, or just type 'replay_modes' to replay for a pre-defined length of time...)
  
  (Alternatively, you can copy the files:
  • /mit/2.010/www/psets/hw4_dir/tutor4_dir/mode_movie.m
  • /mit/2.010/www/psets/hw4_dir/tutor4_dir/movie4.m
  • /mit/2.010/www/psets/hw4_dir/tutor4_dir/movie5.m
  • /mit/2.010/www/psets/hw4_dir/tutor4_dir/replay_modes.m
  and then run your copy of the m-file in MATLAB.)
  
  The script will create a MATLAB MOVIE showing masses moving in some of their natural mode shapes. (It takes a while to create all the frames, but I wanted to run many cycles to emphasize that the different modes have, of course, different frequencies...)
  -Note the familiar plot under each picture showing the position, theta, along the whole axis. Hopefully, the movies clarify what these simple mode-shape plots represent.
  
  
• There were some good questions in class on Thursday (Oct. 5) about why the pop quiz question (about mounting a pendulum on a rotating wheel) resulted in different natural frequencies (of oscillation) when the pendulum was allowed to swing in the plane of the rotating disk vs perpendicular to this. I came up with an explanation that just uses basic trigonometry and an evaluation of the WORK done in either case to move the pendulum by a tiny angle 'a'.
  • In one scenario, the FORCE (as seen by the mass on the pendulum) seems always "vertical", so the work done is approximately the FORCE at radius 'L' times the 'change in height'. [Just like lifting a box onto a shelf, for instance; and very much like a pendulum swinging familiarly in the earth's gravitational field] In the reference frame of the MASS, the force vector just always seems vertical.
  • In the other scenario, the FORCE (as seen by the mass) always points radially inward, and the 'change in height' must be measured (using basic trig) as the change in distance with respect to the point at the center of rotation of the wheel.
  In either case, it's basically measuring the length of a normal dropped to the line of the wheel's axis, but depending on the plane in which the pendulum swings, this results in either a normal-seeming, vertical force 'field' or a less-commonly-modelled radial force field.
  
  Anyway, here are a few figures (pdf or ps file...) that hopefully demonstrate how you would calculate the 'change in height' for a given angle 'a' in either case. Plug in some numbers (in Matlab, for instance) to see that (for small 'a'), you need to do twice the work in the vertical case as in the radial one.
  
  Here is a mini-session in MATLAB which you can cut and paste to compare the work done in each case:
  
  >> a=.01; L=1;
  >> d_vert=L*(1-cos(a))
  
  d_vert =
  
  5.0000e-05
  
  >> d_rad=2*L - (sqrt((L*sin(a))^2 + (L+L*cos(a))^2))
  
  d_rad =
  
  2.5000e-05
  
  The idea to note is that moving through the same angle 'a' requires a factor of two less WORK in the radial case (and thus 'stores half the potential energy'). Since half the work is required as for the 'vertical' pendulum case, it's as though the radial pendulum only has an 'effective vertical-pendulum-length' of L/2 instead of length L.
  The period, T, of a pendulum is proportional to sqrt(L), so frequency varies as 1/(sqrt(L)). So:
  freq_rad = sqrt(2) * freq_vert
  
  
• Tutorial is now set for Friday, 1-2pm, Rm. 2-132.
  - On Friday, Oct. 6th, I will also be available in Rm. 2-132 from 2-3pm.
  
  
• The hand-out "Comments on Proportional (P) Controller Analysis (HW #3)" [given out in class, Oct-5] will have a link here as well [once I put it on the web].
  - THAT HAND-OUT IS NOW ON THE WEB: pdf-file or ps-file
  - Its purpose is to reiterate some concepts covered in last week's HW #3 Tutorial Notes [on gain margin and stability analysis for a closed-loop system] for systems of the type seem in HW #3.