Class Summary Page

for

MATH 570

Mathematical Modeling

  • Tuesday, September 2. We spent the lecture part of the period discussing the basics of modeling. A list of modeling aphorisms was read. We gave a "definition" of a model and of a mathematical model, and in the context of studying aircraft wings gave examples of both non-mathematical models (a modern example might be a scale model in a wind tunnel, while a 300 year old model would be Sir George Cayley's study of trout cross-sections) and mathematical models (a formula for the lift force was given). We discussed how the construction or selection of a mathematical model compared to other methods of studying some real-world phenomenon (e.g., real-world observations, experiments, and simulations). We looked at a schematic for the mathematical modeling process. In the laboratory, we began to work in groups on deciphering a mock-ancient tablet written in some unknown language.

  • Thursday, September 4. We discussed different types of mathematical models: Predictive vs. Descriptive (or Axiomatic). Within the realm of predictive models, we saw that models could be further classified as Deterministic or Probabilistic, and as Continuous/Analytic or Discrete/Combinatorial. The example of studying population growth indicates that there is no one "correct" way to model a particular problem, and that the modeler has to make some choices about how (s)he is going to proceed. A handout desribing four modeling methodologies was distributed. We began to illustrate the modeling process with a pseudo-historical treatment of Galileo's theory of gravity. Of particular interest, in re-creating an early unsuccessful attempt of his, we carefully solved the exponential growth-decay equation: x'=kx. At the end of this discussion, we discussed limits on the utility of Galileo's model that x'=32t for a body released from rest at time 0. In the laboratory, we returned to the problem of deciphering the tablet that we had started on Tuesday. Homework set #1 was assigned. Finally, we chose the first two MCM problems that we would study; the assignments of papers to students can be found here.

  • Tuesday, September 9. We continued modeling falling objects. A stroboscopic photo of a falling golf ball was analyzed and found to be consistent with the hypothesis that its velocity gained 32 feet/second in every additional second it fell; we used Maple to plot the data points together with a parabolic curve (a handout with these plots was distributed on Thursday 9/11). We then turned to rain drops. The model does not seem to work so well here; it predicts that raindrops fall much faster than they do in practice. (For example, the impact velocity of a drop falling 1000 feet would be 175 miles/hour!!) We improved this model by adding a term (proposed by Stokes) that takes into account the deceleration due to the drag of the air, and obtained results for small droplets that seemed to be consistent with the experimental data in a table that was distributed. In the laboratory, we began work in pairs on a new modeling problem concerning the proper placement of lights in order to maximize illumination.

  • Thursday, September 11. We completed our modeling of falling objects. We first noticed that the Stokes correction predicted a terminal velocity of 668 miles per hour for a raindrop of diameter .01 feet. A velocity-squared drag correction was proposed for large droplets, and a quick "back-of-the-envelope" calculation was performed for a drop of diameter .02 feet to compare the predicted time to fall 1000 feet with the observed time. The agreement was "reasonable;" a more careful analysis was assigned for homework. A general air drag model was introduced and we commented (see the handout) on how it reduced to the Stokes model and the velocity-squared model for certain sizes of drops. Up until this point we had spent the week studying the implications and limitations of Galileo's assumption that gravity was the only force at work. We concluded by discussing another assumption: that the body fall from a "moderate height." Introducing Newton's law of Gravitation, we first showed that one could safely ignore the motion of the Earth toward a raindrop, and instead just focus on the motion of the raindrop towards the Earth. We also showed that the change in the gravitational acceleration (due to the change in the distance between the bodies) was negligible for a body falling from 1000 feet. In the laboratory, we resumed working on our illumination modeling exercise. After class, Homework set #2 was posted on the web.

  • Tuesday, September 16. The class period today was devoted to a continuation of the illumination modeling problem. A handout was provided which pointed out that how to incorporate an aspect-angle correction to the inverse-square law approach that the groups had been using.

  • Thursday, September 18. During the "lecture" portion of the class, we began discussing the problem of a flagpole subject to vibrations such as might be experienced during an earthquake (see section 2.1 in the course text). We started with a boundary value problem (BVP) for the "beam equation," which is a fourth-order partial differential equation (PDE). At first sight, it appeared that there were 6 parameters in the problem: E (Young's modulus), rho (density of the pole), k (radius of gyration), l (length of the pole), a (amplitude of the vibration), and omega (frequency of the vibration). After we scaled the two spatial variables (distance along the pole and the displacement there) and the time to convert these to dimensionless variables, we saw that the solutions to the equuation would depend only on a single dimensionless parameter: J = E*k^2/(rho*omega^2*l^4). In advance of solving the BVP we are already able to make some observations about the behavior of solutions to this problem with different choices of the underlying parameters. For instance, we see that in order to keep the same behavior it is necessary to scale the radius of gyration (which is proportional to the diameter of the pole) as the square of the length -- so while short poles can be slender, tall poles need to be thick. In the laboratory, we continued working on the illumination modeling exercise.

  • Tuesday, September 23.

  • Thursday, September 25. After class, Homework set #3 was posted on the web.

  • Tuesday, September 30. We began our study of winning solutions to the MCM. Dan Curtis reported on the Cornell University solution to the Coal Tipple problem, and Linn Damon presented the West Point solution to this problem.

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    This page last modified 10/30/97