This Project LINKS module gives a comprehensive overview of the process of modeling a spring mass system without outside forces. As such it can also be used as an outstanding example of modeling processes in general. Mathematical models are justified and derived. Subsequently the prediction of the model is compared with actual measured data. Discrepancies are then explained and improvements in the determination of the parameters are used to obtain better accuracy for the model. Java applets show the predicted behavior as well as a set of 10 different real measurements. This module perfectly exemplifies the Project LINKS ideal of bridging the gap between mathematical theory and real-world applications.
Users interested in forced harmonic oscillators should consider the module on forced spring mass systems on the same site.
Please also see the reviews of the following individual Project LINKS modules:
The modules are intended to be used in a studio classroom setting with instructor guidance.
Technical Requirements:
Should run in any Java-enabled browser.
The modules are intended to be used in a studio classroom setting for a day or two, and with a professor and TA available. Thus many questions are open-ended. There are parts that are appropriate for independent home-study. Self study is not impossible, but the user should be aware of this underlying idea. Also see ASSUMPTIONS on the front page. href="http://links.math.rpi.edu/index.html">http://links.math.rpi.edu/index.html
Identify Major Learning Goals:
From the module objectives:
Understand the importance and flow of the dynamic system investigation process.
Enhance your modeling and engineering judgment by gaining experience in making simplifying assumptions. Understand the effects of these assumptions on the predictive capabilities of the resulting model.
Apply the dynamic system investigation process to the study of a spring mass system. Obtain an appropriate differential equation describing the motion of the system,
solve the equation, and simulate the motion.
Compare the measured behavior of a physical system to the behavior predicted by the mathematical model. Based on the comparison, determine whether the model captures the important characteristics of the actual behavior.
Target Student Population:
Students taking physics, engineering or mathematic classes that involve the modeling of a harmonic oscillator.
Prerequisite Knowledge or Skills:
From the module prerequisites:
? Newton's Laws
? Concepts of modeling and simulation
? Second order ordinary differential equations with constant coefficients
? Using numerical methods for solving ordinary differential equations
Content Quality
Rating:
Strengths:
This module provides the most comprehensive description of spring mass systems that the reviewers have ever seen. Great care is taken to connect the theory with experiments to check the strengths and weaknesses of the model. Moreover, many possible modeling assumptions are presented and shown how to be used or discarded. The following may not describe all the high points, but should capture many.Good explanation of the modeling process, its starting point and its goals. style="mso-spacerun: yes">
2. style='font:7.0pt "Times New Roman"'> In the determination of the spring constant the authors explain how to actually measure the constant, taking care of pretension, which theorists might ignore. Moreover the spring constant is found as the slope of a line for which multiple points are found, which gives greater accuracy and connects to data analysis.
3. style='font:7.0pt "Times New Roman"'> Assumptions made for derivations are stated very carefully and in great detail.
4. style='font:7.0pt "Times New Roman"'> Gravity is kept as an outside force until specifically eliminated by re-setting the origin.
5. style='font:7.0pt "Times New Roman"'> Various types of damping and how they are typically modeled are described.
6. style='font:7.0pt "Times New Roman"'> Excellent Java applet to show the behavior of a spring mass system for various choices of the parameters (in the ?predicted behavior? section).
Concerns:
The presentation of the numerical solution looks as if it is badly out of context.
In the first description of the actual dynamical behaviour (undamped track), one reviewer missed the description of how the behaviour was determined; in particular, that it was determined by an actual experiment. This might be stated more prominently, since one of the strengths of the site is the careful comparison between actual experimental results and theoretical predictions.
The abstract descriptions of the modeling process can be confusing on first use. A teacher?s hand may be needed here.
Potential Effectiveness as a Teaching Tool
Rating:
Strengths:
Applications connect the lab-type experiments and derivations to important real-life situations. Derivation as well as analysis are as complete as can be.
Excellent questions accompany the derivations. Students who wish to ?see it done? can click on the answer. In a more active setting the instructor can however stop the class at that point and make students answer the questions by themselves.
Concerns:
The comprehensiveness of the site can overwhelm a user that does not have the guidance of an instructor. This is however not a problem of the site. It?s real life that is complex.
In one case (the Mathematical Model page in the undamped track),
clicking on the link labelled ?derive? leads to a solution of the derivation, not the derivation activity. (Similar links on other pages lead to the relevant activity, so this is presumably not intended by the authors.)
Ease of Use for Both Students and Faculty
Rating:
Strengths:
Well-structured presentation, java applet for the harmonic oscillator works well.
Concerns:
In the applications section, clicking on the pictures (unlike what is promised) does not show videos.
The java applet in the ?trig identity? section did not work.
The ?assumptions? link on the Physical Model page (both tracks) seems to lead to a Shockwave file. Netscape showed the reviewers a blank space, and IE got as far as showing the Shockwave icon, but wouldn?t play the animation.
There seems to be a large variety of display and animation formats used in the module. This makes the module more cumbersome to use than it should be.
Creative Commons:
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