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We place here all the materials in support of the SIMIODE Remote Teaching Module - Spring Design to Meet Specs at Minimum Costs. This Module is about second order, linear, ordinary differential equations.The class lesson starts with building a model of a spring-mass using Newton's Second Law of Motion and a Free Body Diagram; develops criteria for choice of spring constant to make the spring overdamped, and then ascertains the spring constant and attendant manufacturing costs to produce that spring constant in order to meet two specifications. The specifications are from same initial condition spring reaches given distance from static equilibrium in a given time at minimum cost.

This module contains

1) (Below and separate file in Supporting Docs) A brief Teaching Guide with an overview of the content and activity, including any necessary prerequisite material

2) Videos and materials to assign to students

3) An explanatory video for instructors on how to implement the module in an online format

We will offer a YouTube Version and SIMIODE Version in Supporting Docs. The YouTube version permits stop and start and bounce around while the SIMIODE version should be downloaded for full mobility.

4) (Included in the brief Teaching Guide below) An assessment guide

Additionally, we will be hosting live Q&A sessions in the future and posting FAQ guides.

For the Q&A session we will use ZOOM meeting software without recording, but we will post FAQ information based on Chat and meeting conversation. We will discuss the lesson experience and issues about teaching, interactions, and online experience.

The file Module-SpringCost is the file used for presentation in the Module while the file Module-Class-SpringCost is the actual lesson material suitable for use with class. The formats for both use LaTeX Beamer structure which permits production of PowerPoint like presentation using the mathematical typesetting capabilities of LaTeX. Both the original .tex file and the produced .pdf files are offered.

The class assignment is for the student to write up an informal flow of the modeling approach and to use the model to determine the spring constant of minimum cost which meets set specifications .

Upon completion of the lesson we will discuss the lesson experience and issues about teaching, interactions, and online experience.

Resources for this Module are in the Supporting Docs with originating Modeling Scenario **3-031-S-SpringCost** (Student Version) and **3-031-T-SpringCost**.

**Teaching Guide for SIMIODE Remote Teaching Module SpringCost**

Prepared by Brian Winkel, Director SIMIODE

**Overview of Content**

Source: Modeling Scenario in SIMIODE at **www.simiode.org**.

- 3-031-S-SpringCost
**https://www.simiode.org/resources/7553**the Student Version of the material in which only the STATEMENT of the problem is offered. - 3-031-T-SpringCost
**https://www.simiode.org/resources/7554**the Teacher Version of the material in which the STATEMENT and COMMENTS by the author are presented discussion solution strategies and pedagogical issues.

The activity involves the construction of the basic spring-mass-dashpot model for a spring with mass suspended by the spring as it experiences resistance to motion from internal friction and its own spring constant. This is done with a Free Body Diagram and an appeal to Newton’s Second Law of Motion.

For a given mass and resistance due to velocity of motion of the spring students are asked to determine the spring constant values which cause overdamping.

There is a cost associated with creating a spring with a given spring constant, the greater the stiffness the more expensive the spring. Students are then asked to create a lowest cost spring to meet certain specifications, namely return from a given stretched position from static equilibrium to a position closer to that static equilibrium in a set time. One can think of this as designing a shock absorber system for a fancy automobile to give a smooth ride and damp out oscillations due to road conditions quickly at lowest cost.

A complete description of the analysis is found in the Teacher Version Source above and in a Mathematica notebook with pdf version supplied.

We offer a **teacher YouTube video** in which we present an introduction and the lesson offered to students. We also offer a **student YouTube video** in which we provide a ready to use lesson for students with assignment as well. We place both videos here in Supporting Docs for download use.

**Prerequisite Material**

- This is quite self-contained, but perhaps an introduction to Free Body Diagrams and Newton’s Second Law of Motion would be appropriate, although this activity would also introduce, enhance, and strengthen understanding and use of these notions.
- Plotting of motion of the spring’s mass because of the solution of the differential equation model.
- Solving second order, linear, homogeneous ordinary differential equation using eigenvalue strategy and characteristic equation or more likely technology, e.g., software which students possess, SAGE, Maple, Mathematics, or Wolfram Alpha (
**https://www.wolframalpha.com/**).

This model could be used to introduce and motivate these prerequisite techniques as well.

**Nature of Activity**

Students create a second order, linear, homogeneous differential equation from some first principles. Then with only one parameter in play, namely the spring constant, they determine the parameter for an otherwise fixed situation which assures the spring will experience overdamping. They are asked to select the spring constant which meets two specifications (a) return to fixed distance from static equilibrium and (b) do so in a fixed time.

**Videos and materials to assign to students**

We direct students to the actual statement of the problem in Source item above: 3-031-S-SpringCost **https://www.simiode.org/resources/7553** - the Student Version of the material in which only the STATEMENT of the problem is offered.

We offer a pdf of the completed modeling activity from start to finish in the file Module-Class-SpringCost.pdf. The file is constructed using LaTeX Beamer and we also enclose the .tex file and all image files for customization, editing, and use by individual faculty.

**Explanatory video for instructors on how to implement the module in an online format.**

In the file Module-SpringCost.pdf, constructed using LaTeX Beamer, we present slides in which we lead the faculty through a development of the modeling activity. We also offer a video walk through of this material with an introduction to the teacher, a middle section from the student perspective, and a closing section as if we were conducting an online session for teachers to engage them and assist them in realizing the potential and use of this material.

**An assessment guide**

Each teacher has to decide how to use this material and hence how and what to assess. We offer here our Assignment (found in this material).

**Assignment**

- Explain what it means for a spring to return to within
*a*m of its static equilibrium of 0 in*b*seconds from*t*= 0. - Using cost function, Cost(
*k*) = 13.70 + 0.01*k*^{2}_find the cheapest spring with spring constant*k*which returns to within*a*= 0.1 m of its static equilibrium of 0 in*b*= 0.5 seconds from time*t*= 0. - Provide others a tool through which they can determine, generally, the cost function for a spring which is underdamped, returns to within
*a*m of its static equilibrium of 0, and does so within*b*seconds from time*t*= 0. - Provide a representation of level curve pricing in which for a fixed cost, say $600, all possible spring configurations (i.e. returns to within
*a*m of its static equilibrium of 0 and does so within*b*seconds from time*t*= 0) are displayed.

This could be an in-class lab in which all the students gain the experience of the full modeling experience to completion and the teacher asks them to organize, communicate, and write-up their understanding of the work as a homework assignment.

So, we would spend about 20 minutes in class as we let them work on the Free Body Diagram and differential equation building from Newton’s Second Law of Motion. The assignment would then be for a several days homework with opportunities for questions during the class of assignment and the due date class.

Questions (1) and (2), along with understanding of the notion of overdamped motion, form the heart of the write-up and we would assign 50% of the grade to these with attention to things like units, proper use of graphics, logical connection and flow, solution strategies, etc.

Question (3) is a bit technical to generalize (2) as they have to solve the equation for *t* when setting their model in *t* equal to specific data points and this would take a bit of technology, probably a solving command in a computer algebra system or something like Wolfram Alpha.

Questions (4) demands understanding of the fact that for many different situations and demands on our spring one could purchase a spring for $600. What are these situations and demands?

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