not so simple harmonic motion

Not So Simple Harmonic Motion

Physics Lesson Plan

by

William G. Daly

Physics Teacher

Collins Hill High School, Gwinnett County, Georgia

Summer Teacher Experience Program -

Utilizing Physics at

Georgia Tech

June 3 - July 26, 2013

PROBLEM

Simple Harmonic Motion is a requirement of all high school physics courses, from algebra based introductory physics through calculus based AP® Physics C. As time dependence is introduced, the ideas of a traveling wave and standing waves enter the curriculum. Such oscillatory motion is generally restricted to pendulums, spring, vibrating strings and open pipes.

Although the importance of wave equations, boundary conditions and mathematical representations are discussed, there is a large gulf between rudimentary example required by the curriculum and many fundamental and interesting real world applications.

Students are often curious about examples and the importance of oscillatory motion beyond the simple examples required of the curriculum. Typical enrichment to begin to answer this curiosity includes musical instruments and their overtones, vibrating structures (airplane wings, skyscrapers and Tacoma Narrows Bridge), wine glasses and Tibetan Singing bowls, just to name a few.

However, when students inquire about how to analytically approach such commonplace examples, they are usually met with how intractable the problem is and that computational methods have to be employed, such as finite element modeling, to analyze many such problems. This gap between simplified examples and real world, relevant applications is often less than satisfying to many students.

The goal of this lesson plan is to not only meet the minimum requirements of the curriculum, but to also enrich the lesson by giving students insight into and experience with commonplace, but mathematically complex, situations involving harmonic motion and wave phenomena. Students should come away from the lesson capable of dealing simple harmonic motion, feeling a little more acclimated to the utility and approaches for considering differential equations and hopefully excited that there are means address mathematically complex physical problems from diverse disciplines. Furthermore, it is natural for students to compartmentalize their academic courses. One of the jobs as an educator is to help students see a broader connection among diverse disciplines. At least some lessons, if not most, should begin with the very narrow topic at hand and let it blossom into diverse connections to as many student experiences and interests as possible.

Abstract

This plan consists of five phases, covering at least 6 days of instruction. It intends to enrich lessons regarding harmonic motion, wave motion and boundary value problems. This is especially appropriate to the curriculum and extended amount of time (that is, depth) available in AP Physics C, and probably for the new AP Physics 1, mechanics taught as a yearlong high school course. In AP Physics C, students are expected to set up, recognize or solve very elementary differential equations representing the introduction of wind resistance and oscillatory motion in the presence of vector fields. Although the new AP Physics 1 will not require differential equations, it is expected that students recognize the importance and meaning of the solutions, as well as the effect of physical constraints such as changes in mass, displacement, boundary conditions and forces. Due to mathematical complexity, only rudimentary cases are presented. These cases stop short of giving students the full importance and a hope of obtaining the means to deal with more complex, but important, problems in their academic and professional futures. Mathematics of many realistic cases go beyond students’ mathematical skill, but removes from discussion all but the most rudimentary cases. Introducing important cases, but saying they are intractable, at best stimulates curiosity, but at worst runs the risk of losing student interest. In the past, computational methods such as finite element analysis (FEA) were mentioned as an alternatives, but without allowing students to grasp what that really means. Advancement of mathematical software and GUIs (graphical user interfaces) provide a low cost (free software) means of bringing commonplace, but mathematically very complex, problems within reach to augment the typical lessons on simple harmonic motion and the mathematical description of waves.

Alignment with Standards

Georgia [1]

A. SCSh4. Students will use tools and instruments for observing, measuring, and manipulating scientific equipment and materials.

B. SCSh5. Students will demonstrate the computation and estimation skills necessary for analyzing data and developing reasonable scientific explanations.

C. SCSh8. Students will understand important features of the process of scientific inquiry.

D. SP4. Students will analyze the properties and applications of waves.

College Board AP® Physics C - Mechanics [2]

I. NEWTONIAN MECHANICS

A. Kinematics

1. Motion in one dimension

c. Students should know how to deal with situations in which acceleration is a specified function of velocity and time so they can write an appropriate differential equation and solve it for v(t) by separation of variables, incorporating correctly a given initial value of v.

2. Motion in two dimensions

a. Students should understand the general motion of a particle in two dimensions so that, given functions x(t) and y(t) which describe this motion, they can determine the components, magnitude, and direction of the particle’s velocity and acceleration as functions of time.

b. Students should understand the motion of projectiles in a uniform gravitational field.

AP® Physics C - Mechanics (continued)

B. Newton’s laws of motion

2. Dynamics of a single particle (second law)

a. Students should understand how Newton’s Second Law applies to an object subject to forces such as gravity, the pull of strings, or contact forces

b. Students should understand the effect of drag forces on the motion of an object, so they can:

1. Find the terminal velocity of an object moving vertically under the influence of a retarding force dependent on velocity.

2. Describe qualitatively, with the aid of graphs, the acceleration, velocity, and displacement of such a particle when it is released from rest or is projected vertically with specified initial velocity.

3. Use Newton’s Second Law to write a differential equation for the velocity of the object as a function of time.

4. Use the method of separation of variables to derive the equation for the velocity as a function of time from the differential equation that follows from Newton’s Second Law.

5. Derive an expression for the acceleration as a function of time for an object falling under the influence of drag force.

AP® Physics C - Mechanics (continued)

F. Oscillations and Gravitation

1. Simple harmonic motion (dynamics and energy relationships) a. Sketch or identify a graph of displacement as a function of time,

and determine from such a graph the amplitude, period and frequency of the motion.

b. Write down an appropriate expression for displacement of the form A sint or A cost to describe the motion.

c. Find an expression for velocity as a function of time.

d. State the relations between acceleration, velocity and displacement, and identify points in the motion where these quantities are zero or achieve their greatest positive and negative values.

e. State and apply the relation between frequency and period.

f. Recognize that a system that obeys a differential equation of the form dx2/dt2 = -x must execute simple harmonic motion, and determine the frequency and period of such motion.

i. Calculate the maximum displacement or velocity of a particle that moves in simple harmonic motion with specified initial position and velocity.

j. Develop a qualitative understanding of resonance so they can identify situations in which a system will resonate in response to a sinusoidal force

AP® Physics 1 [3]

Enduring Understanding 1.A: The internal structure of a system determines many properties of the system.

Essential Knowledge 1.A.1: A system is an object or a collection of objects. objects are treated as having no internal structure.

Enduring Understanding 3.B: Classically, the acceleration of an object interacting with other objects can be predicted by using SF=ma

Essential Knowledge 3.8.3: Restoring forces can result in oscillatory motion. When a linear restoring force is exerted on an object displaced from an equilibrium position, the object will undergo a special type of motion called simple harmonic motion. Examples should include gravitational force exerted by the Earth on a simple pendulum, mass-spring oscillator.

Enduring Understanding 6.D: Interference and superposition lead to standing waves and beats.

Essential Knowledge 6.D.1: Two or more wave pulses can interact in such a way as to produce amplitude variations in the resultant wave. When two pulses cross, they travel through each other; they do not bounce off each other. Where the pulses overlap, the resulting displacement can be determined by adding the displacements of the two pulses. This is called superposition.

Essential Knowledge 6.D.2: Two or more traveling waves can interact in such a way as to produce amplitude variations in the resultant wave.

Essential Knowledge 6.D.3: Standing waves are the result of the addition of incident and reflected waves that are confined to a region and have nodes and antinodes. Examples should include waves on a fixed length of string, and sound waves in both closed and open tubes.

Essential Knowledge 6.D.4: The possible wavelengths of a standing wave are determined by the size of the region to which it is confined.

Objectives

This plan is primarily based on the standards for AP® physics C. Mastering most of the material in this section is expected of AP® physics C students, except for italicized topic, which are enrichment. However, this material is suitable for other physics courses, for students without a calculus background. Solutions of equations, models and lab will still provide the desired enrichment by removing reference to or mastery of differential equations.

The overall objective is to Scaffold the lessons on simple harmonic motion and wave phenomena toward the use of differential equations to handle important real life examples of resonance, standing waves and eigenmodes. This lesson has components ranging from the standard curriculum content, an enrichment component and a lab component.

The first simple differential equation students are exposed to in AP® physics C Mechanics deals with the exponential solution to problems that do not neglect wind resistance. Although students should be mathematically equipped to solve the actual case where drag depends on the square of velocity, they are typically only required to set up this differential equation and evaluate initial and final conditions (rest and terminal velocity). Although students should be able to solve this equation for velocity as a function of time, doing so requires a time consuming partial fraction expansion, so students are typically asked to solve for velocity versus time using separation of variables ("u" substitution) where the simplifying assumption is made, however unrealistic, that drag is directly proportional to velocity.

Beyond their first exposure to differential equations with a decaying exponential solution, student are confronted with writing and solving second order differential equations for physical situations such as springs, pendulums, physical pendulums and torsional pendulums for which there are oscillatory solutions.

Wave phenomena, boundary conditions and mathematical solutions are also taught, but without having to solve partial differential equations involving waves. At this point, enrichment in this lesson is a departure from the required curriculum for even the simplest cases, such as a standing wave on a stretched string. Calculus students should be able to following the standard development of this equation [4].

It is at this point where a level of curiosity or frustration may set in, in which students realize the possible complexity of oscillatory motion. It is often asked that if even a simple vibrating string is this complex, what hope is there of dealing with a complex shape. In the past, a reply to this question is that circular boundaries and more complex boundaries are often considered using computational means, such as Finite Element Modeling (FEM) and the conversation often stops there with students not having the slightest grasp of what that means.

The conclusion of this lab seeks to use recently available opens source FEM to model the eigenmodes of a circular membrane, accompanied by a lab demonstration of such a structure. While not part of the core curriculum, students should still recognize the effect of such a membrane constrained by boundary condition, have an intuition about the effect of mass and tension on the vibrations and be able to discuss deviation of the demonstration from the idealized FEM model. To help insure success for both the student and instructor, a detailed outline of the FEM process is included in appendix A.

In summary the objectives are:

Present first order differential equations o Write the equation from the physical description o Evaluate initial and final conditions o Solve for physical parameters as a function of time

Present second order differential equations o Write the equation from the physical description o Solve for physical parameters as a function of time

Literature review of the development of a simple wave equation Present topics on more complex shapes and boundaries

o Lab demo - Eigenmodes of an irregular boundary o FEM tools to evaluate complex problems

Assessment / Rubrics

Assessment rubrics for differential equations at the required level are well documented at the College Board website, AP Central at the following link:

http://apcentral.collegeboard.com/apc/members/exam/exam_information/8039.html

Questions and rubrics are listed within each topic may be download from AP Central. Additional rubrics for the enrichment phase of this lesson are listed as required.

1. First Order Differential Equation: Write and/or solve a first order differential equation for situations in which wind resistance may not be neglected. Recognize initial / final conditions

and solve for equation of motion as a function of time for equations of the form + 𝑘𝑣 = 0.

Evaluate initial and final conditions for equations of the form + 𝑘𝑣 = 0.

Assessments and rubrics:

2013 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTION: MECH2

(block sliding to a stop-idealized).

2010 AP® PHYSICS C: MECHANICS FREE-RESPONSE QUESTIONS: MECH1

(coffee filter lab-realistic)


(skier - friction - idealized)


(wind resistance - idealized)


(wind resistance - realistic)


(ballistic pendulum - bullet drag - idealized)



http://apcentral.collegeboard.com/apc/members/exam/exam_information/8039.html











2. Second Order Differential Equation: Write and/or solve a second order differential equation for situations which result in oscillatory motion. Recognize the solutions and meaning

of leading coefficients for equations of the form + 𝑘𝜃 = 0 and − 𝑘𝜃 = 0.



(simple harmonic motion - spring - integration)


(torsional pendulum - differential equation)


(physical pendulum - differential equation)


(simple harmonic motion)


(simple harmonic motion-nonlinear spring)







3. Literature review of wave equation for strings and solutions for open pipes.


Students should, without necessarily being able to solve a given differential equation, understand that a standing wave:

- Arises due to boundary conditions

- Is an interference of waves travelling in opposing directions

- Has different representations depending on the boundary conditions.

- Students should be able to write these solutions and calculate wave characteristics

- Students should recognize the importance and effect of different boundary conditions , all harmonics for identical boundaries, odd harmonics for opposing boundaries.

4. Finite Element Model of Vibrating Membrane (see Appendix A):

Apart from modeling complex surfaces, students should recall:

- the relationship among mass, volume and density

- the effects of physical constants and forces on the oscillatory motion of pendulums, spring, pipes and strings.

Although students are not responsible for any the numerous differential equations used by the FEA programs such as CSC Elmer, COMSOL Multiphysics and CST Design Studio, students should know which of these differential equations is being used in the finite element solver. For example, mechanical deformation used during the Step Up 2013 research uses the Navier

equation 𝜌 ̅ − ∇𝜎 = 𝑓 ̅, where 𝜌 is density, 𝜎 is the stress tensor, �̅� is displacement, and 𝑓 ̅is

for per volume. Even though students are not responsible for this equation, given prior

knowledge along with a brief description of what the stress tensor gradient term means, students should be able to recognize this equation as Newton's second law.

5. Demonstration

This phase of the lesson is almost purely enrichment. Students are not responsible for any of the solutions, other than to know in intuitive terms, an intuition that should have been developed in the simpler cases:

- objects vibrate at more than one frequency

- standing wave arise out of constraining boundaries

- the ways physical characteristics such as mass and tension effect these vibrations.

Materials and Supplies

This lesson has five phases:

1. Physical systems requiring first order differential equations, resulting in exponential solutions.

2. Physical systems requiring second order differential equations, resulting either in exponential or oscillatory solutions.

3. Literature review of simple case of a wave equation.

4. Computational analysis of complex physical system.

5. Lab / demonstration of complex physical system, at instructor discretion:

- Lab demonstration

- Extracurricular student centered project (analysis and demonstration of alternate shapes)

This plan assumes lab demonstration.

Materials and supplies for:

1. First order differential equations: No supplies are required. Instruction and guided practice through example problems (see Assessment / Rubric section).

2. Second order differential equations: No supplies are required. Instruction and guided practice through example problems (see Assessment / Rubric section).

3. Literature review: Literature sources for wave equation solutions. The enrichment aspects should be made clear to student; that is, they are not responsible for applying this equation at their level. But students should recognize the similarities to Newton's second Law and recognize this as the source of the solutions that they are responsible for.

Possible source:

Fourier Series [4]

Using Differential Equations to Model a Vibrating String [5]

The second source may be more favorable because of the historical context offered in the introduction along with example MATLAB code for simulations.

4. Computational analysis:

Windows PC(s) with administrative rights o Number of computers depends on the extent to which the analysis is instructor

centered (1 PC) or student centered (multiple). o Administrative rights are required to install open source software. o Windows For example, the work displayed here was running on a low cost

laptop, Windows8, 64bit, AMD quad core processor, 8GB RAM. That the open source software runs fine on , Windows8, 64bit seems to provide some confidence regarding its stability on alternate, earlier operating systems.

Open Source Software (cost $0.00) and associated online documentation: o CAD model development and meshing: gmsh

http://geuz.org/gmsh/ o Grid conversion, Graphical User Interface, Solver, Postprocessing: Elmer7

Elmer7.0-rev6064-2013-02-15.exe, from http://sourceforge.net/projects/elmerfem/files/latest/download?source=files

o Visualization software: Paraview http://www.paraview.org/

5. Lab / demonstration

1 each - PC with Sound Card Oscilloscope installed ($0 for academic use) o http://www.zeitnitz.de/Christian/scope_en

1 each - 5 Gallon plastic bucket 2 each 3.5" x 0.75" x 12" particle board (1.75" x 0.75" notch in center) 1 each - 5" speaker (the size is not critical, use pc speaker or recycle from old AV

equipment - the hardware show scavanged the speaker and amplifier from a discarded "boom box")

1 each - amplifier (unless amplified pc speakers are being used. 1 each - cable with 3.5mm jack to plug into PC audio output. 1 each - garbage bag (approximately 39 Gallon) 1 each - duct tape 4 each - ball point pen barrels for spacers 4 each - Deck screws (3.5") 1/4 cup - sand

http://sourceforge.net/projects/elmerfem/files/latest/download?source=files

http://www.paraview.org/

Plan

1. First order differential equations, solve for velocity in the presence of "drag":

Ideally, students will have executed the College Board "coffee filter" lab. Free Body Diagram with drag force, Fd=kv (simplified), F=kv2 (realistic) Second Law: F=ma Populate the sum: mg - Fd = ma

Simple case, force proportional to velocity: 𝑚 = 𝑚𝑔 − 𝑘𝑣

Show that the mks units of 𝑘 must be , be sure they are clear that this is not the same

thing as the "k" in Hooke's law. Terminal velocity, vT constant, a=0: 0 = 𝑚𝑔 − 𝑘𝑣 , so 𝑣 =

Find velocity as a function of time using "u" substitution: 𝑣(𝑡) = (1 − 𝑒 )

Check initial and final conditions for agreement with 𝑣 = 0 and 𝑣 =

Sketch and label graph of 𝑣(𝑡) Integrate / differentiate to find displacement / acceleration as functions of time: 𝑥(𝑡) =

𝑡 + 𝑒 − 1 / 𝑎(𝑡) = 𝑔𝑒

Check that initially 𝑥(0) = 0, 𝑎(0) = 𝑔 and that finally, 𝑥(𝑡 → ∞) → ∞ and 𝑎(𝑡 → ∞) → 0

Sketch and label graphs of displacement and acceleration.

2. Second order differential

Student should realize first that the first order differential equation for velocity is the second order equation for displacement, so they have already dealt with one second order equation

By inspection, + 𝑘𝑥 = 0, must have a solution of the form 𝑥(𝑡) = 𝐴𝑐𝑜𝑠(𝜔𝑡) or

𝑥(𝑡) = 𝐴𝑠𝑖𝑛(𝜔𝑡). I always ask, what are the only functions that can be twice differentiated and become the opposite of themselves, so that they can sum to zero?

By inspection − 𝑘𝑥 = 0, must have a solution of the form 𝑥(𝑡) = 𝐴𝑒± , where the

negative exponent are the usual physical solutions. A similar question applies, what are the only functions that can be twice differentiated and become the themselves, so that they can subtract to zero?

Students should substitute the sin and cos solutions into + 𝑘𝑥 = 0 to discover for

themselves what the required values of A and k are (A is irrelevant, k=𝜔 ).

mg

Fd

+y

Students should analyze an ideal spring: o starting from the equilibrium FBD: o Find displacement as a function of time

Emphasize that o k in Hooke's law is not the same k as in the above differential equation o in equilibrium, 𝑚𝑔 = 𝑘𝑥

Stretching from equilibrium, xo, to some new displacement, x: o 𝐹 = 𝑘(𝑥 − 𝑥 ) o ∑𝐹 = 𝑚𝑎

o 𝑚𝑔 − 𝑘(𝑥 − 𝑥 ) = 𝑚

o Since 𝑚𝑔 = 𝑘𝑥 , −𝑘𝑥 = 𝑚

o Students should solve, assuming a solution of the form 𝑥(𝑡) = 𝐴𝑐𝑜𝑠(𝜔𝑡) (or

𝑥(𝑡) = 𝐴𝑠𝑖𝑛(𝜔𝑡)), and discover, not surprisingly, that 𝜔 = .

3. Literature review

Divide the literature into manageable steps (one to two steps), assign to student groups of two or three and have students present their steps to the class.

Class discussion o What assumptions were made? Are they reasonable? o What are similarities between the wave equation and differential equations

encountered so far? o What are the similarities between the wave and the Navier Equation in section 4,

𝜌 ̅ − ∇𝜎 = 𝑓.̅ This may be the appropriate point at which to discuss the

meaning of the gradient of the stress tensor and the similarities between the Navier equation and Newton's second law.

o What are the differences between the wave equation and the Navier equation. o What are the similarities and difference between solutions of the equations from

section 2 ( + 𝑘𝑥 = 0) and the wave equation (both are periodic, but only one

is travelling).

4. Computational analysis [6,7]:

It should be apparent to students that it does not take much of a deviation from idealized cases to quickly increase the complexity of differential equations to an enormous, often intractable degree, even for cases of great interest. How does one answer a student's curiosity about how real life problems of this type are handled, whether one finds empirical

+y

mg

F=kxo xo

solutions or just gives up when the mathematical complexity grows to excess. Empirical solutions are often unacceptable; for example, the first F-22 raptor cost $67 billion, with a low price of $150 million for successive copies, not allowing much room for experimentation. One answer is to resort to finite element analysis, without students having a grasp of what that really means. Either as an instructor centered presentation, or as a student long term project, open source software is now available to execute such analyses with easily available hardware.

So can we solve for stresses and vibrations for all but the simples structures? Just as a Tibetan singing bowl screams at given pitch due to the scraping of a wooden mallet, would an F-22's wing vibrate wildly due to the roar of its engines? It would be nice to have a hint of this before building the first one. We have studies the standing waves travelling on a vibrating string. Is it possible to study the possible standing waves, the eigenmodes, for a two dimensional membrane? A circular membrane requires the use of Bessel functions to describe the vibration state. A brief look at how this is done mathematically can be discouraging. Finite element analysis yields to this complexity by breaking up the problem into very small elements, numerically solving the problem on a much smaller, simpler scale, and then aggregating the result as an approximation of the actual solution. The process is as follows (see Appendix A for detailed steps for a circular membrane):

Computer Aided Design File (CAD)

Create/Refine Mesh

GMSH.exe

Convert Mesh

ElmerMesh.exe

Set Up Solver Parameters

ElmerGUI

Run Solver

ElmerGUI

Visualization

Paraview

Paraview displaying the first 9 eigenmodes

n=1 f1=82Hz

n=2 f2=353Hz

n=3 f3=354Hz

n=4 f4=943Hz

n=5 f5=945Hz

n=6 f6=1215Hz

n=7 f7=2011Hz

n=8 f8=2013Hz

n=9 f9=2790Hz

5. Lab / demonstration

Setup:

The bottom of the bucket needs to be mostly open for acoustical venting, except for cross braces. Ideally, the speaker should be mounted securely inside the bucket with the entire bucket assembly mounted to a solid open frame. Angle Styrofoam absorbers / reflectors beneath the bucket appear to enhance the modes if the bucket is set on a table top. It is advisable to tension the membrane shortly before the demonstration, unless an actual drum is used that will maintain tension indefinitely.

Laptop running Soundcard

Oscilloscope Program (or Signal Generator)

Zeitnitz Soundcard

Oscilloscope

Headphone Output

AUDIO AMPLIFIER

5" SPEAKER

OR ... AMPLIFIED PC SPEAKER

Pull trash bag taut, attach to sides with duct tape

Thin layer of sand

Using either a signal generator, or the sweep feature within the signal generator of the sound card oscilloscope, sweep the pitch from 10Hz to 1KHz and observe the motion of the sand, alternating between idle and oscillating in patterns similar to FEA output. With a random scattering of sand on the member, the following are the patterns that emerge at f=42Hz, f=200Hz, f=223Hz. The first image show the sand distribution before resonance.

f=0Hz f=42Hz f=200Hz f=223Hz The still photos are not as dramatic as the live demonstration. At most frequencies, the sand is completely idle, but be very animated at very specific frequencies. For example, activity is seen from 222Hz to 224Hz, but is very idle outside of this range. Follow up discussion:

Students should discuss what they expect if tension is changed. What would happen if the size or density of the membrane is altered. Speculate do the n=2 and n=3 modes seem more prominent than the others (Hint: the

bucket is open at one end and is about 40cm in length).

Summary

This plan seeks to enrich lessons on equations of motion when simplifying assumptions have been remove, such as incorporating drag forces and making no assumptions about where harmonic equations originate from a mathematical view point. The lesson is broken into 5 broad areas.

The first two areas are most suitable for AP® Physics C Mechanics. It is also suitable to AP® Physics C Electricity and Magnetism in the sense that the mathematical form of some of the differential equation apply to those encountered when dealing with circuits.

The third topic, is a literature review to see similarities between harmonic motion differential equations and the wave equation. This topic is purely enrichment and is offered as a preview of more advanced mathematics. As such it is suitable only for AP® Physics C.

The fourth topic seeks to answer questions regarding how scientists and engineers proceed due to the fact that differential equations for all but the simplest become increasingly difficult to solve. While computational methods are one stock answer to this question, students do not have a good grasp of what that means. Recent advances in open sources software put such finite elements analysis tools within reach of the high school classroom, either as an instructor led example, or student centered project.

The final topic is a lab demonstration that parallels the finite element example, showing the eigenmodes of a circular membrane with readily available, low cost materials.

As added enrichment, the topic of oscillatory motion touches far reaching disciplines from rocket science to biological research. Although following are just a few topics as suggested further study, students should be encouraged to find and present areas involving oscillatory motion that mirror their personal interests.

Music - Numerical Solution of Nonlinear Differential Equations in Musical Synthesis [8].

Biology / Medicine - Differential Equations and Mathematical Biology [9].

Photography / Arts - Applied partial differential equations: visualization by photography (other chapters of this book touch on many of the other topics listed here) [10]. Electrical and Computer Engineering - A Practical Guide to 3D Electromagnetic Software Tools [11].

Civil Engineering - Tacoma Narrows Bridge: Lessons From the Failure of a Great Machine [12].

Architecture - Engineering the World’s Tallest – Burj Dubai [13].

Aeronautical Engineering - NASA Experience with Pogo in Human Spaceflight Vehicles [13].

Acknowledgements

Prof. Madhavan Swaminathan, John Pippin Chair in Electromagnetics - Advisor

Mixed Signal Design Group Graduate Students - Mentors Munmun Islam Kyu Hwan Han Jianyong Xie

Dr. Leyla Conrad, Outreach Director Nirvana Edwards, Outreach Coordinator Georgia Tech for support of the Step-Up Program National Science Foundation for support of STEM initiative grant # CMMI-1129918

References

[1] "Physics Curriculum." Georgia Performance Standards. Georgia Department of Education, 13 July 2006. Web. 17 July 2013. <https://www.georgiastandards.org/standards/Pages/BrowseStandards/ScienceStandards9-12.aspx>. [2] "Physics B, Physic C: Mechanics, Physics C: Electricity and Magnetism - Course Description."AP Central. College Board, Sept.-Oct. 2012. Web. 17 July 2013. <http://apcentral.collegeboard.com/apc/public/repository/ap-physics-course-description.pdf>. [3] "AP Physics 1 and AP Physics 2." Advanced Placement Program. N.p., n.d. Web. 18 July 2013. <http://advancesinap.collegeboard.org/math-and-science/physics>. [4] Tolstov, Georgi P., and A. Silverman. Richard. Fourier Series. New York: Dover, 1962. Print.

[5] Hegdal, Boden. "Using Differential Equations to Model a Vibrating String."Http://online.redwoods.edu/instruct/darnold/DEProj/sp05/bodenmike/Paper3.pdf. Math, Science, and Engineering, College of the Redwoods, 19 May 2005. Web. 17 July 2013. <http://online.redwoods.edu/instruct/darnold/DEProj/sp05/bodenmike/Paper3.pdf>. [6] C. Geuzaine and J.-F. Remacle, Gmsh: a three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering, Volume 79, Issue 11, pages 1309-1331, 2009 [7] "CSC Elmer Tutorials." CSC Elmer Documentation. IT Center for Science, 30 May 2013. Web. 17 July 2013. <http://www.csc.fi/english/pages/elmer/documentation>. [8] Yeh, David. "Numerical Solution of Nonlinear Differential Equations in Musical Synthesis."Numerical Solution of Nonlinear Differential Equations in Musical Synthesis. Center for Computer Research in Music and Acoustics, Stanford University, 11 Mar. 2008. Web. 17 July 2013. <http://chess.eecs.berkeley.edu/pubs/403/chesssem20080311.pdf>. [9] Jones, D. S., and B. D. Sleeman. Differential Equations and Mathematical Biology. London: Allen & Unwin, 1983. Print. [10] Emmer, Michele, and Alfio Quarteroni. Mathknow: Mathematics, Applied Sciences and Real Life. Milan: Springer, 2009. Print. [11] Vandenbosch A. E., A Practical Guide to 3D Electromagnetic Software Tools, Microstrip Antennas, Prof. Nasimuddin Nasimuddin (Ed.), 0, InTech, 2011 <hGuy A. E. Vandenbosch and Alexander Vasylchenko (2011). <http://www.intechopen.com/books/microstrip-antennas/a-practical-guide-to-3d-electromagnetic-software-tools>

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Appendix A: Detailed Input for Eigenmode Analysis of Circular Membrane Before engaging in student center activities, to avoid student frustration, it may be wise to have students execute this analysis verbatim to gain efficacy with the process before adaptation to a unique design.

Open gmsh

File - new - bucket

Geometry - Elementary entities - Add - Point

(0 , 0 , 0) press e

(0 , 0.133 , 0) press e, press q

Geometry - Elementary entities - Add - Circle arc

click (0 , 0.133 , 0),

click ((0 , 0 , 0),

click (0 , 0.133 , 0), press q.

Geometry - Elementary entities - Add - Plane surface

select circular boundary, press e, then press q

Geometry - Physical Groups - Add - Line

select circular boundary, press e, then press q

Geometry - Physical Groups - Surface - Line

select dotted line in circle, press e, then press q

Mesh - 2D

Your should have a figure that looks like this:

For simple model like this, and because I we are looking for higher order modes, use a finer mesh.

Tools - Options - Mesh - General - Element size factor = 0.02

Click 2D again to get this mesh:

save mesh

Exit gmsh

There should now be a file bucket.msh

Next use ElmerGrid to create Elmer mesh files:

open a command window, go to the directory containing ElmerGrid (a command line application) and type the following command, which happens to also be the location of my mesh file:

ElmerGrid 14 2 bucket - autoclean

(14 means GMSH input, 2 means Elmer output). A directory should have been created with the four elmer grid files - you may need to have a directory created ahead of time, directory name "bucket" in this case).

Next open ElmerGUI (a windows application)

We first need to append the elastic plate solver:

File - Definitions - Append ... navigate to C:\Elmer7\bin\edf-extra\ elasticplate.xml ... open

Now load the mesh:

file - load mesh - navigate to the "bucket" directory, click OK

Finally, we need to set up the type of analysis and model parameters: Model Setup Simulation Type = Steady state Steady state max. iter = 1 Apply

Model

Equation Add Name = Bucket Apply to bodies = 1 Elastic Plates Active = on Edit Solver Settings Solver Specific Options Eigen Analysis = on Eigen System Values = 10 Linear System Direct = on Umfpack Add OK

Model Material Add Name = Ideal Apply to bodies = 1 General Density = 1000.0 Elastic Plates Youngs Modulus = 1e9 Poisson ratio = 0.3 Thickness = 0.001 Tension = 0.0 Add OK

Model BoundaryCondition Add Elastic Plates Deflection 1 = 0.0 Deflection 2 = 0.0 Deflection 3 = 0.0 Name = Fixed Apply to boundaries = 1 Add OK Model Equation Bucket Results Output Active

Body 1 Edit Solver Settings General After simulation ] Solver specific options Output Format = vtu Binary Output checked Save Geometry Ids checked Sif - Generate File - Save project - bucket Start Solver This simulates vibrations of a circular membrane, radius 0.133m, with a density of 1000 kg/m3, under no tension. The results window should display simulation run parameters and, when the solver completes, the following eigenmodes: EigenSolve: 1 ( 82.535912503774910 , 0.0000000000000000 ) EigenSolve: 2 ( 353.62052488275327 , 0.0000000000000000 ) EigenSolve: 3 ( 353.97403898850945 , 0.0000000000000000 ) EigenSolve: 4 ( 942.99893575509623 , 0.0000000000000000 ) EigenSolve: 5 ( 945.39603372242141 , 0.0000000000000000 ) EigenSolve: 6 ( 1215.5710658910809 , 0.0000000000000000 ) EigenSolve: 7 ( 2011.2506691836024 , 0.0000000000000000 ) EigenSolve: 8 ( 2012.7389961644169 , 0.0000000000000000 ) EigenSolve: 9 ( 2790.4219895460215 , 0.0000000000000000 ) EigenSolve: 10 ( 2797.6033445963662 , 0.0000000000000000 )

Open Paraview (windows application)

Open file case0001.vtu (or whatever name was selected in the ElmerGUI Solver specific options)

A list of deflection modes should appear in the Properties window on the left.

Hit Apply (upper left in Properties window)

In the lower left under coloring, select which of the eigenmodes are to be displayed.

not so simple harmonic motion

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