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    Modeling Scenario
    280

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    167

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    3-045-RampBounce-ModelingScenario
    Students build two projectile motion models (1) a one-dimensional model for a vertically falling ball from a fixed distance until it hits an inclined ramp and (2) a two-dimensional projectile motion model of the ball bouncing off the ramp.
    projectile motioninclined rampcoefficient of restitutionoptimal anglebounce
    Potential Scenario
    122

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    39

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    2008-Luis_San_Andrés-Dynamic Response of Second Order Mechanical Systems with Viscous Response Forces
    Walk through the cases in context of second order linear constant coefficient differential equation with driving function
    driving functionviscousmechanical system
    Article or Presentation
    169

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    47

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    2015-Brian_Winkel-Informed_Conjecturing_of_Solutions_for_Differential_Equations_In_a_Modeling_Contex
    We examine (1) first order exponential growth or decay and (2) second order, linear, constant coefficient differential equations, and show the advantage of learning differential equations in a modeling context for informed conjectures of their...
    exponential growthsolutionslinear systemsdecayconjecture
    Modeling Scenario
    247

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    208

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    3-110-MilitarySpringMassApplication-ModelingScenario
    The is a collection of different scenarios for the shock system of a trailer. In each scenario, students will transform the shock system of a trailer into a second-order differential equation, solve, and interpret the results.
    dampingresonancebeatsspring-mass systemLaplace TransfomDirac Delta function
    Modeling Scenario
    293

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    183

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    3-075-RLCCircuits-ModelingScenario
    We introduce the basics of RLC circuits, defining the terms of inductance, resistance, and capacitance in a circuit in which an induced voltage created a current running through these devices.
    resistancecircuitvoltageinductancecapacitancecurrentKirchhoff's Law
    Modeling Scenario
    243

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    123

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    3-040-FirstPassageTime-ModelingScenario
    We apply the notions of dampedness to second order, linear, constant coefficient, homogeneous differential equations used to model a spring mass dashpot system and introduce the notion of first passage time through 0 value with several applications.
    oscillatordampedunderdampedfirst passagefirst passage timespring mash dashpot
    Modeling Scenario
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    356

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    4-039-FallingDarts-ModelingScenario
    we develop, solve, and analyze a second order differential equation model for free fall incorporating air resistance. Students solve the model using two methods -- reduction of order and separation of variables, and method of undetermined...
    optimizationresistanceterminal velocitymuzzle velocitydrag coefficientfree fallair drag
    Modeling Scenario
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    406

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    3-060-DataToDifferentialEquation-ModelingScenario
    Students use knowledge of second-order linear differential equations in conjunction with physical intuition of spring-mass systems to estimate the damping coefficient and spring constant from data.
    Association & Data Fittingmassspring-mass systemspring\total distancenumerical differentiation
    Modeling Scenario
    211

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    239

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    4-036-AltitudeDependentGravity-ModelingScenario
    When projectiles are way above Earth's surface gravity's changes become important when dealing with projectiles at high altitudes. We lay out an approach for such a case which is a second-order differential equation.
    agravityprojectilelinearizationltitufealtitude
    Modeling Scenario
    221

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    168

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    4-035-ParEstSteadyState-ModelingScenario
    Students estimate parameters in a second order, linear, ordinary differential equations through analysis of the steady state solution. By applying a driver we can collect data in terms of the parameters and estimate these parameters
    driversteady statesum of square errors
    Modeling Scenario
    381

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    472

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    4-060-CircuitTuner-ModelingScenario
    We present essential definitions and laws for the study of simple RLC electrical circuits and build a differential equation model using these notions. We describe how such a circuit can be used to tune a radio to a certain input frequency.
    resistanceciircuittuned mass tunerfrequencyKirchhoffvoltagegainfrequency responseamplitudes Voltage LawinductancecapacitorI am currently working in redesining an existing microbiology course. I would like to be part of ed
    Modeling Scenario
    333

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    162

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    3-102-SpringMassDamped-ModelingScenario
    Students build a model based on their perceptions of what the solution should look like for a simple spring mass system with damping.
    frequencymassoscillationstiffnessspring constantspring\dampedSingle Degree of Freedom Systemcoefficient
    Modeling Scenario
    990

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    2110

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    3-030-SecondOrderIntro-ModelingScenario
    We outline the solution strategies involved in solving second-order, linear, constant coefficient ordinary differential equations, both homogeneous and nonhomogeneous and offer many application and modeling activities.
    parameter estimationnon-homogeneousSpanishfrequencybuoyancyinverse problemthrustunderdampedfirst passagespring mash dashpotrealcomplexhomogeneoussuperpositionparachuteoverdampedcritically dampedbuoyquadratic equationrootsrocket flight
    Article or Presentation
    174

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    40

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    2008-Keith_Landry-Brian_Winkel-Peak_Frequency_Responses_and_Tuned_Mass_Dampers
    We present two applications of systems of ordinary differential equations concerning excitation of a structure through resonance and (1) peak frequency responses with stabilization of the structure by means of a (2) Tuned Mass Damper (TMD).
    vibrationresonancetuned mass damperpeak frequency responseexcitation
    Article or Presentation
    143

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    50

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    2014-C_W_Groetsch-S_A_ Yost-Vertical Projection_in_a_Resisting_Medium_Revelations_on_Observations_of_Mersenne
    This article, inspired by a 17th-century woodcut, validates empirical observations of Marin Mersenne (1588–1648) on timing of vertically-launched projectiles for a general mathematical model of resistance.
    resistanceprojectile motionprojectilefluidMersenne
    Potential Scenario
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    1989-R_Blickhan-Spring Mass Model For Running-Hopping
    A simple spring—mass model consisting of a massless spring attached to a point mass describes the interdependency of mechanical parameters characterizing running and hopping of humans as a function of speed.
    spring massjumpjumping
    Potential Scenario
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    55

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    2003-Givens-Bonfima-Direct observation of normal modes in coupled oscillators
    We propose a simple and inexpensive method to directly observe each normal mode of a system of coupled oscillators, as well as to measure its corresponding frequency, without performing Fourier analysis or using expensive apparatus.
    frequencyresonancecoupled oscillator
    Potential Scenario
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    52

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    2007-Bellomo-De_Angelis-Delitala-Lecture Notes on Mathematical Modelling in Applied Sciences
    The Lectures Notes correspond to the first part of the course devoted to modelling issues to show how the application of models to describe real world phenomena generates mathematical problems to be solved by appropriate mathematical methods.
    spring massLegendre polynomialsintegro-differential equationskinetic theory
    Potential Scenario
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    44

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    1981-Geoffrey_Berresford-Differential Equations and Root Cellars
    This is a classic module from UMAP in which the heat equation in one dimension is fully developed by using the standard technique of measuring the heat flow in and out of a small element of mass and equating them at equilibrium.
    temperaturethermal conductivityheat flowthermal diffusivityroot cellar
    Article or Presentation
    184

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    56

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    2007-Joseph_Myers-Brian_Winkel-Modeling_a_Mystery_Circuit_in_Mathematics_Class_Using_Differential_Equations_Laplace_Transforms_and_Transfer_Functions
    We discuss the introduction, student discovery, and teaching of a system of differential equations which model a circuit.
    circuitLaplace transformtransfer functionmystery circuitvoltage frequencytwo loop circuit