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    Modeling Scenario
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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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    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
    Modeling Scenario
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    3-031-SpringCost-ModelingScenario
    We assume students are familiar with overdamping and underdamping of a spring-mass-dashpot system. Students will apply this knowledge to model the interplay between spring constant, tolerance, and cost.
    designunit costspring constantstatic equilibriumspring masss dashpot
    Modeling Scenario
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    5-040-TunedMassDampers-PartII-ModelingScenario
    Studentsbuild mathematical models to mitigate dangerous swaying in structures using structural improvements called Tuned Mass Dampers (TMD). We model the motion of the original structure as a spring-mass-dashpot with stiffness replacing spring...
    structureresistancespring-mass-dashpotearthquakevibrationcontroldampingtuned mass damperstiffnessreduction of vibration
    Potential Scenario
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    2016-Pennell-Avitabile-White-Engineering Applications Differential Equations
    General discussion of an engineering based differential equations course with examples from RC Circuit, mass spring dashpot.
    circuitvoltagecapacitornondimensionalization
    Modeling Scenario
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    3-034-CarSuspension-ModelingScenario
    We examine the spring-mass-dashpot that is part of a car suspension, how the ride is related to parameter values, and the effect of changing the angle of installation. We model a ``quarter car'', meaning a single wheel.
    designspring-mass-dashpotspring constantunderdampingcar suspensionsuspension tolerancestatic equilibrium
    Modeling Scenario
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    4-050-ResonanceBeats-ModelingScenario
    We study what can happen when a pure oscillator (no damper) is driven by a forced vibration function which has the same or close to the same natural frequency as the system it is driving.
    Spring 2020 workshopvibrationtuningdriverresonanceforced vibrationundampedbeatsnaturl frequency
    Modeling Scenario
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    210

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    3-105-FrequencyResponse-ModelingScenario
    We describe the frequency response to a second order differential equation with a driving function as the maximum steady state solution amplitude and perform some analyses in this regard.
    transient dynamicscircuitspring massdriverfrequency responseamplitudesteady state
    Modeling Scenario
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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
    Article or Presentation
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    410

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    2020-TeachingModule-SpringDesignToMeetSpecsAtMinimumCosts
    We discuss a Modeling Scenario in which students are asked to design a spring mass system at minimum costs give relative costs of features of the spring.
    moduleoptimizationspring massmassspring\costTeaching Moduleminimal cost
    Modeling Scenario
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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
    Potential Scenario
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    2003-Fay-Graham-Coupled spring equations
    Coupled spring equations for modelling the motion of two springs with weights attached, hung in series from the ceiling are described.
    spring massoscillationHooke's Lawcoupled springsperiodicforcing functionphase portrait
    Potential Scenario
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    2003-Fay-Graham-Coupled spring equations
    Coupled spring equations for modelling the motion of two springs with weights attached, hung in series from the ceiling are described.
    massoscillationspring\coupled springsperiodicfporcing
    Modeling Scenario
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    3-140-TwoSpringsOneMassFixedEnds-ModelingScenario
    Students build a model of a two spring, single mass with fixed end configuration and then plot solutions to experience the motion.
    massundampedFree Body Diagramspringstwo springsoscillationspring constantfixed ends
    Modeling Scenario
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    3-090-OneSpringMass-ModelingScenario
    We lead students through building a mathematical model for a single mass (bob)-spring system that is hanging vertically. We also lead the students, using data that they collect together with their model to approximate the value of the spring...
    undampedsum of square errorsmass springHooke's Lawoscilationleast-squares approximation
    Modeling Scenario
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    3-101-SpringMassFirstTry-NoResistance-ModelingScenario
    Students build a model based on their perceptions of what the solution should look like for a simple spring mass system with no damping.
    frequencymassoscillationundampedspring constantspring\Single Degree of Freedom System
    Modeling Scenario
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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
    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
    Modeling Scenario
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    3-001-SpringMassDataAnalysis-ModelingScenario
    We offer data on position of a mass at end of spring over time where the spring mass configuration has damping due to taped flat index cards at the bottom of the mass. Modeling of a spring mass configuration and estimation of parameters are the core.
    datadampingspring-mass systemNewton's Second Law of Motionspring constantHooke's Law
    Modeling Scenario
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    3-010-EnergyInSpringMassSystem-ModlingScenario
    As a way to synthesize the effects of damping and forcing terms, this activity is meant to encourage students to explore how different forcing terms will change the total energy in a mass-spring system.
    energymass-spring systemkinetic energypotential energytotal energy