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Modeling Scenario

3-102-SpringMassDamped-ModelingScenario

Author(s): Keith Landry, Brian Winkel1

SIMIODE - Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations

Keywords: frequency mass oscillation stiffness spring constant spring\ damped Single Degree of Freedom System coefficient

Abstract

Students build a model based on their perceptions of what the solution should look like for a simple spring mass system with damping.

Citation

Researchers should cite this work as follows:

Description

Consider a mass suspended by a spring. If we let the mass hang still, thus extending the spring from its natural length, we will see that the mass comes to rest at what is called static equilibrium.

In attempting to model the vertical motion of this mass we impose a coordinate system for y(t), the vertical displacement of the mass from the static equilibrium. Engineers refer to such a system as a Single Degree Of Freedom System (SDOFS), as we are tracking only one variable, namely, y(t), vertical displacement from the static equilibrium.

For consistency, let us say y = 0 is the spring's vertical displacement at the spring's static equilibrium and is the distance of the displacement of the mass from that static equilibrium and denote positive in the downward direction and negative in the upward direction. That is, if we extend the spring 3 cm downward then y = 3, while if we compress the spring upward 2 cm then y = -2.

We are going to use a Free Body Diagram to depict all the vertical forces acting on the mass in this case. We show the mass pulled down just a bit beyond the static equilibrium position and moving downward, with the two forces acting on it being (1) the restoring force of the spring which acts in an upward vertical direction and (2) the force upward from the damping due to friction in the spring and the resistance of the media on the moving mass.

Authors

Author(s): Keith Landry, Brian Winkel1

SIMIODE - Systemic Initiative for Modeling Investigations and Opportunities with Differential Equations