Lab 2M242

Springs and Beams

The damped mass-spring differential equation MATH is one of the most important models in engineering and physics. Because it is linear, it can be solved exactly for the position $x$ of the mass as a function of time $t$. A closely related system is an upright flexible beam. Let $x$ represent the distance from vertical at the top of the beam, so that $x=0$ represents the beam standing straight up. The differential equation MATH then provides a model for the distance $x$ as a function of time $t$.


Someone has already written a paper on the mass-spring differential equation. You are going to write a companion piece about the upright beam equation. You may assume that your readers have read the first paper.


Part A: The undriven system.

You are going to investigate the beam equation with $a=k=1$, and with $f(t)=0,$ so you now have MATH.

1. For $c=0$ (no damping), what types of oscillations are possible? Explain in terms of the beam. Use time plots to estimate the period for the different types of oscillations. Use a few different initial conditions (to make it simpler, always choose the initial velocity to be zero). Does the period depend on the initial conditions? Make sure to use some initial conditions that are close to the fixed points.

2. Create phase portraits for various values of the damping constant $c$. Start $c$ at zero, and slowly increase it past the point where oscillations stop. Use $-2<x<2$ and $-2<y<2$ (where $y=x^{\prime }$). Discuss similarities and differences with the undriven mass-spring system. Find and identify on the plots the fixed points, and determine the stability of each (stable or unstable). Explain the stability in terms of the beam.


Part B: The driven system.

3. Use $f(t)=0.05\sin (at)$ where $a$ is a parameter that will range from $0.5$ to $1.5.$ The damping constant $c$ will be fixed at zero (no damping). Also, make both initial conditions correspond to a fixed point. This time look at time plots only (phase plots are not very useful for nonautonomous equations). Vary $a$ in the range given, and create $x$ versus $t$ time plots for each $a$ value. Discuss similarities and differences with the driven mass-spring system. In particular note whether or not beats occur, at what $a$ values they occur, and whether or not resonance occurs. If not resonance, then what? Relate all behavior to the beam system.

This document created by Scientific Notebook 4.1.