Differential Equations Day 15 -- Project 4 -- Mechanical Vibrations of Spring Systems

Directions

Follow the steps below and do your work in the provided GeoGebra windows. NOTE to VSC students taking this class with me: be sure you are accessing this through a GeoGebra Classroom Link! The URL of this page should have the word “classroom” in it. If not, then go back to Canvas and be sure to access this page from the Project 4 assignment. Also: I strongly recommend you login to GeoGebra.org with a free account so your work is saved, and you can come back later to review or modify it. For external readers: these "project" activities are meant to be taken as part of my course, so these sections of the GeoGebra book my not be as intelligible as others. My apologies.

Learning Goals

The primary goal of this project is to learn about Mechanical Vibrations of spring systems, a classical application of second order homogeneous constant coefficient differential equations. A secondary goal is that you will practice plotting the slope field of the systemification of a second order differential equation.

Content Overview

When:

an object is suspended by a spring; and
the spring is firmly attached to an anchor; and
damping is present due to immersion of the system in a viscous fluid (including "air resistance"),

then the motion of the object away from equilibrium is well modeled by a second order homogeneous constant coefficient differential equation. Explore the applet below which illustrates the motion of such an object. You can read more about the mathematics of such a system here. You can explore by adjusting 4 parameters:

, the mass of the object (in kilograms)
, a damping coefficient
the spring constant of the spring -- see Hooke's Law for linear springs
the initial displacement (extension or compression) of the spring/mass from equilibrium (in meters)

Press "Play/Pause" to run a 20 second experiment by putting the object into motion. The app keeps a traced record of each experiment you run. Click "Clear Trace" to clear all records.

Part 1 Steps

In the applet below for Part 1 set the mass to 9, the damping coefficient to 6, the spring constant to 4, and the initial displacement to 12.
Trace the motion of the object for the full time period (20).
Increase the damping coefficient to 8. Hold the other settings constant. Reset the time to 0. DO NOT CLEAR THE TRACE.
Trace the motion of the object for the full time period (20).
Continue adjusting the damping coefficient in steps of 2 up to 12. Reset the time, and trace the motion of the object for the full time period for each step.
Study the impact of the damping coefficient on the motion of the object. Summarize your findings in the text box below for Part 1. Be sure your written response uses complete sentences, and is concise.

Part 1 Applet

Part 1 Text Box

Put your written response here.

Part 2 Steps

Find the specific solution of the following second order homogeneous constant coefficient differential equation with initial conditions on u and u'. Plot it in the Part 2 applet below.

Part 2 Applet

Part 3 Steps

Consider the following differential equation with initial conditions on u and u'.

Use the GeoGebra applet below for Part 3 to verify that the following function is a solution of the differential equation and satisfies the initial conditions.

Part 3 Applet

Part 4 Steps

Consider again the following second homogeneous order constant coefficient differential equation with initial conditions on u and u'. This is the same differential equation from Part 2.

Review the method of systemification from earlier and confirm that this is the systemification of above differential equation:

In the GeoGebra applet below for Part 4: 1. Plot the slope field of the systemification with

slopefield(((-4/9)*x-(6/9)*y)/(y))

2. Enter your solution function from Part 2 again in the GeoGebra applet below (for Part 4). Be sure to declare the function as u(x) 3. Calculate the derivative of u(x) with GeoGebra. 4. Hide both u(x) and u'(x) by clicking the dot next to their declaration in the Algebra Pane 5. Generate the parametric curve that illustrates the solution function in the context of the slope field with

curve(u(t),u'(t),t,0,20)

6. Plot the initial condition on u(x) and u'(x) with InitialCondition=(12,0) 7. Verify that the curve passes through InitialCondition as expected by passing 0 to the curve object. This is most likely accomplished with the code a(0) However, if the curve you created with the code in step 5 was called something else, then you'll need to pass 0 to that.

Part 4 Applet

Closing Note on Applicability

There are two primary applications of the theory of Mechanical Vibrations of spring systems:

In mechanical engineering it is often advantageous to build a spring system that damps vibrations. Solutions such as we saw in Part 3 (where there were no oscillators, sine or cosine, in the general solution) are known as critically damped, and are often the goal in designing systems that need to handle oscillations.
In fluid mechanics, the theory of mechanical vibrations can be applied to measure the viscosity of a fluid. For instance, a researcher would know the mass of an object, the spring constant of a spring, but not the viscosity of a fluid they immerse the system in. By observing the motion of the object, and pattern matching a theoretical spring system to the observed motion by holding m and k constant, but adjusting , they can goal obtain an estimate of the measure of the viscosity, .