Differential Equations

Method of Undetermined Coefficients
Resonance

TOOLS: Applet Einmassenschwinger
http://www.zfm.ethz.ch/meca/applets/ems/ems.htm

2D grapher from DESMOS is the best; you can use t as the independent variable and zoom and scroll easily: https://www.desmos.com/calculator

Attach your work for the following on separate paper.
Each person in your group must turn in this work.
You may use printouts for the graphs if you like or copy them onto graph paper accurately with scales.

Use 2 decimal places for your coefficients in the solution.

Part I:

Use your groups' parameters in the applet Einmassenschwinger:
(In the applet, ω = “frequency of harmonic excitation”)

Group	Mass	k	ß	y(0)	y'(0)	sinωt ω=
1	1	1	0	0.5	0	1
2	1	4	0	1	0	2
3	1	9	0	-0.5	0	3
4	1	1	0	1	0	1
5	1	4	0	-0.5	0	2
6	1	9	0	0.5	0	3
7	1	1	0	-0.5	0	1
8	1	4	0	0.5	0	2
9	1	9	0	1	0	3

Write your partner and group number here:

a. Write the differential equation of motion for your mass.

b. Solve this differential equation using undetermined coefficients and write your solution here:

y(t) =

c. Graph this motion, y vs. t on your calculator or the computer with Quickmath or other program.
(Use a large enough tto get a good graph.)

d. Estimate the period and frequency of your graph. Describe the behavior of the graph as t approaches infinity.

e. Sketch the applet Einmassenschwinger graph. Describe any differences between this graph and your graph from part c.

Part II - with damping: Use these parameters and redo parts a,b,c,d,e.

Group	Mass	k	ß	y(0)	y'(0)	sinωt ω=
1	1	1	0.1	0.5	0	1
2	1	4	0.2	1	0	2
3	1	9	0.3	-0.5	0	3
4	1	1	0.1	1	0	1
5	1	4	0.2	-0.5	0	2
6	1	9	0.3	0.5	0	3
7	1	1	0.1	-0.5	0	1
8	1	4	0.2	0.5	0	2
9	1	9	0.3	1	0	3

Part III - Phase Planes

a. Write the 2nd order differential equation in part I as a system of two first order equations. Then use phase plane plotter yv to draw the phase plane plot.
Note that you may have to increase t to get a good plot. Also, make sure the approximation method is RK4, not Euler.

b. Write the 2nd order differential equation in part II as a system of two first order equations. Then use phase plane plotter yv to draw the phase plane plot.
Make time long enough to see the pattern well.