Exercise EPC4 7.5.11.
Edwards, Penney, Calvis: Differential Equations and Linear Algebra,
Edition 4, Chapter 7, section 5.
The mass parameter assignments are not correct but do not affect the exercise.
You need the following assignments to get the 2 mass 3 spring system:
m1 = 1, m2 = 2/3, k1 = 32, k2 = 8, k3 = 32
This corresponds to a weak spring in the middle between two strong springs, and a slightly bigger mass on the left.
These values are easily found by setting m1 = 1 and solving the remaining 4
equations on the remaining 4 parameters in the general coefficient matrix for
the 2 mass three spring system.
The online variation of this problem ignores the initial conditions and does not ask for an initial value problem solution, where one sees that in every online variation, the natural mode with one of the eigenvalues (the faster mode, bigger eigenvalue) is missing in the solution. Thus the original intentions of the founding authors was missed by this online implementation.
To respond online, one only needs to find the steady state response (the
particular solution) by the method of undetermined coefficients, and identify
the natural frequencies for the tandem and accordian modes from the eigenvalues
and eigenvectors of the coefficient matrix.
When you find the natural frequencies for the textbook parameters 6 and 8, you see why the driving frequency of 7 was chosen. In fact it is close to both (it is the average, the faster beat frequency, while the amplitude beat frequency is the difference over 2, just 1) and if you resolve this problem with the system at rest at equilibrium, you see the beating clearly that results with period 2 Pi. This is also lost in the online implementation, though perhaps an unintended consequence of the 3 frequencies chosen close together. Note that the common period of the combined motion 2 π simply because this is the common frequence of all integer frequency oscillations.
See the Maple worksheet epc4-7-5-11.mw at
http://www34.homepage.villanova.edu/robert.jantzen/
exported to HTML:
https://www34.homepage.villanova.edu/robert.jantzen/courses/mat2705/handouts/epc4-7-5-11-web.html