Coupled oscillators,
jiggled from one end
A one-dimensional system of n masses connected by springs. The far end of the last spring, the spring connected to the right 
, is fixed. The first spring, the spring connected to the left of 
, can be "jiggled" from the left with displacement 
. The system of equations to solve is :
X = A X +F
where, for an example of 3 masses,
We can make this system dimensionless by using 
as the time scale: 
. The dimensionless system will be written as
,
where now all the 
and 
should be interpreted as the ratio of the dimensional value to 
and 
, respectively. For example, if all 
and 
, then
Also,
Let the eigenvectors of A be written as 
and the eigenvalues be 
.
Construct a matrix P with the columns being the eigenvectors 
.
Let X=P Y, which means Y is the amplitude of the eigenvectors that
are summed to represent the displacment. Here is the clever way
to solve the coupled problem:
where D is a diagonal matrix of the eigenvalues. Let
. This is really cool: there are uncoupled O.D.E.s for the amplitudes of each of the eigenvectors:
Finding a particular solution of the above equation is rather elementary.
We next present a technique for finding the homogeneous solution,
and in particular an efficient way to satisfy the initial conditions
and 
. The general homogeneous solution is:
where
Let 
And let C be a column vector of the 
.
As a matrix problem, this gives
and
where Q=P D.
I like to turn off warnings about variable names being similar:
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Create a matrix using "Input"-> "Create Table/Matrix/Pallette".
Then check "Matrix". The matrix is stored (and can also be input)
as a list of lists, with the inner lists being the rows of A. Comment
one of these A matrices as text, "uncomment" as needed. Note the
use of one decimal number forces the use of numerical computation, as opposed to
symbolic.
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First, enjoy the exercise of finding eigenvalues without using the "canned" routine.
Note from the graph we expect negative roots for λ.
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Or use the canned routine:
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We will solve the homogeneous equation 
by proposing 
. Obviously, 
. Let's make a list of these "eigenfrequencies":
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Let Mathematica find the eigenvectors:
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Store the eigenvectors are columns in P:
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Calculate 
:
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Have a look at 
:
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Have a look at 
, just to see that it yields the expected diagonal matrix (except for round-off error):
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In this particular example, we set 
. In doing so, we have implicitly nondimensionalized displacement in terms of the amplitude of the "jiggle".
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Project the forcing onto the eigenvectors:
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A little "pencil and paper math" yields the amplitudes of the
eigenvectors as caused by the jiggling:
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Let's consider a particular value of Ω, so that we can have something
to plot out:
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Now with all the masses assumed to have no displacement and no velocity at t=0, we have
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It is amazing what Mathematica can do with a list. Exp[list] makes a list. list*list makes a list of the product of the elements...
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Check that we really did satisfy initial conditions:
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Check close up to see that initial conditions are really satisfied:
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Here we plot all the X, with a shift added to the displacements to
make them easier to see. The shift makes a plot look like the
displacement from a common origin, as it might naturally appear
if the masses are hanging from a ceiling.
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