Linear Algebra, 2020a

484 Chapter Five. Similarity
To see how the interaction can produce the dramatic behavior that we see
in a normal mode imagine that the mass is rotating in the direction that will
make the spring longer. If at the same moment the vertical motion is that the
spring is getting shorter, then superimposing the two could result in their almost
canceling. The bob ends up not moving vertically much at all, just twisting.
With a properly adjusted device this could last for a number of seconds.
“Properly adjusted” means that the period of the pure vertical motion is the
same as, or close to, the period of the pure rotational motion. With that, the
cancellation will go on for some time.
The interaction between the motions can also produce the other normal
mode behavior, where the bob moves mostly vertically without much rotation,
if the spring’s motion x(t) produces a twist that opposes the bob’s twist θ(t).
The bob will stop rotating, almost, so that its motion is almost entirely vertical.
To get the equations of motion in this two degrees of freedom case, we make
the same assumption as in the one degree case, that for small displacements
the restoring force is proportional to the displacement. But now we take that
assumption both for the vertical motion and for the rotation. Let the constant
of proportionality in the rotational motion be κ. Similarly we also use Newton’s
Law that force is proportional to acceleration for the rotational motion as well,
and take the constant of proportionality to be I.
Most crucially, we add a coupling between the two motions, which we take
to be proportional to each, with constant of proportionality ɛ/2.
That gives a system of two differential equations, the first for vertical motion
and the second for rotation. These equations describe the behavior of the coupled
system at any time t.
m · d2 x(t)
dt 2 + k · x(t)+ ɛ · θ(t) =0
2
I · d2 θ(t)
dt 2 + κ · θ(t)+ ɛ (∗∗)
2 · x(t) =0
We will use them to analyze the system’s behavior at times when it is in a
normal mode.
First consider the uncoupled motions, as given by the equations without the
ɛ terms. Without those terms these describe simple harmonic functions, and
we write ω 2 x for k/m, and ω 2 θ
for κ/I. We have argued above that to observe
the stand-still behavior we should adjust the device so that the periods are the
same ω 2 x = ω 2 θ . Write ω 0 for that number.
Now consider the coupled motions x(t) and θ(t). By the same principle, to
observe the stand-still behavior we want them in in sync, for instance so that
the rotation is at its peak when the stretch is at its peak. That is, in a normal
mode the oscillations have the same angular frequency ω. As to phase shift, as