Three springs
This is an elaboration of the classic coupled oscillators
problem that appears in many diffeq texts (problem 7 in chapter 6 of our
textbook). The example illustrates the section about “Harmonic
Oscillators” (§6.2) in the textbook. See also the following pages:
The two oscillator problem
Two masses are connected to two walls and to each other by springs;
gravity plays no role. If $x_1(t)$ and $x_2(t)$ are the displacements of the
masses from their equilibrium positions, then Newton's “$F=ma$” law,
and Hooke's law for the force exerted by a spring imply the following system of
second order differential equations for $x_1$, $x_2$:
\[
\begin{aligned}
m_1 \frac{d^2x_1}{dt^2} &= -k_1x_1 + k_2(x_2-x_1) \\
m_2 \frac{d^2x_2}{dt^2} &= -k_3x_2 + k_2(x_1-x_2)
\end{aligned}
\]
where $k_1$, $k_2$, $k_3$ are the spring constants of the three springs.
Assumptions and non dimensionalization
Assume $m_1=m_2$ and $k_1=k_3$. After nondimensionalizing we can assume
$m_1=m_2=1$, and
\[
k_1 = 1-\epsilon, \qquad k_2 = \epsilon.
\]
where the new parameter $\epsilon$ lies between $0$ and $1$. If the spring in
the middle is much weaker than the two outer springs, then $0<\epsilon \ll 1$.
To get a first order system of differential equations we introduce the
velocities $y_1=x_1'$ and $y_2=x_2'$, and we get this system
\[
\begin{aligned}
x_1' &= y_1 & x_2'&=y_2 \\
y_1' &= -x_1 +\epsilon x_2 &
y_2' &= -x_2 +\epsilon x_1
\end{aligned}
\]
Introduce the vector
\[
X =
\begin{pmatrix}
x_1 \\x_2 \\y_1\\y_2
\end{pmatrix}.
\]
Problems
- Write the differential equations in the form $X'(t) = AX(t)$ and find the
$4\times4$ matrix $A$.
- Find the eigenvalues and eigenvectors of $A$.
- $A$ has complex eigenvalues. For each complex eigenvalue describe the
corresponding real solutions. What would the motion of the masses look
like?
- Find the real solution with initial conditions
\[
x_1(0) = 1, \quad x_1'(0) = 0, \quad x_2(0)=0, \quad x_2'(0)=0.
\]
