Three springs
This is an elaboration of the classic coupled oscillators problem that appears
in many diffeq texts. In Arnold’s book it relates to
- Spherical pendulum example on page 34/35, Chapter 1,
§1#18
- Small oscillations in mechanical systems on page 227, Chapter
3, §25#6.
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 $x_1$, $x_2$:
\[
\begin{aligned}
m_1 \ddot x_1 &= -k_1x_1 + k_2(x_2-x_1) \\
m_2 \ddot x_2 &= -k_3x_2 + k_2(x_1-x_2)
\end{aligned}
\]
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
$0\lt \epsilon \ll 1$.
To get a first order system of differential equations we introduce
the velocities $y_1=\dot x_1$ and $y_2=\dot x_2$, and we get this system
\begin{equation}
\begin{aligned}
\dot x_1 &= y_1 &
\dot x_2&=y_2 \\
\dot y_1 &= -x_1 +\epsilon x_2 &
\dot y_2 &= -x_2 +\epsilon x_1
\end{aligned}\tag{sys}
\end{equation}
Problems
- What is the phase space for the system of
differential equations (sys)?
- Find constants $p,q,r$ so that the quantity
\[
E = \tfrac12 y_1^2 + \tfrac12 y_2^2 + p x_1^2 + q(x_1-x_2)^2 + rx_2^2
\]
is constant along solutions of (sys).
- Write the differential equations in the form
$\dot\vx(t) = A\vx(t)$ 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.
Describe the motion of the masses for each of these two real solutions.
- Find the real solution with initial conditions
\[
x_1(0) = 1, \quad x_1'(0) = 0, \quad x_2(0)=0, \quad x_2'(0)=0.
\]
