A mass is suspended from a spring. A second mass is suspended from the first mass on a second spring. How does the system behave?
This problem is example 1 on page 293 of Edwards & Penney (1996).
The state equations can be written as
x1´ = v1where mi is the mass at the end of spring i, xi is the displacement of spring i, vi is the velocity of mass i and ki is the spring constant for spring i.
x2´ = v2
m1 v1´ = -k1 x1 + k2 (x2 - x1)
m2 v2´ = -k2 (x2 - x1)
To slow the system down, the spring constants used to generate the animations are 1/10 of the values used in Edwards & Penney; the constants used are m1 = 2, m2 = 1, k1 = 10 and k2 = 5. The motion of the masses was calculated using a Runge-Kutta method.
There are two natural modes of oscillation. The first occurs when the masses start at rest with displacements x2(0) = -x1(0).
The second natural mode occurs when the masses start at rest with displacements x2(0) = 2 x1(0).
The next animation shows what happens when the masses start at rest with x1(0) = 0 and x2(0) > 0.
Finally, here is what happens when the masses start at rest with x1(0) > 0 and x2(0) = 0.
