Here we provide a few simple examples to help you get started.
For each of these, the default initial conditions \(t_0 = 0\), \(x_0 = -4\), and \(v_0 = 0\) are used.
Example 1: Let's begin by using \(m = 2\), \(d = 0\), \(k = 8\), and no external forcing term
(\(f(t) = 0\)). This is a simple example of undamped motion. Notice that with undamped
motion, the spring oscillates forever. This is because there is no damping to reduce the motion over time.
In the next examples, we will see what effect damping has on the spring.
Example 2: This time we will use \(m = 5\), \(d = 1\), \(k = 10\), and again no external forcing
term (\(f(t) = 0 \)). The spring now has under damped motion. The oscillation is no longer
constant, and you can see that the spring's motion gets smaller and smaller over time until it has only a
small "ringing" motion. The damping constant is not yet big enough to stop the spring entirely, but it does
have a noticeable effect. The reason for this is that the discriminant (\(\sqrt{d^2 - 4(m)(k)}\)) when using
the quadratic formula to find the roots is less than \(0\), so the roots are complex which means that the
solutions to the differential equation involve sines and cosines still. These cause the ringing with their
oscillatory nature. In the next two examples, we will see what happens when the discriminant is greater than
or equal to \(0\).
Example 3: Use \(m = 2\), \(d = 4\), and \(k = 2\), and again no external forcing
term (\(f(t) = 0 \)). The spring now has critically damped motion. There is no longer
any oscillation. Instead, the spring simply settles into its equillibrium once released. This is because
\(\sqrt{d^2 - 4(m)(k)} = 0\). The solutions to the differential equation are real and repeated, and they
don't have sines or cosines to allow for osciallation of the spring.
Example 4: Using \(m = 1\), \(d = 4\), \(k = 3\), and no external forcing terms, we get a
spring with over damped motion. Like with critically damped, the spring simply
moves to its equillibrium point once released. The discriminant is greater than \(0\), so the
solutions to the differential equation are real and distinct. Therefore, there are no sines or
cosines to cause any oscillation.