Lesson 13.1 Mass Spring Systems Damping and Worked Examples

Let's briefly discuss free vibrations with damping. A mass spring system can be damped using a viscous fluid, such as oil, to cause the system's energy to slowly dissipate over time.

There are three types of damped motion that can be described:

1. Underdamped (or Oscillatory) motion.
2. Critically damped motion.
3. Overdamped motion.

Consider $m{x}^{″}+b{x}^{\prime }+kx=0$.

In a damped oscillatory system, we analyze the damping parameter $b^2$ in relation to $4mk$, where $m$ is the mass of the object and $k$ is the spring constant. The comparison of these two quantities helps to classify the system's damping behavior into one of the three categories.

If $b^2<4mk$, then the system is described as underdamped motion. This creates an oscillatory motion in which the energy dissipates very slowly over time.

If $b^2=4mk$, then the system is described as critically damped motion. This system has energy that dissipates the fastest.

If $b^2>4mk$, then the system is described as overdamped motion. The energy of this system dissipates relatively quickly, but not as much as a critically damped system.

Examples:

Let's analyze these systems and categorize them as underdamped, critically damped, or overdamped. Let's also find the general solutions for each.

1. $x^{″}+x^{\prime }+x=0$
2. $x^{″}+3x^{\prime }+x=0$

For the first system:

$m=1$
$b=1$
$k=1$

$b^2=1$
$4mk=4(1)(1)=4$
$1<4$

Therefore, this system is underdamped, or oscillatory.

$x^{″}+x^{\prime }+x=0$

Let's set up the characteristic equation:

$r^2+r+1=0$

Solving using the quadratic formula:

$r=\frac{-1\pm \sqrt{3}i}{2}$

$x(t)=e^{\frac{-1}{2}t}(C_1\cos \left(\frac{\sqrt{3}}{2}t\right)+C_2\sin \left(\frac{\sqrt{3}}{2}t\right))$

Try the second system for yourself.

Another Example (Try for Yourself)

A $\frac{1}{8}$kg mass is attached to a spring with stiffness $k=16\frac{N}{m}$. The mass is displaced $\frac{1}{2}$m to the right of the equilibrium point and given an outward velocity to the right of $\sqrt{2}$ m/s. Assuming there is no damping or external forces, determine the equation of motion of the mass along with its amplitude, period, angular frequency and natural frequency (see Lesson 13 - Mass Spring Systems).

What's Next?

So far, we’ve looked at several ways to solve differential equations, like using differentiation, integration, and substitution. Now, we’re going to switch gears and start solving them using series. But before we dive into that, let’s take a moment to go over some key series definitions that will help us use this method effectively.

Next Lesson: Lesson 14 - Infinite Series

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