Modeling: Free Oscillations (undamped)

Oscillations play a tremendous role in our life, be it just in order to measure time (and I am not only talking about the old mechanical clock Greatgrandfather used to have that was sold to the antique dealer just recently), or in order to describe mechanical systems, electrical circuits, and even so called static structures such as buildings, bridges, and many more.

Often we are using oscillations for our purposes (e.g. circuits), but – perhaps even more often – we need to know about oscillations in order to avoid them (your car suspension, for example!) As a professional you are expected to understand this issue, and often, too often, lives will depend on your decisions and calculations!

 

This should be sufficient motivation to you to study the following sections seriously. And, as it turns out, most of these problems aren’t that difficult to solve! However, some work will be involved – after all there has to be a reason why engineers still make quite a good living …

 

We begin with the most simple case of an oscillating system: An undamped system.

For example a mass hanging on a spring which again is attached to a ceiling:

The physical laws of this system are quite simple:

Spring force: is the displacement of the spring

The force necessary to displace such a spring by s is proportional to the displacement.

(This is called Hooke’s law).

Spring at rest (no acceleration )

Now, if the spring is at resting position with this mass m attached to it will not move and for sure it will not accelerate either. (therefore: resting position). According to Newton’s law then the sum of all forces must equal to zero.

Let’s use the resting position of our mass as a new (and very convenient) reference point for our following considerations: We will measure the location of our mass (y) in respect to this resting position – call y the "displacement then.

Now displace the spring (stretch it) by a length of y:

additional force (gives the total force)

In order to displace the mass we have to apply additional force, that, according to Hooke’s law is proportional to the displacement…

The total force is also given by Newton's 2nd law:

Therefore:

, or

, or

And just by applying Newton’s second law we immediately obtain this very simple looking differential equation for our displacement function y!

The characteristic equation of it is given as and leads …

, complex root!

to the very familiar case of a complex root! From this we now directly (spare me the detour through the complex exponential function!) find a general solution

Gives us a solution

with

Yes, this is the time function of a harmonic oscillation ….

Actually, an oscillation with a frequency of

Interesting: That means if I increase the mass the frequency will decrease (the oscillation becomes slower) but not linear. Let’s see, to cut the frequency in half I will need 4 times the mass ….

Now think about what will happen if we make the spring "stiffer", means increase the value of k ? Well, you answer this one.

Back to our problem: we found (very fast!!!) a general solution but we do not know the coefficients A and B yet.

And, nothing easier than this, again we are using initial conditions in order to obtain them:

i.c. (1) i.c.(2)

means, we know the displacement and the velocity at time t=0.

Gives two equations (just put these (y,t) values into the general solution) to determine A and B.

1.2.

A

nd – voila! We have solved the equation of a harmonic oscillator. Once and forever!

The highlight of countless science classes in high school solved in less than a minute!

(Don’t tell you old teacher about it – keep this triumph for yourself….)