You’re getting very sleepy…

Fear not, we’re not going to be getting deep into the metaphysical or the occult at this point. We’re instead going to spill some electronic ink talking about things that rock back and forth – or, as we in the engineering world call it… simple harmonic motion. In order to introduce this concept, we’ll consider one of the most basic and familiar of all machines – the spring (or, more technically, the spring-mass version of the harmonic oscillator).

Let us consider the following very simple system…

A simple spring, mass system

For small displacements of the mass (resulting in either extension or compression of the spring), it is found that the force required to displace the spring is directly proportional to the displacement of the spring (which we will call x). In other words…

F = k x

Here, the spring stiffness constant K has the units of N/m and relates the internal restoring or resisting force within the spring directly to the displacement of the spring. A spring that behaves in this manner is known as an ideal or linear spring.

According to Hooke’s Law, the restoring force for an ideal spring is given by

F = -k x

… where the minus sign indicates that the restoring force is always in the opposite direction of the displacement. In other words, if the displacement is down for our example spring above, the restoring force would be in the opposite direction, or… up.

I’m operating under the assumption that this is at least reasonably familiar to all y’all. If it’s not, go read some of the linked articles above and come on back when you’re ready. We’ll wait…

Now, let’s extend the spring/mass system above by a distance +x in the vertical direction and release it in an environment free of friction (in other words, we’re not going to consider, for example, drag force on the mass through the air surrounding the system). There is a restorative force of magnitude K x on the mass that wants to pull it back up. So, when we release the mass, with nothing to keep it in place this force acts on the mass and it starts to move upward.

At the point when the system is in equilibrium (in other words, x=0) there is no longer any force on the mass. However, as the mass is in motion it has inertia associate with it and therefore continues to travel up; as the mass travels upward the spring exerts a force trying to push the mass back down. The mass continues to travel upward until it reaches a displacement of -x, where it comes (very briefly) to rest.

However, as you may have guessed, the spring is now exerting a force in the downward direction of magnitude K x and starts pushing the mass back downward. In an idea system, the mass would bob up-and-down under the influence of the spring and it’s own inertia (notice we’re not talking about other forces, such as gravity, in this example)… forever. This type of motion is what is known as Simple Harmonic Motion.

The travel from original position and back to the starting position is what is called a cycle. For any object undergoing simple harmonic motion, the time required to complete once cycle of motion is known as the period (given the symbol T), and the frequency of such motion is given by

f = 1/T

… where frequency has the units of Hertz (Hz) or s^-1. We can define the angular frequency of the motion (a vibration) as

omega = 2 pi f

To describe the motion of the mass we need to make use of trigonometric functions. For an object traveling in a circular motion around an axis, we can mathematically describe this displacement by:

x = A cos(theta) = A cos(omega t)

… where theta is the angular displacement at any given time. Using our previously discussed definitions, we can then calculate the velocity of this circularly traveling body by:

v = dx/dt = -A omega sin(omega t)

… and the acceleration by:

a = dv/dt = -A omega^2 cos(omega t)

What may not yet be obvious, but will likely become so is you think about it for a bit, is that the motion of the mass in our ideal linear spring follows precisely the same motion as this theoretical “orbiting” body. We thus have equations that can be used to describe the harmonic (vibratory) motion of our body.

Now, the stretching or compressing of the spring is a means of storing energy in the spring; the kinetic energy of the moving mass is being transferred into the spring, which is storing that energy as potential energy – in this case, what we call elastic potential energy. Because there is no other forces acting in our system, we assume that all the force is in the direction of the displacement, and so the work done by the spring as the mass moves by a distance of x is proportional to the average force resisting the motion. Mathematically, the elastic work is:

W_e = F_avg x = (1/2 k x) x = 1/2 K x^2

The elastic potential energy of the spring is thus given by PE_e = 1/2 K x^2. Using a similar approach, we can determine the kinetic energy of the mass by

KE = 1/2 m v^2

Assuming a body that can undergo rotational (circular) as well as translational motion under the effects of a gravitational body force, the total mechanical energy of the system can be described as…

E_total = KE + KE_rot + PE_grav + PE_e = 1/2 m v^2 + 1/2 I omega^2 + mgh + 1/2 K x^2

… where I is the mass moment of inertia, omega is the angular velocity (ain’t it just awesome how we just re-use symbols like that!!), and g is the gravitational constant (on earth = 9.81 m/s^2).

We’ll pick this up in a couple of weeks with a discussion of simple pendulums, and how all of this can be extended to describe (in a macro sense) the one-dimensional motion of solid bodies. It will be my next post, but I’m going to be on travel for my day-job next week and won’t likely be able to post until the last week in September.

I’m sure you’ll be waiting with baited breath for my follow-up, which I promise will be… shorter. Until then…

Leave a comment

Your email address will not be published. Required fields are marked *