Vibratory Modes of an Aluminum Ruler

The Guru recently performed some research work on how an aluminum ruler (or any solid object for that matter) vibrates in space as a function of changing the excitation frequency. To do this, he setup a small electromechanical shaker, a signal generator, and a high-speed video camera to capture the action. Take a look at the test setup below.

 

The Test Setup showing the Dynamic Shaker, Signal Generator, and Video Camera Setup

 

Believe it or not, things just don't vibrate in any old random way. The random or chaotic motion exhibited by, say, a structure during an earthquake or an aircraft in flight is not random at all. It is composed of the superposition of many many different types of vibrational shapes (collectively called vibratory modes). Take a look at the video below which is an edge-on view of the ruler vibrating.

First Transverse Mode of Aluminum Ruler, One Node Near End Point (5 Hz) 

As you can see, the electromechanical shaker (the big round black cylinder at the bottom of the video) is shaking the ruler at its center point (center of gravity) in an up-down fashion. The box tot the left of the shaker is the power amplifier which drives the shaker. The other black box in the photo above (the one with the big center knob and lit light) is a variable frequency generator feeding an input into the shaker amplifier. 

So what do you see?

There is a lot of ruler motion with the ruler flapping like the wings of a bird. This type of motion is unique to the ruler (it is a function of the length of the ruler and what it is made of) and occurs in the photo right around five (5) cycles-per-second (of Hertz, Hz).

Leaving the amplitude alone for all the tests (i.e., we set it at two volts and never adjust it again) it can be observed that as we move the frequency knob (the big knob) away from 5 Hz, the motion dramatically drops-off. At either 3 or 7 Hz, the ruler pretty much moves up and down being pushed by the shaker, the flapping motion disappears. It only occurs right around 5 Hz.

This condition is known as resonance. It is the frequency point for the ruler being examined where it naturally 'wants to vibrate'. Frequencies away from the resonance point and the ruler is sluggish and doesn't respond with much motion at all. In mathematical terms this natural vibration frequency point is known as an Eigenvalue (this is German for ''proper value'). The shape that any object takes at a natural frequency (Eigenvalue or Eigenfrequency) is known as the Eigenfunction.

There is a misconception that there is only one resonance point for an object (like a bridge or building). This is not true. In fact, for a continuous object like our friend the ruler here, there are an infinite number of resonance points each having a specific frequency and each producing a different deflected shape.

Now, take a look at the second video below.

Second Transverse Mode of Aluminum Ruler, Two Nodes at Approximate 1/3 Points (38 Hz) 

This is the observed motion at the rulers second resonance point (which occurs at approximately 38 Hz). This resonance is just as real as the first one and it would be observed by slowly turning the big frequency knob from the first resonance point (5 Hz) to the second point (38 Hz) that not much exciting occurs between these two points. It is therefore observed that the ruler has a second natural point where is wants to vibrate and the frequency of this vibration is 38 Hz. 

But what about the shape?

It is indeed different from the first. Whereas the first motion looked more-or-less like the half of a sinusoid, this mode looks like a full sinusoid reflected about the point where the shaker connects to the ruler. In order for the ruler to accomplish this, the physics of the problem dictate that the ruler must form the shape of a sinusoid and we all know that a sinusoid has two points that are zero for every cycle. 

Take another look at the ruler video. There are indeed two points (excluding the excitation point) which do not move. These points are known as 'nodes' and describe the stationary positions of the eigenfunction. Now, what do you think will happen as we increase the frequency from the second resonance to the next? Again, not much until we get close to 270 Hz. Look at the video below to see the third natural frequency and mode.

Third Transverse Mode of Aluminum Ruler, Three Nodes at Approximate 1/4 Points (270 Hz) 

There are now three stationary points per side of the ruler and the ruler is taking on a more warped shape. Since we have said previously that any object has an infinite number of natural frequencies and modes, we could repeat this test all day. If we did continue,  we would find out that for each successive mode, we would add another node to the ruler and the ruler would be forced to bend through these points. Ultimately, there would be so many nodes that the relative motion would be unobservable to the naked eye, but it would still be there and would be easily measurable using a device known as an accelerometer.

 

As you can see from the photo here showing the first mode shape, the motion of the ruler is quite real. It is bent a considerable amount as it oscillates at it's resonance point. This is the reason that resonance is a phenomenon which is typically to be avoided as these large amplified motions (which are independent of the input energy) can readily damage structures and cause them to catastrophically fail.
		Photo of Deflected First Mode Shape

 

So, the moral to the story here is that what appears to be a random vibration (such as a suspension bridge moving in the wind or a building shaking during an earthquake or a rattle within you car's engine at a certain speed) is actually a composite of multiple eigenvalues and eigenvectors. There is actually quite an order to this type of chaos.