Vibratory Modes of an Aluminum Ruler

Written by The Guru. Posted in Experiments.

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. 

The Analog Computer Project

Written by The Guru. Posted in Experiments.

The Guru has always been fascinated with different ways of approaching a problem - and here's a unique example. Long before laptops and PC's, before the Apple 1 and the MITS ALTAIR 8800, before the VAX, the UNIVAX, the TRS-80 and the CDC Cyber, there was the Heathkit EC-1 analog computer. A desktop computer weighing close to 50 pounds and having a 30-minute warm-up time. This was a big deal more than a half-century ago.



  • A view of the EC-1 resting atop a lab bench. The unit is in standby mode.


  • A front view of the EC-1 . The unit is operating in a simple feedback loop.


  • Another view of the unit performing the unitary feedback test.


  • Top view of the unit showing the amplifier tubes. We are using original Heathkit tubes.


  • The operational schematic for the EC-1 analog computer.

  • A view of the unit zeroed-out and ready for programming.



  • Interior shot of the operational amplifier circuits in standby mode.


  • A simple analog computer program showing the multiplication of 'six' times any value.


  • The result of 6·(-1.5)=-9.

The Guru's EC-1 was manufactured in 1959. It falls into a very early class of computational devices known as 'analog computers'. As the name implies, an analog computer works by analogy, that is, it draws a comparison between an electric circuit and some mechanical device by applying what is known as the electro-mechanical analogy of physics. It is at heart a differential equation solving machine (specifically ones of the second-order constant coefficient nonhomogeneous variety).

The EC-1 originally retailed for 400 dollars in 1959 and was available as a 'low cost' computer for teaching engineering, physics and mathematics. It was sold primarily in kit form to universities for use in complex problem solution (remember, there were no calculators at that time, just pencil, paper, and slide-rules). It is programmed through a switchboard wiring interface and has no microprocessor or digital logic of any kind for that matter. In an analog computer there are no 'yes' (binary 1) or 'no' (binary 0) answers as is the case with modern digital computers. The answer is provided as a continuously variable voltage output. In this regard, the analog computer represents a problem solution exactly as shown in nature (digital computers always show quantized representations of the same thing). The EC-1 utilizes nine DC operational amplifiers (op-amps) for its computational engine allowing for up to three initial conditions and five independent coefficients on the system of equations being modeled. Output information is provided via the panel voltmeter or an oscilloscope (in our case, a modern Tektronix 3014 Data Acquisition unit). The photographs show the unit in a simple feedback test configuration transferring the output of one op-amp into the next.

- - -

One example problem we recently performed for an educational demonstration was to simply have the computer multiply two numbers together. To do this we created a so-called 'broken key calculator' which would only calculate the value of six (the broken key) times any other value.

Why?  Simple, we had a 6 volt lantern battery hanging around the lab which would provide the number 'six' times any input we wanted to adjust on the computer. We could have used any battery type, or a variable power supply for that matter, but it just did not have the charm or simplicity of our one-sided calculator.  

The photo shown here contains the completed program where we are feeding the 6V input into the third coefficient bank via the seventh operational amplifier (the choice is arbitrary, we chose these two to physically separate the electrical connections making them easier to see in the photographs). The fifth coefficient potentiometer is acting as a reset button on a calculator (allowing us to zero out the result at any voltage level).

  

The 'Broken Key' Calculator

The answer is then sent via the remaining red wire to the volt meter display on the computer. The ground of the meter was wired back to the negative terminal of the battery completing the electrical connection. The results of the calculation of 6 x (-1.5) = -9 is shown in the slideshow above.

Synthesizing the 'Stuff' of Comets

Written by The Guru. Posted in Experiments.

Here's a simple way to produce something only seen in nature in the deep reaches of interstellar space ...

We started out with a two-liter (2 L) beaker of liquid nitrogen (N2) boiling at a temperature of 321 degrees Fahrenheit below zero (i.e., -321 °F) as shown in the photo below. This beaker is essentially filled with liquid air since most of Earth's atmosphere is composed of nitrogen gas. The condensation you see on the external sides of the beaker is a mixture of frozen water vapor (i.e., ice) and liquid oxygen (which starts to condense at temperatures greater than that of nitrogen).

 

Initial 2 L Beaker of Liquid Nitrogen at -321 °F

 

The liquid nitrogen was then transferred into a smaller 500 mL beaker and placed inside a vacuum chamber as shown in the two photos below. Make note of the vigorous boiling action occurring inside the beaker.  

  

 
Liquid Nitrogen Boiling in a Vacuum     Close-up of Boiling Action

  

It took approximately 90 minutes under the application of a vacuum for the temperature to transition the mere -26 degrees Fahrenheit from boiling liquid nitrogen (-321 °F) to solid {frozen} nitrogen gas (-346 °F), with the length of the process being solely a function of the small two-stage 3 CFM vacuum pump being used in the lab. The entire process is shown in the time-lapse video below where we have compressed the 90 minute process into two-minutes (this is a large video, you might have to wait a minute for it to fully load).

Time Lapse Video of Nitrogen Gas Freezing into a Solid  


Initially we observe a rapid boiling of the liquid nitrogen with significant condensation vapor present. After the liquid level drops to about 50% of the initial 500 mL volume, the rate of boiling slows down considerably and the liquid is within a few degrees of the freezing point. Once the liquid nitrogen reaches the freezing point, we can see rapid 'ice' crystal formation and the remaining liquid freezes within a couple of minutes. The condensation observed is frost {water ice} forming on the outside of the vacuum bell jar, which can be seen in the first photo below. The second photo shows the finished product, instant 'comet stuff'.

 

 
 Photo of the Frozen Vacuum Bell Jar     Frozen Nitrogen Gas at -346 °F


Outside of a laboratory setting, solid nitrogen is only predominately seen in interstellar space within comet fragments and within the craters of asteroids. It is a transient substance for once we remove the applied vacuum, it rapidly melts back to liquid nitrogen and from there to its gaseous state. The sample shown in the above photo is held in place at a temperature of 114 degrees Fahrenheit above absolute zero (absolute zero is approximately -460 °F)

 

Liquid Nitrogen Fog

Written by The Guru. Posted in Experiments.

Here's a quick way to make a miniature fog bank...

We took an 800 ml beaker of liquid nitrogen outside on a cold winter morning, tossed it up into the air, and recorded the whole thing at 1/10th normal speed (i.e., 300 frames-per-second). As you can see from the video below, the results produced a small tule fog bank as the liquid nitrogen rapidly condensed the moisture out of the air.

Creating a Small, and Very Cold, Fog Bank in Slow Motion 

So, what's going on here???

First off, liquid nitrogen is cold - really cold... Liquid nitrogen boils at 321 degrees Fahrenheit below zero (i.e., -321 °F) and as can be seen in the photo below rapidly coats the outside of the beaker in a thick layer of ice.

 

A Very Cold Beaker...

 

From the viewpoint of the liquid nitrogen, a cold winter's morning is a blisteringly-hot day. Our roughly 50 °F blacktop parking lot might as well be a hot frying pan as far as the nitrogen is concerned. The droplets instantly freeze the moisture out of the surrounding air, and rapidly evaporate the liquid upon contact with the ground, creating a very cold white fog that rolls along the ground. A good example of this is shown in the photo below.

 

Falling Liquid Nitrogen Freezing the Surrounding Air

 

The air surrounding the liquid nitrogen droplets is so cold, in fact, that you can easily see the denser air falling towards the ground (and also the fact that atmospheric air is roughly 78% nitrogen anyway, so there is very little buoyant force present). 

Pretty cool - literally...

 

Newton's Cradle Demonstration

Written by The Guru. Posted in Experiments.

If you've taken a high school physics class, then the odds are pretty good that you've seen one of these gizmos commonly called a "Newton's Cradle", which demonstrates the interaction of kinetic energy and momentum. Essentially the device consists of a small wooden frame with five pinball machine balls suspended from fishing line. As one ball is lifted and then released, it strikes the remaining four balls and causes the energy to be transferred to each successive ball ultimately causing the ball on the other end to fly off.

Take a look at the video below as a demonstration. You are looking at the motion at 1/10th speed or 300 frames per second.

Slow Motion Video of Momentum Transfer for 1x Case

So, what's going on here???

Well, the kinetic energy of the falling ball strikes the first stationary ball of equal mass. Being held in place by the inertia of the other balls, the second stationary ball transfers its energy (and momentum) to the next ball. On down-the-line this goes until the transferred momentum is equal to (or less than) the remaining mass in front of the ball, at which point the energy produces motion (i.e., the end ball, with the same mass as the first ball moves upward with the same amount of energy imparted by the first ball earlier on).

Well sort of... In a perfect world absent of pesky little things like friction and heat, the transfer of energy would be complete (i.e., momentum would be conserved) and we would expect that the first ball would strike with a certain amount of momentum, all intermediary balls would remain motionless, and the last ball would fly off the end up to a height equal to the starting height of the first ball (i.e., conservation of kinetic energy). This is not the case, and even though at a real time speed the process looks pretty good, we can see that when we effectively slow time down by a factor of 10, there is a bit of wobble in the intermediate balls caused by their masses not being exactly equal, frictional losses in the connection points, and the inevitable 'clack' sound as they hit each other converting energy into sound which heats the air. Alas, the motion of our system will decay (and become more chaotic) right in front of your eyes.

So, if the momentum of our system in the above example is, say '1x', what will happen if we drop two balls so that our starting momentum is '2x'? Take a look below.

Slow Motion Video of Momentum Transfer for 2x Case


Momentum is still conserved for the most part, although the kinetic energy of the system is dropping rapidly. The same holds true for three balls...

 

Slow Motion Video of Momentum Transfer for 3x Case

This principle holds true regardless of the size of the system or the physical constraints imposed on it. In fact, the scientific principle demonstrated here is one of the basic mechanisms that holds the universe together. 

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