(4/4) Operation: The details of setting up the Harmonic Analyzer @engineerguyvideo

Uploaded November 2014 | Updated September 2026, 2 weeks ago

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(1/4) Introduction: youtube.com/watch?v=NAsM30MAHLg
(2/4) Synthesis: youtube.com/watch?v=8KmVDxkia_w
(3/4) Analysis: youtube.com/watch?v=6dW6VYXp9HM

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This series on Albert Michelson’s Harmonic Analyzer celebrates a nineteenth century mechanical computer that performed Fourier analysis by using gears, springs and levers to calculate with sines and cosines—an astonishing feat in an age before electronic computers. Check out the series companion book and learn how to get a free PDF of the entire book at engineerguy.com/fourier.

(4/4) Operation: The details of setting up the Harmonic Analyzer

$The masterful design of the two-liter plastic soda bottle Bill explains how the two-liter plastic soda bottle begins as a plastic tube, called a preform, which is heated and inflated with air in a bottle-shaped mold. He explains how the stretching of the preform creates a crystalline regions in the bottles plastic (polyethylene terephthalate) that create a bottle with great strength, low permeability to carbon dioxide, but which is also lightweight—some 35 times lighter than a glass bottle of the same size. Bill explains key features of the bottles design, including: why the bottle looks like it does, why the neck has gaps in its threads, and how the tamper-proof ring works. He also discusses hot-fill bottles used for sports drinks and plastic juice bottles, noting the panels molded into the bottles to accommodate temperature changes. Lastly, he discusses briefly the recycling of PET bottles, although noting that about 75% of the 500 billion PET bottles manufactured annually end up in landfills or are incinerated. _Video summary_ 00:04 Preform Plastic soda bottles begin as preforms (small tubes) created by injection molding. This preform is heated slightly, then inflated to create a bottle. (See Bills video on injection molding: https://www.youtube.com/watch?v=RMjtmsr3CqA) 00:31 Inflation in a mold In manufacturing the preform is inflated in a mold in two steps — by a blast of air at 100 PSI initially, then a blast at 580 PSI — to form the bottle in a fraction of a second. 01:13 Cost of air The energy to generate the compressed air is the largest cost after the plastic itself. 01:31 Stretching strengths plastic A plastic soda bottle is strong enough that it will neither burst nor deform when dropped from two meters onto a concrete floor 02:19 Pressure from carbonation The bottle must also be strong to contain the carbon dioxide dissolved in the soda, which exerts a pressure of about 70 psi/4.8 bar. 02:37 Strength created by stretching the plastic A plastic soda bottle is made from PET, polyethylene terephthalate. When the bottle is stretched small crystalline regions are formed, which are oriented in the two directions of stretch: along the axial and hoop directions. This orientation of the crystalline regions (about 25% of bottles wall) increases the strength of the plastic. 03:36 Other plastics too weak The plastic used in a soda bottle is strong enough to keep the carbonation of the soda from distorting it, unlike a plastic milk jug, which is made from a different type of plastic (high density polyethylene). 03:58 Barrier to permeation of CO2 The stretch induced crystallization reduces carbon dioxide leakage: a 2 liter plastic soda bottle loses 15% of its carbonation in six weeks. Although this isnt as good as a glass bottle, its good enough. 04:40 Shape of the bottle The bottle has a hemispherical shape at the top so the pressure is distributed evenly on the wall. Bottles from the 1970s had hemispherical bottoms, which were covered with a plastic cap so the bottle could sit on the table. 05:20 Petaloid bottom To get rid of the cap, manufacturers invented a petaloid bottom — meaning like a flower. This petaloid bottom provides stability while maintaining strength. 05:36 Neck of the bottle The bottle neck features specific engineering elements: a lip for the mold, a beveled lip for the tamper-proof ring, and discontinuous threads to release pressure. Engineers have reduced the neck size to save plastic (about 2g per bottle). 06:59 Tests on neck To engineer a new neck for the bottle engineers used trial and error. They subjected the neck to many tests. 07:39 Hot fill bottles Hot-fill bottles are designed for beverages requiring higher fill temperatures. They have special panels to accommodate contraction and expansion. 09:39 Patent disputes Patent disputes in the 1990s delayed implementation of the petaloid and hot-fill bottle bottom designs. 11:07 Recycling & disposal Approximately 500 billion PET bottles are manufactured yearly; about 75% end up in landfills or are incinerated. Yet, PET is recyclable and can be converted into fibers for fabrics used in sports jerseys and car upholstery. 12:03 End credits & blooper$

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(4/4) Operation: The details of setting up the Harmonic Analyzer