## Posts Tagged ‘pulleys’

Wednesday, July 5th, 2017
Last time we introduced a scenario involving a hydroponics plant powered by a gas engine and multiple pulleys. Connecting the pulleys is a flat leather belt. Today we’ll take a step further towards determining what width that belt needs to be to maximize power transmission efficiency. We’ll begin by revisiting the two T’s of the *Euler-Eytelwein Formula *and introducing a *formula* to determine a key variable, *angle of wrap.*
__The Angle of Wrap Formula__
We must start by calculating *T*_{1, }the tight side tension of the belt, which is the maximum tension the belt is subjected to. We can then calculate the width of the belt using the manufacturer’s specified safe working tension of 300 pounds per inch as a guide. But first we’ll need to calculate some key variables in the *Euler-Eytelwein Formula, *which is presented here again,
*T*_{1} = T_{2}×* e*^{(μ}^{θ)} (1)
We determined last time that the coefficient of friction, μ, between the two interfacing materials of the belt and pulley are, respectively, leather and cast iron, which results in a factor of 0.3.
The other factor shown as a exponent of *e* is the angle of wrap*,** θ*, and is calculated by the *formula,*
*θ = *(180* – *2*α*) ×* *(*π ÷** *180) (2)
You’ll note that this *formula* contains some unique terms of its own, one of which is familiar, namely *π*, the other, *α*, which is less familiar. The unnamed variable *α* is used as shorthand notation in equation (2), to make it shorter and more manageable. It has no particular significance other than the fact that it is equal to,
*α = sin*^{-1}((*D*_{1} – D_{2}) ÷* *2*x*) (3)
If we didn’t use this shorthand notation for *α*, equation (2) would be written as,
*θ = *(180* – *2*(sin*^{-1}((*D*_{1} – D_{2}) *÷** *2*x*))) ×* *(*π ÷** *180) (3a)
That’s a lot of parentheses!
Next week we’ll get into some trigonometry when we discuss the diameters of the pulleys, which will allow us to solve for *the angle of wrap.*
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: angle of wrap, belt, belt tension, coefficient of friction, Euler-Eytelwein Formula, mechanical power transmission, pulley, pulleys

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Monday, March 20th, 2017
They say necessity is the mother of invention, and today’s look at an influential historical figure in engineering bears that out. Last week we introduced Leonhard Euler and touched on his influence to the science of pulleys. Today we’ll introduce his contemporary and partner in science, *Johann Albert Eytelwein*, a German mathematician and visionary, a true *engineering trailblazer* whose contributions to the blossoming discipline of engineering led to later studies with pulleys.
__Johann Albert Eytelwein, Engineering Trailblazer__
*Johann Albert Eytelwein’s* experience as a civil *engineer* in charge of the dikes of former Prussia led him to develop a series of practical mathematical problems that would enable his subordinates to operate more effectively within their government positions. He was a *trailblazer* in the field of applied mechanics and their application to physical structures, such as the dikes he oversaw, and later to machinery. He was instrumental in the founding of Germany’s first university level *engineering* school in 1799, the Berlin Bauakademie, and served as director there while lecturing on many developing *engineering* disciplines of the time, including machine design and hydraulics. He went on to publish in 1801 one of the most influential *engineering* books of his time, entitled *Handbuch der Mechanik* (Handbook of the Mechanic), a seminal work which combined what had previously been mere *engineering *theory into a means of practical application.
Later, in 1808, *Eytelwein* expanded upon this work with his *Handbuch der Statik fester Koerper* (Handbook of Statics of Fixed Bodies), which expanded upon the work of Euler. In it he discusses friction and the use of pulleys in mechanical design. It’s within this book that the famous Euler-*Eytelwein* Formula first appears, a formula *Eytelwein* derived in conjunction with Euler. The formula delves into the usage of belts with pulleys and examines the tension interplay between them.
More on this fundamental foundation to the discipline of *engineering* next time, with a specific focus on pulleys.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: bel, engineering, friction, Johann Albert Eytelwein, mechanical power transmission, pulley, pulleys

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Thursday, March 9th, 2017
Last time we ended our blog series on *pulleys* and their application within engineering as aids to lifting. Today we’ll embark on a new focus series, *pulleys *used in mechanical devices. We begin with some *history**,* a peek at Swiss scientist and mathematician *Leonhard Euler,* *a historical figure* credited to be perhaps the greatest mathematician of the 18^{th} Century.
__Leonhard Euler, a Historical Figure in Pulleys__
*Euler* is so important to math, he actually has two numbers named after him. One is known simply as *Euler’s* Number, 2.7182, most often notated as *e*, the other *Euler’s* Constant, 0.57721, notated *γ*, which is a Greek symbol called gamma. In fact, he developed most math notations still in use today, including the infamous function notation, *f(x)*, which no student of elementary algebra can escape becoming intimately familiar with.
*Euler *authored his first theoretical essays on the science and mathematics of *pulleys* after experimenting with combining them with belts in order to transmit mechanical power. His theoretical work became the foundation of the formal science of designing pulley and belt drive systems. And together with German engineer Johann Albert Eytelwein, *Euler *is credited with a key formula regarding pulley-belt drives, the Euler-Eytelwein Formula, still in use today, and which we’ll be talking about in depth later in this blog series.
We’ll talk more about Eytelwein, another important historical figure who worked with *pulleys,* next time.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: belt and pulley drive systems, belts, engineering, Euler's Constant, Euler's Number, Johann Albert Eytelwein, Leonhard Euler, mechanical power transmission, pulleys

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Saturday, January 28th, 2017
In our blog series on *pulleys* we’ve been discussing the effects of *friction,* subjects also studied by Leonardo *da Vinci,* a *historical* figure whose genius contributed so much to the worlds of art, engineering, and science. The *tribometre* shown in his sketch here is one of history’s earliest recorded attempts to understand the phenomenon of *friction*. Tribology, according to the Merriam-Webster Dictionary, is “a study that deals with the design, friction, wear, and lubrication of interacting surfaces in relative motion.” Depicted in *da Vinci’s* sketch are what appear to be *pulleys *from which dangle objects in mid-air.
__da Vinci’s Tribometre; a Historical Look at Pulleys and Friction__
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: engineering, friction, Leonardo Da Vinci, pulleys, tribometre

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Monday, December 19th, 2016
Tags: engineering expert, engineering expert witness, pulleys

Posted in Courtroom Visual Aids, Engineering and Science, Expert Witness, Forensic Engineering, Innovation and Intellectual Property, Personal Injury, power plant training, Product Liability, Professional Malpractice | Comments Off on Pulleys Make Santa’s Job Easier

Friday, July 8th, 2016
Lifting heavy objects into position always presents a challenge, whether it’s a mom *lifting* a toddler to her hip or a construction worker *lifting* work materials to great heights. During my career as an engineering expert I’ve dealt with similar challenges, some of which were handled quite nicely by incorporating a simple pulley, which we introduced last time, into my design. But sometimes, due to certain restrictions, the addition of a *simple pulley* into the works isn’t enough to get the job done. We’ll take a look at one of the restrictions working against the use of a *simple pulley* today.
The *simple pulley* is believed to have first been used by the Greeks as far back as the 9^{th} Century BC. Back then it would have come in handy to lift cargo aboard ships, hoist sails on masts, and *lift *building materials high off the ground to supply workmen during the construction of temples and other marvels of ancient architecture. In other words, *pulleys* literally saved ancient workers thousands of steps when it came to *lifting* things off the ground.
Let’s return to ancient times for a moment to get an understanding of the mechanics behind the workings of the *simple pulley* as put to use in a basic *lifting* application.
**The Simple Pulley Gives Us a Lift**
With a *simple pulley,* the tension force *F*_{1} applied to the rope by the pull-er is equal to the tension force *F*_{2} exerted upon the object, the pull-ee. Once lifted off the ground, these forces are also equal to the object’s weight, *W, *which gravity works upon to return the lifted object to its previous position on the ground. All these forces come to bear upon whatever’s doing the pulling. If this pull-er happens to be a human, then the *simple pulley’s* effectiveness to *lift* things is directly proportionate to that human’s strength. In the case of the toga’d figure above, that would be about 10 pounds. It’s this caveat that limits the usefulness of the *simple pulley* when relying on human power alone, particularly when it’s employed to *lift* extremely heavy objects like marble pillars. A single human isn’t up to the task.
Next time we’ll see how ancient Greeks overcame this limitation of the *simple pulley* by managing to cut in half the amount of brute force required to *lift* heavy objects.
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: cable, construction, engineering expert, force, hoist, lifting, pulleys, simple pulley, weight force

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Friday, June 17th, 2016
Ever seen that old movie where they’re lifting a grand piano to the top floor of a tall building with ropes? The huge piano dangles precariously in mid air by the ropes, which are attached to a rather simple looking wheeled device that’s situated at the top of the building. As men on the ground tug on the ropes, they hoist the piano higher and higher by increments of inches as the wheeled device the rope is threaded through spins madly. The piano’s formidable size appears to *magically* levitate off the ground, like in the famous *magician’s* trick. That object with the spinning wheel is a *pulley, *a rather simple device which I as an engineering expert have often made use of in my designs.
__So Where’s The Pulley?__
We’ll be talking about the various types of *pulleys* and their uses in future blogs, beginning with an exploration of a simple *pulley.*
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: engineering expert, pulley, pulleys, types of pulleys

Posted in Engineering and Science, Expert Witness, Forensic Engineering, Innovation and Intellectual Property, Personal Injury, Product Liability | Comments Off on Magic as Performed by Pulleys