May 5th, 2017
Last time we introduced the Pulley Speed Ratio Formula, a *Formula* which assumes a certain amount of friction in a pulley-belt assembly in order to work. Today we’ll introduce another *Formula,* one which oversees how friction comes into play between belts and pulleys, the *Euler-Eytelwein Formula*. It’s a *Formula *developed by two pioneers of engineering introduced in an earlier blog, *Leonhard Euler* and *Johann Albert Eytelwein*.
Here again is the Pulley Speed Ratio Formula,
*D*_{1} × *N*_{1} = D_{2} × *N*_{2}
where, *D*_{1} is the diameter of the driving pulley and *D*_{2} the diameter of the driven pulley. The pulleys’ rotational speeds are represented by *N*_{1} and *N*_{2}.
This equation works when it operates under the assumption that friction between the belt and pulleys is, like Goldilock’s preferred bed, “just so.” Meaning, friction present is high enough so the belt doesn’t slip, yet loose enough so as not to bring the performance of a rotating piece of machinery to a grinding halt.
Ideally, you want no slippage between belt and pulleys, but the only way for that to happen is if you have perfect friction between their surfaces—something that will never happen because there’s always some degree of slippage. So how do we design a pulley-belt system to maximize friction and minimize slip?
Before we get into that, we must first gain an understanding of how friction comes into play between belts and pulleys. To do so we’ll use the famous *Euler-Eytelwein Formula, *shown here,
**A First Look at the Euler-Eytelwein Formula**
where, *T*_{1 }and* T*_{2} are belt tensions on either side of a pulley.
We’ll continue our exploration of the *Euler-Eytelwein Formula* next time when we discuss the significance of its two sources of tension.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: belt, belt slippage, belt tension, drive belt, engineering, Euler-Eytelwein Formula, friction, mechanical power transmission, pulley, pulley belt system

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April 21st, 2017
Last time we saw how pulley diameter governs speed in engineering scenarios which make use of a belt and pulley system. Today we’ll see how this phenomenon is defined mathematically through *application* of the** ***Pulley Speed Ratio Formula*, which enables precise pulley diameters to be calculated to achieve specific rotational speeds. Today we’ll apply this *Formula* to a scenario involving a building’s ventilating system.
The *Pulley Speed Ratio Formula* is,
*D*_{1} × *N*_{1} = D_{2} × *N*_{2} (1)
where, *D*_{1} is the diameter of the driving pulley and *D*_{2} the diameter of the driven pulley.
__A Pulley Speed Ratio Formula Application__
The pulleys’ rotational speeds are represented by *N*_{1} and *N*_{2}, and are measured in revolutions per minute (*RPM*).
Now, let’s apply Equation (1) to an example in which a blower must deliver a specific air flow to a building’s ventilating system. This is accomplished by manipulating the ratios between the driven pulley’s diameter, *D*_{2}, with respect to the driving pulley’s diameter, *D*_{1}. If you’ll recall from our discussion last time, when both the driving and driven pulleys have the same diameter, the entire assembly moves at the same speed, and this would be bad for our scenario.
An electric motor and blower impeller moving at the same speed is problematic because electric motors are designed to spin at much faster speeds than typical blower impellers in order to produce desired air flow. If their pulleys’ diameters were the same size, it would result in an improperly working ventilating system in which air passes through the furnace heat exchanger and air conditioner cooling coils far too quickly to do an efficient job of heating or cooling.
To bear this out, let’s suppose we have an electric motor turning at a fixed speed of 3600 *RPM* and a belt-driven blower with an impeller that must turn at 1500 *RPM* to deliver the required air flow according to the blower manufacturer’s data sheet. The motor shaft is fitted with a pulley 3 inches in diameter. What pulley diameter do we need for the blower to turn at the manufacturer’s required 1500 *RPM*?
In this example known variables are *D*_{1} = 3 *inches*, *N*_{1} = 3600 *RPM*, and *N*_{2} = 1500 *RPM*. The diameter *D*_{2} is unknown. Inserting the known values into equation (1), we can solve for *D*_{2},
(3 *inches*) × (3600 *RPM*) *=* *D*_{2} × (1500 *RPM*) (2)
Simplified, this becomes,
*D*_{2} *=* 7.2 inches (3)
Next time we’ll see how friction affects our scenario.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: belt, blower, blower impeller, cooling coils, drive belt, driven pulley, driving pulley, electric motor, engineering, heat exchanger, mechanical power transmission, pulley, pulley speed, Pulley Speed Ratio Formula, RPM, ventilating system

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April 8th, 2017
Soon after the first *pulleys* were used with belts to transmit mechanical power, engineers such as Leonhard Euler and Johann Albert Eytelwein discovered that the *diameter* of the *pulleys* used *determined* the *speed* at which they rotated. This allowed for a greater diversity in mechanical applications. We’ll set up an examination of this phenomenon today.
Last time we introduced this basic mechanical power transmission system consisting of a driving pulley, a driven pulley, and a belt, which we’ll call Situation A.
__A Driven Pulley’s Larger Diameter Determines a Slower Speed__
In this situation, the rotating machinery’s driven *pulley diameter *is larger than the electric motor’s driving *pulley diameter*. The result is the driven *pulley* turns at a slower speed than the driving *pulley.*
Now let’s say we need to *speed *the rotating machinery up so it produces more widgets per hour. In that case we’d make the driven *pulley* smaller, as shown in Situation B.
__A Driven Pulley’s Smaller Diameter Determines a Faster Speed__
With the smaller *diameter *driven *pulley*, the rotating machinery will operate faster than it did in Situation A.
Next week we’ll introduce the *Pulley Speed Ratio Formula*, which mathematically defines this phenomenon.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: belt, driven pulley, driving pulley, electric motor, engineering, Johann Albert Eytelwein, Leonhard Euler, pulley diameter, Pulley Speed Ratio Formula, rotating machinery, rotational speed

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March 31st, 2017
Last time we introduced two historical legends in the field of engineering who pioneered the science of mechanical power transmission using belts and *pulleys*, Leonhard Euler and Johann Albert Eytelwein. Today we’ll build a foundation for understanding their famous *Euler-Eytelwein Formula *through our example of a simple mechanical power transmission system consisting of two pulleys and a belt, and in so doing demonstrate *the difference between driven and driving pulleys*.
Our example of a basic mechanical power transmission system consists of two *pulleys *connected by a drive belt. The *driving pulley* is attached to a source of mechanical power, for example, the shaft of an electric motor. The *driven pulley*, which is attached to the shaft of a piece of rotating machinery, receives the mechanical power from the electric motor so the machinery can perform its function.
**The Difference Between Driven and Driving Pulleys **
Next time we’ll see how *driven pulleys* can be made to spin at different speeds from the *driving pulley,* enabling different modes of operation in mechanical devices.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: belt, driven pulley, driving pulley, electric motor, engineering, Euler, Euler-Eytelwein Formula, Eytelwein, mechanical power, mechanical power transmission, pulley, pulley speed, rotating machinery

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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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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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February 28th, 2017
For some time now we’ve been analyzing the helpfulness of the engineering phenomena known as *pulleys* and we’ve learned that, yes, they can be very helpful, although they do have their limitations. One of those ever-present limitations is due to the inevitable presence of *friction* between moving parts. Like an unsummoned gremlin, *friction* will be standing by in any mechanical situation to put the wrench in the works. Today we’ll calculate just *how much friction is present* within the example *compound pulley *we’ve been working with. * *
__So How Much Friction is Present in our Compound Pulley?__
Last time we began our numerical demonstration of the inequality between a compound pulley’s work input, *WI*, and work output, *WO*, an inequality that’s due to friction in its wheels. We began things by examining a friction-free scenario and discovered that to lift an urn with a weight, *W*, of 40 pounds a distance, *d*_{1}, of 2 feet above the ground, Mr. Toga exerts a personal effort/force, *F*, of 10 pounds to extract a length of rope, *d*_{2}, of 8 feet.
In reality our compound pulley must contend with the effects of friction, so we know it will take more than 10 pounds of force to lift the urn, a resistance which we’ll notate *F*_{F}. To determine this value we’ll attach a spring scale to Mr. Toga’s end of the rope and measure his actual lifting force, *F*_{Actual}, represented by the formula,
*F*_{Actual} = *F + F*_{F }(1)
We find that *F*_{Actual} equals 12 pounds. Thus our equation becomes,
12 *Lbs = *10 *Lbs + F*_{F }(2)
which simplifies to,
2 *Lbs =* *F*_{F }(3)
Now that we’ve determined values for all operating variables, we can solve for work input and then contrast our finding with work output,
*WI = *(*F ×* *d*_{2}) *+ *(*F*_{F} *×* *d*_{2}) (4)
*WI = *(10* Lbs ×** *8* feet*) *+ *(2* Lbs ×** *8* feet*) (5)
*WI = *96* Ft-Lbs *(6)
We previously calculated work output, *WO* to be 80 *Ft-Lbs, *so we’re now in a position to calculate the difference between work input and work output to be,
*WI – WO =* 16 *Ft-Lbs* (7)
It’s evident that the amount of work Mr. Toga puts into lifting his urn requires 16 more Foot-Pounds of work input effort than the amount of work output produced. This extra effort that’s required to overcome the pulley’s friction is the same as the work required to carry a weight of one pound a distance of 16 feet. We can thus conclude that work input does not equal work output in a *compound pulley*.
Next time we’ll take a look at a different use for pulleys beyond that of just lifting objects.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: compound pulley, engineering, friction, pulley, work, work input, work output

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February 15th, 2017
Last time we began *work* on a *numerical* demonstration and engineering analysis of the inequality of work input and output as experienced by our example persona, an ancient Greek lifting an urn. Today we’ll get two steps closer to demonstrating this reality as we *work a compound pulley’s numerical puzzle, *shuffling equations like a Rubik’s Cube to arrive at values for two variables crucial to our analysis, *d*_{2}, the length of rope he extracts from the *pulley* while lifting, and *F, *the force/effort required to lift the urn in an idealized situation where no friction exists. * *
__A Compound Pulley’s Numerical Puzzle is Like a Rubik’s Cube__
We’ll continue manipulating the work input equation, *WI,* as shown in Equation (1), along with derivative equations, breaking it down into parts, and handle the two terms within parentheses separately. Term one, (*F *× *d*_{2}), corresponds to the force/effort/work required to lift the urn in an idealized no-friction world. It’ll be our focus today as it provides a springboard to solving for variables *F* and *d*_{2}.
*WI = *(*F ×* *d*_{2}) *+ *(*F*_{F} *×* *d*_{2}) (1)
Previously we learned that when friction is present, work output, *WO,* is equal to work input minus the work required to oppose friction while lifting. Mathematically that’s represented by,
*WO = WI – *(*F*_{F} ×* d*_{2}) (2)
We also previously calculated *WO* to equal 80 *Ft-Lbs*. To get *F* and *d*_{2} into a relationship with terms we already know the value for, namely *WO*, we substitute Equation (1) into Equation (2) and arrive at,
80 *Ft-Lbs = *(*F ×* *d*_{2}) *+ *(*F*_{F} *×* *d*_{2})* – *(*F*_{F} *×* d_{2}) (3)
simplified this becomes,
80 *Ft-Lbs =* *F ×* *d*_{2} (4)
To find the value of *d*_{2}, we’ll return to a past equation concerning compound pulleys derived within the context of *mechanical advantage, MA*. That is,
*d*_{2} ÷ *d*_{1 }= _{ }MA_{ }(5)
And because in our example four ropes are used to support the weight of the urn, we know that *MA* equals 4. We also know from last time that* d*_{1} equals 2 feet. Plugging these numbers into Equation (5) we arrive at a value for *d*_{2},
*d*_{2} ÷ 2* ft*_{ }= _{ }4 (6)
*d*_{2} *= *4 *×* 2* ft *(7)
*d*_{2} = 8* ft *(8)
Substituting Equation (8) into Equation (4), we solve for *F,*
80 *Ft-Lbs =* *F ×* 8* ft *(9)
*F = 10 Pounds *(10)
Now that we know *F* and *d*_{2} we can solve for *F*_{F}, the amount of extra effort required by man or machine to overcome friction in a *compound pulley* assembly. It’s the final piece in the *numerical puzzle* which will then allow us to compare work input to output.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: compound pulley, engineering, friction, friction force, lifting force, work input, work output

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February 6th, 2017
Last time we performed an engineering analysis of a *compound pulley* which resulted in an equation *comparing* the amount of true work effort, or *work input,* *WI*, required by machine or human to lift an object, in our case a toga’d man lifting an urn. Our analysis revealed that, in real world situations, work input does not equal *work output,** WO,* due to the presence of friction. Today we’ll begin to numerically demonstrate their inequality by first solving for work output, and later *work input.*
__Comparing Work Input to Output in a Compound Pulley__
To solve for the *work output* of our compound pulley, we’ll use an equation provided previously that is in terms of the variables *W* and *d*_{1},
*WO = W ×* *d*_{1} (1)
In our example Mr. Toga lifts an urn of weight, *W,* equal to 40 pounds to a height, or distance off the floor, *d*_{1}, of 2 feet. Inserting these values into equation (1) we arrive at,
*WO = *40* pounds ×* * *2* feet* *= 80 Ft-Lbs* (2)
where, *Ft-Lbs* is a unit of *work* which denotes *pounds* of force moving through *feet* of distance.
Now that we’ve calculated the* ***work output****,** we’ll turn our attention to the previously-derived equation for *work input**,* shown in equation (3). Interrelating equations for *WO* and *WI* will enable us to solve for unknown variables, including the force, *F,* required to lift the urn and the length of rope, *d*_{2}, extracted during lifting.** **Once *F* and *d*_{2} are known, we can solve for the additional force required to overcome friction, *F*_{F, }then finally we’ll solve for *WI*.
Once again, the equation we’ll be working with is,
*WI = *(*F ×* *d*_{2}) *+ *(*F*_{F} *×* *d*_{2}) (3)
To calculate *F*, we’ll work the two terms present within parentheses separately, then use knowledge gained to further work our way towards a numerical *comparison* of *work input* and *work output*. We’ll do that next time.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: compound pulley, engineering analysis, friction, weight, work input, work output

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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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