July 17th, 2017
Sometimes things which appear simple turn out to be rather complex. Such is the case with the Euler-Eytelwein Formula, a small formula with a big job. It computes how friction, an omnipresent phenomenon in mechanical assemblies, contributes to the transmission of mechanical power. Today we’ll *determine *the value of one of the Euler-Eytelwein Formula’s variables, the *angle of wrap*.
__Determining Angle of Wrap__
Here again is the basis for our calculations, the Euler-Eytelwein Formula.
*T*_{1} = T_{2 ×}* e*^{(μ}^{θ)} (1)
To recap what we’ve discussed thus far, *T*_{1 }is the tight side tension, the maximum the belt can endure before breaking. *T*_{2} is the loose side tension. It’s just going along for the ride. The term *e* is Euler’s Number, a constant equal to 2.718, and the coefficient of friction, *μ*, for contact points between the belt and pulleys is 0.3 based on their materials.
The *formula* introduced last time to calculate the angle of wrap, *θ*, is,
*θ = *(180* – *2*α*) × (*π ÷** *180) (2)
where,
*α = sin*^{-1}((*D*_{1} – D_{2}) *÷** *2*x*) (3)
By direct measurement we’ve determined the pulleys’ diameters, *D*_{1 }and *D*_{2}, are equal to 1 foot and 0.25 feet respectively. The term *x* is the distance between the two pulley shafts, 3 feet. The term *sin*^{-1 }is a trigonometric function known as *inverse sine*, a button commonly found on scientific calculators.
Inserting our known values into equation (3) we arrive at,
*α = sin*^{-1}((1.0* foot – *0.25* feet*) *÷** *2 ×* *(3* feet*)) (4)
*α = *7.18 (5)
We can now incorporate equation (5) into equation (2) to solve for *θ*,
*θ = *(180* – *(2 × 7.18)) ×* *(*π ÷** *180) (6)
*θ = *2.89 (7)
Inserting the values for *m* and *θ* into equation (1) we arrive at,
*T*_{1} = T_{2 }*× *2.718^{(}^{0.3 ×}^{ 2.89)} (8)
*T*_{1} = 2.38T_{2} (9)
We have at this point solved for over half of the unknown variables in the Euler-Eytelwein Formula. We still can’t solve for *T*_{1}, because we don’t know the value of *T*_{2}. But that will change next time when we introduce yet another *formula,* this one to determine the amount of mechanical power present in our pulley-belt system.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: angle of wrap, belt, coefficient of friction, Euler-Eytelwein Formula, Euler's Number, friction, loose side tension, mechanical assemblies, mechanical power, pulley, pullies, tight side tension

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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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June 23rd, 2017
*Belts* are important. They make fashion statements, hold things up, keep things together. Today we’re introducing a scenario in which the *Euler-Eytelwein Formula *will be used to, among other things, determine the ideal *width* of a *belt* to be used in a mechanical power transmission system consisting of two pulleys inside a *hydroponics plant*. The ideal *width belt* would serve to maximize friction between the *belt* and pulleys, thus controlling slippage and maximizing belt strength to prevent belt breakage.
An engineer is tasked with designing an irrigation system for a *hydroponics plant.* Pulley 1 is connected to the shaft of a water pump, while Pulley 2 is connected to the shaft of a small gasoline engine.
__What Belt Width does a Hydroponics Plant Need?__
Mechanical power is transmitted by the *belt* from the engine to the pump at a constant rate of 4 horsepower. The *belt* material is leather, and the two pulleys are made of cast iron. The coefficient of friction, *μ*, between these two materials is 0.3, according to *Marks Standard Handbook for Mechanical Engineers*. The *belt* manufacturer specifies a safe working tension of 300 pounds force per inch *width* of the *belt*. This is the maximum tension the *belt* can safely withstand before breaking.
We’ll use this information to solve for the ideal *belt width* to be used in our *hydroponics* application. But first we’re going to have to re-visit the two T’s of the *Euler-Eytelwein Formula.* We’ll do that next time.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: belt, belt breakage, coefficient of friction, engine, engineer, horsepower, leather belt, mechanical power transmission, pulley, pump

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June 13th, 2017
Last week we saw how *friction coefficients *as used in *the Euler-Eyelewein Formula,* can be highly specific to a *specialized application,* U.S. Navy ship capstans. In fact, many diverse industries benefit from aspects of the Euler-Eytelwein Formula. Today we’ll introduce *another* engineering *application* *of the* *Formula, *exploring its use within the irrigation system of a hydroponics plant.
__Another Specialized Application of the Euler-Eyelewein Formula__
Pumps conveying water are an indispensable part of a hydroponics plant. In the schematic shown here they are portrayed by the symbol ⊗.
In our simplified scenario to be presented next week, these pumps are powered by a mechanical power transmission system, each consisting of two pulleys and a belt. One pulley is connected to a water pump, the other pulley to a gasoline engine. A belts runs between the pulleys to deliver mechanical power from the engine to the pump.
The width of the belts is a key component in an efficiently running hydroponics plant. We’ll see how and why that’s so next time.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: belt, belt width, coefficient of friction, engineering, Euler-Etytelwein Formula, gasoline engine, hydroponics, mechanical power transmission, power, pulley, pump

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June 4th, 2017
We’ve been talking about pulleys for awhile now, and last week we introduced the term friction coefficient, numerical values derived during testing which quantify the amount of *friction* present when different materials interact. *Friction coefficients* for common materials are routinely presented in engineering texts like *Marks’ Standard Handbook for Mechanical Engineers*. But there are circumstances when more specificity is required, such as when the U.S. *Navy*, more specifically the Navy Material Command, tested the interaction between various synthetic ropes and ship *capstans* and *developed* their own *specialized friction coefficients* in the process.
__Navy Capstans and the Development of Specialized Friction Coefficients__
*Capstans* are similar to pulleys but have one key difference, they’re made so rope can be wound around them multiple times. When the *Navy *set out to determine which synthetic rope worked best with their *capstans,* they did testing and *developed highly specialized friction coefficients* in the process. This research was at one time Top Secret but has now been declassified. To read more about it, follow this link to the actual handbook:
https://archive.org/stream/DTIC_ADA036718#page/n0/mode/2up
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: capstan, engineering, friction, friction coefficient, pulley, rope

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May 25th, 2017
Last time we learned that the two T’s *in the* *Euler-Eytelwein Formula* correspond to different belt tensions on either side of a pulley wheel in a pulley-belt assembly. Today we’ll see what the remaining variables in this famous *Formula *are all about, paying special attention to the *angle of wrap* that’s formed by the belt *wrapping* around the pulley wheel.
__Angle of Wrap in the Euler-Eytelwein Formula__
Here again is the *Euler-Eytelwein Formula*,
*T*_{1} = T_{2}* × e*^{(μ}^{θ)}
The tight side tension, *T*_{1}, is equal to a combination of factors, namely: loose side tension *T*_{2} ; the friction that exists between the belt and pulley, denoted as *μ *; and the length of belt coming in direct contact with the pulley, namely, *θ*. These last two terms are exponents of the term, *e*, known as Euler’s Number, a mathematical constant used in many circles, including science, engineering, and economics, to calculate a wide variety of things, from bell curves to compound interest rates. It’s a rather esoteric term, much like the term π that’s used to calculate values associated with circles.
Euler’s Number was discovered in 1683 by Swiss mathematician Jacob Bernoulli, but oddly enough was named after Leonhard Euler. Its value, 2.718, was determined while Bernoulli manipulated high level mathematics to calculate compound interest rates. If you’d like to learn more about that, follow this link.
The term *μ* is known as the friction coefficient. It quantifies the degree of friction, or roughness, present between the belt and pulley where they make contact. It’s a specific number that remains constant for a given combination of materials. For example, according to *Marks’ Standard Handbook for Mechanical Engineers, *the value of *μ* for a leather belt operating on an iron pulley is 0.38. The numerical values for these coefficients were determined over the last few centuries by engineers conducting laboratory testing on various belt and pulley materials. They’re now routinely found in a variety of engineering texts and handbooks.
Finally, the term *θ* denotes the *angle of wrap* that the belt makes while in contact with the face of the pulley. In our example illustration above, *θ* measures the arc that’s formed by the belt riding along the surface of the pulley between points A and B, as shown by dotted lines. The *angle of wrap* is important to overall functionality of the assembly, because the proper amount of friction will allow the pulley-belt assembly to operate efficiently and without slippage.
Next time we’ll present an example and use the *Euler-Eytelwein Formula* to calculate optimal belt friction within a pulley-belt assembly.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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May 14th, 2017
Last time we introduced some of the variables in *the Euler-Eytelwein Formula,* an equation used to examine the amount of friction present in pulley-belt assemblies. Today we’ll explore its two tension-denoting variables, *T*_{1 }and* T*_{2}.
Here again is *the Euler-Eytelwin Formula,*where, *T*_{1 }and * T*_{2} are belt tensions on either side of a pulley,
*T*_{1} = T_{2} × *e*^{(μ}^{θ)}
*T*_{1} is known as the *tight side tension* of the assembly because, as its name implies, the side of the belt containing this tension is tight, and that is so due to its role in transmitting mechanical power between the driving and driven pulleys. *T*_{2} is the *loose side tension* because on its side of the pulley no mechanical power is transmitted, therefore it’s slack–it’s just going along for the ride between the driving and driven pulleys.
Due to these different roles, the tension in *T*_{1} is greater than it is in *T*_{2}.
__The Two T’s of the Euler-Eytelwein Formula__
In the illustration above, tension forces *T*_{1 }and* T*_{2} are shown moving in the same direction, because the force that keeps the belt taught around the pulley moves outward and away from the center of the pulley.
According to the *Euler-Eytelwein Formula*, *T*_{1} is equal to a combination of factors: tension *T*_{2 }; the friction that exists between the belt and pulley, denoted as *μ*; and how much of the belt is in contact with the pulley, namely *θ*.
We’ll get into those remaining variables next time.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
____________________________________ |

Tags: belt, belt tension, driven pulley, driving pulley, engineering, Euler-Eytelwein Formula, mechanical power transmission, pulley

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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
____________________________________ |

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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