## Posts Tagged ‘friction’

Monday, 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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Sunday, 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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Friday, 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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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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Tuesday, 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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Wednesday, 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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Monday, 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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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, January 16th, 2017
We left off last time with an engineering analysis of energy factors within a compound pulley scenario, in our case a Grecian man lifting an urn. We devised an equation to quantify the amount of work effort he exerts in the process. That equation contains two terms, one of which is beneficial to our lifting scenario, the other of which is not. Today we’ll explore these two terms and in so doing show how there are situations when* work input does not equal work output.*
__Work Input Does Not Equal Work Output __
Here again is the equation we’ll be working with today,
*WI = *(*F ×* *d*) *+ *(*F*_{F} *×* *d*) (1)
where, *F* is the entirely positive force, or *work,* exerted by human or machine to lift an object using a compound pulley. It represents an ideal but not real world scenario in which no friction is present within the pulley assembly.
The other force at play in our lifting scenario, *F*_{F,} is less obvious to the casual observer. It’s the force, or *work,* which must be employed over and above the initial positive force to overcome the friction that’s always present between moving parts, in this case a rope moving through pulley wheels. The rope length extracted from the pulley to lift the object is *d*.
Now we’ll use this equation to understand why *work input*, *WI,* *does not equal* *work output*, *WO*, in a compound pulley arrangement where friction is present.
The first term in equation (1), (*F ×* *d*), represents the *work input* as supplied by human or machine to lift the object. It is an idealistic scenario in which 100% of energy employed is directly conveyed to lifting. Stated another way, (*F ×* *d*) is entirely converted into beneficial *work* effort, *WO*.
The second term, (*F*_{F} *×* *d*), is the additional *work input* that’s needed to overcome frictional resistance present in the interaction between rope and pulley wheels. It represents lost *work* effort and makes no contribution to lifting the urn off the ground against the pull of gravity. It represents the heat energy that’s created by the movement of rope through the pulley wheels, heat which is entirely lost to the environment and contributes nothing to *work output*. Mathematically, this relationship between *WO, WI,* and friction is represented by,
*WO = WI –* (*F*_{F} *×* *d*) (2)
In other words, *work input* is *not equal* to *work output* in a real world situation in which pulley wheels present a source of friction.
Next time we’ll run some numbers to demonstrate the inequality between *WI* and *WO*.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: compound pulley, engineering, friction, heat energy, pulley, work input, work output

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Saturday, January 7th, 2017
Last time we saw how the presence of *friction* reduces mechanical advantage in an engineering scenario utilizing* a compound pulley*. We also learned that the actual amount of effort, or force, required to lift an object is a combination of the portion of the force which is hampered by *friction* and an idealized scenario which is *friction*-free. Today we’ll begin our exploration into how *friction* *results* in reduced *work input*, manifested as *heat* energy *lost* to the environment. The net result is that *work* input does not equal *work output* and some of Mr. Toga’s labor is unproductive.
__Friction Results in Heat and Lost Work Within a Compound Pulley__
In a past blog, we showed how the actual force required to lift our urn is a combination of *F*, an ideal *friction*-free work effort by Mr. Toga, and *F*_{F} , the extra force he must exert to overcome *friction* present in the wheels,
*F*_{Actual} = *F + F*_{F} (1)
Mr. Toga is clearly *working* to lift his turn, and generally speaking his *work* effort, *WI*, is defined as the force he employs multiplied by the length, *d*, of rope that he pulls out of the compound pulley during lifting. Mathematically that is,
*WI = F*_{Actual} × *d* (2)
To see what happens when *friction* enters the picture, we’ll first substitute equation (1) into equation (2) to get *WI* in terms of *F* and *F*_{F},
*WI = *(*F + F*_{F} ) × *d* (3)
Multiplying through by *d*, equation (3) becomes,
*WI = *(*F ×* *d *)*+ *(*F*_{F} × *d*) (4)
In equation (4) *WI* is divided into two terms. Next time we’ll see how one of these terms is beneficial to our lifting scenario, while the other is not.
Copyright 2017 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
____________________________________ |

Tags: compound pulley, engineering, friction, heat energy, lost work, mechanical advantage, pulley, reduced work, work input, work output

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