## Posts Tagged ‘engineering’

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

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Tuesday, December 13th, 2016
The presence of *friction* in *mechanical *designs is as guaranteed as conflict in a good movie, and engineers inevitably must deal with the conflicts *friction* produces within their *mechanical* designs. But unlike a good movie, where conflict presents a positive, engaging force, *friction’s *presence in *pulleys* results only in impediment, wasting energy and *reducing* *mechanical advantage*. We’ll investigate the math behind this phenomenon in today’s blog.
__Friction Reduces Pulleys’ Mechanical Advantage__
A few blogs back we performed a work input-output analysis of an idealized situation in which no *friction* is present in a compound pulley. The analysis yielded this equation for *mechanical advantage*,
*MA = d*_{2} ÷* d*_{1} (1)
where *d*_{2} is the is the length of rope Mr. Toga extracts from the *pulley* in order to lift his urn a distance *d*_{1} above the ground. Engineers refer to this idealized frictionless scenario as an *ideal mechanical advantage*, *IMA*, so equation (1) becomes,
*IMA = d*_{2} ÷* d*_{1} (2)
We also learned that in the idealized situation *mechanical advantage* is the ratio of the urn’s weight force, *W,* to the force exerted by Mr. Toga, *F,* as shown in the following equation. See our past blog for a refresher on how this ratio is developed.
*IMA = W ÷** F* (3)
In reality, friction exists between a *pulley’s* moving parts, namely, its wheels and the rope threaded through them. In fact, the more *pulleys* we add, the more *friction *increases.
The actual amount of lifting force required to lift an object is a combination of *F*_{F }, the friction-filled force, and *F*, the idealized friction-free force. The result is *F*_{Actual} as shown here,
*F*_{Actual} = *F + F*_{F} (4)
The real world scenario in which *friction* is present is known within the engineering profession as *actual mechanical advantage*, *AMA, *which is equal to,
*AMA = W ÷** F*_{Actual} (5)
To see how *AMA* is affected by *friction* force *F*_{F}, let’s substitute equation (4) into equation (5),
*AMA = W ÷** *(*F + F*_{F}) (6)
With the presence of *F*_{F} in equation (6), *W* gets divided by the sum of *F *and *F*_{F} . This results in a smaller number than *IMA,* which was computed in equation (3). In other words, *friction* *reduces* the actual* ***mechanical advantage** of the compound *pulley*.
Next time we’ll see how the presence of *F*_{F} translates into lost work effort in the compound *pulley,* thus creating an inequality between the work input, *WI *and work output *WO*.
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: actual mechanical advantage, AMA, compound pulley, engineering, friction, friction force, ideal mechanical advantage, IMA, mechanical advantage, mechanical design, pulley, pulley friction, pulley work input, pulley work output, weight force

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Sunday, November 6th, 2016
In our last blog we saw how adding extra *pulleys* resulted in *mechanical advantage *being doubled, which translates to a 50% decreased lifting effort over a previous scenario. *Pulleys* are engineering marvels that make our lives easier. Theoretically, the more *pulleys* you add to a compound pulley arrangement, the greater the mechanical advantage — up to a point. Eventually you’d encounter undesirable tradeoffs. We’ll examine those tradeoffs, but before we do we’ll need to revisit the engineering principle of *work* and see how it applies to compound *pulleys* as a *work input-output device.*
__Pulleys as a Work Input-Outut Device__
The compound *pulley* arrangement shown includes distance notations, *d*_{1} and *d*_{2}. Their inclusion allows us to see it as a *work* *input-output device.* *Work* is *input* by Mr. Toga, we’ll call that *WI*, when he pulls his end of the rope using his bicep force, *F*. In response to his efforts, *work* is *output *by the compound *pulley *when the urn’s weight, *W, *is lifted off the ground against the pull of gravity. We’ll call that *work output* *WO.*
In a previous blog we defined *work* as a factor of force multiplied by distance. Using that notation, when Mr. Toga exerts a force *F* to pull the rope a distance *d*_{2} , his *work input* is expressed as,
*WI = F ×** d*_{2}
When the compound *pulley* lifts the urn a distance *d*_{1} above the ground against gravity, its *work output* is expressed as,
*WO = W ×** d*_{1}
Next time we’ll compare our *pulley’s* *work input to output* to develop a relationship between *d*_{1} and *d*_{2}. This relationship will illustrate the first undesirable tradeoff of adding too many *pulleys*.
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: compound pulley, distance, engineering, engineering principle, force, mechanical advantage, pulley, weight, work, work input-output device, work of lifting

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Thursday, October 27th, 2016
Last time we saw how compound *pulley**s* within a dynamic lifting scenario result in *increased* *mechanical advantage *to the lifter, *mechanical advantage* being an engineering phenomenon that makes lifting weights easier. Today we’ll see how the *mechanical advantage *increases when *more* fixed and movable *pulleys *are added to the compound pulley arrangement we’ve been working with.
__More Pulleys Increase Mechanical Advantage__
The image shows a *more* complex compound *pulley* than the one we previously worked with. To determine the *mechanical advantage* of this pulley, we need to determine the force, *F*_{5}, Mr. Toga exerts to hold up the urn.
The urn is directly supported by four equally spaced rope sections with tension forces *F*_{1}, *F*_{2}, *F*_{3}, and *F*_{4}. The weight of the urn, *W,* is distributed equally along the rope, and each section bears one quarter of the load. Mathematically this is represented by,
*F*_{1} = F_{2} = F_{3} = F_{4 }= W ÷* *4
If the urn’s weight wasn’t distributed equally, the bar directly above it would tilt. This tilting would continue until equilibrium was eventually established, at which point all rope sections would equally support the urn’s weight.
Because the urn’s weight is equally distributed along a single rope that’s threaded through the entire *pulley* arrangement, the rope rule, as I call it, applies. The rule posits that if we know the tension in one section of rope, we know the tension in *all* rope sections, including the one Mr. Toga is holding onto. Therefore,
*F*_{1} = F_{2} = F_{3} = F_{4 }= F_{5 }= W ÷* *4
Stated another way, the force, *F*_{5} , Mr. Toga must exert to keep the urn suspended is equal to the weight force supported by each section of rope, or one quarter the total weight of the urn, represented by,
*F*_{5 }= W ÷* *4
If the urn weighs 40 pounds, Mr. Toga need only exert 10 pounds of bicep force to keep it suspended, and today’s compound *pulley* provides him with a *mechanical advantage, MA,* of,
*MA = W ÷** F*_{5}
*MA =* *W ÷** (W ÷* 4*) *
*MA = 4*
It’s clear that adding the two extra *pulleys* results in a greater benefit to the man doing the lifting, decreasing his former weight bearing load by 50%. If we added even more *pulleys,* we’d continue to increase his *mechanical advantage,* and he’d be able to lift far heavier loads with a minimal of effort. Is there any end to this *mechanical advantage?* No, but there are undesirable tradeoffs. We’ll see that next time.
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: compound pulley, engineering, fixed pulley, lifting, mechanical advantage, movable pulley, rope section, tension force

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