## Posts Tagged ‘pulley’

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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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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Wednesday, November 30th, 2016
We’ve been discussing the *mechanical advantage* that *compound pulleys* provide to humans during lifting operations and last time we hit upon the fact that there comes a point of diminished return, a reality that engineers must negotiate in their *mechanical* designs. Today we’ll discuss one of the *undesirable tradeoffs* that results in a diminished return within a *compound pulley *arrangement when we compute the length of rope the Grecian man we’ve been following must grapple in order to lift his urn. What we’ll discover is a situation of *mechanical overkill* – like using a steamroller to squash a bug.
__Mechanical Overkill__
Just how much rope does Mr. Toga need to extract from our working example compound pulley to lift his urn two feet above the ground? To find out we’ll need to revisit the fact that the compound pulley is a work input-output device.
As presented in a past blog, the equations for work input, *WI*, and work output, *WO*, we’ll be using are,
*WI = F ×** d*_{2}
*WO = W ×** d*_{1}
Now, ideally, in a compound pulley no friction exists in the wheels to impede the rope’s movement, and that will be our scenario today. Our next blog will deal with the more complex situation where friction is present. So for our example today, with no friction present, work input equals output…
*WI = WO*
… and this fact allows us to develop an equation in terms of the rope length/distance factors in our *compound pulley* assembly, represented by *d*_{1} and *d*_{2}, …
*F ×** d*_{2} = W ×* d*_{1}
*d*_{2} ÷* d*_{1 }= W ÷* F*
Now, from our last blog we know that *W* divided by *F* represents the *mechanical* advantage, *MA,* to Mr. Toga of using the *compound pulley, *which was found to be 16, equivalent to the sections of rope directly supporting the urn. We’ll set the distance factors up in relation to *MA*, and the equation becomes…
*d*_{2} *÷* d_{1 }= MA
*d*_{2} = MA ×* d*_{1}
*d*_{2} = 16 *×** *2* feet = *32* feet*
What we discover is that in order to raise the urn 2 feet, our Grecian friend must manipulate 32 feet of rope – which would only make sense if he were lifting something far heavier than a 40 pound urn.
In reality, *WI* does not equal *WO, *due to the inevitable presence of friction. Next time we’ll see how friction affects the *mechanical* advantage in our *compound pulley.*
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: compound pulley, engineer, force times distance, lift, mechanical advantage, mechanical design, pulley, rope length, work, work input, work output

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Friday, November 18th, 2016
We’re all familiar with the phrase, “too much of a good thing.” As a professional engineer, I’ve often found this to be true. No matter the subject involved, there inevitably comes a point when undesirable *tradeoffs* occur. We’ll begin our look at this phenomenon in relation to *compound pulleys* today, and we’ll see how the pulley arrangement we’ve been working with encounters a *rope length tradeoff*. Today’s arrangement has a lot of pulleys lifting an urn a short distance.
We’ll be working with two distance/length factors and observe what happens when the number of *pulleys* is increased. Last time we saw how the compound pulley is essentially a *work* input-output device, which makes use of distance factors. In our example below, the first distance/length factor, *d*_{1}, pertains to the distance the urn is lifted above the ground. The second factor, *d*_{2}, pertains to the *length* of rope Mr. Toga extracts from the *pulley* while actively lifting. It’s obvious that some *tradeoff *has occurred just by looking at the two lengths of rope in the image below as compared to last week. What we’ll see down the road is that this also affects *mechanical advantage*.
The *compound pulley* here consists of 16 pulleys, therefore it provides a mechanical advantage, *MA*, of 16. For a refresher on how *MA* is determined, see our preceding blog.
**Rope Length Tradeoff in a Compound Pulley**
With an *MA* of 16 and the urn’s weight, *W*, at 40 pounds, we compute the force, *F,* Mr. Toga must exert to actively lift the urn higher must be greater than,
*F >* *W ÷** MA*
*F >* 40 *Lbs. ÷** *16
* **F > *2.5* Lbs.*
Although the force required to lift the urn is a small fraction of the urn’s weight, Mr. Toga must work with a long and unwieldy length of rope. How long? We’ll find out next time when we’ll take a closer look at the relationship between *d*_{1 }and *d*_{2}.
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: compound pulley, effor, force, mechanical advantage, professional engineer, pulley, rope length, weight force, work

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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, September 29th, 2016
In this blog series on *pulleys* we’ve gone from discussing the simple pulley to the improved simple pulley to an introduction to the complex world of compound pulleys, where we began with a static representation. We’ve used the engineering tool of a free body diagram to help us understand things along the way, and today we’ll introduce another tool to prepare us for our later analysis of dynamic *compound pulleys*. The tool we’re introducing today is the engineering concept of *mechanical advantage*, *MA*, as it applies to a *compound pulley *scenario.
The term *mechanical advantage* is used to describe the measure of force amplification achieved when humans use tools such as crowbars, pliers and the like to make the work of prying, lifting, pulling, bending, and cutting things easier. Let’s see how it comes into play in our lifting scenario.
During our previous analysis of the simple pulley, we discovered that in order to keep the urn suspended, Mr. Toga had to employ personal effort, or force, equal to the entire weight of the urn.
*F = W * (1)
By comparison, our earlier discussion on the static compound pulley revealed that our Grecian friend need only exert an amount of personal force equal to 1/2 the suspended urn’s weight to keep it in its mid-air position. The use of a compound pulley had effectively improved his ability to suspend the urn by a factor of 2. Mathematically, this relationship is demonstrated by,
*F = W ÷* 2 (2)
The factor of *2* in equation (2) represents the *mechanical advantage* Mr. Toga realizes by making use of a *compound pulley*. It’s the ratio of the urn’s weight force, *W*, to the employed force,* F*. This is represented mathematically as,
*MA =* *W* *÷* *F * (3)
Substituting equation (2) into equation (3) we arrive at the *mechanical advantage* he enjoys by making use of a *compound pulley,*
*MA =* *W* *÷* (*W ÷* 2) = 2 (4)
__Mechanical Advantage of a Compound Pulley__
Next time we’ll apply what we’ve learned about *mechanical advantage* to a *compound pulley* used in a dynamic lifting scenario.
* *
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: compound pulley, engineering, force, lifting, mechanical advantage, pulley, simple pulley, static analysis, tools, weight

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Tuesday, August 2nd, 2016
Last time we introduced the free body diagram, applied it to a simple pulley, and discovered that in so doing lifting objects required 50% less effort. As an engineering expert, I’ve sometimes put this *improved* version of a *simple pulley* to work for me in designs. We’ll do *the math behind the* *improvement* today.
Here again is the *free body diagram* showing the *improved* *simple pulley* as introduced last week.
__The Math Behind the Improved Simple Pulley__
The illustration shows the three forces, *F*_{1}, *F*_{2}, and *W,* acting upon the *simple* *pulley *within the highlighted free body diagram. Forces *F*_{1} and *F*_{2} are exerted from above and act in opposition to the downward pull of gravity, represented by the weight of the urn, *W*. Forces *F*_{1} and *F*_{2 }are produced by that which holds onto either end of the rope that’s threaded through the *pulley.* In our case those forces are supplied by a man in a toga and a beam. By engineering convention, these upward forces, *F*_{1} and *F*_{2,} are considered positive, while the downward force, *W,* is negative.
In the arrangement shown in our illustration, the *pulley’s* rope ends equally support the urn’s weight, as demonstrated by the fact that the urn remains stationary in space, neither moving up nor down. In other words, forces *F*_{1} and *F*_{2} are equal.
Now, according to the basic rule of *free body diagrams,* the three forces *F*_{1}, *F*_{2}, and *W* must add up to zero in order for the *pulley* to remain stationary. Put another way, if the *pulley* isn’t moving up or down, the positive forces *F*_{1} and *F*_{2} are balancing the negative force presented by the urn’s weight, *W*. *Mathematically* this looks like,
*F*_{1} + F_{2} – W = 0
or, by rearranging terms,
*F*_{1} + F_{2} = W
We know that *F*_{1} equals *F*_{2}, so we can substitute* F*_{1} for *F*_{2} in the preceding equation to arrive at,
*F*_{1} + F_{1} = W
or,
*2 ×** F*_{1} = W
Using algebra to divide both sides of the equation by 2, we get:
*F*_{1} = *W* ÷ 2
Therefore,
*F*_{1} = *F*_{2} = *W* ÷ 2
If the sum of the forces in a free body diagram do not equal zero, then the suspended object will move in space. In our situation the urn moves up if our toga-clad friend pulls on his end of the rope, and it moves down if Mr. Toga reduces his grip and allows the rope to slide through his hand under the influence of gravity.
The net real world benefit to our Grecian friend is that the urn’s 20-pound weight is divided equally between him and the beam. He need only apply a force of 10 pounds to keep the urn suspended.
Next time we’ll see how the improved *simple pulley* we’ve discussed today led to the development of the compound pulley, which enabled heavier objects to be lifted.
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: engineering expert, forces, free body diagram, gravity, pulley, simple pulley, weight

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