## Posts Tagged ‘belt’

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

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

Tags: belt, driven pulley, driving pulley, electric motor, engineering, Euler, Euler-Eytelwein Formula, Eytelwein, mechanical power, mechanical power transmission, pulley, pulley speed, rotating machinery

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Tuesday, June 28th, 2016
*Pulleys* are *simple* devices with many uses, and as an engineering expert, I’ve often incorporated them into mechanical designs. They’re used in machinery to transmit mechanical power from electric motors and engines to devices like blowers and pumps. Another common usage for *pulleys *is to aid in lifting. There are two types of *pulleys *for this purpose, *simple *or compound. We’ll start our discussion off by looking at the *simple* type today.
The* simple pulley* may have been an advanced application of the wheel. It consists of a furrowed wheel on a shaft with some device for pulling threaded through it. The *pulley* wheel supports and guides the movement of a rope, cable, or other pulling device around its circumference. The pulling device runs between a pull-ee and pull-er, that is, the object to be moved and the source of pulling power, with the *pulley* itself situated somewhere between them.
**Simple Pulley**
*Pulleys* are believed to have first been used by the Greeks as early as the 9^{th} Century BC. We’ll look into how they put them to use next time.
Copyright 2016 – Philip J. O’Keefe, PE
Engineering Expert Witness Blog
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Tags: belt, blower, cable, compound pulley, electric motors, engineering expert, engines, mechanical design, pulley, pumps, simple pulleys, transmit mechanical power

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