Posts Tagged ‘Euler-Eytelwein Formula’

The Mechanical Power Formula in Pulley and Belt Assemblies

Saturday, July 29th, 2017

     Last time we determined the value for one of the key variables in the Euler-Eytelwein Formula known as the angle of wrap.   To do so we worked with the relationship between the two tensions present in our example pulley-belt assembly, T1 and T2.   Today we’ll use physics to solve for T2 and arrive at the Mechanical Power Formula, which enables us to compute the amount of power present in our pulley and belt assembly, a common engineering task.

    To start things off let’s reintroduce the equation which defines the relationship between our two tensions, the Euler-Eytelwein Formula, with the value for e, Euler’s Number, and its accompanying coefficients, as determined from our last blog,

T1 = 2.38T2                            (1)

    Before we can calculate T1 we must calculate T2.   But before we can do that we need to discuss the concept of power.

 The Mechanical Power Formula in Pulley and Belt Assemblies

 The Mechanical Power Formula in Pulley and Belt Assemblies

   

    Generally speaking, power, P, is equal to work, W, performed per unit of time, t, and can be defined mathematically as,

P = W ÷ t                               (2)

    Now let’s make equation (2) specific to our situation by converting terms into those which apply to a pulley and belt assembly.   As we discussed in a past blog, work is equal to force, F, applied over a distance, d.   Looking at things that way equation (2) becomes,

P = F × d ÷ t                         (3)

    In equation (3) distance divided by time, or “d ÷ t,” equals velocity, V.   Velocity is the distance traveled in a given time period, and this fact is directly applicable to our example, which happens to be measured in units of feet per second.   Using these facts equation (3) becomes,

P = F × V                               (4)

    Equation (4) contains variables that will enable us to determine the amount of mechanical power, P, being transmitted in our pulley and belt assembly.

   The force, F, is what does the work of transmitting mechanical power from the driving pulley, pulley 2, to the passive driven pulley, pulley1.   The belt portion passing through pulley 1 is loose but then tightens as it moves through pulley 2.   The force, F, is the difference between the belt’s tight side tension, T1, and loose side tension, T2.   Which brings us to our next equation, put in terms of these two tensions,

P = (T1 – T2) × V                   (5)

    Equation (5) is known as the Mechanical Power Formula in pulley and belt assemblies.

    The variable V, is the velocity of the belt as it moves across the face of pulley 2, and it’s computed by yet another formula.   We’ll pick up with that issue next time.

 

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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Determining Angle of Wrap

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

Determining Angle of Wrap

   Here again is the basis for our calculations, the Euler-Eytelwein Formula.

T1 = T2 × e(μθ)                                                                 (1)

   To recap what we’ve discussed thus far, T1 is the tight side tension, the maximum the belt can endure before breaking.   T2 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 – 2α) × (π  ÷ 180)                                             (2)

where,

α = sin-1((D1 – D2) ÷ 2x)                                                 (3)

   By direct measurement we’ve determined the pulleys’ diameters, D1 and D2, 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,

T1 = T2 × 2.718(0.3 × 2.89)                                               (8)

T1 = 2.38T2                                                                    (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 T1, because we don’t know the value of T2.   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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The Angle of Wrap Formula

Wednesday, 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

The Angle of Wrap Formula

    We must start by calculating T1, 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,

T1 = T2× e(μθ)                                                       (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 – 2α) × (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((D1 – D2) ÷ 2x)                                       (3)

If we didn’t use this shorthand notation for α, equation (2) would be written as,

θ = (180 – 2(sin-1((D1 – D2) ÷ 2x))) × (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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The Two T’s of the Euler-Eytelwein Formula

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, T1 and T2.

   Here again is the Euler-Eytelwin Formula,where, T1 and  T2 are belt tensions on either side of a pulley,

T1 = T2 × e(μθ)

    T1 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.   T2 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 T1 is greater than it is in T2.

The Two T’s of the Euler-Eytelwein Formula

The Two T’s of the Euler-Eytelwein Formula

    In the illustration above, tension forces T1 and T2 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, T1 is equal to a combination of factors:  tension T2 ; 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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A First Look at the Euler-Eytelwein Formula

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,

D1 × N1 = D2 × N2

where, D1 is the diameter of the driving pulley and D2 the diameter of the driven pulley. The pulleys’ rotational speeds are represented by N1 and N2.

   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

A First Look at the Euler-Eytelwein Formula

   

where, T1 and  T2 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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The Difference Between Driven and Driving Pulleys

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

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

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