Archive for May, 2017

Angle of Wrap in the Euler-Eytelwein Formula

Thursday, May 25th, 2017

    Last time we learned that the two T’s in the Euler-Eytelwein Formula correspond to different belt tensions on either side of a pulley wheel in a pulley-belt assembly. Today we’ll see what the remaining variables in this famous Formula are all about, paying special attention to the angle of wrap that’s formed by the belt wrapping around the pulley wheel.

Angle of Wrap in the Euler-Eytelwein Formula

Angle of Wrap in the Euler-Eytelwein Formula

    Here again is the Euler-Eytelwein Formula,

 T1 = T2 × eθ)

    The tight side tension, T1, is equal to a combination of factors, namely:  loose side tension T2 ; the friction that exists between the belt and pulley, denoted as μ ;  and the length of belt coming in direct contact with the pulley, namely, θ.   These last two terms are exponents of the term, e, known as Euler’s Number, a mathematical constant used in many circles, including science, engineering, and economics, to calculate a wide variety of things, from bell curves to compound interest rates.   It’s a rather esoteric term, much like the term π that’s used to calculate values associated with circles.

    Euler’s Number was discovered in 1683 by Swiss mathematician Jacob Bernoulli, but oddly enough was named after Leonhard Euler.   Its value, 2.718, was determined while Bernoulli manipulated high level mathematics to calculate compound interest rates.   If you’d like to learn more about that, follow this link.

    The term μ is known as the friction coefficient.  It quantifies the degree of friction, or roughness, present between the belt and pulley where they make contact.   It’s a specific number that remains constant for a given combination of materials.  For example, according to Marks’ Standard Handbook for Mechanical Engineers, the value of μ for a leather belt operating on an iron pulley is 0.38.   The numerical values for these coefficients were determined over the last few centuries by engineers conducting laboratory testing on various belt and pulley materials.   They’re now routinely found in a variety of engineering texts and handbooks.

    Finally, the term θ denotes the angle of wrap that the belt makes while in contact with the face of the pulley.   In our example illustration above, θ measures the arc that’s formed by the belt riding along the surface of the pulley between points A and B, as shown by dotted lines.   The angle of wrap is important to overall functionality of the assembly, because the proper amount of friction will allow the pulley-belt assembly to operate efficiently and without slippage.

    Next time we’ll present an example and use the Euler-Eytelwein Formula to calculate optimal belt friction within a pulley-belt assembly.

 

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

____________________________________

 

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 × eθ)

    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 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 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

____________________________________

 

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

____________________________________