## Posts Tagged ‘Euler’s Number’

### 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         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 ____________________________________

### 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    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 ____________________________________

### Leonhard Euler, a Historical Figure in Pulleys

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 18th 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 ____________________________________