Archive for October, 2017

Two Types of Velocity Associated With a Spinning Flywheel

Thursday, October 19th, 2017

    Anyone who has spun a potter’s wheel is appreciative of the smooth motion of the flywheel upon which they form their clay, for without it the bowl they’re forming would display irregularities such as unattractive bumps.   The flywheel’s smooth action comes as a result of kinetic energy, the energy of motion, stored within it.   We’ll take another step towards examining this phenomenon today when we take our first look at calculating this kinetic energy.   To do so we’ll make reference to the two types of velocity associated with a spinning flywheel, angular velocity and linear velocity, both of which engineers must negotiate anytime they deal with rotating objects.

    Let’s begin by referring back to the formula for calculating kinetic energy, KE.  This formula applies to all objects moving in a linear fashion, that is to say, traveling a straight path.   Here it is again,

KE = ½ × m × v2

where m is the moving object’s mass and v its linear velocity.

    Flywheels rotate about a fixed point rather than move in a straight line, but determining the amount of kinetic energy stored within a spinning flywheel involves an examination of both its angular velocity and linear velocity.   In fact, the amount of kinetic energy stored within it depends on how fast it rotates.

    For our example we’ll consider a spinning flywheel, which is basically a solid disc.   For our illustrative purposes we’ll divide this disc into hypothetical parts, each having a mass m located a distance r from the flywheel’s center of rotation.   We’ll select a single part to examine and call that A.

Two Types of Velocity Associated With a Spinning Flywheel

Two Types of Velocity Associated With a Spinning Flywheel

   

    Part A has a mass, mA, and is located a distance rA from the flywheel’s center of rotation.   As the flywheel spins, part A rides along with it at an angular velocity, ω, following a circular path, shown in green.   It also moves at a linear velocity, vA, shown in red.   vA represents the linear velocity of part A measured at any point tangent to its circular path.  This linear velocity would become evident if part A were to become disengaged from the flywheel and cast into the air, whereupon its trajectory would follow a straight line tangent to its circular path.

    The linear and angular velocities of part A are related by the formula,

vA = rA × ω

    Next time we’ll use this equation to modify the basic kinetic energy formula so that it’s placed into terms that relate to a flywheel’s angular velocity.   This will allow us to define a phenomenon at play in the flywheel’s rotation, known as the moment of inertia.

 

 

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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Radians and the Angular Velocity of a Flywheel

Tuesday, October 10th, 2017

    Last time we introduced angular velocity with regard to flywheels and how a fixed point riding piggyback on a moving flywheel travels the same circular path as its host at a pace that’s measured in units of degrees per second.   Today we’ll introduce another unit of measure, the radian, and see how it’s uniquely used to measure angles of circular motion in units of radians per second.

Radians and the Angular Velocity of a Flywheel

 Radians and the Angular Velocity of a Flywheel

   

    Back in elementary school we worked with protractors and measured angles in degrees, and we were all too familiar with the fact that the average protractor maxed out at 180, or half the degrees present in a complete circle.   But in the grownup worlds of physics and engineering, angles of circular motion are measured in units called radians, an international standard equal to 57.3 degrees that’s used to measure objects rotating in circular motion.

    If we divide a circle’s value of 360 degrees by the 57.3 degrees that represent a radian, we find there are 6.28 radians in a circle, and oddly enough, it just so happens that 6.28 is equal to 2 × π.   Anyone who stayed awake during math class can’t help but remember that π represents a constant value of 3.14, a number that pops up anytime you divide the circumference of a circle by its diameter.   No matter the circle’s size, π will always result when you perform this operation.

   Applying these facts to radians, we find that during one complete revolution of a flywheel the measure of the angle θ increases from 0 radians to 2π radians.

    Suppose we have a flywheel spinning at N revolutions per minute, or RPMs.   To calculate the angular velocity, ω, of any point on the flywheel, or the whole wheel for that matter, we use the following formula which provides an answer in radians per second,

ω = [2 × π × N ] ÷ 60 seconds/minute                             (1)

    If a flywheel spins at 3000 RPM, its angular velocity is calculated as,

ω = [2 × π × (3000 RPM)] ÷ 60 seconds/minute               (2)

ω = 314.16 radians/second                                             (3)

    Next time we’ll see how angular velocity is used to determine the kinetic energy contained within a flywheel.

 

 

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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Angular Velocity of a Flywheel

Wednesday, October 4th, 2017

   We introduced the flywheel in our last blog and the fact that as long as it’s spinning it acts as a kinetic energy storage device.   Today we’ll work our way towards an understanding of how this happens when we discuss angular velocity.

   Angular velocity is represented in engineering and physics by the symbol, ω, the Greek letter Omega.   The term angular is used to denote physical quantities measured with respect to an angle, especially those quantities associated with rotation.

Angular Velocity of a Flywheel

Angular Velocity of a Flywheel

   

   To understand how angular velocity manifests let’s consider a fixed point on the face of a flywheel, represented in the illustration as A.   When the flywheel is at rest, point A is in the 12 o’clock position, and as it spins A travels clockwise in a circular path.    An angle, θ, is formed as A’s position follows along with the rotation of the flywheel.   The angle increases in size as A travels further from its starting point.   If A moves one complete revolution, θ will equal 360 degrees, or the total number of degrees present in a circle.

   As the flywheel  spins through its first revolution into its second, point A travels past its point of origination, and in two complete revolutions it will travel 2 × 360, or 720 degrees, in three revolutions 3 × 360, or 1080 degrees, and so forth.   The degrees A travels continue to increase with each revolution of the flywheel.

   Angular velocity represents the total number of degrees A travels within a given time period.  If we measure the flywheel’s rotational speed with a tachometer and find it takes one second to make 50 revolutions, point A will have traveled the circumference of its path fifty times, and A’s angular velocity would be calculated as,

ω = (50 revolutions per second) × (360 degrees per revolution)

ω = 18,000 degrees per second

   Next time we’ll introduce a unit of measurement known as radians which is uniquely used to measuring the angles of circular motion.

Copyright 2017 – Philip J. O’Keefe, PE

Engineering Expert Witness Blog

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