flywheel experiment report

MOMENT OF INERTIA OF FLYWHEEL

This experiment is an introduction to some basic components of rotational dynamics to develop an understanding of Moment of Inertia

Torque and Mass Moment of Inertia

If a body is free to rotate about a fixed axis, then a torque is required to initiate or change the rotational motion of the body.

The torque τ ⃗ of a force about an axis is given by the cross-product of the force F ⃗ and the distance from the axis of rotation

The net torque is proportional to the angular acceleration α ⃗ of the body and shall exist during the entire time the torque acts. The equation is given as

Where I is the constant of the body known as the Mass Moment of Inertia about the specified axis of rotation. Mass moment of inertia (also known as rotational inertia) is a measure of a body’s resistance to a change in its rotation direction or angular momentum. The moment of inertia depends not only on the mass but also the distribution of the mass around the axis. Just as the mass is a measure of resistance of linear acceleration, mass moment of inertia is a measure of resistance to angular acceleration.

The experiment consists of estimating the mass moment of inertia of the flywheel system. A flywheel is a heavy thick circular discs designed for storing rotational energy. It is generally made of cast iron or steel along is mounted on an axle free to rotate on ball bearings. In other words, it’s a kind of system that needs a large force to start or stop spinning. The capacity of storing / shedding of kinetic energy depend on the rotational inertia of the flywheel.

In real life, flywheels come in all shapes and sizes. For obtaining the maximum moment of inertia per volume, most flywheels have a heavy outer circular rim with spokes. They may be mounted on the crankshaft of machines such as turbines, steam engines, diesel engines etc. This makes the engine run smoothly by storing kinetic energy when the machines are on higher loads and maintains that constant angular velocity during idle conditions.

Mass Moment of Inertia of Flywheel

The Mass Moment of Inertia of cylindrical objects about an axis passing through the centre can be given by the equation

I = (mr^2)/2.

The flywheel in this experiment is a solid disc of mass M1 and radius R attached to a shaft of mass M2 and radius r. So the moment of inertia of the flywheel system is given as

I = Σ (mr^2)/2= (M_1 R^2)/2+(M_2 r^2)/2

For complex geometries, the mass moment of Inertia of the flywheel can be estimated by measuring the approximate mass of different simplified geometrical components and adding the Mass Moment of Inertia about the central axis (from the known equations of MI of rings, cylinders, rods, etc).

Experimental Setup and Theory

In the experiment, a hanging mass m attached to the end of a spring, the remainder of which is wrapped around the axle is allowed to fall initiating the necessary torque τ ⃗ to the flywheel system initially at rest. Suppose that the string is wrapped around the axle n times and that a mass m is suspended from its free end and the system is released at time t = 0. As the mass accelerates downward, the flywheel attains an angular acceleration α ⃗. Because of the friction in the bearings, there will be an additional torque in the direction opposite to the motion of the flywheel. This frictional torque (α_f ) ⃗ depends upon a number of factors such as speed of rotation, coefficient of friction, etc but shall be assumed to be a constant value for simplicity.

If T is the tension in the string, then the net torque exerted by the wheel is

The net force on the mass m is

If the frictional torque is constant, then the angular acceleration of the system, (α_f ) ⃗, is also constant The flywheel will achieve a maximum angular velocity at the instant when the string detaches from the axle. The axle will continue to rotate until all the work is used to overcome the friction in bearings. Finally, the axle will stop rotating against the frictional forces.

Theoretical Calculations

As the slotted weight falls a particular height, it loses its potential energy. The loss in potential energy during unwinding is converted into its translation kinetic energy and rotational kinetic energy of flywheel. Some of the energy is lost in overcoming frictional forces in the bearings. Applying the law of conservation of energy at the instant the mass hits the ground.

(P.E)m = (R.K.E.)F + (L.K.E.)m + Frictional losses

The loss of potential energy (P.E)m of the slotted weights as it hits the ground is given as

(P.E)m = m g h = mg (2 π r n)

Note that we have neglected the thickness of the cord since radius of elastic cord cannot be determined experimentally. Another source of error is the slipping of the cord from the axle during unwinding.

The rotational kinetic energy of the flywheel (R.K.E)f can be given by

(R.K.E.)m = ½ Iω^2

The gain in linear Kinetic Energy (L.K.E)m of the slotted weights just before the mass touches the ground is given as (L.K.E.)m = ½ mv^2

If ω is the angular speed of the disc just as the mass hits the ground, then the final velocity of slotted weights is given by

The frictional losses are mainly due to friction in the axle and bearing assembly of the apparatus. We assume that the bearing frictional losses per unit rotation to be a constant value Wf. The total bearing friction depends on the number of wounds of cord around the axle

Bearing friction at the end of n1 rotations = n Wf

It is worth mentioning that air friction acting on the surface of the rotating disc as well as the moving weights may also result in losses which are ignored here.

Applying the individual equations in the law of conservation of energy, we obtain

Now, even after the mass detaches from the axle, the flywheel will continue to rotate. The angular velocity of the flywheel would decline gradually and finally come to a rest when all is rotational kinetic energy of flywheel (R.K.E)f is spent to overcome the frictional forces. If N is the number of rotation made by the flywheel after the string has left the axle then

Substituting the values of v and Wf in the equation,

Solving the equation for I, we obtain the following equation for mass moment of inertia of a flywheel which is,

The maximum angular velocity ω in the above equation can be found out by calculating the average velocity ωa as the flywheel comes to a final stop.

If N revolutions take a time t, then the average angular is given by

The above two equations give us a direct relationship between maximum angular velocity, number of rotations after detachment and the time required to complete that revolution

The experimental moment of inertia calculated by the equation may be slightly different from the theoretical moment of inertia because of the following criteria The thickness of the cord is assumed to be negligible. The bearing friction per rotation was assumed to be a constant value throughout the rotation. The air frictional losses are ignored. Any slip between the cords and the axle during unwinding is ignored

Experiment: Determination of Moment of Inertia of a Fly Wheel

Experiment: Determination of moment of Inertia of a Fly Wheel

Theory: The flywheel consists of a weighty round disc/massive wheel fixed with a strong axle projecting on either side. The axle is mounted on ball bearings on two fixed supports. There is a little peg on the axle. One end of a cord is loosely looped around the peg and its other end carries the weight-hanger.

Suppose, the angular velocity of a wheel is ω and its radius r. Then lineal velocity of the wheel is, v = ωr. If the moment of inertia of a body is I and the wheel is rotating around an axle.

Then its rotational kinetic energy, E = ½ Iω 2 .

Apparatus: An iron axle, a heavy wheel, some ropes, a mass, stopwatch, meter scale, slide calipers.

Description of the apparatus:

The flywheel was set as shown with the axle of the flywheel straight or parallel. A polystyrene tile was placed on the floor to avoid the collision of the mass on the floor.

(1) First of all, let us measure the radius of the axle by a slide caliper.

(2) Then for the determination of a number of rotation a mark by chalk is put on the axle and a rope is wound on the axle. At the other end of the rope a mass m is fastened and if it is dropped from position R, the wheel after rotating a few times, the weight with the rope will fall to position S. The wheel makes m 1 number of rotation to touch the point S and time for this drop is noted from the stopwatch.

Now the rope is again wound on the axle and the mass is fastened on the other end of the rope. From position R the mass is allowed to fall to the ground and as soon as it touches the ground, the stopwatch is started. When the axle comes to rest the stop wealth is stopped. Total time and the number of rotation of the wheel before it comes to rest are noted i.e., a total number of rotation (n 2 ) as noted.

Table 1: radius (r) of the axle B

Table 2: Determination of time and number of rotation

Calculation : If the axis takes time t for n 2 number of rotation, the average angular velocity,

ω 2 = (2πn 2 )/t

The axle acquires zero velocity with uniform retardation from angular velocity ω, so its average angular velocity,

ω 2 = (ω + 0) / 2 = ω/2

or, (2πn 2 )/t = ω/2

or, ω = 4πn 2 rad S -1

Then, I = (2mgh – mω 2 r 2 ) / ω 2 (1+ n 1 / n 2 ) = ….. g.cm 2 = ….. Kg.m 2

By inserting the value of n 2 , ω can be found out. By increasing the values of m, ω, r, h, n 1 , n 2 and g in an equation; the moment of inertia of the heavy wheel can be found out.

Precautions:

In the axle, a rope is to be wounded in such a way that while unwinding from the wheel it can easily drop on the ground.

• There should be the least friction in the flywheel.
• A number of rotation n and time t is to be unwired correctly.
• The length of the string should be less than the height of axle from the floor.
• Height ‘h’ is to be measured from the mark on the axle.
• ‘h’ is to be measured correctly.
• There should be no kink in string and string should be thin and should be wound evenly.
• The stopwatch should be started just after detaching the loaded string.

Applications: The main function of a flywheel is to maintain a nearly constant angular velocity of the crankshaft.

• A small motor can accelerate the flywheel between the pulses.
• The phenomenon of precession has to be considered when using flywheels in moving vehicles.
• Flywheels are used in punching machines and riveting machines.

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Flywheel experiment

1. INTRODUCTION

A flywheel is a mechanical device with a significant moment of inertia used as a storage device for rotational energy 1 . The rotational energy stored enables the flywheel to accelerate at very high velocities, and also to maintain that sort of velocity for a given period of time. The force that enables the flywheel to attain such velocities also produces energy to slow down the flywheel’s motion.

The objectives of the experiment are;

• To determine the friction torque due to the bearings, T f  
• To determine, experimentally, the moment of inertia, I, for the flywheel.
• To estimate the moment of inertia, using simple equations.
• To compare the experimental value of I with the estimate and suggest reasons for any discrepancies.

To calculate friction torque, it is assumed that the energy lost due to bearing friction is equal to the potential energy lost by the mass during unwinding and rewinding:

               Mg(H 1 -H 2 ) = T f  θ                                                                   . . . . . (1)

Where, m        = applied mass (kg)

               H 1         = original height of mass above some arbitrary datum (m)

               H 2         = final height of mass above the same datum (m)

               T f            = friction torque (Nm)

               θ                    = total angle turned through during unwinding and rewinding (rads)

To calculate the angular acceleration, (α),

              S = u t + a t 2 /2                                                                  . . . . . . . .(2)

This is a preview of the whole essay

and       α =a/r                                                                                . . . . . . . . (3)

where, s             = distance travelled by mass during decent (m)

             u             = initial velocity of mass (=0)

             t              = time to travel distance s (s)

             a             = linear acceleration of mass (m/s 2 )

             r              = effective radius of the flywheel axle (m)

To determine, experimentally, the moment of inertia (I exp );

     T – T f = (I + m r 2 ) α                    where T = m g r             . . . . . . . . . (4)

To calculate a theoretic value for I. The equation is;

     I = MR 2 /2                                                                              . . . . . . . . (5)

Where M            = mass of flywheel (kg)

              R            = radius of flywheel (m)

3 . EXPERIMENTAL PROCEDURES

3.1     DESCRIPTION OF THE TEST EQUIPMENT

• Flywheel (disc and axle)
• Pencil (to mark distance)
• Mass (known) and mass holder

                                 Flywheel

                         (disc + Axle)

         A        

             C          H 1

         String

        H 2

A known mass          B

Figure 3.1a

3.2   PROCEDURE

• The string is wrapped around the flywheel in a clockwise direction, which in turn lifts the known mass that is attached to the bottom of the string to a point close to the flywheel (point A on fig 3.1).
• The string, with the mass attached to it, is then allowed to wind down the flywheel until the mass reaches its lowest point (point B on fig 3.1), which is timed with a stop watch.
• The distance between points A and B is measured as H 1 .
• After reaching its lowest point, the mass then bounces back and starts to travel in the opposite direction, but then stops at a particular point (point C on fig 3.1).
• The distance between points B and C is measured as H 2 .
• The experiment is then repeated again, so as to improve reliability and accuracy of the supposed result.

3.3   RESULT                                                                                                        Table 3.3

H 1  = Original height of mass after it unwinds from the flywheel

H 2  = Final height of mass after bouncing back in opposite direction

 θ = Total angular displacement (rads)

r   = Effective radius of the axle = 13.75x10 -3 m.

Radius of shaft and rope (r) = 0.01375m

Mass of flywheel = 6.859kg

Radius of flywheel = 0.1m

Radius of axle = 0.0125m

4.  ANALYSIS OF RESULTS

To calculate the moment of inertia of the flywheel;

             T – T f = (I + m r 2 ) α                    where T = m g r  

Make ‘I’ the subject of the formula;

            I exp = (T – T f  )/α – (m r)

then, the value of T(applied torque) is;

   = 0.1 x 9.81 x (13.75x10 -3 )

   = 13.49x10 -3  Nm

To calculate T f (frictional torque);

T f  = mg (H 1  – H 2 )/θ

    = (0.1 x 9.81 x 0.77)/ 56

    = 1.35x10 -2  Nm

To calculate the angular acceleration (α);

α = 2H 1 / (r x t 2 )

    = (2 x 0.98)/ (13.75x10 -3  x 22.88 2 )

    = 0.27ms -2

I exp  = (13.49x10 -3  – 1.35x10 -2 )/0.27 - (0.1 x [13.75x10 -3 ] 2 )

       = 3.7x10 -5  – 1.89x10 -5

       = 1.81x10 -5  kgm 2

To calculate the theoretic value for the moment of inertia;

I theory  = MR 2 / 2

            = 6.859 x (0.1) 2  / 2

            = 3.43 x 10 -2  kgm 2

% error  = [(Expected Value – Actual value)/ Expected Value] x 100

               = [3.4x10 -2 / 3.43x10 -2 ] x 100 = 99.13%

5. DISCUSSIONS/CONCLUSION

Following the analysis of my results, the values of I experiment  and I theory  differ by fairly a significant amount i.e. (a percentage error of 99.13%). The errors that led to the difference in the two values can be categorize into two sub-groups called “Measurement errors” and “Procedural errors”.

Measurement Errors.

• Errors may perhaps have crept up while measuring the distances of H 1  and H 2 . These distances could have possibly been marked incorrect if the points were not marked at eye level, which could have lead to errors in the final value. However, these errors could have been minimised by taking more repeated readings, or even recording the experiment with the use of a video camera in order to help in checking for these kind of errors.
• Furthermore, another error that could have affected the final value was the timing of the stopwatch while measuring H 1  and H 2 . This human error can be significantly reduced via total concentration of everyone involved in the experiment.

Procedural Errors.

The motion of the mass that was attached to the spring could have been affected by factors, such as the air resistance and friction, which would lead to easy energy loss during the experiment. This could have also led to some errors in the final value.                                                                                

This error could have been minimised by doing the experiment in a closed system, which would have not just minimised errors, but also increase the accuracy and reliability of the result.

• Lynn White, Jr., “Theophilus Redivivus”, Technology and Culture , Vol. 5, No. 2. (Spring, 1964), Review, pp. 224-233 (233) 1  
•  Lynn White, Jr., “Medieval Engineering and the Sociology of Knowledge”, The Pacific Historical Review , Vol. 44, No. 1. (Feb., 1975), pp. 1-21 (6)

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EXPERIMENT U4 MOMENT OF INERTIA OF FLYWHEELS

In this experiment, the moment of inertia of flywheel is being studied by variating the point of mass of flywheel. The experiment is conducted by recording the time taken for the respective point of mass to being rotated by a fixed load until the point where the load is escaping from the flywheel and the number of rotations done after be independent from the load. The moment of inertia then is calculated by substituting the data obtained from the experiment and the experimental value is calculate and compared to the experimental one.

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This paper presents the details of construction and implementation of an automated data acquisition (DAQ) system for an undergraduate laboratory experimental setup that is intended to measure the moment of inertia of a flywheel (without disassembling the setup), using the falling mass method. The developed DAQ system is a microcontroller-based system which has facilities to calculate a value for the moment of inertia of the flywheel directly, using the acquired data.It employs optical sensors to detect the position of a known mass attached to one end of a string wound around the flywheel axel, count the number of turns made by the flywheel before releasing the mass and after releasing the mass and measure the time taken for the mass to fall through a known distance.Measurements were taken under nine different conditions by changing the mass and its fall-through height with both manually operated and automated experimental setup. The average of the measured values of the moment of inertia of the flywheel under the two operation modes are found to be manua 2 l 0.348 0.005 kg m

IJMER Journal

Abstract: In present investigation, to counter the requirement of smoothing out the large oscillations in velocity during a cycle of a I.C. Engine, a flywheel is designed, and analyzed. By using Finite Element Analysis are used to calculate the stresses inside the flywheel, we can compare the Design and analysis result with existing flywheel

International Journal of Engineering Research and Technology (IJERT)

IJERT Journal

https://www.ijert.org/analysis-of-flywheel-used-in-petrol-engine-car https://www.ijert.org/research/analysis-of-flywheel-used-in-petrol-engine-car-IJERTV3IS051286.pdf A flywheel used in machines serves as a reservoir which stores energy during the period when the supply of energy is more than the requirement and releases it during the period when the requirement of energy is more than supply. For example, in I.C. engines, the energy is developed only in the power stroke which is much more than engine load, and no energy is being developed during the suction, compression and exhaust strokes in case of four stroke engines. The excess energy is developed during power stroke is absorbed by the flywheel and releases its to the crank shaft during the other strokes in which no energy is developed, thus rotating the crankshaft at a uniform speed. The flywheel is located on one end of the crankshaft and serves two purposes. First, through its inertia, it reduces vibration by smoothing out the power stroke as each cylinder fires. Second, it is the mounting surface used to bolt the engine up to its load. The aim of the project is to design a flywheel for a multi cylinder petrol engine flywheel using the empirical formulas. A 2D drawing is drafted using the calculations. A parametric model of the flywheel is designed using 3D modeling software Pro/Engineer. The forces acting on the flywheel are also calculated. The strength of the flywheel is validated by applying the forces on the flywheel in analysis software ANSYS. Structural analysis, modal analysis and fatigue analysis are done on the flywheel. Structural analysis is used to determine whether flywheel withstands under working conditions. Fatigue analysis is done for finding the life of the component. Modal analysis is done to determine the number of mode shapes for flywheel Analysis is done for two materials Cast Iron and Aluminum Alloy A360 to compare the results. Pro/ENGINEER is the standard in 3D product design, featuring industry-leading productivity tools that promote best practices in design. ANSYS is general-purpose finite element analysis (FEA) software package. Finite Element Analysis is a numerical method of deconstructing a complex system into very small pieces (of user-designated size) called elements. INTRODUCTION:

I. Introduction Flywheel is a heavy rotating body which serves as an energy reservoir. The flywheel stores the energy in the form of kinetic energy during the period when the supply of energy from the prime mover is more than the requirement of energy by the machine, and releases it during the period when the requirement of energy by the machine is more than the supply of energy by the prime mover. In this work stress induced in a flywheel was studied and considered different parameters such as speed, material, outer diameter of flywheel, diameter of spoke and number of spoke.

Abdallah Abdallah

Rustem Kushtayev

IJSRD - International Journal for Scientific Research and Development

— Flywheels serve as kinetic energy storage and retrieval devices with the ability to deliver high output power at high rotational speeds as being one of the emerging energy storage technologies available today in various stages of development, especially in advanced technological areas. There are many causes of flywheel failure among them one of is the non-linear behavior of the flywheel. Hence in this work evaluation of non-linear stresses in the flywheel for different material is done. The design of flywheel is used by solid work software. The software used for analysis and apply forces for validation of flywheel is ANSYS. The FEA of flywheel is considering centrifugal forces on its comparative non-linear analysis is done for von-mises stress, shear stress and deformation of the flywheel made of Cast iron and aluminium alloy. The paper also gives a topology optimization approach in reducing the mass of flywheel.

https://www.ijert.org/a-review-paper-on-structural-and-parametric-analysis-of-composite-flywheel https://www.ijert.org/research/a-review-paper-on-structural-and-parametric-analysis-of-composite-flywheel-IJERTV3IS120185.pdf Flywheel energy storage (FES) can have energy fed in the rotational mass of a flywheel, store it as kinetic energy, and release out upon demand. For example, in I.C. engines, the energy is developed only in the power stroke which is much more than engine load, and no energy is being developed during the suction, compression and exhaust strokes in case of four stroke engines. The excess energy is developed during power stroke is absorbed by the flywheel and releases its to the crank shaft during the other strokes in which no energy is developed, thus rotating the crankshaft at a uniform speed. The current paper is focused on the analytical design of flywheel & FEM analysis flywheel used in Press. Different types of forces acting on flywheel & design parameters has taken into consideration for optimizing design of flywheel By using ANSYS stresses obtained & compared with analytical calculations, also weight is compared.

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• Moment Of Inertia

Moment Of Inertia Of Flywheel

Moment of inertia of a flywheel is calculated using the given formula;

Where I = moment of inertia of the flywheel.Here, the symbols denote;

m = rings’ mass.

N = flywheel rotation.

n = number of windings of the string.

h = height of the weight assembly.

g = acceleration due to gravity.

r = radius of the axle.

Or, we can also use the following expression;

Flywheels are nothing but circular disc-shaped objects which are mainly used to store energy in machines.

Determining The Moment Of Inertia Of Flywheel

To determine the moment of inertia of a flywheel we will have to consider a few important factors. First, we have to set up a flywheel along with apparatus like a weight hanger, slotted weights, metre scale and we can even keep a stopwatch.

Then we make some assumptions. We will take the mass as (m) for the weight hanger as well as the hanging ring. The height will be (h). Now we consider an instance where the mass will descend to a new height. There will be some loss in potential energy and for which we write the equation as;

P loss = mgh

Meanwhile, there is a gain in kinetic energy when the flywheel and axle are rotating. We express it as;

K flywheel = (½) Iω 2

I = moment of inertia

ω = angular velocity

Similarly, the kinetic energy for descending weight assembly is expressed as;

K weight = (½) Iv 2

Here, v = veocity

We also have to take into account the work that is done in overcoming the friction. This can be found out by;

W friction = nW f

In this case,

n = number of windings of the string

W f = work done in overcoming frictional torque

If we state the law of conservation of energy then we obtain;

P loss = K flywheel + K weight + W friction

We will substitute the values and the equation will now become;

mgh = (½)Iω 2 + (½) mv 2 + nW f

Moving on to the next phase, we look at the flywheel assembly’s kinetic energy that is used in rotating (N) number of times against the frictional torque. We get;

NW f = (½ ) Iω 2 and W f = (1 / 2N) Iω 2

Further, we establish a relation between the velocity (v) of the weight assembly and the radius (r) of the axle. The equation is given as;

We have to substitute the values for W f and v.

mgh = (½) Iω 2 + (½ )mr 2 ω 2 + (n / N) x ½ Iω 2

If we solve the equation for finding the moment of inertia, we obtain;

\(\begin{array}{l}I = \frac{Nm}{N+n}(\frac{2gh}{\omega ^{2}}-r^{2})\end{array} \)

⇒ Check Other Object’s Moment of Inertia:

• Moment Of Inertia Of Circle
• Moment Of Inertia Of A Quarter Circle
• Moment Of Inertia Of Semicircle
• Moment Of Inertia Of A Sphere
• Moment Of Inertia Of A Disc

Parallel Axis Theorem

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The Flywheel

• January 2019

• University of Baghdad, College of Science,

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