Moment of Inertia of Flywheel

CHAPTER - 1

INTRODUCTION

The mass moment of inertia is an important concept in rotational motion. The mass moment of inertia also called as the rotational inertia of a body is a measure of how hard it is to get it rotating about some axis. Rotational inertia is one indicator of the ability of rotating body to store kinetic energy. It is also an indicator of the amount of torque that will be needed to rotationally accelerate the body. Just as the mass is a measure of resistance to linear acceleration, moment of inertia is a measure of resistance to angular acceleration. In the design of devices that are intended for the storage and transfer of large amount of energy, maximum moment of inertia of the rotating parts is desirable. Flywheels used in automobiles are a good example for this. Flywheels are designed to have very large moment of inertia. A large moment of inertia will require a large driving torque. The flywheel is set in rotation when the vehicle is started.  It stores energy given to it as energy of rotation.  Because of its large moment of inertia, it prevents the engine from stopping when the pistons are not supplying energy.  The flywheel prevents the engine from accelerating too much when energy is supplied to it.   In this way, it keeps the engine turning through its cycles even when the piston is not supplying power (pistons supply power for only a fraction of the total cycle time).  The function of the flywheel is to make the motion of the vehicle smooth and jerk-free. Flywheels are not only used in automobiles but also in generators, electric motors etc.

Since the moment of inertia of flywheels is of high importance, its measurement is very relevant. There are theoretical formulas and expressions for the calculation of moment inertia. But most of these empirical formulas are restricted to simple geometries. In practical applications the shape and geometry of flywheels employed will be complex and hence such formulas will not serve the purpose. Here arises the need of some method for the determination of rotary inertia of flywheels.

In this project we deal with an experimental method for the determination of moment of inertia of a flywheel. A particular experimental setup is fabricated such that the moment of inertia of the flywheel can be calculated using the measurable or known parameters such as number of rotations of the flywheel, radius of the axle, the weight applied, time for rotation etc.

CHAPTER - 2

METHODOLOGY

·         Detailed study of the experiment.

·         Study of the requirements.

·         Experimental setup is designed.

·         Components are fabricated and assembled.

·         Testing of the setup is done, for different loads.

CHAPTER - 3

LITERATURE REVIEW

v Moment of Inertia

In classical mechanics, moment of inertia, also called mass moment of inertia, rotational inertia, polar moment of inertia of mass, or the angular mass, (SI units kg·m²) is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum and angular velocity, torque and angular acceleration, and several other quantities. The symbol I and sometimes J are usually used to refer to the moment of inertia or polar moment of inertia.

While a simple scalar treatment of the moment of inertia suffices for many situations, a more advanced tensor treatment allows the analysis of such complicated systems as spinning tops and gyroscopic motion.

The concept was introduced by Leonhard Euler in his book ‘Theoria motus corporum solidorum seu rigidorum’ in 1765. In this book, he discussed the moment of inertia and many related concepts, such as the principal axis of inertia.

The moment of inertia of an object about a given axis describes how difficult it is to change its angular motion about that axis. Therefore, it encompasses not just how much mass the object has overall, but how far each bit of mass is from the axis. The farther out the object's mass is, the more rotational inertia the object has, and the more force is required to change its rotation rate. For example, consider two hoops, A and B, made of the same material and of equal mass. Hoop A is larger in diameter but thinner than B. It requires more effort to accelerate hoop A (change its angular velocity) because its mass is distributed farther from its axis of rotation: mass that is farther out from that axis must, for a given angular velocity, move more quickly than mass closer in. So in this case, hoop A has a larger moment of inertia than hoop B.

The moment of inertia of an object can change if its shape changes. Skaters who begin a spin with arms outstretched provide a striking example. By pulling in their arms, they reduce their moment of inertia, causing them to spin faster (by the conservation of angular momentum).

The moment of inertia has two forms, a scalar form, I, (used when the axis of rotation is specified) and a more general tensor form that does not require the axis of rotation to be specified. The scalar moment of inertia, I, (often called simply the "moment of inertia") allows a succinct analysis of many simple problems in rotational dynamics, such as objects rolling down inclines and the behavior of pulleys. For instance, while a block of any shape will slide down a frictionless decline at the same rate, rolling objects may descend at different rates, depending on their moments of inertia. A hoop will descend more slowly than a solid disk of equal mass and radius because more of its mass is located far from the axis of rotation. However, for (more complicated) problems in which the axis of rotation can change, the scalar treatment is inadequate, and the tensor treatment must be used (although shortcuts are possible in special situations). Examples requiring such a treatment include gyroscopes, tops, and even satellites, all objects whose alignment can change.

The moment of inertia is also called the mass moment of inertia (especially by mechanical engineers) to avoid confusion with the second moment of area, which is sometimes called the area moment of inertia (especially by structural engineers). The easiest way to differentiate these quantities is through their units (kg·m² as opposed to m⁴). In addition, moment of inertia should not be confused with polar moment of inertia (more specifically, polar moment of inertia of area), which is a measure of an object's ability to resist torsion (twisting) only, although, mathematically, they are similar if the solid for which the moment of inertia is being calculated has uniform thickness in the direction of the rotating axis, and also has uniform mass density, the difference between the two types of moments of inertia is a factor of mass.

CHAPTER - 4

DESCRIPTION OF EQUIPMENT

The main components of the experimental setup are

1)      Flywheel

2)      Axle on which flywheel is mounted

3)      End bearings to support the axle

4)      The base and stands on which bearings are fixed

5)      Thread to hang the slotted masses to the flywheel

6)      A set of slotted masses with a hanger

v Flywheel

Fig. 1

           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. In case of steam engines, internal combustion engines, reciprocating compressors and pumps, the energy is developed during one stroke and the engine is to run for the whole cycle on the energy produced during this one stroke. For example, in I.C. engines, the energy is developed only during power stroke which is much more than the engine load, and no energy is being developed during suction, compression and exhaust strokes in case of four stroke engines and during compression in case of two stroke engines. The excess energy developed during power stroke is absorbed by the flywheel and releases it to the crankshaft during other strokes in which no energy is developed, thus rotating the crankshaft at a uniform speed. A little consideration will show that when the flywheel absorbs energy, its speed increases and when it releases, the speed decreases. Hence a flywheel does not maintain a constant speed, it simply reduces the fluctuation of speed. In machines where the operation is intermittent like punching machines, shearing machines, riveting machines, crushers etc., the flywheel stores energy from the power source during the greater portion of the operating cycle and gives it up during a small period of the cycle. Thus the energy from the power source to the machines is supplied practically at a constant rate throughout the operation. The flywheel does not maintain a constant speed; it simply reduces the fluctuation of speed. In other words, a flywheel controls the speed variations caused by the fluctuation of the engine turning moment during each cycle of operation. It does not control the speed variations caused by the varying load.

v Axle

Fig. 2

An axle, though similar in shape to the shaft, is a stationary machine element and is used for the transmission of bending moment only. It simply acts as a support for some rotating body such as hoisting drum, a car wheel or a rope sheave. Bearings or bushings are provided at the mounting points where the axle is supported.

v Bearing

A bearing is a device to allow constrained relative motion between two or more parts, typically rotation or linear movement. Bearings may be classified broadly according to the motions they allow and according to their principle of operation as well as by the directions of applied loads they can handle. Plain bearings use surfaces in rubbing contact, often with a lubricant such as oil or graphite. A plain bearing may or may not be a discrete device. It may be nothing more than the bearing surface of a hole with a shaft passing through it, or of a planar surface that bears another (in these cases, not a discrete device); or it may be a layer of bearing metal either fused to the substrate (semi-discrete) or in the form of a separable sleeve (discrete). With suitable lubrication, plain bearings often give entirely acceptable accuracy, life, and friction at minimal cost. Therefore, they are very widely used.

However, there are many applications where a more suitable bearing can improve efficiency, accuracy, service intervals, reliability, and speed of operation, size, weight, and costs of purchasing and operating machinery.

Thus, there are many types of bearings, with varying shape, material, lubrication, principle of operation, and so on. For example, rolling-element bearings use spheres or drums rolling between the parts to reduce friction; reduced friction allows tighter tolerances and thus higher precision than a plain bearing and reduced wear extends the time over which the machine stays accurate. Plain bearings are commonly made of varying types of metal or plastic depending on the load, how corrosive or dirty the environment is, and so on. In addition, bearing friction and life may be altered dramatically by the type and application of lubricants. For example, a lubricant may improve bearing friction and life, but for food processing a bearing may be lubricated by an inferior food-safe lubricant to avoid food contamination; in other situations a bearing may be run without lubricant because continuous lubrication is not feasible, and lubricants attract dirt that damages the bearings.

v Pulley

  A pulley, also called a sheave or a drum, is a mechanism composed of a wheel on an axle or shaft that may have a groove between two flanges around its circumference. A rope, cable, belt, or chain usually runs over the wheel and inside the groove, if present. Pulleys are used to change the direction of an applied force, transmit rotational motion, or realize a mechanical advantage in either a linear or rotational system of motion. It is one of the six simple machines. Two or more pulleys together are called a block and tackle.

v Cord

A rope is a length of fibers, twisted or braided together to improve strength for pulling and connecting. It has tensile strength but is too flexible to provide compressive strength (i.e. it can be used for pulling, but not pushing). Rope is thicker and stronger than similarly constructed cord, line, string, and twine. Common materials for rope include natural fibers such as Manila hemp, hemp, linen, cotton, coir, jute, and sisal.

Fig. 4

 

v The base and stands on which bearings are fixed

The base is made up of plywood. The stands on which bearings are fixed is made up of teak wood. The two stands are joined to the base and they are fastened using screws.

                                                          

Fig. 5

v Hanging slotted weight

The slotted weight consists of four 50gram weights which can be added or removed from the default 50gram weight provided with the hook.

Fig. 6

CHAPTER - 5

PRINCIPLE

The flywheel is mounted in its bearings with its axle horizontal and at a suitable height from the ground, and a string carrying a suitable mass m at its one end is wrapped completely and evenly round the axle. When the mass m is released, the string unwinds itself, thus setting the flywheel in rotation. As the mass m descends further and further the rotation of the flywheel goes on increasing till it becomes maximum when the mass touches the floor.

Let h be the distance fallen through by the mass and let v and w be the linear velocity of the mass and angular velocity of the flywheel respectively at the instant the mass drops off. Then, as the mass descends a distance h, it loses potential energy (mgh) which is used up: (i) partly in providing kinetic energy of translation 1/2mv² to the falling mass itself, (ii) partly in giving kinetic energy of rotation 1/2Iω² to the flywheel (where I is the moment of inertia of the flywheel about the axis of rotation) and (iii) partly in doing work against friction. If the work done against friction is steady and F per turn, and, if the number of rotations made by the flywheel till the mass detaches is equal to n₁, the work done against friction is equal to n₁F. Hence by the principle of conservation of energy, we have                                                    

After the mass reaches the floor, the flywheel continues to rotate for a considerable time t before it is brought to rest by friction. If it makes n₂ rotations in this time, the work done against friction is equal to n₂F and evidently it is equal to the kinetic energy of the flywheel at the instant the mass touches the floor. Thus

Fig. 7

After the mass has detached, its angular velocity decreases on account of friction and after some time t, the flywheel finally comes to rest. At the time of detachment of the mass the angular velocity of the wheel is ω and when it comes to rest its angular velocity is zero. Hence, if the force of friction is steady, the motion of the flywheel is uniformly retarded and the average angular velocity during this interval is equal to ω/2,

Thus   

...(4)

              

Thus observing the time‘t’ and counting the rotations n₁ and n₂ made by the flywheel its moment of inertia can be calculated from equation (3) and (4).

For a flywheel with large moment of inertia (I), 1/2mv² may be neglected, the equation (3a) becomes

                  

Substituting the value of ω from Eq. (4) in above equation, we get

                                     

This formula is used to calculate the moment of inertia of a flywheel.

v ALTERNATIVE METHOD

Fig. 8

Moment of inertia is calculated by plotting the graph and obtaining its slope.

CHAPTER – 6

DESIGN AND DRAWING

Fig. 9

 

Fig. 10

 

CHAPTER - 7

PROCEDURE

1.      Attach a mass m to one end of a thin thread.

2.      The thread is wrapped evenly round the axle of the wheel.

3.      Allow the mass to descend slowly and count the number of revolutions n₁ during descent.

4.      When the thread has unwound itself and the mass reaches the floor after n1 turns, start the stopwatch.

5.      Count the number of revolutions n₂ before the flywheel comes to rest and stop the stop watch. Thus n2 and t are known.

6.      Repeat the experiment with three different masses.

7.      Calculate the value of I using the equation derived earlier.

v Alternative method

1.      Attach a mass m to one end of a thin thread.

2.      The thread is wrapped evenly round the axle of the wheel.

3.      The mass is released from fixed height h unwinding the thread around the axle.

4.      Record the time t for the load to reach the ground.

5.      Vary m, and determine the corresponding value of t.

6.      Plot a graph of α against T.

7.     Calculate the moment of inertia of the flywheel.

CHAPTER -8

OBSERVATION AND CALCULATIONS

Total load applied m ( kg)	Height of fall of the mass h (m)	No. of revolutions of flywheel before the mass touches the floor n1	No. of revolutions of flywheel after the mass touching the floor n2	Time for n2 revolutions t    (seconds)	Moment of inertia I (kg-m2)
0.15	0.5	6	7	4.75	2.31×10-3
0.2	0.5	7	8	5.46	3.086×10-3
0.25	0.5	8	12	6.76	2.957×10-3

                                                                                Mean I = 2.78×10^-3 kg-m²

Total load applied m (in kg)	Height of fall of the mass h (m)	Time taken by the mass to touch the floor t (s)	Linear acceleration (m/s2)	Tension (N)	Angular acceleration (rad/ s2)
0.15	0.5	5.38	0.0345	1.466	3.622
0.2	0.5	4.15	0.0580	1.9504	6.089
0.25	0.5	3.20	0.097	2.428	10.1837

SAMPLE CALCULATION

We have the expression for moment of inertia,

                                        

Sl. No. 1

m = 0.15 kg                               g = 9.81 m/s²            h = 0.5 m                                

 t = 4.75 s                                       n₁ = 6                        n₂ = 7

I = (0.15×9.81×0.5) _/ (8×3.14×3.14×7×7_/4.75×4.75) × (1+6/7)  

= 2.3127×10^-3 Kg-m²

Alternative method

The graph plotted with T in X-axis and α in Y-axis

Moment of inertia, I = R/s where R = radius of axle and s =slope from the graph

We have,

R = 9.525×10^-3  

s = 3.5 from the graph

I = 9.525×10^-3 / 3.5

   = 2.72×10^-3 Kg-m²

 Angular acceleration (α) vs. Tension (T) Graph

T

 

CHAPETR - 9

SOURCES OF ERRORS AND PRECAUTIONS

1.       The loop should be sufficient so that loop remains even after the mass after reach the floor and there may be no tendency for it to rewind in the opposite direction and stop the free rotation of flywheel.

2.      The string should be evenly wound on the axle, i.e., there should be no overlapping of, or a gap left between, the various coils of the string.

3.      The string used should be of very small diameter compared with the diameter of the axle. If the string is of appreciable thickness half of its thickness should be added to the radius of the axle to get the effective value of r.

4.      The friction at the bearings should not be great and the mass tied to the end of the string should be sufficient to be able to overcome the bearing-friction and so to start falling of its own accord.

5.      Take extra care to start the stop-watch immediately the string starts to move.

CHAPTER - 10

CRITICISM OF THE METHOD

In this method the exact instant at which the mass drops off cannot be correctly found out and hence the values of n₁, n₂ and t cannot be determined very accurately. The angular velocity ω of the flywheel at the instant the mass drops off has been calculated from the formula ω = (4Πn₂)/t on the assumption that the force of friction remains constant while the angular velocity of the flywheel decreases from w to zero. But as the friction is less at greater velocities, we have no justification for this assumption. Hence for more accurate result, ω should be measured by a method in which no such assumption is made e.g., with a tuning fork.

CHAPTER - 11

CONCLUSION

After the detail study about moment of inertia and related concepts the experimental method for determination of moment of inertia of a flywheel was designed. The experimental setup for this purpose was fabricated successfully. The result obtained upon conducting the experiment was very satisfactory and nearly accurate. The project gave us more confidence that we will be able to put into practice whatever theoretical knowledge we gained during our course of study till now.