Machine Brakes, Clutches, Flywheels,Couplings

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    MODULE III

    Brakes, Clutches, Flywheel, CouplingsBrakes

    A brake is a device by means of which artificial resistance is applied on to a

    moving machine member in order to retard or stop the motion of the member or

    machine

    Types of Brakes

    Different types of brakes are used in different applications

    Based on the working principle used brakes can be classified as mechanical

    brakes, hydraulic brakes, electrical (eddy current) magnetic and electro-magnetic

    types.

    Mechanical Brakes

    Mechanical brakes are invariably based on the frictional resistance principles In

    mechanical brakes artificial resistances created using frictional contact between the

    moving member and a stationary member, to retard or stop the motion of the moving

    member.

    Design and Analysis

    To design, select or analyze the performance of these devices knowledge on the

    following are required.

    The braking torque

    The actuating force needed

    The energy loss and temperature rise

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    At this beginning stage attention will be focused mainly on some preliminary analysis

    related to these aspects, namely torque, actuating force, energy absorbed and

    temperature rise. Torque induced is related to the actuating force, the geometry of the

    member and other contact conditions. Most mechanical brakes work on the frictional

    contact basis and classified based on the geometry. The figure shows a brake shoe

    mounted on a lever, hinged at O, having an actuating force Fa, applied at the end of the

    lever. A normal force Fn is created when the shoe contacts the rotating drum.

    Brakes Classification

    Various shoe configurations are illustrated. Each consists of a body whose motion is

    braked together with a shoe which can swing freely about a fixed hinge H. A lining is

    attached to the shoe and contacts the braked body. The actuation force P applied to the

    shoe gives rise to a normal pressure and corresponding braking friction distributed over

    the area A of contact between lining and braked body, Shoes are classed as being

    either short or long. A short shoe is one whose lining dimension in the direction of

    motion is so small that contact pressure variation is negligible, i.e. the pressure is

    uniform everywhere, at pm say.

    When the area of contact becomes larger, the contact may no longer be with a uniform

    pressure, in which case the shoe is termed as long shoe. The shoes are either rigid or

    pivoted, pivoted shoes are also some times known as hinged shoes. Rigid shoe brakes

    -rigid because the shoes with attached linings are rigidly connected to the pivoted posts.

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    Basic Mechanism of Braking

    The illustration below explains the working of brakes in more detail. An

    element dA of the lining is shown with the braked body moving past

    at velocity v.

    The moment of the frictional force relative to the point of motion

    contributes to the retardation of motion and braking. The basic

    mechanism of braking is illustrated above. Some basic concepts of

    braking are now highlighted using a short shoe analysis.

    Preliminary Analysis

    And a frictional force Ff of magnitude f.Fn, f being the coefficient of friction, develops

    between the shoe and the drum develops. Since the shoe is short we assume the we

    assume the pressure is uniformly distributed over the contact area. Consequently the

    equivalent normal force Fn = p .A, where = p .A, where A is the surface area of the shoe.

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    Self- energizing

    Leading and trailing shoe

    For a given direction of rotation the shoe in which self energization is present is known

    as the leading shoe

    When the direction of rotation is changed, the moment of frictional force now will be

    opposing the actuation force and hence greater magnitude of force is needed to create

    the same contact pressure. The shoe on which this is prevailing is known as a trailing

    shoe.

    Leading and Trailing Shoe

    At certain critical value of f.c the term (b-fc) becomes zero. i.e no actuation force need

    to be applied for braking. This is the condition forself-locking. Self-locking will not occur

    unless it is specifically desired.

    Self Locking

    Short and Long Shoe Analysis

    Foregoing analysis is based on a constant contact pressure p.

    In reality constant pressure may not prevail at all points of contact on the shoe.

    In such case the following general procedure of analysis can be adopted

    General Procedure of Analysis

    Estimate or determine the distribution of pressure on the frictional surfaces

    Find the relation between the maximum pressure and the pressure at any point

    For the given geometry, apply the condition of static equilibrium to find the actuating

    force, torque and reactions on support pins etc.

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    Drum Brakes

    Among the various types of devices to be studied, based on their practical use, the

    discussion will be limited to Drum brakes of the following types which are mainly used

    in automotive vehicles and cranes and elevators.

    Drum Brake Types:

    Rim types with internal expanding shoes

    Rim types with external contracting shoes

    Internal expanding Shoe

    The rim type internal expanding shoe is widely used for braking systems in automotive

    applications and is generally referred as internal shoe drum brake. The basic approach

    applied for its analysis is known as (pivoted) long shoe brake analysis.

    Long Shoe Analysis

    In this analysis, the pressure at any In this analysis, the pressure at any point is

    assumed to be proportional to point is assumed to be proportional to the vertical

    distance from the hinge pin, the vertical distance from the hinge pin, which in this caseis proportional to which in this case is proportional to sine of the angle and thus, sine of

    the angle and thus,

    The normal force dN is computed as the product pressure and area and the frictional

    force as the product of normal force and frictional coefficient i.e. f dN. By integrating

    these over the shoe length in terms of its angle the braking torque T, actuating force F

    and the pin reactions and are computed. Rx Ry

    pp sina or p pasin sin sina a

    = =

    ( )p bra 1 1aM (sin 2 sin 2 )N 2 1 2 1sin 2 4

    a

    =

    ( )aa

    fp br a 2 2M r cos cos (sin sin )f 1 2 2 1sin 2

    =

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    p braR (B fA) Fy xsin a

    =

    ( )1 2 2A sin sin2 12

    =

    1 1 1B ( ) sin 2 sin 2

    2 2 42 1 2 1

    =

    The braking torque T on the drum by the shoe is of the frictional forces fDN times the

    radius of the drum and resulting equation is the actuating force F is determined by the

    summation of the moments of normal and frictional forces about the hinge pin and

    equating it to zero. Depending on the direction of drum rotation,

    M MN fFc

    =

    (-sign for self energizing) where,

    MN and Mfare the moment of the normal and frictional forces respectively, about the

    shoe pivot point.

    Double Shoe Brakes

    Note that our foregoing analysis is for a single shoe In most practical application the

    brakes will contain two shoes symmetrically positioned around the brake drum as

    shown earlier

    If both the shoes are arranged such that both are leading shoes in which self

    energizing are prevailing, then all the other parameters will remain same and the

    total braking torque on the drum will be twice the value obtained in the analysis. However in most practical applications the shoes are arranged such that one

    will be leading and the other will be trailing for a given direction of drum rotation

    If the direction of drum rotation changes then the leading shoe will become

    trailing and vice versa.

    Thus this type of arrangement will be equally effective for either direction of

    drum rotation.

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    AUTOMOTIVE DRUM BRAKES

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    However the total braking torque will not be the twice the value of a single shoe.

    This is because the effective normal pressure (force) on the trailing shoe will not

    be the same, as the moment of the friction force opposes the normal force, there

    by reducing its actual value though in most applications the same normal force is

    applied or created at the point of force application on the brake shoe.

    Consequently we may write the actual or effective pressure prevailing on a

    trailing shoe

    F.a'p p .a a (M M )n f

    =

    +

    Resulting torque

    p2 aT f .w.r . (cos q - cosq )(p p ')B 1 2 a asinqa

    = +

    External Contracting Shoe

    The same analysis can be extended to a drum brake with external contracting type of

    shoes, typically used in elevators and cranes.

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    Corresponding contact geometry is shown in the figure. The resulting equations for

    moment of normal and frictional force as well as the actuating force and braking torque

    are same as seen earlier. For convenience they are reproduced here again

    ( )2fbp r cos cosa 1 2T

    sina

    =

    M MN fFc

    =

    ( )p bra 1 1aM (sin 2 sin 2 )N 2 1 2 1sin 2 4

    a

    =

    ( )aa

    fp br a 2 2M r cos cos (sin sin )f 1 2 2 1sin 2

    =

    Pivoted block brake with Long shoe

    Twin Shoe Brakes

    Behavior of a single shoe has been discussed at length. Two such

    shoes are combined into a complete practical brake unit, two being

    used to minimize the unbalanced forces on the drum, shaft and

    bearings.

    Brake with long Pivoted Shoe

    When the shoe is rigidly fixed to the lever, the tendency of the

    frictional force (f.Fn) is to unseat the block with respect to the lever.

    This is eliminated in the case of pivoted or hinged shoe brake and it

    also provides some additional advantages.

    Long Hinged Shoe

    This is a hinged shoe brake - the shoes are hinged to the posts. The

    hinged shoe is connected to the actuating post by the hinge, G, which

    introduces another degree of freedom - so the shoe tends to assume

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    an optimum position in which the pressure distribution over it is less

    peaked than in a rigid shoe.

    As wear proceeds the extra degree of freedom allows the linings to

    conform more closely to the drum than would be the case to rigid

    shoes. This permits the linings to act more effectively and also

    reduces the need for wear adjustment.

    The extra expense of providing another hinge is thus justified on the

    grounds of more uniform lining wear and consequently a longer life.

    This is the main advantage of the pivoted shoe brake

    This is possible only if the shoe is in equilibrium.

    For equilibrium of the shoe: -

    MG=T+Fxby-Fybx=0 where bx = b.cos G

    by = b.sinG

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    This needs that the resultant moment due to the frictional force (and

    due to the normal force) about the pivot point should be zero, so that

    no rotation of the shoe will occur about the pivot point.

    The actuating force P is applied to the post HG so the shoe itself is

    subject to two contacts only - the (ideal) at pin G and the distributed

    contact with the drum.

    To facilitate this location of the pivot is to be selected carefully.

    The location is in such a way that the moment of frictional force (and

    the normal force) about the pivot is zero. i.e the actual distributed

    contact leads to the ideal contact at the hinge or pivot to facilitate this

    the location of the pivot is to be selected carefully. The location is in

    such a way that the moment of frictional force (and the normal force)

    about the pivot is zero.

    i.e the actual distributed contact leads to the ideal contact at the hinge

    or pivot Further it is desirable to minimize the effect of pin reaction for

    which the shoe pivot and post pivot points are made con current.

    Let us now look how this can be met and satisfying the conditions set

    above and consequently the derive the equations relating the location

    of the pivot from the center of the drum

    In the previous section, a block brake with short shoe was discussed.

    The angle of contact between the block and the brake drum in such

    cases is usually small and less than 45o. It is therefore, reasonable to

    assume that the normal reaction (N) and frictional force ( N ) are

    concentrated at the midpoint of the shoe. This assumption is not

    applicable for the brake with the long shoe. When the block is rigidly

    fixed to the lever, the tendency of the frictional force ( N ) is to unseat

    the block with respect to the lever as shown in the figure. In case of

    the pivoted shoe brake, the location of the pivoted can be selected in

    such as way that the moment of frictional force about the pivot is zero.

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    This is the main advantage of the pivoted shoe brake. A double block

    brake with two symmetrical and pivoted shoes is show in figure.

    An element of friction lining located at an angle and subtending

    an angle d is shown in figure. The area if the element is ( )Rd w ,

    where w is the width of the friction lining parallel to the axis of the

    brake drum. If the intensity of pressure at the element is p, the normal

    reaction dN on the element is given by

    dN (Rd w)p=

    Distribution of pressure

    If the shoe is long then the pressure will not be uniform

    We need to determine the distribution of pressure along the lining; the

    pressure distribution should be conducive for maintaining a uniform

    wear, Since the brake drum is made of a hard material like cast iron or

    steel, the wear occurs on the friction lining, which is attached to the

    shoe. As shown in fig the lining will retain the cylindrical shape of the

    brake drum when wear occurs. After the radial wear has take place, a

    point such as X moves to X in order to maintain contact on the lining

    with the brake drum. In figure x is the wear in the X direction and r

    is the wear in the radial direction. If it is assumed that the shoe is

    constrained to move towards the brake drum to compensate to wear,

    x should be constant because it is same for all points. Therefore,

    rx

    cos

    =

    = constant (b)

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    The radial wear x is proportional to the work done by the frictional

    force. The work done by the frictional force depends upon the

    frictional force ( dN ) and the rubbing velocity. Since the rubbing

    velocity is constant for all points on friction lining,

    r dN

    Or ( )r Rd wp

    Therefore r p (c)

    From the expression (b) and (c)

    pcons tan t or p C cos1cos

    = =

    (d)

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    Where C1 is the constant of proportionality. The pressure is maximum

    when 0 = .

    Substituting,

    p Cmax 1=

    (e)

    From Eqs (d) and (e),

    p p cosmax=

    Substituting this value in Eq. (a

    dN (Rd w)p cosmax= (f)

    The forces acting on the element of the friction lining are shown in

    Figure. The distance h of the pivot is selected in such a manner that

    the moment of frictional force about it is zero.

    Therefore,

    M fdN(h cos R) 0f

    = =

    Substituting dN from Eq. (f),

    ( )

    ( )

    2

    0

    0 0

    0

    0

    h cos R cos d 0

    1 cos 2or h d R cos d 0

    2

    1sin2

    2or h R sin 02

    4R sinh

    2 sin 2

    =

    + =

    + =

    =

    +

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    The elemental torque of frictional force N about the axis of brake

    drum is dNR . Therefore

    t

    0

    M 2 dNR

    =

    Substituting the value of dN from Eq.(f)

    2t max

    0

    2t max

    M 2 R wp cos d

    M 2 R wp sin

    =

    =

    The reactionXR can be determined by considering two components

    ( ) ( )dNcos and dNsin .

    Due to symmetry,

    dNsin 0 =

    Therefore,

    x

    0

    2max

    0

    max

    x max

    R 2 dN cos

    2Rwp cos d

    2 sin 22Rwp

    4

    1or R Rwp (2 sin 2 )

    2

    =

    =

    + =

    = +

    The reactionyR can be determined by considering two components

    ( ) ( )dN sin and dN cos

    Due to symmetry,

    dNsin

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    Therefore,

    y

    0

    2max

    0

    y max

    R 2 dN cos

    2 Rwp cos d

    1or R Rwp (2 sin 2 )

    2

    =

    =

    = +

    Pivoted shoe brakes are mainly used in hoists and cranes. Their

    applications are limited because of the physical problem in locating

    pivot so close to the drum surface.

    Energy Consideration

    Kinetic energy is absorbed during slippage of either a clutch or brake, and this energy

    appears as heat. If the heat generated is faster than it is dissipated, then the

    temperature rises.

    The capacity of a clutch or brake is therefore limited by two factors:

    1. The characteristics of the material and,

    2. The ability of the clutch/brake to dissipate heat.

    Energy to be Absorbed

    If t is the time of brake application andav the average angular velocity then

    the energy to be absorbed in braking E

    E = T. av.t

    If a constant deceleration is assumed the av= /2 as 1= and2 = 0

    Temperature Rise

    The temperature rise of the clutch or the temperature rise of the clutch or brake

    assembly can be approximated brake assembly can be approximated by the classic

    expression, by the classic expression,

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    . = E T Cm

    Where is temperature rise in ,C is the specific heat of the brake drum material - 500J/Kg

    for steel or Cast Iron m is the mass (kg) of the brake parts dissipating the heat into the

    surroundings T C

    A brake or clutch friction material should have the following characteristics to a degree,

    which is dependent upon the severity of the service.

    A high and uniform coefficient of friction.

    Imperviousness to environmental conditions, such as moisture.

    Frictional Material

    The ability to withstand high temperatures together with good thermal conductivity.

    Good resiliency.

    High resistance to wear, scoring, and galling.

    An improved lining material is being tried on an existing passenger car drum brake

    shown in Figure. Quality tests on the material indicated permissible pressure of1.0 MPa

    and friction co-efficient of 0.32. Determine what maximum actuating force can be

    applied for a lining width of 40 mm and the corresponding braking torque that

    could be developed.

    Linings

    The choice of lining material for a given application is based upon criteria such as the

    expected coefficient of friction; fade resistance, wear resistance, ease of attachment,

    rigidity or formability, cost, abrasive tendencies on drum, etc. The lining is sacrificial - it

    is worn away. The necessary thickness of the lining is therefore dictated by the volume

    of material lost - this in turn is the product of the total energy dissipated by the lining

    throughout its life, and the specific wear rate Rw (volume sacrificed per unit energy

    dissipated) which is a material property and strongly temperature dependent. The

    characteristics of Ferodo AM 2, typical moulded asbestos, are illustrated. The coefficient

    of friction, which may be taken as 0.39 for design purposes, is not much affected by

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    pressure or by velocity - which should not exceed 18 m/s. The maximum allowable

    temperature is 400C.

    Linings traditionally were made from asbestos fibers bound in an organic matrix,however the health risks posed by asbestos have led to the decline of its use. Non-

    asbestos linings generally consist of three components - metal fibers for strength,

    modifiers to improve heat conduction, and a phenolic matrix to bind everything together.

    Brake Design Section

    The braked system is first examined to find out the required brake capacity that is thetorque and average power developed over the braking period. - The brake is then

    either selected from a commercially available range or designed from scratch ff a drum

    brake has to be designed for a particular system (rather than chosen from an available

    range) then the salient brake dimensions may be estimated from the necessary lining

    area, A, together with a drum diameter- to- lining width ratio somewhere between 3:1

    and 10:1, and an angular extent of 100 C say for each of the two shoes.

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    E T. .tav

    1 27.7 78.24441.329 .

    2 0.125 27.7

    138206

    138.2KJ

    =

    =

    =

    =

    Clutch

    Clutch Introduction

    A Clutch is ia machine member used to connect the driving shaft to a driven shaft, so

    that the driven shaft may be started or stopped at will, without stopping the driving shaft.

    A clutch thus provides an interruptible connection between two rotating shafts

    Clutches allow a high inertia load to be stated with a small power.

    A popularly known application of Clutch is in automotive vehicles where it is used to

    connect the engine and the gear box. Here the clutch enables to crank and start the

    engine disengaging the transmission Disengage the transmission and change the gear

    to alter the torque on the wheels. Clutches are also used extensively in production

    machinery of all types.

    Mechanical Model

    Two inertias and traveling at the respective angular velocities and one of which may

    be zero, are to be brought to the same speed by engaging. Slippage occurs because

    the two elements are running at different speeds and energy is dissipated during

    actuation, resulting in temperature rise.

    To analyze the performance of these devices the following are required.1. The torque transmitted

    2. The actuating force.

    3. The energy loss

    4. The temperature rise

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    As in brakes a wide range of clutches are in use wherein they vary in their are in use

    their working principle as well the method working principle of actuation and application

    of normal of actuation and application of normal forces. The discussion here will be

    limited to mechanical type friction clutches or more specifically to the plate or disc

    clutches also known as axial clutches

    Frictional Contact axial or Disk Clutches

    An axial clutch is one in which the mating frictional members are moved in direction

    parallel to the shaft.

    Single Dry plate Clutch

    The torque that can be transmitted by a clutch is a function of its geometry and the

    magnitude of the actuating force applied as well the condition of contact prevailing

    between the members together with a uniform pressure all over its contact area and the

    consequent analysis is based on uniform pressure condition.

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    Uniform Pressure and wear

    However as the time progresses some wear takes place between the contacting

    members and this may alter or vary the contact pressure appropriately and uniform

    pressure condition may no longer prevail. Hence the analysis here is based on uniform

    wear condition

    Elementary Analysis

    Assuming uniform pressure and considering an elemental area of

    dF 2 prdr=

    The normal force on this elemental area is

    r0 2 2F 2 prdr p(r r )0 iri

    = =

    The frictional force on this area is therefore

    r ro o riF 2 prdr 2 p rdr 2 p (r r )max i max 0 irr ri i

    = = =

    Now the torque that can be transmitted by this elemental are is equal to the frictional

    force times the moment arm frictional force times the moment arm about the axis that is

    the radius r

    The total torque that could be transmitted is obtained by integrating this equation

    between the limits of inner radius ri to the outer radius ro

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    ro 22 3 3T 2 p r dr p (r r )o i3ri

    = =

    Integrating the normal force between the same limits we get the actuating force that

    need to be applied to transmit this torque.

    Uniform Wear Condition

    According to some established theories the wear in a mechanical system is proportional

    to the PV factor where P refers the contact pressure and V the sliding velocity. Based

    on this for the case of a plate clutch we can write

    p.r = constant for wear to be constant ir

    r

    Hence pressure at any point in the contact required

    i

    max

    rp p

    r=

    In the previous equations substituting this value for the pressure term p and integrating

    between the limits as done earlier we get the equation for the torque is transmitted and

    the actuating force to be transmitted and the actuating force to be applied. The clutch

    used in automotive applications is generally a single plate dry clutch. In this type the

    clutch plate is interposed between the flywheel surface of the engine and pressure plate.

    Single Clutch and Multiple Disk Clutch

    The clutch used in automotive applications is generally a single plate dry clutch

    In this type the clutch plate is interposed between the fly wheel surface of the engine

    and pressure plate. As both side surfaces of the clutch plate is used for transmitting the

    torque, a term z is added to include the number of surfaces used for transmitting the

    torque

    The pressure plate is also used to apply the actuating force supplied from a series of

    helical compression spring placed around the periphery of the plate or a single plate

    type of spring.

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    By rearranging the terms the equations can be modified and a more general form of the

    equation can be written as

    T Z.f .F .Ra m=

    T is the torque (Nm). Z is the number of frictional disks in contact.

    f is the coefficient of friction

    is the actuating force (N).

    is the mean or equivalent radius (m).

    Values of the actuating force F and the mean radius mr for the two conditions of

    analysis are summarized and shown in the table

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    Flywheel

    A flywheel is an inertial energy-storage device. It absorbs mechanical

    energy and serves as a reservoir, storing 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 the

    supply.

    Flywheels-Function need and Operation

    The main function of a fly wheel is to smoothen out variations in the

    speed of a shaft caused by torque fluctuations. If the source of the

    driving torque or load torque is fluctuating in nature, then a flywheel is

    usually called for. Many machines have load patterns that cause the

    torque time function to vary over the cycle. Internal combustion

    engines with one or two cylinders are a typical example. Piston

    compressors, punch presses, rock crushers etc. that have Flywheel

    absorbs mechanical energy by increasing its angular velocity and

    delivers the stored energy by decreasing its velocity. And time varying

    loads need flywheels

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    Design Approach

    There are two stages to the design of a flywheel.

    First, the amount of energy required for the desired degree of

    smoothening must b found and the (mass) moment of inertia needed

    to absorb that energy determined.

    Then flywheel geometry must be defined that caters the required

    moment of inertia in a reasonably sized package and is safe against

    failure at the designed speeds of operation.

    Small fly wheels are solid discs of hollow circular cross section.

    As the energy requirements and size of the flywheel increases the geometry changes to

    disc of central hub and peripheral rim connected by webs and to hollow wheels with

    multiple arms.

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    Design Parameters

    Flywheel inertia (size) needed directly depends upon the acceptable

    changes in the speed.

    The change in the shaft speed during a cycle is called the speed

    fluctuation and is equal to max- min

    We can normalize this to a dimensionless ratio by dividing it by the

    average or nominal shaft speed (ave) desired.

    Flmax min

    =

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    Speed Speculation

    This ratio is termed as coefficient of speed fluctuation Cfand it is defined as

    max min

    Cf

    =

    Where is nominal angular velocity, given by, by the average ornominal shaft speed (ave) desired. where avg is nominal angular

    velocity. This coefficient is a design parameter to be chosen by the

    designer.

    Coefficient of Speed Speculation

    The smaller this chosen value, the larger the flywheel have to be and

    more the cost and weight to be added to the system. However the

    smaller this value more smoother the operation of the device

    It is typically set to a value between 0.01 to 0.05 for precision

    machinery and as high as 0.20 for applications like crusher

    hammering machinery.

    Design Equation

    The kinetic energy Ek in a rotating system

    = ( )1 2I2

    Hence the change in kinetic energy of a system can be given as,

    hammering machinery.

    ( )min2 2K m max1

    E I2

    =

    K 2 1E E E=

    ( )max minavg 2

    + =

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    Torque Time Relation without Flywheel

    Torque Time Relation without Flywheel

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    Geometry of Flywheel

    The geometry of a flywheel may be as simple as a cylindrical disc of

    solid material, or may be of spoked construction like conventional

    wheels with a hub and rim connected by spokes or arms

    The latter arrangement is a more efficient of material especially for

    large flywheels, as it concentrates the bulk of its mass in the rim which

    is at the largest radius. Mass at largest radius contributes much more

    since the mass moment of inertia is proportional to mr2

    For a solid disc geometry with inside radius ri and out side radius ro

    , the mass moment of inertia I is

    m2 2 2I mk (r r )m o i2= = +

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    The mass of a hollow circular disc of constant thickness t is

    ( )W 2 2m r r to ig g

    = =

    Combing the two equations we can write

    ( )4 4I r r tm o i2 g

    =

    Where is material weight density The equation is better solved by

    geometric proportions i.e by assuming inside to out side radius ratio

    and radius to thickness ratio.

    Stresses in Flywheel

    Flywheel being a rotating disc, centrifugal stresses acts upon its

    distributed mass and attempts to pull it apart. Its effect is similar to

    those caused by an internally pressurized cylinder

    2 2 2 2t i o

    2 22 2 2 2i o

    r i o 2

    3 v 1 3vr r r

    g 8 3 v

    r r3 vr r r

    g 8 r

    + + = + +

    + = +

    Analogous to a thick cylinder under internal pressure the tangential

    and radial stress in a solid disc flywheel as a function of its radius r is

    given by

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    :

    The point of most interest is the inside radius where the stress is a

    maximum. What causes failure in a flywheel is typically the tangential

    stress at that point from where fracture originated and upon fracture

    fragments can explode resulting extremely dangerous consequences,

    Since the forces causing the stresses are a function of the rotational

    speed also, instead of checking for stresses, the maximum speed at

    which the stresses reach the critical value can be determined and safe

    operating speed can be calculated or specified based on a safety

    factor. Generally some means to preclude its operation beyond this

    speed is desirable, for example like a governor.

    Consequently

    F.O.S (N) = Nos yield

    =