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5. CAMS A Camis an irregular-shaped mechanical member used fortransmitting a desired motion to another element, known as follower, bydirect contactThe cam and follower have line contact and constitute a higher pair.A cam may:

stationary translate, or

rotatewhile the follower:

translate ,orrotate

usually rotate at a constant angular speed Provide a means of achieving any desired

follower motion are used in many machines

Inlet and exhaust valves of internal combustion enginesMachine tools, mechanical computers, instruments, etc

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5.1. Classification of Followers

Followers may be classified based on the following:

1. Construction of the surface of contact

2. Types of follower motion

3. Location of line of motion with respect to center of cam

5.1.1 Classification of Followers Based on Surface of Contact

a. A knife-edge follower:

a sharp, knife-edge is in contact with the cam Produce excessive cam wear

They are of little practical use

b. A roller follower:

A cylindrical roller is in contact with the cam

At low speeds pure rolling contact is possible but at high speeds somesliding can occur

Reduce wear of the cam surface at high peripheral speeds

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c. A flat-faced follower:

Flat face is in contact with the cam

Cause high surface stress

To reduce the surface stress the flat face is modified to a sphericalsurface with a large radius

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5.1.2 . Classification of followers based on types of followermotion

a) Translatory followers: as the cam rotates the follower

reciprocates in guides.

b) Oscillatory follower: for a uniform rotary motion of the cam,the follower oscillates through a certain angle.

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5.1.3. Classification of followers based on follower line ofmotion

a) Radial follower: translate along an axis passing through thecam center of rotation

b) Off-set follower: the axis of follower movement is displacedfrom the cam center of rotation

c) Oscillating follower: oscillate about the axis of the followerthrough a certain angle.

5.2. CLASSIFICATION OF CAMS Cams are classified based on:a) Cam shape : as disc cams, translation cams, cylindrical cams,

globoid cams, etc.b) Follower motion: eg. Dwell-rise-dwell-return, dwell-rise-return,

etc.

c) Cam constraint :as spring or pre-loaded cams and positivereturn cams. In pre-loaded cams the follower is held in contact by an external

force provided by spring, gravity, etc. In positive return cams no external force is required to keep the

cam and the follower in contact.

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5.3. GRAPHICAL DESIGN OF CAM CURVES

5.3.1 Disc cam with a flat-faced radial follower:

The cam rotates at a constant angular velocity

The follower moves upwards a distance of a specified rise.

The return motion is assumed to be through the samedisplacement

To determine the cam contour graphically, the mechanism isinverted by holding the cam stationary and making the follower

move around it (relative motion is not affected).

Procedures of construction

i. Rotate the follower about the cam center in opposite direction tocam rotation

ii. Move the follower radially outward (or inward in case of returnmotion) the correct amount for each division of cam rotation.

iii. Draw the cam surface tangent to the polygon that is formed by thevarious position of the follower.

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5.3.2 . Disc cam with a radial roller follower

The center of the roller moves with a prescribed motion.

Same procedure of construction as for a cam with a flat-faced

follower, with the exception that the cam contour is drawn tangentto the various positions of the roller of the follower

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5.3.3. Disc cam with oscillating follower

The follower is rotated about the cam center and at the same timeit rotates about its center through the required displacement angle

for each position.

Procedure of construction

The centers of the follower rotation are indicated by 0, 1, 2,, 11

With this points as centers and radius 00 arcs are drawn to

intersect with other curves drawn with O as center and therequired follower displacement

The intersection of this arcs give points 0, 1, 2, , 11.

Drawing tangent lines through these points to the circles drawnwith centers 0, 1, 2, , 11, determines the polygon which

circumscribes the cam

From the polygon the cam profile can be drawn by passing acurve tangent to the sides of the polygon. See figure below.

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5.3.4. Positive Return Cams

No external force is required to keep the cam and follower incontact.

The displacement for the rise and return motions must be thesame and in opposite directions.

As can be noted from the figure below, the width bis equal tothe diameter of the base plus the total rise of the follower.

Points 1, 2, .., 12 are located on the flat face in such a waythat lines formed by diametrically opposite points are of equallength,

i.e. 17 = 28 = = 612 = b.

this condition provides a return motion which is the same as the

outward motion.

Construction of a positive-return cam follows the sameprocedure as cam construction for disc cams with flat-facedradial followers.

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5.4. NOMENCLATURE

Base Circle:

is the smallest circle that can be drawn about the center of cam

rotation and through the cam surface. Its size determines the size of the cam.

Trace Point:

A theoretical point on the cam follower which corresponds to thepoint of a knife-edge follower

It is used to generate the pitch curve. In a roller follower, the center of the roller represents the trace

point.

Pressure Angle:

Is the angle b/n the direction of follower motion & a normal to thepitch curve.

Indicates that the force existing b/n the cam and follower are not inthe direction of the follower motion

In which case, if the pressure angle is too large, the translatingfollower may jam in its bearings.

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Important factor in determining the size of the cam. a small cam has a higher pressure angle and is steeper than a

large cam with a smaller pressure angle. large cam is undesirable b/c it uses more space and may produce

unbalance For good performance of the cam, the pressure angle should not

exceed 30o. For a given displacement of the follower, the pressure angle can be

reduced by the following methods: increase the size of the base circle

Increase the cam angle for a given follower displacement Change amount of follower offset.

Pitch point: is a point on the pitch curve which designates thelocation of maximum pressure angle.

Pitch Circle: The circle about the center of cam rotation and throughthe pitch point.

Prime Circle :is the smallest circle about the center of cam rotationand through the pitch curve.

Lift or Rise:is the maximum travel of the follower from the lowest tothe top most position.

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5.5. Displacement Diagram Has an abscissa of the cam rotation angle, and the ordinate is the

follower travel. The displacement diagram identifies the following motion

characteristics. The rise: motion of the follower away from the cam center, The dwell: those periods during which the follower is at rest, and The return: motion of the follower toward the cam center.

5.6. TYPES OF FOLLOWER MOTION The first step in the design of a cam curve consists in constructing

the associated displacement diagram. Some of the follower standard motions are

Uniform, modified motion, simple harmonic, parabolic andcycloidal.

The cams rises, dwells and returns must suitably be distributedaround the periphery of the cam.

The periods of these cams motions occupy the time of onerotation of the cam.

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5.6.1 . Uniform Motion

A follower has uniform motion when its velocity is constant.

The follower moves through the same distance for each equal

interval of time or cam rotation angle.

Simplest possible cam motion, but shock results from the changesin velocity from zero to some finite value or vise-versa.

Infinite accelerations and decelerations occur at the beginning andend of the rise and return motions.

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Uniform motion can be represented by a simple equation like:

Where y = follower displacement corresponding to the cam angle ;c = constant to be determined from the boundary conditions.

If d = the total distance through which the follower is to rise

= the angle in radians through which the cam is to rotate toproduce the required rise.

Then

the equation that represents the follower displacement is:

this is an equation of a straight line or uniform motion.

)1.5(cy

)3.5(

)2.5(

dc

orcd

)4.5(

dy

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The velocity and acceleration are obtained by differentiation.

Due to the infinite acceleration b/n the dwell and rise periods, the

force transmitted are very large and shock and other secondaryeffects result.

)6.5(0

)5.5(

2

2

dt

dd

dt

yda

d

dt

dd

dt

dyv

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5.6.2 . Modified Uniform Motion The uniform-motion curve is modified to reduce the shock at the

beginning and end of the motion. This is done by using circular arcs at the beginning and end of

the motion which are tangent to the dwell and rise lines.

Poor acceleration characteristics limits the use of this motion to lowcam speeds.

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5.6.3. Parabolic Motion:

It is a constant acceleration motion.

For a given cam speed and follower rise, parabolic motion has the

lowest or maximum acceleration.

It is recommended for low or moderate speeds.

To construct the displacement diagram:

Use even number of time divisions with least number of division equalto six.

Through the origin of the displacement diagram, construct any line atany angle to the y-axis or displacement axis.

Divide this line into parts proportional to:

1, 3, 5, 5, 3, 1 for six divisions

1, 3, 5, 7, 7, 5, 3, 1 for eight divisions, and so on.

Connect the last division with the point marked on the displacementaxis corresponding to the rise.

Through the other divisions draw lines parallel to this line to obtain thefollower displacement for each division of cam rotation.

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The parabolic motion of the follower can be represented by theequation:

This equation is valid only up to the inflation point where the followerrise is d/2 and the cam angle is /2.

The velocity and the acceleration of the follower is respectively:

)7.5(2cy

)9.5(2

)8.5(2

2

2

dy

and

dc

)11.5(4

)10.5(44

2

2

2

2

22

d

dt

yda

d

dt

dd

dt

dyv

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The velocity is maximum for =/2and is given by

For the second half of the motion the displacement equation canbe given as:

Where c1, c2, and c3are constants to be evaluated from the boundaryconditions:

The boundary conditions are:

when =, y = dwhich yields

when =, v= 0which yields

)12.5(2

2

42max

ddv

)13.5(2

321 cccy

)14.5(2

321 dccc

)15.5(02 32 cc

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and, when =/2, v=vmax

solving equation (14), (15), and (16) for the unknownssimultaneously,

Substituting for the constants, the displacement equation becomes

)16.5(2

22 32

dcc

23

2

1

2

4

dc

dc

dc

)18.5(121

,

)18.5(24

2

2

2

bdy

or

add

dy

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The velocity and acceleration equations, respectively, are given by

Here we note that jerk, which is proportional to the rate of changeof force (acceleration), is infinite at the beginning and end of riseand at the inflation point.

)20.5(4

)19.5(1

4

2

2

2

2

d

dt

yda

d

dt

dy

v

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5.6.4. Simple Harmonic Motion

Obtained graphically as shown in the figure below.

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The simple harmonic motion of follower can be represented by theequation:

The constants C1 and C2 can be obtained from the boundaryconditions:

y = 0 for= 0 and

y = d for=.

Applying this conditions. We get

Solving for the constants

)21.5(cos21 CCy

)23.5(cos

)22.5(00cos

21

21

dCC

CC

221

dCC

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the equation of the follower displacement is given by

where the angular velocity of the rotational radius, r is taken to be

constant.

To relate the angular velocity of the rotational radius r to the camangular velocity ,

N.B. when=, the corresponding cam rotation is

Which yields:

)24.5(cos12

cos1

2

tdd

y r

tort

r

,

)26.5(

)25.5(

r

r

and

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Substituting for r , the displacement equation is

But the cam angle = t. Hence the displacement equationbecomes

The velocity and acceleration are given by time derivatives ofdisplacement.

)27.5(cos1

2

t

dy

)28.5(cos12

dy

)30.5(cos2

)29.5(sin2

2

da

d

v

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5.6.5. Cycloidal Motion

Often employed to avoid infinite jerk between rise, return and dwellmotions.

Motion which has a zero acceleration at the beginning and end ofthe rise.

its displacement diagram is obtained from cycloid, which is definedas the locus of a point on a circle that rolls on a straight line.

for cam rotation angle , the displacement of the cycloidal motion

is the rise d, which must be equal to the circumference of therolling circle.

From which the radius of the rolling circle is obtained to be

)31.5(2 dr

)32.5(2

dr

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The construction of the displacement diagram for the cycloidalmotion is as shown in the figure (a) below.

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It can be shown that the ordinate of a point Pon the displacementdiagram is:

Since the rolling circle makes one revolution for the rise (return) dof the follower, the displacement of the rotational radius can berelated to the cam angular displacement:

The displacement y of the follower with cycloidal motion is thengiven by

Substituting for r,

)33.5(sin ry

)34.5(2

)35.5()2sin(2

ry

)36.5()2sin(2

1

dy

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The velocity and acceleration of the cycloidal motion are then:

)38.5()2

sin(2

)37.5()2

cos(1

2

da

dv

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5.6.6. Advanced CAM Curves

for high-speed cams the selection of the motion of the cam followermust not be based only on the displacement but also on forcesacting on the system as a result of the motion selected.

Dynamic loading produces impact loads, which may cause: Structural damage

Undesirable vibrations

Jerk is an indication of the impact characteristics of the loading;

Impact might have jerk equal to infinity

This may be improved by using: Cycloidal motion, or

Combining portions of several basic curves, which can give lowpeak acceleration (force).

When basic curves are combined, the displacement curve must betangent at the junction and acceleration should be equal.

By so selecting the acceleration at the junction, infinite jerk isavoided.

At junctions the velocity also must match.

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Another method of eliminating the deficiencies of basic curves isto use a polynomial curve. The polynomial equation is:

where yis the follower rise, is the cam angle as before, and C0, C1,C2, , Cnare constants, which depend on the boundaryconditions.

As an example lets select the following boundary conditions.

for

)39.5(...3

3

2

210

n

nCCCCCy

0,0,

0,0,00

yydy

yyy

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For this particular case, the polynomial equation to be used is:

The first and second derivatives are:

)(5

5

4

4

3

3

2

210 aCCCCCCy

)(201262

)(5432

2

3

5

2

2

4

2

3

2

2

4

5

3

4

2

321

cdt

d

Cdt

d

Cdt

d

Cdt

d

Cy

bdt

dC

dt

dC

dt

dC

dt

dC

dt

dCy

)(201262

)(5432

32

5

22

4

2

3

2

2

4

5

3

4

2

321

eCCCCy

dCCCCCy

or

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Substituting the boundary conditions in equations (a), (d) and (e)and solving for the constants, we obtain

The equations of motion are obtained by substituting for theconstants.

554433

210

6,

15,

10

,0,0,0

dC

dC

dC

CCC

)(12018060

)(306030

)(61510

3

5

2

2

4

2

3

2

4

5

3

4

2

3

5

5

4

4

3

3

hddd

y

gddd

y

fddd

y

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This is called 3-4-5 polynomial, because of the remaining terms inthe follower displacement equation.

This type of cam will begin and end its motion slower than other

types.

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5.6.7. Nonstandard Cam Curves

Motion characteristicsother than the simple dwell-rise-return canbe obtained by combining various curves like the following analyticfunctions. (see figures given separately).

a. Harmonic and half harmonic

b. Cycloidal and half cycloidal, and

c. Eighth-power polynomial

These can be used to generate the required motion of the followerand are developed for cam design to avoid infinite jerk.

It should be noted that in using these functions,

the rise or return cam angle should start at the position where the

cam rotation angle = 0, and matching of angles should be carefully handled.

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By observing the velocity characteristics of the above mentionedfunctions the following remarks can be made.

i. Harmonic motion characteristics can be applied with or without any

dwell after the rise or return motion.ii. Cycloidal motion characteristics can also be applied in similar manner

to harmonic motion characteristics.

iii. The eighth-power polynomial motion is good for rise-dwell-returnmotion with a dwell after the rise motion because it provides zerovelocity condition after the rise and avoid infinite jerk.

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5.7. ANALYTICAL CAM DESIGN

Graphical method of cam design is limited to slow-speedapplications

The analytical design approach is adapted for high speed cams.

5.7.1. Disc Cam with Radial Flat-Faced Follower In analytical method of cam design, three valuable characteristics of

the cam are determined.i. Parametric equation of the cam contour,ii. Minimum radius of the cam to avoid cusps or sharp points, and

iii. Location of point of contact which gives the length of the follower face. Consider a cam with flat-faced radial follower, in which the cam

rotates with a constant angular velocity. The point of contact b/n the follower and the cam is at (x, y), which

is at a distance of lfrom the radial center-line of follower. Then the displacement of the follower from the origin is given by

where C = minimum radius of the cam, andf() = the desired motion of the follower, which is a function

of the angular displacement of the cam angle.

)40.5()(fCr

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From geometr of the cam and follo er arrangement the radial

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From geometry of the cam and follower arrangement, the radialdisplacement is

And the length of the contact is given to be

The right side of equation (5.42) is the derivative with respect toof equation (5.41).

And substituting for r,

The cam contour is determined by solving equation (5.41) and(5.42) simultaneously from which we obtain

)43.5(ddrl

)44.5()())((

ffcd

dl

)45.5(cos)(sin)(

sin)(cos)(

ffCy

ffCx

)41.5(cossin xyr

)42.5(sincos xyl

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Minimum Radius c to Avoid Cusps

A cusp occurs when both dx/dand dy/dare equal to zero.

a point is formed on the cam as can be seen in the figure below.

At this point, for the follower rotation by d, the point of contact (x,y) does not change; i.e.

Differentiating equation (5.45)

Equation (5.47) can become zero simultaneously if

)47.5(

cos)()(

sin)()(

ffCd

dy

ffCd

dx

)48.5(0)()( ffC

)46.5(0 d

dyd

dx

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to avoid cusps, the condition that must be satisfied is

If the sum is positive, Cbecomes negative and doesnot have any physical significance.

In this case the minimum diameter of the cam is determined from

the hub of the cam.

5.7.2. Disc Cam with Radial Roller Follower The displacement of the center of the follower from the cam center

is given by

where Ro= minimum radius of the pitch circle of the cam;f() = radial motion of the follower as a function of the cam angle.

Let = radius of curvature of the pitch surface

c= radius of curvature of the cam surfaceRr= radius of the roller.

)51.5()(fRr o

)()( ff

)50.5()()( ffC

or

)49.5(0)()( ffC

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Keeping constant if R is increased decreases

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Keeping constant, if Rr is increased, cdecreases.

If Rris increased until Rr= , c equals zero and the cambecomes pointed.

So to prevent pointed cams, Rrmust be less than minwhere min

is the minimum value of over a particular segment of profile beingconsidered.

The radius of curvature at a point, expressed in polar coordinates(r,), is given by:

where r() and the first two derivatives are continuous.

From equation (5.52) mincan be determined that prevent pointedcam surface (but complex to solve)

min is determined from curves developed for the ratio min/Ro forvarious L/Rovalues, where L is follower rise.

)52.5(

22

22

2

2

3

2

2

d

rdr

d

drr

d

dr

r

Harmonic motion

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Harmonic motion

Harmonic motion

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Harmonic motion

Cycloidal motion

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Cycloidal motion

Cycloidal motion

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Cycloidal motion

Eighth power polynomial

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Eighth-power polynomial

Eighth-power polynomial

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Eighth-power polynomial

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5.7.3 Maximum Pressure Angle

Determination of the

maximum pressure angle isoften difficult because of thealgebraic manipulationsinvolved.

Kloomokand Maffleyhave

developed a monogram, fordetermining the maximumpressure angle as a functionof the cam angle and L/Roratio for harmonic, cycloidal

and eighth-power motions.(see the figure above)

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5.8.Tangent Cam with

Reciprocating Roller Follower

A camin which the flanks arestraight and tangential to thebase and nose circles is

known as a tangential Cam. This type of cams are used in

engine cam shafts to operatethe inlet and exhaust valves.

5 8 1 Velocity and Acceleration Analysis of Tangent Cams

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5.8.1. Velocity and Acceleration Analysis of Tangent Cams

a) When the Roller is in Contact with the Straight Flanks.

The figure on the slid above shows the roller having contact withthe straight flank.

In its lowest position, the roller center lies at B.

When the cam has rotated through the angle , the center of theroller moves to C.

i.e. for the cam angle , the follower rise from its lowest position isgiven by BC.

From the figure it can be observed that OB = OG.

)55.5(1cos

1

cos

)54.5(cos

)53.5(

OGOGOG

y

OGOC

But

OGOCy

Since OG = OE + EG where OE = r2 the radius of the base circle

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Since OG = OE + EG, where OE = r2, the radius of the base circle,and EG = r1 , radius of the roller,

The velocity of the follower is obtained by differentiating equation

(5.57) with respect to time:

)57.5(1cos

11cos

1

cos

)56.5(21

o

o

ROGOGOGy

RrrOG

)59.5(cos

sin

)58.5(1cos

1

2

o

o

Rv

or

Rdt

dyv

The acceleration of the follower is given by

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The acceleration of the follower is given by

b) When the Roller is in Contact with the Nose. A roller follower in contact with nose is as shown in the figure

below.

B is the center of the roller when it is in the top-most position.

When the cam turns through an angle , the displacement of theroller, measured from the top position of the roller, is given by

)61.5(cos

cos2

)60.5(cos

sin

3

2

2

2

o

o

Ra

or

Rdt

dy

dt

dva

)62.5(OCOBBCy

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But, OC =OE + EC,

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But, OC OE + EC,

the displacement of the follower is

N.B. that CK = L , OK = r

Again from the above figure

From which we obtain

)63.5()coscos()( CKOKOBECOEOBy

)64.5()coscos()( LrOBECOEOBy

)65.5(sinsin rLEK

)67.5(sincos

)66.5(sin)cos1(sin

222

222222

rLL

or

rLL

Substituting the value of L cos, we get the displacement of the

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Subst tut g t e a ue o L cos, e get t e d sp ace e t o t efollower to be

Differentiating equation (5.68) we obtain the velocity andacceleration equations, respectively,

)68.5(sin11cos1 22

L

rLry

)70.5(

sin1

sin)2cos(cos

)69.5(

sin12

)2sin(sin

2

3

2

2

432

2

2

L

rL

rrLra

L

rL

rrv

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