fin th machines 4

8/13/2019 Fin Th Machines 4

1/16

Theory Machines 4

CAMS

A Cam is a mechanical element used to drive another element, a follower, through a specified motion

by direct contact. Follower motions having almost any desired characteristics are not difficult todesign. By desired characteristics are typically meant the following: displacement the height or

distance through which the follower is moved for one revolution of the cam; velocity the speed withwhich the cam moves the follower; acceleration the rate of change of velocity of the follower

remember force mass ! acceleration "; #er$ the rate of change of acceleration. %he problem in camdesign is how to determine a cam contour that will ultimately deliver a specified motion with

acceptable velocity, acceleration and #er$.

CLASSIFICATION OF CAMS AND FOLLOWERS:

Disk or Plate Ca !ith Ra"ial Roller Follo!er

&ost common type of cam. Follower may not be radial in some applications, ie, the centre of the roller,

the centre of cam rotation and direction of follower travel may not coincide.

'iagram ()

Translation Ca #We"$e Ca%

*ot very commonly used. %he cam moves over and bac$, reciprocating motion, which drives thefollower vertically. Cam reciprocating motion may be due to a pneumatic or hydraulic cyliner, rac$ and

pinion, or some other linear output device.


2/16

'iagram (+

Cylin"rical &'arrel Ca

Again not a very common device but may be used where it is reuired to have the a!is of rotation ofthe of the cam parallel to the direction of follower motion

'iagram ((

Face Ca

A!is of rotation of the cam and follower direction of motion are parallel but not collinear. %he cam isessentially a cylinder cut at an angle to the a!is of rotation. %he follower rides on the perimeter of the

cylinder as the cam rotates.'iagram (-


3/16

FOLLOWER T(PES

%here are a range of follower types which are used but all effectively perform the same function theyfollow the cam profile as designed. hat is important is that they must remain in contact with the cam

surface at all times. Failure to do so will lead to non/predictable motion, ie. something other than thecam profile may determine the follower displacement, vel. accel and #er$ at a given instant. 0mpact,

noise, e!cessive wear or damage to cam and follower surfaces may result from the follower losingcontact with the cam surface. 1ravity, springs and mechanical constraint are used to ensure that the

follower is pressed tightly against the cam surface as it rotates.

O))set reci*rocatin$ kni)e+e"$e )ollo!er

'iagram (2

Reci*rocatin$ ra"ial )lat )ace" )ollo!er

Follower stem lines up with the cam a!is of rotation

'iagram (3

Oscillatin$ Roller Follo!er

'iagram (4


4/16

Oscillatin$ s*herical+)ace or c,r-e"+shoe )ollo!er

'iagram (5

DISPLACEMENT DIA.RAMS

%he input motion 6 t" is derived from the angular velocity of the shaft, 7.%he output displacement ofthe follower 8t" consists of rises, dwells and falls. hat is typically reuired is to design a cam to

provide an output motion 8 t " for a given angular motion input. %he diagram below shows a typicaldisplacement diagram for a disc cam with one rise, one fall and two dwells occurring within a cam

rotation of (39 degrees. A cam may have multiple rises and falls, no dwells, or whatever configurationis necessary for the desired follower motion.

'iagram (

CAM /OCA'0LAR( AND DEFINITIONS

Pitch C,r-es: ocus generated by the trace point for a $nife/edge follower the pitch curve is the cam

surface. For a roller follower the trace point is the centre of the roller therefore the pitch curve isoutside the cam surface

Prie Circle:%his is the smallest circle tangential to the pitch curve and with its centre at the cam

centre of rotation.

'ase Circle:%his is the smallest circle tangential to the cam surface and with its centre at the cam

centre of rotation.


5/16

D!ell Perio"s:here no follower displacement occurs even though the cam is rotating.

'iagram -9

T(PES OF FOLLOWER MOTION

'well periods are typically decided by the application, ie. in a motor car engine the valve must remainclosed or open for a fi!ed period. 'esign a dwell is not difficult.

?ise and return have many possible follower motions / uniform motion, parabolic, harmonic,

cycloidal, etc. so a choice available to the designer>

0ni)or Motion

%his is where the follower moves at a constant velocity. %he rise or fall on a displacement diagram is

thus plotted as a straight line. @roblems arise where the uniform motion meets a dwell. As can be seenbelow this gives rise to discontinuities in the velocity, acceleration and #er$ curves. er$ is the rate of

change of acceleration and is very important in determining the smoothness of the displacement curve"


6/16

'iagram -)

Mo"i)ie" 0ni)or Motion

&odified uniform motion is where uniform or constant velocity motion is lin$ed to a dwell using a

parabolic curve. %he displacement diagram below shows a dwell lin$ed to a constant velocity using aparabolic rise 6)and further on the constant velocity section 6+is lin$ed bac$ to the dwell at the top of

the rise using another parabolic section 6(.

'iagram -+


7/16

Para1olic Motion%he displacement of a body moving from rest with a constant acceleration is

D At+ displacement.

A acceleration.t time.

%his is the euation for a parabola !+ $y where $ constant" so that motion involving constantacceleration is often called parabolic motion. %he following shows how a parabolic motion may be

used to lin$ a dwell and a constant velocity section in a displacement diagram

For a curve where displacement, h, and 6 are the coordinates, h D B 6+and 6 7t where 7 isconstant, B A>7. Accel>Ang Eel.

'iagram -(

/& @arabolic rise h occurring in angle 6)

&/* Constant Eelocity ?iseEel E" with rise h occuring in angle 6+

h D B.6)+which implies B +h> 6)

+

E B.6) +h> 6) Gn. )

For constant velocity rise to &, assuming no parabolic section.

h E.6+ or E h> 6+ Gn. +

6) +. 6+ From euations ) and +

o that the point of inflection of the parabolic and constant velocity curves is twice the distance of thepoint of intersection of the constant velocity portion alone.


8/16

'iagram --

As can be seen above the solution is not ideal as, while velocity and acceleration remain within finite

limits, #er$ reaches infinite values at the start and end of each parabolic section.

Si*le 2aronic Motion

0n pure harmonic motion the )st, +ndand (rdderivatives of displacement all remain finite. As can be seen

for the circular cam, radius ? and rotating about a centre a distance b from the true centre, and flat facefollower shown below a set of euations to describe follower motion in terms of cam angle of rot can

be derived.'iagram -2


9/16


10/16

uppose


11/16

A cycloid is the locus of a point on a circle as the circle rolls on a straight line.

'iagram -

%o create a cycloidal motion in cam design involves superimposing cycloidal motion on constantvelocity motion. ee the graphical method shown below for creating a cycloidal curve.

'iagram 29

0n designing a cam profile with cycloidal motion the total displacement, h, needs to be eual to thecircumference of the circle creating the cycloid. A circle of radius r h>+I ma$es one revolution in

going from =ero displacement to ma!imum displacement h. J traces a cycloid and the graph of J as itmoves vertically generates the displacement diagram.

'iagram 2)


12/16

At point J the displacement for the centre of the circle describing the cycloid is rK.

%he actual displacement of point J is given by

J rK rsinK r K sinK"

As regards cam rotation the rise s occurs inside an angle L and total rise h occurs inside an angle M

%herefore K +I L>M" +I ! fraction of total rotation for rise h"

Also r h>+I

Combining the above gives the following

s h L>M" h>+I" sin +I L>M" 'isplacement

ds>dt 7 h>M ) cos +IL>M" Eelocity

d+s>dt+ 7++Ih>M+" sin +IL>M" Acceleration

d(s>dt( 7(-I+h+>M(" cos +IL>M"er$

'iagram 2+


13/16

As can be seen above when a cycloidal rise is plotted both velocity and acceleration are =ero at the start

and end of the rise. %his means that the curve is suitable for lin$ing with a dwell transition from =erovelocity and acceleration is therefore smooth and #er$ remains finite.

COMPARISON OF MOTION C0R/ES

Parabolic motiongives constant acceleration. ne advantage is that for a given rise in given time this

curve gives the lowest value of acceleration when compared with other methodsModifier constant velocity/ going from a constant velocity to constant acceleration implies infinite

#er$.Simple harmonic motion/ acceleration curves are smooth when rise and fall periods are )59N, dwell or

lin$ing with constant velocity motion creates problems.

Cycloidal motion/ connects smoothly with dwell, but gives the highest pea$ accelerations. But thecycloidal motion acceleration curve can connect with the acceleration curve for any other cycloidalmotion or dwell without #er$.

PRESS0RE AN.LE

%his is the angle which the common normal for the com and follower ma$e with the path of thefollower.

'iagram 2(


14/16

oading system on a disc cam with a radial roller follower is as shown above. %he force which the cam

imposes on the follower @ acts at an angle M to the follower as shown. %his force splits into two

components, one tangential to the follower direction of travel and one at right angles to it. %he angle Mis $nown as thepressure angle.

@ @t H @n

@n is undesirable as it e!erts a side thrust on the follower guides or bearings. 0f large enough it cancause wear or ma$e follower bind but in any case will cause loss of power due to friction. ?educing the

pressure angle reduces side thrust. 0t is common practice in cam design to try and $eep the pressureangle below (9 degrees.

ays to reduce pressure angle:

). 0ncrease prime circle diameter

+. ?educe follower total rise.(. 0ncrease the amount of cam rotation for a given follower displacement.

-. Change follower motion type.2. Change follower offset ?educe generally".

Pro1le 'erive euations to describe the displacement diagram of a cam which rises with parabolicmotion from one dwell to another, such that the total lift is and cam rotation angle is M. @lot the

displacement diagram and its first three derivatives velocity, acceleration and #er$ w.r.t. cam rotation

'iagram 2-


15/16

%wo parabolas used with inflection point at mid/range.

et y aO+ H bO H c @arabolic curve

8P +aO H b

8PP +a

8PPP 9

For the first parabolic curve 9 Q O>M Q 9.2

O>M 9 R y9" 9 R c 9

yP9" 9 R b 9

%hus y aO+

At point of inflection O>M 9.2 and y >+

>+ a M+>- giving a +>M+

%herefore: y +O>M"+

yP -O>M+

yPP ->M+

yPPP 9 Accel. Const

%his is #ust true go to the point of inflection.

&a!imum slope occurs at the point of inflection O>M 9.2

yP ma! -O>M+ +>M

*ow ta$e from the point of inflection on: For the second parabolic curve 9.2 Q O>M Q ) and the

euation of the curve is the same as before: y aO+ H bO H c

At the end of the rise O>M ) R y)" aO+ H bO H c Gn. )

yP)" 9 +aO H b Gn. +


16/16

&atching with the slope at the point of inflection: O>M 9.2 this gives

yP +>M +aO H b a M H b Gn. (

olving ). +. and ( simultaneously gives:

a / +>M+

b - >M

c /

'isplacement euations for the second parabolic section are:

y ) +)/ O>M"+" 'isplacement

yP ->M ) O>M" Eelocity

yPP /->M+ Acceleration

yPPP 9 er$

%hroughout this e!ample when differentiating to obtain velocity, acceleration and #er$ they have beenconsidered with respect to angle and not with respect to time. %hey should be scaled by 7, 7+ and 7( to

ma$e them correct w.r.t. time. 0n general for a constant speed shaft, the $inematic derivatives of thefollower motion are eual to the time derivatives of the follower if the $inematic derivatives are scaled

by 7, 7+, and 7( respectively.

fin th machines 4

Documents