mechanics of machines dr. mohammad kilani class 5 cams

Mechanics of Machines

Dr. Mohammad Kilani

Class 5CAMS

Introduction to Cams

Cam-follower systems are frequently used in many kinds of

machines. Common examples of cam follower systems are the

automobile engine valves, which are opened by cams.

Machines used in the manufacture of many consumer goods are

full of cams. Compared to linkages, cams are easier to design to

give a specific output function, but they are much more difficult

and expensive to make than a linkage.

Cams are a form of degenerate fourbar linkage in which the

coupler link has been replaced by a half joint.

Cams Terminology

Cam-follower systems can be classified in several ways:

by type of follower motion, either translating or rotating (oscillating);

by type of cam, radial, cylindrical, three-dimensional;

by type of joint closure, either force- or form-closed;

by type of follower, curved or flat, rolling or sliding;

by type of motion constraints, critical extreme position (CEP), critical path motion

(CPM); by type of motion program, rise-fall (RF), rise-fall-dwell (RFD), rise-dwell-fall-

dwell (RDFD).

Types of Follower Motion

Follower motion can be an oscillation or

translation. An oscillating follower

rotates around a pivot point. A rotating

follower moves in usually rectilinear

translation.

The choice between these two forms of

the cam-follower is usually dictated by

the type of output motion desired.

Type of Joint Closure

Force closure requires an external force be applied to the joint in order to

keep the two links, cam and follower, physically in contact. This force is

usually provided by a spring. This force, defined as positive in a direction

which closes the joint, cannot be allowed to become negative. If it does, the

links have lost contact because a force-closed joint can only push, not pull.

Form closure, closes the joint by geometry. No external force is required.

There are really two cam surfaces in this arrangement, one surface on each

side of the follower. Each surface pushes, in its turn, to drive the follower in

both directions.

Type of Follower

Follower, in this context, refers to that part of the follower link which contacts

the cam. Three types are available, flat-faced, mushroom (curved), and roller.

The roller follower has the advantage of lower (rolling) friction than the sliding

contact of the other two but can be more expensive.

Flat-faced followers can package smaller than roller followers and are often

favored for that reason as well as cost in automotives.

Roller followers are more frequently used in production machinery where their

ease of replacement and availability from bearing manufacturers' stock in any

quantities are advantages.

Type of Cam

The direction of the follower's motion relative to the axis

of rotation of the cam determines whether it is a radial

or axial cam.

In radial cams, the follower motion is generally in a

radial direction. Open radial cams are also called plate

cams.

An axial cam is one whose follower moves parallel to the

axis of cam rotation. This arrangement is also called a

face cam if open (force-closed) and a cylindrical or barrel

cam if grooved or ribbed (form-closed).

Type of Motion Constraints

There are two general categories of motion constraints which determine the shape of the cam,

critical extreme position (CEP; also called endpoint specification) and critical path motion (CPM).

Critical extreme position refers to the case in which the design specifications define the start and

finish positions of the follower (i.e., extreme positions) but do not specify any constraints on the

path motion between the extreme positions. This case is the easier of the two to design as the

designer has great freedom to choose the cam functions which control the motion between

extremes.

Critical path motion is a more constrained problem than CEP because the path motion, and/or

one or more of its derivatives are defined over all or part of the interval of motion.

Dwells and Type of Motion Program

A dwell is a period of time for which no change in output motion appears for a

changing input motion. Dwells are an important feature of cam-follower systems

because it is very easy to create exact dwells in these mechanisms.

The motion programs rise-fall (RF), rise-fall-dwell (RFD), and rise-dwell-fall-dwell

(RDFD) all refer mainly to the CEP case of motion constraint and in effect define how

many dwells are present in the full cycle of motion, either none (RF), one (RFD), or

more than one (RDFD)

The cam-follower is the design type of choice whenever a dwell is required. Cam-

follower systems tend to be more compact than linkages for the same output motion.

Dwells and Type of Motion Program

If your need is for a rise-fall (RF) CEP motion, with no dwell, then you

should really be considering a crank-rocker linkage rather than a cam-

follower to obtain all the linkage's advantages over cams of reliability, ease

of construction, and lower cost.

If your needs for compactness outweigh those considerations, then the

choice of a cam-follower in the RF case may be justified. Also, if you have a

CPM design specification, and the motion or its derivatives are defined

over the interval, then a cam-follower system is the logical choice in the RF

case

S V A J DIAGRAMS

The first task faced by the cam designer is

to select the mathematical functions to be

used to define the motion of the follower.

The easiest approach to this process is to

"linearize“ the cam, i.e., "unwrap it" from

its circular shape and consider it as a

function plotted on Cartesian axes.

S V A J DIAGRAMS

We plot the displacement functions, its first

derivative velocity v, its second derivative

acceleration a, and its third derivative jerk}, all on

aligned axes as a function of camshaft angle.

Note that we can consider the independent

variable in these plots to be either time t or shaft

angle θ, as we know the constant angular velocity

(ω) of the camshaft and can easily convert from

angle to time and vice versa.

Cam Design Procedure

A cam design begins with a definition of the

required cam functions and their svaJ diagrams.

Functions for the nondwell cam segments should

be chosen based on their velocity, acceleration,

and jerk characteristics and the relationships at the

interfaces between adjacent segments including

the dwells..

Double-Dwell Cam Design Choosing S V A J Functions

The double-dwell case is quite common design requirement for cams. A double-dwell cam design specifications are

often depicted on a timing diagram which is a graphical representation of the specified events in the machine cycle

cycle. A machine's cycle is defined as one revolution of its master driveshaft.

In a complicated machine. The time relationships among all subassemblies are defined by their timing diagrams

which are all drawn on a common time axis.

Example 1. Uniform Speed S V A J Diagram

Consider the following cam design CEP specifications:

dwell at zero displacement for 90 degrees of cam rotation (low dwell)

rise 1 in (25 mm) in 90 degrees of cam rotation

dwell at 1 in (25 mm) for 90 degrees of cam rotation (high dwell)

fall 1 in (25 mm) in 90 degrees of cam rotation

cam speed (ω) 1 rev/sec 2π rad/sec


A uniform velocity cam design merely "connect the dots" on the timing diagram by straight lines to create the displacement diagram. This

approach ignores the effect on the higher derivatives of the resulting displacement function.


The acceleration is zero during the rise and fall intervals, but

becomes infinite at the boundaries of the interval, where rise

meets low dwell on one side and high dwell on the other.

Note that the velocity function is multivalued at these point,

creating discontinuities at these boundaries.

The effect of these discontinuities is to create a portion of the

velocity curve which has infinite slope and zero duration. This

results in the infinite spikes of acceleration shown at those

points


Clearly the dynamic forces will be very large at these

boundaries and will create high stresses and rapid

wear. This is an unacceptable design.

In fact, if this cam were built and run at any

significant speeds, the sharp comers on the

displacement diagram which are creating these

theoretical infinite accelerations would be quickly

worn to a smoother contour by the unsustainable

stresses generated in the materials.

The Fundamental law of Cam Design

Any cam designed for operation at other than very

low speeds must be designed with the following

constraints:

The cam function must be continuous through the

first and second derivatives of displacement across

the entire interval (360 degrees).

Corollary: The jerk function must be finite across the

entire interval (360 degrees).


In any but the simplest of cams, the cam motion program

cannot be defined by a single mathematical expression,

but rather must be defined by several separate functions,

each of which defines the follower behavior over one

segment, or piece, of the cam.

These expressions are sometimes called piecewise

functions, and must have third-order continuity at all

boundaries. The displacement, velocity and acceleration

functions must have no discontinuities.


If any discontinuities exist in the acceleration function,

then there will be infinite spikes, or Dirac delta functions,

appearing in the derivative of acceleration, jerk.

Thus the corollary merely restates the fundamental law of

cam design. In the uniform velocity cam design example,

the low-degree (linear) polynomial as selected for the

displacement function, resulted in discontinuities in the

upper derivatives.

Simple Harmonic Motion S V A J Diagrams

Simple harmonic functions remain continuous throughout any number of

differentiations. Differentiation of a harmonic functions only amounts for a phase

shift by 90 degrees.

The equations of a simple harmonic rise of magnitude h over a cam angle β that starts

at a cam angle θi are as follows:

i

i

i

i

hj

ha

hv

hs

sin2

cos2

sin2

cos12

3

2

θi

θi + β


i

i

i

i

hj

ha

hv

hs

sin2

cos2

sin2

cos12

3

2

θi

θi + β

Example 2: Simple Harmonic Motion S V A J Diagrams

Consider the cam design CEP specifications of Example 1:






Plot the S V A J diagrams resulting from simple harmonic motion for the rise and fall periods.









rise:

h = 1 in., θi = π/2, β = π/2

22sin4

22cos2

22sin

22cos15.0

j

a

v

s

Fall:

h = -1 in., θi = 3π/2, β = π/2

232sin4

232cos2

232sin

232cos15.0

j

a

v

s

i

i

i

i

hj

ha

hv

hs

sin2

cos2

sin2

cos12

3

2


rise:

h = 1 in., θi = π/2, β = π/2

22sin4

22cos2

22sin

22cos15.0

j

a

v

s

Fall:

h = -1 in., θi = 3π/2, β = π/2

232sin4

232cos2

232sin

232cos15.0

j

a

v

s

0 π/2 π 3π/2 2π


The velocity function resulting from a SHM is continuous but the acceleration has discontinuities at its starting and ending points.

The acceleration is not zero at these points, and thus do not match the zero acceleration of the dwell period.

The simple harmonic motion displacement function, when joint with dwells, does not satisfy the fundamental law of cam design.

0 π/2 π 3π/2 2π


The only case in which the simple harmonic motion

displacement functions satisfy the fundamental law of

cam design is in which the rise cam angle βr is equal to

the fall cam angle βf

with no dwells in between.

A common example is the non-quick return (equal period)

rise-fall (RF), in which the follower rises in 180° and falls in

180°. βr

= βf

= 180° in this case.

0 π 2π

Cycloidal Cam Motion

The cycloidal motion starts with a full wave sine wave defined as the

acceleration function for the cam follower through the cam angle β

ii

iii

iii

ii

i

iiiii

hj

ha

hvhs

hCk

hkhsvs

kkCdvs

kCdav

hssvvCa

2cos4,2sin2

2cos1,2sin2

1

2,0,,,0,0

2sin4

2cos2

,0)(,0)(,2sin

32

2

221

212

2

1

gives using

Example 3:Cycloidal Motion S V A J Diagrams

i

i

i

ii

hj

ha

hv

hs

2cos4

2sin2

2cos1

2sin2

1

32

2








rise:

h = 1 in., θi = π/2, β = π/2

24cos32

24sin8

24cos12

224sin42

1

j

a

v

s

rise:

h = 1 in., θi = π/2, β = π/2

24sin32

24sin8

24cos12

224sin42

1

j

a

v

s

Polynomial Cam Motion

The displacement equation for general polynomial motion can be written as:

where s is the follower displacement, θ is the angular position of the cam, and θi is the initial cam angle at the beginning of the

polynomial motion. The integer N is referred to as the degree of the polynomial

The velocity, acceleration and kerk are obtained by successive differentiation as

N

k

kik

NiNii

Cs

CCCCs

0

2210

N

k

kik

N

k

kik

N

k

kik

NiNi

Ckkkdt

d

d

da

dt

datj

Ckkdt

d

d

dv

dt

dvta

kCtv

NCCCdt

d

d

ds

dt

dsv

0

33

0

22

0

1

121

11

1

2

Example 3:Polynomial Motion S V A J Diagrams

Determine the polynomial equation that will satisfy the following

conditions

1. s = 0 when θ = θi

2. v = 0 when θ = θi

3. a = 0 when θ = θi

4. s = h when θ = θi + β

5. v = 0 when θ = θi + β

6. a = 0 when θ = θi + β

Substituting the six required conditions , we get the following set of

equations

352

4322

45

34

2321

55

44

33

2210

201262

5432

iii

iiii

iiiii

CCCCa

CCCCCv

CCCCCCs

02

0

0

22

1

0

C

C

C

0

0

0

2

1

0

C

C

C

0201262

054323

52

4322

45

34

2321

55

44

33

2210

CCCC

CCCCC

hCCCCCC

53

44

33

6

15

10

hC

hC

hC

Solving the equations produces

2

3323

3

2

2

222

432

543

36036060

12018060

306030

61510

ii

iii

iii

iii

hhhj

hhha

hhhv

hhhs

The equations for s, v, a, and j are therefore

Example 3:Polynomial Motion S V A J Diagrams

Determine the polynomial equation that will satisfy the following

conditions

1. s = 0 when θ = θi

2. v = 0 when θ = θi

3. a = 0 when θ = θi

4. s = h when θ = θi + β

5. v = 0 when θ = θi + β

6. a = 0 when θ = θi + β

Because of the form of the displacement function, this particular

follower motion is referred to as 3-4-5 polynomial motion.

2

3323

3

2

2

222

432

543

36036060

12018060

306030

61510

ii

iii

iii

iii

hhhj

hhha

hhhv

hhhs

Polynomial Motion S V A J Diagrams

There are no jumps in the s, v or a curves,

resulting in good dynamic characteristics.

The curves are similar in many respected to

cycloidal motion, and the two motions are

comparable. For the same values of h, ω, and

β , cycloidal motion produce higher values of

maximum acceleration, but lower values of

maximum jerk than does the 3-4-5 polynomial

motion.

Polynomial Motion S V A J Diagrams

Using the same approach, polynomial

motions can be derived for a wide range of

different boundary conditions for

displacement and derivatives.

Manufacturing precision requirements,

however, increase as the number of

boundary conditions increase.

Combined Functions

Instead of selecting a single function to represent the rise or fall

period of the follower, one may specify a number of interconnected

segments, each of which is represented by a different function.

As seen earlier, the uniform speed rise produced infinite acceleration

at the ends of the rise segment. Instead, one can combine a

uniformly increasing speed rise for a portion of the rise segment,

followed by a uniformly decreasing speed for the other portion. The

resulting follower motion is referred to as parabolic motion.

Combined Functions – Parabolic Motion

Let β be the cam angle turned through which the follower is given a total

rise h. When the follower displacement is h/2, the cam has turned by

β /2. The constant acceleration motion specifications are:

The follower displacement for the first half of the parabolic motion rise is

2

1

2

212

1

4

22

000

0002

,

0)0(,0)0(,

hC

hs

kv

ks

kkC

skCv

svCa

gives using

gives using gives using

22

22

4,

4,

2

h

ah

vh

s

Combined Functions – Parabolic Motion

During the 2nd half of follower motion, the cam has turned by an

additional β /2. The constant acceleration motion specifications are:

Integrating as before, and applying the boundary conditions

hsv

hs

hv

hCa

)(,0)(

2)2(,

2)2(

,42

hkhh

hkhh

hkkh

s

hkk

hv

kkh

skh

v

4

42

2

432

2

332

432

232

2

2

2,0

2

,2

using

using

22

22

4,

4,

2

h

ah

vh

s

Graphical Construction of Radial Cam ProfileKnife-Edge Follower with Uniform Speed Rise and Fall

A cam is to give the following motion to a knife-edged follower :

1. Outstroke during 60° of cam rotation

2. Dwell for the next 30° of cam rotation

3. Return stroke during next 60° of cam rotation

4. Dwell for the remaining 210° of cam rotation.

The stroke is 40 mm and the minimum radius of the cam is 50 mm. The follower moves with uniform velocity during both the outstroke and

return strokes.

Draw the profile of the cam when (a) the axis of the follower passes through the axis of the cam shaft, and (b) the axis of the follower is offset

by 20 mm from the axis of the cam shaft.

Graphical Construction of Radial Cam Profile

Graphical Construction of Radial Cam ProfileKnife-Edge Follower with Simple Harmonic Rise and Fall

A cam is to be designed for a knife edge follower with the following data :

1. Cam lift = 40 mm during 90° of cam rotation with simple harmonic motion.

2. Dwell for the next 30°.

3. During the next 60° of cam rotation, the follower returns to its original position with simple harmonic motion.

4. Dwell during the remaining 180°.

Draw the profile of the cam when (a) the line of stroke of the follower passes through the axis of the cam shaft, and (b) the line of stroke is offset 20

mm from the axis of the cam shaft. The radius of the base circle of the cam is 40 mm. Determine the maximum velocity and acceleration of the

follower during its ascent and descent, if the cam rotates at 240 r.p.m.

Graphical Construction of Radial Cam ProfileKnife-Edge Follower with Simple Harmonic Rise and Fall

Graphical Construction of Radial Cam ProfileRoller Follower with Simple Harmonic Rise and Fall

A cam, with a minimum radius of 25 mm, rotating clockwise at a uniform speed is to be designed to give a roller follower, at the end of a valve rod,

motion described below :

1. To raise the valve through 50 mm during 120° rotation of the cam ;

2. To keep the valve fully raised through next 30°;

3. To lower the valve during next 60°; and

4. To keep the valve closed during rest of the revolution i.e. 150° ;

The diameter of the roller is 20 mm and the diameter of the cam shaft is 25 mm. Draw the profile of the cam when (a) the line of stroke of the valve

rod passes through the axis of the cam shaft, and (b) the line of the stroke is offset 15 mm from the axis of the cam shaft.

The displacement of the valve, while being raised and lowered, is to take place with simple harmonic motion. Determine the maximum acceleration of

the valve rod when the cam shaft rotates at 100 r.p.m.

Draw the displacement, the velocity and the acceleration diagrams for one complete revolution of the cam.

Graphical Construction of Radial Cam ProfileRoller Follower with Simple Harmonic Rise and Fall

Graphical Construction of Radial Cam ProfileFlat Follower with Simple Harmonic Rise and Fall

A cam drives a flat reciprocating follower in the following manner :

1. During first 120° rotation of the cam, follower moves outwards through a distance of 20 mm with simple harmonic motion.

2. The follower dwells during next 30° of cam rotation.

3. During next 120° of cam rotation, the follower moves inwards with simple harmonic motion.

4. The follower dwells for the next 90° of cam rotation.

The minimum radius of the cam is 25 mm. Draw the profile of the cam.

Graphical Construction of Radial Cam ProfileFlat Follower with Simple Harmonic Rise and Fall

mechanics of machines dr. mohammad kilani class 5 cams

Documents

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type of follower follower

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type of motion constraints

type of motion program

cycle of motion

motion programs