§9 － 2 Follower Motion Curves

§9 － 3 Graphical Cam Profile Synthesis

§9 － 4 Cam Size and Physical Characteristics

§9 － 1 Applications and Classification of Cam Mechanisms

Chapter 9 Cam MechanismsChapter 9 Cam Mechanisms

一、 Applications

internal combustion engine coiler Cylindrical cam conveyer

二、 Characteristics Cam mechanisms are the simplest mechanisms to transfer a simple motion i

nto any desired complicated motion. Cam mechanism can not transmit heavy lo

ads because of the point or line contact between the cam and the follower.

§§99 －－ 1 1 Applications and Classification of Cam Mechanisms

Plate cam (or disc cam)

三、 Classifications of Cam Mechanisms

1. By the cam shape

Translating cam Cylindrical cam

2. By the shape of the follower end

Characteristics ： Knife-edge follower——Although it is simple, the stresses at the line of contact

are excessive and cause rapid wear on the follower and the cam surface.

Roller follower——The roller follower greatly reduces wear because the contact

is almost entirely rolling rather than sliding.

Flat-faced follower——This follower is used often in high-speed cam

mechanisms since a dynamic pressure oil film can be formed easily.

3. By the manner of keeping the cam and the follower in contact ：

Form-closed cam

mechanism

Force-closed cam mechanism

（ by a preloaded spring, or by gravity ）

B’

B

0'

0'D

O

O

s

r0

ω

h

B/

01

01

A

0

0

B

C

02 02

一、 Motion curve of the follower1. Technology termsPrime circle, Prime circle radius: r0

Cam angle for rise: δ0

Cam angle for inner dwell: δ02

Total follower travel ： h

Cam angle for outer dwell: δ01

Cam angle for return:0'

Follower Motion Curves:s = s ( t )

v = v ( t )

a = a ( t )

= s (δ )

= v (δ )

= a (δ )

§§99 －－ 2 2 Follower Motion Curves

s

ｏ δv

ｏ δ

a

ｏ δ

2. Polynomial Motion Curve

The standard polynomial equation s=C0+ C1δ+ C2δ2+…+Cnδn

where s —— displacement of the follower, δ——rotation angle ， Ci—— the constants.

（ 1 ） Constant Velocity Motion Curve

0

h

s = C0 + C1v = ds / dt = C1ωa = dv / dt = 0

h

δ0

Depend on the boundary conditions:

δ=0 ， s=0δ=δ0 ， s=h

Therefore 　 C0= 0 ， C1= h/δ0

Equations for rise s ＝ hδ/δ0

v ＝ hω /δ0 a ＝ 0

Rigid impulse——The consequent large inertia force will result in

a very large amount of shock.

（ 2 ） Constant Acceleration and Deceleration Motion Curve The follower is given a constant acceleration during the first half of the rise and a constant deceleration during the second half of the rise.

Constant acceleration motion equations for rise:

s = 2hδ2 /δ0

2

v = 4hωδ /δ02

a = 4hω2 /δ02

1 2 3 4 5 61

9

4

４10

s

ｏ δ

v

ｏ δa

ｏ δ

h/2

A

h/2

2hω/δ0

4hω2/δ02

Constant deceleration motion equations for rise:

s =h-2h(δ0 –δ)2/δ02

v = -4hω(δ0-δ)/δ02

a = -4hω2 /δ02

δ0

BC

Soft impulse—— Since every link in a earn mechanism has elasticity, an abrupt change in the magnitude and/or direction of inertia force would initiate undesired vibration.

There are three more abrupt changes in the

acceleration of the follower(points A 、 B 、 C)

（ 3 ） Polynomial Motion Curve

s=10h(δ/δ0)3 － 15h (δ/δ0)4+6h (δ/δ0)5

There is neither rigid impulse nor soft impulse.

v =ds /dt = C1ω+ 2C2ωδ+ 3C3ωδ2+ 4C4ωδ3+5C5ωδ4

a =dv/dt = 2C2ω2+ 6C3ω2δ+12C4ω2δ2+20C5ω2δ3

Expressions :

Depend on the boundary conditions:

s = 0 ， v = 0 ， a = 0 Whenδ= 0;

s = h, v = 0 ， a = 0 Whenδ =δ0.

Results in C0 = C1 = C2 = 0, C3 = 10h/δ03

C4 =15h/δ04 , C5 = 6h/δ0

5

s =C0+ C1δ+ C2δ2+ C3δ3+ C4δ4+C5δ5

Displacement equation

v

a h

ssva

O δ(t)

3. Triangle Function Motion Curve

(1) Cosine Acceleration Motion Curve

(or Simple Harmonic Motion

Curve)motion equations for rise:

s=h[1-cos(πδ/δ0)]/2

v =πhωsin(πδ/δ0)δ/2δ0

a =π2hω2 cos(πδ/δ0)/2δ02

Although the motion curve avoids the sudden reverse of

acceleration at the middle-point of the travel, the acceleration

will still change abruptly at the beginning and end points. That

would produce soft impulses.

v

O δ

s

O δ

a

O δ

h

δ0

amax= 4.93hω2/δ02

vmax=1.57hω/2δ0

v

O δ

s

O δ

a

O δ

h

δ0

amax=6.28hω2/δ02

vmax= 2hω/δ0

（ 2 ） Sine Acceleration Motion

Curve

(or Cycloid Motion Curve)motion equations for rise:

s = h[δ/δ0 - sin(2πδ/δ0)/2π]

v = hω[1-cos(2πδ/δ0)]/δ0

a =2πhω2 sin(2πδ/δ0)/δ02

In this motion curve, the velocity and the acceleration always begin from zero, and change smoothly. There is no sudden change in velocity and acceleration at either end of the stroke, even if there are dwell periods at the beginning or the end of the travel. Therefore there is neither rigid impulse nor soft impulse.

一、 The Fundamental Principle of Cam Profiles Design

-ω

12

3

ω

O

e

r0

Cam: rotationFollower: reciprocation

Cam mechanism: in the direction opposite

to ω (-ω )

Cam ： stationary

Follower : reciprocation + rotation

The relationship between the follower travelling distance along

its guide-way and the inverse angle is exactly the same as the des

ired motion curve. This is called the principle of inversion.

Principle of inversion

§§99 －－ 3 3 Graphical Cam Profile Synthesis

r0

-ω

ω

80°

二、 Disk Cam with reciprocating Follower

1. Plate cam with translating offset knife-edge follower

Given: prime circle radius r0, angular velocityω, offset e and motion curve of follower. Design the cam contour.

120°

60°100°

1 2 3 4

5 6

1’

6’

2’3’

4’

5’ 7

8 9

10

12 11

7’

11’

8’

9’

10’

eProcedu

re1) Select scaleμl ， draw the prime circle and the offset circle .2 ） Divide the offset circle into a number of equal segments in the direction opposite to ω, and assign station numbers to the

boundaries of these segments.

4 ） The corresponding position of the knife

edge can be located.

3 ） Draw line from these points, making them all tangent to the offset circle.

5)The pitch curve is obtained by drawing a smooth curve.

r0

-ω

ω

80°

120°

60°100°

1 2 3 4

5 6

1’

6’

2’3’

4’

5’ 7

8 9

10

12 11

7’

11’

8’

9’

10’

e

Given: prime circle radius r0, angular velocityω, offset e, roller

radius rr and motion curve of follower. Design the cam contour.

2. Plate Cam with Translating Roller Follower

1)The motion curve of the knife-edge is the same as that of the roller center, or that of the translating roller follower. The locus of the roller center, relative to the cam can be designed according to the methods mentioned in the last section.

Procedure

2 ） The locus is called the pitch curve of the cam.

3 ） Many roller circles are drawn with roller radius rr and center at the points on the pitch curve. Since the roller is tangent to the cam contour at all times, the cam contour(cam surface) is the envelope of the family of the roller circles.

1 2 3 4

5 6 7

8 9

10

12 11

r0

-ω

ω

80°

120°

60°100°

6’

2’

3’

4’

5’

11’

8’

9’

10’

1’

7’

3.Disk Cam with Reciprocating Flat-faced Follower

Given: prime circle radius r0, angular velocityω, and motion c

urve of follower. Design the cam contour.

1 ）  Similar to the roller follower, the inters

ection of the centerline of the follower stem a

nd the face of the follower is treated as the kn

ife-edge of a virtual knife-edge follower whic

h is fixed to the flat faced follower.

Procedure

2 ） A line is drawn at each of these points

perpendicular to the center line of the

follower to represent the position line of the

flat face of the follower during inversion.

3 ） So the cam contour may be drawn as a

smooth curve tangent to the family of the

position lines of the flat face.

5) Plate Cam with Oscillating Knife-edge Follower The following data should be provided: the radius of prime ci

rcle radius r0 , the distance lOA between the two pivots of the cam

and the follower,the length l (lAB )of the follower arm, the initial s

tatus of the follower (left or right),

the direction of ω , the motion

curve φ-δ.

Design the cam contour. ω

-ω

90°

180°

φ1

φ2

φ3φ4

φ6

φ5

φ7

φ8

A8

A9

A6

A7

A3

A1

A4

A2

A5

A

Br0B8

B6

B5

B7

B2

B1

B3B4

B9

The motion programs discussed in section 9-2 dealt only with the motion of the cam follower and not with the nature of the cam that produced that motion.

一、 Pressure Angle

Pressure Angle——the angle between the force that the

cam applies to the follower ( neglecting friction) and

the direction in which the follower is free to translate.

ω

n

n

t

t

G

d

b

l

B

Fφ1

φ2

φ2

1RF

2RF

∑Fx=0 ，∑ Fy=0 ，∑ MB=0

F=cos(α+φ1 )- (1+2b/l) sin(α+φ1 )tanφ2

G

α↑,F↑； whenα ↑, →denominator=0 ， F → ∞ ， → mechanism self-lock

§§99 －－ 4 4 Cam Size and Physical Characteristics

Pressure angle a changes during motion so it is necessary to control the value of the maximum pressure angle amax.

αmax ≤[α]

Rise stroke translating follower [α] ＝ 30°

oscillating follower [α] ＝ 35° ～ 45°

Return stroke [α]’ ＝ 70° ～ 80°

αc= arctan [1 / (1 + 2b / l) tanφ2 ] -φ1

The allowable pressure angle [a] in the return for a force -closed earn mechanism is quite large. The oscillating follower works more smoothly than the translating follower.

The allowable pressure angle[α] for cam mechanism ：

n

n

E

vpω

Or0

S

The smaller the value of r0 is, the larger the value of αmax will be.

90

ds / d

S 0

PｅC

二、 Radius of the prime circle

OP= v/ω=[ds/dδ]

In △BCP ,

s0= r02

- e2

For the translating roller follower, an increase in r0 will definitely reduce α and hence αmax However, larger cams require more sp

ace and this means more mass and greater inertia forces which may lead to unwanted vibrations when speeds are high.

tanα =(ds / dδ-e) / [( r02 - e2 ) 1/2+ s ]

20 [(d d )/tan[ ]- ]r s / e s e

Depend onα≤[α]， therefore

三、 Radius of roller

ρa

ρ

Pitch curve

Cam surfacearr

ρ

a rr

r

a r

r

r

ρ

0r

a r

r

r

rr

rr

rr

a

0r

a r

r

r

In a words ρmin> rr , and ρamin ≥(1 ～ 5)mm.

四、 Width Requirements for a Translating Flat-faced Cam Follower

l = 2lmax+ ( 5 ~ 7 ) mm