CHAPTER 3 MODIFIED CAM CURVES - CAD.de · In Chap. 2, the characteristics ... Curve development and selection is one of the primary steps in the design of any cam- ... parison of

CHAPTER 3MODIFIED CAM CURVES

Harold A. Rothbart, D.Eng.

3.1 INTRODUCTION 56

3.2 FUNDAMENTALS 57

3.3 MODIFIED CONSTANT VELOCITYCURVE 57

3.4 TRAPEZOIDAL CURVE 61

3.5 MODIFIED TRAPEZOIDAL CURVE 61

3.6 SKEWED MODIFIED TRAPEZOIDALCURVE 70

3.7 MODIFIED SINE CURVE 73

3.8 MODIFIED CYCLOIDAL CURVE 76

3.9 DWELL-RISE-RETURN-DWELL MOTION 78

3.10 COUPLED CURVE SIMPLIFICATION 81

SYMBOLS

h = total rise of the follower, inh¢ = maximum rise of follower segment, inh = maximum follower displacement for full or half curve, int = time, sec.y = follower displacement

follower velocity, dimensionless

follower acceleration, dimensionless

follower jerk, dimensionless

follower acceleration

follower velocity

follower jerk

A = follower acceleration, in/sec2

Vmax = maximum velocity, ipsV0 = follower initial velocity, in/seca = follower acceleration, dimensionlessb = cam angle for rise h, radiansb = angle for maximum follower displacement, radiansb1, b2 = periods during positive and negative accelerations respectively, radiansq = cam angle rotation, radiansw = cam speed, rad/secv0 = follower initial velocity, in/radians

˙yd y

dty= = ¢¢¢ =

3

33w

ydy

dty= = ¢ =w

˙yd y

dty= = ¢¢ =

2

22w

¢¢¢ = =yd y

d

3

3q

¢¢ = =yd y

d

2

2q

¢ = =ydy

dq

55

Source: CAM DESIGN HANDBOOK

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3.1 INTRODUCTION

In Chap. 2, the characteristics of displacement, velocity, acceleration, and jerk of basicsymmetrical curves were presented. These curves were employed because of their sim-plicity of mathematical analysis and ease of construction. However, for many machineperformance requirements, as when the cam requires either special functional motions ormust operate at high speeds, the basic symmetrical curves are inadequate and modifica-tions in curve selection are necessary. These modifications can consist of blending,skewing, or combining sectors of the cubic curves, simple harmonic curves, cycloidalcurves, constant velocity curves, and constant acceleration curves.

Last, it should be mentioned that cam curve development (not shown) can be accom-plished by starting with the fourth derivative (dify/dyif) curve with numerical trial and errorcombined with past experience to find the ultimate desired cam shape. Computers areemployed to perform the increment integration in determining the displacement velocity,acceleration, and jerk curves.

Curve development and selection is one of the primary steps in the design of any cam-follower system. Later chapters include investigation of the pressure angle, cam curva-ture, cam torques, lubrication, materials, and necessary fabrication tolerances, among otherthings. A typical design evolutionary process will proceed as a series of trade-offs toproduce the final design.

This chapter has two parts:

• complete mathematical development for popular DRD curves

• a simplified procedure for combining sectors of various basic curves

The dwell-rise-dwell (DRD) and dwell-rise-return-dwell (DRRD) curves will be ana-lyzed. The rise-return-rise (RRR) curve is not presented since the eccentric mechanism ofChap. 15 satisfies the action in a simple, reliable, less expensive way. The RRR curve isalso best for high-speed requirements because it provides a motion curve having continu-ity in all of its derivatives.

Note that the DRD cam curve is a portion of the total action which could be a part ofthe dwell-rise-dwell-return-dwell (DRDRD) cam. Figure 3.1 shows the complete cycle ofblended DRD cycloidal curve producing a DRDRD cam. The period of rise is smaller thanthe period of fall, producing higher rise maximum acceleration than the maximum fallacceleration.

56 CAM DESIGN HANDBOOK

++

––

2pb

Dwell

Dwell

Dwell

Displacement

Dis

plac

emen

tac

cele

ratio

n

Cycloidolacceleration

Cam Angle q

Time, t

FIGURE 3.1. Dwell-rise-dwell-return-dwell cam with cycloidal acceleration curves.



MODIFIED CAM CURVES

3.2 FUNDAMENTALS

In this section, the fundamental conditions for shaping and combining curves are presented, and typical combinations of simple curves are shown. It is not practical to show all the possibilities, but it should be noted that any combination of sections of basiccurves may be utilized to fulfill design requirements. The control conditions are:

• The sum of the displacements of the combining sectors shall equal the total rise of thefollower.

• The sum of angles rotated of the combining sectors shall equal the total cam angle.

• The velocities at each sector junctions shall be equal.

• High-speed action requires that acceleration at all sector junctions be equal, that is,having no discontinuity in the acceleration and no infinite jerk value. Also, the acceler-ation curve should have the lowest maximum value with the value of jerk not too large.

• Sometimes special design requirements dictate the proportions of the acceleration curve.An example may be the controlling of the ratio of positive and negative accelerationperiods and shapes. An asymmetrical acceleration curve, with the maximum positiveacceleration larger than the negative maximum acceleration (ratio about 3 :1) would bea good choice for spring-loaded high-speed cams. Smaller springs, larger cam curva-tures, and longer surface life result.

On rare occasions limitations in the available manufacturing facilities may dictate thecam profile developed. For convenience, we have presented Fig. 3.2, which gives a com-parison of important cam curves. Note that the velocity, acceleration, and jerk curves pre-sented in this figure are all normalized (i.e., they have a unit total displacement h in a unitcam displacement b).

To illustrate terminology we see that the trapezoidal acceleration curve (discussed later)is a continuous function whereas its derivative (jerk curve) has many discontinuities. Notethat continuity of the jerk curve is of little value due to the usual tolerance limitations ofcam profile machine tool fabrication.

3.3 MODIFIED CONSTANT VELOCITY CURVE

In Chap. 2, we saw that the simplest curve is the constant velocity curve. It has a straight-line displacement at a constant slope. It also has the smallest cam for a given rise and provides a long stroke action. In this section we will blend any acceptable curve at thedwell ends for proper rise. The cycloidal curve or parabolic curves have been utilizeddepending on the cam speed, mass of the follower, and work performed by the machine.

As an example let us combine the parabolic motion curve blended with the straight-line displacement curve.

EXAMPLE A cam having a rigid heavy-mass follower rotates at 300rpm with a total DRDrise of 4 inches in 130 degrees of cam movement. As a preliminary study, use a paraboliccurve modified with the constant velocity curve. The action is as follows: (a) for the first40 degrees a positive parabolic curve acceleration; (b) for the next 30 degrees a straight-line displacement; and (c) for the last 60 degrees a negative parabolic curve accelera-tion. Find the ratio of accelerations, and plot all characteristic curves indicating pertinentvalues.

MODIFIED CAM CURVES 57



MODIFIED CAM CURVES


ParabolicSimple

harmonic Cycloidal TrapezoidalV

eloc

ityA

ccel

Jerk

Vel

ocity

Acc

elJe

rk

2.00

2.00

6.284.

00

4.93

1.57

40Modified

trapezoidalModified

Sine3–4–5*

polynomial4–5–6–7*polynomial

2.00

1.76

1.89

2.19

7.52

5.77

5.53

30

5242606961

4.89

23

8 8

8

8 8 16

2.00

5.3

5.3

4.4

4.4

b8

b8

*In Chap. 4

FIGURE 3.2. Comparison of important DRD cam curves.



MODIFIED CAM CURVES

Solution The time for a revolution = 60/300 = 0.2 sec/rev. Let us divide the total actioninto three parts, as shown in Fig. 3.3 with T and Q the tangent points of the curves, givingthe times

From the last chapter, we have the constant acceleration displacement

(2.24)

where V0 = initial velocity, in./sect = time, sec

A = acceleration, in./sec2

Therefore, the displacements and the velocity for the parabolic motion of parts 1 and3 are

(3.1)

For constant velocity of part 2

Combining yields

Substituting yields

or

Also, the displacement of part 2 is

We are given that the total displacement

y y y1 2 3 4+ + = .

y v tT2 2= .

A A

A A

1 3

1 3

0 0222 0 0333

11

2

. .

.

( ) = - ( )

= -

A t A t1 1 3 3= - .

v vQ T= .

y A t

y v t A t

v A t

v A t

Q

T

Q

1 1 12

3 3 3 32

1 1

3 3

1

21

2

=

= +

== -

y V t At= +02

1

2 in

t

t

t

1

2

3

0 240

3600 0222

0 230

3600 0167

0 260

3600 0333

= ÊË

ˆ¯ =

= ÊË

ˆ¯ =

= ÊË

ˆ¯ =

. . sec

. . sec

. . sec




MODIFIED CAM CURVES

Substituting gives

Again substituting,

Thus

The velocity at the points of intersection from Fig. 3.3 is

The displacements for each part are

We can now plot the curves of displacement, velocity, and acceleration as shown in Fig. 3.3.

y A t

y A t t

y

1 1 12 2

2 1 1 2

3

1

2

1

24050 0 0222 1

4050 0 0222 0 0167 1 5

1 5

= ( )( ) =

= ( )( )( ) ==

.

. . .

.

in

in

in

v v A tT Q= = = ( )( ) =1 1 4050 0 0222 90. ips.

A

A1

3

4050

2700

== -

in sec

in sec

2

2

1

2

1

34

1

20 0222 0 0222 0 0167 0 0222 0 0333

1

30 0333 4

1 12

1 1 2 1 1 3 1 32

12

1 1 12

A t A t t A t t A t

A A A A

+ + - =

( ) + ( )( ) + ( )( ) - ( ) =. . . . . . .

1

2

1

241 1

22 3 3 3

2A t v t V t A tT Q+ + + = .


Acceleration4050 in/sec2

Velocity90 ips

-2700 in/sec2

FIGURE 3.3. Example modified constant velocity curve.



MODIFIED CAM CURVES

3.4 TRAPEZOIDAL CURVE

The trapezoidal acceleration curve is a combination of the cubic and parabolic curves. Itmodifies the parabolic curve by changing its acceleration from a rectangular to a trape-zoidal shape. It is an early composite that was first recognized by Neklutin (1969). Heshowed that the trapezoidal acceleration curve is an improvement over the parabolic curveand that it offers good dynamic response under high-speed operation. It is a slight improve-ment over the cycloidal curve with its lower maximum acceleration.

In trapezoidal curve motion, the fraction of the total rise angle used for the initial cubic segment is known as the b value for the motion. In Fig. 3.4 we see a trapezoidalacceleration curve (DRD cam) where b = 1/8. This choice of b = 1/8 yields satisfactory cam-follower performance.

3.5 MODIFIED TRAPEZOIDAL CURVE

A combination cam curve (Chen, 1982) that has been used in lieu of the trapezoidal acceleration curve is the modified trapezoidal curve. It is composed of a parabolic motioncombined with the cycloidal curve. This combination reduces the maximum accelerationat the expense of somewhat higher jerk values.

The modified trapezoidal curve is popular in industry. However, it has one objection-able characteristic: the torque (discussed in later chapters) goes from positive maximumto negative maximum in 20 percent of the travel time. If dynamic forces represent a sig-nificant part of the load on the cam, this sudden release of energy may be detrimental tothe cam-follower system performance and limit the operating speeds. Much better torquecharacteristics can be obtained with the modified sine curve (see Sec. 3.7).

Figure 3.5a shows the basic cycloidal curve from which the modified trapezoidal curveis developed. The displacement and acceleration diagrams of the modified trapezoidal arealso shown. The variables pertaining to the cycloidal curve are denoted by the primedsymbols. At the start of the rise from A to B (Fig. 3.5b) the follower acceleration is aquarter sine wave; from B to C the acceleration is constant; and from C to D the acceler-ation decreases to zero with a quarter sine wave. After D, the follower has negative accel-eration in the same way that it was positively accelerated.


Dwell

Cam angle q

b

Dwell

8b

4b

8b

2b

Acc

eler

atio

n, y

" or

y

FIGURE 3.4. Trapezoidal acceleration curve—DRD cam.



MODIFIED CAM CURVES

Similar to the trapezoidal acceleration curve, we use b = 1/8 in the derivation.The equations of the curve from A to B are

(3.2)

When point B is reached, Substituting this value of q into Eq. (3.2) provides the

characteristic equations at point B.

The general expression for displacement under constant acceleration has a displacement

(2.24)

where n0 = follower initial velocity, dimensionlessa = follower acceleration, dimensionlessq = cam angle rotation, radians

y a= +n q q02

1

2

y h

yh

yh

1

1

1 2

1

4

1

22

8

= ¢ -ÊË

ˆ¯

¢ =¢

¢¢=¢

p

bpb

.

qb

=8

.

Displacement

Velocity

Acceleration =8

where h = maximum rise for cycloidal segment

y h

yh

yh

= ¢ -ÊËÁ

ˆ¯

¢ =¢

-ÊËÁ

ˆ¯

¢¢¢

¢

2 1

24

2 2 4

42

qb p

pqb

bp

qb

pb

pqb

sin ,

cos ,

sin ,


A'B' h'

h

Cam Angle q

Acceleration, y1''

Displacement, y1

Acceleration, y''

Displacement, y

BC

D

A

b2

b8

b8

b8

b4

Dis

plac

emen

t, y

Acc

eler

atio

n, y

''

(a) Basic cycloidal curve. (b) Modified trapezoidal curve.

FIGURE 3.5. Modified trapezoidal acceleration curve.



MODIFIED CAM CURVES

Therefore, the general equations of the curve from B to C are

To get displacement, velocity, and acceleration to match at the junction B, it is necessarythat

Therefore, the equations from B to C are

(3.3)

When point C is reached, Substituting in Eq. (3.3), we obtain

The cycloidal displacement is the sum of a constant velocity displacement and a quartersine wave displacement. The displacement equation of the curve from C to D is

Hence,

(3.4)

¢ = +-

¢¢ = --

yC

C

y C

23

3

2

2

44 2

164 2

bpb

pq b

b

pb

pq b

b

cos

sin

y y y C C C= + + +-

+-Ê

Ë

ÁÁÁ

ˆ

¯

˜˜1 2 1 2 3

38 4 2

q b

bp

q b

bsin .

y hh h

y y yh h

= ¢ -¢

+¢

\ = - =¢

-¢

3

4 2 4

2 42 1

pp

p

q b=3

8.

y hh h

yh h

yh

= ¢ -ÊË

ˆ¯ +

¢-Ê

Ëˆ¯ +

¢-Ê

Ëˆ¯

¢ =¢

+¢

-ÊË

ˆ¯

¢¢ =¢

1

4

1

2

2

8

4

8

2 8

8

8

2

2

2

2

p bq

b pb

qb

bpb

qb

pb

.

vh

ah

0

2

2

8

=¢

=¢

bpb

y y v a

y v a

y a

= + -ÊË

ˆ¯ + -Ê

Ëˆ¯

¢ = + -ÊË

ˆ¯

¢¢ =

1 0

2

0

8

1

2 8

8

qb

qb

qb

.




MODIFIED CAM CURVES

where C1, C2, and C3 are undetermined coefficients that can be obtained as follows:

An acceleration match at point requires

thus giving

A velocity match at point C requires

from which we obtain

The total displacement at point C is

or equivalently

from which C1 can be obtained

Substituting the values of C1, C2, and C3 in the displacement equation gives the curve fromC to D

or

(3.5)y h= ¢ - + +( ) --È

Î

ÍÍÍ

˘

˚

˙˙˙

pp

qb p

pq b

b22 1

1

24 4sin .

yh h h h h

hh

=¢

-¢Ê

Ëˆ¯ +

¢+

¢ÊË

ˆ¯ +

¢+ ¢ +( )

--

¢ -

4 2 2 4 22 1

38

24 4

pp

pp

q b

b pp

q b

bsin

Ch

1 2=

¢p

.

C hh

1

38

2 1

38

24 2 0= - ¢ +( )

-È

Î

ÍÍÍ

-¢ - ˘

˚

˙˙˙

=

=

pq b

b pp

q b

bq b

sin ,

y y y= +1 2

C h2 2 1= ¢ +( )p .

2 8

8 2

44 2

2 38

2

38

¢+

¢-Ê

Ëˆ¯

ÈÎÍ

˘˚

= +¢ -È

Î

ÍÍÍ

˘

˚

˙˙˙=

=

h h C h

bpb

qb

b ppb

pq b

bq b

q b

cos

Ch

3 2=

¢p

.

8 164 2

2 3

2

2

38

pb

pb

pq b

bq b

¢= -

-È

Î

ÍÍÍ

˘

˚

˙˙˙

=

hC sin

at

C when q b=ÊË

ˆ¯

3

8




MODIFIED CAM CURVES

At point the total displacement can be found from this equation to be

The displacement of the final segment is

or

From the relationship

we establish the relationship between h¢ and h

(3.6)

Therefore, the displacement equations of the first three segments are

(3.7)

Evaluating all constants, characteristic curve equations are:

(3.8)

y h

yh

yh

yh

= -ÊËÁ

ˆ¯

¢ = -ÊËÁ

ˆ¯

¢¢ =

¢¢¢ =

0 09724613 41

4

0 3889845 1 4

4 888124 4

61 425975 4

2

3

. sin

. cos

. sin

. cos

qb p

pqb

bp

qb

bp

qb

bp

qb

for qqb

£ £1

8,

yh

yh

yh

=+

-ÊËÁ

ˆ¯ £ £

=+

- + -ÊË

ˆ¯

ÈÎÍ

+ -ÊË

ˆ¯

˘˚

£ £

=+

- + +( )ÈÎÍ

+- ˘

˚

˙

2

2 1

24 0

8

2

1

4

1

2

2

8

4

8 8

3

8

2 22 1

1

24 2

2

2

pqb p

pqb

qb

p p bq

b pb

qb b

q b

pp

pqb p

pq b

b

sin

sin ˙˙

£ £3

8 2b q

b.

¢ =+

hh

2 p.

y y yh

1 2 3 2+ + = ,

y h3

1

4 4

1

2= ¢ + +Ê

Ëˆ¯

pp

.

y y y y3 1 2= - -

y h= ¢ +ÊË

ˆ¯1

2

p.

D i.e., when =2

qbÊ

Ëˆ¯




MODIFIED CAM CURVES

y h

yh

yh

y

= - ÊËÁ

ˆ¯ -

ÊËÁ

ˆ¯

¢ = -ÊËÁ

ˆ¯

¢¢ = -( )

¢¢¢ =

4 6660917 2 44406188 1 2292650

4 6660928 4 888124

4 888124

0

2

2

. . .

. .

.

qb

qb

bqb

b

for5

8

7

8£ £

qb

,

y h

yh

yh

yh

= + ÊËÁ

ˆ¯ -Ê

ËÁˆ¯

¢ = + ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

¢¢ = - ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

¢¢¢ = - ÊËÁ

ˆ¯

1 6110155 0 0309544 4 0 3055077

1 6110154 0 3889845 4

4 88124 4

61 425975 4

2

3

. . sin .

. . cos

. sin

.

qb

pqb

bp

qb

bp

qb

bp

qb

for1

2

5

8£ £

qb

,

y h

yh

yh

yh

= - ÊËÁ

ˆ¯ -Ê

ËÁˆ¯

¢ = - ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

¢¢ = ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

¢¢¢ = - ÊËÁ

ˆ¯

1 6110155 0 0309544 4 0 3055077

1 6110155 0 3889845 4

4 88124 4

61 425975 4

2

3

. . sin .

. . cos

. sin

. cos

qb

pqb

bp

qb

bp

qb

bp

qb

for3

8

1

2£ £

qb

,

y h

yh

yh

y

= - +ÊËÁ

ˆ¯

¢ = -ÊËÁ

ˆ¯

¢¢ = ( )

¢¢¢ =

2 444016188 0 22203094 0 00723406

4 888124 0 222031

4 88124

0

2

2

. . .

. .

.

qb

qb

bqb

b

for1

8

3

8£ £

qb

,




MODIFIED CAM CURVES

A computer solution is employed to establish the incremental displacement value and thecharacteristic curves of the action.

The modified trapezoidal curve has the following peak values

(3.9)

The nondimensional factors of the displacement, the velocity, and the acceleration of thiscurve are given in App. B.

EXAMPLE A cam rotates at 300rpm. A symmetrical modified trapezoidal accelerationcurve (parabolic motion combined with the cycloidal curve) is to be drawn with the ratiob = 1/8. The total rise is 4 inches in 160 degrees of cam rotation. Find pertinent values ofall the characteristics and plot the curves without the use of Eqs. (3.7) through (3.9).

Solution In Fig. 3.6 we see the basic cycloidal curve from which the combination curve is developed. This figure also shows the modified trapezoidal acceleration curve.The variables pertaining to the cycloidal sector will be denoted by the primed symbols(Fig. 3.6a). In Fig. 3.6b, let us divide one-half of the rise into its three component parts. Since b = 1/8 and the angle b/2 is 80 degrees, we see that the cam angle for parts 1and 3 is q1 = q3 = 20 degrees = b¢/4. This gives q2 = 40 degrees. The angular velocity ofthe cam is

The characteristics of the cycloidal curve from Eqs. (2.58), (2.59), and (2.60) are

Displacement in

Velocity ips

Acceleration in sec2

y h

y yh

y yh

= -ÊËÁ

ˆ¯

= ¢ = -ÊËÁ

ˆ¯

= = ÊËÁ

ˆ¯

qb p

pqb

wwb

pqb

ww p

bpqb

1

2

2

12

2 22 1

2

2

sin

˙ cos

˙ sin

w p= ¥ =300 60 2 31 4. rad sec

¢ =

¢¢ =

¢¢¢ =

yh

yh

yh

2

4 888

61 43

2

3

b

b

b

.

. .

y h

yh

yh

yh

= + - ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

¢ = - ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

¢¢ = - ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

¢¢¢ = ÊËÁ

ˆ¯

0 6110155 0 3889845 0 0309544 4

0 3889845 0 3889845 4

4 888124 4

61 425975 4

2

3

. . . sin

. . cos

. sin

. cos .

qb

pqb

bp

qb

bp

qb

bp

qb

for7

81£ £

qb

,




MODIFIED CAM CURVES

90.0

ips

72.5

ips

17.5

ips

2460

in./s

ec2

–246

0 in

./sec

2

0.07

3''

0.89

6'' 0.99

8''

FIG

UR

E 3

.6.

Exa

mpl

e m

odifi

ed t

rape

zoid

al a

ccel

erat

ion

curv

e.



MODIFIED CAM CURVES

At point A substituting

The displacement of part 2, the parabolic curve, in Sec. 2.6

Substituting, we obtain

Also, the velocity at point B

The displacement of part 3

we know that

Substituting,

Let us now find the displacements:

y

y

y

1

2

3

0 091 0 7767 0 071

1 285 0 7767 0 998

1 191 0 7767 0 896

= ( ) == ( ) == ( ) =

. . .

. . .

. . .

in

in

in

0 091 1 285 1 199 2

0 7767

. . .

.

¢ + ¢ + ¢ =¢ =

h h h

h in

y y y1 2 3 2+ + =

˙ ..

.y vh

hh

h3 33

293 3

20 180

31 4 21 199= +

¢= ¢ÊË

ˆ¯ +

¢= ¢q

w pp

p

˙

..

.

y v A

h h h

B A= +

= ¢ + ¢ÊËˆ¯ = ¢

22

22 5 317040 180

31 493 3

qw

p

yh h

h h

hh h

22

2

22

2

2 22

2

1

2

2

2

1

41 285

=¢¢

+¢¢( )

ÊË

ˆ¯

=¢

¢+

¢¢( )

=¢

+ ¢ ÊË

ˆ¯ = ¢

wb

qw

p wb

qw

qb

pqb

p .

y v AA22

22

21

2= + Ê

Ëˆ¯

qw

qw

y yh h

h

yh h

h

y Ah

hh

A

A

A

= =¢

-¢

= ¢

=¢¢

=¢( )

= ¢

= =¢¢( )

=¢( )

ÊË

ˆ¯

= ¢

1

2

2

2

2

2

4 20 091

31 480

180

22 5

2

2 31 4

80180

3170

pw

b p

p wb

p

p

.

˙ ..

˙

.




MODIFIED CAM CURVES

Substituting to find the velocity

Also, the maximum acceleration

The curves may now be plotted in Fig. 3.6b. If they were to be compared with a trape-zoidal acceleration curve, we would find that this curve has a slightly lower maximumacceleration and the advantage of lower required cutting accuracy in the initial and finalrise portions. Also, the vibrations induced at high speeds should be slightly smaller thanthose of the trapezoidal curve.

3.6 SKEWED MODIFIED TRAPEZOIDAL CURVE

Occasionally, the follower requires a particular velocity and acceleration at some criticalpoints in the machine motion. This can be accomplished by skewing the accelerationprofile as seen in Fig. 3.7. Neklutin (1969) has treated the modified trapezoidal curve withunequal periods of acceleration, positive and negative. Ragsdell and Gilkey (1969) haverelated the skewed acceleration to a correspondingly symmetrical one. Before skewing isconsidered, the follower rise h and angle b have been determined and will be consideredas constants.

Let b1 and b2 be the periods during positive and negative acceleration, respectively,

and let be the skew ratio (Fig. 3.7). Thus

The velocity match at the transition point that

Substituting and equating from Eq. (3.9)

(3.10)

2 2

1

1

1

1

2

2

1

1

h h

p

p

hp

ph

b b

b b

=

\ =+

=+

¢( ) = ¢( )y ymax max1 2

b b b1 2

1 2

+ =+ =h h h.

p =bb

1

2

AA = ( ) =3170 0 7767 2460. in sec2

v

v

v

v v

A

B

T

B B T

= ( ) == ( ) ==

+= + =

22 5 0 7767 17 5

93 3 0 7767 72 5

72 5 17 5 900

. . .

. . .

. .

ips

ips

maximum velocity

=

ips to D




MODIFIED CAM CURVES

and

(3.11)

The relationship between the skewed and corresponding symmetrical acceleration, sincethe maximum velocity is the same in both skewed and symmetric cases, is:

Thus using Eq. (3.10)

(3.12)

Similarly

(3.13)ap

a2

1

2=

+ÊË

ˆ¯ sym .

ap

pa1

1

2=

+ÊË

ˆ¯ sym .

2

2

1 1

11

b bbb

a a V

a a

= =

\ =

sym max

sym

b b2

2

1

1

=+

=+

p

p

hp

Ph.


AsymA2

h

h2

h1

Cam angle q

AsymA1Skewed acceleration

Symmetrical acceleration

h/2

Skewed displacement

Symmetrical displacement

Transitionpoint

Dis

plac

emen

tac

cele

ratio

n

b1 b2

b

b

b /2 b /2

Cam angle q

1

2

FIGURE 3.7. Skewed modified trapezoidal curve.



MODIFIED CAM CURVES


1.0

1.0

0.5

00.5

b1/b 2 = p

b1/b 2 = p

0.250

1.000

4.000

Dis

plac

emen

t rat

iohy

hy'

bqRatio p =

(a) Displacement.

bqRatio p =

1.0

1.0

0.5

00.5

0.250

1.000

4.000

Vel

ocity

rat

io

(b) Velocity.

FIGURE 3.8. Skewed modified trapezoidal curve comparison (normalized h = 1, b = 1).



MODIFIED CAM CURVES

Figure 3.8a shows the normalized displacement plot and Fig. 3.8b shows the normalizedvelocity plot where h = 1, b = 1.

3.7 MODIFIED SINE CURVE

The modified sine curve (Chen, 1982; Schmidt, 1960) is a combination of quarter sinewave curves. In terms of its torsional action, the change from positive to negative torqueoccurs in over 40 percent of the travel time. This attribute makes this curve attractive asa choice in moving large masses such as indexing intermittent turrets. Its lower torque andpower demand make the modified sine curve one of the best choices of curves.

Figure 3.9a shows the basic cycloidal curve from which the combination curve is devel-oped, and Fig. 3.9b shows the displacement and the acceleration diagram of the modifiedsine curve. The primed symbols used in the drawing refer to the basic cycloidal curve.One-half of the rise is divided into the following segments; the follower is accelerated

from A to B: with a quarter sine wave, and the acceleration decreased

to zero from B to again with a quarter sine wave. The equa-

tions of cycloidal motion from A to B, given that b/8 is the length of the initial quartersine wave, are:

y h

yh

yh

= ¢ -ÊËÁ

ˆ¯

¢ =¢

-ÊËÁ

ˆ¯

¢¢ =¢

2 1

24

2 2 4

84

2

qb p

pqb

bp

qb

pb

pqb

sin

cos

sin .

C from =8

to =2

qb

qbÊ

Ëˆ¯ ,

from = 0 to =8

q qbÊ

Ëˆ¯


(a) Cycloidal curve. (b) Modified sine curve.

A'B' h' B C

A

h

8b

8b

2b 4

3b8b

b

Displacement

AccelerationAcceleration

Displacement

Dis

plac

emen

tac

cele

ratio

n

Cam angle q

FIGURE 3.9. Modified sine curve.



MODIFIED CAM CURVES

At the end of the first segment and equations at point B are

(3.14)

The sine curve from B to C characteristics are:

(3.15)

The acceleration boundary conditions give:

The velocity boundary conditions give:

The displacement boundary conditions give:

C hh

Ch

1

8

1

2 8 9

2

4

3 30

9

2

+ ¢-

-¢

+ÊËÁ

ˆ¯

È

Î

ÍÍÍ

˘

˚

˙˙˙

=

=¢

=

q b

b pp q

bp

p

q b

sin

2 9

2

4

3

4

3 3

2

2

8

2

¢= -

¢ ÊËÁ

ˆ¯ +Ê

ËÁˆ¯

ÈÎÍ

˘˚

= ¢

=

h C h

C h

b b ppb

p qb

p

q bcos

8 16

9

4

3 3

9

2

2 3

2

2

8

3

pb

pb

p qb

p

p

q b

¢= - +Ê

ËÁˆ¯

ÈÎÍ

˘˚

= -¢

=

hC

Ch

sin

y y C C C

yC

C

y C

= + +-

+ +ÊËÁ

ˆ¯

¢ = + +ÊËÁ

ˆ¯

¢¢ = - +ÊËÁ

ˆ¯

1 1 2 3

23

3

2

2

8 4

3 3

4

3

4

3 3

16

9

4

3 3

q b

bpqb

p

bpb

pqb

p

pb

p qb

p

sin

cos

sin

y h

yh

yh

1

1

1 2

1

4

1

22

8

= ¢ -ÊË

ˆ¯

¢ =¢

¢¢=¢

p

bpb

.

qb

=8

,




MODIFIED CAM CURVES

Then

(3.16)

When the rise is

Hence,

Finally, from the relationship

we obtain

Therefore, the displacement equations of the modified sine curve are

(3.17)

Evaluating all constants the characteristics equations for the modified sine curve are:

for

y h

yh

yh

yh

= -ÊËÁ

ˆ¯

¢ = -ÊËÁ

ˆ¯

¢¢ =

¢¢¢ =

0 43990 0 35014 4

0 43990 1 4

5 52796 4

69 4664 4

2

3

. . sin

. cos

. sin

. cos .

qb

pqb

bp

qb

bp

qb

bp

qb

01

8£ £

qb

y h

y h

y h

=+

-+( )

ÊËÁ

ˆ¯

ÈÎÍ

˘˚

£ £

=+

++

ÈÎÍ

-+( )

+ÊËÁ

ˆ¯ £ £

=+

++

ÈÎÍ

-+( )

pp

qb p

pqb

qb

pp

pqb p

p qb

p bq b

pp

pqb p

pq

4

1

4 44 0

8

2

4 4

9

4 4

4

3 3 8

7

8

4

4 4

1

4 44

sin

sin

sinbb

b q bÊËÁ

ˆ¯˘˚

£ £7

8.

¢ =+( )

h hp

p2 4.

y y yh

= + =1 2 2

y y y h2 1

3

4

9

2= - = +Ê

Ëˆ¯ ¢

p.

y h= ¢ +ÊË

ˆ¯1

4

p

qb

=2

,

yh h h

hh

h

=¢

-¢Ê

Ëˆ¯ +

¢+ ¢

-Ê

Ë

ÁÁÁ

ˆ

¯

˜˜ -

¢+Ê

ËÁˆ¯

= ¢ + - +ÊËÁ

ˆ¯

ÈÎÍ

˘˚

4 2

9

22 8 9

2

4

3 3

42

9

2

4

3 3

p p

q b

b pp q

bp

pqb p

p qb

p

sin

sin .




MODIFIED CAM CURVES

for

for

A computer solution is employed to establish the incremental displacement values and thecharacteristic curves of the action. The maximum velocity of the modified sine curve is

the maximum acceleration is and the maximum jerk is

The nondimensionalized displacement, velocity, and acceleration factors

are given in Table A-4, App. B. Figure 3.10 indicates the comparison (Erdman and Sandor,1997) of the cycloidal, modified trapezoidal, and modified sine curves. The data shown isfor a 3-inch pitch diameter cam having a 2-inch rise in 6 degrees of cam rotation.

3.8 MODIFIED CYCLOIDAL CURVE

In this section we will reshape the cycloidal curve to improve its acceleration character-istics. This curve is the modified cycloidal curve that was developed by Wildt (1953).Figure 3.11 indicates the acceleration comparison between the true cycloidal curve andthe Wildt cycloidal curve. The basic cycloidal curve equation for the displacement

(2.58)

From this equation it is seen that the cycloidal curve is a combination of a sine curve and a constant velocity line. Figure 3.12a shows the pure cycloidal curve with point A

y h= -ÊËÁ

ˆ¯

qb p

pqb

1

2

2sin

¢¢¢ =yh

max . .69 473b

¢¢ =yh

max . ,5 5282b

¢ =yh

max . ,1 760b

y h

yh

yh

yh

= + -ÈÎÍ

˘˚

¢ = -ÊËÁ

ˆ¯

ÈÎÍ

˘˚

¢¢ =

¢¢¢ =

0 56010 0 43990 0 03515006 4

0 43989 1 4

5 52796 4

69 4664 4

2

3

. . . sin

. cos

. sin

. cos .

qb

pqb

bp

qb

bp

qb

bp

qb

7

81£ £

qb

y h

yh

yh

yh

= + - -ÊËÁ

ˆ¯

ÈÎÍ

˘˚

¢ = + +ÊËÁ

ˆ¯

ÈÎÍ

˘˚

¢¢ = +ÊËÁ

ˆ¯

¢¢¢ = +

0 28005 0 43990 0 3515064

3 3

0 43990 1 319704

3 3

5 527964

3 3

23 15554

3 4

2

3

. . . cos

. . sin

. sin

. cos

qb

p qb

p

bp q

bp

bp q

bp

bp q

bpÊÊ

ËÁˆ¯ .

1

8

7

8£ £

qb


(3.18)



MODIFIED CAM CURVES


Cam rotation (deg)

18

16

14

12

10

8

6

4

2

0 10 20 30 40 50 60

y V

eloc

ity (

in./s

ec.)

5.0

4.8

4.6

4.4

4.2

4.0

3.8

3.6

3.4

3.2

y D

ispl

acem

ent (

in)

Dis

tanc

e of

Cen

ter

of R

adia

lly T

rans

latin

g R

olle

r F

ollo

wer

from

Cen

ter

of C

am

0 10 20 30 40 50 60

(a) Displacement.

(b) Velocity.

Cam rotation (deg)

.

FIGURE 3.10. Comparison of popular curves (cam has 2-inch rise in 60-degree rotationand 3-inch pitch diameter). (Erdman and Sandor (1997) with permission by Prentice Hall,Upper Saddle River, N.J.)

the beginning of motion, point B the end of motion, and point P the mid-stroke transition point. APB is the constant velocity line. M is the midpoint between A andP. The sine amplitudes are added to the constant-velocity line in this true cycloidal case.

The modified cycloidal curve was developed to maximize the orientation of the superimposed sine wave amplitude on the straight line; see geometric construction in

Fig. 3.12b. In this figure, a point D equal to 0.57 the distance is the first chosen and

then is joined to M by a straight line. The base of the sine curve is then constructed

b2



MODIFIED CAM CURVES


Cam rotation (degrees)

250

200

150

100

50

0

–50

–100

–150

–200

–250

10 2030

40 50

60

(c) Acceleration.

y A

ccel

erat

ion

(in/s

ec)2

Cycloidal

Modified Sine

Modified Trapezoidal

..

FIGURE 3.10. Continued

perpendicular to DM. This procedure results in the modified cycloidal curve having amaximum acceleration about 7 percent lower than that of the basic cycloidal curve.

3.9 DWELL-RISE-RETURN-DWELL MOTION

Now, let us consider the dwell-rise-return-dwell curve, using a combination of curves toimprove the high-speed action. To analyze the action, we shall use the symmetricalcycloidal curve (Fig. 3.13), although any of the high-speed shapes, such as trapezoidal andmodified trapezoidal, introduce the same problem. A difficulty arises that did not prevailin the dwell-rise-dwell action, that is, an abrupt change or dip in the acceleration curveoccurs at the maximum rise point. This dip is undesirable, because it produces suddeninertia loads and vibrations.

In Fig. 3.13, the problem is eliminated by blending a parabolic curve to the cycloidalcurve portions. A modified total curve is produced. Also, a smoother, more desirable acceleration curve is developed with lower peak accelerations. An alternative solution,necessitating less mathematical work, is to employ polynomial equations as shown inChap. 4.



MODIFIED CAM CURVES


True cycloidal curve

Wildt modified cycloidal curve

Cam angle qA

ccel

erat

ion

FIGURE 3.11. Comparison of cycloidal accelerationcurves.

A

M

P

B

B

P

M

AD b

b

(a) Pure cycloidal curve.

(b) Modified cycloidal curve.

Cam angel q

Cam angel q

b/2

0.57b/2

Dis

plac

emen

t, y

Dis

plac

emen

t, y

FIGURE 3.12. Modified cycloidal curve construction.



MODIFIED CAM CURVES


+ +

– –

Displacement curves

Dis

plac

emen

tac

cele

ratio

n

Cycloidal acceleration

Cycloidalacceleration

Parabolic acceleration

DwellDwell

FIGURE 3.13. Dwell-rise-return-dwell curves, symmetrical rise-fall.

1 2

Rise

y

Fall

h

A0

Dwell

Fol

low

er y

,y

Cam angle 3 4I II III IV

b1 b2

q2 q3 q4

q0

b3 b4

q

y

FIGURE 3.14. Example of asymmetrical dwell-rise-fall-dwell camcurve.

EXAMPLE Derive the relationship for the dwell-rise-fall-dwell cam curve shown in Fig. 3.14 having equal maximum acceleration values. Portions I and III are harmonic;portions II and IV are horizontal straight lines.

Solution Let the q’s and b’s be the angles for each portion shown. Note that for veloc-ity and acceleration one should multiply values by w and w2, respectively, and the bound-ary conditions are y(0) = 0, (0) = 0, (b4) = 0, and y(b4) = total rise h. Use basictrigonometric relationships for

Portion I

˙ sin˙ cos

sin˙

y A

y A

y A A

y A

y A

= ( )= ( ) - ( )[ ]= - ( ) ( ) + ( )== ( ) -( )

pq bb p pq b

b p pq b b p qb pb p p

2

2 1 2

2 2 2

2

2 1 2

1

1 1

12

1 1

1 1

1 12

yy



MODIFIED CAM CURVES

Portion II

Portion III

Portion IV

Total Rise

Substituting

For a given total rise h for angle q0 and any two of the b angles given one can solvefor all angles and all values of the derivative curves.

3.10 COUPLED CURVE SIMPLIFICATION

This section presents a convenient method for combining segments of basic curves toproduce the required design motion. This procedure was developed by Kloomok andMuffley in Mabie and Ocvirk (1979), who selected three analytic functions of the simpleharmonic, the cycloidal, and the eighth-degree polynomial (the latter described in Chapter4). These curves, having excellent characteristics, can be blended with constant accelera-tion, constant velocity, and any other curve satisfying the boundary conditions stated inSec. 3.2. Figures 3.15, 3.16, and 3.17 show the three curves including both half curve seg-ments and full curve action in which

h = total follower displacement for half curve or full curve action, andb = cam angle for displacement h, in

Ah

=- + + + ( ) + + + +[

+( ) -( )]b b b b b p b p b b b b p b b b p

b p p42

2 4 1 4 32

2 3 1 3 22

1 2

12

2 2 2 2 2 2

2 1 2

h y A A A y= = -( ) + +( ) +4 42

2 1 4 32 2b b b p b

˙

˙ ˙

˙

˙ ˙

y A

y A y

y A y y

y A y A A A

= -= - +

= - ( )[ ]+ ¢¢¢ += - + = - + + == +

q

q qb b b b p

b b b p

4 3

42

3 3

4 2 3 4 2 1

4 2 1

2

2 0

2

˙ cos˙ sin ˙

cos ˙

˙ ˙

y A

y A y

y A y y

y y

y A A A A A A

= ( )= ( ) +

= - ( ) - ¢¢( )[ ]+ ¢¢ +=

= ( ) + + + + +

pq bb p pq b

b p pq b q

b p b b b b p b b b p b p

3 3

3 3 3 2

32

3 2 2

3 2

3 32

2 3 1 3 22

1 2 12

1

2 2 2 2 2(( ) -( )1 2 p

˙ ˙

˙ ˙

˙

˙

y A

y A y

y A y y

y A A

y A A A

== +

= ( ) + ¢ += += + + -( )

q

q qb b pb b b p b p p

2 1

22

1 1

2 2 1

2 22

1 2 12

2

2

2 2 2 1 2




MODIFIED CAM CURVES

The displacement, velocity, and acceleration characteristics of the curves are indicated.The simple harmonic curve (Fig. 3.15) has a low maximum acceleration for a given rise.Its acceleration discontinuity at the ends of the DRD cycle can be matched with thecycloidal segments (Fig. 3.16) to produce the popular cycloidal coupling. The cycloidalsegments are excellent choices to eliminate the acceleration curve discontinuities of anyDRD curve by its blending segment. The eighth polynomial, Fig. 3.17, is a good choice


y

y'

y"

H – 1

q

q

q

b

y = h 1 – cos2bpq

y' = sin2bpq

2bph y' = cos

2bpq

2bph

2bpqy" = cos

4b 2p2h

2bpqy" = – sin

4b 2p2h

y

y'

y"

H – 5

q

q

q

b

y = 1 – cos bpq

2h y = 1 + cos

bpq

2h

y' = sinb

pq2bph y' = – sin

bpq

2bph

y" = cosb

pq2b 2p2h y" = – cos

bpq

2b 2p2h

y

y'

y"

H – 6

q

q

q

b

y

y'y"

H – 4

q

q

q

b

y = h 1 – sin2bpq

y

y'

y"

H – 2

q

qq

b

y = h sin2bpq

y

y'

y"

H – 3

q

q

q

b

y = h cos2bpq

y' = – sin2bpq

2bph y' = – cos

2bpq

2bph

2bpqy" = – cos

4b 2p2h

2bpqy" = sin

4b 2p2h

h h

h h

h h

ÊËÁ

ÊËÁ

ÊËÁ

ˆ¯

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

FIGURE 3.15. Harmonic motion characteristics.Note: h = total follower displacement for half-curve or full-curve action, and

b = cam angle for displacement h, in.



MODIFIED CAM CURVES

for blending with curves in general, especially with DRRD cam action. Chapter 4 showsan elaborate treatment of polynomials.

EXAMPLE A spring-loaded textile machine has a cam that rotates at 720rpm in whichthe follower rises 13/4 inches in 120 degrees. To keep the spring size small, the maximumpositive acceleration is twice the maximum negative acceleration. Design (a) the riseportion of the DRRD system with harmonic and cycloidal coupling using the simplified


y

y'

y"

C – 1

q

q

q

b

y = h – sin p bq

bq

bq

bq

p1 y = h + sin p

bq

bq

p1

y' = 1 – cos pbh

bq

y' = 1 + cos pbh

y" = sin p b 2ph

bqy" = – sin p

b 2ph

y

y'

y"

C – 2

q

qq

b

y

y'

y"

C – 3

q

q

q

b

y = h 1 – – sin p bq

bq

p1y = h 1 – + sin p

bq

bq

p1

y' = 1 – cos pbh

y" = – sin pb 2ph

bqy" = sin p

b 2ph

y

y'

y"

C – 4

q

q

q

b

y' = – 1 + cos p bq

bh

y

y'

y"

C – 5

q

q

q

b

y = h – sin 2pbq

bq

2p1

y = h 1 – + sin 2p bq

bq

2p1

y' = 1 – cos 2p bq

bh y' = – 1 – cos 2p

bq

bh

bqy" = sin 2p

b 22ph

bqy" = – sin 2p

b 22ph

y

y'y"

C – 6

q

q

q

b

h h

h h

h h

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

bqÊ

ËÁˆ¯

ÊËÁ

ˆ¯

bqÊ

ËÁˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

ÊËÁ

ˆ¯

FIGURE 3.16. Cycloidal motion characteristics.Note: h = total follower displacement for half-curve or full curve action, and

b = cam angle for displacement h.



MODIFIED CAM CURVES


y

y'

y"

q

q

q

b

h

P –1y

y'

y"

q

q

q

b

h

P –2

y = h 6.09755 3 – 20.78040 5 + 26.73155 6

– 13.60965 7 + 2.56095 8

bq

bqÊ

ËÁˆ¯

ÊËÁ

ˆ¯

bqÊ

ËÁˆ¯ b

qÊËÁ

ˆ¯

bqÊ

ËÁˆ¯

ÈÎÍ

˘˚

y' = 18.29265 2 – 103.90200 4 + 160.38930 5

– 95.26755 6 + 20.48760 7

bh

bqÊ

ËÁˆ¯ b

qÊËÁ

ˆ¯ b

qÊËÁ

ˆ¯

bqÊ

ËÁˆ¯ b

qÊËÁ

ˆ¯

ÈÎÍ

˘˚

y" = 36.58530 – 415.60800 3 + 801.94650 4

– 571.60530 5 + 143.41320 6

b 2

hbqÊ

ËÁˆ¯

bqÊ

ËÁˆ¯ b

qÊËÁ

ˆ¯

bqÊ

ËÁˆ¯ b

qÊËÁ

ˆ¯

ÈÎÍ

˘˚

y = h 1.00000 – 2.63415 2 + 2.78055 5

+ 3.17060 6 – 6.87795 7 + 2.56095 8

bqÊ

ËÁˆ¯ b

qÊËÁ

ˆ¯

bqÊ

ËÁˆ¯ b

qÊËÁ

ˆ¯ b

qÊËÁ

ˆ¯

ÈÎÍ

˘˚

y' = –5.26830 + 13.90275 4 + 19.02360 5

– 43.14565 6 + 20.48760 7

bh

bqÊ

ËÁˆ¯

bqÊ

ËÁˆ¯ b

qÊËÁ

ˆ¯

bqÊ

ËÁˆ¯ b

qÊËÁ

ˆ¯

ÈÎÍ

˘˚

y" = –5.26830 + 55.61100 3 + 95.11800 4

– 288.87390 5 + 143.41320 6

b 2

hbqÊ

ËÁˆ¯ b

qÊËÁ

ˆ¯

bqÊ

ËÁˆ¯ b

qÊËÁ

ˆ¯

ÈÎÍ

˘˚

FIGURE 3.17. Eighth-degree polynomial motion characteristics.Note: h = total follower displacement.

b = cam angle for displacement h, in.

method of this section. Also find (b) at point the characteristic values of displacement,velocity, and acceleration.

Solution (a)

Angular velocity

wp

bp

=¥

=

=( )

=

720 2

6075 398

1 75

2

3602 094

.

.

.

rad sec

in.

=120

rad

h



MODIFIED CAM CURVES

From Fig. 3.16 for cycloidal motion (C1 segment), the characteristic curves are

From Fig. 3.15 for harmonic motion (H2 segment) the characteristic curves are

We need velocity continuity at the end of the C1 curve and the beginning of the H2 curve(Fig. 3.18)

h h1

1

1

1

2

2 2

12

0

2bpbb

pb

pb

-ÊËÁ

ˆ¯ =cos cos

y h

yh

yh

=

¢ =

¢¢ = -( )

22

2

2 2

22

22

2

2

2 2

4 2

sin

cos

sin

pqb

pb

pqb

pb

pqb

y h

yh

yh

= -ÊËÁ

ˆ¯

¢ = -ÊËÁ

ˆ¯

¢¢ =( )

11

1

1 1

1

12

1

1

1

qb p

pqb

bpqb

pb

pqb

sin

cos

sin


y, y

', y"

0

Dwell

b1 b2

Acceleration, y"

Velocity, y'

Displacement, y

h2

h1

(C1)

q

(H2)

FIGURE 3.18. Cycloidal harmonic compling example.



MODIFIED CAM CURVES

which yields

For the C1 curve the acceleration is

And for the H2 curve, the acceleration is

It is given that y≤max for C1 curve = 2 ¥ y≤max for H2 curves.Therefore

Now we have four equations to solve for h1, h2, b1, b2.

This solves to yield

By following the above principles, the designer may complete the DRRD curve (utilizingH3 and C4) and modify it to suit the design conditions of the machine.

Solution (b)

The 15 degrees falls in the cycloidal curve C1 region

The displacement

y = -( )È

ÎÍ˘˚

=0 4930 262

0698

1 0 262

0 6980 400.

.sin

..

.p

p in

15 0 262 deg =15

180 rad

p= .

h

h1

2

0 4934

1 2566

40180

0 6981

80180

1 3963

==

= ¥ =

= ¥ =

.

.

.

.

in

in

deg rad

deg rad

1

2

bp

bp

h h

h

h

h h

1 2

2

1

2

1

2

2

12

22

22

1 75

2 094

2

+ =+ =

=

= -

.

.

in

rad

41b b

pbb

pb

pb

pb

pb

h h1

12

22

222

= -

¢¢ =yh

max

pb

22

224

¢¢ =yh

max

pb

1

12

4 1

2

1

2

h

hpbb

= .




MODIFIED CAM CURVES

The velocity

The acceleration

REFERENCES

Chen, F.Y., Mechanics and Design of Cam Mechanisms, Pergamon Press, New York, 1982.

Erdman, A.G., and Sandor, G.N., Mechanism Design, Vol. 1, third edition, Prentice Hall, Upper SaddleRiver, N.J., 1997.

Mabie, H.H., and Ocvirk, F.W., Mechanisms and Dynamics of Machinery, John Wiley & Sons, NewYork, 1979.

Neklutin, C.N., Mechanisms and Cams for Automatic Machines, American Elsevier, New York, 1969.

Ragsdell, K.M., and Gilkey, H.E., “Optimal Cam Design Using the Skewed Modified TrapezoidalProfile,” Proceedings of the Applied Mechanism Conference, Paper 28, Oklahoma State Univer-sity, Stillwater, 1969.

Schmidt, E., “Continuous Cam Curves,” Machine Design 32 (1): 127–132, 1960.

Wildt, P., “Zwanglaufige Triebkurvenherstellung,” Dusseldorf, VDJ-Trungungsheft, 1: 11–20, 1953.

¢¢ =( )

ÊË

ˆ¯ =

¢¢ ( ) ( ) =

y

y

0 493

0 698

0 262

0 6982 938

6700

22

2 2

..

sin..

.

˙

p p

w

in rad

= y = 75.398 2.938 in sec2

¢ = -ÈÎÍ

˘˚

=

¢ ( ) =

y

y

0 493

0 6981

0 262

0 6980 437

32 93

.

.cos

..

.

˙ .

p

w

in

= y = 75.398 0.437 in sec




MODIFIED CAM CURVES



MODIFIED CAM CURVES

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