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MENG 372
Chapter 8
Cam Design

All figures taken from Design of Machinery, 3rd ed. Robert Norton
2003
1

1
Cams
• Function generator
• Can generate a true dwell

2

2
Cam Terminology
•
•
•
•
•

Type of follower motion (rotation, translation)
Type of joint closure (force, form)
Type of fol...
Type of Follower Motion
Oscillating follower

Translating follower
4

4
Type of Joint Closure
Force and form closed cams
• Force closed cams
require an external
force to keep the
cam in contact ...
Type of Joint Closure
• Form closed cams are
closed by joint geometry
• Slot milled out of the
cam

6

6
Types of Followers
Roller
Follower

Mushroom
Follower

Flat-Faced
Follower

• Roller Follower
• Mushroom Follower
• Flat-F...
Direction of Follower Motion
• Radial or Axial
Radial Cam

8

Axial Cam

8
Cam Terminology (review)

•
•
•
•

Type of follower motion (rotation, translation)
Type of joint closure (force, form)
Typ...
Type of Motion Constraints
• Critical Extreme Position (CEP) – start and end
positions are specified but not the path
betw...
Type of Motion Program
• From the CEP cam profile
• Dwell – period with no output motion with input
motion.
• Rise-Fall (R...
SVAJ Diagrams

• Unwrap the cam
• Plot position (s),
velocity (v),
acceleration (a) and
jerk (j) versus cam
angle
• Basis ...
RDFD Cam Design
• Motion is between two dwells

13

13
RDFD Cam, Naïve Cam Design
• Connect points using
straight lines
• Constant velocity
 Infinite acceleration
and jerk
 No...
Fundamental Law of Cam Design
Any cam designed for operation at other than
very low speeds must be designed with the
follo...
RDFD Cam Sophomore Design
Simple Harmonic Motion

• Sine function has
continuous derivatives
s=

 πθ
h
1 - cos 

2
 ...
RDFD Cam, Cycloidal

Start with acceleration & integrate:
 2πθ 
a = C sin 
β ÷


 2πθ
Cβ
v=−
cos 
2π
 β

Since
k1 ...
RDFD Cam, Cycloidal

h

2

 2πθ 
Cβ
 β 
s=
θ − C  ÷ sin 
÷+ k 2
2π
 2π 
 β 

• Since s=0 at θ=0, k2=0
• Since s=...
RDFD Cam, Cycloidal

(

)

s = h θ − h sin 2πθ
β
2π
β
Equation for a cycloid.
Cam has a cycloidal displacement
or sinusoid...
RDFD Cam, Trapezoidal
• Constant acceleration gives infinite jerk
• Trapezoidal acceleration gives finite jerk, but
the ac...
RDFD Cam, Modified Trapezoidal
• Combination of sinusoidal and constant acceleration
• Need to integrate to get the magnit...
RDFD Cam, Modified Trapezoidal
• After integrating, we get the following curves
• Has lowest magnitude of peak acceleratio...
RDFD Cam,
Modified Sine
• Combination of
a low and high
frequency sine
function
• Has lowest
peak velocity
(lowest kinetic...
RDFD Cam, SCCA Family
The cam functions discussed so far belong to the SCCA
family (Sine-Constant-Cosine-Acceleration)

24...
RDFD Cam, SCCA Family
• Comparison of accelerations in SCCA family
• All are combination of sine, constant, cosine family
...
Polynomial Functions
• We can also choose polynomials for cam functions
• General form:
2

3

4

n

s = C0 + C1 x + C2 x +...
3-4-5 Polynomial
• Boundary conditions
 @θ=0, s=0,v=0,a=0
 @θ=β, s=h,v=0,a=0

• Six boundary conditions, so
order 5 sinc...
3-4-5 Polynomial
θ
s = C0 + C1 
β



θ
 + C2 

β



2

3

4


θ 
θ 
θ 
 + C3   + C4   + C5  

...
3-4-5 and 4-5-6-7 Polynomial

• 3-4-5 polynomial
– Similar in shape to cycloidal
– Discontinuous jerk
 θ
s = h 10

 ...
Acceleration Comparisons

• Modified trapezoid is the best, followed by modified
sine and 3-4-5
• Low accelerations imply ...
Jerk Comparison
• Cycloidal is lowest, followed by 4-5-6-7 polynomial
and 3-4-5 polynomial
• Low jerk implies lower vibrat...
Velocity Comparison
• Modified sine is best, followed by 3-4-5 polynomial
• Low velocity means low kinetic energy

32

32
Position Comparison
• There is not much difference in the position curves
• Small position changes can lead to large
accel...
Table for Peak VAJ for Cam Functions
• Velocity is in m/rad, Acceleration is in m/rad 2, Jerk
is in m/rad3.

34

34
Single Dwell Cam Design, Using
Double Dwell Functions
• The double dwell cam functions have an
unnecessary return to zero ...
Single Dwell Cam Design, Double
Harmonic function

• Large negative acceleration

 θ  1 
 θ  
h 


s = 1 − c...
Single Dwell Cam Design, 3-4-5-6
Polynomial
• Boundary conditions
@θ=β s=v=a=0

@θ=0
@θ=β/2

 θ
s = h 64

 β



...
Unsymmetrical RFD Cams
• If the rise has different time than the fall, need more
boundary conditions.
• With 7BC’s

38

38
Unsymmetrical RFD Cams
• If you set the velocity to zero at the peak:

39

39
Unsymmetrical RFD Cams
• With 3 segments, segment 1 with 5BC’s, segment 2
with 6BC’s get a large peak acceleration

40

40
Unsymmetrical RFD Cams
• Best to start with segment with lowest acceleration
with 5BC’s then do the other segment with 6BC...
Critical Path Motion (CPM)
• Position or one of its derivatives is specified
• Ex: Constant velocity for half the rotation...
Critical Path Motion (CPM)
• Segment 1 has
4BC’s
• Segment 2 has
2BC’s (constant V)
• Segment 3 has
4BC’s
• Last segment h...
Resulting Curves

44

44
Constant Velocity, 2 Segments
• The divisions on the previous approach are not
given, only one segment of constant velocit...
Resulting SVAJ diagram
• 2 segment design has better properties
• 4 segment design had ∆s=6.112, v=-29.4, a=257

46

46
Sizing the Cam, Terminology
• Base circle (Rb) – smallest circle that can be drawn
tangent to the physical cam surface
• P...
Cam Pressure Angle
• Pressure Angle (φ)
– the angle between
the direction of
motion (velocity) of
the follower and the
dir...
Cam Eccentricity
• Eccentricty (ε) – the
perpendicular distance
between the follower’s
axis of motion and the
center of th...
Overturning Moment
For flat faced follower,
the pressure angle is
zero
There is a moment on
the follower since the
force...
Radius of Curvature
• Every point on the cam has an associated radius of
curvature
• If the radius of curvature is smaller...
Radius of Curvature – Flat Faced
Follower
• We can’t have a negative
radius of curvature

52

52
Cam Manufacturing Considerations
•
•
•
•

53

Medium to high carbon steels, or cast ductile iron
Milled or ground
Heat tre...
Actual vs.
Theoretical Cam
Performance
• Larger acceleration
due to
manufacturing
errors, and
vibrations from jerk

54

54
Practical Design Considerations
• Translating or oscillating follower?
• Force or Form-Closed?
– Follower Jump vs. Crossov...
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Chapter 8

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Chapter 8

  1. 1. MENG 372 Chapter 8 Cam Design All figures taken from Design of Machinery, 3rd ed. Robert Norton 2003 1 1
  2. 2. Cams • Function generator • Can generate a true dwell 2 2
  3. 3. Cam Terminology • • • • • Type of follower motion (rotation, translation) Type of joint closure (force, form) Type of follower (roller, mushroom, flat) Direction of follower motion (radial, axial) Type of motion constraints (critical extreme position(CEP) and critical path motion (CPM)) • Type of motion program (rise-fall (RF), rise-falldwell (RFD), rise-dwell-fall-dwell (RDFD) 3 3
  4. 4. Type of Follower Motion Oscillating follower Translating follower 4 4
  5. 5. Type of Joint Closure Force and form closed cams • Force closed cams require an external force to keep the cam in contact with the follower • A spring usually provides this force 5 5
  6. 6. Type of Joint Closure • Form closed cams are closed by joint geometry • Slot milled out of the cam 6 6
  7. 7. Types of Followers Roller Follower Mushroom Follower Flat-Faced Follower • Roller Follower • Mushroom Follower • Flat-Faced Follower 7 7
  8. 8. Direction of Follower Motion • Radial or Axial Radial Cam 8 Axial Cam 8
  9. 9. Cam Terminology (review) • • • • Type of follower motion (rotation, translation) Type of joint closure (force, form) Type of follower (roller, mushroom, flat) Direction of follower motion (radial, axial) 9 9
  10. 10. Type of Motion Constraints • Critical Extreme Position (CEP) – start and end positions are specified but not the path between • Critical Path Motion (CPM) – path or derivative is defined over all or part of the cam 10 10
  11. 11. Type of Motion Program • From the CEP cam profile • Dwell – period with no output motion with input motion. • Rise-Fall (RF) – no dwell (think about using a crank-rocker) • Rise-Fall-Dwell (RFD) – one dwell • Rise-Dwell-Fall-Dwell (RDFD) – two dwells 11 11
  12. 12. SVAJ Diagrams • Unwrap the cam • Plot position (s), velocity (v), acceleration (a) and jerk (j) versus cam angle • Basis for cam design 12 12
  13. 13. RDFD Cam Design • Motion is between two dwells 13 13
  14. 14. RDFD Cam, Naïve Cam Design • Connect points using straight lines • Constant velocity  Infinite acceleration and jerk  Not an acceptable cam program 14 14
  15. 15. 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°). Corollary: • The jerk must be finite across the entire interval (360°). 15 15
  16. 16. RDFD Cam Sophomore Design Simple Harmonic Motion • Sine function has continuous derivatives s=  πθ h 1 - cos   2  β h  ÷   πθ  ds hπ = sin  ÷ dθ 2 β  β   πθ  dv hπ 2 a= = cos  dθ 2 β 2 β ÷   v=  πθ  da −hπ 3 j= = sin  ÷ dθ 2β 3  β  ∞ ∞  Acceleration is discontinuous  Jerk is infinite (bad cam design) 16 16
  17. 17. RDFD Cam, Cycloidal Start with acceleration & integrate:  2πθ  a = C sin  β ÷    2πθ Cβ v=− cos  2π  β Since k1 = v=0 Cβ 2π Cβ v= 2π at  ÷+ k1  θ =0   2πθ 1 − cos   β  h then:  ÷  2  2πθ  Cβ  β  s= θ − C  ÷ sin  ÷+ k 2 2π  2π   β  17 17
  18. 18. RDFD Cam, Cycloidal h 2  2πθ  Cβ  β  s= θ − C  ÷ sin  ÷+ k 2 2π  2π   β  • Since s=0 at θ=0, k2=0 • Since s=h at θ=β,  Cβ h=  2π 2π h  ÷β ⇒ C = 2 β  s = h θ − h sin  2πθ  β 2π  β   v = h 1 − cos 2πθ    β β     a = 2πh 2 sin  2πθ   β   β h( 2π ) 2 j= 18 β  2πθ  β   3 cos 18
  19. 19. RDFD Cam, Cycloidal ( ) s = h θ − h sin 2πθ β 2π β Equation for a cycloid. Cam has a cycloidal displacement or sinusoidal acceleration h Valid cam design (follows fundamental law of cam design) Acceleration and velocity are higher than other functions General procedure for design is to start with a continuous curve for acceleration and integrate. 19 19
  20. 20. RDFD Cam, Trapezoidal • Constant acceleration gives infinite jerk • Trapezoidal acceleration gives finite jerk, but the acceleration is higher 20 20
  21. 21. RDFD Cam, Modified Trapezoidal • Combination of sinusoidal and constant acceleration • Need to integrate to get the magnitude 21 21
  22. 22. RDFD Cam, Modified Trapezoidal • After integrating, we get the following curves • Has lowest magnitude of peak acceleration of standard cam functions (lowest forces) 22 22
  23. 23. RDFD Cam, Modified Sine • Combination of a low and high frequency sine function • Has lowest peak velocity (lowest kinetic energy) 23 23
  24. 24. RDFD Cam, SCCA Family The cam functions discussed so far belong to the SCCA family (Sine-Constant-Cosine-Acceleration) 24 24
  25. 25. RDFD Cam, SCCA Family • Comparison of accelerations in SCCA family • All are combination of sine, constant, cosine family 25 25
  26. 26. Polynomial Functions • We can also choose polynomials for cam functions • General form: 2 3 4 n s = C0 + C1 x + C2 x + C3 x + C4 x +  + Cn x where x=θ/β or t • Choose the number of boundary conditions (BC’s) to satisfy the fundamental law of cam design 26 26
  27. 27. 3-4-5 Polynomial • Boundary conditions  @θ=0, s=0,v=0,a=0  @θ=β, s=h,v=0,a=0 • Six boundary conditions, so order 5 since C0 term θ  θ s = C0 + C1   + C2  β β    3 θ  θ + C3   + C4  β β    27 4     2  θ   + C5    β    5 27
  28. 28. 3-4-5 Polynomial θ s = C0 + C1  β   θ  + C2   β   2 3 4  θ  θ  θ   + C3   + C4   + C5    β β β        5 2 3 4 θ  θ  θ  θ   1 v = C1 + 2C2   + 3C3   + 4C4   + 5C5    β  β β  β β            2 3 θ  θ  θ   1  a = 2 2C2 + 6C3   + 12C4   + 20C5    β  β  β β           @θ=0, @θ=β, s=0=C0 v=0=C1/β a=0=2C2/β2 C0=0 C1=0 C2=0 s=h= C3+C4+C5, v=0=2C3+3C4+5C5 a=0= 6C3+12C4+20C5   θ 3 θ s = h 10  − 15 Solve the 3 equations to get     β   28 4 5  θ    + 6    β β      28
  29. 29. 3-4-5 and 4-5-6-7 Polynomial • 3-4-5 polynomial – Similar in shape to cycloidal – Discontinuous jerk  θ s = h 10   β  3  θ  − 15  β   4-5-6-7 Polynomial 5 4  θ   + 6    β       • 4-5-6-7 polynomial: set the jerk to be zero at 0 and β  θ s = h 35   β  4  θ  − 84  β   5 6  θ  θ  + 70  − 20  β β         7    • Has continuous jerk, but everything else is larger 29 29
  30. 30. Acceleration Comparisons • Modified trapezoid is the best, followed by modified sine and 3-4-5 • Low accelerations imply low forces 30 30
  31. 31. Jerk Comparison • Cycloidal is lowest, followed by 4-5-6-7 polynomial and 3-4-5 polynomial • Low jerk implies lower vibrations 31 31
  32. 32. Velocity Comparison • Modified sine is best, followed by 3-4-5 polynomial • Low velocity means low kinetic energy 32 32
  33. 33. Position Comparison • There is not much difference in the position curves • Small position changes can lead to large acceleration changes 33 33
  34. 34. Table for Peak VAJ for Cam Functions • Velocity is in m/rad, Acceleration is in m/rad 2, Jerk is in m/rad3. 34 34
  35. 35. Single Dwell Cam Design, Using Double Dwell Functions • The double dwell cam functions have an unnecessary return to zero in the acceleration, causing the acceleration to be higher elsewhere. 35 35
  36. 36. Single Dwell Cam Design, Double Harmonic function • Large negative acceleration  θ  1   θ   h    s = 1 − cos π  − 1 − cos 2π   for rise  β 4  β 2           θ  1   θ   h    s = 1 + cos π  − 1 − cos 2π   for fall  β 4  β 2          36 36
  37. 37. Single Dwell Cam Design, 3-4-5-6 Polynomial • Boundary conditions @θ=β s=v=a=0 @θ=0 @θ=β/2  θ s = h 64   β      3 4 • Has lower peak acceleration (547) than cycloidal (573) or double harmonic (900) 37 5  θ  θ  θ  − 192  + 192  − 64  β β β       s=v=a=0 s=h 6    37
  38. 38. Unsymmetrical RFD Cams • If the rise has different time than the fall, need more boundary conditions. • With 7BC’s 38 38
  39. 39. Unsymmetrical RFD Cams • If you set the velocity to zero at the peak: 39 39
  40. 40. Unsymmetrical RFD Cams • With 3 segments, segment 1 with 5BC’s, segment 2 with 6BC’s get a large peak acceleration 40 40
  41. 41. Unsymmetrical RFD Cams • Best to start with segment with lowest acceleration with 5BC’s then do the other segment with 6BC’s 41 41
  42. 42. Critical Path Motion (CPM) • Position or one of its derivatives is specified • Ex: Constant velocity for half the rotation • Break the motion into the following parts: 42 42
  43. 43. Critical Path Motion (CPM) • Segment 1 has 4BC’s • Segment 2 has 2BC’s (constant V) • Segment 3 has 4BC’s • Last segment has 6BC’s (almost always) 43 43
  44. 44. Resulting Curves 44 44
  45. 45. Constant Velocity, 2 Segments • The divisions on the previous approach are not given, only one segment of constant velocity 45 45
  46. 46. Resulting SVAJ diagram • 2 segment design has better properties • 4 segment design had ∆s=6.112, v=-29.4, a=257 46 46
  47. 47. Sizing the Cam, Terminology • Base circle (Rb) – smallest circle that can be drawn tangent to the physical cam surface • Prime circle (Rp) – smallest circle that can be drawn tangent to the locus of the centerline of the follower • Pitch curve – locus of the centerline of the follower 47 47
  48. 48. Cam Pressure Angle • Pressure Angle (φ) – the angle between the direction of motion (velocity) of the follower and the direction of the axis of transmission • Want φ<30 for translating and φ<35 for oscillating followers 48 φ 48
  49. 49. Cam Eccentricity • Eccentricty (ε) – the perpendicular distance between the follower’s axis of motion and the center of the cam • Aligned follower: ε=0 49 ε b 49
  50. 50. Overturning Moment For flat faced follower, the pressure angle is zero There is a moment on the follower since the force is not aligned with the direction of follower motion. This is called the overturning moment 50 50
  51. 51. Radius of Curvature • Every point on the cam has an associated radius of curvature • If the radius of curvature is smaller than the radius of the follower the follower doesn’t move properly • Rule of thumb: ρmin =(2→3) x Rf 51 51
  52. 52. Radius of Curvature – Flat Faced Follower • We can’t have a negative radius of curvature 52 52
  53. 53. Cam Manufacturing Considerations • • • • 53 Medium to high carbon steels, or cast ductile iron Milled or ground Heat treated for hardness (Rockwell HRC 50-55) CNC machines often use linear interpolation (larger accelerations) 53
  54. 54. Actual vs. Theoretical Cam Performance • Larger acceleration due to manufacturing errors, and vibrations from jerk 54 54
  55. 55. Practical Design Considerations • Translating or oscillating follower? • Force or Form-Closed? – Follower Jump vs. Crossover Shock • • • • • 55 Radial or Axial Cam? Roller or Flat-Faced Follower? To Dwell or Not to Dwell? To Grind or not to Grind? To Lubricate or Not to Lubricate? 55

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