Review of Selected Topics in Gear and Cam Systems

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Review of Selected Topics in Gear and Cam Systems

• Fundamentals of standardized gears
• Gear types (how to select the type of gears to
  use based on functionality)
• Fixed axis and planetary gear trains analysis
  and synthesis
• Fundamentals of cam systems
• Cam system types
• Analytical synthesis of cams using Conjugate
  Geometry

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Fundamentals of Standardized Gears

• Angular velocity ratio
• p pitch (diametral pitch) number of
  teeth/inch of Diameter (pitch)
• Circles Pitch, Base, addendum, dedendum
• Angles Pressure!
• (standards are 14.5, 20, 22.5, 25 deg.)

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Fundamentals (cont.)

• For two gears to mesh
• 1) They must have the same p
• 2) They must have the same f
• 3) They must be standard gears
• Look at catalogs

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Gear Types

• Gears are classified in terms of the geometry of
  the shafts they connect
• Parallel axis shafts
• Spur gears (teeth parallel to axis of rotation)
• No axial thrust, lower performance
• Helical gears (teeth lie at an angle to axis of
  rotation)
• Higher capacity and performance, create axial
  thrust
• Herringbone gears (2 helicals of opposing hand
  side by side)
• Expensive

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Gear types (cont)

• Intersecting axes
• Bevel gears Pitch surface is a cone with fixed
  pitch angle
• Straight bevel like a spur gear
• Spiral bevel like a helical gear
• Non-intersecting, non-parallel shafts
• Hypoid gears Pitch surface is a hyperboloid of
  revolution
• High performance, sliding motion b/n teeth
• Example Auto differential, square baler
• Crossed helical gears 2 helicals with
  non-equivalent helix angle
• Example distributor
• Worm Gearing gear with very large helix angle
• Worms can have 1, 2, or 4 teeth resulting in very
  high reduction

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Gear types

• Rotary to Linear
• Rack and Pinion gears (Rack is a gear with
  infinite pitch diameter)

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Gears in design, Rules of thumb

• Maximum velocity ratio for 1 stage 101 (more
  like 51) (except for worm gears)
• Minimum number of gear teeth to avoid
  undercutting is 15-18
• Gear efficiencies
• Spur, helical, bevel 97-99
• Hypoid 85-98
• Crossed helical 50-95
• Worm gears 35-50
• Good references Mabie and Reinholtz, Norton
• Worm gears can be non-backdrivable

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Fixed axis and planetary gear trains analysis
and synthesis

• w1/w2N2/N1
• wla/wfa(wl-wa)/(wf-wa)

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Fundamentals of Cam Systems

• Know your cams (and followers)
• Disk cam, cylindrical cam
• oscillating or translating
• knife edge, flat-faced, roller
• Inline or offset

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Synthesis of Cams via Conjugate Geometry

• Two surfaces are conjugate if they obey the
  condition of contact the velocity of the
  contact point b/n 2 surfaces must be in the
  direction of the common tangent of the 2 surfaces
• This equation is used to solve for a desired cam
  profile

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Procedure

• Specify cam and follower type
• Specify the displacement function, sf(q)
• Invert the system, fix the cam
• Attach an x-y coordinate system to the cam, use
  this to express all vectors in the system
• Find the vector expression for the contact point,
  P (which gives the cam profile) using loop
  closure
• Find the velocity vector of the contact point
  w.r.t the cam (dP/dq)
• Find the common normal vector, N

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Procedure (cont.)

• Apply the condition of contact N . dP/dq 0
• Solve for the scalar unknown
• Solve for the cam profile

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Example 1 Disc cam with oscillating flat-faced
follower

• Figure
• Given C and ff(q), Unknowns l, P
• P Ceiq lei(qf)
• N1ei(qf90) iei(qf)
• dP/dqCieiq lei(qf) li(1f) ei(qf)
• N. dP/dqiei(qf). ( Cieiq lei(qf) li(1f)
  ei(qf) ) 0
• Ccos(f)lcos(90)l (1f)cos(0)0
• l-Ccos(f)/ (1f)
• Solve P

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Example 2 disc cam with translating, offset
roller follower

• Figure
• Given r, d, and Rf(q), Unknowns l, P
• P Reiq dei(q90) rei(qy)
• N1ei(qy) iei(qy)
• dP/dqReiq Rieiq di2eiq ri(1y) ei(qy)
• N. dP/dqiei(qy). (Reiq Rieiq di2eiq
  ri(1y) ei(qy)) 0
• (R-d)cos(y)Rcos(y -90)r (1y)cos(90)0
• tan(y)(d-R)/R
• Solve P

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Finding the displacement function

• Displacement function must at least be continuous
  in the first and second derivatives of motion
• Standard technique, choose from Kloomak and
  Muffley curves.