Mechanisms Final

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Mechanisms Final
2011-05-09 23:29:56

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  1. The cam-follower train is a degenerate form of:
    • a pure fourbar linkage (oscillation)
    • a fourbar slider-crank (translation)
  2. Force Closure
    • requires an external force to keep the cam in contact with the follower
    • a spring usually supplies this force
  3. Form Closure
    • closed by joint geometry
    • slot milled out of cam
    • no external force required
  4. In cam design constant velocity is UNACCEPTABLE because:
    produces infinite acceleration and infinite jerk
  5. In cam design Constant Acceleration (Parabolic Displacement) is UNACCEPTABLE because:
    produces infinite jerk
  6. In cam design Simple Harmonic Motion (SHM) is UNACCEPTABLE because:
    produces infinite jerk
  7. Acceptable Double Dwell Functions
    • Cycloidal Displacement
    • Sinusoidal Acceleration
    • Modified Trapezoidal Acceleration
    • Modified Sine Acceleration
  8. Cycloidal Displacement
    B.C.: v=0 at theta=beta (to match zero velocity of the dwell)
  9. Choosing Cam Functions
    • lower peak acceleration better
    • lower peak velocity better
    • smoother jerk means lower vibrations
    • acceleration and velocity are higher than other functions
  10. Trapezoidal Acceleration
    • finite jerk
    • higher accceperation
  11. Modified Trapezoidal Acceleration
    Advantage: lowest magnitude of peak acceleration of standard cam functions (lowest forces)
  12. Modified Sinusoidal Acceleration
    • lowest peak velocity (lowest kinetic energy)
    • smoother jerk
  13. Polynomial Functions:
    • General form: s = C0 +C1x +C2x2 +...+Cnxn
    • where x = theta/beta or t
    • the value of the rise and fall functions at their boundaries with the dwells must match with no discontinuities
  14. 3-4-5 Polynomial
    • similar in shape to cycloidal
    • discontinuous jerk because jerk unconstrained
  15. 4-5-6-7 Polynomial
    • similar in shape to cycloidal disp
    • set jerk to zero at 0 and beta
    • Continuous and smooth jerk but everything else is larger
  16. Jerk Comparison (Lowest to Highest Jerk)
    • Cycloidal
    • 4-5-6-7 Poly
    • 3-4-5 Poly
    • Low jerk implies lower vibrations
  17. Acceleration Comparison (Lowest to Highest Acceleration)
    • Modified Trapezoid
    • Modified Sine
    • 3-4-5 Poly
    • Low accelerations imply low forces
  18. Velocity Comparison (Lowest to Highest Velocity)
    • Modified Sine
    • 3-4-5 Poly
    • Low velocity means low kinetic energy
  19. Double-dwell cam-follower, design to minimize accelerations
    use Modified Trap!
  20. Base Cicle Rb
    Smallest circle that can be drawn tangent to the physical cam surface.
  21. Prime Circle Rp
    smallest circle that can be drawn tangent to the locus of the centerline of the follower
  22. Pitch Curve
    locus of the centerline of the follower
  23. Pressure Angle Phi
    • phi < 30 for translating follower to avoid excessive side load on the sliding follower
    • phi < 35 for oscillating followers (on a pivot arm) to avoid undesirable levels of pivot friction
    • Increasing the prime circle radius (Rp) will reduce phi
  24. Eccentricity epsilon
    • perpendicular dist. btw follower's axis of motion and center of cam
    • can be used to correct asymmetry in max and min phi
  25. Procedure to Choose the Prime Circle
    • Start with Rp = 3*h , h= max lift
    • compute phi for all theta
    • iterate to acceptable condition
    • for translating roller follower maximum pressure angle should be < = 30 deg
    • eccentricity can be introduced to correct asymmetry in max and min phi
  26. Radius of Curvature rho
    • minimum radius of curvature occurs near the point of minimum acceleration (maximum negative acceleration)
    • rho can only be controlled with Rp once s, v, a are defined
  27. Undercutting
    • If | rho | < Rf : Undercut due to small negative rho => BAD!
    • Undercut due to small positive radius of curvature creates a cusp => ALSO BAD!
    • If |rho| = Rf : Undercutting => BAD!
  28. Flat Faced Follower
    Can't have a negative radius of curvature
  29. Involute Curve
    • The involute is a curve that can be generated by unwrapping a taut string from a cylinder (called the evolute).
    • The string is always tangent to the cylinder
    • The center of curvature of he involute is always at the point of tangency of the string with the cylinder
    • A tangent to the involute is always normal to the string, the length of which is the instantaneous radius of curvature of the involute curve.
  30. Meshing Gears
    size of teeth must be the same for both gears
  31. Involute Tooth Form/ Involute Gears
    • center distance errors do not affect the velocity ratio
    • as the center distance increases so will the pressure angle and vice versa
    • in involute gears the pressure angle remains constant between the point of tooth engagement.