Mechanisms Final
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The camfollower train is a degenerate form of:
 a pure fourbar linkage (oscillation)
 a fourbar slidercrank (translation)

Force Closure
 requires an external force to keep the cam in contact with the follower
 a spring usually supplies this force

Form Closure
 closed by joint geometry
 slot milled out of cam
 no external force required

In cam design constant velocity is UNACCEPTABLE because:
produces infinite acceleration and infinite jerk

In cam design Constant Acceleration (Parabolic Displacement) is UNACCEPTABLE because:
produces infinite jerk

In cam design Simple Harmonic Motion (SHM) is UNACCEPTABLE because:
produces infinite jerk

Acceptable Double Dwell Functions
 Cycloidal Displacement
 Sinusoidal Acceleration
 Modified Trapezoidal Acceleration
 Modified Sine Acceleration

Cycloidal Displacement
B.C.: v=0 at theta=beta (to match zero velocity of the dwell)

Choosing Cam Functions
 lower peak acceleration better
 lower peak velocity better
 smoother jerk means lower vibrations
 acceleration and velocity are higher than other functions

Trapezoidal Acceleration
 finite jerk
 higher accceperation

Modified Trapezoidal Acceleration
Advantage: lowest magnitude of peak acceleration of standard cam functions (lowest forces)

Modified Sinusoidal Acceleration
 lowest peak velocity (lowest kinetic energy)
 smoother jerk

Polynomial Functions:
 General form: s = C_{0} +C_{1}x +C_{2}x^{2} +...+C_{n}x^{n}
 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

345 Polynomial
 similar in shape to cycloidal
 discontinuous jerk because jerk unconstrained

4567 Polynomial
 similar in shape to cycloidal disp
 set jerk to zero at 0 and beta
 Continuous and smooth jerk but everything else is larger

Jerk Comparison (Lowest to Highest Jerk)
 Cycloidal
 4567 Poly
 345 Poly
 Low jerk implies lower vibrations

Acceleration Comparison (Lowest to Highest Acceleration)
 Modified Trapezoid
 Modified Sine
 345 Poly
 Low accelerations imply low forces

Velocity Comparison (Lowest to Highest Velocity)
 Modified Sine
 345 Poly
 Low velocity means low kinetic energy

Doubledwell camfollower, design to minimize accelerations
use Modified Trap!

Base Cicle R_{b}
Smallest circle that can be drawn tangent to the physical cam surface.

Prime Circle R_{p}
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

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 (R_{p}) will reduce phi

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

Procedure to Choose the Prime Circle
 Start with R_{p} = 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

Radius of Curvature rho
 minimum radius of curvature occurs near the point of minimum acceleration (maximum negative acceleration)
 rho can only be controlled with R_{p} once s, v, a are defined

Undercutting
 If  rho  < R_{f} : Undercut due to small negative rho => BAD!
 Undercut due to small positive radius of curvature creates a cusp => ALSO BAD!
 If rho = R_{f} : Undercutting => BAD!

Flat Faced Follower
Can't have a negative radius of curvature

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.

Meshing Gears
size of teeth must be the same for both gears

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.