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Mechanisms Final
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The cam-follower train is a degenerate form of:
a pure fourbar linkage (oscillation)
a fourbar slider-crank (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 C
onstant 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
3-4-5 Polynomial
similar in shape to cycloidal
discontinuous jerk because jerk unconstrained
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
Jerk Comparison (Lowest to Highest Jerk)
Cycloidal
4-5-6-7 Poly
3-4-5 Poly
Low jerk implies lower vibrations
Acceleration Comparison (Lowest to Highest Acceleration)
Modified Trapezoid
Modified Sine
3-4-5 Poly
Low accelerations imply low forces
Velocity Comparison (Lowest to Highest Velocity)
Modified Sine
3-4-5 Poly
Low velocity means low kinetic energy
Double-dwell cam-follower, 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.
Author
sep293
ID
84889
Card Set
Mechanisms Final
Description
Conceptual
Updated
2011-05-10T03:29:56Z
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