PARABOLIC, CONSTANT ACCELERATION, OR GRAVITY CURVE (DWELL_RISE-DWELL CAM)

This curve of the polynomial family has the property of constant positive and negative accelerations .No other curve will produce a given motion from rest to rest in a given time with so small a maximum acceleration. This is probably the reason that the curve is erroneously known as the best curve .It is in many ways the worst of the all curves

With perfectly rigid members having no backlash or clearance in the system, the constant acceleration curve would give excellent performance .However, all members are somewhat elastic and clearance or backlash always exists, especially in the positive drive-roller groove-type follower. The curve’s abrupt change of acceleration at the dwell ends and the transition point produces noise, vibrations, wear, and requires a large spring size. Thus the parabolic curve should be used only at moderate or lower speeds. One of the reason for the popularity of this curve is the ease of the determining the inertia forces, which are proportional to constant accelerations.

Characteristics:

Cam angle from θ from zero to β/2:

The equation for displacement is

y=2h(θ/β)²

For the velocity and the acceleration, we differentiate, yielding,

v=dy/dt=4hθω/β²

a=dv/dt=4h(ω/β)²

Cam angle θ from β/2 to β:

The equation for displacement is

y=h[1-2(1- θ/β)²]

Differentiating we find the velocity

v=dy/dt=4hω/β(1- θ/β).