If a circle rolls along a straight without slipping, a point on the circumference traces a curve that is called a cycloid This curve can be drawn by drawing a circle with center C on the line OO’. In the literature, one can also find “skewed constant acceleration”, where the cam rotation angles for the acceleration and deceleration periods are not equal. This motion curve has the lowest possible acceleration. Displacement, velocity and acceleration curves are as shown. However the third derivative, jerk, will be infinite at the two ends as in the case of simple harmonic motion. In this case the velocity and accelerations will be finite. The equations relating the follower displacement, velocity and acceleration to the cam rotation angle are:įor the range 0 < q < b/2 For the range b/2 < q < b Point of intersection of these lines with the corresponding vertical lines yield points on the desired curve as shown This curve can be graphically drawn by dividing each half displacement into equal number of divisions corresponding to the divisions on the horizontal axis and joining these points with O and O’ for the first and second halves respectively. The resulting motion curve will be two parabolas. Noting that the velocity must be zero at the two ends, we can assume a constant acceleration for the first half and a constant deceleration in the second half of the cycle. Parabolic or Constant Acceleration Motion Curve: In cases where the motion curve is composed of rise-return only, if the rise and return takes place for 180 0 crank rotation each, simple harmonic motion curve results with a circular cam eccentrically pivoted (eccentricity = H/2, half the rise), as shown.ģ. This curve will not be suitable for high or moderate speeds. Hence the third derivative, jerk, will be infinite at the start and end of the rise portion. Note that even though the velocity and acceleration is finite, the maximum acceleration is discontinuous at the start and end of the rise period. The maximum velocity and acceleration values given by equations: In figure below the displacement, velocity and acceleration curves are shown. The equations relating the follower displacement velocity and acceleration to the cam rotation angle are: The curve is the projection of a circle about the cam rotation axis as shown in the figure. Simple harmonic motion curve is widely used since it is simple to design. One basic rule in cam design is that this motion curve must be continuous and the first and second derivatives (corresponding to the velocity and acceleration of the follower) must be finite even at the transition points. Due to infinite accelerations, high inertia forces will be created at the start and at the end even at moderate speeds. Note that the acceleration is zero for the entire motion (a=0) but is infinite at the ends. The motion curve and velocity and acceleration curves are as shown. when these boundary conditions are applied to the linear equation: a 0=0 and a 1=H w/b. Later we shall see how a full motion curve can be constructed.Įquation describing a linear motion with respect to time is:Īssuming constant angular velocity for the input cam ( w), since t = q/w :ī= angular rotation of the cam corresponding to the total rise of the follower.Īlso assume when s = 0 q = 0 (rise is to start when t=0). We shall consider the rise portion of the motion curve only. In this section we shall discuss the basic philosophy in the selection of motion curves will be discussed and some well known motion curves will be explained.
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