Simple-Harmonic Cam Follower Displacement
Simple-Harmonic Cam Follower Displacement
Cams convert rotational motion into a prescribed reciprocating follower motion, and the simple-harmonic-motion (SHM) profile is one of the most common rise curves used because it produces smooth, continuous velocity with no abrupt changes at the start and end of the rise — unlike a naive constant-velocity or constant-acceleration profile, which can produce jarring accelerations at the transition points.Cam profile selection directly affects follower dynamics — vibration, wear, noise, and even follower jump at high speed — so knowing the follower's displacement (and, by extension, its velocity and acceleration) at any point through the rise is fundamental to both mechanism design and to checking that the follower will track the cam surface properly across the full operating speed range.
The simple-harmonic-motion follower displacement is y_disp = (h_lift/2)·(1 - cos(pi·theta_c/beta_c)), tracing a smooth half-cosine rise from zero to full lift. where h_lift is the total follower rise (lift), theta_c is the cam angle traveled from the start of the rise, beta_c is the total angle over which the rise occurs, and y_disp is the resulting follower displacement at that point in the rise.
Inputs
h_lift := 20 mm=20 mm
theta_c := 0.5=0.5
beta_c := 1.2=1.2
The half-cosine term traces the follower smoothly from zero displacement at the start of the rise, through this specific angle, toward the full lift at the end of the rise interval — the hallmark shape of simple harmonic follower motion.
y_disp := (h_lift/2)·(1 - cos(pi·theta_c/beta_c))=7.41 mm
Results
At this point, roughly 42% through the rise angle (theta_c/beta_c ≈ 0.42), the follower has risen about 7.4 mm out of its total 20 mm lift — noticeably less than the 42% of full lift a linear (constant-velocity) profile would give at the same fraction of the angle, which is the expected signature of the SHM profile's smooth, ease-in start. This displacement equation is normally differentiated (with respect to cam angle and machine speed) to get follower velocity and acceleration, which are what actually govern spring force requirements and whether the follower can stay in contact with the cam at the design operating speed. Simple harmonic motion has a known limitation at high speed: unlike cycloidal motion, its acceleration does not go to zero at the very start and end of the rise, producing a small but nonzero jerk (rate of change of acceleration) at those transition points that can excite vibration in high-speed cam-follower systems.
Assumptions & limitations
- Rigid follower and cam assumed — no elastic deformation, backlash, or follower mass dynamics included.
- theta_c and beta_c are treated as dimensionless angle values (radians) rather than a physical unit.
- Displacement only is calculated here; velocity and acceleration require differentiating this expression with respect to time.
- Simple harmonic motion is one of several standard cam profiles (others include cycloidal and modified trapezoidal) chosen based on speed and dynamic requirements.