SIMPLE HARMONIC MOTION OR CRANK CURVE (SHM) (DWELL-RISE-DWELL CAM)

This curve of the trigonometric family is one of the most popular primarily because of its simplicity in layout and understanding. It provides acceptable performance at moderate speeds. The basis for the harmonic curve is the projection (on a diameter) of the constant angular velocity movement of a point on the circumference of a circle. This circle is called the harmonic circle. The resulting motion of the follower on such a cam is simple harmonic movement similar to that of a swinging pendulum. The simple harmonic curve is a definite improvement over the previous curves. Shock is reduced so that it is no longer serious at moderate speeds.

Characteristics:

The displacement diagram is a cosine curve plotted from points projected from the harmonic circle of radius h/2, giving

y= (h/2) (1-cosφ)

Since the cam rotates β radians while the harmonic circle vector turns through π radians.

Φ/π= θ/β

Solving yields

Φ= πθ/β radians

Where

φ=angle of rotation of harmonic-circle vector, radians.

Substituting yields the displacement

y=h/2(1-cos πθ/β)

Differentiating gives the velocity

V=dy/dt=hπω/2β sin πθ/β.

We observe that the velocity curve is a sine curve with points plotted from a rotating vector hπω/2β in length.

Differentiating again for the acceleration

A=dv/dt=h/2(πω/β)²cos πθ/β.

We see that the acceleration is a cosine function with a rotating vector (h/2) (πω/β) ² in length. The displacement and velocity curves are smooth and continuous .However, at the ends where the dwell meets the simple harmonic curve, there is a sudden acceleration a discontinuity in the acceleration curve .This is undesirable for high-speed cams since noise, vibration and wears result.