This post is similar to the post on light clocks *here*. Simple harmonic motion (SHM) is like the spring below:

The equation for describing the period is

where *T* is the period, *m* is the mass, and *k* is the spring constant.

The displacement for each cycle is zero since it returns to its starting point. In a stationary frame, the displacement, *x*, is a function of the amplitude, *A*, the angular velocity *ω*, and the time:

The total distance traversed in each cycle is 4*A* and the total time for each cycle is *T*, so the average speed for a cycle is 4*A*/*T* in a stationary frame.

To determine the average speed relative to an inertial frame moving with velocity *v* parallel to the SHM, break the cycle into the part moving in the positive direction (I) and the negative direction (II).

(I) the average speed is (4*A*/*T*) + *v*.

(II) the average speed is −((−4*A*/*T*) + *v*) = (4*A*/*T)* − *v*.

Then the average speed for each cycle is again 4*A*/*T*.

Thus the average speed for each cycle of SHM is constant for all inertial frames of reference.