`    You are correct in pointing out that in swing the pendulum bob does not pass over the point the pendulum bob hangs above when it is not swinging. (Actually, the image on the right, that I manufactured and uploaded about a year ago, shows precisely that.) So the sampling point would have to be the point during the swing where the velocity of the pendulum bob relative to the Earth is greatest. (Alternative formulation: the point during the swing where the angle between the pendulum wire and the plumb line is smallest)  As you point out there is a range where separation between ``precession frequency`` and ``swing frequency`` is meaningless. So it is better to keep the explorations in the range where the swing frequency is much higher than the rotation frequency.  The rate of change of direction of the pendulum bob is not constant, it is proportional to the velocity of the pendulum bob with respect to the rotating system, and I want a solution to the equation of motion that incorporates that. In the image the successive swings aren't sections of a circle: close to the extremal points the curvature is close to zero, the lines curve strongest where the velocity is largest.  I know differential equations for damped harmonic oscillators, and how solving that kind of equations is facilitated by the relation eit = cos(t) + isin(t) (I'm reproducing from memory here, so I may have a detail wrong.) In the differential equation for a damped harmonic oscillation the dampening force is parallel to the direction of the velocity, so I take it that a force perpendicular to the direction of the velocity is represented as an imaginary factor.  |  `