Is it possible to find a "replacement pendulum" for a system of two equal but perpendicular pendulums?

Question

I ask this question, because at the end of this long day I'm just too dazed to derive the proofs myself (even though I know that I should feel ashamed for this).

So, the question:

Given two simple gravity pendulums, attached to the same hinge point. Both masses and lengths of the strings (or rods) are equal. The only difference between the two is that the motion of one pendulum takes place in a plane perpendicular to the motion of the other.

So, supposing that initial conditions are chosen such that these pendulums never collide, can the position of the center of mass of both pendulums be described by the motion of a single pendulum with twice the mass, the motion of which is not restricted to any single plane?

Intuitively, I'd say: perhaps only for small angles. For example, I can easily imagine one pendulum going one-way (spinning a full 2$\pi$ per period), while the other is at near-standstill. Obviously, the motion of the COM can not be described by a single pendulum without significant, non-physical changes to that model (the periods of both are unequal in that case, generally meaning the center of mass moves along non-periodic Lissajous patterns in $x$,$y$,$z$.).

So, what I'm really asking is: under what conditions is this possible, if at all? What are the constraints and/or adjustments you'd need to make to this "replacement pendulum" model for it to work?

And how does this all translate to physical pendulums?

Answer 1

A rather elegant approximation of what you describe may be set up by tying a string between two horizontally-separated mounting points so that the center hangs down a slight distance, and then tying a length of string from the center of that string to a weight. If one sets the weight in motion at a diagonal to the first string, one will observe that the swinging weight traces out Lissajous figures. The motion is similar to what the center of gravity of the two pendulums would do in your proposed configuration, except the two pendulums created by the string configuration interfere with each other slightly while in your example they're completely independent. Still, the level of interference is not huge, so the behavior of the system is dominated by the behavior of the individual pendulums.

Answer 2

What you are proposing is just the inverse of a 3D pendulum (non constrained to a plane). The movements in $x$ and $y$ are completely independent, and they happen to be described by the gravity pendulum equation. The solutions are a subset of the Lissajoux figures.