Gyroscope Equations

Gyroscopes are well-known for their ability to maintain stability and resist
 changes in orientation. Their behavior is governed by precession, a
 principle that describes how a spinning object responds to external forces.
 If you drop a spinning gyroscope alongside a regular object, the gyroscope
 will not simply fall straight down.  It will  follow a slower spiraling
 path and land after the other object.
To test this idea, imagine a heavy wheel mounted on an axle, spinning
 rapidly in a vertical plane. If you rotate the axle in a horizontal plane
 while the wheel is still spinning, the wheel will either float upward or
 sink downward, depending on the direction of rotation.  This is a 90 degree
 movement up or down.
We can describe this with math.
d is the diameter of the wheel.
L is the length of the axle
We calculate the total distance traveled by a point on the wheel as it
 rotates once, while the wheel spins around the axle once.
The axle describes a circular path of radius L, and the wheel describes a
 circular path of radius d/2.The distance traveled is the sum of these two
 circular paths.
D1=π * d * sqrt(2)+2π * L
This equation combines the motion of both the wheel and the axle. The 2π*L
 term represents the circumference of the circular path made by the  axle
If the wheel also moves 90 degrees vertically during the rotation, then we
 also add the vertical movement, which is simply the length of the axle, L,
 because the wheel moves up by half its diameter in the vertical direction.  (or down)
D2=π * d * sqrt(2)+2π*L+L
Here, 2π*L represents the circular motion of the axle, and L represents the
 vertical distance the wheel moves during the rotation.
You can watch the experiment here:
https://youtu.be/GeyDf4ooPdo?si=qrxh4EmBG1IhxzkD
The question is where the additional energy comes from to move L 90 degrees.