Peter A. Taylor

May 21, 2003

Here are some estimates for some of the lowest natural frequencies of vibration of the Ringworld and their vibration mode shapes in a coordinate system that rotates with the Ringworld. Note that these are not shock waves or other high frequency waves. These are the lowest frequencies. If you think of the Ringworld as being like a curved violin string with no endpoints, I am trying to estimate the lowest pitch it can make, not the sound of it being smashed by a punk rocker.

If you want to skip a longwinded explanation, scroll down to the link labelled "mode1." Otherwise....

In order to get a feel for what I am trying to calculate, and the effect
of the rotating reference frame, forget the violin string analogy for a moment
and imagine that you're riding a merry-go-round while trying to play
tetherball. If you pull the tetherball outward from its equilibrium position
(away from the merry-go-round's center) and push it forward (in the direction
the merry-go-round is moving), the tetherball will appear to move in a circle.
This is shown in the figure,

tetherball1
.

The large dotted circle shows the tetherball's equilibrium point moving relative to the merry-go-round's pivot. The long red line connects the pivot to the equilibrium point. The short red line connects the equilibrium point to the ball's current position, the arrow shows the ball's current velocity, and the small circle shows the ball's path around its equilibrium point. As is true of all of these figures, the displacements have been scaled to make them look good, so don't try to read anything into the scale; it is arbitrary.

In this "pure" mode of oscillation, the ball does not pass through its
equilibrium position, but circles around it (it is not "harmonic"). This
makes it complicated to draw the ball's path. The way the math works out, I
get pairs of solutions that represent the ball's position at two different
times in its cycle, 90 degrees out of phase. In some ways it is easier to
think of these two solutions as different phases of the same mode of
vibration, but in other ways it is easier to think of them as a pair of
different modes that happen to share the same frequency. I show this in my
next figure,

tetherball2
.

We could just as easily push the ball backwards, opposite to the
merry-go-round's rotation. But there is a difference. In the forward case,
the ball's rotation and the merry-go-round's rotation add. In the backward
case, they subtract. So the periods of the motions in the two directions will
be different. For this "tetherball" example (actually assuming linear springs
rather than a dangling rope), the frequency of motion in the posigrade
direction already shown was 8.198 radians/second and in the retrograde
direction was 12.198 radians/second. The second pair of vectors is shown in
the next figures,

tetherball3 and

tetherball4
.

In real life, we usually don't see pure modes like these circles, but
combinations of several modes at once. An event like the Fist of God impact
will put energy into many modes simultaneously. You'll notice that to get
these pure "tetherball" modes, I had to displace the ball and then push it
forward or back. If I had just pushed it without first displacing it, I would
have excited two modes at once, and would have gotten a more complicated
motion that did swing back through the equilibrium point, but didn't quite
repeat itself on the next swing. The two modes vibrate at different
frequencies, and so the tendencies to swing clockwise and counterclockwise
don't quite cancel. We then see, in an exaggerated form, the familiar motion
of a Foucault pendulum mounted well away from the Earth's equator:

pendulum
.

Now let's talk about the Ringworld. I made a number of important assumptions:

- Damping has a negligible effect on the lowest frequencies and mode shapes.
- Deflections are small compared to the Ringworld radius (linear response).
- The radius is 1 astronomical unit (AU). ( Andy Love's presentation says 1.02 is the official number.)
- The central star's mass is exactly that of our sun.
- The artificial gravity level is exactly 1 standard gravity.
- The total mass of the Ringworld is the same as that of the planet Jupiter. (This ends up not mattering because the restoring forces scale with mass.)
- The motion is two-dimensional, entirely within the Ringworld's plane of rotation.
- Because the Ring is so thin relative to its diameter, the Ring's bending
stiffness is negligible compared to the violin string effect. It holds its
shape
*only*because it is under load, rotating to provide artificial gravity. - The scrith is rigid along its length (it bends, but doesn't stretch).
- A discrete model with 24 straight segments is close enough to being a circle for present purposes. You can think of my model of the Ringworld as a well-oiled bicycle chain with 24 links in it.
- Tidal effects and Coriolis forces are included.

All frequencies are in radians/second. The ring rotation rate (omega) corresponds to a period of 9 days. The period of the first mode is 21.4 days.

First 32 natural frequencies in radians/second (omega = 8.101607E-0006):

1 2 3 4: | 3.391208E-6 | 3.391208E-6 | 1.494779E-5 | 1.494779E-5 |

5 6 7 8: | 1.648368E-5 | 1.648368E-5 | 2.501163E-5 | 2.501163E-5 |

9 10 11 12: | 2.579653E-5 | 2.579653E-5 | 3.396396E-5 | 3.396396E-5 |

13 14 15 16: | 3.650952E-5 | 3.650952E-5 | 4.345375E-5 | 4.345375E-5 |

17 18 19 20: | 4.747975E-5 | 4.747975E-5 | 5.356643E-5 | 5.356643E-5 |

21 22 23 24: | 5.889438E-5 | 5.889438E-5 | 6.429545E-5 | 6.429545E-5 |

25 26 27 28: | 7.069410E-5 | 7.069410E-5 | 7.543140E-5 | 7.543140E-5 |

29 30 31 32: | 8.243244E-5 | 8.243244E-5 | 8.638629E-5 | 8.638629E-5 |

The corresponding periods in days are (rotation = 8.976250):

1 2 3 4: | 21.44429 | 21.44429 | 4.865072 | 4.865072 |

5 6 7 8: | 4.411762 | 4.411762 | 2.907530 | 2.907530 |

9 10 11 12: | 2.819063 | 2.819063 | 2.141153 | 2.141153 |

13 14 15 16: | 1.991866 | 1.991866 | 1.673551 | 1.673551 |

17 18 19 20: | 1.531644 | 1.531644 | 1.357605 | 1.357605 |

21 22 23 24: | 1.234788 | 1.234788 | 1.131061 | 1.131061 |

25 26 27 28: | 1.028686 | 1.028686 | 0.9640820 | 0.9640820 |

29 30 31 32: | 0.8822019 | 0.8822019 | 0.8418240 | 0.8418240 |

The mode shapes: The red near-circles (24-sided figures) are the undeformed shapes. The green figures are exaggerated deformed shapes. The dotted ellipses are the paths of the deformed nodes. Short red lines connect the undeformed node positions to the deformed positions. Red arrows show the velocities of the nodes.

The first mode (two phase angles): It looks elliptical, but notice that
it rolls around the Ring like a tetherball rather than having the eccentricity
grow or shrink. My "mode 2" is 90 degrees behind "mode 1."

mode1

mode2

Here's what they look like with phase angle offsets of 45 degrees:

mode1-45

mode2-45

Here's the next mode pair. Oddly, this three-cornered posigrade mode has
a lower frequency than the retrograde elliptical mode that I was expecting
would be next:

mode3

mode4

The other shoe finally drops. Here's that retrograde elliptical
mode:

mode5

mode6

And here is the expected retrograde three-cornered mode pair.

mode7

mode8

Now they start to come in a predictable pattern. Here are the
four-cornered posigrade and retrograde modes:

mode9

mode10

mode11

mode12

Here is a pentacle for Carol (mode 13 with a 190 degree phase angle
offset):

mode13-190

mode13

mode14

mode15

mode16

Here is some striking visual evidence that the "Tnuctipun plot" that has
been bandied about on the Niven list is wrong. The Ringworld appears to have
been built by the lost tribe of Israel as some sort of signalling
device.

mode17-270

Or maybe by the Sheriff's Department?

mode21

A predictable eight-pointed star:

mode25-240

Or was it built by the Ba'hai?

mode29

As you can see, 24 nodes isn't enough to get good representations of these more complicated mode shapes. But it seems good enough for the first handful.

Or maybe not. You know, I think we should build one and test it....