For the record, ta0 unleashed this monster on the forum. Couldn't resist taking it for a ride.Simple models can be dangerous when it comes to real top behavior, but sometimes they work quite well...
Interactive spin decay, critical speed, and spin time simulatorIacopo, ta0, and Alan have been kind enough to post some useful (time, RPM) data sets or empirical spin decay curves (SDCs) over the years. I've plotted empirical SDCs from all but ta0's raw data (not posted) in Excel, and lo and behold:
I can fit a purely exponential decay to each and every one with excellent closeness of fit -- all the way to the bitter end in some (like Iacopo's Nr. 25) and at all but the lowest recorded speeds in the rest. The low-speed SDC tails in the latter group become more linear than exponential as they approach the horizontal -- just what you'd expect as speed-dependent air resistance finally gives way to mostly weight-dependent tip resistance.
This interactive simulator graphs strictly exponential SDCs from key parameters adjusted with sliders. The good news: Many of the empirically linear SDC tails are still pretty close to exponential, with ta0's 2 peg tops and Iacopo's Nr. 14 finger top being the most notable exceptions. The bad news: Projected spin times can be rather sensitive to tail shape. So don't take them too seriously.
The graphs1. Horizontal (x) axis = time
t after release in seconds. Ignore everything below.
2. Vertical (y) axis = angular speed
w in rad/s. Ignore everything to left. Remember, 1 rad/s = 9.54 RPM.
3.
Solid curve
w(
t) = purely exponential SDC crossing the vertical (
t = 0) axis at the specified release speed
w0. The release speed slider maxes out at
w0 = 500 rad/s = 4,775 RPM.
4.
Dashed horizontal line = speed
w(
T) at one top "lifetime"
T after release. The time spent above this line is
T, which is akin to a half life. Lifetime grows with the top's absolute axial moment of inertia (AMI) and shrinks mainly with its air resistance. Importantly, a top with an exponential SDC will lose 63.2% of its speed over
any time interval
T, not just the one starting at release (
t = 0).
5.
Dotted horizontal line marks the critical speed
wC for stability against gravity, a property of the top's mass distribution alone. The time spent above this line is the top's
minimum spin time. Real tops generally begin their rapid death spirals shortly thereafter, but some manage to hang on longer now and then.
Be sure to fiddle with the
w0 and
T sliders controlling the solid SDC and the
J3,
J1, and
H sliders controlling the dotted critical speed line (see below). The limits on these sliders cover posted data for Iacopo's tops, Alan's 3-inch disk top, and most of mine as well. Best to have a dotted line showing above the horizontal time axis before experimenting with
w0 and
T.
Things to tryA. Set release speed at
w0 = 100 rad/s (954 RPM) and tap the solid SDC at any point of interest. The displayed y-value for that point will then be the percentage of
w0 remaining.
B. Vary release speed
w0 alone. Note how little that affects spin time when the lifetime
T is very short. Exactly what happens in my highest-drag tops: Paltry percentage spin-time gains from huge percentage release speed bumps.
C. Vary lifetime
T at constant starting and critical speeds (
w0 and
wC). Note the very strong effect on spin time. Streamlining pays!
D. Raise and lower the dotted critical speed line at constant
w0 and
T. Note how much spin time you can gain by shaving off a little critical speed.
E. Vary CM height
H alone. Note how easy it is to push critical speed
wC above release speed
w0. A top in this situation
never stays up. Critical speed is most sensitive to this parameter.
F. Vary central transverse moment of inertia per unit mass
J1 alone. Critical speed is least sensitive to this parameter, but it's still pretty easy to push critical speed
wC above release speed
w0. Note that
J1 is about the top's CM, not its tip.
G. Vary AMI per unit mass
J3 alone. The effect on critical speed is much stronger than in (F).
The implications for top design are pretty clear.
NB: The simulator's main flaw is its failure to link lifetime
T to AMI per unit mass
J3. Purely exponential spin decay implies a total braking torque proportional to current speed. But the resulting deceleration is inversely proportional to the absolute AMI
I3 =
M J3, where
M = total mass. And
T is an inverse measure of deceleration rate.
Equations using notation above in SI unitsi.
w(
t) =
w0 exp(-t /
T)
ii.
T = (
t2 -
t1) / ln(
w1 /
w2) for any two time marks
t2 >
t1.
iii.
w(
t +
T) =
w(
t) exp(-1) = 0.368
w(
t) for every
t.
iv.
wC = 2 sqrt[9.81 (
J1 +
H²)
H] /
J3, where
J3 = AMI per unit mass,
J1 = central transverse moment of inertia per unit mass (about CM),
H = CM height at 0° tilt, and 9.81 is the acceleration of gravity near sea level.