TREP™
- Time Related and Equivalent Performance
Copyright ©
2000 Richard Boehmer
Time
related and equivalent performance, TREP™, is a graphic and/or mathematic
technique that presents an object's performance not as a single speed but as a
set of maximum speeds coupled with the respective time periods over which these
speeds were either measured or calculated.
When plotted on either semilog or log-log paper with time period on the
x-axis always being logrithmetic, the points representing a set of best or
maximum speeds of various objects have been observed to lie on a nearly
strainght line. Likewise, when a
set of speeds coupled with the time periods are subjected to a bivariate,
best-fit, regression analysis, an equation is obtained which better defines the
speed of the object rather than simply one indescriminant value.
Therefore, a TREP™ analysis consists of either plotting and/or
regressing a bivariate set of speed/time data.
The
following Figure 1. presents different types of
animate and inanimate objects for which their maximum speeds have been measured
over various periods of time. This
plot shows that sailing vessels are just one of many objects for which its
maximum speed declines as the period over which it is measured increases.
As
one works over the years refining a method of analysis and interpreting the
results, one's perception of the correct procedure may change.
Such has been the case for my work with TREP™ analysis.
From the start, this work has consisted of subjecting sets of
sailing vessels' best speeds coupled to the time periods over which they
were ascertained to a bivariate regression analysis.
Over the years the resulting type of equation has changed with the
analysis from geometric to logarithmic and back to geometric again.
Initially,
I used a geometric regression (1,2,3,4,8,11) simply because it gave a better fit
(higher correlation coefficient) to the data than did a linear, an exponential,
or a logarithmic regression. Riegel
(5,6) likewise presented a geometric equation to express his findings.
He and Chapman (10) also plotted their data on log-log paper to get
straight lines.
Then
for about two years starting in 1982, I examined not just the best but all -
particularly the worst - of the speed/time couplets of the ARGOS data from many
racing boats. At this time I
decided to switch to a logarithmic regression to allow for zero speeds.
Although this particular work was never published, a logarithmic revision
(9) of my previous work was published, and it was used as part of my analysis of
transatlantic sailing speed records (12).
The
following year I switched back to a geometric analysis (13) having reexamined
the implications of using a logarithmic regression.
The foremost problem with the logarithmic regression is than it produces
an equation that will yield zero and negative speeds for very long periods of
time. This is an impossibility
because once an object moves its average speed will forever be greater than zero
even if the object never moves again. Also,
although negative velocity exists by reversing direction, negative speed is
meaningless. TREP™ examines speed not velocity. A geometrically regressed TREP equation will always yield a
positive speed therefore it is more realistic.
I've
been asked of what use is TREP™ other than to simply view speed in a unique
way. Aside from my original use of
TREP™ to settle the arguement of whether or not American built clipper ships
could out sail the larger and stronger iron sailing ships which followed them
(1-3,8), I've since used TREP™ to suggest that a speed record may have been
surpassed (14,15) and what records might be easy targets as compared to others
(16), to question or verify speed claims (17), and to compare the record
performances of various ocean racing multihulls (18-21).
I'd welcome others to apply TREP™ to their interest in speed where
appropriate.
Several
questions about sailing speed remain to be investigated which involve TREP™.
As shown in Figure 2, rather abrupt changes in
slope can occure with different boats at different times (4,13).
The causes of these changes need further investigation. Also a nagging question remains as to how small must
the time period become for speed to level off because it cannot keep increasing
without a limit. Another
observation is that the slopes (decay coefficients in the equations) of the
lines appear to flatten as the set of speeds increase, i.e. the TREP™ line for
the SR-71 at the top is flatter than those below in
Figure 1.
References:
1.
Richard Boehmer, "Will a multihull ever break the clipper ship
record?", Multihulls 3:4 (Sum'77), 20-21;
2.
Richard Boehmer, "TREP analysis of CHAMPION OF THE SEAS' one day
record run", Sailing Facts and Figures [AYRS 89] (Oct'77), 36-40; also by the same title in Nautical Research Journal 24:2
(Jun'78),69-71;
3.
Richard Boehmer, "A new look at some old sailing speed
records", The Mariner's Mirror 65:2 (May'79), 163-167.
4.
Richard Boehmer, "TREP analysis of WILLIWAW", Shallow Draft
Craft [AYRS 94] (Apr'80), 27-29.
5.
Peter Riegel, "Athletic records and human endurance", American
Scientist (M/J'81), 285-290.
6.
Peter Riegel, " A statistical analysis of ultramarathoning
records", Ultrarunning (Sep'81), 12-13.
7.
Jean Forester, "L'optimum", Union National des Multicoques
Bulletin (Sum'87), 37-42.
8.
Richard Boehmer, "465 milles en 24 heures!", Union National des
Multicoques Bulletin 40 (Aut'81), 40-43.
9.
Richard Boehmer, "TREP analysis: revised, revisited, reaffirmed, and
revealed", Sail Rigs and Hydrofoils [AYRS 97] (Jun'83), 30-36.
10.
George Chapman, "TREP Analysis for durations below 24 hours",
Sail Rigs and Hydrofoils, [AYRS 97] (Jun'83), 36-37.
11.
Richard Boehmer, "In the wake of the CHAMPION", Nautical
Research Journal 30:3 (Sep'84), 152-158.
12.
Richard Boehmer, "W-E transatlantic sailing records: a speed
analysis", Proceedings of the Ancient Interface XIV Symposium on Sailing (1984), 1-17. [partial reprint in Multihull Inter.
205 (Feb'85), 30-31.]
13.
Richard Boehmer, "Transat en Double: an analysis of on-wind and
off-wind sailing performance", Proceedings of the Ancient Interface XV
Symposium on Sailing (1985), 109-126.
14.
Richard Boehmer, "Who's the
fastest of them all?", Multihull International 207 (Apr'85), 87.
15.
Richard Boehmer, "The fastest of them all", Yacht Racing and
Cruising 24:7 (Jul'85), 58-59.
16.
Richard Boehmer, "Setting your sights on a record", Sail 20:8
(Aug'89), 45-48.
17.
Richard Boehmer, "Maintaining the truth in multihull speed
claims", Multihull International 358 (Nov'97), 300-301.
18.
Richard Boehmer, "The 1998 Route du Rhum: a summary and
analysis", Multihull International 372 (Feb'99), 46-48.
19.
Richard Boehmer, "Tri, Tri, Tri, Tri Again:
A commentary on Formula 60 trimarans", Multihull International 377/8
(July/August'99) 206-208.
20.
Richard Boehmer, "2000 STAR: results & comments", Multihull
International 389/390 (July/Aug'00), 196-198.
21.
Richard Boehmer, "Facts & figures: two cats, two BDRs, two
transats", Multihull International 389/390 (July/Aug'00), 188-191.