FREE INTERNET ROWING MODEL
(FIRM)
EXAMPLES: Coxed Eights
March 25, 2015
FIRM IS RESEARCH CODE!
Please check all estimates generated by the program
against experimental results before committing any
time or funds to your project as no liability can be
accepted by Cyberiad.
c
2015
Cyberiad
All Rights Reserved
Contents
1 INTRODUCTION
1
2 M8+: Men’s Coxed Eight (Normal Rig)
2.1
M8+ exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
4
3 W8+: Women’s Coxed Eight (Normal Rig)
3.1
W8+: Women’s Coxed Eight (German Rig)
3.2
W8+: Women’s Coxed Eight (Italian Rig) .
3.3
W8+: Women’s Coxed Eight (Rig A) . . . .
3.4
W8+: Women’s Coxed Eight (Rig B) . . . .
1
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9
10
11
12
13
INTRODUCTION
Six examples of coxed eights are included in this version of FIRM. More will be added in future versions.
The men’s eight is for a real crew.
The five examples for women’s eights are for an idealised crew. These examples show how to set up rowers in various
arrangements such as Italian, German, and three other “zero yawing moment” configurations.
1
2
M8+: Men’s Coxed Eight (Normal Rig)
The on-water trial for this example was conducted over 500 metres. The rowers named Adam, Bob, Conan and Doug
row on the starboard (i.e. bow) side; Wally, Yuri, Xave and Zeke row on the port (i.e. stroke) side.
Rigging details, oar angles, gate normal forces, and anthropometry were recorded and they have all been used as input
to FIRM. Body angle regimes were not recorded. Average angle were estimated for the crew as a whole using an elaborate
fitting process, and these were used as input to FIRM.
Table 1: Summary of experimental results for this simulation: number of strokes, stroke rate, non-dimensional pull phase duration
(tp /ts ), minimum hull velocity (Umin ), maximum hull velocity (Umax ), and mean hull velocity (U ).
Item
Nstrokes
Rate (spm)
tp /ts
Umin (ms−1 )
Umax (ms−1 )
U (ms−1 )
Value
37.808
0.453
4.715
7.345
6.249
14
±0.389
±0.009
±0.056
±0.056
±0.057
Table 1 summarises the main quantities relating to the simulation for this crew. Values are given ± one standard deviation.
Table 2: Experimental oar-related values for this simulation: Minimum and maximum oar angles, and maximum gate normal force.
Name
Seat
Adam
Wally
Bob
Xave
Conan
Yuri
Doug
Zeke
1
2
3
4
5
6
7
8
Min. Angle
(degrees)
Port Oar
Max. Angle
(degrees)
-55.2±0.64
31.9±0.47
931.7±61.1
-55.6±0.92
31.7±0.72
1220.3±35.4
-54.5±0.61
33.6±0.45
1006.8±39.5
-71.3±1.33
28.1±0.62
1304.2±55.5
Max. FGn
(N)
Min. Angle
(degrees)
-53.5±0.64
Starboard Oar
Max. Angle
Max. FGn
(degrees)
(N)
34.2±0.53
814.2±40.8
-58.1±0.91
34.0±0.37
1174.3±65.1
-64.0±1.20
31.2±0.55
1280.7±43.8
-50.8±1.07
35.7±0.29
988.1±71.7
A blade loss factor of kloss = 0.02 has been applied to each rower’s gate normal force to bring FIRM predictions in line
with the experimental mean speed of U = 6.249ms−1 . Given the very many uncertainties, the small (2.0%) reductions seem
quite acceptable. Note that the adjustments are well within the standard deviations for the maximum gate normal forces
shown in Table 2.
Instead of using a blade loss factor, we could have adjusted, for example, the oarhandle centres of effort, or the hull
viscous form factor, however, as we have discussed previously, it is probably better to show this “fudge factor” in the (single)
main input file.
The hull propulsive acceleration is shown in the left panel of Fig. 1. Interestingly, there is no “dip” around t/ts ≈ 0.15
apparent in the acceleration curve. It seems that the dip is more pronounced in singles, pairs and doubles.
There is good agreement between measured and predicted hull propulsive velocities in the plot at the right of Fig. 1.
The forces in the equations of motion are shown in the left panel of Fig. 2. Drag components during the stroke are in
the right panel of Fig. 2.
Experimental oar azimuth angles shown in Fig. 3 have been shifted so they are referenced to the centre of the pin. Values
for starboard-side rowers are shown in the right side panel of the figure; port-side rowers are at the left. Curves are values
used as input to FIRM. See the oarangles.csv input files for each rower.
Note the scatter for some of the oar angles for Seat 3 (Bob) and Seat 4 (Xave). The faulty data was removed for
calculations of means and standard deviations, but it was left in the plots to illustrate that on-water signals are not only
scattered because of normal variations, but they can also be the result of equipment glitches and failures.
Gate normal forces are shown in the two parts of Fig. 4. Curves are values used as input to FIRM. See the gateforces.csv
input files for each rower.
Blade propulsive forces for the starboard-side and port-side rowers are shown in the two parts of Fig. 5. It is fairly
obvious who the greatest contributors to propulsion are from these graphs, however, they are also among the heaviest rowers
in the crew and so they contribute more to the total drag. Deciding on whether there are better candidates for a crew is best
done by experienced coaches who can take into account intangibles, such as commitment, grit and esprit de corps, factors
that no computer program can ever hope to model.
2
0.8
7.5
7
0.4
6.5
a (g)
U (ms-1)
0
6
-0.4
5.5
M8+: AWBXCYDZ
Exp.
Exp. Mean
± SD
Pred.
Crew
-0.8
-1.2
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
M8+: AWBXCYDZ
Exp.
Exp. Mean
± SD
Pred.
Crew
5
4.5
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
Figure 1: Hull propulsive acceleration and crew cg acceleration (left); hull velocity and crew cg velocity (right).
3000
800
M8+: AWBXCYDZ
Fprop
Fboat
Fcrew
-Fdrag
Fsys
2000
M8+: AWBXCYDZ
Air
Viscous
Wave
Total
700
600
500
Drag (N)
Force (N)
1000
0
400
300
200
-1000
100
-2000
0
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
Figure 2: Equation of motion forces (left) and drag components (right).
The oarlever ratios in Fig. 6 include the effect of variations in the location of the OBCP during the stroke, which is why
they are not constant, as they are in most other rowing models.
For this example, body angles were not recorded. The regimes shown for the two rowers in Fig. 7 are identical, as they
are for all other members of the crew. The body angles were found through a fitting process (not described in this manual)
that minimised the difference between measured and predicted values of the hull propulsive acceleration and hull velocity.
In Fig. 8, yawing moment lever arms and yawing moments are shown only for the drive phase. FIRM convention is
that positive yawing moments tend to turn the hull bow to port (i.e. the stroke side). Thus, the total yawing moment for
this (conventionally-rigged) 8+, with these particular rowers, tend to move the boat to starboard (i.e. the bow side) during
the early part of the pull phase. Note how Zeke, the rower in Seat 8 (i.e. the “stroke” seat) tends to pull the hull bow to
starboard during the early part of the stroke, and then to port for the remainder of the stroke. Kleshnev reported the same
phenomenon in his newsletter of Nov. 2009 [?].
Trajectories of the OBCP are almost identical for all rowers, as shown in Fig. 9. The minimum and maximum values for
the vertical oar angles were chosen so that oars would enter the water near the catch, and exit at the release. As discussed
previously, vertical angles do not play a significant role in FIRM performance predictions. Training a crew to display such
exquisite blade harmony would be a considerable challenge for coaches!
The trajectories of the OBCP for all eight oars are shown in the two parts of Fig. 10. The traces of each oar are relative
to the individual rowers’ ankles (or heel cups). The path of Zeke’s oar is quite different to those of the other crew. The
on-water trial for this crew was conducted at an early stage of their campaign, before they had become fully coordinated.
The OBCP trajectories are shown on the correct side of the hull and relative to the boat in Fig. 11.
3
40
20
20
0
0
ψxy (degrees)
ψxy (degrees)
40
-20
-40
-20
-40
M8+: AWBXCYDZ
(Seat 2) Wally
(Seat 4) Xave
(Seat 6) Yuri
(Seat 8) Zeke
-60
-80
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
M8+: AWBXCYDZ
(Seat 1) Adam
(Seat 3) Bob
(Seat 5) Conan
(Seat 7) Doug
-60
-80
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
Figure 3: Oar azimuth angles. Port side: seats 2,4,6,8 (left); Starboard side: seats 1,3,5,7 (right).
1400
1200
M8+: AWBXCYDZ
(Seat 1) Adam
(Seat 3) Bob
(Seat 5) Conan
(Seat 7) Doug
1200
1000
1000
800
800
FGn (N)
FGn (N)
1400
M8+: AWBXCYDZ
(Seat 2) Wally
(Seat 4) Xave
(Seat 6) Yuri
(Seat 8) Zeke
600
600
400
400
200
200
0
0
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
Figure 4: Gate normal forces. Port side: seats 2,4,6,8 (left); Starboard side: seats 1,3,5,7 (right).
The thick grey lines are the OBCP trajectories of Seats 1 and 2 during the previous stroke. It is obvious that the crew
have managed to “clear the puddles” at this stroke rate and hull speed. If they had not, then the OBCP trajectories of Seats
7 and 8 would have intersected the previous OBCP traces of Seats 1 and 2.
2.1
M8+ exercises
Exercise M8+ 1.0: The on-water trial was held in winter when the air and water temperatures were about 8◦ C. The Olympics
in Brazil will be held during the middle of August in 2016. Look up the expected water temperature for Lagoa Rodrigo
de Freitas in Appendix ?? and make an estimate of the predicted mean hull velocity for the temperature expected for
mid-August. How does that compare with the world’s best time for the M8+ class shown in Appendix ???
Exercise M8+ 1.1: This on-water trial was held over 500 metres. It is unlikely that the crew could sustain the same effort
over a 2000 metre course. Reduce each rower’s maximum gate normal force by 5%. Would they still achieve a world’s best
time in the warm water expected for Rio 2016?
Exercise M8+ 1.2: What would be the expected mean boat velocity if the lagoon will be filled with salt water instead of
fresh?
4
350
300
M8+: AWBXCYDZ
(Seat 1) Adam
(Seat 3) Bob
(Seat 5) Conan
(Seat 7) Doug
300
250
250
200
200
FBx (N)
FBx (N)
350
M8+: AWBXCYDZ
(Seat 2) Wally
(Seat 4) Xave
(Seat 6) Yuri
(Seat 8) Zeke
150
150
100
100
50
50
0
0
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
Figure 5: Blade propulsive forces. Port side: seats 2,4,6,8 (left); Starboard side: seats 1,3,5,7 (right).
2.4
2.4
M8+: AWBXCYDZ
(Seat 2) Wally
(Seat 4) Xave
(Seat 6) Yuri
(Seat 8) Zeke
2.39
2.39
2.38
Dynamic Oarlever Ratio
2.38
Dynamic Oarlever Ratio
M8+: AWBXCYDZ
(Seat 1) Adam
(Seat 3) Bob
(Seat 5) Conan
(Seat 7) Doug
2.37
2.36
2.37
2.36
2.35
2.35
2.34
2.34
2.33
2.33
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
Figure 6: Dynamic oarlever ratios. Port side: seats 2,4,6,8 (left); Starboard side: seats 1,3,5,7 (right).
180
150
M8+: AWBXCYDZ
Adam
Knee
Hip
Neck
Shoulder
150
120
Joint Angle (degrees)
120
Joint Angle (degrees)
180
M8+: AWBXCYDZ
Wally
Knee
Hip
Neck
Shoulder
90
60
30
90
60
30
0
0
-30
-30
-60
-60
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
Figure 7: Body angle regimes for seat 2 (left) and seat 1 (right).
5
0.5
t/ts
0.6
0.7
0.8
0.9
1
8
4
2
M8+: AWBXCYDZ
(Seat 1) Adam
(Seat 3) Bob
(Seat 5) Conan
(Seat 7) Doug
(Seat 2) Wally
(Seat 4) Xave
(Seat 6) Yuri
(Seat 8) Zeke
Total
1000
Yawing Moment (Nm)
6
Yawing moment lever arm (m)
1500
M8+: AWBXCYDZ
(Seat 1) Adam
(Seat 3) Bob
(Seat 5) Conan
(Seat 7) Doug
(Seat 2) Wally
(Seat 4) Xave
(Seat 6) Yuri
(Seat 8) Zeke
0
-2
500
0
-500
-4
-1000
-6
-8
-1500
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0.2
0.2
0.1
0.1
zobcp (m. above water)
zobcp (m. above water)
Figure 8: Yawing moment lever arms (left); yawing moments (right).
0
-0.1
M8+: AWBXCYDZ
Waterplane
(Seat 2) Wally
(Seat 4) Xave
(Seat 6) Yuri
(Seat 8) Zeke
-0.2
-0.3
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
0
-0.1
M8+: AWBXCYDZ
Waterplane
(Seat 1) Adam
(Seat 3) Bob
(Seat 5) Conan
(Seat 7) Doug
-0.2
-0.3
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
Figure 9: OBCP trajectories in the yz-plane. Port side: seats 2,4,6,8 (left); Starboard side: seats 1,3,5,7 (right).
3.5
3.5
Direction of
Direction of
Boat Travel
Lateral distance from hull centreline (m)
Lateral distance from hull centreline (m)
Boat Travel
3
Release
2.5
Catch
2
M8+: AWBXCYDZ
(Seat 2) Wally
(Seat 4) Xave
(Seat 6) Yuri
(Seat 8) Zeke
1.5
-3
-2.75
3
Release
2.5
Catch
2
M8+: AWBXCYDZ
(Seat 1) Adam
(Seat 3) Bob
(Seat 5) Conan
(Seat 7) Doug
1.5
-2.5
-2.25
-2
-1.75
-3
x (m)
-2.75
-2.5
-2.25
-2
x (m)
Figure 10: OBCP trajectories in the xy-plane. Port side: seats 2,4,6,8 (left); Starboard side: seats 1,3,5,7 (right).
6
-1.75
4
Lateral distance from hull centreline (m)
3
2
M8+: AWBXCYDZ
(Seat 2) Wally
(Seat 4) Xave
(Seat 6) Yuri
(Seat 8) Zeke
(Seat 1) Adam
(Seat 3) Bob
(Seat 5) Conan
(Seat 7) Doug
1
0
-1
-2
-3
-4
0
2
4
6
8
10
12
14
x (m)
Figure 11: OBCP trajectories in the xy-plane. Thick grey lines are possible location of puddles formed by seats 1 and 2 during the
previous stroke.
7
M8+: AWBXCYDZ
Rate 37.8 spm
Speed 6.25 m/s
Work on
Oarhandles
A.
MUSCULAR 4585 W
EFFORT
100 %
Dead Mass 154.0 kg
Moving Mass 778.0 kg
Total Mass 932.0 kg
Net
Kinetic Energy
B.
HANDLES 3654 W
B/A
E.
SYSTEM
931 W
NOTE: B+F=D+H and C+E=D+G
MOMENTUM
80 %
E/A
20 %
Blade Efficiency
Mom. Efficiency
C/B = 80.5 %
F/E = 61.5 %
C.
F.
PROPULSION 2942 W
FOOT BOARDS 572 W
(External)
C/A
64 %
F/A
12 %
D.
DRAG
Propelling Efficiency
D/(D+H) = 83.2 %
H.
BLADE 712 W
LOSSES
H/A
16 %
Lost to water
Work done
on shell
3514 W
D/A
77 %
Transferred to air and water
I=D+G+H.
TOTAL 4585 W
LOSS
I/A
100.0 %
Net Efficiency
D/(D+H)-G/A = 75.3 %
Figure 12: Power flow chart.
8
Air 11 %
Visc. 83 %
Wave 6 %
Velocity Efficiency
1-G/A = 92.2 %
G.
BODY FLEX 359 W
(Internal)
G/A
8%
Lost as heat, breath etc.
3
W8+: Women’s Coxed Eight (Normal Rig)
To run this example double-click on the icon for the batch file w8 normal.bat.
This example, and all other W8+ examples, use clones of a rower named “Arale”. Her rigging details, anthropometry,
and her body angle, oar angle and gate normal force regimes are identical, except that four of the clones row on the port
side, and four on the starboard side.
8
4
2
M8+: Arale Clones
(Seat 1) Star Clone
(Seat 2) Port Clone
(Seat 3) Star Clone
(Seat 4) Port Clone
(Seat 5) Star Clone
(Seat 6) Port Clone
(Seat 7) Star Clone
(Seat 8) Port Clone
Total
1000
Yawing Moment (Nm)
6
Yawing moment lever arm (m)
1500
M8+: Arale Clones
(Seat 1) Star Clone
(Seat 2) Port Clone
(Seat 3) Star Clone
(Seat 4) Port Clone
(Seat 5) Star Clone
(Seat 6) Port Clone
(Seat 7) Star Clone
(Seat 8) Port Clone
0
-2
500
0
-500
-4
-1000
-6
-8
-1500
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
Figure 13: Yawing moment lever arms Lyaw (left); yawing moments Myaw (right).
Yawing moment lever arms and yawing moments are shown in the two parts of Fig. 13. The plot at the right shows that
the total yawing moment is negative for the first two thirds of the drive phase. This means (in our convention) that the bow
will tend to be pushed to the staboard side of the hull. During the last third of the drive phase, the total yawing moment is
positive, so that the bow will be pushed back towards the port side. The nett effect is that a small rudder correction will be
required to keep the boat going on a straight course.
Table 3: Oar arrangements used in W8+ examples.
Seat
1
2
3
4
5
6
7
8
Normal
S
P
S
P
S
P
S
P
German
P
S
P
S
S
P
S
P
Italian
P
S
S
P
P
S
S
P
Rig A
P
P
S
S
S
S
P
P
Rig B
S
P
P
S
P
S
S
P
Table 3 summarises the oar arrangements used in this example and those to follow. A “P” indicates that the oar is on
the port side; an “S” is for a starboard side oar.
The last four cases are examples of “zero nett yawing moment” oar arrangements recently investigated by Barrow [?]
although they have been known for several decades. As we have already seen in the case of Men’s Fours in Section ??, zero
nett yawing moment is difficult to achieve when the crew have different body sizes and each rows slightly differently. The
only zero yawing moment arrangement for the 4- class is the Italian Rig in Section ??. For the 8+ class, there are four zero
yawing moment arrangements.
9
3.1
W8+: Women’s Coxed Eight (German Rig)
To run this example double-click on the icon for the batch file w8 german.bat.
8
4
2
M8+: Arale Clones
(Seat 1) Port Clone
(Seat 2) Star Clone
(Seat 3) Port Clone
(Seat 4) Star Clone
(Seat 5) Star Clone
(Seat 6) Port Clone
(Seat 7) Star Clone
(Seat 8) Port Clone
Total
1000
Yawing Moment (Nm)
6
Yawing moment lever arm (m)
1500
M8+: Arale Clones
(Seat 1) Port Clone
(Seat 2) Star Clone
(Seat 3) Port Clone
(Seat 4) Star Clone
(Seat 5) Star Clone
(Seat 6) Port Clone
(Seat 7) Star Clone
(Seat 8) Port Clone
0
-2
500
0
-500
-4
-1000
-6
-8
-1500
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
Figure 14: Yawing moment lever arms Lyaw (left); yawing moments Myaw (right).
Yawing moment lever arms and yawing moments are shown in the two parts of Fig. 14.
This oar arrangement has been used for several years by very successful German Men’s 8+ crews.
10
0.7
0.8
0.9
1
3.2
W8+: Women’s Coxed Eight (Italian Rig)
To run this example double-click on the icon for the batch file w8 italian.bat.
8
4
2
M8+: Arale Clones
(Seat 1) Port Clone
(Seat 2) Star Clone
(Seat 3) Star Clone
(Seat 4) Port Clone
(Seat 5) Port Clone
(Seat 6) Star Clone
(Seat 7) Star Clone
(Seat 8) Port Clone
Total
1000
Yawing Moment (Nm)
6
Yawing moment lever arm (m)
1500
M8+: Arale Clones
(Seat 1) Port Clone
(Seat 2) Star Clone
(Seat 3) Star Clone
(Seat 4) Port Clone
(Seat 5) Port Clone
(Seat 6) Star Clone
(Seat 7) Star Clone
(Seat 8) Port Clone
0
-2
500
0
-500
-4
-1000
-6
-8
-1500
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
Figure 15: Yawing moment lever arms Lyaw (left); yawing moments Myaw (right).
Yawing moment lever arms and yawing moments are shown in the two parts of Fig. 15.
11
0.7
0.8
0.9
1
3.3
W8+: Women’s Coxed Eight (Rig A)
To run this example double-click on the icon for the batch file w8 riga.bat.
8
4
2
M8+: Arale Clones
(Seat 1) Port Clone
(Seat 2) Port Clone
(Seat 3) Star Clone
(Seat 4) Star Clone
(Seat 5) Star Clone
(Seat 6) Star Clone
(Seat 7) Port Clone
(Seat 8) Port Clone
Total
1000
Yawing Moment (Nm)
6
Yawing moment lever arm (m)
1500
M8+: Arale Clones
(Seat 1) Port Clone
(Seat 2) Port Clone
(Seat 3) Star Clone
(Seat 4) Star Clone
(Seat 5) Star Clone
(Seat 6) Star Clone
(Seat 7) Port Clone
(Seat 8) Port Clone
0
-2
500
0
-500
-4
-1000
-6
-8
-1500
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
Figure 16: Yawing moment lever arms Lyaw (left); yawing moments Myaw (right).
Yawing moment lever arms and yawing moments are shown in the two parts of Fig. 16.
12
0.7
0.8
0.9
1
3.4
W8+: Women’s Coxed Eight (Rig B)
To run this example double-click on the icon for the batch file w8 rigb.bat.
8
4
2
M8+: Arale Clones
(Seat 1) Star Clone
(Seat 2) Port Clone
(Seat 3) Port Clone
(Seat 4) Star Clone
(Seat 5) Port Clone
(Seat 6) Star Clone
(Seat 7) Star Clone
(Seat 8) Port Clone
Total
1000
Yawing Moment (Nm)
6
Yawing moment lever arm (m)
1500
M8+: Arale Clones
(Seat 1) Star Clone
(Seat 2) Port Clone
(Seat 3) Port Clone
(Seat 4) Star Clone
(Seat 5) Port Clone
(Seat 6) Star Clone
(Seat 7) Star Clone
(Seat 8) Port Clone
0
-2
500
0
-500
-4
-1000
-6
-8
-1500
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
0.7
0.8
0.9
1
0
0.1
0.2
0.3
0.4
0.5
t/ts
0.6
Figure 17: Yawing moment lever arms Lyaw (left); yawing moments Myaw (right).
Yawing moment lever arms and yawing moments are shown in the two parts of Fig. 17.
13
0.7
0.8
0.9
1