NSST SPECIAL EVOLUTIONS TRAINING
INSTRUCTIONAL MODULE
MODULE – MATH FOR THE OOD (I)
(APPLICATIONS FOR UNREP OPS)
REVISION DATE: 6 NOVEMBER 2015
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BEST PRACTICES / SHIP INFORMATION SHEET
MODULE – MATH FOR THE OOD (I)
STUDENT HANDOUT
Some Math for the OOD basic concepts and best practices:
1. The Radian Rule and its Applications. From high school math, we recall there
are 2 pi radians in a circle. If we use a value of 3.14 for pi, through division
we can calculate that there are 57.3 degrees in a radian or 60 degrees by
rounding up. The Radian Rule, simply put, states that for every angle of one
degree, the ratio of the long side of a triangle to the short side, within a circle,
is 60 to 1. The basic equation or relationship in analyzing the one-degree
segment of a circle is R/60. R equals the radius, or range for our purposes. If
the long side of a triangle/radius of a circle is 600, a one degree angle
subtends an arc that has a chord length of 10.
Example: A contact 3 degrees off your starboard bow at 4000 yards is DIW.
How close will you pass if present course in maintained? Three degrees,
divided by 60, or 1/20 of a radian, is 1/20th of the range to the contact, or 200
yards. Another way to do the math is to return to the mental circle diagram.
We can state what we already know in slightly different terms; an angle of one
degree will subtend an arc on the circle that has a length of 1/60th of the
radius, or R/60. In this example, the range divided by 60 (4000/60) is equal to
66.66 yards. This distance times the 3 degrees equals 200 yards.
2. The Radian Rule can be applied to several different operations, such as
contact management, underway replenishment approach, and handling the
ship alongside. The most common application of the Radian Rule is in
UNREP operations. By calculation, desired lateral separation (from the
delivery ship) can be used to build a table that provides angular offset for
given ranges. This information can be used during an UNREP approach to
determine the approaching/receiving ship’s position relative to the desired
approach path. The worksheet in this lesson’s handout can be used to
perform the calculations and construct a table that provides angular offset and
true bearing to the delivery ship for given ranges.
3. During the alongside phase of UNREP operations, the 1-minute rule (speed in
knots times 100 equals distance traveled in feet) can be combined with the
Radian Rule to determine lateral closure/opening rate. For a replenishment
(ROMEO) speed of 13 knots, the ship will travel 1300 feet in one minute.
Let’s return to the earlier discussion of analyzing a one-degree segment. For
a radius/range of 1300 feet, the chord of a one-degree angular segment is
equal in length to R/60, or 1300/60 = 21.66 feet. Let’s round this value to 22
feet. At 13 knots, for every course adjustment of 1 degree, the ship will move
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laterally 22 feet in one minute. As conning officer, if I feel I am too wide on
the delivery ship and adjust my course by 2 degrees towards the oiler, after 1
minute, I will have reduced the lateral separation by 44 feet.
4. Surge Rate and its Application in Shiphandling.
 Surge rate refers to the distance a ship travels between the time a speed
change is initiated and the ship is actually at the new speed.
 The surge rate can vary widely by ship class and by initial speed. Surge rates
can be derived from the ship’s acceleration tables. For gas turbine surface
combatants, 25 yards per knot (of speed differential) can be used as a surge
rate rule of thumb during UNREP ops.
 Based on the information from the acceleration tables or by using an
established rule of thumb, the distance traveled while effecting a speed
change can be determined by multiplying the surge rate (expressed in yards
per knot) by the magnitude of the speed change (in knots). This figure can be
used to calculate the relative position of the “speed cut” when making an
approach on an UNREP ship.
 Proper utilization of surge rate enhances sharp and concise ship-handling,
and results in effective underway replenishment approaches or the arrival on
station in any maneuver requiring a speed change.
5. The Five Degree Maneuver and its Use.
 The five degree maneuver is used to expedite opening the distance between
ships that have gotten too close during underway replenishment.
 As ships that are alongside get closer, the low pressure area between the
hulls becomes more of a factor exerting greater attraction between the ships.
 At close separation, attempts to open the distance by changing course will
cause the stern to swing even closer to the other vessel, increasing the
possibility of a collision. For a six hundred foot ship moving ahead with its
pivot point relatively forward, one degree of course change will cause the
stern to swing in by 10 feet, but also add 22 feet to the opening vector as we
discussed in paragraph 3 above.
 A five degree course change will cause the stern to swing in 50 feet, but also
add 110 feet per minute to the opening vector. As such, the stern will not
complete the swing through 50 feet as the opening vector takes effect quicker
and the ship opens rapidly. The key step is to execute the five degree
maneuver before the two ships reach 50 feet of lateral separation.
 Ordering the five degree maneuver. In this type of situation with a master
helmsman who has been briefed on executing the maneuver, the five degree
maneuver should be executed by giving a course order (i.e. “Come right (or
left) smartly to XXX (ROMEO CORPEN +/- 5 degrees).” The helmsman will
know what is required to smartly execute the course change.
 The challenge lies in returning to replenishment course before the ships have
opened too far.
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STUDENT NOTETAKING GUIDE
MODULE – MATH FOR THE OOD (I)
The Radian Rule and its Applications
From high school math class, we know there are 2 pi radians in a circle.
From this value, we can calculate that one radian is equal to _____
degrees. To keep things simple, we round this number to 60.
To analyze a one-degree segment, i.e., determine the chord length, we
use the equation: R/60 where “R” represents ________________.
The Radian Rule has two applications in UNREP operations. The first
application is used to assist the conning officer in making the approach on
the delivery ship. A table can be constructed (see the handout) that
allows us to use a numerical constant based on desired lateral separation
to provide angular offset (converted to true bearing) to the oiler for set
ranges. The second application is used when alongside to fine tune
lateral separation. By using the one-minute rule in conjunction with a
standard replenishment speed of 13 knots, we know that for each course
adjustment of one degree, our ship will move laterally _____ feet per
minute.
Surge Rate for my ship is: ________________. Surge Rate is used to:
The Five Degree Maneuver and its Application
The Five Degree Maneuver is used to prevent a collision during alongside
operations.
The critical “timing” decision in executing this maneuver is:
The proper order to the master helmsman is, “Come right / left smartly to
(course).”
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HANDOUT, MATH FOR THE OOD (I)
UNREP APPROACH, APPLYING THE RADIAN RULE
1. Desired lateral separation: __________ ft. / __________ yds.
2. Translate the desired lateral separation into what the angular offset will be at a range
of 600 yards from the delivery ship. NOTE: At a range of 600 yards, one degree of
angular offset equals 10 yards / 30 feet of lateral separation. Divide the desired
lateral separation by 30 feet (or 10 yards). This gives you the angular offset in
degrees. The angular offset at a range of 600 yards will be __________ degrees.
3. Multiply 600 yards by the angular offset value determined in step 2. This value
serves as a numerical constant that can be used to determine the angular offset for
any given range. This constant can be expressed as the “rule of __________.” For
example, for a desired lateral separation of 180 feet / 60 yards, the angular offset at
a range of 600 yards to the delivery ship is 6 degrees. Multiply 600 by 6 to obtain a
“rule of 3600.” Using the rule of 3600 (remember, it applies only for a desired lateral
separation of 180 feet), the angular offset for any range can be calculated by dividing
3600 by the range: 3 degrees for 1200 yards, 4 degrees for 900 yards, 6 degrees for
600 yards, etc.
4. Now, a table can be created that will serve as a ready reference for range versus
angular offset and true bearing to the delivery ship. Divide the value of the constant
by the ranges listed below. This will give the angular offset for each range. Apply
the angular offsets to ROMEO CORPEN to give a desired true bearing to the oiler for
each range. NOTE: When using own ship’s port side during replenishment, subtract
the angular offset from ROMEO CORPEN to obtain true bearing. For evolutions
using own ship’s starboard side, add the angular offset to ROMEO CORPEN to
obtain true bearing. VERY IMPORTANT POINT – this true bearing is what the outer
edge/outer most object on the delivery ship’s engaged side will be if own ship is on
the approach path, i.e., at the desired lateral separation. If the observed true bearing
is too large or too small, make a slight course adjustment to correct this. Also, this
observation must be made from your engaged side’s bridge wing.
ROMEO CORPEN: __________
RANGE (YDS)
Own Ship’s Engaged Side: P / S
VALUE OF THE CONSTANT
ANGULAR OFFSET
TRUE BRG
1200
__________
__________
__________
1000
__________
__________
__________
800
__________
__________
__________
600
__________
__________
__________
400
__________
__________
__________
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UNREP APPROACH, CHECKING STATION-TO-STATION LINE-UP
APPLYING THE SURGE RATE
1. Checking Station-to-Station Line-up:
a. Line-up provided by the CLF ship:
Delivery ship stations:
Receiving ship stations:
b. Fore and aft separation between stations:
Delivery ship stations:
Receiving ship stations:
2. Applying the surge rate:
a. Receiving ship’s surge rate: ____________ yards/knot of speed
differential
b. Romeo speed: 13 knots
c. Desired approach speed: ______________ knots
d. Predicted distance traveled during deceleration: _____________ yards
e. Predicted distance traveled during deceleration: _____________ feet
f. Delivery ship’s LOA: _______________ feet
g. Speed cut reference point (on delivery ship): _________________ feet
aft of bow
h. Speed cut reference point (on delivery ship): _________________ feet
forward of stern
i.
Speed cut reference point visual cue (on delivery ship):
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