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Old 04-27-2004
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Dead Reckoning Calculations


Dead reckoning begins at a known point, like this red nun above.
Any sailor who has ever used dead reckoning (DR) knows that it's a simple system which allows you to determine your present position by plotting the course and speed from a known past position. A navigator can also determine the boat's future position by projecting the present course and speed from the present position. Of course it's important to remember that a DR position is only approximate because it does not allow for the effect of leeway, current, or steering error.

All DR calculations take one of three different forms; (1) How long will it take to go a certain distance? (2) How far will I go in a certain amount of time? and (3) How fast should I go to travel a certain distance in a given amount of time? To answer any of these questions, a navigator needs to know three things: (1) the true course, (2) the boat's speed, and (3) the amount of time the boat has traveled at that course and speed. Thus, in addition to the need for a compass course, you will need a knowledge of boat speed, an accurate watch, and a chart on which to plot the information.

To measure courses, use the compass rose nearest to the chart section currently in use. Transfer course lines to and from the compass rose using parallel rulers, rolling rulers, or triangles. You can also measure direction at any convenient place on a Mercator chart because the meridians are parallel to each other, and a line making an angle with any one makes the same angle with all others. Compass roses give both true and magnetic directions. However, for consistency, and by convention, use only true directions on the chart.

"Measure distances using the latitude scales which run up and down the sides of the chart. Here, one minute of latitude equals one nautical mile."
Measure distances using the latitude scales which run up and down the sides of the chart. Here, one minute of latitude equals one nautical mile. Since the Mercator's latitude scale expands as latitude increases, measure distances on the latitude scale closest to the mid-area of the course line. On large-scale charts, such as harbor charts, use the distance scale provided in the margin. To measure long distances on small-scale charts, break the distance into a number of segments and measure each segment at its mid-latitude to insure accuracy.

Time is always recorded in military format, i.e., using four digits in a 24 hour format. This means that each day starts at midnight with a time reading of (0000) proceeds to 0100, 0200, 0300 and so on to noon (1200). After noon (1200) the time is additive, 1300, 1400, 1500, continuing up to midnight, which is 2400. Remember that 1300 is 1:00 p.m. and 0100 is 1:00 a.m.


Sailors can use the latitude scale on the chart (never the longitude scale) for measuring distance. One degree of latitude equals 60 nautical miles. One minute equals one nautical mile. The distance between the 30 and 35 above is five nautical miles.
For DR work, always use the boat's speed through the water, which is taken from the knot meter. Do not use the boat's Speed Over Ground (SOG) which you can get from the GPS or Loran. I will cover this difference in more detail in later articles.

After knowing what direction you are going, the second part of DR calculations is the use of boat speed and time. In order to calculate (reckon) where you are, you must have all three pieces of the puzzle. For example, if you are on a compass course of 170 degrees, where will you be in one hour if your boat is making six knots through the water? Using a variation of 10 west and deviation of 0 degrees, first you will convert the compass course of 170 to a true course of 160 degrees. Then you will compute the distance traveled in one hour (60 minutes) at a speed of six knots. In this case, the computation is very easy since you will travel six nautical miles in one hour at six knots.

Remember that a knot is a unit of speed equal to one nautical mile per hour. So, in this example your DR position will be plotted six nautical miles (NM) in a true direction of 160 degrees on your chart.


A handbearing compass is a useful tool to check the accuracy of where you think you should be, and where you end up--which further enables calculations of set and drift.
While this may sound complicated, it is really quite simple. In fact, you make these types of calculations mentally every time you drive your automobile. If you need to be at a place 180 miles away from your house at 1200 (noon), and you expect to average 60 mph, you almost automatically calculate that you need to depart home at 0900 (9:00 a.m.). First you deduce that it will take three hours to travel 180 miles at 60 mph, and then you subtract the three hours from 1200 to arrive at that 0900 departure time. If you have been on the road for two hours at an average speed of 60 mph, how far have you traveled? The answer is easily figured out to be 120 miles and you know that you have 60 miles left to travel on your 180 mile trip. If the distance remaining is 60 miles and you need to be there in one hour, how fast should you drive? The answer, of course, is 60 mph.

In order to simplify things I deliberately used a speed and time that would make the calculations easy. If I had said that your average speed was 49 mph, the distance was 176 miles, and you needed to be there at 1145, you would have needed some extra time to do the calculations. For this reason, you can always use a boat speed of some multiple of six knots, i.e. three knots, six knots, 12 knots or 18 knots, for planning purposes. On a sailboat, the speed will vary greatly and inevitably complicate your calculations.

Fortunately, the actual formulas to make these calculations are easy. It is 60D = ST, where D is the distance in nautical miles, S is the boat speed, and T is the time in minutes. Many navigators have been taught to remember the mnemonic as the address 60 "D" Street. By algebraic manipulation, we can rewrite the main equation to determine the answers to the three navigational questions of distance, speed, and time.

D =      S x T solves for Distance
 60
S = 60 x D solves for Speed
 T
T =  60 x Dsolves for Time
S

Here is an example for the more complex problem above where the speed was 49mph, the distance was 176 miles, and you needed to arrive at 1145. Since we are solving for time, the formula is:

T =60 x D
S

Fill in the numbers as:

T= 60 x 176  T=10,560T= 215.51 minutes,
or 3 hours, 36 minutes
 49 49

If you need to be at your destination by 1145, you will have to leave by 0809 to make it on time.


With a proper grasp of dead-reckoning fundamentals, a crew can expect a safe and enjoyable nautical adventure.
Draw a line from each DR position and extend this line from the DR or fix position in the direction of the course steered. Above the course line, place a capital C followed by the true course in degrees to the nearest whole degree (do not use tenths or fractions of a degree). Below the course line, place a capital S followed by the speed through the water in knots to the nearest knot (do not use tenths of a knot). Enclose a fix from two or more Lines Of Position (LOPs) or GPS by a small circle and label it with the time to the nearest minute. Mark a DR position with a semicircle and the time. Express the time using four digits without punctuation (military format). Use either zone time or Greenwich Mean Time (more on time in future articles). By convention label the fix time horizontally and the DR time diagonally in relation to the bottom of the chart, at that position. Make sure that you write all of these labels neatly, succinctly, and clearly so that you can read them later.

Practice your DR techniques every time you go out until it becomes second nature and you gain confidence in your navigational ability. The effort you expend on those sunny days will be returned tenfold on the foggy days. Your reward will be a safe voyage.

THE RULES OF DEAD RECKONING

Keep these fundamental rules in mind when plotting the vessel's DR position. You should plot:

1.  At least every hour on the hour.

2.  After every change of course or speed.

3.  After every fix or running fix.

4.  After plotting any line of position.



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