One night I was running a
freight up hill at 7 mph with a Dash 944CW on the point. I had previously calculated
that we should have gone up the hill at 11 mph, so why were we only doing 7 mph?
The rail was slightly frosty. I punched up the loco monitor screen on the computer.
It showed that this supposedly 4400 HP unit was only putting out 2930 HP!!! It had
derated to prevent slipping in spite of the sanders being on. So the adhesion factor
of this loco at that time was not the touted 3643% but instead only 22%. The railroad
had paid for a 4400 HP locomotive with 36% adhesion but was only getting a 2930
HP locomotive with 22% adhesion. The common SD402 would have done as good as or
better in this situation than the hitech wonder. This was not a one time occurrence.
I have seen similar performances on many occasions,
Horsepower is Speed
Up to now we have assumed
that a locomotive has enough power to slip its wheels. That is true only at low
speeds. Note that HP is TE times speed. If the speed remains the same and the TE
(pull) increases then the HP requirement increases. If the TE remains the same and
the speed increases then the HP requirement increases. If you have a fixed maximum
Hp, such as a loco has, then as speed increases the TE must come down. The
product of the two must remain a constant and is directly related to the HP rating
of the loco. On the 12,000 HP coal train above why can't we go faster than 15 mph
on the 1% grade? Because 15 mph times the required 300,000 lb of drawbar pull divided
by 550 lbft per second (the definition of a HP) equals the total locomotive HP.
If the train went faster the product of the speed times the pull would be higher
and thus the required HP would be higher. But we are limited to 12,000 hp on this
consist so it trudges along at 15 mph. Similarly if we throttle down a notch or
two, reducing the HP, then the speed is going to drop because there is now less
than 12,000 HP available. The drawbar pull account of the grade remains the same
at 300,000 lbs and the lower HP means a lower pull x speed figure so the speed must
drop until the product is proportional to the new lower HP. This why I like to say
HP is speed.
Steeper Grades
We are proceeding at 15 mph
up the 1% grade with our 15,000 ton train powered by four 4 axle GP40s. What happens
when we encounter a steeper grade with this train? The drawbar pull needed to hold
or move a train on a grade is 20 lbs per ton per grade percent. That is where the
300,000 lb figure came from for the 1% grade of the example above. If this train
were to roll onto a 1.5% grade what happens? The drawbar pull needed now is 450,000
lbs.
(15,000 tons) x (20 lbs per ton) x (1.5 grade) = 450,000 lbsWe still have only 12,000 hp available. Since HP
equals pull times speed, if the pull goes up then the speed must come down. The
train speed will drop to 10 mph, the point where the product of the new 450,000
pounds pull and the speed divided by 550 (HP definition) equals the available HP.
But we are in real serious trouble here folks. Our four 4 axle locos can only deliver
336,000 lbs of pull because of their 1.12 million lbs of weight on drivers and the
30% adhesion factor. So our locos are going to slip and stall on the hill. Although
we have enough HP to pull this train up the 1.5% grade at 10 mph we do not have
enough traction to do so. And, again, you had better set the train air brakes as
you stall or the train will drag you back down the hill.
Slugs and SDs
So how do we proceed? Well
we can't increase the weight on drivers by adding weight to our existing four 4
axle locos because they are already weighted to the max for the rail. The only other
way to increase traction, weight on drivers, is to increase the number of drivers.
Add more units. We will need to add two more units (8 more axles) to get the weight
up to at least 1.5 million lbs so the 30% adhesion factor gives us 450,000 lbs of
adhesion. We do not need the added HP of those two units however, they could
be just engineless slugs. Without additional HP the train will go up the hill at 10 mph. At
this point I would like to point out that if we simply add two 4 axle slugs, which
get the power for their traction motors through electrical cables from the original
4 locos that we can do the same thing by switching from GP models to SD models.
From 4 axle units to 6 axle units. An SD40 is simply a GP40 with two more axles (with traction motors) and 50%
more weight. In other words we've added "half a slug" to each of the 4
units. In this manner we once again have the required weight on drivers by using
just four SD40s. These are the same weight and number of drivers as 4 four axle
units plus two 4 axle slugs. If we want to go up the hill at 15 mph instead of 10
mph however, we must add the HP. Slugs or converting to SDs will not do. If the
additional units added are 3000 hp like the rest then we will again go up the hill
at 15 mph. HP is speed.
Helpers
In either case we will not
go up the hill very far, probably not at all! Why not? The figures say we will.
The one word is Kapow! You are going to break in two. We are now trying to
put 450,000 lbs of pull into a coupler rated at 390,000 lbs, it is going to break.
So what can we do? Well we could double the hill. Take half the train up to the
top and leave it there then come back with the engines and get the second half.
When you get both halves to the top, recouple them, make an air test, and proceed.
By taking half the train up the hill at a time the required coupler pull is only
half that of the entire train or only 225,000 lbs. Well under the 390,000 lb strength
of our couplers. This method also requires no more slugs, SDs, or other units. The
original 4 GP40s have enough traction to haul half the train up at a time. Unfortunately
doubling the hill requires a lot of time. The line is blocked while it is being
done and this train and others are delayed for the duration. Alternatively we could put the added two units on
the rear of the train to PUSH. We would need another engineer (a helper engineer)
or a distributed power setup (radio controlled slaves). The physics are the same,
the coal train and grade could care less where you apply the power just so you have
the right amount to move it. But the couplers do care where you put all that power.
If you try to put it all thru one coupler, the first one, it is going to say "Screw
you, I ain't gonna take this abuse" and it will break to prove its point (you
never knew couplers were so animated did you).
Some things left out
Now for you purists, I know
I have left out a few things. (If you know enough about this to know that I left
things out then you sure don't need to be reading this document).
·
As the grade gets steeper
less and less of the loco weight is felt as pressing directly down on the rails
so effective weight on drivers decreases slightly. (Do the geometry yourself if
you want).
·
I neglected the weight of
the locomotives. They don't go up hill for free.
·
I neglected the efficiency
of the locomotive's mechanical and electrical transmission.
·
I neglected rolling resistance
of the train. At low speeds such as these, on straight rail, rolling resistance
for a loaded coal train is only 1015% relative to the grade resistance. However
that will increase the total pull and HP required.
·
I neglected acceleration.
The figures given are for steady state running. To accelerate requires more pull
than steady speed.
The Weight of the Locomotive
First we'll look at the loco
weight. Four 280,000 lB GP402s weigh 1,120,000 lbs. That 1 million pounds of locos
does not go up the hill for free. It takes just as much HP to move each of those
pounds up the hill as each of the train's pounds. So you should add their weight
to the train when calculating traction, speeds, & HP required. In fact this
is one reason for 4 axle high HP locos. The more the locos weigh the more of their
HP is required to just to move the loco upgrade.
Heavy Haul vs. High Speed.
You may have noticed that
most railroads tend to use 6 axle power on heavy trains such as coal and grain while
they use 4 axle power on their high speed lighter weight intermodal trains.
 If a particular railroad
has a good mix of high speed and heavy haul trains its locomotive roster will be
a mixed bag of 4 axle and 6 axle power.
 If a railroad, such as BN, has a preponderance
of coal and grain trains and/or operates its trains in mountainous territory where
grades are steep and speeds are low its roster will be dominated by heavy 6 axle
power.
 If a railroad, such as ATSF,
has a majority of lighter weight high speed intermodal trains and a lot of relatively
flat territory its roster will reflect that with lots of high HP 4 axle power and/or
lighter weight 6 axle power.
Let’s look at the difference between 6 axle SD40s
and 4 axle GP40s. Suppose we want to run a 5200 ton priority manifest train up a
1% grade at 30 mph. This requires 8320 HP. TE is not a consideration because even
three 280,000 lb GP40s will have 252,000 lbs of adhesion. Our train only requires
104,000 lbs of adhesion on this grade. Three GP40s weigh a total of 840,000 lbs
or 420 tons. These locos require 672 HP just to move themselves up the grade at
30 mph. So our 5200 tons of train requires 8320 HP and the 420 tons of locos require
672 HP to go up this grade at 30 mph, total 8,992 HP. The three GP40s produce 9,000
HP. What happens if we use three SD40s instead of GP40s? Same HP at 3000 each but
the SDs weigh much more. Ours are ballasted for lots of TE needed on coal &
grain trains. Our SD40s weigh 420,000 lbs each. Three of them weigh 630 tons! To
move these SDs up the grade at 30 mph requires 1008 HP. This means we only have
7992 HP left for the train. That means we can only haul 4995 tons at 30 mph instead
of the original 5200 that the GP40s hauled. While this may not seem like much difference
it is over 4% and a 4% efficiency improvement is a big deal when you are burning
1.6 billion gallons of fuel per year. Trains that run at high speeds don't need
heavy locos with lots of Tractive Effort. What they need are light weight high horsepower
locos.Here is a table showing the theoretical Tractive
Effort a 3000 HP loco produces at various speeds.
Speed

Tractive
Effort

60 mph

18,707 lbs

40 mph

28,060 lbs

30 mph

37,415 lbs

25 mph

44,898 lbs

15.0 mph

75,000 lbs

13.4 mph

84,000 lbs

8.9 mph

126,000 lbs

Using a 30% adhesion factor,
a locomotive that weighs 280,000 lbs has 84,000 lbs of adhesion. From the above
chart we can see that such a loco can operate as slow as 13.4 mph without slipping.
Suppose our railroad has a large proportion of service sensitive intermodal trains
and we want to operate those trains no slower than 30 mph on our worst grades. At
30 mph a 3,000 HP loco is only capable of producing 37,415 lbs of Tractive Effort.
Therefore as long as we put enough HP on our trains to maintain 30 mph on our worst
grades we do not need to make them weigh 280,000 lbs. In fact they only need to
weigh about 125,000 lbs because 30% of 125,000 is 37500 lbs of adhesion and we only
need 37,415 lbs. We could save a lot of fuel by using these light weight locos instead
of the 280,000 lbs heavy weights. If we have four such locos on a train we are saving
310 tons of wasted weight and that translates into saved fuel.A loco that cannot use full throttle below 30 mph
without slipping would be rather restricted in its service. So railroads tend to
compromise. If instead of 280,000 lb locos or 125,000 lb locos we use a loco weighing
250,000 lbs we still get some of the fuel savings and the loco becomes much more
versatile since it can now operate as low as 15 mph in full throttle without slipping.It is in high speed freight service where locos with
high HP to weight ratios shine. This is why ATSF had 3800 HP 4 axle GP60s and 4,000
HP 4 axle B408s. Since BN had a preponderance of heavy coal & grain trains
and even its freights had to contend with steep grades it had 3,800 HP heavy SD60s
and 4,000 HP heavy SD70MACs instead of the GP60s and B408s that the ATSF had.Let’s look at this HP to loco weight ratio from another
angle. The following chart shows the maximum Tractive Effort of various loco models
and the speed at which that maximum TE is achieved. All units are 3,000 HP and we
assume a 30 % adhesion factor.
Model

Weight

Max
TE

Speed

Total
tons on 1% grade

Trailing
tons

Light GP40

250,000

75,000

15.0

3750

3625

Heavy GP40

280,000

84,000

13.5

4200

4060

Light SD40

380,000

114,000

9.8

5700

5540

Heavy SD40

420,000

126,000

8.9

6300

6090

A first glance at the table
looks as if the heavy SD40 is the best loco. It can pull the most trailing tonnage
up the 1% grade. But we have to ask ourselves "What is the job?” If the job
is to haul as much tonnage up the grade as is possible, then indeed the heavy SD40
is the loco we want. But if the job is to haul as much trailing tonnage up the grade
at 15 mph then the SD40 is not the best choice. The following chart shows
how much tonnage each of these locos can haul up the 1% grade at 15 mph. Since all
the locos are 3,000 HP they all produce the same 75,000 lbs of TE at 15 mph. But
the weight of the loco uses up some of that TE. What is left over can pull the freight
that is paying the bills.
Model

Weight

TE

Speed

Total
tons on 1% grade

Trailing
tons

Light GP40

250,000

75,000

15

3750

3625

Heavy GP40

280,000

75,000

15

3750

3610

Light SD40

380,000

75,000

15

3750

3590

Heavy SD40

420,000

75,000

15

3750

3540

The light weight GP40s can
haul 85 more tons of paying freight per unit up the grade at 15 mph than the heavy
SD40 can.
Those Superpower units
If you haven't been paying
attention you might think that the new 6000 HP single unit locos are destined for
heavy haul service. True they are all heavy 6 axle units. But that is because the
weight is needed to put that 6,000 HP to the rail without slipping. A 6,000 HP unit
that weighs 420,000 lbs and can attain a 43% adhesion factor has an adhesion of
180,600 lbs. The 6,000 HP diesel engine can deliver that 180,600 lbs of Tractive
Effort at a speed of 13 mph. Below that speed you cannot use full throttle on these
locos because they will slip. That was for an astounding adhesion factor of 43%.
What if they cannot maintain that extreme level of adhesion? What if they "only"
get 36%? 36% of 420,000 lbs is 151,200 lbs of TE. The 6000 hp diesel can deliver
that TE at 15 mph so the loco cannot operate below 15 mph in full throttle without
slipping. At an adhesion factor of 30% the lowest full throttle speed is 18 mph.
If the rail is wet or frosty can these modern marvels maintain even a 30% adhesion
factor? My experience with 4400 HP units is a definite no. The C44s often have trouble
maintaining 22% adhesion with bad rail conditions. If a 6,000 HP unit gets down
to 22% adhesion it can only operate at full throttle above 24 mph! Thus if you want
these behemoths to reliably move your trains over the hills in all kinds of weather
you had better dispatch them with trains light enough that they can maintain 24
mph or greater on your steepest hills. That means they are only useful for trains
such as intermodals which get a high HP to tonnage ratio. When it is frosty they
won't work on heavy freights or coal or grain trains which routinely pull up the
hills at 1012 mph.
The railroad I work for uses 12,000 HP on their coal
trains through here and we go up the hills at about 1213 mph. Note that you can
replace the 12,000 HP of 3 SD70MACs, or the 12,000 HP of 4 SD402s, with the 12,000
HP of just two SD90s. You have the same HP so you should go up the hills at the
same 1213 mph. But it will be awfully iffy. That is because the minimum speed these
6,000 HP units can operate at full throttle is 13 mph even with an adhesion factor
of 43%. If anything causes the train speed to fall below 13 mph even momentarily,
you will never regain the lost speed. The train might be temporarily slowed for
various reasons. Perhaps the SD90s temporarily lost that 43% adhesion factor and
slipped or reduced HP to prevent slipping. Perhaps a wind came up and increased
train resistance. At 12 mph the 6,000 HP locos cannot operate in full throttle even
if they regain that 43% factor of adhesion. They will slip. Operating at reduced
throttle the locos are not producing the 12,000 HP this train needs to travel up
the hill at 13 mph. So the train will never accelerate back up to 13 mph where it
could again operate at full throttle. Four SD40s or 3 SD70MACs would have no difficulty
reaccelerating the train back up to 13 mph.
That is because they are not operating
at the limit of their adhesion as the SD90s are. The 4 SD40s have 12,000 HP just
like the two SD90s but the SD40s have a total weight of 1,680,000 lbs and even at
a 30% factor of adhesion can operate in full throttle down to 9mph! The 3 SD70MACs
weigh 1,260,000 lbs and with only a 30% factor adhesion they can operate at full
throttle down to 11.9 mph. If they achieve a 36% factor of adhesion they can operate
at full throttle down to 9.9 mph. So either the SD40s or the SD70s have enough reserve
adhesion they can operate at full throttle after being temporarily slowed. That
allows them to accelerate the train back up to the 13 mph.Thus on an equal total HP basis these high HP units
are not equal to their lower HP cousins when used in heavy haul service. And heaven
help you (more like helpers help you) if the factor of adhesion on these brutes
ever falls below 36% because you won't have enough adhesion to pull that 15,000
ton train up that 1% grade, period. You had better hope that it does not rain, frost,
or snow.Keep the high HP units in high speed freight service
where they do the most good. You are trading 8 axles of weight on two 3000 HP GP40s
or 12 axles of weight on two SD40s for the 6 axles of the new units and you have
2550% less Hpwasting weight with the two high HP units. Remember that TE decreases
as speed increases, so as long as they keep the HP per ton ratio of the trains high
enough to maintain high speeds then the TE will be low enough that these high HP
single units won't slip. But try to use them in low speed drag service and they
will slip as noted in the coal train discussion above. The slower the train goes
up a hill the closer these high powered 6,000 HP wonders perform like the good old
3,000 HP SD40.
The Efficiency of the Locomotive
Next we'll look at the efficiency of the locomotives transmission. Their
transmission consists of the generator, traction motors, and gearing. My experience
is that the loco's transmission efficiency normally runs in the 80% range. This
means that if the physics of the train, grade, and speed dictate X HP then you really
need X / .80 HP. If the physics say 12,000 HP then you really need 12,000 /.80 which
is 15,000 HP. Another full unit!
Put another way....The 15,000 ton coal train going up a 1% grade at 10 mph requires
9564 HP. That is 8442 HP for the speed up that grade and 1122 HP for the rolling
resistance at that speed. (We'll get to rolling resistance in a minute). But that
assumes 100% efficiency. At 80% efficiency this train would need 9564 HP / .80 which
is 11,995 HP. SURPRISE! That is three 4,000HP SD70MACs or four 3,000HP SD402s to
get a 15,000 ton coal train up a 1% grade at 10 mph. Sound familiar?
The Rolling Resistance
of the Train
Now we'll look at rolling
resistance. Assume the same train as above, i.e., 15,000 tons plus 840 tons of locos
(4 SD402s) rolling at 10 mph on a 1.0% grade. Using the well known Davis formula
we get the following values: (note: the Davis formula )is explained at the end of this article)
Resistance

Pull

HP

Grade

316,800 LBs

8447 HP

Rolling

41,880 LBs

1116 HP

Total

358,680 LBs

9563 HP

The calculated HP required is 9563 HP. Since our
locos are only about 80% efficient this means we need a HP rating of 12,000 to actually
deliver the required 9563 HP.Put the train on a 2000 ft long 3 degree curve and
you get:
Resistance

Pull

HP

Grade

316,800 LBs

8447 HP

Rolling

41,880 LBs

1116 HP

Curve

15,840 LBs

422 HP

Total

374,520 LBs

9985 HP

Using a loco efficiency of 80% the required 9985
HP becomes 12,481 HP. The 12,000 HP of four SD40s is not going to be able to pull
this train up the hill and around the curve at 10 mph. The speed will drop until
the rolling resistance and grade HP drops enough that the actual HP required equals
80% of 12,000 HP (9600 Hp). That speed is 9.6 mph.
At 9.6 mph we get the following values:
Resistance

Pull

HP

Grade

316,800 LBs

8109 HP

Rolling

41,557 LBs

1063 HP

Curve

15,840 LBs

405 HP

Total

374,197 LBs

9577 HP

Acceleration
Let’s look at accelerating
trains. The force of acceleration is mass times acceleration. Force = mass x acceleration (F = m * a).
A coal train is a very big mass!
So even small acceleration s need a lot of force. That force adds to the drawbar
pull account of the grade alone and it can break the train in two. If you keep the
acceleration low by notching out one notch at a time and allowing speed to increase
slowly you can minimize the force of acceleration. If you are reckless and try to
accelerate quickly you may end up in two pieces.
Let’s say we want to accelerate a 15,000 ton train
at a rate of 1 mph per minute. In other words we want to be going 1 mph faster at
the end of one minute than we are going now. To accelerate at that rate requires
a steady force of 23,450 lbs. Note that it doesn't matter whether we are going uphill
or on the level. We need to supply an additional 23,450 LBs of drawbar pull to accelerate
at 1 mph per minute. Horsepower is pull times speed. Since the force to accelerate
this train at 1mph/min is a constant, the HP required to accelerate the train varies
according to speed. At 10 mph the HP needed is 625 HP, at 40 mph the HP needed is
2500 HP.
An acceleration of 1 mph/minute is slow. It would
take a train 60 minutes to go from 0 to 60 mph. But if we want to accelerate at
10 mph per minute it requires 10 times the force and 10 times the HP at each speed.
At an acceleration rate of 10 mph per minute the drawbar force needed is 234,520
lbs. At 10 mph that requires 6,250 HP. At 20 mph it requires 12,500 HP. At 40 mph
it requires a whopping 25,000 HP. Note that if we have only 12,000 HP then we run
out of HP before we reach 20 mph. We can no longer accelerate at 10 mph per minute
and will fall back to lower and lower acceleration rate as speed increases.
Keep in mind that these values of drawbar pull and
HP are ONLY for acceleration. You still need to supply the normal pull and HP for
any grade and rolling resistance. Let’s look at that. A 15,000 ton train on a 1%
grade going 8 mph requires 357,104 lbs of pull and 7617 HP. If we have 4 SD402s
we have 12,000 HP times 80% efficiency = 9600 HP available. 7620 HP is what you
get in throttle 7. We have one more throttle notch and 1980 HP (96007620) remaining
that we can use for acceleration. At the stated 8 mph that equates to 97,345 lbs
of additional drawbar pull available. This additional force will accelerate the
train at a rate of almost 4 mph per minute. Yeehaw! Put'em in number 8 throttle
and we'll be doing 12 mph at the end of the next minute.
Well not quite. A few problems crop up with that
assumption. One is that as the train speed increases so does the horsepower required
for both the grade and the rolling resistance. So as the speed begins to increase
we have less "left over HP" for acceleration. The rate of acceleration
will drop; we cannot maintain that 4 mph / minute rate we started with. In fact
when we reach 10 mph all of the loco's HP is being used to pull the train and none
is left over for acceleration. The speed will level out at 10 mph and stay there.
Ain't physics neat?
The second problem is that you just broke the train
in two so you are actually stopped. Why? Because the train traveling at 8 mph required
357,104 lbs of drawbar pull to maintain that speed on this grade. When you opened
the throttle from notch 7 to notch 8 to accelerate you just put the additional 92,602
lbs of available loco tractive effort into that same drawbar. 357,104 lbs + 92,602
lbs equals a total of 449,706 lbs. Since drawbars are only good for about 390,000
lbs you just pulled one in two. Moral: When you are moving slowly you'd better handle
that throttle gently if you want the train to remain in one piece. Acceleration
can break trains in two.
Now for the purists, those 4 SD402s are not going
to develop that 454,449 lbs of TE. That would mean an adhesion factor of 27% and
SD402s can rarely if ever achieve that. They would most likely slip. But they can
develop the 390,000 lb rating of the drawbars either continuously or by slipping
and jerking. So either way the train is going to be in two pieces.
Note that the above train theoretically can go up
this hill at 10 mph based upon Hp, efficiency, grade, and drawbar strength. Whether
it actually can or not is in doubt. If we rolled onto this hill at a speed higher
than 10 mph then all would be OK. As the train rolled onto the grade in number 8
throttle it would simply slow down to 10 mph and proceed up the hill. The drawbar
force would be that 357,104 lb figure. Well within the rating of the drawbars. But
if the train had stopped on this grade or had started from a stop on a lesser grade
and was not yet up to 10 mph then we may be in trouble. Under these circumstances
we may find ourselves in the situation above where we are only going 8 mph when
the entire train is on the hill. We cannot go from notch 7 to notch 8 because the
drawbar force will exceed their rating. Thus we cannot get to 10 mph. The only recourse
is to slug it out all the way up the hill in the lower throttle position and a lower
speed. It is very annoying to know you have the HP to go faster but you can't use
it. If you are a real good engineer and really know what you are doing you can get
around this obstacle in some cases. How? By applying some independent brakes to
the locomotives drivers. Those brakes will absorb some of the extra HP you get when
you go from #7 to #8. Therefore that amount of the extra HP and its attendant TE
never reaches the train's drawbars. In that manner you can keep the total drawbar
force lower than the drawbar rating. As the speed increases you feather off more
and more of the independent brake until finally you are at the 10 mph physical limit
and the brake is fully released. But make one mistake during that process; fail
to coordinate the independent brake just right with the increasing HP as the locos
rev up and increase their load....or feather it off too quickly.......and Kapow!
You are in two pieces. You have let enough extra TE reach the drawbars that their
rating was exceeded. Going 10 mph in #8 vs. 8 mph in #7 saves you 15 minutes on
the hour. But if you break it in two attempting to reach 10 mph then you are delayed
2 hours while you chain up a car and set it out and double the hill. If you are
not sure of your expertise maybe it is better to just go up the hill at 8 mph in
number 7 instead of trying for 10 mph in number 8.
The main point of all this is to hopefully dispel
the myth that high HP means lots of pull. It does not. Higher HP means higher pull
at higher speeds but the total maximum pull is strictly related to weight on drivers.
No HP required. None! Therefore a switch engine which only operates at low speeds
does not need, nor can it use, high HP. It needs to be heavy. (but not too heavy
that it breaks or turns over light industrial or yard rails). Life is a compromise.
The Davis Formula
Substantial research early in the 20th century led to the
development of a general formula for train resistance. Developed by W.J. Davis, it is still sometimes referred to as the
“Davis” equation.
 Ro = resistance in lbs. per ton
 w = weight per axle (= W/n)
 W = weight of car
 b = an experimental friction coefficient for flanges, shock, etc.
 A = crosssectional area of vehicle
 C = drag coefficient based on the shape of the front of the train and
other features affecting air turbulence etc.
 V = velocity
 n = number of axles
The Davis Equation has been substantially updated to reflect
modern developments, but its basic form is the same as the original equation.
Definitions
(as used in this document)
·
Axle  Two
wheels and an axle with a traction motor geared to it. All "axles" are
powered, there are no idler axles.
·
GP40  A 4
axle locomotive of 3,000 HP that weighs 280,000 lbs.
·
SD40  A 6
axle locomotive of 3,000 HP that weighs 420,000 lbs.
·
Slug  A 4
axle unit that has traction motors but no diesel engine. Its traction motors get
their electrical power from adjacent units. A concrete weight ballasts the slug
to 280,000 lbs.
·
C44 A 6 axle
locomotive of 4,400 HP that weighs 420,000 lbs. Actual model designation is Dash
944CW.
·
SD70MAC 
A 6 axle locomotive of 4,000 HP that weighs 420,000 lbs and has AC traction motors.
·
SD90  A 6
axle locomotive of 6,000 HP that weighs 420,000 lbs and has AC traction motors.
·
Grade
Pull or Grade Resistance
 The force required on a grade to prevent a train from rolling back down the hill.
It is expressed as 20 lbs per ton per percent of grade.
·
Adhesion 
The ability of the steel wheels of a locomotive to "stick" to the steel
rails to prevent spinning or sliding of the wheels. The amount of force required
to slide the wheels of a locomotive.
·
Adhesion
Factor  The ratio of the adhesion
to the weight of a locomotive. A good ballpark figure for steel wheels on steel
rails is 30%. IE, it requires a force equal to 30% of the loco's weight to slide
its wheels.
·
Tractive Effort
 The pull developed by a locomotive. The maximum tractive effort value is directly
proportional to the weight on drivers and the adhesion.
·
Horsepower
 Any combination of pull and speed that equals 550 lbft per second. Examples:
Pulling with a force of 1 pound for 550 feet and accomplishing that in one second.
Pulling with a force of 550 lbs for 1 foot and accomplishing that in 1 second. Pulling
with a force of 225 lbs for 2 feet and accomplishing it in 1 second. Etc.
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