TRACTIVE EFFORT AND TRAIN RESISTANCE

• We will be begin by discussing Tractive Effort or Tractive Force, which is the amount of force at the wheels available for moving a train.

1. Energy Utilization

• To move any object, a force must be applied to overcome the resistance or inertia of the resting object. The object remains at rest until sufficient force is applied. Once movement occurs, the object will continue to move so long as the applied force remains greater than the resisting forces reacting against the force moving the object.

• Energy Efficiency is a measure of how well energy in forms such as diesel fuel is converted into useful work.

• Trains require are very energy efficient moving loads over level ground, but the need significantly more energy to go uphill, even on even a slight grade. At 15 mph, the extra energy required to for a train to climb uphill to an elevation of 200 feet, would move the same train a distance of about 21 miles at the same speed if it were on level ground.

• Further, a train powered at 1.5 horsepower per ton, which could travel at a speed of 60 mph on level track, will slow to about 22 mph on a 1% grade, and slow to 10 mph on a 2% grade. The same train will eventually stall to a stop as the grade gets steeper.

• Railways actually seldom use much more than 0.5 hp per ton to move their heavy trains, though.

• For example, a 100-car unit train of coal weighing 143 tons per car are often powered with only two locomotives producing 3600 horsepower each.

2. Train Resistance

• For our purposes, resistance is simply the combination of forces work against a train’s movement.

• We describe the combined resistant force against movement in terms of “pounds of applied force required to move 1 ton of weight”.

• The primary resistant force that trains must overcome is friction.

• In the diagram, once enough weight “w” is applied to overcome the friction, the car will move.

• The set-up shown was once the primary way to calculate actual train resistance.

• Note that this simple set-up neglects the effect of speed, and that resistance increases significantly as the speed increases.

3. Horsepower Tractive Effort

• Horsepower is a measure of the rate of doing work.

• One horsepower = 550 pounds at the rate of one foot per second or 375 pound-miles per hour.

• When a train is at rest, its horsepower is zero.

• The maximum horsepower of most diesel engines is developed between 800 and 1000 rpm.

• In a diesel-electric locomotive, part of the horsepower produced powers the cooling fans, blowers, air brake compressor, etc.

• After that, the available crankshaft horsepower is converted to electricity to power the traction motors driving the wheel axles.

• With the modern diesel electric locomotives, normally 82%-85% of the diesel horsepower is available for traction.

• The tractive effort (in pounds-force) available from a locomotive can be roughly calculated as:

• Tractive Effort (pounds-force) = Horsepower X 308 / Speed (mph)

• The “308” is 82% of 375 pound-miles per hour per horsepower.

• For example, a 3000 hp locomotive will have approximately 74,000 pounds-force tractive effort at 12.5 mile-per-hour or an AC 6,000 horsepower locomotive with over 150,000 pounds-force tractive effort at 12.5 miles per hour.

4. Traction Forces

• The locomotive produces a force, which acting through the drawbar pulls the train attached to it.

• While a locomotive produces “draw-bar pull”, its wheels exert a force against the rails at their point of contact with the rail

• Thus the mechanism of the locomotive’s working, both in traction and braking, is accomplished through adhesion between wheel and rail.

• There is a limit beyond which the tractive or braking force cannot be increased.

• If “TE” is the maximum Tractive Effort that can be applied without causing the wheel to slip, “W” is the effective weight carried by the wheel, and μ (mu) is the adhesion. TE= μ x W

• Similarly during braking, the coefficient of rolling friction (adhesion) is the ratio of maximum braking effort that can be applied on the rolling wheel without causing the wheels to lock-up and slide.

• With braking, failure of adhesion is far more serious for safety reasons. Hence a much lower value of COA (μ) is adopted to reduce the chance of failure. In conventional braking, every vehicle is equipped with brakes; thus weight on all axles comes into play.

• With locomotive dynamic braking, (the equivalent of a truck "Jake Brake") only the weight of the locomotive comes in to play, thereby stretching the ratio between braking effort and weight.

• Hence very careful management of adhesion is even more important.

• In most urban mass transit vehicles, the production of tractive as well as braking effort is distributed among a large number of axles.

• Remember slipping wheels is the failure result of both traction and braking efforts.

5. Adhesion

• Adhesion is the tractive force at the locomotive driving wheels (drivers) at the rail that starts and moves tonnage.

• The maximum tractive force that can be developed at the rail is equal to the weight on driving wheels, multiplied by the adhesion of the wheels on the rail.

• The biggest factors, among others, affecting adhesion are rail condition and speed.

• Adhesion decreases as the speed increases.

• At about 10 mph, adhesion varies from less than 10% on slimy, wet rail to about 40% on dry, sanded rail.

• In general, with the aid of the sanders, approximately 25% adhesion is usually available.

• As all the wheels on most diesel locomotives are driving wheels, the weight of the locomotives must be about four times the tractive force to be developed.

• Most High-horsepower Direct Current units used for mainline service weigh about 195 tons and have 6 axles. The maximum tractive effort is therefore approximately 97,000 lb. per locomotive or 16,000 lb. per axle.

• New Alternating Current locomotives weigh significantly more and develop higher tractive forces.

6. Factors Affecting Adhesion

• Many factors affect Adhesion

• Adhesion is affected by Wheel-Rail Contact factors, Track conditions, and Vehicle factors.

• Wheel/Rail Factors

• The Coefficient of Adhesion (μ) of steel on steel is generally rated as being between 0.3 - 0.4; but in reality is often much lower.

• Hertzian Stresses regarding two elastic bodies in contact with each other depend on the ratio of load carried by the wheel vs. its radius.

• The phenomenon of “Rail Creep” and its utilization has opened new areas for improving adhesion.

• Rail Creep is longitudinal sliding movement of rails in track under traffic or due to temperature changes. The effect of rail creeping is resisted by rail anchors.

• Track Factors

• The Coefficient of Adhesion is higher for well-maintained tracks such as mainline tracks compared to yard tracks.

• The surface condition of the rail is perhaps the most important factor in causing this. Unfortunately the rail surface is rarely clean. Adhesion is adversely affected by

presence of contaminants such as rust, water, and oil. Also metal corrugations and uneven rail wear can reduce adhesion.

• Tangent track also has better cohesion than curved track.

• Vehicle Factors

• The strongest influence on adhesion is mechanical in nature and comes from weight transfer from the train cars to the rails.

• As the speed of the train increases, the adhesion decreases.

• Differently-sized wheels rotate at different rates, and for trains with wheels of different diameters the adhesion is less.

• Also, today’s smarter locomotives can determine when wheels lose adhesion and slip and can adjust power accordingly to regain adhesion

7. Train Resistance

• Train resistance, the force required to move a train, may be categorized into two elements:

• Rolling resistance

• Grade and curve resistance

• Rolling resistance is friction related, including the resistance of the wheels to rolling, friction in the axle bearings, and wind resistance.

• Rolling resistance varies with the weight of the vehicle and increases as the speed increases – at 30 mph, it is about 4 lb/ton.

• Empties, particularly gondolas, have significantly higher rolling resistance than loads – at 30 mph, the rolling resistance for empty gondola car is about 7 lb/ton.

• Grade and curve resistance is related to track alignment and elevation changes. It is affected by the weight of the train but it is independent of speed.

8. Rolling Resistance

• Rolling Resistance is the net of the forces that must be overcome by the tractive effort of the locomotive in order to move a train on level tangent track, in still air, at a constant speed.

• These resistive forces include:

• Rolling friction between wheels and rail that depends mainly on the quality of track.

• Bearing resistance which varies with the weight on each axle and, at low speed, the type, design and lubrication of the bearing.

• Train dynamic forces that include the effects of friction and impact between the wheel flanges against the gauge side of the rail and those due to sway, concussion, buffing and slack-action.

• The rail-flange forces vary with speed and quality of the wheel tread and rail as well as the tracking effect of the trucks.

• Air resistance that varies directly with the cross-sectional area, length and shape of the vehicle and the square of its speed.

• In general, rolling resistance of a train, R (in lb.), can be calculated using the Davis Formula which has the following general form:

• R = A + B V + C D V2

• where A, B, C & D are coefficients defining the different resistive forces that are either independent, dependent or affected by the square of the train speed V.

• For standard gage, 4’-8 1/2” railroad, the formula is listed in AREMA manual Volume 4, Chapter 16 is found in Section 2.1.2.

9. Freight Train Speed V Resistance

• The chart depicted above provides a relative indication of how each of the coefficients of the Davis Equation are affected by speed.

• At low speeds, bearing resistance (A) Note: the A variable equals 1.3 + 29/w dominates but remains constant independent of speed.

• Flange resistance (B) increases somewhat linearly as speed increases.

• Air resistance is relatively insignificant at low speeds, but becomes a major factor once speeds of 30 mph or better are reached.

• Typical values of (B) range from 0.03 for locomotives to 0.045 for freight cars and 0.03 for passenger cars. B = b X V

• The air resistance value (C) in the chart above is actually (c X A) where the typical Drag Coefficient values range from 0.0024 for locomotives to 0.0017 for freight cars (streamlining effect) to 0.00034 for passenger cars.

• Cross-sectional area, ranges from 120 ft2 for locomotives and passenger cars to 85 – 90 ft2 for freight cars.

• Testing has indicated that the drag factor should be reduced 15% for conventional freight equipment and multiplied by 1.05 for Trailers-on-Flat-Cars, 1.20 for empty covered auto racks, 1.3 for loaded auto racks and 1.9 for empty uncovered auto racks.

10. Davis Equation Modifications

• The Davis Equation has been substantially updated to reflect modern developments, but its basic form is the same.

• Ro = 1.5 + 18N/W + 0.03V + CaV2/10,000W (CN Equation 1990)

where:

Ro = resistance in lbs. per ton

N = Number of Axles

W = Total Weight in Tons of Locomotive or Cars

V = Velocity of Train (MPH)

a = Cross-sectional Area of Vehicle in ft2

C = Canadian National Streamlining Coefficient

See AREMA Manual for Railway Engineering Chapter 16 - 2.1.3 for latest version of formula

The original train resistance formula still uses the same variables, but over the years, different coefficients have been developed to reflect changes such as higher speeds, more modern equipment and today's truck and track design.

In 1990, the Canadian National Railway developed the most recent version of the Davis Formula presented above.

When used with the coefficients shown, many of which were developed in dynamometer car tests, the formula has given reliable results in train performance calculator programs or similar applications.

The AREMA Manual For Railway Engineering, provides tables containing recommended coefficient “C” values, as well as cross-sectional frontal areas for various pieces of equipment.

11. Resistance Versus Speed

• Train resistance is calculated by multiplying the resistance per ton at each speed, by the total tonnage of the train.

• In the case above, a 10,000 ton coal train of 100 loaded 100-ton hoppers represents about 12,000 pounds of resistant force at 10 mph but jumps to 38,000 pounds for the same train at 60 mph.

• The above chart is derived on tangent track at 0% grade in still air – not very representative of real world conditions.

• The Davis Equation does not consider the gravitational effects on a train ascending a grade (added resistance) or descending a grade (helpful resistance).

• It further neglects the effect of curvature as well. The binding forces of the flange of the wheel riding up against the gage corner of the rail are significant and increase dramatically as the curvature increases.

12. Grade Resistance

• Railroad grades in the US and Canada are typically measured in percent (feet of rise per horizontal 100 feet of run).

• Our European friends measure grade as a slope ie., a rise of 1 meter in 40 meters, etc. Gradient then is the rate of grade in percent.

• Grade Resistance is the force required to overcome gradient and is equal to 20 lb. per ton per percent grade.

• This force is derived from the resolution of force vectors and is independent of train speed.

• Going uphill produces a resistive force while a downgrade produces an accelerating (negative resistive) force.

• A train moving up a long tangent of 1% grade at 10 mph, a speed that most tonnage trains slow down to at ruling grade locations, will have a train resistance coefficient of 22.4 to 23.5 lb. per ton with the grade resistance accounting for over 85% of the total.

• After a portion of the tractive force is used to overcome the grade resistance, the excess is used to accelerate the train. Rolling resistance increases as the speed increases.

• As the excess tractive force decreases, the accelerating rate drops.

• For a train operating on a long stretch of consistent grade, there is an equilibrium point when the total tractive force is equal to the total train resistance.

• At this point or speed, the train will accelerate no more.

• This is the “balance speed” (or top speed) of the particular train on that particular grade.

• If the grade resistance increases after the balance speed is reached, the train will slow down.

• If the grade keeps on increasing, the train will slow to a speed that the engine can not sustain and it will stall and stop.

• On any particular section of a railway, there is a point at which the combined grade resistance makes the train pull hardest and, therefore, rules how much tonnage can be hauled by a locomotive.

• This is the “Ruling Grade” for that railway section that the designer must not exceed.

13. Curve Resistance

• Curve Resistance is an estimate of the added resistance a locomotive or car must overcome when operating through a horizontal curve.

• The exact details of the mechanics contributing to curve resistance are not easy to define.

• It is generally accepted in the railway industry, that curve resistance is approximately the same as a 0.04% up grade per degree of curvature (which equals 0.8 lb. per ton per degree of curvature) for standard gauge tracks.

• In other words, for a 100 ton car on a 2 degree curve a resistance of 160 lb-f is produced. At very slow speeds, say 1 or 2 mph, the curve resistance is closer to 1.0 lb. (or 0.05% up grade) per ton per degree of curve.

• Curve resistance is substantially contributed to by the flanging effect or flange contact to rail of the outer leading wheel of a truck and the slip/slide over the railhead during curving of the two wheels on the same fixed axle.

• The sharper the curvature the higher the resistance of the portion of the train that is on the curve.

• When gradients and curves are involved the resistance for the portion of the train length on the curve or gradient is added to the mentioned train resistance on level tangent track to produce a total train resistance.

14. Compensated Grade

• It is a common practice to describe curvature and grade together as compensated grade.

• Compensated Grade is the algebraic total of the track gradient and the equivalent grade of the curve or Gc = G + Dc * 0.04

• Where Gc = compensated grade in %

• G = track gradient in %

• Dc = degree of curvature in decimal form

• The track gradient “G” is positive for up grade and negative for down grade.

• The equivalent grade of a curve is always positive; i.e., at +0.04% per degree of curve with tangent tracks as 0.00%.

• The combined resistance due to track geometry can thus be calculated by converting the compensated grade at 20 lb. per ton per percent grade as shown below.

• The total train resistance that must be overcome by the locomotive on a given grade with given curvature is then Train Resistance (lb/ton) + Compensated Grade Resistance (lbs/ton)

15. Typical Freight Consist

• As all the wheels on most diesel locomotives are driving wheels, the weight of the locomotives must be about four times the tractive force developed.

• High-horsepower units for mainline service, weigh about 195 tons each, and have 6 axles.

• The maximum tractive force is therefore approximately 97,000 lb. per locomotive or 16,000 lb. per axle.

• After some of the tractive effort is used to move the locomotive itself, the remaining effort, in the form of “drawbar pull”, is used to move the rest of the train.

• As the train speed increases, the tractive effort from the locomotives decreases and the drawbar pull available to move the train also decreases.

• Due to the limited strength of drawbars and coupler knuckles, the number of locomotives or motorized axles that can be used in the head end of a train is restricted.

• Although rated with a minimum strength of 350,000 lb. (general service coupler made of Grade B steel), coupler knuckle failure may happen at 250,000 lb. due to age and wear.

• Grade E knuckles used on some captive services may have an ultimate strength of 650,000 lb.

• To avoid the risk of drawbar failure enroute, railroads generally limit the number of motorized axles in a locomotive consist to 18 (three 6-axle units).

• If more tractive effort is required to move the tonnage of a train, pusher locomotives at the rear of the train or remote controlled locomotives in the middle of the train are used.

16. Speed/Tractive Effort Curve

• Shown here is a “Speed and Tractive Effort Curve”.

• As you can see, at low speeds, the tractive effort is limited by wheel adhesion to the rails, not by locomotive power.

17. Maximum Possible Train Speed?

• Available tractive effort is typically not the controlling criteria for locomotives operating at low speeds.

• Adhesion (coefficient of friction X Weight on axles) must be large enough to overcome the train

resistance, else wheel spin will occur.

• As speeds begins to increase, the available tractive effort also drops.

• The main interest is in the amount of tractive effort needed to overcome the total train resistance for the speed at which we wish trains to operate over a given territory.

• Let’s examine the chart.

• Train resistance and tractive effort are both measurements of force (typically in pounds in North America) so we can simply graph both tractive effort and resistance curves.

• The blue line represents the available tractive effort at any given speed for this locomotive consist.

• The red line is the total train resistance calculated in lb./ton multiplied by the total train tonnage.

• The maximum speed of this consist is the point at which total train resistance equals available tractive effort (approximately 38,000 pounds or in this case at 58 mph – this is called the “balance speed”.

• On any particular section of railway, the ruling grade determines how much tonnage can be hauled by a train.

• This is the particular point on the section at which the combined grade and curve resistance makes the train pull hardest and, therefore, rules how much tonnage can be hauled by a locomotive consist.

• It is not at the same location for both directions, and may not be the same location for all trains.

• So when considering balancing speed, you must use the total resistance produced at the ruling grade, as the controlling criteria.

18. Resistance vs Acceleration

• It takes about 100 lb. force to accelerate a mass of 1 ton at the rate of 1 mph per second.

• The total tractive force, "F" (lb.) required to accelerate a train of "W" tons (locomotive and cars) at the rate of "A" mph per sec. can thus be calculated approximately as:

F (lb.) = 100 W (ton) A (mph/sec)

• After a portion of the drawbar pull is used to overcome the train resistance, the excess is used to accelerate the train.

• Rolling resistance for a train increases as the speed increases.

• At the same time, the tractive effort of the locomotive decreases as the speed increases.

• As the available drawbar force decreases, the accelerating rate drops.

• For a train operating on a long stretch of consistent grade, there is an equilibrium point when the total drawbar pull is equal to the total train resistance.

• At this point or speed, the train will accelerate no more – This is the “balance speed” of the particular train on that particular grade.

• If the grade resistance increases after the balance speed is reached, the train will slow down to another balance speed for the increased grade.

• If the grade keeps on increasing, the train will slow to a speed that the locomotive cannot sustain and will stall.

• At any given speed that the train is to maintain, there is a maximum tonnage that a locomotive can pull up a specified grade.

• This is the tonnage rating of the locomotive for the specified grade.

19. What If We Need More Power

• Our previous example is based on the available tractive effort produced by one locomotive.

• To increase the total tractive effort we add additional locomotives to the train.

• Nearly all modern locomotives are capable of “multiple unit” control, which means the control circuits are electrically connected and a single operator controls multiple locomotives.

• Doing so allows power to be matched to the requirements of any size train to maximize either efficiency or speed.

• This “Distributed power” is particularly useful for heavy freight trains because it enables longer trains and better control by pushing a train, particularly on grades.

• It reduces the potential for coupler or drawbar failure due to tractive effort produced only at the front of the train from exceeding the ultimate strength of the drawbars.

• Most railways publish “Tonnage Ratings” for their locomotive fleet, which indicate the maximum tonnage that a specific locomotive can haul over a given territory, at a minimum speed.

• No single rating can be used for assigning maximum tonnage where the number of cars and their weights vary from train to train.

• So a system has been developed and it is used on most railways, that makes it possible to express tonnage ratings without regard to the weight of the cars in a train.

20. Freight Time/Speed Graph

• The successful operation of a railway is based on getting trains over a given territory within an acceptable time.

• Railroad-customer service contracts are often heavily penalized for late delivery.

• A slow train occupying a segment of track ultimately affects higher priority trains as well, particularly if the track is timetabled for a higher speed.

• Trains other than high-priority container trains rarely run consistently at time-tabled speeds.

• Stops for red signals, meets with other trains and a variety of other factors may mean frequent stops or speed restrictions.

• It may not be enough to ensure that the locomotive consist can provide sufficient tractive effort to simply keep the train moving.

• The ability to accelerate to a given speed within a reasonable time period is also important.

• This single locomotive requires 15 minutes to reach 40 MPH – on many railway subdivisions this would cause unacceptable delays to all traffic.

• However, if the maximum timetabled speed is 30 MPH, 8 minutes may not be unacceptable.

21. Time/Speed Graph

• The addition of a second locomotive dramatically reduces the time required to reach 40 MPH.

• It is the responsibility of the Operations Department to assign the number locomotives necessary to a train with a given consist so that that it has adequate capability to accelerate and maintain speeds needed.

22. Freight Distance/Speed Graph

• Curves are also produced for speed distance relationships.

• Speed and distance are crucial for momentum grades.

• The ruling grade may not be the steepest grade on the territory.

• A short grade does not affect the whole train length at the same time.

• A short incline may be run as a momentum grade, if conditions are such that trains can get a good run for the hill.

• If the velocity head of the train at the foot of the grade is higher than the actual rise, the incline is a momentum grade.

• Velocity head, “H” is described in “feet” and it can be calculated as:

• H = 0.03 times V-squared

• where V = train speed in mph at foot of grade

• Conversely, if the velocity head, h, is less than the actual rise in feet, the grade is considered as a ruling grade.

• The effects of train length must be considered in the above calculation to ensure a good portion of the train is over the hill when the velocity head is depleted.

• In the above graph, one determines the distance required to achieve the desired speed in order to ensure sufficient velocity head is available to crest and bring the entire train up the momentum grade.

23. 2 Unit Distance/Speed Graph

• In this graph, the addition of a second locomotive dramatically reduced the distance required to obtain the desired speed.

24. Horsepower: Trailing Tonnage

• The type of train consist determines the ratio of locomotive horsepower-to-trailing railcar tonnage pulled.

• Moderate-tonnage freight trains operating long distances with reasonable gradients and curvature and with minimal stopping may comfortably operate with a ratio of 0.8 horsepower/ton.

• High-speed trains, with rapid-acceleration of double stack equipment or commuter rail, and long distance passenger trains with frequent station stops will require that top speed is reached quickly.

• So the required horsepower-per-ton of weight moved will be significantly greater – in the 2-to-4 horsepower-per-trailing ton range.

• Highway trucks typically have a mass-power ratio of 0.15 ton-per-horsepower.

• Railways usually haul a 10,000-ton freight train with two 3,000 horsepower locomotives.

• A typical freight train therefore has a mass/power ratio of 1.7 ton-to-horsepower, and they max out at 2.1 ton-to-horsepower for bulk commodities.

• What this all means is that Railway tracks need gentler grades to handle trains with mass/power ratio 10 times higher than highway truck traffic.

25. Stopping the Train

• When planning train operations, safety is most important, so stopping ability and distances can be more important than tractive force, bigger cars, or stronger couplers.

• The same low coefficient of friction between steel wheel and steel rail that allows low rolling resistance, also limits braking ability.

• Although trains can be decelerated in less distance than they can be accelerated, stopping a train requires considerably more distance and time than a motor vehicle.

26. Stopping Distance V Speed

• Signal spacing is determined by the stopping distances of a representative train consist.

• It is not unusual on dry rail with 0% grade for a drag freight train operating at 40 MPH to take up to a mile to come to a complete stop.