### Train Resistance

**Aim -** This page is dedicated to describing the different types of resistance that impact a trains and how they have been modelled within Open Rails.

If you wish to provide any feedback on this page, please use the contact page. It would be great to have some feedback, as this helps to ensure the accuracy of the information and model.

For information on how to apply resistance settings within Open Rails, refer to the Resistance Setting page.

#### Index

Resistance on straight and level track

#### Introduction

As described in the Train Movement section, the movement of the train is opposed by a number of different forces which are collectively grouped together to form the "train resistance".

In other words the the principal forces opposing the movemnet of the train will include:

**Resistive Force = F**

_{l}+ F_{g}+ F_{c}+ F_{t}Where:

i) **F _{l}** = Resistance (or Friction) on straight and level track

ii)

**F**= Gradient resistance - trains travelling up hills will experience greater resistive forces then those operating on level track.

_{g}i11)

**F**= Curve resistance - applies when the train is traveling around a curve, and will be impacted by the curve radius, speed, and fixed wheel base of the rolling stock.

_{c}iv)

**F**= Tunnel resistance - applies when a train is travelling through a tunnel.

_{t}As noted in the Train Movement section, and highlighted in the first diagram in that section, the tractive effort must exceed the resistance force of the train to ensure that the train moves. The point where the tractive effort crosses the train resistance force curve will typically indicate the maximum speed that the train can travel under the prevailing conditions, such as gradient, curve radius, etc, that the train is currently experiencing.

Typically resistance force is expressed in lbf (Imperial) or N(ewtons) (metric). Resistance force is often also called "friction".

This page describes these different types of resistance values, as well as describing a track geometry designed to minimise the impacts and effects of the above resistances on the trains performance.

#### Resistance on straight and level track

On straight and level track, the forces that oppose the movement of the train are:

i) **Bearing resistance** - the type of bearing used on the stock represents some of the force. Older style stock is typically fitted with journal type bearings, whereas newer stock is fitted with roller bearings, which come in standard models or low friction models. The journal type of bearings tended to be cheaper, but presented a higher value of resistance to train movement.

ii) **Train dynamic friction losses** - These include flange effects which are associated With lateral motion and the resulting friction and impact of the wheel flanges against the gage side of the rail. They vary with speed, rail alignment, track quality, the surface condition of the rail under load, the horizontal contour of the railhead, contour and condition of the wheel tread, and the tracking effect of the trucks. Also there are miscellaneous losses due to sway, concussion, buffing and slack-action.

iii) **Atmospheric resistance** - resistance due to both still air and wind. Typically this resistance was influenced by the cross sectional area of the train and its speed. At high speeds air resistance became especially important as it was based upon the square of the speed.

iv) Flange resistance - Rolling friction between wheel and rail. This can be considered a constant for a given quality of track.

Typically to calculate train resistance, an empirical approach is adopted, ie the resistance calculation are based upon tests undertaken on the train. For example, a series of test was undertaken by Edward Schmidt and Harold Dunn as part of their research into Passenger Train Resistance. As part of this work the test results were plotted onto a graph, and a line of "best fit" was determined as shown on the following diagram. Typically the resistance of the complete train was measured, and then the weight of the train was used to calculate a resistance figure per ton. Over the years many researchers and railway companies have followed a similar approach to estimate the resistance of a train.

In 1926, W. J. Davis undertook a similar series of tests to measure resistance on moving trains. He then devloped a "best fit" curve, and proposed an empirical formula for computing the resistance of a train moving on a straight and level section of track. This formula took the form shown below, and was correlated to various parameters of the train, such as the weight of the wagon, number of axles, and cross sectional area.

**Resistance = A + B * V + C * V^2**

where:

**Resistance** = train resistance in lbf (often expressed in lbs per ton)

**A** = rolling resistance component independent of train speed, this term can also contain an allowance for "internal" mechanical friction due to the valve gear on a steam locomotive.

**B** = coefficient used to define train resistance dependent on train speed

**C** = streamlining coefficient used to define train resistance dependent on the square of the train speed, can include information about the frontal surface area of the wagon, and an allowance for the drag created by the wagon.

**V** = train speed in mph (or m/s depending upon the A, B & C values)

A number of other researchers, including Schmidt, Tuthill, and Koffman, have also undertaken investigation to determine appropriate measurements for resistance, and these also have been described in form to the above.

As a broad correlation, the Davis formula parameters equate to the different types of friction as follows:

- Coefficient A - bearing type - varies depending upon whether a journal or roller bearing type.
- Coefficient B - Flange resistance and train dynamic losses
- Coefficient C - Air resistance

##### Original Davis Formula

The original formula proposed by Davis was as follows:

**Resistance (lb/ton) = (1.3 + 29/w) + 0.045 * V + ((0.0005 * A) / (w * n)) * V^2**

where:

**w**= Weight per axle (tons US)

**V**= train speed (mph)

**A**= Car cross sectional area (square feet)

**n**= Number of axles per car

The original Davis formula has given satisfactory results for older equipment with journal (or friction) bearings within a speed range between 5 and 40 mph. Over the years the formula have been adjusted for higher speeds, roller bearings, increased dimensions and heavier loading of freight cars and changes in the track structure have made it desirable to modify the constants in the Davis equation.

The above formula is for a standard freight car, and can be adjusted by replacing the B and C values with those from the following table as appropriate, and using the approximate cross section areas provided.Wagon Type |
A (sq ft) |
B |
C |
---|---|---|---|

Locomotive - 50 tons |
105 |
0.03 |
0.0024 |

Locomotive - 70 tons |
110 |
0.03 |
0.0024 |

Locomotive - > 100 tons |
120 |
0.03 |
0.0024 |

Freight Cars |
85-90 |
0.045 |
0.0005 |

Passenger Car |
120 |
0.03 |
0.00034 |

Motor Cars - in train |
80-110 |
0.045 |
0.0024 |

Motor Cars - single |
80-110 |
0.045 |
0.0024 |

##### Modified Davis Formula

Based upon work done by Committee 16 of the AREA, in 1970, the following modified formula was developed to allow for improvements in technologies, and operating practices.

**Resistance (lb/ton) = (0.6 + 20/w) + 0.01 * V + ((K / (wn)) * V^2**

Where the parameters are the same as above with the following parameter added:

**K**= air resistance (drag) coefficient. Typical values are 0.07 for conventional equipment, 0.0935 for containers, and 0.16 for trailers on flat cars.

##### Adjusted Davis Formula

In more recent years it has been suggested that the original Davis equations tended to overstate the resistance value, and thus an adjustment factor is sometimes applied.

**Resistance**

_{adj}= K * R_{D}Where:

**Resistance**= adjusted train resistance

_{adj}**K**= an adjustment factor to "modernise" Davis resistance values

**R**= original Davis formula

_{D}The K (adjustment factors) are based upon testing in controlled conditions, and they are as follows:

**1.00** - for pre-1950 equipment

**0.85** - for conventional post-1950 equipment

**0.95** - for container on flatcar

**1.05** - for trailer-on-flatcar and hopper cars

**1.20** - for empty covered auto racks

**1.30** - for loaded auto racks

**1.90** - for empty, uncovered auto racks

The large value for empty rack cars reflects the turbulence created by the wagon frame, and the pockets of air encapsulated within the framework. Loaded rack cars achieve some level of streamlining from the cars loaded on the wagons.

To assist in understanding the different outcomes that can occur, a number of curves have been plotted on the following diagram based upon a number of different "Davis" type formula. A 32 ton uk freight bogie wagon (4 axles) with a cross sectional area of 90 ft^{2} has been used for comparison purposes. The properties of the wagon modelled by the Koffman curve are not known.

The diagram shows the original Davis formula, as well as the modified and adjusted values, and also curves based upon Canadian National Railways and British Rail information. It can be seen that the curves diverge significantly at higher speeds. It should however be noted, that whilst the diagram speed axis has been extended to 125mph, it is unlikely that freight stock would reach this type of speed. Therefore a more realistic view can be obtained by comparing the curve differences around the 60mph point.

For information on setting the Davis values in the WAG or ENG file, refer to Level Resistance setting and Resistance Calculation page.

#### Gradient Resistance

The gradient resistance is the extra force required to lift a train up a gradient, or conversely is the extra force "pushing" the train down a hill.

It is a positive quantity when going up the grade and a negative quantity for going down the grade.

As shown in the diagram below the work of lifting a weight W up a distance represented by BC is equal to the work required to overcome the resistance R (due to grade only) through a distance AB, ie W x BC = R x AB.

Thus the following formula can be used to calculate grade resistance:

**Grade Resistance = W x (vertical rise) / (length of incline)**

or alternatively

**Grade Resistance = W Sin(TR)**

where:

**Grade resistance** = Grade resistance in lbf (often expressed in lbs per ton)

**W** = load in tons

**TR** = (vertical rise / length of incline)

This is calculated in Open Rails automatically.

#### Curve Resistance

When a train travels around a curve, due to the track resisting the direction of travel (ie the train wants to continue in a straight line), it experiences increased resistance as it is "pushed" around the curve.

Over the years there has been much discussion about how to accurately calculate curve friction. The calculation methodology presented (and used in OR) is meant to be representative of the impacts that curve friction will have on rolling stock performance.

##### Factors Impacting Curve Friction

A number of factors impact upon the value of resistance that the curve presents to the trains movement, and these are as follows:

i) Curve radius - the tighter the curve radius the higher the higher the resistance to the train

ii) Rolling Stock Rigid Wheelbase - the longer the rigid wheelbase of the vehicle, the higher the resistance to the train. Modern bogie stock tends to have shorter rigid wheelbase values and is not as bad as the older style 4 wheel wagons.

iii) Speed - the speed of the train around the curve will impact upon the value of resistance, typically above and below the equilibrium speed (ie when all the wheels of the rolling stock are perfectly aligned between the tracks). See the section below "Impact of superelevation".

The impact of wind resistance on the curve is ignored.

##### Impact of Rigid Wheelbase

The length of the rigid wheelbase of rolling stock will impact the value of curve resistance. Typically rolling stock with longer rigid wheelbases will experience a higher degree of "rubbing" or frictional resistance on tight curves, compared to stock with smaller wheelbases.

Steam locomotives usually created the biggest problem in regard to this as their drive wheels tended to be in a single rigid wheelbase as shown in the figure below. In some instances on routes with tighter curve the "inside" wheels of the locomotive was sometimes made flangeless to allow it to "float" across the track head. Articulated locomotives, such as Shays, tended to have their drive wheels grouped in bogies similar to diesel locomotives and hence were favoured for routes with tight curves.

The value used for the rigid wheelbase is shown as W in the diagram below.

##### Impact of SuperElevation

On any curve whose outer rail is super-elevated there is, for any car, one speed of operation at which the car trucks have no more tendency to run toward either rail than they have on straight track, where both rail-heads are at the same level (known as the equilibrium speed). At lower speeds the trucks tend constantly to run down against the inside rail of the curve, and thereby increase the flange friction; whilst at higher speeds they run toward the outer rail, with the same effect. This may be made clearer by reference to the figure below, which represents the forces which operate on a car at its centre of gravity. With the car at rest on the curve there is a component of the weight **W** which tends to move the car down toward the inner rail. When the car moves along the track centrifugal force **Fc** comes into play and the car action is controlled by the force **Fr** which is the resultant of **W** and **Fc**. The force **Fr** likewise has a component which, still tends to move the car toward the inner rail. This tendency persists until, with increasing speed, the value of **Fc** becomes great enough to cause the line of operation of **Fr** to coincide with the centre line of the track perpendicular to the plane of the rails. At this equilibrium speed there is no longer any tendency of the trucks to run toward either rail. If the speed be still further increased, the component of **Fr** rises again, but now on the opposite side of the centre line of the track and is of opposite sense, causing the trucks to tend to move toward the outer instead of the inner rail, and thereby reviving the extra flange friction. It should be emphasised that the flange friction arising from the play of the forces here under discussion is distinct from and in excess of the flange friction which arises from the action of the flanges in forcing the truck to follow the track curvature. This excess being a variable element of curve resistance, we may expect to find that curve resistance reaches a minimum value when this excess reduces to zero, that is, when the car speed reaches the critical value referred to. This critical speed depends only on the super-elevation, the track gauge, and the radius of track curvature. The resulting variation of curve resistance with speed is indicated in the diagram below.

##### Calculation of Curve Resistance

**R=WF(D+L)/2r**

where:

**R** = Curve resistance

**W** = vehicle weight

**F** = Coefficient of Friction,

**u** = 0.5 for dry, smooth steel-to-steel, wet rail 0.1 - 0.3

**D** = track gauge

**L** = Rigid wheelbase

**r** = curve radius

**Source**: The Modern locomotive by C. Edgar Allen - 1912

##### Curve Speed

As has been described in the proceeding section, the permissible speeds that a train can travel around a curve becomes critical, and the three key speed limits are as follows:

- Equilibrium Speed - maximum speed around a curve which ensures that the load of the wagon is balanced across all the wheels, and that the forces between the wheels and track are equal. Speeds in excess of this value may result in uneven track wear and tear. This speed limit is mainly influenced by track superelevation.
- Unbalanced Speed - maximum speed around a curve which allows a certain amount of load unbalance between the wheels and the track. Typically the amount of unbalance is determined by the of the amount of unbalanced superelevation (cant deficiency) which is allowable by individual railway companies. The higher this value the greater the impact felt by passengers, and their level of discomfort in rounding the curve.
- Critical Speed - the speed at which the rolling stock is likely to overturn. Mostly impacted by the Centre of Gravity.

A formula in imperial units to calculate the __Equilibrium Speed__ and __Unbalanced Speed__ is:

**V = √ Sgr/G**

**V**= speed ft/s

**S**= Superelevation in inches (includes both Balanced and Unbalanced values of superelevation, for Equilibrium speed only use the track superelevation, for Unbalanced speed sum the Balanced and Unbalanced superelevation values.)

**g**= Acceleration due to gravity (32.2 ft/s

^{2})

**r**= Radius of curve in feet

**G**= Track gauge (note the distance between rail centres, and not the inside rail measurement)

To calculate the __Critical Speed__, the Centrifugal Force (**Fc** above) balances the overturning weight (**W** above)of the wagon:

**V = √ Fgr/W**

**V**= speed ft/s

**F**= Centrifugal force in tons (acting on the point of centre of gravity)

**g**= Acceleration due to gravity (32.2 ft/s

^{2})

**r**= Radius of curve in feet

**W**= Weight of the Vehicle in tons

To gain a sense of how the speed around a curve will vary under different conditions, refer to the calculators shown on the Curve Speed Test page. Also refer to this page when testing rolling stock to see how it will perform for different curve scenarios.

##### Calculation of Speed Impact on Curve Resistance

As described above the least value of resistance will occur at the equilibrium speed, and as the speed increases or decreases from the equilibrium speed, the value of curve resistance will increase accordingly. This concept is shown pictorialy in the following graph.

Open Rails uses the following formula to model the impact of speed upon the curve resistance:

**Change in Resistance(due to speed) = ABS (((Equilibrium Speed-Train Speed))/((Equilibrium Speed) )) x ResistanceFactor @ start**

##### Typical Rigid Wheelbase Values

The following values can be used as defaults where actual values are not readily available from rolling stock plans.

Rolling Stock Type |
Typical value |
---|---|

Freight Bogie type stock (2 wheel bogie) |
5' 6" (1.6764m} |

Passenger Bogie type stock (2 wheel bogie) |
8' (2.4384m) |

Passenger Bogie type stock (3 wheel bogie) |
12' (3.6576m) |

Typical 4 wheel rigid wagon |
11' 6" (3.5052m) |

Typical 6 wheel rigid wagon |
12' (3.6576m) |

Tender (6 wheel) |
14' 3" (4.3434m) |

Diesel, Electric locomotives |
Similar to passenger stock |

Steam locomotives |
Dependent on# of drive wheels, Can be up to 20'+ , eg large 2-10-0 locos |

Modern publications suggest that an allowance of approximately 0.8 lb. per ton (us) per degree of curvature for standard gauge tracks. 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.

For information on setting the curve resistance in the WAG or ENG file, refer to Curve Resistance setting page.

#### Tunnel Resistance

##### Introduction

When a train travels through a tunnel it experiences increased resistance to the forward movement.

Over the years there has been much discussion about how to accurately calculate tunnel resistance. The calculation methodology presented (and used in OR) is meant to provide a indicative representation of the impacts that tunnel resistance will have on rolling stock performance.

##### Factors Impacting Tunnel Friction

In general, the train aerodynamics are related to aerodynamic drag, pressure variations inside train, train-induced flows, cross-wind effects, ground effects, pressure waves inside tunnel, impulse waves at the exit of tunnel, noise and vibration, etc. The aerodynamic drag is dependent on the cross-sectional area of train body, train length, shape of train fore- and after-bodies, surface roughness of train body, and geographical conditions around the traveling train. The train-induced flows can influence passengers on a subway platform and is also associated with the cross-sectional area of train body, train length, shape of train fore- and after-bodies, surface roughness of train body, etc.

A high speed train entering a tunnel generates a compression wave at the entry portal that moves at the speed of sound in front of the train. The friction of the displaced air with the tunnel wall produces a pressure gradient and, as a consequence, a rise in pressure in front of the train. On reaching the exit portal of the tunnel, the compression wave is reflected back as an expansion wave but part of it exits the tunnel and radiates outside as a micro-pressure wave. This wave could cause a sonic boom that may lead to structural vibration and noise pollution in the surrounding environment. The entry of the tail of the train into the tunnel produces an expansion wave that moves through the annulus between the train and the tunnel. When the expansion pressure wave reaches the entry portal, it is reflected towards the interior of the tunnel as a compression wave. These compression and expansion waves propagate backwards and forwards along the tunnel and experience further reflections when meeting with the nose and tail of the train or reaching the entry and exit portals of the tunnel until they eventually dissipate completely.

The presence of this system of pressure waves in a tunnel affects the design and operation of trains, and they are a source of energy losses, noise, vibrations and aural discomfort for passengers. These problems are even worse when two or more trains are in a tunnel at the same time. Aural comfort is one of the major factors determining the area of new tunnels or the maximum train speed in existing ones.

##### Importance of Tunnel Profile

As described above, a train travelling through a tunnel will create a bow wave of air movement in front of it, which is similar to a "piston" effect. The magnitude and impact of this effect will principally be determined by the **tunnel profile, train profile and speed**.

Typical tunnel profiles are shown in the diagrams below.

As can be seen from these diagrams the smaller the tunnel cross sectional area compared to the train cross sectional area, the less air that can "escape" around the train, and hence the greater the resistance experienced by the train. Thus it can be understood that a single train in a double track tunnel will experience less resistance then a single train in a single track tunnel.

**Source**: Reasonable compensation coefficient of maximum gradient in long railway tunnels by Sirong YI, Liangtao NIE, Yanheng CHEN, Fangfang QIN

For information on setting the curve resistance in the WAG or ENG file, refer to Tunnel Resistance setting page.

#### Track Geometry

Most railway companies have established track geomety standards to ensure that the construction and maintenance of tracks is undertaken within safe limits, prevent excessive forces on rolling stock and tracks to minimise wear and tear, ensure passenger comfort and finally to ensure that trains are able to perform at their optimal level of efficiency. The track geometry has typically been improved over the years as the need for higher speed, heavier loads, and passenger comfort have been identified. The table and notes below summarises the principle track geometry parameters currently used for New South Wales mainline routes. Branchlines and older routes may be built or operate on more onerous standards. Additional parameters and a more detail explanation can be found in ESC 210 - Track Geometry and Stability.

For the sake of comparison with older standards, imperial figures have been included in barackets after each of the parameters below, and they have been rounded to the relevant imperial value.

Parameter | Mixed Passenger / Freight Main Line |
Maximum (or Minimum) Limits | ||
---|---|---|---|---|

Track Gauge | 1435mm (4ft 8.5in) | 1435mm (4ft 8.5in) | ||

Speed | Normal | 115km/h (70mph) | 115km/h (70mph) | |

XPT | 160km/h (100mph) | 160km/h (100mph) | ||

Minimum Curve Radius | 400m (20chs or 1312ft) | 160m (8chs or 525ft) | ||

Superelevation (or Cant) | Mainline | 125mm (5.0in) | 140mm (5.5in) | |

Platform (New) | 60-100mm (2.5-4.0in) | 75-100mm (3.0-4.0in) | ||

Turnout | 50mm (2.0in) | 50mm (2.0in) | ||

Superelevation (or Cant) Deficiency(+)/Excess(-) | Plain Track | Normal | +/-75mm (3.0in) | +80/-75mm (+3.1/-3.0in) |

XPT (T1) | +110/-75mm (+4.25/-3.0in) | +110/-75mm (+4.25/-3.0in) | ||

Turnout Track (Conventional) | Normal | 75mm (3.0in) | 140mm (5.5in) | |

XPT (T1) | 75mm (3.0in) | 140mm (5.5in) | ||

Diamond Crossing | Normal | 0mm (0.0in) | 25mm (1.0in) | |

XPT (T1) | 0mm (0.0in) | 25mm (1.0in) | ||

Gradient (compensated) | Mainline | 1 in 100 (1.0%) | Ruling Gradient | |

Platform | 1 in 150 (0.67%) | 1 in 100 (1.0%) |

**Cant Deficiency** - the train is allowed to exceed the Equilibrium Speed by an amount determined by the Cant Deficiency. In this scenario the outer wheels on the bogie have more apparent weight upon them.

**Cant Excess** - determines the minimum ideal speed around the curve, i.e. the speed less then the Equilibrium Speed. In this instance the Cant is deemed to be in Excess. In this scenario the inner wheels on the bogie have more apparent weight upon them.

**Ruling Gradient** - is typically the steepest gradient on a section of line in a particular route or section thereof. Normally train loads were calculated so that a single steam locomotive could successfully haul the load up the gradient. Sometimes, where it was operationally expedient and efficient to do so a bank (or helper) locomotive was employed to assist the train locomotive in hauling the train to the top of the grade.

**Gradient (Compensated)** - As suggested in the section above on curve resistance, when traveling around a curve a train will encounter greater resistance then on a straight and level track. Railways often equate this to an amount of gradient that a train will face, thus if a train is traveling up a grade and encounters a sharpe curve then the equivalent gradient that the train is facing will be greater then that experienced on the level graded track. On modern designed tracks the gradient on a curve can be eased by a relevant amount to ensure that the ruling gradient on the slope is consistent throughout the climb. On older lines, where the curves were often laid on the same gradient as the straight track, the ruling gradient for section will be increased by the amount of impacted of the sharpest curve on the section.

The compensated gradient can be found from the formula:

**Grade compensation = 60/R %**

Thus for example, if a 1 in 100 (1.0%) gradient has a 300m curve located on it, then a grade compensation of 0.2% must be applied. Hence to maintain the 1 in 100 ruling grade, the curve can be laid on a 1 in 125 (0.8%) grade, or alternatively if the curve is laid on a 1 in 100 grade then the ruling gradient must be changed to 1 in 83 (1.2%). Train loads will need to be calculated accordingly.