### Calculating Wagon or Locomotive Resistance

**Aim -** this section describes how to calculate the Davis resistance parameters for wagons or locomotives.

If you wish to provide any feedback, or suggest corrections, please use the Contact page. Please provide appropriate references.

Where published data does not exist for the rolling stock being set up, it is recommended that **FCalc2** is used. This makes the calculation of friction parameters very easy.

#### Index

Example for the use of Published Formula

Example for the use of Published Test Graphs

Tools for Calculating Resistance

#### Example for the use of FCalc

**It should be noted that FCalc appears to predominately uses the Original Davis formulas, and the 1970 variations formulas. For more modern stock using the 1992 Canadian variations manual calculations may require to be done using the relevant formulas.**

To calculate **default** resistance settings it is recommended that FCalc2 is used. This is the easiest way to calculate default resistance values. For those who have obtained relevant railway resistance information for the stock that they are modelling, may need to persue the following sections.

FCalc can calculate the ABC Davis resistance values for various rail vehicle types. The ABC values generated with the bearing set as "friction", will align approximately with the original Davis formula. Variations to the formula within FCalc cater for roller and low torque bearings. The diagram below shows, for example, the different curves for a 32.5 ton uk bogie (4 axle) freight wagon with a cross sectonal area of 90 ft^{2} (8.36 m^{2}).

In most instances the default FCalc values should sufficice, with stock using friction or journal bearings, and more modern stock using the roller or low torque setting. The C value of the Davis equation has the greatest impact on resistance as the speed increases, and thus using the correct cross sectional area will provide the most accurate resistance curve. For those interested in experimenting with the Davis curves values, it might be valuable to use the following Excel spreadsheet to plot and compare user defined curves. (See tools section below.)

FCalc can also accept Metric or US units of measure, however these options need to be selected. It is recommended that metric units of measure are used as the outputs can be entered directly into Open Rails. To ensure FCalc is set to do this, undertake the following steps:

- Under "File" menu, and select "Metric Units"**Units of Measure**- Under "View" select "Davis co-efficiencts".**Davis co-efficiencts**

Consider the following example for an Austerity 2-10-0 locomotive, which has the following characteristics relating to FCalc:

**Type:**Steam - Standard - select the relevant rolling stock type.**Bearings:**Roller - Select the relevant bearing types.**Speed:**60 mph (100km/h) - There is limited value in identfying values under 60mph, as the resistance does not vary significantly.**Axles:**6 - This is the total number of wheel axles on the rolling stock.**Front Area (Cross section):**110 ft2 (10.2 m2) - This ideally should be calculated from the end profile of the rolling stock (the width and height statements from the WAG size statement in the general properties can be used), or alternatively the area values indicated in this table can be used. However these values reflect American stock, and may not be appropriate for other countries.**Weight:**78.3 tons uk ( 79.6 tonnes metric) - Total steaming weight of the locomotive**DrvWeight:**67.15 tons uk ( 68.2 tonnes metric) - Total weight on the locomotive drive wheels only.**Drag:**1 - most typical rolling stock will have drag figures of 1.0, whereas streamlined stock maybe reduced to values around 0.5. You may want to read this section on air drag, but be wary adjusting these figures.

In the case of the above example, the FCalc would look like the following screen:

The resulting Davis ABC co-efficients can be entered directly into the relevant WAG OR parameters. If the units of measure are in metric, then no units need to be specified as OR assumes that they are metric.

#### Example for the use of Published Formula

In some instances railway companies or researchers publish Davis type equations for specific rolling stock. An example of some published Davis equations can be found on 5AT Project web site. Others can sometimes be found by Googling with appropriate search terms.

These published formulae can often be in different units compared to what we need to input into OR. To see how published data can be entered into OR we will select a published curve from the above website, and translate it into the correct units to be applied in OR. For our example we will choose a BR passenger car weighing 32 tons-uk, and use the Koffman formula for passenger cars, which is as follows:

**Resistance = 1.1 + 0.021 * V + 0.000175 * V^2**

Thus giving ABC values of, A = 1.1, B = 0.021 and C = 0.000175

We need to note that it will be necessary to convert these ABC values as follows:

- The formula needs to be applied to a wagon of a certain weight, thus each of the values will need to be multiplied by the wagon weight.
- The speed used in the formula is in km/h, whereas OR uses m/s as the default, thus it will need to be converted.
- The calculated resistance is in kg/tonne, whereas OR uses values of Newtons by default, thus it will need to be converted.

**Step 1**: Adjust to weight of wagon. As our ABC values are in metric we will need to convert our weight to tonnes, and then adjust our values accordingly.

The equivalent metric weight of our car is 32.51 tonnes (1 ton-uk = 1.01605 tonnes)

Thus A = 35.761, B = 0.68271 and C = 0.00568925

**Step 2**: Adjust the equation for speed in m/s.

To do this we must multiply the B value by 3.6, and the C value by 3.6^{2}.

Thus now, A = 35.761, B = 2.457756 and C = 0.07373268

**Step 3**: Adjust the values to be in Newtons.

The conversion factor is 1 kgf = 9.80665 Newtons.

Thus now, A = 350.695, B = 24.10235 and C = 0.772307

To compare the published formula with the various Davis equations use the spreadsheet in the tools section below.

##### Useful Conversion Values

The following conversion factors can be applied to the more common published Davis formula.

A (lbs) = A * 4.44822 (Newtons)

B (lbs/mph) = B * 4.44822 / 0.44704 = B * 9.950389 (Newtons/m/s)

C (lbs/mph^{2}) = C * 4.44822 / 0.44704 / 0.44704 = C * 22.25838 (Newtons/m/s^2)

A (kgf) = A * 9.80665 (Newtons)

B (kgf/kmh) = B * 9.80665 / 0.277778 = B * 35.3039 (Newtons/m/s)

C (kgf/kmh^{2}) = C * 9.80665 / 0.277778 / 0.277778 = C * 127.094 (Newtons/m/s^2)

#### Example for the use of Published Graph

In other instances, railway companies publish test results showing a graph of train or rolling stock resistance. For example the folloing graph is from a British Rail test report for the Standard Class 8 Express Passenger Locomotive, and shows the resistance of coaches used in the test. This publication also has a curve for the locomotive resistance.

This graph needs to be translated into a form that can be entered into OR, and the following approach can be used to achieve this. For explanation purposes, we will use the above BR test results graph, and assume a 32 ton uk coach with 4 axles, and a cross section area of 8.3m^{2}. We will need to use the spreadsheet found in the tools section below.

**Step 1**: Plot the relevant graph for the rolling stock required on the excel spreadsheet. Note we need to multiple the above graph by the weight of the car, in this instance 32 tons.

**Step 2**: Use Fcalc to plot the relevant equivalent Davis formula as a reference value. Typically this may either be the friction or roller bearing type. Enter these values into the first line of the Davis formula section on the excel spreadsheet. Also enter them into the second line, but with a different graph title.

**Step 3**: Extrapolate the curve on the BR curve above until it crosses the 0 mph axis, and read the resistance value. In our graph above the resistance at 0 mph,by guestimation, is approximately 89lbf. Convert this value to Newtons, which in this case is 395.892N. Thus we have approximated our A value for the above graph. This value can be entered into the relevant second Davis equation on the spreadsheet. Note, in reality the curve does not go below 5 mph as the curve losses its accuracy at values less then this.

**Step 4**: Adjust the B & C values of our user defined curve as required, until the Davis type equation sits over the top of our test graph curve. Note increasing the B value will increase the bend in the curve upwards, decreasing it will straighten the line out. Increasing the C value will lift the end point of the curve higher, and decreasing it will drop it lower.

The outcome of these steps on the excel spreadsheet is as shown in the diagram below. Note how the BR Test curve crosses the Davis equation at approximately 80mph. Thus depending on the speed of operation of the train, the resistance could be better or worse then that specified in the default Davis formaula. We have now calculated the matching ABC values for the BR coaching stock test results as A = 395.9, B = 15.25, C = 0.8, and these can be entered directly into OR.

#### Tools for Calculating Resistance

##### Spreadsheet for Plotting and Comparing Curves

To use the above spreadsheet, enter values obtained from the relevant Davis formula values into the appropriate cells with the red figures in them. The graph names can also be changed by simply typing over the titles in left hand column.

##### Parameter Conversion

#### Useful References

Fcalc2 is available for download from TrainSim and then by typing in "fcalc.zip " into the File Name field of the search page.