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DESIGN OF A SYSTEM TO MEASURE TRACK MODULUS FROM A MOVING RAILCAR

Chris Norman*, Shane Farritor*, Richard Arnold*, S.E.G. Elias*, Mahmood Fateh†, Magdy El-

Sibaie†

*College of Engineering and Technology University of Nebraska – Lincoln Lincoln, NE, 68588-0656, USA

†Federal Railroad Administration

Office of Research and Development Washington, DC 20590, USA

ABSTRACT

Track modulus is an important parameter in track quality or performance. Modulus is

defined as ratio between the rail deflection and the vertical contact pressure between the rail base

and track foundation. This paper describes the design of a system for on-board, real-time, non-

contact measurement of track modulus.

Measuring track modulus from a moving rail car is non-trivial because there is no stable

reference for the measurements. The proposed system is based on measurements of the relative

displacement between the track and the wheel/rail contact point. A laser-based vision system is

used to measure this relative displacement. A mathematical model is then used to estimate track

modulus.

A mathematical analysis is presented to evaluate the design and sensitivity of the

proposed system. A simulation of a moving railcar is used to show the effectiveness of the

system. Finally, the results of field tests are presented for a slow (< 10 mph ~ 16.1 km/hr)

moving railcar over various sections of track including road crossings, rail joints, and bridges.

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INTRODUCTION

Railroad safety is highly dependent on track quality. Track modulus is one of many

accepted indicators of the quality and safety of railroad track. Modulus is defined as the

coefficient of proportionality between the rail and the vertical contact pressure between the rail

base and track foundation (1). More simply, it is the supporting force per unit length of rail per

unit deflection (2). Track modulus is influenced by many factors such as the quality of rails, ties,

rail joints, ballast, and sub-grade and the use of the word ‘track’ in this paper refers to all these

components. Modern rail systems traveling at higher speeds require increasingly stiffer track.

Estimating track modulus is difficult. Current methods include sending a work crew to a section

of track with special equipment to apply known loads and measure the resulting deflection. This

is an expensive, cumbersome, and a time consuming process. The result is knowledge of track

modulus for only a single section of track.

This paper explains the preliminary design of a system that will provide real-time

measurements of track modulus from a moving railcar. Such a system could provide nearly

continuous knowledge of track modulus for large sections of track. Track modulus could be

monitored over time and could lead to preventative maintenance eliminating problems before

they occur.

Measurement of track modulus from a moving railcar is difficult since the moving car

provides no absolute frame of reference for the measurement. The proposed system estimates

modulus by measuring the relative displacement between the wheel/rail contact point and the

rail. Deflection of the track is caused by the weight of the railcar. A heavy car on soft track will

“sink” into the track. The proposed system uses analytical models of both the railcar and the

track to estimate track stiffness based on these deflection measurements.

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A mathematical analysis is presented to evaluate the sensitivity of the system. The

analysis considers the behavior of the track as well how the system will measure track modulus.

A simulation of the system is presented as well. Finally, results from the simulation and field-

testing are presented for a slow moving (< 10 mph ~ 16.1 km/hr) railcar. The tests in include

tangent track as well as a short bridge.

Future work includes testing at higher speeds and over longer distances.

BACKGROUND

There are several theoretical methods for determining track modulus. Some include the

Deflection-Area, Pyramid Load Distribution, and the Beam-on-Elastic-Foundation method (1).

The Defection Area method is based on the vertical equilibrium of an infinitely long rail

under an applied load (1). In this method the shape of the rail is measured so that the deflection

curve of the rail is known. The deflection curve is then integrated along the length of the track

providing the area between the curve and the unloaded rail. The track modulus is then the ratio

of the applied load to the supporting area.

The Pyramid Load Distribution Method assumes the pressure beneath the rail seat is

uniform at each depth across the area of an imaginary pyramid zone spreading through the

ballast layer (1). In this theory the effective stiffness of each component is determined and then

related to the overall stiffness for the entire support system. Track modulus is then the ratio of

the stiffness of the entire support system to the tie spacing.

The Winkler method or beam-on-elastic-foundation method is a common method to

describe track shape under various loading conditions. This method describes the rail as an

infinitely long beam on an elastic foundation (1, 3, 4). From this model the track modulus is

defined as the vertical stiffness of the rail foundation (2). To determine the track modulus from

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this method the load applied to the rail and the resulting deflection must be measured. Tests

have shown that the beam-on-elastic-foundation method is an acceptable method for determining

track modulus (3).

Several methods have been used in the field to measure track modulus. All these

methods use static loading and require deflection measurements to be made before and after the

loads are applied. The methods for measuring track modulus include the Deflection Basin Test,

Single Load Point Test, and the Multiple Axle Vehicle Load Test (2). Tests have been

conducted showing that measuring track modulus can be used to determine track quality (5). It

is also recommended that track modulus be measured continuously over the track (5, 6). Some

work has been done to make these measurements low speeds (5 mph ~ 8.05 km/hr).

Track inspection vehicles are becoming more equipped to measure the quality of track.

One such vehicle is the Federal Railroad Administration’s (FRA) T-16 research car. The T-16 is

capable of measuring track geometry, rail head profile, ride quality and wheel-rail forces at high

speeds (up to 150 mph or 241.5 km/hr). Currently there is not a system to measure the track

modulus.

A vehicle, which is capable of measuring track modulus, is the Track Loading Vehicle

(TLV) developed by Transportation Technology Center, Inc. (TTCI). This vehicle functions at

speeds up to 10 mph. The TLV is capable of indicating sections of track that have low modulus

or have large variations in modulus. The TLV has a center load bogie and laser reference beam,

as does an accompanying empty tank car. During testing the center load bogie on the TLV can

apply wheel loads ranging from 1 to 60 kips and the tank car center load bogie can apply wheel

loads up to 3 kips. The empty tank car serves as an undeflected or reference measurement. The

TLV does two runs over the track, first applying a 10 kip load and on the second pass a 40 kip

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load. This two-pass method allows for the TLV to determine if the problem lies in the rail, tie,

and ballast or in the subgrade (7).

A couple other methods for measuring track stiffness from a moving railcar have been

proposed. The first method is the mid-chord offset (MCO), in which the vertical offset of the rail

is measured at the mid-chord. This method requires two measurements, one with a light load and

the second a heavy load. Also the chord must be of long enough length so that the wheel loads

do not affect the ends. The second method is the instantaneously fixed reference system. This

method requires three deflection measurements at each instant. With these measurements a

single reference length is determined and compared with measurement at the next measurement

interval. A system has been proposed that uses the instantaneously fixed reference system,

which would be made from a railcar truck. This method also requires measurements from a light

and heavy load before track stiffness can be determined (8).

THE PROPOSED APPROACH

The proposed measurement system estimates track modulus by measuring the deflection

of the rail relative to the railcar. The modulus is estimated using an analytical expression that

relates the shape of the rail to the applied loads. The model used is referred to as the Winkler

model (9). It is a continuum mechanics model for a point load applied to an infinitely long

elastic beam mounted on an elastic foundation. The applied load is perpendicular to the length

of the beam (i.e. vertical). The vertical deflection, y, for a given point load, P, as a function of

the distance from the load, x, is:

( ) ( ) ( )[ ]xxeu

Pxy x βββ β sincos2

+−= ⋅− Equation 1

Where:

6

41

4

=

zEIuβ Equation 2

Here, E is the modulus of elasticity of the beam (i.e. rail) and I is the second moment of

area of the beam. The variable, u, is the estimate of the track modulus. This model linearly

relates rail deflection to a single applied point load and shows a non-linear relationship to track

modulus. Rail deflection under a planar railcar is calculated with the Winkler model using the

superposition of four point loads at each wheel contact point. Figure 1 shows the results of this

model. Here the deflection of the rail relative to the ground is shown for a fully loaded by a coal

hopper for various track moduli. The maximum absolute deflection occurs at the wheel/rail

contact points for high modulus (stiff) track and at the midpoint between the wheels for low

modulus (soft) track.

Figure 1: Winkler Model of Rail Deflection

If the absolute deflection of the rail, y, could be measured at any point, x, the above

equations could be solved for the track modulus, u, giving an estimate of track modulus. The

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above model gives a one to one (non-linear) relationship between absolute deflection and track

modulus. Such absolute measurements are extremely difficult from a moving railcar because of

the lack or an absolute reference frame (the car bounces, rotates, and bends). For this reason the

proposed approach uses relative measurements between the railcar truck and the rail. The

proposed method continuously measures the relative deflection of the track and these

measurements are used to estimate the relative track modulus.

The proposed method is a non-contact measurement system that uses two line lasers and

a camera mounted to the railcar truck, Figure 2. As seen from the camera view (Figure 2 right),

each laser generates a curve across the rail head. The exact shape of the curve depends on the

shape of the rail. The distance, d, between the curves is found using imaging software. The

change in the distance d, as the train moves along the track represents a change in the relative

displacement between the railcar truck and the track (Assuming the shape of the rail and applied

loads is constant). So as the distance between the camera and rail decreases the measured

distance, d, will also decrease. The opposite is also true, as the distance between the camera and

rail increases the measured distance, d, will increase. Two lasers are used to increase the

resolution of the measurement system and the overall sensitivity of the instrument.

Figure 2: Proposed Measurement System

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Measuring the distance between the laser curves gives the relative displacement of the

rail and the railcar truck. This relative displacement along the length of the car is shown in

Figure 3. This curve is very similar to the absolute measurements shown in Figure 1, except the

curve is drawn relative to the rail/wheel contact points (zero displacement). This figure is

important because it shows that the location of the measurement along the length of the car (x) is

important to the sensitivity of the sensor. For example, a measurement taken close to the

rail/wheel contact point would be near zero for any value of track modulus. The maximum

sensitivity to track modulus occurs at a location approximately 150 in. (3.81 m) from the

rail/wheel contact point.

Figure 3: Track Deflection Relative to the Wheel/Rail Contact Point

The stated approach is based on some fundamental assumptions. First, that the Winkler

model is an accurate prediction of rail shape. Or, more precisely, that the Winkler model has

high resolution and can accurately predict relative changes in track modulus. A second

assumption is that there are no dramatic changes in the shape of the rail head over the distances

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where relative measurements are taken. Also, that the applied load is constant between two

measurements. Dynamic loads (bouncing of the car) can affect this assumption. However, at the

low speeds (~ 4 mph or 6.44 km/hr) that the system is currently being tested, dynamic loading is

not a concern. The final assumption is that the wheel/truck system is rigid. This assumption

does not include the car’s suspension. The validity of the approach with respect to these

assumptions is evaluated in the following sections in simulation and in field tests.

MODELING AND SIMULATION

The system that has been developed uses two lasers, each with optics to generate lines

across the head of the rail, and a camera to create an image of these lines (Figure 2). This image

is then analyzed with computer software and the distance between the two laser lines is

measured. Since the rail head is not flat the lasers appear as curves in the image. The minimum

distance between the lasers is measured and a mathematical model relates this measurement to

track modulus. The mathematical model is discussed in this section, along with a simulation of

the system.

System Model

The mathematical model relates the measured distance between the lasers to the track

modulus. The system that is being modeled is shown in Figure 4. The model consists of the

instrumented coal hopper truck and a locomotive truck. The rail deflection measured by the

sensor is dependent on these four wheel loads.

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Figure 4: System Schematic

The sensor will measure the relative rail displacement between the rail and wheel/rail

contact point. This measurement can be made if it is assumed that the instrument beam, truck,

and wheels are rigid. With this assumption the distance between the sensor system and

wheel/rail contact point can be assumed constant. Only rotation of the truck will cause this

distance to change, but this rotation can be taken into account and corrected.

Figure 5: Rail Deflection/Sensor Measurement

Figure 5 illustrates that the fixed distance between the wheel/rail contact point and

sensor, H, relates the relative rail displacement, yr, to the measured height of the sensor above the

rail surface, h. The height of the camera/lasers above the rail surface is:

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ryHh −= Equation 3

Where the relative rail displacement is:

wheelcamerar yyy −= Equation 4

Here ycamera is the deflection of the rail at the location underneath the camera/lasers and

ywheel is the deflection of the rail at the wheel nearest the sensor. These rail deflections are

calculated using the Winkler model and superposing the deflections caused by each of the four

wheel loads. The deflections are negative in value because the positive axis is defined upwards.

The relative rail displacement, yr, will always be a positive number.

The sensor reading, which is the measured distance between the lasers, is geometrically

related to the height of the sensor above the rail. The sensor in effect measures its height above

the rail by measuring the distance between the lasers. As the sensor moves closer or farther from

the rail surface the distance between the lasers changes. Below is a schematic of the sensor

(Figure 6).

Figure 6: Sensor Geometry

From the above figure the following equations can be written:

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( ) hlL =+ 111 tanθ Equation 5

( ) hlL =+ 222 tanθ Equation 6

21 lld += Equation 7

Where L1 and L2 are the horizontal displacement of the lasers from the camera, θ1 and θ2

are the angles between the lasers and the horizontal, l1 and l2 are the horizontal distance between

the center of the camera and laser/rail intersection, h is the vertical distance between the

camera/lasers and the surface of the rail, and d is the distance between the lasers on the rail

surface. Solving these equations results in:

( )2121 tantan

LLhhd +−+=θθ

Equation 8

Using Equation 3, Equation 4, and Equation 8 a sensor reading can be calculated for any

given value of track modulus. The sensor reading that will be calculated is highly dependent on

the geometry of the sensor and the fixed height of the sensor above the wheel/rail contact point,

H. Figure 7 shows how track modulus and the sensor reading are related. Changing the height

of the sensor above the wheel/rail contact point will shift the curve along the abscissa, which is

also shown in Figure 7. By curve fitting the appropriate curve in Figure 7, a mathematical model

that relates track modulus to the sensor reading is determined.

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Figure 7: Track Modulus vs. Sensor Measurement

Studying the middle (solid) curve in Figure 7, it is visible that the curve is rather flat at

the left end and grows rapidly at the right end. Since the ends of the curve are extremes, the

system should be calibrated so that most measurements fall in the mid-section of the curve. In

doing so, the system will have good resolution for measuring both low and high values of track

modulus.

System Simulation

Up to this point the mathematical model for relating track modulus to the sensor

measurement has been discussed. The model is based on the system geometry and track

behavior, which is described by the Winkler model. The computer simulation uses Equation 3,

Equation 4, and Equation 8, which were developed earlier, to determine what sensor readings

would result from the test vehicle traveling over track with a given track modulus.

In the simulation the track modulus is known for each position along the track. As the

loads caused by the four wheels are moved along the track, the relative rail displacement

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(Equation 4) is calculated along with the angle of the instrumented truck (Equation 9). After

finding the relative rail displacement and truck angle, the height of the sensor above the rail is

found (Equation 10), taking into account that the distance between the sensor and wheel/rail

contact point is affected by the truck angle. Finally, the sensor reading is calculated using

Equation 8. The sensor reading can then be converted to track modulus using the mathematical

model (curve fit) found earlier.

( ) ( ) ( )

−= −

70sin 4,3,1 xyxy

x wheelwheelθ Equation 9

Here the truck angle is θ, the deflection of the wheels on the coal hopper are ywheel,3 and

ywheel,4. The wheel closest the measurement system is ywheel,4.

( )[ ] ryDDDHh −−+−= 221 cossin θθ Equation 10

Where D1 is the horizontal offset of the camera from the center of the nearest wheel, and

D2 is the vertical offset of the sensor from the center of the nearest wheel.

TESTING

Field testing included both ground based track deflection measurements and on-board

track modulus measurements. The ground based measurements were needed to determine if the

developed sensor could accurately determine track modulus.

Trackside Measurements

Trackside measurements of track modulus were needed to verify that the sensor system

could be used to determine track modulus. These trackside measurements are considered to be

the ‘ground truth’ and are compared to the on-board sensor measurements. These measurements

serve as a method to calibrate the system as well as validate that it works.

Trackside measurements were made using linear variable differential transformers

(LVDTs) to measure the rail deflection as a coal train passed. The LVDTs were securely

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clamped to steel rods that were placed approximately 3 ft (0.914 m) into the ballast and subgrade

near the rail. These steel rods created a grounded reference frame from which the deflection

measurements could be made. Figure 8 shows two LVDTs mounted near the rail.

Figure 8: LVDT Setup

The LVDTs output a voltage signal that is proportional to the position of the plunger.

The voltage signal is converted to a digital signal and recorded with a laptop computer. The

voltage signal can then be processed to show the rail deflection versus time. The Winkler model

is compared to data so that the track modulus can be estimated. Figure 9 below shows trackside

data compared to the Winkler model.

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Figure 9: Trackside Data and Winkler Model

In Figure 9 above, the track modulus is adjusted until the Winkler model (dotted) fits the

LVDT data (solid). Once the curves are reasonably close to the same, the track modulus at the

location that the data was taken is estimated to be that of the Winkler model fit.

On-board Track Modulus Measurements

The test vehicle for on-board measurements was a locomotive and coal hopper. The

measurement system aboard the coal hopper was a digital video camera and two line lasers. The

camera and lasers were mounted on a beam that was bolted to the left side frame of the lead

truck on the coal hopper. Figure 10 is a picture of the system mounted to the coal hopper truck.

The beam is bolted down to the truck in two locations and extends down and away from the

truck. The lasers and camera are mounted on the inside of the beam above the rail.

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Figure 10: Instrumented OMAX 92

Testing was performed over approximately 20 miles (32.2 km) of track at low speeds (~ 4

mph or 6.44 km/hr). The locomotive pulled the coal hopper and the digital video camera

recorded the position of the lasers on the surface of the rail. The location of the coal hopper was

measured and displayed to the camera also, so that the data could be compared to trackside

measurements. Below are some resulting test images.

Figure 11: Test Images

After the testing was completed the video had to be processed to determine the distance

between the lasers. Two steps must be performed in the video processing. The first is

Camera/Lasers

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determining which part of the image is a part of either of the laser curves. The second step is to

determine the distance between the two laser curves at each pixel across the surface of the rail.

The laser curves were identified using filtering methods and thresholds. Edge detecting

algorithms were then used to locate the positions of the laser curves in the image. The data points

found with the edge detector then underwent curve fitting to better determine the shape of the

laser curve. The distance between the lasers was then calculated from the two curve fits for each

point along the rail surface. The minimum value of the distance is selected to be the ‘sensor

measurement’ for that image and location on the track.

Test and Simulation Results

Results for two test sections will be presented in the following section. Trackside

measurements were made in five places along this section of track. These measurements are

shown in Figure 12. The measurements were taken approximately 100 in. (2.54 m) apart and

ranged from 1200 lbf/in./in. (8.27 MPa) to 3400 lbf/in./in. (23.4 MPa). These measurements are

used as the input to the simulation described earlier and are also compared to the measurements

made by the on-board system.

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Figure 12: Trackside Data at MP31 West

The simulation uses the track modulus as an input to the system at each point along the

track. Since trackside measurements were only taken in a few locations, the modulus between

these locations is assumed to fall on a line connecting the measurements. After simulating the

sensor system traveling over this section of track, the trackside measurements and simulation

results are compared to the sensor readings. The figure below shows trackside measurements

(dotted line), simulation results (dashed line), and the sensor measurements (solid line).

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Figure 13: Trackside Data, Simulation Results, and Sensor Measurements at MP31 West

The simulation (dashed) follows the input track modulus (dotted) very closely. The

simulation predicts that if the actual track modulus was the input shown in Figure 13, then the

on-board system would be able to measure the track modulus very accurately. The actual track

modulus is unknown between the trackside measurements (vertical dotted lines) though and is

assumed to be linear.

The measurements from the on-board system (solid) match the trackside measurements

(vertical dotted lines) very well. Since the actual track modulus between measured points is

unknown, it is not expected that the sensor measurements perfectly match the assumed track

modulus. The measurements are all within 709 lbf/in./in. (4.89 MPa) of the assumed track

modulus and the standard deviation of the error is approximately 283 lb/in./in. (1.95 MPa).

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Figure 14: Bridge Test Location

The second test location that will be discussed is a bridge. There is no related LVDT data

for the bridge because of the difficulty involved in making the measurements. A picture of the

bridge is shown in Figure 14. In this picture it is visible that the bridge is supported on the ends

and at its center. Sensor measurements of the test car crossing the bridge are shown in Figure 15.

The sensor readings show that the track is ‘soft’ at the approaches to the bridge. The sensor also

was able to detect the location of the bridge supports where the track is stiffer. The three peaks

in Figure 15 correspond to the bridge supports.

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Figure 15: Sensor Data from Bridge

CONCLUSIONS AND FUTURE WORK

Track failure is a major factor in many railroad accidents. Track modulus, or stiffness, is

an important parameter in track quality. This paper describes the preliminary design of a system

for on-board, real time, non-contact measurement of track modulus.

The Winkler model for track deflection was explained. A model of the sensor system

was presented and simulation results for a section of track were given. The simulation showed

that the system should be able to measure the track modulus very accurately. The results from

dynamic (moving) railcar testing were presented. Early testing at low speeds (~ 4 mph or 6.44

km/hr) shows the system is capable of measuring track modulus.

The tests support the initial sensor design presented in this paper, although much future

work is required to implement the system in the field. Currently, a more robust system, capable

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of higher speeds is being designed and testing of the system is being planned. The improved

system will be tested on a tank car at a variety of speeds and over longer distances. Again,

absolute measurement of track deflection will be used to confirm the results of the slow speed

tests. Higher speed tests will follow.

ACKNOWLEDGEMENTS

This work is sponsored under contract from the Federal Railroad Administration. Special

thanks to Richard Kotan of Omaha Public Power District, John Leeper of the Burlington

Northern Santa Fe Railroad, David Connell of the Union Pacific Railroad, and William

GeMeiner of the Union Pacific Railroad.

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