Modelling the pulse of a double wheel sensor

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MASTER’S THESIS

Universitetstryckeriet, Luleå

Esko Allinen

Modelling the pulse of

a double wheel sensor

MASTER OF SCIENCE PROGRAMME Engineering Physics

Luleå University of Technology

Department of Applied Physics and Mechanical Engineering Division of Physics

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Abstract

Axle counting system is an application to monitor track section vacancy and to ensure railway safety. The aim of this study is to develop a mathematical model for a pulse from train wheel detection sensor for a simulator. The simulator simulates a train wheel detection sensor known as double wheel sensor. The double wheel sensor is attached to the railway track, and it sends pulses each time a train wheel passes it. The simulator is designed to send similar pulses as the original double wheel sensor to the counting system.

The research for this study is carried out at Mipro Oy in Mikkeli, Finland. It is a medium sized company, which specializes in customised automation. One of its main business fields is safety related systems in railways. Mipro Oy has two test locations for axle counting testing with real train traffic. The need for a proper simulator has arisen to ensure the safety of an axle counting system, and to make its further development possible in a cost efficient way.

If a simulator described in this study were to be built, it would be possible to test the axle counting system more efficiently, since a real train can be replaced by a simulator. By having a better ability to test the system, it becomes more reliable, and safety level of the axle counting system increases.

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Preface

This paper is the last part of my Master of Science program in Engineering Physics at Luleå University of Technology, Sweden. The project was performed at Mipro Oy in Mikkeli, Finland. I would like to thank all the people in Mipro Oy, especially the head manager Raimo Laine for giving me this great opportunity as well as my supervisor Marko Varpunen, who has supported me by giving good advice and point of views during the whole project.

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List of Abbreviations

OY Osakeyhtiö (Finnish), means the limited company, Ltd EN 5012x Group of international standards in railway transportation IEC International Electrotechnical Commission

IEC 61508 International standard “functional safety of electrical/electronic/programmable electronic safety-related systems”

SIL Safety Integrity Level MTTF Mean Time To Failure

RHK Ratahallintokeskus (Finnish), The Finnish Rail Administration

TDP Train Detecting Peripheral, interface unit in axle counting system, product of Mipro Oy

GmbH Gesellschaft mit beschränkter Haftung (German) company with limited liability

DC Direct Current

MiSO Product name of Mipro Oy´s product

TÜV Technischer Überwachungs-Verein (German), Technical Monitoring Association in Germany and Austria.

HIMA A German company

VR Valtion Rautatiet (Finnish), a Finnish Railway company owned by the Finnish government

RPM Rotations per minute EMF Electromotive Force SSE Error sum of squares

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List of Symbols

d Diameter r Radius ω Angular velocity v Velocity h Height

E Sum of kinetic and potential energy K Kinetic energy

U Potential energy m Mass

g Acceleration of gravity M Mass of the body

CM

v Speed of the mass center

CM

I Moment of inertia respect to the mass center R Radius of the sphere

λ Wave length

f Frequency t Time

s Distance v

δ Uncertainty of the speed s

δ Uncertainty of the distance t

δ Uncertainty of the time

s v

∂ ∂

Partial derivate of the speed respect to distance, partial uncertainty about distance

t v

∂ ∂

Partial derivate of the speed respect to time, partial uncertainty about time

S

d Distance between sensors (coils)

M

d Distance between magnetic fields

L

X Inductive reactance

L Self inductance of the coil

V Voltage

I Current B Magnetic field

0

μ Permeability of free space

a Radius of the circular conducting loop x Distance x

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1 Introduction

1.1 Background

Railway safety has become an important part of modern railway system. There has become many improvements in the field and new devices have been provided with new technology. Whole new systems and more safe tracks have been developed. Still accidents and errors occur every now and then. Material damages are always unfavourable and should be avoided as often as possible. But when it comes to human lives, every single fatal accident is too much, as zero tolerance in death accidents in the railways is tried to reach.

One of the worst train accidents ever happened in 21st of August 2006 in Egypt [1], when two passenger trains crashed. “One train, travelling at around 50mph (80km/h) slammed into the rear of

a slowly moving train after it passed through a stop sign, according to police at the scene.”

According to the local authority at least 58 people died and 143 were injured. It was said to be a human error. This accident could have been avoided if a better train detecting system with integrated speed limitation system had been used.

Mipro Oy is a Finnish medium sized company, which is specialized in customised automation. One of its main business fields is safety related systems in railways. At the moment Mipro has over 2500 kilometres railway line under its control in Finland [2]. There is 5741 km railway tracks in total [3]. Mipro Oy has already several years experience in developing the safety systems.

1.2 Improving Safety Level of the Axle Counting System

There are some important safety standards in industry as well as in other transport branches. All devices, and systems must fulfill those requirements. In railway branch an international standard set EN 5012x is used. It defines the Safety Integrity Levels (SIL) in railways. One definition for SIL according to Gulland [4] is following; the Safety Integrity Levels are a measure of the quality or dependability of a system, which has a safety function. In other words it is a measure of the confidence with which the system can be expected to perform that function. There are four Safety Integrity Levels defined, which are SIL 1, SIL 2, SIL 3 and SIL 4. The SIL 4 is the most dependable and the SIL 1 is the least. The difference between different levels is based on failure rate.

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Following table from Gulland explains how the failure rate is determined.

Table 1: Numerical Definitions of SILs for High Demand / Continuous Mode from BS EN 61508 For functions which have a high demand rate or operate continuously the accident rate is the failure rate λ. MTTF, Mean Time to Failure of the function is an alternative measure. Railway systems are continues mode systems.

SIL

Range of λ (failures per

hour) ~Range of MTTF (years) 2

4 10-9_{≤ λ ≤ 10}-8_100,000_{≥ MTTF > 10,000}

3 10-8_{≤ λ ≤ 10}-7_10,00_{≥ MTTF > 1,000}

2 10-7_{≤ λ ≤ 10}-6_1,000_{≥ MTTF > 100}

1 10-6_{≤ λ ≤ 10}-5₁₀₀_{≥ MTTF > 10}

Finnish Rail Administration, RHK is a part of the Finnish government, Ministry of Transport and Communications. It controls the safe of the equipments, devices and systems that are used in Finish railway system. Mipro Oy´s axle counting system is currently approved to SIL 3 level. In the Finnish railway system for all high density tracks SIL 4 level is required, and SIL 3 for low density [5].

The axle counter simulator helps to test the axle counter unit and gives a possibility to develop a better axle counter system, which has safety level of SIL 4. The importance of developing the axle counter is huge for Mipro. It increases the safety of those track parts where the SIL 3 system is currently used, because it gives a possibility to upgrade those system from SIL 3 to SIL 4. This gives Mipro Oy a possibility to market the product widely where ever the SIL 4 is demanded.

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2 Project Plan

The researcher´s role in inventing and building a mathematical model of the pulse created by double wheel sensor, and investigating the possible simulator types had following steps:

Step 1 Studying the type of simulator that should be built. Considering the positive and negative aspects in different simulator types. Giving an own recommendation or plan of a possible simulator type.

Step 2 Understanding the properties of the double wheel sensor. Step 3 Finding reliable data concerning train speeds.

Step 4 Building a mathematical model, which combines train speed and the trace of the train in sensor.

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3 Axle Counter System

Mipro Oy has built a railway safety system from different devices and components. Its main purpose is to count the axles of the train each time it passes a measuring device. Another purpose is to determinate the direction of the train. The axle counter system has following devices; a double wheel sensor, a TDP device and a counting system kernel. Figure 1 below illustrates the system.

Figure 1: Illustration of axle counting system with essential devices and the direction of an information flow.

3.1 Double Wheel Sensor

Double wheel sensor is built by Tiefenbach GmbH. As the device is not Mipro Oy´s own product, Mipro does not have the licence of the product. This is why opening the device box, measuring forces of its magnetic and electric fields, and other systematic examination of the product can not be done. This is a restricting factor for a researcher to perform his research. The only information available is based on Mipro Oy´s own documents [6-7].

The model of the sensor has the following type number: 2N59-1R-400-40. It has a two 2-wired DC inductive proximity sensors. It has been especially designed and constructed to use measuring wheel flange of the train at the speed up to 250 km/h. The sensor generates a current output. Each sensor consists of an oscillator, demodulator and trigger circuit. The high frequency oscillator has an open magnetic circuit. Sensor does not need a separate tuner. It produces an alternating current, which in turn generates an electromagnetic field. A metal object, (i.e. a railwheel flange) entering the airspace, affects this field, dampening the oscillator and reducing the magnitude of the output current [6-7].

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3.2 TDP Device

TDP device is Mipro Oy´s own product, which name is MiSO TDP. TDP increases safety and security by recognising the moving object in the track. The main function of the device is to continuously check the condition of the circuitry from the TDP device to the counting sensor of the double wheel sensor. TDP device is also used to recognise the metal object over the wheel sensor [6-7].

3.3 Counting System Kernel

Counting equipment is based on Safety Logic components produced by HIMA. Kernel is approved by TÜV, German and Austrian Technische Überwachungsvereine, technical inspection association [8]. HIMA is one of the biggest corporations, which offers safety-related automation solutions [9]. The devices used are approved up to safety level SIL3 [6-7].

The system integrates counting, monitoring and communication functions of the MiSO AXLE 3 Counting System. The counting system kernel has three specific devices: a rack system, a counter card and an I/O card.

3.3.1 Rack Systems

There are two different kinds of rack systems in a counting system kernel, either H41q or H51q. The H41q Counting System Kernel version is built in one 19” rack. The system has 2 integrated RS 485 communication channels. The system may contain safety redundant counter units up to 6 axle counting points, optional availability redundancy for central unit and communication channels [6-7].

H51q Counting System Kernel version is built in several 19” racks. The system may contain safety redundant counter units for up to 8 axle counting points per 19” rack. The whole system can contain up to 4 racks. Total amount of axle counting points is then 32 / central unit. Up to 6 additional RS485 communication channels or one ethernet communication channel or profibus (a combination is possible as well). There is optional availability redundancy for the central unit and communication channels [6-7].

3.3.2 Counter Card

Product number of counter card is F5220. It is a safety related 2 channel counter card for safety related proximity switches input signals 24V and 5V. Its counting range is from 0 Hz up to 1 MHz [6-7].

3.3.3 I/O Card(s)

System requires at least one safety related input card F3236 for signals from TDP device. Potential free output for relay control is made of safety related output card F3330. Both card types are automatically tested during operations for safety related errors [6-7].

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4 Determination of the Simulator Type

4.1 Definition of Simulated System

A simulator should simulate train wheel movements when it passes the double wheel sensor. There are two main possibilities how to perform the simulation. The first option is to simulate the train wheel and its actual movement over the double wheel sensor. The second option is to simulate the current that the double wheel sensor sends to TDP device after the train wheel has passed the double wheel sensor.

4.2 Limitations and Requirements of the Simulator

There are few limitations that need to be considered. The base of limitations are the physical properties of the wheel after it has passed the double wheel sensor and the properties of the double wheel sensor itself.

4.2.1 Speed Limitations

The simulator should simulate the real train speed, any speed from 0 km/h up to 300 km/h. According to VR the fastest Finnish trains, pendolinos drive at the speed of 220 km/h [10], which is the highest possible speed in Finnish railways. Nowadays there are only few track sections where the speed of 220 km/h is allowed. VR is a Finnish Railway company that is owned by the Finnish Government [11]. The fastest non official speed record in Finnish railways was achieved in summer 2006 in test drives of direct line Kerava-Lahti in Finland. The non official record was 242 km/h according to Suomen Kuvalehti magazine [12]. The foundations of the direct line Kerava-Lahti are build to manage trains driving up to 300 km/h. The Finnish railway tracks, which have normal geometry and inclination can be used for both passenger and freight train traffic. The special geometry and inclination of the tracks would only serve passenger trains, because due to their heavy mass, freight trains can not use those tracks [12].

4.2.2 Time Limitations

According to definition of demands of the axle counter documentation, the maximum speed for Tiefenbach double wheel sensor is 300km/h. It has to manage up to 800 axles per one continuous counting [13]. As it was mentioned earlier the maximum speed limit in Finnish railway is 220 km/h. As the highest possible speed for the simulator to manage, as a researcher I chose 300 km/h. It is the maximum speed where the double wheel sensor is documented to work properly. There is a possibility that axle counting system may be used at track sections with higher speed in the future.

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4.3 Mechanical Simulators

The purpose of a mechanical simulator is to simulate the physical properties of the wheel movement, when it has passed the double wheel sensor. Three ideas of some of the mechanical simulators were studied more deeply. Those were rotating disc, pendulum and sphere in the channel.

4.3.1 Rotating Disk

One possible simulator type is a rotating disc. Material of the disc should be non-magnetic, except for the smaller disc on the side, which is made of magnetic material, see the figure 3. Diameter of the smaller circle has to be approximately the same as the diameter of a real train wheel. The diameter of a Finnish Pendolino wheel is 890mm. In this research the diameter of the rotating disc was chosen to be 2,8 meters. The only requirement for diameter of the disc is that it should come to sensor and leave from it as horizontally as possible. That is the reason why the diameter is about three times bigger than the diameter of a train wheel.

Figure 3: Illustration of rotating disc simulator.

The maximum necessary velocity for simulator to rotate is 300 km/h. The angular velocityω can be calculated by using the relation between linear and angular speed,

r

v=ω (4.1)

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Formula for angular speed can be obtained from equation (4.1)

r v

=

ω (4.2)

Putting the number values d/2= 1,4 m and maximum speed v=300 km/h into equation (4.2) the value for angular speed becomes

s rad m s m h km / 5 , 59 4 , 1 3600 1000 300 ≈ ⋅ = ω (4.3)

The corresponding value for rotations per minute, rpm can be calculated by multiplying the angular speed (4.3) with min 1 2 60 ⋅ π s .

The following angular velocity was obtained

rpm s s rad 568 min 1 2 60 5 , 59 ≈ ⋅ ⋅ = π ω (4.4)

4.3.2 Pendulum with Metallic Mass

Another simulator type is pendulum, where a metallic disc or a disc, which has a metallic surface hangs in a string. The disc should have the same diameter as the real train wheel, d=890mm. The disc is lifted to height h and then it is let swing freely. Velocity at the bottom of the swing can be approximated as same way as dropping weight from the height. It can be calculated by using the conservation of the energy, the sum formula of kinetic and potential energy, see Young and Freedman [15]. The pendulum starts from rest, and the maximal kinetic energy is achieved at the bottom of the swing.

The first calculation is suggestive. The model of the system is simple. A mass hanging on the string. The model does not take any friction force, shape of the disc, mass of the disc or mass of the string into consideration. Instead it considers the disc as point mass. Model is presented in figure 4.

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Figure 4: Illustration of pendulum double wheel sensor simulator.

If the results of this simulation show that the simple model has to be studied more deeply, then a more accurate model will be calculated.

The sum formula of kinetic and potential energy, when only gravitation does the work [15] is 2 2 1 1 U K U K E = + = + (4.5)

Potential energy in the beginning is mgh

U₁ = . (4.6)

Potential energy at the bottom of the swing is zero 0

2 =

U . (4.7)

Kinetic energy in the beginning is 0

1 =

K . (4.8)

Kinetic energy at the bottom of the swing is 2 2 2 2 1 mv K = . (4.9)

Putting equations (4.6),(4.7),(4.8) and (4.9) into equation (4.5) 2 2 1 2 1 mv mgh E = = (4.10)

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Height where pendulum has to rise can be calculated from equation (4.10)

g v h 2 2 2 1 = (4.11)

Setting the number values v₂ =300km/h and g =9,81m/s2 into equation (4.11) an approximated

value of the height h₁ is received.

m s m s h km m h km h 354 / 81 , 9 2 3600 1 1 1000 300 2 2 1 _⋅ ≈ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ = (4.12)

4.3.3 Metallic Sphere in a Channel

The metallic sphere has radius about the same as the train wheel. The goal is to calculate height of the channel. The system was considered as ideal system, which does not have air resistance or any slowing down, pushing down forces, or friction force. The idea was to get approximation of the height of the channel that is needed to get the required velocity for the sphere at the bottom of the channel as it rolls down. After the calculation it will be possible to estimate the reasonable height that is required to build a simulator. See figure 5.

When the sphere rolls freely down without pushing force of any kind in the channel, the total mechanical energy has the same value at all points during the motion. This process is known as conservation of the energy. See Young and Freedman [15].

Figure 5: Illustration of metallic sphere in a channel sensor simulator, where the height of the channel is marked as h.

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E=K+U=constant (if only gravity works) E=total mechanical energy of the system K= total kinetic energy

U=total gravitational potential energy

The calculations were done by using the sum formula of kinetic and potential energy. 2

2 1

1 U K U

K

E = + = + (only gravity does work) [mentioned earlier in equation (4.5)] More information about the kinetic energy formula with rigid body with both translation and rotation and the condition for rolling without slipping, see Young and Freedman [16].

The kinetic energy formula with rigid body with both translation and rotation 2 2 2 1 2 1 _ω cm cm I Mv K = + (4.13)

The condition for no slipping is

r

v=ω . (4.14)

The moment of inertia for solid sphere is 2

5 2

MR

I = . (4.15)

The moment of inertia for thin-walled hollow sphere is 2

3 2

MR

I = . (4.16)

More information for the kinetic energy formula with rigid body with both translation and rotation, the condition for rolling without slipping and moments of inertia of various bodies, see Young and Freedman [16].

Using condition (4.14) and assuming that sphere is solid (4.15), putting these into equation (4.13) Kinetic energy for solid sphere is

2 2 2 5 2 2 1 2 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⋅ ⋅ + = r v MR Mv K cm (4.17) Assuming that R r= (4.18)

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18 and

v

v_cm = (4.19)

Putting assumptions (4.18) and (4.19) into equation (4.17) a kinetic energy for solid sphere is obtained. 2 10 7 cm Mv K = (4.20)

Using moment of inertia analogously in thin-walled hollow sphere (4.16), condition (4.14) Assumptions (4.18) and (4.19) in equation (4.13)

Kinetic energy for thin-walled hollow sphere is 2 6 7 cm Mv K = (4.21)

Potential energy in the beginning is mgh

U₁ = (4.22)

Kinetic energy in the beginning is 0

1 =

K , sphere starts from rest. (4.23)

Potential energy at bottom is zero, 0

2 =

U (4.24)

The height of the channel when the sphere is solid can be calculated by putting equations (4.20), (4.22), (4.23) and (4.24) into equation (4.5).

2 2 1 10 7 mv mgh = (4.25)

The height from equation (4.25)

g v h 10 7 2 2 1 = (4.26)

Changing km/h to m/s and putting numerical values to g and v₂ into equation (4.26)

m s m s h km m h km h 496 / 81 , 9 10 3600 1 1 1000 300 7 2 2 1 ≈ ⋅ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ = (4.27)

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The height of the channel when the sphere is thin-walled hollow can be calculated analogously by putting equations (4.21), (4.22), (4.23) and (4.24) into equation (4.5).

2 2 1 6 7 mv mgh = (4.28)

The height from equation (4.28)

g v h 6 7 2 2 1 = (4.29) m s m s h km m h km h 826 / 81 , 9 6 3600 1 1 1000 300 7 2 2 1 _⋅ ≈ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⋅ ⋅ ⋅ = (4.30) 4.3.4 Results

Results of pendulum (4.12) and sphere in the channel, (4.27) and (4.30) show that the heights of simulator types are absurd. There is no sense to start building a machine higher than few hundred meters. Even if the disc or sphere was pushed down in the pendulum or in the channel the height would still be at least tens of meters. These are the main reasons why both pendulum and the sphere in channel were not accepted as a simulator model.

The simulator where the disc rotates is more compact. The total height of the machine is approximately 2-3 meters. It can fit in a normal room. One problem with a rotating disc is the acceleration to desirable speed. See result (4.4). It is a fact that speed of the train is almost constant in every part of the train when it passes the measurement point. Because the train has huge mass the change of the speed is minimal. The speed has to be the same in all axles. There are at least two options how these conditions can be achieved.

1. The motor accelerates the disc during the first rotation into maximum speed 300 km/h. 2. The rotating disc is first accelerated freely to desirable speed and after that the whole motor and disc system is moved over the sensor.

Both options have big problems. The engine of the simulator can not be affordable if it satisfied the condition 1, because of the enormous acceleration. See appendix 1. The condition 2, lifting of the disc system and putting the disc in right phase into the sensor is also problematic. An accurate measurement and controlling of the system is quite expensive. These are the reasons why also the rotating disc is not acceptable as a simulator type.

As the result of the research of the simulator type the voltage simulator seemed the easiest and most affordable simulator type. It simulates the double wheel sensor to TDP device by sending pulses to TDP device. The simulator can be connected to pc for data recording or analysing. Also different types of pulses and pulse strings can easily be created with voltage simulator.

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5. Measuring the Train Speed

It was necessary to measure the reference speed of a train to be able to compare the speed of a double wheel sensor with reliable speed. Trains have speedometer, but in this measurement it was decided not to use it. Such a measurement would have been too difficult to implement as each time when the train passed the measuring point, the speed of the train would have had to be asked. There is train traffic 24 hours per day and there simply was not possibility nor allowance to observe the traffic all that time. This would only have been possible by calling the driver, which was practically impossible to arrange.

5.1 Location of the Test Center

The measurements were done in Mipro`s Vuolinko Test Center. It is located between Mikkeli and Mäntyharju, about 5 km south from Mikkeli railway station.

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5.2 Measurement Devices

The actual speed measurements were done with two Sick Laser photoelectric switches. Its model name was Sick WL12L-2B530.

The WL 12 L-2 photoelectric reflex switch is an optoelectronic sensor that is used to optically detect objects, animals and persons. For the operation a reflector is needed. Its wavelength is

nm

650 =

λ .The maximum scanning range is 0-18 meters. Signal sequence is f =2500Hz[19].

When the signal sequence is known, it is possible to calculate the pulse duration, see Nordling et al [20]. f t = 1 (5.1) s Hz t 0,0004 2500 1 ₌ = (5.2)

According to result (5.2) of equation (5.1) the pulse duration of the Sick Laser is 4 μs.

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5.3 Measurement Environment

All the measurements were implemented close to the railway track. Using laser and prism turned out to be an appropriate way to measure. The wireless equipment allowed a more simple way to study the phenomena. Without it a whole tunnel for wires and cables would have had to be built and train traffic would have had to be stopped due the construction work. The distance between switches was 86 meters, and the distance between switch and prism about 8 meters. See figure 8 and 9. The switches were connected to computer clock system. Measurement was following: When the train crossed the first laser, a timer started to run, and as the train hit the other laser, the timer stopped. The computer clock system measured the time when the train had travelled 86 meters. After the time and distance were measured it was easy to calculate the speed of the train v_photoe_._switch

by using the formula of average velocity,

t s v= ,

see Nordling et al [21].

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Figure 9: Sick photoelectric switch and prism in Vuolinko test place in April 2008. 5.4 Collecting Data

The train traffic, which went through the test center, could be divided in two different groups: 1. passenger traffic and 2. freight traffic. Data for the research was collected between 3rd and 23rd of April 2008. There were about 15 passenger trains and 15 freight trains per day passing the test center. The photoelectric switch measured all together 640 observations.

About 100 observations from the total amount of 640 were not acceptable. The photoelectric switch had registered some of these observations but the TDP device was unable to receive the pulse from double wheel sensor. Another explanation is that the photoelectric switch was very sensitive for changes in the environment such as wind, rain and smelting of ground frost. This is why the photoelectric switch registered trains that actually never passed.

In the calculation 240 observations were used. The maximum speed was about 145 km/h and the minimum speed was about 50 km/h. The place where the measurements where done has a speed limit of 140 km/h. See appendix 2 for used observation.

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5.5 Error Calculation of the Train Speed

There are always some uncertainties present in a measurement. One way to control the reliability of a measurement is to make an error calculation. In this measurement system there were few error sources. First was the measured distance between the photoelectric switch, which was 86 meters. It was measured with measuring coil and an approximated error of reading a measuring coil is ± 0,1 meters. Readings on the measuring coil were located every ten centimetres.

The second error source was a computer clock, which was programmed to check the status of laser measurement output, in other words if there was something between the laser and the prism, every 0,01 seconds. The system had two pairs of photoelectric switches, which means that the total error of time was ± 0,02 seconds.

The measured distance and time of the counting unit clock was assumed to be independent and random. The variables time and distance were independent, because they were measured with separate devices. Variables are random variables because the time and distance will vary from measurement to measurement as the experiment is repeated.

The best way to calculate the uncertainty of the speed v is to calculate the total uncertainty δ , v which is the quadratic sum of the partial uncertainties due to each of the separate uncertainties δ s and δ , see John R. Taylor [22]. t

Uncertainty in a Function of Several Variables according to John R. Taylor [21] is 2 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = z z q x x q q δ δ δ _L (5.3)

Train speed can be calculated with the formula of average velocity [20], which is defined as

t s

v= (5.4)

Setting variables (5.4) into equation (5.3) 2 2 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ∂ ∂ = t t v s s v v δ δ δ (5.5)

Total uncertainty of the speed is δ . v

Uncertainty of the distance is

m s=0,1

δ . (5.6)

Uncertainty of the time is

s t =0,02

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25 Partial uncertainty about distance is

t s v 1 = ∂ ∂ . (5.8)

Partial uncertainty about time is

2 t s t v − = ∂ ∂ . (5.9)

Error of the speed was calculated when the speed was 144 km/h. Distance is m s=86 . (5.10) Time is s t =2,15 . (5.11)

Setting value (5.11) into equation (5.8)

15 , 2 1 = ∂ ∂ s v (5.12)

Setting values (5.10) and (5.11) into equation (5.9)

2 15 , 2 86 − = ∂ ∂ t v (5.13)

Setting (5.6), (5.7), (5.12) and (5.13) into equation (5.5)

s m v 0,02 0,375 / 15 , 2 86 1 , 0 15 , 2 1 2 2 2 ≈ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛₋ _⋅ + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ _⋅ = δ h km s m/ 1,35 / 375 , 0 ≈

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26 ±0,0 ±0,2 ±0,4 ±0,6 ±0,8 ±1,0 ±1,2 ±1,4 ±1,6 50 55 60 65 70 75 80 85 90 95 100 105 110 115 120 125 130 135 140 145 speed (km/h) er ro r o f t h e sp eed ( k m/ h )

Error of the speed

Figure 10: The error of the speed versus speed shows how the error of the speed increases when the speed increases.

The figure 10 shows that as the speed increases also the amount of error increases. The error was still unsignificant, when the speed was 145 km/h, the error was only about 1% of the measured speed. It can be concluded that observed speed can be used as reference speed.

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6. Examination of the Pulses

FlukeView® ScopeMeter® is a commercial software, which helps to get more data and information out of portable oscilloscopes. With FlukeView® Scopemeter® it is possible to gather measurement data and analyze slow moving signals and other related events [23]. Mipro Oy uses FlukeView® ScopeMeter® version 3 program to collect data about the trains that pass the double wheel sensor. The program has been used years in Mipro Oy with satisfying results.

FlukeView® ScopeMeter® program has an interface based on windows. It has an “application window”, which contains the toolbar and menus, and “workbook windows” with oscilloscope picture and data summary of the pulses. Voltage level is placed on y-axel and time on x-axel.

Double wheel sensor has two coils, both of them create a magnetic field. The left coil is the main part of the left sensor. This sensor was named as sensor 1. The right sensor was named as sensor 2. See figure 11. Magnetic fields do not affect each other, because of their different frequency. Magnetic fields partly overlap each other. When Flukeview® oscilloscope is connected to double wheel sensor, the effect of the wheel as it passes the double wheel sensor can be seen from an oscilloscope. Pulses can be copied from the oscilloscope to a computer after which they can be monitored with FlukeView® ScopeMeter® program.

Figure 11: Illustration of double wheel sensor and its magnetic fields and connection between magnetic field of the sensors and pulse pair in Flukeview® ScopeMeter® program.

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6.1 Pulse Pictures

Double wheel sensor sends two pulses. The Flukeview® oscilloscope adds two extra pulses to the original pulses by creating two pulse pairs. Those pulse pairs can be seen in FlukeView® ScopeMeter® program. In the beginning both pulse pairs have a voltage level of about 6 V. As the wheel penetrates the magnetic field, the voltage level rises to maximum value that is between 9,2 V – 9,5 V. Pulses maintain the maximum voltage level as long as the wheel stays inside the magnetic fields. When the wheel leaves from the magnetic field, voltage level drops back to the starting level of about 6 V. See figure 11.

Pulses start to rise and fall at different time point. The direction where the train comes from determinates, which pulse will rise first. From figure 11 can be seen that pulse pairs have different maximum voltage values. Pulse pairs have approximately the same duration.

Figure 11: Picture of FlukeView® ScopeMeter® user window. There are two pair of pulses, totally four

pulses

6.2 Direction of the Train

Vuolinko test center is located in a track section where train traffic is coming from both directions. The main rule for train direction in Finland is that the trains, which are heading away from the Helsinki railway station are said to be going to north and all the trains, which are heading to Helsinki railway station are going to south. There are horizontal track sections, which go from west to east or from east to west, and it is agreed that these track sections are also said to go in south-north direction.

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6.2.1 North Heading Train

In this research by north heading train it is meant a train that comes first to sensor 1, and after that to sensor 2. On condition that the sensor 1 points to south and the sensor 2 to north. See figure 12. The direction of the train can also be seen in pulse figures. To be able to compare the pulse pairs, both of them were lifted to the same voltage level about 6 V, as normally they have a difference about 0,2 V – 0,3 V in the beginning of the pulse. The pulse pairs of the north heading trains had following properties: The first pulse pair always had a higher maximum voltage value than the second pulse. The voltage value of the first pulse was about 9,5 V as the one for the second pulse was 9,2 V. See figure 13.

Figure 12: Illustration of a north heading train with location of the nearest railway station and test center [24].

5,5 6,0 6,5 7,0 7,5 8,0 8,5 9,0 9,5 10,0 0,025 0,030 0,035 0,040 0,045 0,050 0,055 0,060 0,065 0,070 0,075 time (s) vol tage ( V )

1.pulse from upper pulse pair 2.pulse from upper pulse pair 1.pulse from lower pulse pair 2.pulse from lower pulse pair

Figure 13: Graph of pulse pairs at the speed of 79,57 km/h. First pulse pair has higher maximum voltage value than the second pulse pair, train is heading to north.

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6.2.2 South Heading Train

In this research by south heading train it is meant a train that comes first to sensor 2, and after that to sensor 1. On condition that the sensor 1 points to south and the sensor 2 to north. See figure 14. The direction of the train can also be seen in pulse figures. To be able to compare the pulse pairs, both of them were lifted to the same voltage level about 6 V, as normally they have a difference about 0,2 V – 0,3 V in the beginning of the pulse. The pulse pairs of the south heading trains had following properties: The second pulse pair always had a higher maximum voltage value than the first pulse. The voltage value of the first pulse was about 9,2 V as the one for the second pulse was 9,5 V. See figure 15.

Figure 14: Illustration of the south heading train with location of the nearest railway station and test center [24]. 5,5 6,0 6,5 7,0 7,5 8,0 8,5 9,0 9,5 10,0 0,025 0,030 0,035 0,040 0,045 0,050 0,055 0,060 0,065 0,070 0,075 time (s) v o lta g e ( V )

1.pulse from upper pulse pair 2.pulse from upper pulse pair 1.pulse from lower pulse pair 2.pulse from lower pulse pair

Figure 15: Graph of pulse pairs at the speed of 79,57 km/h. First pulse pair has lower maximum voltage value than the second pulse pair, train is heading to south.

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7. Preparing the Pulses for Modelling

7.1 Outlining the Observation Data

Observation data was imported from FlukeView® ScopeMeter® to Microsoft® Office Excel® 2003. The reason for importing was to make the data easier to handle. Also the calculation could be done more efficiently when observation data was collected to a spreadsheet. In Office Excel® data was first outlined and scaled. Outlining was implemented by following steps:

Step 1 Measuring devices saved the pulses each time a train passed the double wheel sensor. The original data was opened in FlukeView® ScopeMeter®.

Figure 16: Picture of Pulses in Scopeview® program.

Step 2 The Scopeview® pulse data was copied to Office Excel® as spreadsheet and was plotted in it. 5,5 6,0 6,5 7,0 7,5 8,0 8,5 9,0 9,5 -0,02 -0,01 0 0,01 0,02 0,03 time (s) vol ta g e ( V )

1. pulse from sensor 1 2. pulse from sensor 1 1. pulse from sensor 2 2. pulse from sensor 2

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Step 3 Extra pulses were removed after consulting the work team in Mipro Oy. It was known that double wheel sensor only sends two pulses, not four. The inner pulses of the pulse pairs were removed. This decision was based on an own interest of the Mipro Oy.

5,5 6,0 6,5 7,0 7,5 8,0 8,5 9,0 9,5 -0,02 -0,01 0 0,01 0,02 0,03 time (s) vo lt ag e ( V )

1. pulse from sensor 1 1. pulse from sensor 2

Figure 18: Extra pulses from pulse pairs were removed.

Step 4 Pulses were in different voltage level, the other pulse was lifted about 0,2 V – 0,3 V by adding a constant A to the all data points. Shape and duration of the pulses did not change after the procedure. The constant A was the mean value of the difference that pulses have in horizontal part at the beginning of the pulse. From figure 18 it can be seen the horizontal part at the beginning of the pulses were located between -0,02 s and -0,01 s. The x-axel was scaled so that both pulses would begin from time zero instead of starting from -0,02.

After these actions were implemented, the pulses could be combined in to one figure having a basic voltage level of around 6 V.

5,5 6,0 6,5 7,0 7,5 8,0 8,5 9,0 9,5 10,0 0,02 0,03 0,04 0,05 0,06 0,07 time (s) vo lt a g e ( V )

1. pulse from sensor 1 1. pulse from sensor 2

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7.2 Double Wheel Sensor as a Speedometer

Mipro was interested to know if it was possible to use a double wheel sensor as a speedometer. The double wheel sensor is not constructed to be used as a speedometer. It was interesting to try to find out whether it could be used also as a speedometer, because information about train speed would be useful. This far the double wheel sensor was only calculated the axles and determinated the direction of the train. To get reliable results the double wheel sensor speed of the double wheel sensor, v_double_wheel was compared to the speed of photoelectric switch v_photoe_._switch.

To be able to calculate the speed of the train with double wheel sensor, few assumptions were needed. The first assumption was that the magnetic fields were equally strong and located symmetrically in axle counting unit. The second assumption was that the wheel would act the same way in both sensors and theirs magnetic fields. The third assumption was that the distance between the sensors, d would be the same as the distance between the edge of the magnetic fields, _S d_M. See figure 20.

2 1

d_S

d_M

Figure 20: Illustration of double wheel sensor. Figure shows the distances d_S and d_M.

The reason why these assumptions were necessary was that Tiefenbach double wheel sensor is sealed and even if Mipro Oy uses these devices it does not have a permission to study it more closely. For example measuring the real distance between the coils, was not possible.

The coils were assumed to locate directly over adjustment tools of coils. The distance between coils is 10,3 cm, which is the same as the distance between the edges of the magnetic fields.

Time could be calculated from the pulse figures. The difference between measured speed and the calculated speed was copied into a spreadsheet.

Below is an example of an event about the calculation and how the comparison was made at the speed of vm =78,3km/hin 7,5 V.

The time points when the pulses from sensor 1 and sensor 2 rose up to the voltage level 7,5 V were saved. The time difference between those times was calculated.

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34 1 2 t t t= − Δ (7.1)

Taking time values from pulses to equation (7.1)

s s

s

t=0,0389 −0,0340 =0,0048

Δ (7.2)

The distance that corresponds the time points was the distance between front edges of the magnetic fields, dM. Now the constant velocity could be calculated. See figure 21.

m

d_M =0,103 (7.3)

The measured speed with double wheel sensor, vdoublewheel was achieved by setting the numerical

values (7.2) and (7.3) to equation of constant velocity,

t s v= . h km s m s m t d v M wheel double 21,458 / 77,25 / 0048 , 0 103 , 0 _≈ _≈ = Δ = (7.4)

The speed difference was calculated and copied to a chart.

h km h km h km v

vphotoe.switch − doublewheel =78,3 / −77,25 / =1,1 /

5,5 6,0 6,5 7,0 7,5 8,0 8,5 9,0 9,5 10,0 0,025 0,030 0,035 0,040 0,045 0,050 0,055 0,060 0,065 0,070 time (s) voltage (V)

pulse from sensor 1 pulse from sensor 2 time difference at 7,5 V

Figure 21: Calculation of the time in pulse pictures for the speed of double wheel sensor. Pulses had speed 78,3 km/h.

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7.3 Result of Comparing Velocities

Speed of the double wheel sensor was calculated in samples. Totally four samples were taken. The table 2 shows the information of the samples.

Table 2: Information of the pulse samples

sample number speed interval sample size train direction

1.sample 76,8 km/h - 86,2 km/h 50 north

2.sample 135,73 km/h - 137,6 km/h 50 north

3.sample 57,44 km/h - 74,48 km/h 30 south

4.sample 135,73 km/h - 138,52 km/h 30 south

The time difference of the pulses was calculated between 7 V and 8 V every tenth parts. See appendix 4.

After calculating 160 observations, a conclusion that there is no clear factor or function was made. That explained why the photoelectric switch speed and double wheel sensor speed differs. There is not any voltage level, which gives good approximation to speed. Speed from north heading trains was lower than original speed. Speed from south heading trains was higher than the original speed. The distance between the edges of the magnetic field was constant, as it was assumed. The difference between speeds can only depend on the fact that the pulses have a different duration. For the north heading trains the time of the pulse duration was longer and for south heading trains it was shorter.

Due to all these observations, it was clear that an axel counting unit could not be used to calculate the exact train speed. There was not an explanation found, why the speed differed almost randomly. But Mipro Oy was interested to find out the possibility to calculate the speed of passing trains with double wheel sensor even if the speeds varied up to 15 % compared to the original speed. One idea was to use double wheel sensor as a speedometer by compartmenting speeds in three categories. Those could be 0 – 60 km/h, 60 km/h – 120 km/h and 120 km/h – 220 km/h.

7.3.1 Pulse Duration

The results from double wheel sensor as a speedometer showed that the duration of the pulse from sensor 1 and sensor 2 differs. Difference of the pulse duration had to be investigated more deeply. Investigation was started by analysing what was known about double wheel sensor.

The double wheel sensor has two coil, that are connected to electric circuit. The coil creates a magnetic field, when there is a current in electric circuit. The coil has an inductive reactanceXL,

which is a self-induced emf that opposes any change in the current through the coil. See Young and Freedman [25]. Inductive reactance depends on alternating current and it can be calculated with formula

L f

X_L =2π (7.5)

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36

Coils in a double wheel sensor have different frequencies, 2

1 f

f ≠ (7.6)

otherwise they would interfere and disturb each others magnetic fields, which can not be allowed because it would prevent the device from working properly. This could cause severe damage as the main function of the device is to ensure railway safety. Because of the frequency difference (7.6), the coils have a different inductive reactance,

L

L X

X₁ ≠ ₂ (7.7)

The magnitude of the current can be calculated according to the formula of amplitude of voltage across an inductor, ac circuit. See Young and Freedman [25].

L

L IX

V = (7.8)

Current I can be solved from equation (7.8)

L L

X V

I = (7.9)

Using the relation that coils have different inductive reactance (7.7) in equation (7.9)

2 1 2 2 1 1 I I X V X V L L L L ≠ ⇒ ≠ _(7.10)

The formula of magnitude of magnetic field at distance x produced by circular loop of radius a. See Young and Freedman [26].

(

₂ ₂

)

3/2 2 0 2 x a Ia B + = μ (7.11)

From equation (7.11) can be seen that the magnitude of current had an effect on magnitude of magnetic field. 2 1 2 1 I B B I ≠ ⇒ ≠ (7.12)

The result (7.12) can be one explanation to why the duration of the pulses differed. But it is necessary to note that the result (7.12) is based on assumptions that sensor 1 and sensor 2 are otherwise identical, except for the difference of the frequencies.

This result lead to a conclusion that using double wheel sensor as a speed detector should be studied more deeply by verifying the assumptions by empiric research. Another possibility to verify the assumptions would be to get the permission from manufacturer to study the double wheel sensor and get more detailed information of the product.

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8 Modelling the Pulse

To make the pulse easier to produce and modify, a mathematical model of the pulse was made. Easy to produce means that the model of the pulse became simpler and there was not noise in the pulse. A mathematical function, which corresponds to the real pulse made the pulse more modifiable. It was possible to create a simulated pulse by changing values of speed variable. Simulated pulse corresponds to the original pulse, which the train creates when it travels at that specific speed.

The actual pulse was divided into three parts to make it easier to concentrate only on one part at the time.

The parts of the pulse were following:

1. Rising part of the pulse 2. Top of the pulse 3. Falling part of the pulse

8.1 The Top of the Pulse

This was a critical phase of the research, because of the relation between duration of the pulse at the top and speed of the train. The formula of constant velocity,

t s

v= , shows that the time is inversely

proportional to speed. As the speed increases, the time decreases.

Time difference between rising and falling pulse was calculated at voltage level of 9 V in sensor 1. In sensor 2 it was calculated analogously at the voltage level of 8,5 V. The duration was not calculated exactly at the top of the pulse, because all the pulses had noise at the top of the pulse, which would have done the estimation of time problematic. There was no need to model the top of the pulse more accurately. Sufficient accuracy was achieved when the duration between the rising and falling parts of the simulated pulse corresponded the duration of the original pulse, and top of the pulse stayed long enough over the voltage level of 9 V. That was the reason why lower voltage levels, 9V and 8,5 V were chosen to measurement points.

An example about the calculations Δt =(x₂,V₂)−(x₁,V₁) ) 0 , 9 , ( ) 0 , 9 , (x₂ x₁ t= − Δ ) 0 , 9 , 0355 , 0 ( ) 0 , 9 , 0541 , 0 ( − = Δt s t=0,0186 Δ See figure 22.

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38 5,5 6,0 6,5 7,0 7,5 8,0 8,5 9,0 9,5 10,0 0,02 0,03 0,03 0,04 0,04 0,05 0,05 0,06 0,06 0,07 0,07 time (s) vo lt a g e ( V )

pulse from sensor 1 pulse duration at 9 V

Figure 22: Pulse from sensor 1 and pulse duration at the voltage level of 9V presented in same graph. Speed of the pulse was 76,2 km/h.

After the pulse duration was calculated in 9 V in sensor 1 and 8,5 V in sensor 2, a plot time versus speed was made. See figure 23.

0,010 0,012 0,014 0,016 0,018 0,020 0,022 0,024 0,026 50 70 90 110 130 150 speed (km/h) ti me (s ) measured points

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39

A function, which explains the connection between time and speed had to be found. The process used to estimate the trend of the outcomes, was curve fitting. The best-fitting curve was obtained by using the method of least-square fit.

8.1.1 Least-Square Fit

To find the best function to describe observation points, an analytical method least-squares fit, also known as linear regression was used. See Milton et al [27].

It can be assumed that there is a set of measured points(x₁,y₁),K,(xN,yN). Those points form a scatter diagram. It is possible to draw straight lines through the diagram. Lines can be written as

x B A

yi = i + i . (8.1)

Parameters A and B, for which the straight line fits the “best” with the data, need to be found. “Best” in a sense of line coming as close as possible to all data points. The difference between actual values and predicted values from estimated model is defined as residual, e . The parameter A _i

and B are chosen so that they minimize the sum of squared residuals

∑

= = − − = n i i i n i i y A Bx e 1 2 1 2 ₍ ₎ _. _(8.2)

Another common name used to mean the same as sum of squared residuals is a sum of squared errors, SSE.

To better understand how the residuals are connected to least-square fit, following example can be illustrated: It is assumed that there is a sample of five data points. The least-square fit selects the

line that causes 2

5 2 4 2 3 2 2 2 1 e e e e

e + + + + to be as small as possible. See figure 24.

1 e 2 e 3 e 4 e 5 e ) , (x₁ y₁ ) , (x2 y2 ) , (x₃ y₃ ) , (x₄ y₄ ) , (x₅ y₅

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40

The minimum of SSE is determined by differentiating this function with respect to variables A and B and setting them to zero.

Differentiating SSE with respect to A and B

∑

= − − − = ∂ ∂ n i i i A Bx y A SSE 1 ) ( 2 (8.3) i n i i i A Bx x y B SSE

_∑

= − − − = ∂ ∂ 1 ) ( 2 (8.4)

Partial derivates (8.3) and (8.4) were set to be equal to zero and after reducing them, the equations could be written as:

∑

= = = + n i i n i i y x B nA 1 1 (8.5)

∑

= = = = + n i i i n i i n i i B x x y x A 1 1 2 1 (8.6)

Equations (8.5) and (8.6) could be solved to obtain variables A and B.

2 1 1 2 1 1 1 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − =

∑

= = = = = n i i n i i n í n i i n i i i i x x n y x y x n B (8.7) x B y A= − (8.8)

8.1.2 First Order Linear Fit

The first order linear least square fit was the first method used when trying to find a function, which would match with data points. Calculation was done by using the formulas (8.5) and (8.6).

Summary of observation statistic of pulse from sensor 1. 92 = n (8.9)

∑

x=8868,73 (8.10)

∑

y= 55051, (8.11)

∑

2 =917164,91 x (8.12)

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41

∑

2 = 0268870, y (8.13)

∑

xy=142,9726 (8.14) 0169 , 0 = y (8.15) 39924 , 96 = x (8.16)

Setting values (8.9), (8.10), (8.11), (8.12), (8.13) and (8.14) into equation (8.7)

0001 , 0 ) 73 , 8868 ( ) 91 , 917164 ( 92 ) 5505 , 1 )( 73 , 8868 ( ) 9726 , 142 ( 92 2 2 ₋ ≈− − = ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ ⎟⎟ ⎠ ⎞ ⎜⎜ ⎝ ⎛ − =

∑

x x n y x xy n B (8.17)

Setting values (8.15),(8.16) and (8.17) into equation (8.8) 0269 , 0 ) 39924 , 96 )( 0001 , 0 ( 0169 , 0 − − = = − = y Bx A (8.18)

Setting results (8.17) and (8.18) into equation (8.1) following estimated regression equation was obtained.

x yˆ=0,0269−0,0001

The regression line can be seen in figure 25.

0,000 0,005 0,010 0,015 0,020 0,025 0,030 0,035 0,040 0 50 100 150 200 250 speed (km/h) tim e (s) measured points 1.order linear fit

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42

8.1.3 Higher Order Polynomial Fits

Higher polynomial fits for observation data were tested. The idea was to get many fit models, which could be ranked. The calculation of the second and higher polynomial fit follows the same basic idea that was introduced with first order linear fit. Calculating higher order polynomial fits the same way as the first order fit, would have required much extra calculation and time. There were many mathematical software available, with which the polynomial fits could be done efficiently. The software that was used in doing fits was Data Master 2003. Data Master 2003 is a free software for automation of test and measurement systems, data acquisition, processing and analysis [28].

The second and third order polynomial fits were implemented. Fourth and higher order polynomial fits were left outside, because a simpler model was required.

8.1.4 Exponential Fit and Power Fit

The next step was to find exponential and power fits for observation points. Exponential function can be written as

Bx

Ae

y= (8.19)

where A and B are constants. Power function can be written as

D

Cx

y= (8.20)

where C and D are constants.

It is not possible to calculate exponential or power fit the same way as polynomial least-square fits, because y is not linear inx. Variables x and y have a nonlinear relation in equations (8.19) and (8.20). The inverse logarithmic function was used to linearize exponential and power data. See Taylor [29]. Following conversion of nonlinear relation to linear relation was done.

Exponential fit of sensor 1 data

A logarithm was taken of both sides of exponential function (8.19) ) ln( ln Bx e A y= ⋅ Bx e A y ln ln ln = + Bx A y= ln + ln (8.21)

Changing the variables of the equation (8.21):

y Y =ln (8.22) A L=ln (8.23) L Bx Y = + (8.24)

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43 Then Y is linear in x, in other words ln is linear in x. y

The observation points were linearized by taking a natural logarithm of time, (y variable). Nothing was done to speed, (variable x). The ln time versus speed scatter diagram was plotted. First order linear fit was made. See figure 26.

-4,6 -4,5 -4,4 -4,3 -4,2 -4,1 -4 -3,9 -3,8 -3,7 0 20 40 60 80 100 120 140 160 speed (km/h) ln [ti m e (s )] measured points

1. order linear regression

Figure 26: Graph of linearized observation points, exponential function was linearized and first order linear fit of linearized observation points.

The equation for linear regression was 478 , 3 0064 , 0 lny=− x− (8.25)

Identifying the constants A and B was done by comparing the linearized exponential function L

Bx

Y = + [mentioned earlier in equation (8.24)]

to linear regression line equation (8.25). Directly from equations (8.24) and (8.25) can be seen that

x Bx=−0,0064 ,

after reducing the following value was obtained 0064 , 0 − = B . (8.26)

Constant A can be calculated from equationL=−3,478after using relation (8.23) the following equation is obtained

478 , 3

lnA=− constant A can be solved 478 , 3 − = e A =0,031 (8.27)

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44

Putting values (8.26) and (8.27) into equation (8.19) the following expression for exponential least-square fit was obtained.

x

e

y=0,03 −0,0064 (8.28)

Power fit of sensor 1 data

A logarithm was taken of both sides of power function (8.20) ) ln( ln D Cx y= D x C y ln ln ln = + x D C y ln ln ln = + (8.29)

Changing the variables of the equation (8.29):

y Y =ln (8.30) C M =ln (8.31) x X =ln (8.32) M X D Y = ⋅ + (8.33)

Then Y is linear in X, in other words ln is linear iny lnx.

The observation points were linearized by taking a natural logarithm of time, (y variable) and speed, (x variable). The ln time versus ln speed scatter diagram was plotted. First order linear fit was made. See figure 27. -4,6 -4,5 -4,4 -4,3 -4,2 -4,1 -4 -3,9 -3,8 -3,7 4 4,2 4,4 4,6 4,8 5 ln [speed(km/h)] ln [ tim e (s )] measured points

1. order linear regression

Figure 27: Graph of linearized observation points, power function was linearized and first order linear fit of linearized observation points.

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45 The equation for linear regression line is

1748 , 1 ln 6448 , 0 lny=− x− . (8.34)

Identifying the constants C and D was done by comparing the linearized power function

M X D

Y = ⋅ + [mentioned earlier in equation (8.33)]

to linear regression line equation (8.34).

Directly from equations (8.33) and (8.34) can be seen that

x X

D⋅ =−0,6448⋅ln ,

after reducing the following value was obtained 6448 , 0 − = D (8.35)

Constant C can be calculated from equationM =−1,1748 after using relation (8.31) the following equation is obtained

1748 , 1 lnC=−

constant C can be solved 309 , 0 1748 , 1 ₌ = − e C (8.36)

Putting values (8.35) and (8.36) into equation (8.20) the following expression for exponential least-square fit was obtained.

6448 , 0 309 , 0 ⋅ − = x y (8.37)

Exponential and Power fit of sensor 2 data

Similar calculations as doing calculations of sensor 1 data, were done to obtain the exponential and power functions to pulses from sensor 2.

The exponential function was

x

e

y=0,03 −0,0065 . (8.38)

The power function was -0,6523 317

,

0 x

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8.1.4 Results of Fits

All the calculation was done with the data from both sensor 1 and sensor 2. The data and the result were almost the same as which is the reason why all the decisions which were made concerning data of pulses from sensor 1, were suitable for pulses from sensor 2. See results from appendix 5. All the three linear polynomial fittings had flaws. The first order linear fit went negative when the speed was about 250 km/h. The distance between the edges of the magnetic fields was constant, it was about 0,1 m. The speed was set to be 250 km/h. Using the formula of average velocity,

v s t= ,

the time at the speed of 250 km/h was

s m s h km m t 0,0014 1000 3600 / 250 1 , 0 _⋅ ₌ = .

The result is above zero.

With the same analogy as the first order polynomial linear fit, the third order polynomial fit went to negative when the speed was about 260 km/h. The second order polynomial fit was a parable, which started to open upwards. It could not be possible that time needed to travel a specific distance would increase as the speed increases. Due to these facts it was easy to decide that the first order, second order and third order fits were not acceptable. See figure 28.

0,000 0,005 0,010 0,015 0,020 0,025 0,030 0,035 0,040 0 50 100 150 200 250 spe e d (km/h) ti m e ( s ) measured points 2. order polynomial fit 3. order polynomial fit 1. order linear fit