Estimation of Radial Runout

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Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Estimation of Radial Runout

Examensarbete utfört i Reglerteknik vid Tekniska högskolan i Linköping

av

Martin Nilsson

LITH-ISY-EX--07/3949--SE

Linköping 2007

Department of Electrical Engineering Linköpings tekniska högskola

Linköpings universitet Linköpings universitet

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Estimation of Radial Runout

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Martin Nilsson

LITH-ISY-EX--07/3949--SE

Handledare: Frida Eng

isy, Linköpings universitet Henrik Jansson Scania CV AB Tor Langhed Scania CV AB Katrin Strandemar Scania CV AB Examinator: Thomas Schön

isy, Linköpings universitet

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Avdelning, Institution Division, Department

Division of Automatic Control Department of Electrical Engineering Linköpings universitet

SE-581 83 Linköping, Sweden

Datum Date 2007-04-12 Språk Language Svenska/Swedish Engelska/English Rapporttyp Report category Licentiatavhandling Examensarbete C-uppsats D-uppsats Övrig rapport

URL för elektronisk version http://www.control.isy.liu.se http://www.ep.liu.se/ ISBN — ISRN LITH-ISY-EX--07/3949--SE Serietitel och serienummer Title of series, numbering

ISSN —

Titel Title

Estimering av orunda hjul Estimation of Radial Runout

Författare Author

Martin Nilsson

Sammanfattning Abstract

The demands for ride comfort quality in today’s long haulage trucks are constantly growing. A part of the ride comfort problems are represented by internal vibrations caused by rotating mechanical parts. This thesis work focus on the vibrations

generated from radial runout on the wheels. These long haulage trucks travel

long distances on smooth highways, with a constant speed of 90 km/h resulting in a 7 Hz oscillation. This frequency creates vibrations in the cab, which can be found annoying. To help out with the vibration diagnosis when a truck enters a mechanical workshop, this work studies methods for radial runout detection using the wheel speed sensors.

The main idea is to represent the varying radius signal with a sinusoid, where the calculations are based on Fourier series. The estimated radial runout value is then the amplitude of the sinusoid. In addition to the detection part, the work also present results regarding how the relative phase diﬀerence between two wheels with radial runout eﬀects the lateral motion of the cab.

This thesis work was performed at Scania CV AB in Södertälje, Sweden and all measurements have been full scale experiments on real trucks.

Nyckelord

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Abstract

The demands for ride comfort quality in today’s long haulage trucks are constantly growing. A part of the ride comfort problems are represented by internal vibrations caused by rotating mechanical parts. This thesis work focus on the vibrations generated from radial runout on the wheels. These long haulage trucks travel long distances on smooth highways, with a constant speed of 90 km/h resulting in a 7 Hz oscillation. This frequency creates vibrations in the cab, which can be found annoying. To help out with the vibration diagnosis when a truck enters a mechanical workshop, this work studies methods for radial runout detection using the wheel speed sensors.

The main idea is to represent the varying radius signal with a sinusoid, where the calculations are based on Fourier series. The estimated radial runout value is then the amplitude of the sinusoid. In addition to the detection part, the work also present results regarding how the relative phase diﬀerence between two wheels with radial runout eﬀects the lateral motion of the cab.

This thesis work was performed at Scania CV AB in Södertälje, Sweden and all measurements have been full scale experiments on real trucks.

Sammanfattning

Kravet på körkomforten i dagens dragbilar ökar ständigt. En del problem vad gäller körkomforten för de här fordonen kommer från interna vibrationer som har sitt ursprung i roterande delar. I det här arbetet ligger fokus på vibrationer gener-erade från orunda hjul. Dessa dragbilar kör på bra vägar och håller en konstant hastighet av 90 km/h vilket resulterar i en 7 Hz vibration. Denna frekvens skapar vibrationer i hytten vilket upplevs som störande. För att hjälpa verkstäderna att hitta orsaken till vibrationerna har det här arbetet undersökt hur man utifrån hjulhastighetssensorerna kan detektera vilka hjul som är orunda.

Idén är att med hjälp av fourierserier representera radievariationerna med en sinuskurva, där amplituden är måttet på orundhet. Förutom själva detekteringen av orunda hjul innehåller arbetet också studier för hur den relativa fasskillnaden mellan två orunda hjul påverkar hyttens vibrationer.

Arbetet har utförts på Scania CV AB i Södertälje och alla mätningar har gjorts på riktiga fordon.

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Acknowledgments

I would like to thank the people at the RTCD division, all included, for being friendly and helpful throughout my work. Special thanks goes out to my super-visors at Scania; Henrik Jansson, Tor Langhed and Katrin Strandemar for their valuable contribution to my thesis work. I would also like to thank the supervisor Frida Eng and the examiner Thomas Schön from Linköping University for their feedback of the work when they visited Scania and for all the help with the report. Marcus Palm deserves a great thank you for his time and discussion concerning this report. Last but not least I would like to thank my family for being there.

Södertälje, April 2007 Martin Nilsson

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Introduction

The demands for ride comfort quality in today’s long haulage trucks are constantly growing. A today existing problem is annoying vibrations in the cab when traveling speed is approximately 90 km/h caused by wheels with radial runout. This is a known problem for Scania as it is for its competing companies. This master thesis study some methods for detecting radial runout wheels, using the trucks already existing wheel speed sensors.

1.1 Background

A typical truck wheel with center oﬀset will cause vibrations of frequency 7 Hz when rotating at approximately 90 km/h. The vibration is transmitted through the chassis up to the cab, where it easily can be felt by the driver. The problem is mostly concentrated to the so called long haulage trucks because they travel long distances on smooth highways with a constant speed of 90 km/h. Of course the problem with radial runout wheels will in this case be disturbing for the drivers.

When a truck with vibration problems enters a mechanical workshop the source of the vibrations is often hard to determine. Internal vibrations not caused by road unevenness can have its source from a lot of diﬀerent things like engine and propeller shaft unbalances or other rotating mechanical parts. Even though the wheels are suspected the equipment for measuring the wheels are quite expensive and the workshops does not always have access to the equipment. This causes the diagnosis to be rather inaccurate and time consuming. With no analysis tools, telling what is wrong the work can end up in changing faultless components which shows no result. Even if the wheels are changed one at the time to ﬁnd the vibration problem it will be a very time consuming work when a standard long haulage truck can easily hold up to six or more tires.

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2 Introduction

1.2 Idea

The main vibration problem arises from the wheel’s ﬁrst order oscillation due to axis of rotation oﬀset δ, illustrated in Figure 1.1. Typical for a long haulage truck is a traveling speed of 90 km/h and considering a truck wheel with a 0.5 m radius this results in a 7 Hz oscillation. If the longitudinal speed v of the wheel contact point with the road surface is considered constant this results in a varying angular velocity. The varying radius can be approximated from the recorded angular velocity ω over one revolution by

r(θ) = v

ω(θ) (1.1)

where r(θ) is the distance from axle center to road contact point for θ∈ [0 2π]. Example 1.1 shows how a typical result from a radial runout calculation could be presented.

Figure 1.1. Illustration of the motion of an axle attached on a wheel with radial runout

(center oﬀset).

Example 1.1: Measured wheel speed signal converted to radius

Figure 1.2 shows the radius (1.1) for ten revolutions of the wheel. Each revolution is calculated for θ∈ [0 2π], which implies that the revolutions are placed on top of each other. The repeatability in the radius signal is signiﬁcant.

The oﬀset δ can be estimated as

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1.3 Goal and Purpose 3 0 50 100 150 200 250 300 350 −0.01 −0.005 0 0.005 0.01 r (θ )[ m ] θ [deg]

Figure 1.2. This plot shows the radius approximation from 10 revolutions on top of

each other. The dashed black line is the approximated sinusoid from the radius, based on Fourier series.

where a and b are the ﬁrst order Fourier series coeﬃcients given by

a = 1 π 2π 0 r(θ) cos(θ)dθ (1.3) b = 1 π 2π 0 r(θ) sin(θ)dθ (1.4)

Figure 1.3. This plot shows the corresponding estimated δ for each revolution in Figure

1.2 calculated according to (1.2).

1.3 Goal and Purpose

The goal of this thesis is to find a method that automatically can detect wheels with radial runout using only the wheel speed sensors of the truck. From a ride comfort point of view, the relative phase difference between radial runout wheels, attached on the same axle have a big effect. For that reason this thesis will also study the effect the radial runout wheels relative phase have with respect to the cab’s lateral vibrations.

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4 Introduction

The purpose of this thesis is to use the information for on-board diagnosis and to minimize the vibration level in the cab by optimizing the relative phase of the wheels.

1.4 Scania

This master thesis has been performed on Scania CV AB in Södertälje, department of Truck Development, group of Vehicle Dynamics, RTCD. Scania is a global company with operations in Europe, Latin America, Asia, Africa and Australia. Scania is the world’s third largest make for heavy trucks and the world’s third largest make in the heavy bus segment. The group RTCD have two major working ﬁelds in vehicle dynamics, handling and comfort, where the problem with cab vibration due to wheels with radial runout concerns the comfort ﬁeld.

1.5 Thesis Outline

Chapter 2 The radial runout results in a varying radius, represented by a

sinu-soid and the amplitude of this sinusinu-soid will be the radial runout value. This chapter studies diﬀerent methods for estimating this amplitude and provides the underlying theory. The main idea is to use Fourier analysis, which has also been used in [2]. Another approach involves adaptive signal processing where an algorithm for continuous adaptation is used, the Kalman ﬁlter.

Chapter 3 The estimation of radial runout for each wheel is based on the varying

angular velocity. This chapter studies the wheel speed sensors of the truck. Since no additional sensors are used, the wheel speed sensors are our main source of information for determining both radial runout and phase diﬀerence between two wheels. These wheel speed sensors are a part of the trucks brake system. In Section 3.2 the way of creating wheels with radial runout is explained followed by laser reference measurements of the wheels. In the end of the chapter, the phase calculation is done.

Chapter 4 An objective in this thesis is to connect phase diﬀerences from wheels

with radial runout to vibration levels of the cab during ride. This chapter studies how to determine the cab vibrations from the windscreen accelerom-eter. To get a measure of the vibration levels, the power spectral density (PSD) is calculated from the accelerometer data. By comparing the magni-tude of this PSD from the same frequency interval in each test ride and at the same time look at the phase diﬀerence, some conclusions regarding ride comfort can be drawn.

Chapter 5 This chapter shows the radial runout results from the measurements

using calculations based on the theory in Chapter 2. The ﬁrst part considers some technicalities in the calculations and in Section 5.1.2 the results from the experiments are shown and discussed. Section 5.2 deals with the sensor

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1.5 Thesis Outline 5

fault, which includes theory of the sensor fault and a method of determine its appearance. In Section 5.2.3 an attempt to explain the diﬀerence between the

δ values from the laser and the calculation with the sensor fault is performed.

In Section 5.3, some words regarding the use of the Kalman ﬁlter is given.

Chapter 6 In an attempt to summarize this thesis, the conclusions that was

drawn throughout the past ﬁve months of work are given in this chapter. As always you do not have enough time to study every aspect of the problem. This leaves a few question marks which need to be studied. This chapter also speciﬁes the areas which could be improved by future work.

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6 Introduction

For Knowledge

———————————————————————————————————– An important thing to mention is that some information used in this project is

strictly confidential and cannot be revealed in the public version of this report.

Due to this requirement some values are changed from the true values without eﬀecting the integrity of the report.

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Chapter 2

Amplitude Estimation of

Periodic Signals

The radial runout results in a varying radius, represented by a sinusoid and the amplitude of this sinusoid will be the radial runout value. This chapter studies diﬀerent methods for estimating this amplitude and provides the underlying theory. The main idea is to use Fourier analysis, which has also been used in [2]. Another approach involves adaptive signal processing where an algorithm for continuous adaptation is used, the Kalman ﬁlter.

2.1 Fourier Series

Periodic signals like the radius (1.1), see Figure 1.2, are suitable to express as Fourier series, see, for example, [3] for background theory. A Fourier series is an expansion of a periodic function, in this case the radius r(t), in terms of an inﬁnite sum of sines and cosines.

r(t) = R0+

∞ k=1

R_ksin(kω1t + ϕk) (2.1)

where ω1 is the fundamental frequency

ω1=

2π

T , T = period (2.2)

and R0 is the mean value of r(t)

R0= 1 T T 0 r(t)dt (2.3) 7

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8 Amplitude Estimation of Periodic Signals

Using standard trigonometric rules we get:

R_ksin(kω1t + ϕk) = Rksin ϕkcos(kω1t) + Rkcos ϕksin(kω1t)

= A_kcos(kω1t) + Bksin(kω1t) (2.4) where A_k = R_ksin ϕ_k (2.5) and B_k = R_kcos ϕ_k (2.6)

By introducing (2.4) in equation (2.1) the alternative Fourier series is given by

r(t) = R0+

∞ k=1

[A_kcos(kω1t) + Bksin(kω1t)] (2.7)

The Fourier series coeﬃcients A_k and B_k can be calculated as follows:

A_k = 2 T T 0 r(t) cos(kω1t)dt (2.8) B_k= 2 T T 0 r(t) sin(kω1t)dt (2.9)

The amplitude R_k is calculated as:

R_k =

A2_k+ B_k2 (2.10)

and the phase

ϕ_k= ⎧ ⎪ ⎨ ⎪ ⎩ arctan Ak Bk if B_k ≥ 0 arctan Ak Bk + π if B_k < 0 (2.11)

2.1.1 Change of Variables

In this case, the radius as a function of the angle θ is preferable because it is actually the radius oﬀset of each revolution that is interesting. The angle θ can however be written as a combination of frequency ω and time t as θ = ωt (dθ =

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2.2 Time Domain Methods 9 r(θ) = R0+ ∞ k=1 [A_kcos(kθ) + B_ksin(kθ)] (2.12) R0= 1 π 2π 0 r(θ)dθ (2.13) A_k = 1 π 2π 0 r(θ) cos(kθ)dθ (2.14) B_k = 1 π 2π 0 r(θ) sin(kθ)dθ (2.15)

2.1.2 Radius Approximation Used for

δ Estimation

For estimation of the wheel radius runout, only the ﬁrst order Fourier series coef-ﬁcients are used, implying that the varying radius is approximated as

˜ r(θ) = R0+ A1cos(θ) + B1sin(θ) (2.16) and δ is δ = A2 1+ B12 (2.17)

2.1.3 Diﬀerence Between

h and δ

In the proceeding chapters both h and δ are used when terms of radial runout is discussed. A short explanation of the difference: when speaking of detected radial runout, δ is referring to the center offset, i.e., the offset value that the calculation shows. h values are related to the different Fourier series coefficients which represents the different shapes of a wheel with radial runout, illustrated in Figure 2.1.

h_i=

A2

i + Bi2 (2.18)

The radial runout (center oﬀset) value δ = h1, i.e., the ﬁrst harmonic oscillation

of the radius as shown in (2.17).

2.2 Time Domain Methods

The Fourier series is a frequency based method. When searching for other methods not involving Fourier series, time based methods were considered. In this study, a recursive ﬁlter was used, the Kalman ﬁlter.

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Figure 2.1. The diﬀerent shapes of a wheel based on the h matching. 1. The study in

this master thesis;h1 orδ (oﬀset wheel) 2. h2 (oval-shaped)3. h3 (out-of-round)4. h4

(square-shaped).

2.2.1 Regression Model

When using the Kalman ﬁlter to estimate the amplitude and phase of a sinusoid it is preferable to represent the signal as a linear regression. The radius (1.1) written in a linear regression form is given by:

r(t) = ϕT(t)θ(t) + e(t), (2.19)

where r(t) is the radius signal and ϕ(t) is the regression vector. The vector θ(t) contains the parameters that characterize the connection between ϕ(t) and r(t). The variable e(t) represents measurement errors. In this study the model will be a sinusoid with a certain amplitude and a certain phase which should reﬂect the center oﬀset value δ and the phase of the radial runout wheel. For further reading on adaptive signal processing, see [1].

Sine Model of the Radius

The varying angular velocity gives a varying radius (r(t)) which is modeled as a sinusoid:

r(t) = h1sin(ωt + φ) = a sin ωt + b cos ωt (2.20)

where

a = h1cos φ (2.21)

b = h1sin φ (2.22)

Written in the form (2.19) gives

ϕ(t) = sin ωt cos ωt (2.23)

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2.2 Time Domain Methods 11 and θ = a b (2.24) The method of least squares is used to minimize the quadratic residual:

V_N(θ) = 1 N N t=1 [r(t)− ϕT(t)θ]2 (2.25) The estimate becomes

ˆ θ_N = arg min θ VN(θ) = R −1 N fN (2.26) where R_N = 1 N N k=1 ϕ(k)ϕT(k) (2.27) f_N = 1 N N k=1 ϕT(k)r(k) (2.28)

From the estimated θ both amplitude and phase can be calculated as:

δ = h1= a2_{+ b}2 _(2.29) φ = arctan b a (2.30)

2.2.2 Kalman Filter

The Kalman ﬁlter is a recursive estimator. This means that only the estimated state from the previous time step and the current measurement are needed to compute the estimate for the current state.

With the Kalman ﬁlter, not only the regression model is used, but also a model of the parameters. Let ω(t) be the change in parameter values θ(t):

θ(t + 1) = θ(t) + ω(t) (2.31) Collect the equations (2.31) and (2.19):

θ(t + 1) = θ(t) + ω(t) (2.32)

y(t) = ϕT(t)θ(t) + e(t) (2.33)

Here, ω(t) and e(t) are assumed to be independent stochastic variables so that,

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Algorithm 2.1: Adaptation with Kalman Filter

The update of the parameters is given recursively by:

ˆ θ(t) = ˆθ(t− 1) + K(t)[y(t) − ϕT(t)ˆθ(t− 1)] (2.34) K(t) = P (t− 1)ϕ(t) R(t) + ϕT_{(t)P (t}− 1)ϕ(t) (2.35) P (t) = P (t− 1) − P (t− 1)ϕ(t)ϕ T_{(t)P (t}_{− 1)} R(t) + ϕT_{(t)P (t}_{− 1)ϕ(t)} + Q(t) (2.36) Detailed information of the equations and the use of Kalman ﬁlter can be read in [1].

The ﬁlter starts at time t = 0 with the initial mean values ˆθ(0) and its initial

covariance P (0). When lacking information of θ(0) a common choice is ˆ

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Chapter 3

Experiment Setup for Radial

Runout and Phase

Determination

The estimation of radial runout for each wheel is based on the varying angular velocity. This chapter studies the wheel speed sensors of the truck. Since no additional sensors are used, the wheel speed sensors are our main source of in-formation for determining both radial runout and phase diﬀerence between two wheels. These wheel speed sensors are a part of the trucks brake system. In Sec-tion 3.2 the way of creating wheels with radial runout is explained followed by laser reference measurements of the wheels. In the end of the chapter, the phase calculation is done.

3.1 Wheel Speed Sensor

The way to determine the speed of each wheel is by using the wheel speed sen-sors. On each wheel axle there is a toothed ferromagnetic ring, see Figure 3.1(a), together with an inductive sensor, see Figure 3.1(b). This inductive sensor gives a pulse of voltage each time a tooth is passing by. To get a good resolution of the signal the sampling frequency is set to 20 000 Hz.

3.1.1 Determination of Wheel Speed

In Figure 3.1(c) the measured voltage pulses from one of the wheel speed sensors is shown. The period of these pulses speciﬁes the time between two neighbour-ing teeth on the ferromagnetic rneighbour-ing and can be determined by ﬁndneighbour-ing the zero transition points in the voltage pulses. Every second zero transition point gives a time stamp. From these time stamps the angular velocity ω of the wheel can be determined as:

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14 Experiment Setup for Radial Runout and Phase Determination

(a) The ferromagnetic ring at-tached to the trucks wheel axles. This model has a hun-dred teeth all around.

(b) Schematic of the inductive sensor. The voltage ei is di-rectly proportional to the peri-odic variations in the magnetic ﬂux. 5.644 5.6445 5.645 5.6455 5.646 5.6465 5.647 5.6475 x 105 −2 −1 0 1 2 3 t_i− t_i−1 v o lt age time

(c) Voltage pulses (ei) generated from the inductive sensor when the teeth are passing by. The period of these pulses is the time diﬀerence

ti− ti−1. With this time diﬀerence the wheel speed is determined as

ω(ti) = _L(t 2π

i−ti−1).

Figure 3.1. The ferromagnetic ring and the inductive sensor together with its pulse

train.

ω(t_i) = 2π

L(t_i− t_i−1) [rad/s] (3.1) where t_i is time stamp i and L is the number of teeth on the ferromagnetic ring.

3.2 Create Wheels with Radial Runout

When collecting measurement data, it is preferable to consider a series of wheels with different radial runout. All wheels have more or less radial runout, but there is no answer of the exact value. Instead of searching for wheels with the different radial runout required for testing, wheels with radial runout are created manually. The reason why different radial runout is considered is for exploring all kinds of probable connections between detected radial runout and real radial runout as well as relative phase difference versus cab vibration.

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3.3 Radial Runout Measuring with Laser 15

3.2.1 Mounting the Wheel

The wheel is mounted on the hub of the axle. The hub has five heels which get the rim centered on the axle. By grinding down two of these heels, the wheel can move in the same direction, shown in Figure 3.2. When doing this the wheel can be mounted with different center offsets before each test drive.

(a) Wheel center mounted. (b) Two heels are grinded down and the wheel can be mounted with a center oﬀset.

Figure 3.2. When two of the ﬁve heels are grinded down the wheel can be mounted

with diﬀerent center oﬀset. This gives the opportunity to study the connection between detected radial runout and real radial runout.

3.3 Radial Runout Measuring with Laser

When a manually created wheel with radial runout is used during a test drive, there is a need for knowing the precise δ value when the calculations are done on the measurement data. To ﬁnd an estimation method based on the wheel speed sensor signal, there must be an answer of the radial runout used in each measurement set. Then a comparison between the real and the detected values can be done. The answer is found by having each wheel measured by a laser equipment before every test drive.

The idea of calculating the radial runout by using Fourier series has its begin-ning in the laser measurements. The laser is used for doing manual measurements of the wheels to determine their shape. By using an angle sensor together with the laser as in Figure 3.3, the shape of the wheel can be measured for each angle. While rotating the wheel for one revolution, the signals from both the angle sensor and the laser are recorded. Then the diﬀerent Fourier series matching the laser signal are calculated, giving their corresponding h1, . . . , h4 values. The result of

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Figure 3.3. A detailed picture of the angle sensor and the laser placement when

mea-suring the shape of a wheel.

Figure 3.4. When a laser measurement is performed the result is presented as this ﬁgure

shows. In the above plot the measurement data together with its diﬀerent Fourier series is shown. The legend shows the corresponding h values of the sinusoids.

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3.4 Phase Calculation 17

3.4 Phase Calculation

As stated in Section 3.1, each wheel axle has an attached ferromagnetic ring (toothed wheel) with a hundred teeth on both left and right side. Between two teeth there are 360/L degrees (here ideal teeth are assumed), L is the number of teeth. For each time t_i a tooth passes the sensor a time stamp will be set and the number of degrees ϕ(t_i) the wheel have rotated for each time stamp is then known. When you know how many degrees both the left and the right wheel have rotated at a certain time an absolute phase diﬀerence Δϕ_a(t_i) can be determined. This is done by subtracting the phase of the left wheel from the phase of the right wheel

Δϕ_a(t_i) = ϕ_r(t_i)− ϕ_l(t_i), (3.2) where the subscripts denote right and left wheel, respectively. Because the absolute phase difference is determined by counting the teeth on the rings (each tooth is a sample in the signal), this is an easy task to manage. The difficulty is to get the initial phase difference, since the above method have both right and left phase starting at zero. When the measurement starts, the left and the right wheel probably have their radial runout faced in different directions, as illustrated in Figure 3.5. This makes the absolute phase difference useless when an optimal phase is to be determined by a certain number. Therefore, the initial phase difference Δϕ_i is needed to calculate a relative phase difference

Δϕ_rel(t_i) = Δϕ_a(t_i) + Δϕ_i (3.3)

Figure 3.5. Illustration of the diﬀerent directions each wheel’s radial runout can be

headed when a measurement begins, shown by the arrows. The diﬀerence between the

initial phase of both wheels, gives the Δϕi. δlandδr are the center oﬀset value for the

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The way the phase is determined in (3.2), is considered to be right according to the absolute phase diﬀerence between the left and the right wheel. By looking at Figure 3.6, a decrease in the phase curve means that the truck turns right and vice versa. Here, a turn of 45 degrees gives a 180 degrees phase change, because of the axle width of the test truck used.

0 5 10 15 20 25 30 35 40 −150 −100 −50 0 50

Absolute phase difference between right and left wheel

phase [deg]

time [s]

Figure 3.6. The absolute phase difference Δϕa of the front wheels from one of the measurements. To get the relative phase difference the curve should be moved up or down in the y axis, by adding the initial phase difference, according to (3.3).

As mentioned before, both the left and the right wheel’s phase are starting from zero. To get the relative phase difference (3.3) between the wheels with radial runout this phase curve, in Figure 3.6, should be moved up or down in the y axis, i.e. the initial phase difference needs to be found. From the Fourier series coefficients, the phase is calculated as (2.11), but unfortunately the result from this calculation varies to much, implying that it cannot be used to find Δϕ_i. Also by using data from the accelerometers attached to the wheel axle, a phase is determined. This phase determining method, based on the axle movements, is depending on how much radial runout there are on the wheels and therefore neither this method is good enough for finding Δϕ_i. So in spite of all efforts using the information from calculations and measured signals no reliable method for determining the relative phase difference has been found, whereupon no optimal phase difference that minimizes the cab vibration is presented.

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Chapter 4

Cab Vibrations

An objective in this thesis is to connect phase diﬀerences from wheels with radial runout to vibration levels of the cab during ride. This chapter studies how to de-termine the cab vibrations from the windscreen accelerometer. To get a measure of the vibration levels, the power spectral density (PSD) is calculated from the accelerometer data. By comparing the magnitude of this PSD from the same fre-quency interval in each test ride and at the same time look at the phase diﬀerence, some conclusions regarding ride comfort can be drawn.

4.1 3D Accelerometers

To collect all necessary information for later analysis, not only the wheel speed sensors were recorded, the truck was also rigged with three accelerometers used for vibration analysis. One accelerometer was placed in the windscreen of the truck to collect data from cab vibrations. There were also two accelerometers on the front axle, one on each side. Figure 4.1 shows the placement of the accelerometers and also deﬁnes the coordinate system used, x (forward), y (to the left) and z (up). The accelerometers attached to the truck measures acceleration in all three direction (x, y, z) and are used to see if there are any correlations between the movement of the front axle and the lateral motion of the cab.

4.2 Vibrations due to Wheels with Radial Runout

When riding a truck with radial runout the vibrations can easily be felt and to really show the interesting vibration at frequency around 7 Hz and its magnitude the PSD from accelerometer data is calculated. The accelerometer attached in the windscreen of the truck provides the information of cab vibration during the test drives. PSD stands for power spectral density and is deﬁned as ”Amount of power per unit (density) of frequency (spectral) as a function of the frequency”.

The PSD is calculated with the MATLAB function pwelch. pwelch calculates the power spectral density using Welch’s method, see [1].

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20 Cab Vibrations

Figure 4.1. Main truck used in this master thesis, Maxima 4x2. Three accelerometers

rigged at diﬀerent places. One in the windscreen and one on each side of the front

axle. The windscreen accelerometer is used to determine the motion of the cab and the accelerometers attached on the front axle are used for comparing the axle and cab motion due to radial runout, to see if there is any correlation between them.

Here, the PSD is calculated over a window with 1024 samples. This window is moving forward 64 samples after every calculation. To focus on the interesting frequency around 7 Hz, the maximum value in an interval of [6 8] Hz from each calculation is stored, see Figure 4.2. The maximum values, representing the power of 7 Hz vibrations as a function of time is then plotted as in Figure 4.3. A PSD is calculated for each direction (x, y, z) of the axle accelerometers and are used in analysis of the connection between axle and cab motion. In Figure 4.3, the PSD from the interesting directions are shown and you can see that both lateral and vertical PSD values from the front axle follow the same pattern as the cab lateral PSD values, but with diﬀerent magnitudes.

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4.2 Vibrations due to Wheels with Radial Runout 21 0 5 10 15 20 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 frequency [Hz] power [(m/s 2) 2/Hz]

Result from one PSD calculation

Figure 4.2. Plot from one PSD calculation. The peak values around 7 Hz is stored and

the values from the whole measurement can be plotted as in Figure 4.3.

0 50 100 150 200 250 300 350

0 5 10 15

Vibration of 7 Hz from PSD calculation

time [s]

power

y−dir windscreen y−dir front axle (left) z−dir front axle (left−right)

Figure 4.3. 7 Hz vibration values from PSD calculation. The windscreen lateral

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22 Cab Vibrations

4.3 Relation Between Phase Diﬀerence and Cab

Vibration

In a real setting, the cab vibrations often come and go. By analyzing the signals from the accelerometers and the wheel speed sensors a clear connection between cab vibration and phase difference of the wheels with radial runout can be seen in most of the measurements. The most apparent connection is seen from a measure-ment shown in Figure 4.4. In this measuremeasure-ment there is a lot of radial runout on both front wheels (2.8 mm and 2.0 mm). The plot tells us that the PSD levels are rather big sometimes and rather small or nearly missing some other times. When looking at the phase difference curve at the same time the conclusion of an obvious connection between these states can be made. Figure 4.4 shows that an interval between 120 and 140 degrees in absolute phase difference is giving big vibrations of the cab in this particular measurement.

0 50 100 150 200 250 300 0 1 2 3 4 5

Max Values from PSD Calculation

[(m/s

2) 2/Hz]

y−dir windscreen y−dir front axle

0 50 100 150 200 250 300

−300 −200 −100 0

Absolute Phase Difference (right−left)

[

°

]

time [s]

Figure 4.4. This plot shows how the PSD levels and the absolute phase diﬀerence

changes during the measurement. There is a very clear connection between the behaviours of both states.

4.4 Summary

Somehow the sum of the wheels radial runout will eﬀect the levels of the cab vibrations. Measurements where one wheel has almost no radial runout, the PSD values varies between 0 and 0.5 [(m/s2₎2_{/Hz] and with an extreme value (}_{≥ 2 mm)}

on the other wheel only a marginal increase of the vibration levels is obtained. When both wheels have much radial runout, the vibration eﬀect comes and goes in greater extension and the magnitude is also larger, which indicates that the changing phase of the radial runout wheels due to the turns of the road makes

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4.4 Summary 23

a difference. Even though it is not as obvious as in Figure 4.4, there are still connections between phase difference and cab vibration. This is also confirmed by Figure 4.5 which are representative of the different measurement configurations.

In few of the measurements, the speed has been too low for hitting the right vibration frequency. This also appears from the PSD plot in the Figures 4.6(a) and 4.6(b), involving speeds of 70 km/h and 50 km/h. Since it is important to travel with the right speed during the measurements for the vibration problem caused by wheels with radial runout to occur, a traveling speed between 85 to 90 km/h have been used in almost every test. As Section 3.4 clearly states, no method for ﬁnding the relative phase diﬀerence have been found. By not having the phase curve plotted right according to the y-axis, no value of an optimal phase can be presented. What you can say is that a certain change in absolute phase gives a change in the magnitude of the cab vibration. This is also a subjective feeling you get when riding a truck with the same condition as the test truck used in this thesis work.

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24 Cab Vibrations 0 50 100 150 200 250 0 0.5 1 1.5

[(m/s

2) 2/Hz]

0 50 100 150 200 250 −150 −100 −50 0 50 100

Phase Difference (right−left)

[ ° ] time [s] (a) 0 50 100 150 200 0 0.5 1 1.5

[(m/s

2) 2/Hz]

0 50 100 150 200 −300 −250 −200 −150 −100 −50 0

[ ° ] time [s] (b) 0 50 100 150 200 250 300 0 0.2 0.4 0.6 0.8

[(m/s

2) 2/Hz]

0 50 100 150 200 250 300 −350 −300 −250 −200 −150 −100 −50 0

[ ° ] time [s] (c) 0 50 100 150 200 250 300 0 1 2 3 4

[(m/s

2) 2/Hz]

0 50 100 150 200 250 300 −150 −100 −50 0 50

[

°

]

time [s]

(d)

Figure 4.5. Four plots showing the connection between phase diﬀerence and cab

vibra-tion. Each plot represents different configurations of radial runout. The connection can be seen but is not specified to certain numbers.

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4.4 Summary 25

0 5 10 15 20 25 30 35 40

0 0.05 0.1

[(m/s

2) 2/Hz]

(a) 0 20 40 60 80 100 0 0.1 0.2 0.3

[(m/s

2) 2/Hz]

(b)

Figure 4.6. When performing these measurements, the speed was set to 70 km/h in the

upper plot and 50 km/h in the other. This means that the axle vibration, due to wheels with radial runout, does not hit the eigen frequency of the cab. Practically no vibration at all is seen in the signal, which also the test drives indicated. The x-axis shows time in [s].

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Chapter 5

Analysis and Results

This chapter shows the radial runout results from the measurements using calcula-tions based on the theory in Chapter 2. The first part considers some technicalities in the calculations and in Section 5.1.2 the results from the experiments are shown and discussed. Section 5.2 deals with the sensor fault, which includes theory of the sensor fault and a method of determine its appearance. In Section 5.2.3 an attempt to explain the difference between the δ values from the laser and the cal-culation with the sensor fault is performed. In Section 5.3, some words regarding the use of the Kalman filter is given.

5.1 Estimation of Radial Runout Using Fourier

Series

The estimation of radial runout is based on a varying angular velocity, obtained from the wheel speed sensor. The aim is to calculate the varying radius according to (1.1), which is done by assuming that the speed v is constant over one revolution. Because the speed itself is not taken from any other sensor, it is calculated as the mean value of the angular velocity over one revolution, multiplied by the nominal radius value r0, see (5.1). Figure 5.1 illustrates the speed calculation.

v(k) = 1 L

L/2−1 i=−L/2

ω(k + i)· r0 (5.1)

With the radius calculated as

r(k) = v(k)

ω(k) (5.2)

where ω(k) is the value in the middle of the calculation frame, seen in Figure 5.1. This radius (5.2) is then split up for each revolution and the Fourier Analysis is

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28 Analysis and Results

applied, see Figure 5.2. One δ value for each revolution is calculated and gathered in a plot, where the change in δ during a measurement can be seen. By using the equations (5.1) and (5.2) to calculate the speed v and the radius r, a typical plot of these signals can be seen in Figure 5.3.

2.125 2.13 2.135 2.14 2.145 2.15 2.155 2.16 2.165 2.17 x 104 44.8 45 45.2 45.4 45.6 45.8 46 46.2 k [sampel]

angular velocity [rad/s]

100 samples

v(k)

Figure 5.1. The speed v(k) is calculated within the frame of a hundred samples, i.e.

one revolution, as the mean value of the rotational velocity multiplied with the nominal radiusr0.

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5.1 Estimation of Radial Runout Using Fourier Series 29

Figure 5.2. When the radius is calculated according to (5.2), it is divided into parts

containing only one revolution each. Then the Fourier series coeﬃcients are calculated

to determine theδ value for every revolution. The r1, . . . , rnis actually containing four

revolutions as stated in (5.6). 0 2 4 6 8 10 x 104 22 23 24 25 26 Speed v [m/s] sample 0 100 200 300 400 500 600 0.515 0.52 0.525 0.53 Radius r [m] sample

Figure 5.3. The speed v calculated as (5.1) gives a relative constant velocity (top plot)

and the radius (5.2) shows a typical variation pattern. Here, six revolutions of the radius are plotted, but the speed is plotted for the whole measurement. Note that 100 samples is one revolution, which goes for both ﬁgures.

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5.1.1 Remarks on the Calculation

In the calculation involving Fourier series actually four revolutions were considered when estimating the radial runout values, i.e. δ = h4. When using one revolution

(period), the radius is approximated as (2.16), but in this case, when four revo-lutions are considered, the radius is approximated as (5.3), and the amplitude is given by the fourth order Fourier series coeﬃcients as (5.4).

˜ r(θ) = R0+ A4sin(4θ) + B4cos(4θ), (5.3) δ = h4= A2 4+ B42, (5.4)

where R0is the same as (2.13), A4 and B4are calculated according to (2.14) and

(2.15). The reason for this was to get a non frequent ﬂuctuating estimate. Because if the amplitude of each revolution of the varying radius diﬀer from each other, as illustrated in Figure 5.4, and only one revolution was considered, the estimate of

δ varies to much, see Figure 5.5. By using four revolutions in the Fourier series

calculation, the resulting sinusoid will try to match all four revolutions with just one amplitude, which is almost the mean value of the amplitude of all revolutions. Let δ_i denote the amplitude of revolution i in Figure 5.4 (bottom left plot). The matching Fourier series, seen in the top plot in the same ﬁgure, have an amplitude,

δ_res, according to:

δ_res≈ 1 4 4 i=1 δ_i (5.5)

To have a δ value for each revolution, in spite of the fact that the calculations is based on four revolutions, the calculation frame will only move forward one revolution at a time. This means that r1, . . . , rn in Figure 5.2, actually contains four revolutions as the implementation is made today.

r1= [r1, r2, r3, r4] r2= [r2, r3, r4, r5] r3= [r3, r4, r5, r6] . . . r_n−3= [r_n−3, r_n−2, r_n−2, r_n] (5.6)

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5.1 Estimation of Radial Runout Using Fourier Series 31 0 5 10 15 20 25 −2 −1 0 1 2 θ [rad] radius fourier series 0 2 4 6 −2 −1 0 1 2 θ [rad] 1st rev 2nd rev 3rd rev 4th rev

Figure 5.4. This plot illustrates that the varying radius can have diﬀerent amplitudes for

each revolution. By using four revolutions in the Fourier series calculation, the resulting

amplitudeδresof the sinusoid (dotted) is almost the mean value of the amplitude for all

revolutions. The bottom left plot shows the different amplitude of each revolution of the radius, shown in the top plot (solid). The amplitude difference between two following revolutions is here magnified to clarify the point.

0 50 100 150 200 250 300 350 400 −1 −0.5 0 0.5 1x 10 −3 angle [deg] amplitude [m]

Fourier Serie, 1 revolution

0 500 1000 1500 −1 −0.5 0 0.5 1x 10

−3 Fourier Serie, 4 revolutions

angle [deg]

amplitude [m]

δ = h1

δ = h4

(a) Fourier series of one and four revo-lutions.

⇒

0 50 100 150 0 2 4 6 [mm]

δ estimation using 1 revolution

δ = h1 0 50 100 150 0 2 4 6 time [s] [mm]

δ estimation using 4 revolutions

δ = h

4

(b) The estimatedδ values using one or four revolutions.

Figure 5.5. Calculations of radial runout over one and four revolutions gives almost

the same result. The advantage of the four revolution approach is the non frequent ﬂuctuation in the detected radial runout as seen in the bottom right ﬁgure.

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5.1.2 Experiment Results

During the work, twentyﬁve measurements with fully rigged truck have been col-lected. The result given in this section is based on the same conﬁgurations of the wheels for all measurements, i.e., the grinded heels are the same. This means that the direction the wheels are changed for various radial runout is always the same relative to the ferromagnetic ring, shown in Figure 3.2. The measurements were performed on real roads to imitate regular driving. By using the calculated mean value of δ from each measurement, the result from the entire measurement series is given in Figure 5.6. There is a clear connection between real radial runout and detected radial runout on the left wheel, see Figure 5.6(a). The right wheel also show a clear connection, but here the values are showing a mirrored behaviour, see Figure 5.6(b). 0 5 10 15 20 25 0 1 2 3 4 5 6

Real Radial Runout and Detected Radial Runout, left wheel

measurement

radial runout [mm]

real detected

(a) Left wheel

0 5 10 15 20 25 0 0.5 1 1.5 2 2.5

Real Radial Runout and Detected Radial Runout, right wheel

measurement

radial runout [mm]

real detected

(b) Right wheel

Figure 5.6. Real radial runout values compared with mean value of detected radial

runout from each set of wheel configuration. It is clear that the detected radial runout differ from the real radial runout on both wheels. The difference is caused by a so called sensor fault.

By looking at Figure 5.6, it is clearly stated that the detected radial runout differ from the real radial runout on both wheels. The probable reason is sensor faults. Assume that the sensor fault gives a sinusoidal signal with a certain am-plitude and a certain phase. The radial runout also gives a sinusoidal signal, but maybe with different amplitude and phase. A reasonable thinking is that these sinusoids can be superpositioned. If the signals have the same phase it increases the amplitude but with different phase it can make the signal go away. With the same amplitude on both signals, but reversed phases the calculation would give no radial runout at all. Then if the real radial runout value is changed the cal-culation may go the opposite direction as shown in Figure 5.6(b). Because of the grinded heels, the relative phase between the radial runout and the sensor fault is approximately the same for all measurements with radial runout more than 1 mm. In the other measurements with radial runout less than 1 mm, the wheel is resting on the non grinded heels and therefore the phase could be changed.

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5.2 Estimation of Sensor Fault 33

5.2 Estimation of Sensor Fault

In this section, the aim is to explain the sensor fault, how it behaves and what the contribution to the radial runout is. The theory of the sensor fault is given in Section 5.2.1. In Section 5.2.2, a test method dealing with the sensor fault is described and, in Section 5.2.3, an attempt to show that the sensor fault can explain why the real radial runout and the detected radial runout diﬀer from each other in Figure 5.6 is done.

5.2.1 Sensor Fault Theory

The theory of why the sensor gives a varying signal is due to center offset assembly of the ferromagnetic ring. When an offset is present the sensor will read the teeth in different places and because the teeth are cut out of a ring, the width of the teeth will increase the further out on the ring you are looking, illustrated in Figure 5.7.

Figure 5.7. The sensor reads the teeth along the dashed circle. Because the ring is

attached with a center oﬀset, the period time between the teeth will vary, i.e. T1= T2.

This yields a varying angular velocity which is the radial runout contribution from the sensor fault.

With the diﬀerent width of the teeth, the period time between them will vary and in the end the varying angular velocity signal will appear, shown in Figure 5.8. The amplitude of this varying rotational velocity is the radial runout contribution from the sensor fault, which is determined in Section 5.2.3.

5.2.2 Free Spinning Wheel

For a free spinning wheel, the angular velocity should decrease linearly with time if the sensor was working properly. Due to fault in the sensor, described in Section 5.2.1, the angular velocity is varying, as illustrated in Figure 5.8. To study the

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sensor fault behaviour, tests based on manually spinning the wheels have been performed. The reason for letting the wheel spin free is to prevent the signal to be inﬂuenced from the wheels radial runout.

Figure 5.8. For a well balanced wheel the angular velocity should more or less look like

the decreasing solid line when a free spinning test is performed. From the measurements, the signal looks more like the sinusoidal curve (dashed), which is the sensor fault eﬀect. By also taking unbalance and friction into account, the signal often behaves like the dotted curve when the wheel speed is slow.

Wheel Spinning

To do the free spinning test the wheel is hoisted up and manually made spinning with a wheel spinning machine, see Figure 5.9. During the test the same signals from the wheel speed sensors are recorded as for the regular test drives. Since the same signals are used, also the same calculations are performed to estimate the amplitude of the varying rotational velocity caused by the sensor fault. By doing the same calculations, the results can be used together in Section 5.2.3.

Unbalance by Adding Weights

As an extension to the wheel spinning test, unbalancing of the wheels by adding weights can be done. The intention of such a test was to determine the phase diﬀerence between the sensor fault and the radial runout and also the sensor fault contribution to the radial runout, i.e., the amplitude of the sensor fault. First of all the wheels were balanced and mounted as close to the center as possible. Then ﬁve measurements were recorded of each wheel. One with no additional weights (referred to as ”balanced wheel” in Figure 5.12) and then four measurements where extra weights were added at one point at a time. This point moved 90 degrees

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(a) Wheel spinning machine (b) Spinning wheel, maximum speed with this machine is approximately 65 km/h.

Figure 5.9. The wheel spinning machine used for testing the wheel speed sensors.

between each measurement, see Figure 5.10. Depending on where the sensor fault acts relative to the added extra weights, the δ estimate should change for every measurement. If the sensor fault increases the δ estimate in the measurement with extra weight on point 1 most, as Figure 5.10(a) should illustrate, then the measurement with extra weight on point 3 would have the lowest δ estimate. Depending on which side of the point 1 the sensor fault acts, the measurements with extra weight on point 2 and point 4 will get the second and third largest δ estimate or vice versa. This is theoretically based on the phase diﬀerence between the two sinusoids representing the sensor fault and the physical radial runout eﬀect (in this case represented by unbalance from the extra weights), as illustrated in Figure 5.11.

The resulting δ estimate of this test can be seen in Figure 5.12, which corre-sponds to where the sensor fault acts for each wheel respectively, as illustrated in Figure 5.10. Because of the symmetry of the added weights in this test, the mean value of the δ estimate should be the same as for the ”balanced wheel”. This is considered to be the answer of how much the sensor fault contribute to the radial runout and is used in the next section. The result in Figure 5.12 also implies that both sensors are reacting pretty much in the same manner. Though the determi-nation of the phase difference did not work as planned (due to miscalculations), the test was only used to determine the amplitude of the sensor fault, which is an equally important step as determining the phase difference. The phase difference is dealt with in Section 5.2.3.

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(a) This is the case for the left wheel. The sensor fault acts near point 1, which can be seen in Figure 5.12(a) where an extra weight on point 1 gets the highestδ estimate.

(b) This is the case for the right wheel. In Figure 5.12(b) you can see that point 2, 1, 3 and 4 comes in order of highest

δ estimate.

Figure 5.10. This is an illustration of the adding weights test. Where the ellipse

symbolize where the sinusoid of the sensor fault has its peak, i.e. the maximum amplitude due to the phase of the sensor fault. The numbers 1,. . . ,4 indicates where the extra weights were added.

0 2 4 6 −2 −1 0 1 2

Added weight at point 1

0 2 4 6 −2 −1 0 1 2

Figure 5.11. Four plots showing how two added sinusoids with diﬀerent phase gives

diﬀerent resulting amplitudes, according to the case in Figure 5.10(a). The solid line is the sensor fault, the sinusoid representing the extra weight is shown in dots and the dash-dotted curve is the superposition of the two. Because the eﬀect from the extra weight is not as big as from the sensor fault, the amplitude was chosen to 0.5.

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5.2 Estimation of Sensor Fault 37 0 5 10 15 20 0 1 2 3 4 5 6 7 speed [m/s] δ [mm]

δ from wheel spinning test with added weights, left wheel

balanced wheel added weight at point 1 added weight at point 2 added weight at point 3 added weight at point 4

(a) Left wheel

0 5 10 15 20 0 1 2 3 4 5 6 7 speed [m/s] δ [mm]

δ from wheel spinning test with added weights, right wheel

balanced wheel added weight at point 1 added weight at point 2 added weight at point 3 added weight at point 4

(b) Right wheel

Figure 5.12. δ estimate from the adding weights test. The solid line shows the result of

using a well balanced wheel, which is also considered to be the contribution of the sensor fault. The reason for the increasing values when the speed is slow is because then the unbalance eﬀect together with the friction is causing the angular velocity to vary a lot.

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5.2.3 Adding Sensor Fault to Real Radial Runout

The δ estimate from the balanced wheel recording is considered to be the amplitude of the sensor fault. As Figure 5.12 shows, the sensor fault is almost the same for both the left and the right sensor. This test together with the earlier tests (wheel spinning, Section 5.2.2), Table 5.1 shows that the sensor fault contributes to radial runout with a value of about 2 mm. By knowing the values from the sensor fault, the real radial runout and the detected radial runout, the next step is to try ﬁtting sinusoids to match up all the values. The estimated radial runout should be the sum of the real radial runout and the sensor fault. Sinusoids representing sensor fault (sf), real radial runout (rrr) and detected radial runout (drr):

y_sf = a sin(x) (5.7)

y_rrr = b sin(x + φ1) (5.8)

y_drr= c sin(x + . . .) = y_sf+ y_rrr (5.9)

where a is the amplitude of the sensor fault. b is given by the laser measurement and c is the calculated mean values, seen in Figure 5.6. φ1, is the phase diﬀerence

between the sensor fault and the radial runout and this is the important parameter to find, because depending on this phase the calculated δ estimate shows different values. For instance, in Figure 5.6, when both wheels have almost the same real radial runout (∼1.2 mm), the estimated δ values shows very different results (left:

∼2.7 mm, right: ∼0.5 mm). This phase is shifting the sinusoids with φ1 degrees,

as Figure 5.13 shows.

Figure 5.13. Two sinusoids; sensor fault (solid) and radial runout (dashed), where one

has a phase shift ofφ1degrees. The amplitude of the superposition depends of this phase

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Truck Detected Radial Runout MAXIMA [mm]

Left wheel (forward) 2.06 2.19 2.23 2.03

Left wheel (backward) 2.01 2.08 1.98 1.96

Right wheel (forward) 1.99 1.78 2.05

Right wheel (backward) 2.17 2.27 2.28

Table 5.1. δ estimate values from wheel spinning tests on Maxima, see Section 5.2.2.

Left Wheel

By minimization of the estimation error, the phase diﬀerence is determined to be

φ1 ≈ 64 degrees in the cases where the real radial runout is more than 1 mm.

Figure 5.14 illustrates the meaning of φ1= 64 degrees. In the case where the real

radial runout is 0.5 mm, the phase φ1 is not uniquely determined. The reason for

this is because the wheel, in this case, is resting on the three non grinded heels, which means that the wheel is centered on the hub, see Figure 3.2(a). The radial runout then comes from the original wheel assembly, i.e., the fact that the rim and tire is not perfectly round. This radial runout can act in any direction and to determine the phase, relative to the phase of the radial runout the wheel had when it rested on the grinded heels, it should have been monitored during the laser measurement. Because this was not done, φ1 is put to 180 degrees, which gets the

best correspondence between real radial runout and detected radial runout in this case. By using the above phases in the equations (5.7)-(5.9) the results are shown in Figure 5.15(a). There is a very good correspondence as Figure 5.16(a) shows and this is probably because the sensor fault contributes to the radial runout instead of counteracts. This is making it easier to determine the phase φ1.

Figure 5.14. An illustration of the phase diﬀerence φ1 = 64 degrees. The ellipse

symbolizes the sensor fault, just as in Figure 5.10. Here, the radial runout is faced

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Right wheel

For the right wheel it is a lot more difficult to find a phase difference φ1 that

match all values. With a sensor fault of 2 mm and real radial runout of 1.3 mm, it is impossible to match a sinusoid with an amplitude of 0.48 as the detected radial runout suggests. The absolute value is in the interval [0.7 3.3] mm. Even though it is impossible to match all values, the phase φ1≈ 206 degrees minimized

the quadratic sum of the estimation error. But as you can see in Figure 5.15(b), the result is not nearly as good as it is for the left wheel. The reason why the correspondence for the right wheel is not as good as for the left wheel, see Figure 5.16(b), has probably to do with the mean value of the calculation. This makes it sensitive to where the measurement is recorded. For instance, Figure 5.17 shows that the δ estimate varies a lot even though the real radial runout is the same. This is maybe why the plot for the right wheel has the jagged behaviour. The fact that the sensor fault counteracts to the radial runout is also eﬀecting the estimate, because of the reversed manner.

1 2 3 4 5 6 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

Residual plot with φ₁ = 64.6 deg

Quadratic sum of est. error = 0.036

3.13 → 3.09 2.66 → 2.74 2.63 → 2.77 3.97 → 3.88 4.14 → 4.12 1.47 → 1.5

(a) Left wheel

1 1.5 2 2.5 3 3.5 4 −0.6 −0.4 −0.2 0 0.2 0.4 0.6

Residual plot with φ₁ = 205.7 deg

Quadratic sum of est. error = 0.393

0.931 → 0.873

0.482 → 1

1.1 → 0.89

1.88 → 1.61

(b) Right wheel

Figure 5.15. This plot shows the deviation between the sensor fault and real radial

runout joined together according to (5.9) and the detected radial runout. The ﬁrst

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5.2 Estimation of Sensor Fault 41 0 5 10 15 20 25 0 1 2 3 4 5 6 Left wheel measurement radial runout [mm] real detected sensor fault + real

(a) Left wheel

0 5 10 15 20 25 0 0.5 1 1.5 2 2.5 3 3.5 4 Right wheel measurement radial runout [mm] real detected sensor fault + real

(b) Right wheel

Figure 5.16. With the determined phase diﬀerence φ1 of both wheels these plots show how well the adding of sensor fault and radial runout match up with the detected radial runout. The correspondence for the left wheel is as good as it could be.

0 50 100 150 200 250

0 2 4 6

1.90 mm real radial runout on right wheel

[mm]

Figure 5.17. The δ estimate, for the right wheel, varies a lot during the measurement

(x-axis shows time [s]). This makes the use of mean value as result sensitive to where the measurement starts and stops.

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5.2.4 Changing Radial Runout Phase

Section 5.2.3 showed that the δ estimate becomes diﬀerent depending on the phase diﬀerence between the varying sensor fault signal and the radial runout. With a change in the relative phase between the sensor fault and the radial runout by 144 degrees, in Figure 5.16(b), the sensor fault should theoretically contribute to the radial runout instead of counteract. Using the same sinusoids as in Section 5.2.3, the sensor fault is represented by:

y_sf = a sin(ωt) (5.10)

and the radial runout is represented by another sinusoid

y_rrr = b sin(ωt + φ1) (5.11)

Then the resulting sinusoid, illustrated in Figure 5.18(b), becomes

y_drr= a sin(ωt) + b sin(ωt + φ1) = c sin(ωt + . . .) (5.12)

Let Figures 5.18(a) and 5.18(b) represent the right wheel, where the phase φ1≈

206 degrees, as determined in previous section. A change of the phase by 144 degrees, illustrated in Figure 5.18(c), should give the larger amplitude of the δ estimate, shown in Figure 5.18(d). By doing the physical change of grinding down two opposite heels, this phase shift is done and with a few test drives the phase theory could be confirmed. The right wheel was mounted with a 2.5 mm offset which should give a detected radial runout between 1 and 1.5 mm with the first configuration, according to Figure 5.6(b). But by changing the phase of the radial runout, the result was more similar to the left wheel, because the detected radial runout exceeded 4 mm. The expected value, by adding two sinusoids with the right phases, should have been 4.4 mm. Though the estimate differed between 4.1 and 4.8 mm for the same configuration, the importance of the relative phase between the radial runout and the sensor fault has been illustrated.

Estimation of Radial Runout

Institutionen för systemteknik

Department of Electrical Engineering

Examensarbete

Estimation of Radial Runout

Estimation of Radial Runout

Examensarbete utfört i Reglerteknik

vid Tekniska högskolan i Linköping

av

Abstract

Sammanfattning

Acknowledgments

Contents

Chapter 1

Introduction

1.1

Background

1.2

Idea

1.3

Goal and Purpose

1.4

Scania

1.5

Thesis Outline

Chapter 2

Amplitude Estimation of

Periodic Signals

2.1

Fourier Series

2.1.1

Change of Variables

2.1.2

Radius Approximation Used for

δ Estimation

2.1.3

Diﬀerence Between

h and δ

2.2

Time Domain Methods

2.2.1

Regression Model

2.2.2

Kalman Filter

Chapter 3

Experiment Setup for Radial

Runout and Phase

Determination

3.1

Wheel Speed Sensor

3.1.1

Determination of Wheel Speed

3.2

Create Wheels with Radial Runout

3.2.1

Mounting the Wheel

3.3

Radial Runout Measuring with Laser

3.4

Phase Calculation

Chapter 4

Cab Vibrations

4.1

3D Accelerometers

4.2

Vibrations due to Wheels with Radial Runout

4.3

Relation Between Phase Diﬀerence and Cab

Vibration

4.4

Summary

Chapter 5

Analysis and Results

5.1

Estimation of Radial Runout Using Fourier

Series

5.1.1

Remarks on the Calculation

⇒

5.1.2