Radial Piston Hydraulic Motors
Niklas Ydebäck
Mechanical Engineering, master's level
2021
Luleå University of Technology
DIV. OF MACHINE ELEMENTS
M.
SC. T
HESIS REPORTMECHANICAL ENGINEERING - MACHINE DESIGN E7020T
G
REY
B
OX
M
ODEL OF
L
EAKAGE IN
R
ADIAL
P
ISTON
H
YDRAULIC
M
OTORS
Preface
In my final work as a M.sc. student in mechanical engineering at Luleå University of Technology I have performed this thesis project regarding the development of a grey box model of leakage trends in radial piston hydraulic motors, performed at the product development section of the engineering department at Bosch Rexroth AB in Mellansel.
The Hägglunds hydraulic direct drive
Bosch Rexroth AB in Mellansel (BRM) develops and sells complete hydraulic drive systems known as the Hägglunds hydraulic direct drive (HDD). The drive system is composed of mainly two compo nents, the motor and the drive unit (DU). The radial piston motors developed at BRM lie in the motor category hydrostatic motors, or in other words displacement motors or hydraulic motors. Hydrostatic motors make use of a static pressurisation of a medium, in this case oil, in order to induce movement of parts in the motor from the applied pressure, or force, see p. 1 in [1]. The drive units main compo nent is this external pump system which is driven by an electrical motor to be able to deliver the flow the hydraulic motor need to rotate at requested speed. In addition to the pump and the motor there are several control systems developed at BRM. An overview of the products delivered by BRM can be seen in Figure 1.
Figure 1: An overview of the products developed and delivered by BRM.
of motor, namely the CB motor, further denoted as the motor. The motor is, just as the other motor families available, used in many different industries and applications. The motor has a recommended maximum working pressure limit of 350 bar [3]. The motor can, depending of motor size, generate a speed of up to 125 revolutions per minute (RPM) and deliver torque up to 370 kNm [3]. A cross section view can be seen in Figure 2 and the explanatory table that describes the notations in Figure 2 can be found in Table 1.
Figure 2: A cross section view of a CB motor [3].
Table 1: Descriptions of the notations in Figure 2 [3].
1. Cam ring 8. Cylindrical roller bearing 2. Cam roller 9. Connection housing 3. Piston 10. Distributor
4. Shrink disc 11. Combined axial and radial bearing 5. Cylinder block, hollow shaft 12. Wear ring
Acknowledgements
I would like to thank my supervisors at Bosch Rexroth AB in Mellansel, Michael Westman and Daniel Svanbäck, along with their colleagues, in particular Jonas Lindberg, for their help and support during this project. I would also like to thank my supervisor and examiner at Luleå University of Technology, Andreas Almqvist, for the help and guidance he has provided.
Abstract
This report covers the work and results of the thesis project in Mechanical Engineering from Luleå university of technology performed by Niklas Ydebäck. The objective of the thesis project is to re search if it is possible, with general principles of fluid flow between components and the corresponding geometric constraints between them and just a few channels of data, to model the leakage of a radial piston hydraulic motor. The model is of the grey box kind which makes use of both numerical and statistical methods together with known physical properties of a system in order to model the system. The unknown parameters of this system that are estimated using the least squares method are the three different gap heights of the system as well as the two different eccentricities in the system. The model contains the physical properties of the system, stated in equations for the leakage in the relevant lubrication interfaces, but no relational properties for the dynamics and affects between the individual lubricating interfaces.
The model developed is due to the model generality together with the data quality accessible not able to model the system with reliable quality. The model is however able to capture the general trend of the leakage in the system over the applied time series datasets.
Sammanfattning
Den här rapporten presenterar arbetsgången och resultatet av examensarbetet för en civilingenjörsex amen i Maskinteknik från Luleå tekniska universitet utförd av Niklas Ydebäck. Målet med exam ensarbetet är att utvärdera och undersöka om det är möjligt, med generella och vedertagna principer av fluidflöde mellan smorda komponenter tillsammans med de geometriska begränsningarna som hör dem till och några få kanaler av data, att modellera läckaget för en radialkolvsmotor. Modellen är en grålådemodell som med hjälp av numeriska och statistiska metoder och kända fysikaliska principer av ett system bildar en modell av systemet.
De okända parametrarna av systemet som estimeras med hjälp av minsta kvadrat metoden är de tre olika typerna av spalthöjderna och de två olika eccentricitetstyperna som finns i systemets smorda kon takter. Modellen består av de fysikaliska egenskaperna i systemet, formerade i ekvationer för läckaget i de relevanta smorda kontakterna, men inga relationella egenskaper för dynamiken och effekterna mellan de olika smorda kontakterna.
Den utvecklade modellen är på grund av den generella karaktären av modellen tillsammans med kvaliteten på den data som finns tillgänglig inte möjlig att modellera läckaget i systemet med till räcklig noggrannhet. Modellen är trots detta kapabel att fånga de generella trender som återfinns i den uppmätta datan på läckaget för de applicerade dataseten.
Acronyms
AIC Akaike Information criterion BIC Bayesian inofrmation criterion BRM Bosch Rexroth AB in Mellansel DU Drive Unit
EHD Elastohydrodynamic
EHL Elastohydrodynamic lubrication FPE Final prediction error
HDD Hägglunds direct drive HDL Hydrodynamic lubrication LSE Least squares error
Nomenclature
Notation Description Unit
α Pressure-viscosity coefficient, step size −
ε Prediction error −
εnoise White noise −
ζ NRMSE between model and data −
η Dynamic viscosity mPa · s
η0 Dynamic viscosity at ambient pressure mPa · s
θ Model parameter set −
ˆθ Estimated model parameter set −
Λ Film thickness ratio −
µ Known datadependent vector −
ν Kinematic viscosity mm 2 s π Archimedes constant − τ Lag s ϕ Regression vector −
ω Revolutions per minute min 1 Ci Loss factor from geometric constraints −
d Number of estimated parameters −
e Eulers number −
ex Eccentricity −
f Search Direction −
h Gap height m
hmin Minimum gap height m
hde f Gap height from deformation m
J Final prediction error (FPE) − l Length of lubrication interface m L Signal to sensor distance m l(ε) Norm on prediction error −
M Model −
N Size of training dataset − ni Amount of interfaces of type i −
P Pressure Pa
ΔP Pressure drop Pa
Qf Fluid flow from flushing min L
Qi Total leakage from lubrication interface of type i min L
Ql Total leakage from motor min L
r Radius of component m
Rai Surface roughness of surface i m
ˆ
R Residual −
Fortsättning från föregående sida
Notation Description Unit
T Temperature ◦C
t Time s
u Model input −
u Floating point accuracy −
v Velocity m s
V Criterion function −
W Amount of starting points − Xi Deformation coefficient for interface of type i −
y Measured output −
yˆ Model output −
5 10 15 20 25 30 35 40
List of Figures
1 An overview of the products developed and delivered by BRM. . . I 2 A cross section view of a CB motor . . . II
3 An example of the stribeck curve . . . 5
4 Visualisations of the different lubrication regimes . . . 6
The simulation results for leakage from the piston ring - cylinder interface . . . 9
6 A model of a distributor . . . 10
7 A model of the balance sleeves and balance pistons . . . 11
8 A description of the objective of the balance piston and balance sleeve . . . 12
9 Results from simulation of the leakage with deformed sleeve and cylinder. . . 13
Description of the piston design in the piston-roller contact area . . . 14
11 The curve for leakage from sliding velocity . . . 15
12 An alternative result from the model developed in [4] . . . 15
13 Example of a visualisation of the bias and variance trade-off . . . 23
14 Alternative in visualisation between the variance and bias trade-off . . . 24
An example of the Residual test results . . . 27
16 The simulated results of Equation 29. . . . 29
17 Comparison between the simulation results in Figure 9 and Equation 11 . . . 30
18 Comparison between results from Equation 31 and simulation results in Figure 9 . . 31
19 Difference between the simulation results and Equation 31 without deformation . . . 32
A comparison for the difference between the simulation results and Equation 31. . . 32
21 The result of the curve fitting for replicating the simulation results in Figure 12. . . . 33
22 Visualisation of the StepTolerance and FunctionTolerance criteria . . . 38
23 Used training dataset for system 1 . . . 43
24 Used validation dataset for system 1 . . . 44
Model fit against the training dataset for system 1 . . . 45
26 Model fit against the validation data for system 1 . . . 45
27 The residual analysis result of the estimated model of system 1. . . . 46
28 The resulting prediction error of the model against the training dataset for system 1. . 47
29 Used training dataset for system 2 . . . 48
Used validation dataset for system 2 . . . 49
31 Model fit against training dataset for system 2 . . . 50
32 Model fit against the validation dataset for system 2 . . . 51
33 The residual analysis result of the estimated model of system 2. . . . 52
34 The resulting prediction error of the model against the training dataset for system 2. . 52
Used training dataset for system 3 . . . 54
36 Used validation dataset for system 3 . . . 55
37 Model fit against the training dataset for system 3 . . . 56
38 Model fit against the validation dataset for system 3 . . . 57
39 The residual analysis result of the estimated model of system 1. . . . 58
List of Tables
1 Descriptions of the notations in Figure 2 . . . II
3 Input parameters of the construction function for constructing an idnlgrey object . . . 34
4 Specification inputs parameter input in the construction of an idnlgrey object . . . . 35
5 Search options of the lsqnonlin search method . . . 38
6 The gradient options used for the estimation . . . 39
7 The amount of every high pressurised components found in a CB1120 motor. . . 42
8 The amount of every high pressurised components found in a CB400 motor. . . 42
9 The amount of every high pressurised components found in a CB560 motor. . . 42
10 The model fit results for system 1. . . . 47
11 The model fit results for system 2. . . . 53
Contents
List of Figures VIII
List of Tables IX
1 Introduction 1
1.1 Purpose and objective . . . 1
1.2 Delimitation . . . 1
1.3 Literature review . . . 2
2 Theory 3 2.1 The origins of Tribology . . . 3
2.2 Fluid film lubrication . . . 3
2.2.1 Hydrostatic lubrication . . . 3
2.2.2 Hydrodynamic lubrication . . . 3
2.2.3 Elastohydrodynamic lubrication . . . 4
2.2.4 Stribeck Curve . . . 4
2.2.5 Effects on fluid viscosity from temperature and pressure . . . 7
2.3 Leakage . . . 7
2.3.1 Piston ring and cylinder . . . 8
2.3.2 Distributor . . . 10
2.3.3 Balance pistons . . . 11
2.3.4 Balance sleeves . . . 12
2.3.5 Piston and roller . . . 13
2.4 Flushing . . . 16
2.5 Grey box modeling . . . 16
2.5.1 Model building . . . 17
2.5.2 Preprocessing of data . . . 19
2.5.3 Parameter optimisation . . . 20
2.5.4 Local and global minimum . . . 21
2.5.5 Accumulated prediction error metrics . . . 22
2.5.6 Training and validation datasets . . . 22
2.5.7 Underfitting and overfitting . . . 22
2.5.8 Comparison of model structures . . . 24
2.5.9 Model validation . . . 25
3 Methodology 28 3.1 Defining loss factors for interface leakage . . . 28
3.1.1 Piston ring and cylinder leakage . . . 28
3.1.2 Distributor leakage . . . 29
3.1.3 Balance sleeves leakage . . . 29
3.1.5 Flushing . . . 34
3.2 Matlabs non-linear grey box model object . . . 34
3.3 Data preprocessing . . . 36
3.3.1 Temperature sample correction . . . 36
3.3.2 Outlier removal and noise smoothing . . . 36
3.3.3 Construction of iddata object . . . 37
3.4 Specification of estimation settings . . . 37
3.4.1 Search method . . . 37
3.4.2 Gradient options . . . 38
3.5 Multiple starting points and parallelisation . . . 39
3.6 Model validation . . . 40
3.6.1 Comparing model to validation data . . . 40
3.6.2 Residual analysis . . . 40
3.6.3 Prediction error visualisation . . . 40
4 Results 41 4.1 Training and validation procedure . . . 41
1 Introduction
The background of the need for the research performed in this thesis lie in the need for a more reliable need for health evaluation of industrial applications. The specific industrial application covered in this thesis is the radial piston hydraulic motor, introduced in the preface of this report. The radial piston hydraulic motor is under the influence of lubrication between internal components with the purpose to increase the efficiency and lower the wear of components. Today the evaluation of wear in the motors is made ad-hoc by the engineers and service technicians at BRM by analysing the data sampled from the systems in use around the world. The analysis criteria is most often to evaluate the characteristics of aggregated data values for different logged information of the system behaviour, e.g. the leakage flow, pressure level etc. This is many times a hard task to get reliable results from and can also be very time consuming as well as having the risk of failures of system appear before the analysis is made. Thus the use of a more automated analysis process is needed.
The external leakage of hydrostatic motors is known to increase with the increase of wear. Since wear is often the final cause for failure of a hydrostatic motor, the leakage is thus chosen as the basis for system behaviour and health.
1.1 Purpose and objective
The aim for this project is to develop and evaluate a grey box model for the leakage characteristics of the radial piston motors developed and manufactured by Bosch Rexroth AB in Mellansel. The main focus is to research how to model the most prominent leakage characteristics of a radial piston motor of the kind that is developed at BRM. Then to research how to connect this model with a system that optimises the unknown parameters of the model to approximate the characteristics of the motor to match collected real data of rotational speed, fluid temperature, fluid pressure, and fluid flow rate of the leakage from the motor. Within this the main objective is to evaluate if such a model, with the data stated above, is enough to be able to model the leakage in a reliable way. If it is found that is is possible, such a model can then be used to evaluate the leakage trend of the motor to be able to identify deviations of leakage in the motor from the increase or decrease of the leakage over time.
1.2 Delimitation
The delimitations of the project is to create a method for developing a grey box model regarding the external leakage of the motor for only one of the available radial piston motor families. The leakage characteristics taken in to consideration is of the five kinds described in the following bullet point list and no other than these will be modeled, this since these areas generate the most fluid leakage in the motor according to information given from representatives at BRM.
• Leakage from balance pistons • Leakage from balance sleeves.
• Leakage between piston ring and cylinder. • Leakage between distributor and cylinder block.
1.3 Literature review
2 Theory
In this chapter the theoretical principles used in this thesis projects are presented.
2.1 The origins of Tribology
The use of tribology has existed through out history, with some of the oldest records dated back to 3500 B.C., see p. 1-3 in [5]. The academic definition as well as the term tribology, containing the sciences of lubrication, wear and friction, was first coined by the esteemed professor Peter Jost in 1966 [6]. The word tribology comes from the combination of the greek word tribo, τριβω [Greek], which means rub or wear and the word ending -logy which in the English language is referring to the subject of study [7]. Even if the field had not been stated as a collected area of study prior to 1966 the fields that it contain has been studied scientifically for many centuries.
For this project the field of lubrication is of most interest and will be introduced further in the following sections.
2.2 Fluid film lubrication
The first detailed theoretical assumptions of the behaviours of fluid film lubrication was first stated by Osborne Reynolds in 1886 through interpretation of the experimental works made by Beauchamp Tower in 1884 [8, 9]. The interpretations made by Reynolds together with the use of the Navier-Stokes equations resulted in a partial differential equation (PDE) which describes the pressure distribution for fluid film lubrication [9].
The field of lubrication studies the effects of manipulation and alteration of contacting surfaces in order to reduce friction and wear. Lubrication comes in two general categories, solid lubrication and fluid film lubrication, see p. 423 in [5]. Since this project focuses on the leakage of lubricant fluid the following introductions to lubrication theory will focus only on fluid film lubrication.
2.2.1 Hydrostatic lubrication
The radial piston motor is a type of hydrostatic motor and is thus, in some parts of the motor, affected by hydrostatic lubrication. Hydrostatic lubrication (HSL) is the kind of lubrication that is the result of a thick film between the contact points from the pressurisation of fluid from an external pump. HSL is common in applications where touching of surfaces is forbidden also in stand-still mode, i.e. at start up or shut down of the application, see p. 424 in [5].
2.2.2 Hydrodynamic lubrication
of external pressurization of the fluid. The hydrodynamic film thickness is normally in the sizes of between 5-500 µm m which is much larger than the normal heights of the asperities of the contacting surfaces used in applications today, more details on HDL can be found in [5] at p. 425-6.
2.2.3 Elastohydrodynamic lubrication
Elastohydrodynamic (EHD) lubrication (EHL) is a subcategory of HDL. The main difference between HDL and EHL is that the elastic deformation of the contacting surfaces is affecting the lubrication process. The film thickness of EHL is even thinner than the HDL film thickness and is normally in the magnitudes of 0.5-5 µm, see p. 427 in [5]. Since the film thicknesses can be very thin there are risks of contacting surfaces; this since the irregularities of the surfaces may for some application components be larger than the film thickness of EHL.
2.2.4 Stribeck Curve
Figure 3: An example of the stribeck curve and the corresponding self-acting lubrication regimes.
As can be seen in Figure 3 there are some lubrication regimes that are presented that have not yet been introduced.
Mixed lubrication is the area of fluid film lubrication where the film thickness heights can for an application make the lubrication regime be in both the HDL and EHL regimes. This may result in surface contacts being more common but this does not apply that the contact is apparent in the whole portion of the contacting surfaces, most of the portion of the surfaces is upheld by HDL. A subarea of the EHL regime is what is called the thin film lubrication regime that is defined as the lower region of EHL, about film thickness between 0.025-2.5 µm, more on mixed lubrication in [5] p. 427-8.
Boundary lubrication is the phenomena of lubrication that has very small fluid film thickness so that the surface-surface contact is dominant mechanism, see p. 428 in [5]. The boundary lubrication regime is present when the load is increased or alternatively the speed or viscosity is increased in the stribeck curve.
Figure 4: Visualisations of the different lubrication regimes. Hydrodynamic lubrication (Top left), Elastohydrodynamic lubrication (Top Right), Mixed lubrication (Bottom left) and Boundary lubrication (Bottom right).
Another way of estimating in which regime a machine element contact is working in is to calculate the film thickness ratio, which is the minimum film thickness divided by the root mean squares of the roughness of the two contacting surfaces [12].
hmin
Λ = �
R2 a1 + R2 a2
where hmin is the minimum film thickness, Rai is the mean surface roughness of surface i and the
different values of Λ represent different lubrication regimes, according to [12] they are • Λ < 1 , Boundary lubrication
• 1 < Λ < 5 , Mixed lubrication
• 3 < Λ < 10 , Elastohydrodynamic lubrication • 5 < Λ < 100 , Hydrodynamic lubrication and according to [13] they are
• Λ « 1 , Boundary lubrication • 1 < Λ < 3 , Mixed lubrication
• 3 < Λ < 10 , Elastohydrodynamic lubrication • Λ » 6 , Hydrodynamic lubrication
regimes. These interpretations give some guideline to the relationship between surface roughness and minimum film thickness which further gives some information on the probable risks in material wear. 2.2.5 Effects on fluid viscosity from temperature and pressure
The viscosity of fluids is dependent on primarily three things, temperature, pressure and shear strain rates. The viscosity of a fluid is dependent on the intermolecular forces of the fluid. If the intermolec ular forces increase the viscosity is increase and vice versa. In this section the used theories for the calculation of fluid viscosity will be presented.
For the pressure dependence on viscosity a common theoretical relationship is the barus equation [14]. Barus equation, formed by C. Barus in 1893, relates the viscosity to pressure comes with the assumption that the fluid is iso-thermal. Equation 1 is the barus equation.
α p
η = η0e (1)
where η is the resulting viscosity in centi Poise (cP), η0 is the fluid viscosity at ambient pressure, e is
the Euler’s number, α is the pressure-viscosity constant that differs between different fluids, normally a value of 2 · 10−8 for mineral oil, and P is the pressure applied on the oil. Since 1893, when C. Barus introduced the relationship expression between pressure and viscosity, vast advancements have been made in the field of tribology and it is suggested that the use of Barus equation and its assumptions is not applicable for all the conditions it is used [15]. However, as introduced in section 2.5.1, from the crude approximations used in the building of the leakage model, together with that the pressures are less than 0.1 GPa for the system, it is assumed to be reasonable to use Barus equation.
The temperature dependence in this project will be related to the oil specification given by the man ufacturer. The viscosity specification for the fluids is given in different kinematic viscosities for two different temperatures. Together with the information by the manufacturer and the use of the Ubbelohde-Walther formula
log(log(ν + 0.7)) = A − Blog(273.15 + T ) (2) the viscosity of the fluid is approximated. Where ν is the sought after viscosity in cSt, A and B is the fluid dependent constants found from the relation between the two reference viscosities given by the manufacturer and T is the temperature in Celsius. The Ubbelohde-Walther formula is a standard, ASTM-D341, in viscosity-temperature relationship.
2.3 Leakage
that BRM develops. The five sources of leakage have different difficulties and complexities that give different levels of possible detail in the modelling of the total leakage. In this section previous research and principles of these leakages will be presented.
The leakage in hydrostatic motors can qualitatively, for laminar flow, be described by Darcy’s law [16]. Darcys law is described as
ΔP
Qi = Ci (3)
η
for fluid flow in an hydrostatic motor, where Qi is the fluid flow for the lubrication interface, ΔP is
the pressure drop over the contact area, η is the dynamic viscosity and Ci is the loss factor for the
volumetric losses due to geometric dimensions of the contact area, see p. 63-4 in [1]. For the span of the modelling of this project this seems to be an appropriate approach to hold as a basis for the leakage of the motor. Thus, with this approach, the remaining objective for the modeling of the leakage, in the different interesting lubrication interfaces, is to find such loss factors that replicates the geometric constraints of the lubrication interfaces.
The total leakage of the motor becomes
Ql = npQp + nsQs + nrQr + noQo + ndQd (4)
where Ql is the total leakage of the motor, n is the amount of interfaces of the type in the motor and
subscript p denotes the balance pistons, subscript s denotes the balance sleeves, subscript r denotes the cam roller, subscript o denotes the piston ring and subscript d denotes the distributor. The fluid flow from flushing is not defined as a source of leakage but is, from the placement of fluid flow sensors, added to the leakage value and has to be dealt with in the model. Thus the total fluid flow over the flow rate sensor becomes
Q = Ql + Qf (5)
where subscript f denotes the flow rate from flushing of the motor, if such fluid flow is present. 2.3.1 Piston ring and cylinder
The piston ring and cylinder contact is located in the motor at notation 3 in Figure 2. The motor contains, depending on how many cam rows it holds, different amounts of pistons and thus different amounts of piston rings. But each cam row of the CB motor holds 16 pistons. Half of the pistons of the motor is at every time in high pressure mode, out-stroke direction, and the other half in low pressure mode, in-stroke direction.
πrh3 ΔP
Qo = (6)
6l η
where r is the ring radius, h is the mean height of the gap between the piston ring and cylinder and l is the length of the lubrication interface, see p. 63-4 in [1].
The ring is of solid material and both the cylinder and piston ring deforms due to pressure increase. This deformation has been researched internally by BRM. The research resulted in a simulation model that make use of a modified version of the Reynolds equation that incorporate the mixed lubrication regime. The simulation was made for stationary conditions without sliding.
The simulation showed that the leakage increases with pressure up to about 40 bar and then decreases again after 40 bar, depending on the gap height simulated. The simulation results for the leakage on different gap heights and different pressures can be seen in Figure 5.
Figure 5: The simulation results for leakage from the piston ring - cylinder interface with different gap heights and different pressures. Where the x-axis is the pressure in Pa, the y-axis is the gap height in m and the colours of the graph is the leakage with the values related to the colour bar on the left side, increased leakage from bottom to top.
πrh3 de f ΔP
Qo = (7)
6l η
where hde f is the gap height with the deformation is included according to the simulation results in
Figure 5.
2.3.2 Distributor
The distributor is the component of the motor that distributes the high pressure and low pressure oil to the cylinder ports. The distributor can be seen as a flat ring with holes parallel to the axis of the ring [17]. In Figure 6 a distributor for a CA-10 motor is shown, but the principles of a distributor is the same.
Figure 6: A model of a distributor from a CA-10 motor [17]. The principles of the distributor is the same for a CB motor.
Figure 7: A model of the balance sleeves and balance pistons from a CA-10 motor [17], which has the same principles as in a CB motor.
The interesting leakage for the distributor, as a component in itself, is assumed to be the lubrication interface between the distributor and the cylinder block, visualised by the yellow colour in Figure 6. The leakage in the interface is a very complex one and thus the leakage is assumed to be the same as the simulation results in [17]. Further simulations have been made at BRM, where the distributor and other parameters has been adjusted for those of a CB motor. The leakage equation for the distributor thus become
ΔP
Qd = Cd (8)
η
where Cd is a term that holds the characteristics of the leakage according to the simulation results from
BRM.
2.3.3 Balance pistons
Figure 8: A description of the objective of the balance piston and balance sleeve [17].
The balance piston in itself is a ring free piston in a cylinder which can be assumed to have the characteristics of leakage as that of a cylindrical piston in a cylinder
πrh3 ΔP 2
Qp = πvrh + (1 + 1.5e ) x (9)
6l η
from the internal engineering handbook of BRM, Powerful Engineering [18]. Where v is the transla tional velocity of the piston, r is the radius of the piston, h is the mean gap height between the smallest and largest gap height due to eccentricity, l is the length of the piston and ex is the eccentricity of the
piston in a value between 0 for when the piston and the cylinder is concentrically aligned and 1 for the opposite.
Equation 9 reduces to
πrh3 ΔP 2
Qp = (1 + 1.5e ) x (10)
6l η
due to the fact the the balance piston practically is in stand-still. 2.3.4 Balance sleeves
In the distributor and in between every cylinder block, if several is present in the motor type, there is a set of balance sleeves, seen in Figure 8. The objective of the balance sleeves is to ensure pressurisation between the distributor and cylinder block as well as for the between the cylinder blocks when the system is not pressurised.
is substantial deformation of the sleeves and cylinders, so the leakage equation for the balance sleeves become πrh3 de f ΔP 2 Qs = (1 + 1.5e ) x (11) 6l η to account for the change in gap height from deformation.
The simulations was made at BRM on the basis of the model in [17]. The results of the leakage can be seen in Figure 9.
Figure 9: Results from simulation of the leakage with deformed sleeve and cylinder.
2.3.5 Piston and roller
The piston-cam roller contact is at notation 3 in Figure 2 and the amount of high pressurised interfaces is of the same amount as for the pistons. For every piston assembly there is a cam roller which objective is to transfer the translational motion delivered on to the roller, by the pressurised fluid, in to a rotary motion of the roller. The lubricational contact between the piston and the cam roller is, in general, a journal bearing contact. But due to the complex shape and design of the piston, in the journal contact, the fluid film and the resulting behaviour differ from a classical journal bearing interface [4].
the roller [4]. The film thickness over the contact will increase due to deformation of the piston mainly because of the material stiffness of the surrounding components [4].
In Figure 10 the design of the piston roller can be found.
Figure 10: Description of the piston design in the piston-roller contact area with the fluid film flow [4].
Figure 11: The curve for leakage from sliding velocity according to simulations in [4].
From further simulations made at BRM other results from the same model presented in [4] was achieved. One of the results was a curve for leakage on rotational speed of the motor which is more applicable to the sensor measurements available for this project. This curve can be seen in Figure 12.
Figure 12: An alternative result from the model developed in [4] with the leakage on basis of rotational speed. The interesting curve is the green one for ”leak_old” which is results for the relevant CB motor piston design. The blue curve is the result from simulations of another piston design. The x-axis holds the revolutions per minute of the motor and the y-axis is the leakage.
ΔP
Qr = f (ω) (12)
η
where f (ω) is a function for the leakage-rotational speed relationship, matching the green curve in Figure 12.
2.4 Flushing
The flushing of a motor is a feature which objective is to, primarily, cool the motor so that the system in which it is applied does not overheat. Measuring of the amount of flushing applied is not available for the analysed motors in this project. The equation for the influence of flushing on the system thus is simply approximated as
Qf = Cf (13)
where Cf is a constant that relates to the amount of flushing applied to the system.
2.5 Grey box modeling
A model is defined as a description of a system that is used to answer questions about that system without the use of experimentation, see p. 13 in [19]. A model can be of different complexity and level of detail but is however still a model that is describing a systems behaviour. There are different categories of models, which mental models, verbal models and mathematical models are a few of them. This thesis will focus on the use of mathematical models. A mathematical model is a model that with the help of mathematical descriptions of the physics and characteristics describes the system. There are two methods for mathematical model building, physical modeling and system identification, see p. 15 in [19]. The physical modeling aim at describing the system with the help of the developed laws of nature that the system is affected by. Identification is performed with the use of observations of the system and its behaviour in order to describe the system. These two methods is in most cases used in combination with each other to construct a model that with the chosen level of detail replicates the behaviours of the system. There are a few different approaches to mathematical modeling in practise and these may in turn take very different forms and shapes depending on the prerequisites of the problem at hand. The main areas of modeling are described as white box, black box and grey box models.
Black box models are parameterized models where the behaviour of the system have not been pre viously modeled but is instead a completely data-driven method, see p. 247 in [19]. These models have no interpretable connection to the system physics that they are describing but are only used to directly describe how the system input corresponds to the system response. Models of this kind is thus describing the behaviour of the system by using mathematical equations or graphs describing the input-output relationship.
from a priori knowledge [20]. White boxes are thus directly connected to the physical behaviour of the system where all the parameters that the model depend on are known.
When trying to describe a large complex system it is often a difficult task to express the different subsystems and details about the physics of these systems, see p. 247 in [19]. It is therefore necessary to describe these with the help of parameters and with the help of statistical methods determine the values of these parameters. The parameters of the mathematically described systems will many times be know to an approximate value by experience and design. But from the uncertainties of the match of the values of the parameters against the expected design the values needs to be adjusted. This is where parameter estimation and grey box models comes in to play.
Grey box models are models that consists of a combination of white and black box models. The two different kinds of grey box models are semi-hybrid models, which consists of a combination of white and black box models in parallel or serial [21]. And the use of white box models with unknown parameter values which are estimated with the use of black box model techniques [20]. This thesis will make the use of the latter.
This type of grey box model can have a different portion of influence from black and white box models [22]. The scale of greyness is very diverse and span from models that are very light grey, i.e. models that are depending very much on physical knowledge of the system but still has some unknown parameters that needs to be estimated. To the very dark grey models where the models mainly depend of the statistical interpretation of the relationship between input and output but with some physical knowledge of the system parameter values are present in the model. The different scales are in theory endless but some examples can be found in [22].
2.5.1 Model building
Making a good description of a system as a mathematical model is many times a hard task to perform. It depends strongly on the constructors knowledge of the system and the complexity of the system. It is also not a trivial task to perform since there is no general steps for all systems that needs to be taken in to constructing a model, see p. 65 in [19]. There are however some phases that can summarise the process of creating a model for a physical system, these are
1. Structuring of problem 2. Formulate basic equations 3. Reformat equations
three handles the reformatting of the previously stated equations and relations between the systems dependent subsystems. The third phase is heavily affected by the choice of calculation tools and these tools method of solving problems as well as the intended use of the model, see p. 66 in [19].
Modelling of a system always contain simplification of the system. These simplifications can be of different order with the main trade-off between computation time and accuracy of replication of the system. The trade-off of the system depends on the systems dependency of different subsystems identified as well as the approximations made of the subsystems. A key part in simplification of a model is that the model is no better than the crudest approximation made, see p. 83 in [19], this may however be influenced by the parallelism of the subsystems where different subsystems may not be affecting each other but still affects the system output by different magnitudes. Three different simplifications are generally discussed, these are
• Small effects are neglected - approximate relationships are used • Separation of time constants
• Aggregation of state variables
The first kind of simplification, small effects are neglected, it is discussed that physical intuition plays a big part in deciding which approximations is suitable according to the physical area of the system and, again, the purpose of the model. Often this is by simplifying the model in terms of neglecting the physical effects that gives a small impact on the system compared to the computational cost. Some examples of physical effects that is common to neglect are air drag, friction and compressibility of liquids.
The second simplification area covers the time dependency of the dynamics of the system, as examples the cases of the dynamics between the acceleration and velocity of a car which is in the order of seconds opposed to the reaction time of a economic system which in some cases might be in the order of several years. If a system is dependent on different time scales the model should then be constructed on the phenomena with the time constants that is most interesting for the use of the model. The recommendation is then to model the phenomena with much smaller time scales as static relationships and the phenomena with much larger time scales modelled as constants in order to reduce the order of the model.
2.5.2 Preprocessing of data
A substantial part of developing a model that can replicate a system well enough depends on the data available and its quality. It is more common than rare that the collected data contains measurements that is not representing the true value of the measurement. The bad data may be missing values, false values or extreme outliers among others [23]. In order to be able to create a good model it is then necessary to evaluate the collected data to identify and handle the flaws of the dataset with the help of different methods and approaches.
Missing values in a dataset is not always a flaw of data collection but may be in fact be caused of the type of data collection approach that has been used, in [23] this is named Legitimate missing data where some examples are presented. The contrary kind is called Illegitimately missing data, which is data that is missing which is not identified as a product of a cause and effect relationship. This kind of data is most common in technical data collection with the use of sensors, where as some examples a sensor can malfunction, is badly calibrated or simply returns a false measurement.
Missing data may also be a result of different time scales of the collected data, which can be related to the second phase of model building. How to handle these missing data values varies from data to data and application to application. It is therefore important to have a good understanding of the kind of data collected as well as the system that the model is to replicate. It is then also important to inspect and analyse the collected data in order to evaluate the usability of the data and what measures that is needed to be taken in order to obtain a set of data that can enable good replication of the system. The first thing to do when preprocessing data is to inspect the data in order to try to identify flaws of the collected data. When flaws of the dataset has been identified, depending of the identified flaws, there are different methods available to clean the data to get a representative dataset. One measure that can be taken is to select segments of the dataset that is considered to be free from flaws and to only proceed with that segment of the original dataset, see p. 464 in [24]. If the dataset consists of many inputs and outputs it may sometimes be difficult to identify such a segment that is considered flawless and then the developer of the model is forced to use data that contain flawed data values. The flawed data values, as mentioned before, for technical data collection is often of the kind Illegitimately missing data.
In a physical system like the one handled in this thesis it is common that there is some time delay between the input and output data. The time delay may come from two main sources which are transport delay and sampling delay. Depending on the medium of the measured data and placement of the sensors in the system the effects may be large or small.
The transportation delay is defined as the distance between the input and the output and the velocity of the signal [25]. A fixed time delay is defined as the physical distance between signal and measurment over the velocity of the signal, but since the velocity of the medium is changing over time the time delay of this system is of the time-varying kind as
L
τ(t) = (14)
v(t)
where τ(t) is the time-delay at time t, L is the signal distance and v(t) is the signal velocity of the signal.
2.5.3 Parameter optimisation
The essence of a good model is the models ability to predict the output from a given input with small prediction errors. Let the prediction error for a certain model, M(θ), be given by
ε(t, θ) = y(t) − yˆ(t|θ) (15) where ε is the error, y(t) is the actual output, ˆy is the model output, t is the time and θ is the parameter set of the model M(θ), details on prediction error and adjacent theory at p. 198 in [24]. The goal is to find a model, M(θ), such that the prediction error, ε, is as small as possible.
A grey box model is, by definition, containing a mixture of known and unknown parameter values. In order to estimate the value of the unknown parameters there exist many different methods. A common method to use is the least squares error (LSE) method.
The LSE is a parameter estimation method that aim to minimise the squared error, essentially the difference, between the actual output data and the output data generated by the model. Utilising linear regression for the prediction of the output
yˆ(t|θ) = ϕT (t)θ + µ(t) (16)
where µ is a known data-dependent vector, and ϕ is the regression vector. Equation 16 then changes Equation 15 to become
The criterion function, VN (θ, ZN ), is a scalar-valued function which is a measure of the validity of the
model, where ZN is the given dataset
VN (θ, ZN ) =
1
∑
(l(ε)) (18)N
where l(ε) is the scalar valued function denoting the applied norm on the prediction error and 1 < t < N. When the function applied to the prediction error is chosen to be squared
l(ε) = ε(t|θ)2 = [y(t) − ϕT (t)θ]2 (19)
Equation 18 together with Equation 19 becomes the least squared criterion
N
1
[y(t) − ϕT (t)θ]2
VN (θ, ZN ) =
∑
(20)N t=1
Equation 20 can in most cases not be solved analytically and thus has to be solved iteratively with the help of some numerical method according to
θˆ(i+1)= θˆ(i)+ α f (i) (21)
where θˆ is the estimated parameter set, i is the current iteration, f is the search direction given from information about V (θ) and α is a positive constant that makes the decrease of V (θ) appropriate, also known as the step size, see p. 327 in [24].
A common group of iterative search methods are called the Newton algorithms which makes use of the gradient and the Hessian of V (θ) in order to decide the search direction
f (i)= −[V ( ˆθ(i))]−1V ( ˆθ(i)) (22)
2.5.4 Local and global minimum
The issue of minimised solutions that has generated a set of estimated parameters that does not provide a good validation of the model still persists. Newtons method, mentioned in section 2.5.3, has good convergence to a minima but does not converge fast far from the minima. It is therefore a good approach to pay some time in order to give a good initial guess in order to lower the number of iterations for convergence which might result in lower total computation cost, see p. 338-40 in [24]. Models constructed from the basis on a physical system it is most natural to use the knowledge and insights about the process or system in order to give a reasonable initial guess.
2.5.5 Accumulated prediction error metrics
To be able to evaluate different models against each other it is necessary to introduce some metric that relates the models performance to each other. The metric used for this project is the normalized root mean squared error (NRMSE) according to
(y − yˆ)
ζ = 1 − (23)
(y − y)
This gives a score, or fit, value between the real output data and the predicted output data in a measure between 0-100%, where 100% denotes a perfect value match between prediction and real data; in practice, however, the value can go below 0% if the match is bad enough but this should be interpreted as a 0% match [26].
2.5.6 Training and validation datasets
When developing a model a crucial part is the ability to validate that the model can predict future output data. That is where different sets of input and correlated output data comes to play. To be able to estimate the parameters the model needs some data to evaluate the new guesses to, this data is the training dataset. But, since the model has been adjusted to this specific dataset, the model will perform best to this set of data. To evaluate if the model will be good enough to predict the output of future input data it is necessary to have a set of, for the trained model, unseen data to assess how well the model performs, see [27] and p. 176 in [28]. This is the validation dataset.
2.5.7 Underfitting and overfitting
When developing predictive models for replicating a systems behaviour the developed model is always exposed to the risks of overfitting and underfitting [29].
The trade-off between underfitting and overfitting is often described by the trade-off between bias and variance in the context between the prediction error and the model complexity. The bias in this context is denoting the error that is induced by the model complexity from trying to replicate a real physical system, i.e. the bias induced error will be high for a simple model and low for a complex model. The variance is the measure for how much the estimated output will change when the model is trained on different datasets, i.e. the variance induced error will be high for a complex model and low for a simple model, see p. 33-6 in [28]. The goal is thus to find a model structure with the complexity that gives the lowest combination between the variance and bias of the model. This trade-off can be visualised by Figure 13. The trade-off relationship can be expressed as in Equation 24 where the lowest mean squared error, MSE, is sought after.
ε(y − yˆ)2 = bias(yˆ)2 + var(yˆ) + var(εnoise) (24)
where var(εnoise) is the variance of the irreducible white noise of the system measurements.
Figure 13: Example of a visualisation of the bias and variance trade-off. The goal is to find the model structure that gives the combination of low bias and low variance.
prediction error in the training dataset.
Figure 14: Example of an alternative, more practically relatable, visualisation between the vari ance and bias trade-off in model complexity.
2.5.8 Comparison of model structures
There are many mathematical metrics for the evaluation of model quality, some of these are the Final Prediction Error (FPE), Akaike Information Criterion (AIC) and the Bayesian Information criterion (BIC). However, the AIC and BIC are used for comparing models with different structures and com plexity and will thus not be used in this thesis.
FPE is a metric for evaluating a models expected performance on validation data. This is done by simulating a situation where the model is tested on a different data set that the estimation dataset, without the actual need of a validation dataset [30].
1 + (dM/N) 1 + (dM/N) 1 N
Jp(M) ≈ VN (θˆN , ZN ) =
∑
ε2(t, θˆN ) (25)N 1 − (dM/N) 1 − (dM/N) t=1
where Jp is the final prediction error, N is the size of the estimation dataset, dM is the number of
2.5.9 Model validation
To be able to confidentially choose a model structure that replicates the system and implement the model for its intended use. It is needed to validate the model in terms of the following aspects, see p. 509-10 in [24].
1. Models agreement with observed data 2. Models performance for the purpose 3. Models description of the "true system"
To answer evaluate the model in these terms it is best done with as much information about the true system as possible with the aid of a prior knowledge, experiment data and experience of using the model.
The most commonly used approach for model validation is the first aspect. With the help of process data or experiment data validate the model performance. The data should then be from the validation dataset, described in section 2.5.6 [27].
A model always has some form of purpose of use, where some categories of purpose might be predic tion, simulation or control. Thus, if a model can sufficiently handle the purposed task that the model was created for it is a good enough model. A cause for concern is, although, that it might often be very costly, impossible or dangerous to test if the models can perform well in their intended use. For example if the models purpose is to handle regulation of a power plant it is very risky to test a model in actual use and thus it is needed to develop confidence in the model with the help of some other metrics.
Another way to acquire confidence in the developed model is to evaluate the feasibility of the physi cal parameters of the model. For a grey box model this should, for the physically related parameters, already be evaluated in the construction of the model in terms of reasonable bounds. But it is still inter esting to evaluate the estimated parameters of the model and to ensure that the parameter distribution is reasonable.
In the case of non-linear models, it is natural to evaluate the models input-output behaviour consistency from different data with the help of simulation or prediction. The numerical fits, such as the metric described in section 2.5.5, is an intuitive way of relating different models performance to each other, see p. 509-10 in [24]. With the help of plotting the prediction or simulation, from different sets of data and comparing the simulated output to a set of measured output data, the performance can easily be examined and possible feature flaws of the developed model can be visualised.
N
RˆN εu(τ) = 1
∑
ε(t)u(t − τ) (26) N t=1where τ is the lag term. And on the other hand evaluate if the prediction errors are correlated or not according to
N
RˆN ε (τ) = 1
∑
ε(t)ε(t − τ) (27) N t=1Checking if the value from Equation 27 for different τ is inside a chosen confidence interval of the prediction errors answers the question of if the prediction errors are correlated or not. This is called a whiteness test. If the values of Equation 27 for some τ is exceeding the bounds of the confidence interval, then part of ε(t) could be predicted from ε(t − τ) and the prediction error does not originate from white noise, see p. 511-6 in [24]. Thus the model could be improved to better predict the true output.
The same approach is done for the test of prediction errors and past inputs as in Equation 26. Likewise, if the value from Equation 26 for different τ is exceeding the confidence interval then the values of the true data y(t) is assumed to be originating from past inputs. Thus, from a test like this show that,
if the covariance is found from past inputs, some dynamics of the system is not captured in the model and the model could be improved in order to better predict the system output. This test is called the independence test, more details on residual analysis can be seen at p. 511-6 in [24].
Figure 15: An example of the Residual test results for a set of data with added simulated white noise. The AutoCorr plot relates to Equation 27 and XCorr(u1) relates to Equation 26 for different lags τ where the blue field is the confidence interval of, in this example, 99%.
3 Methodology
In this section the performed steps and practical executions that has been made during the project that enabled the results received from it are presented.
3.1 Defining loss factors for interface leakage
From section 2.3 there are some missing details and pieces in the established loss factors of the inde pendent leakage interfaces to be able to good enough represent the system behaviour. In the following subsections the measures taken to approximate the characteristics of the lubrication interfaces is pre sented.
3.1.1 Piston ring and cylinder leakage
As stated in section 2.3.1 the piston ring is expected to deform due to pressurisation of the ring. To replicate the simulation results found in Figure 5 an attempt to find an expression that could be applied to the gap height parameter in Equation 7 was made. By analysing the simulation results in Figure 5 it can be seen that the leakage is the highest for a pressure difference of about 40 bar. To replicate this behaviour an expression for the gap height dependent on the pressure difference was found to be
hde f = he−ΔP/Xr (28)
where h is the gap height without deformation, e is the natural number and Xr is a constant. The new
expression thus becomes
3
πr ΔP
Qo = he−ΔP/Xr (29)
6l η
Figure 16: The simulated results of Equation 29.
3.1.2 Distributor leakage
The leakage from the distributor was from the start of the project expected to be the lowest contributor to the total leakage of the motor. The leakage, as introduced in section 2.3.2, is a very complex one and was thus only represented by a constant in Equation 8. Results from simulations by BRM shows that the leakage indeed is expected to be very small as well as varying very little from the pressure difference and it is therefore assumed to be negligible.
3.1.3 Balance sleeves leakage
The leakage from the balance sleeves was in section 2.3.4 introduced as dependent on deformation on the sleeve and cylinder from increased pressure. From simulation results by BRM, seen in Figure 9, the deformation varies from pressure. The procedure was to try a similar approach as that in section 3.1.1 and resulted in an expression for the gap height as
hde f = he−ΔP/XS (30)
which makes Equation 11 become
πr ΔP 2 3
Qs = 1 + 1.5ex 1.1he−ΔP/Xs (31)
6l η
where XS is a constant. The resulting leakage from Equation 31 compared to the simulated result in
shows the simulation results from Figure 9 compared to the leakage from Equation 11 with a gap height parameter without consideration for deformation (hde f = h) a substantial improvement of the
match between results can be seen with the added deformation term in Equation 31.
Figure 17: Comparison between the simulation results in Figure 9 and the leakage equation, Equation 11 for balance sleeves without consideration for deformation, i.e. hde f = h. The trans
Figure 18: Comparison between results from Equation 31 and simulation results in Figure 9. The transparent surface is the equation results and the underlying solid surface is the simulation results.
Figure 19: A comparison for the difference between the simulation results and Equation 31 without the deformation term as hde f = h.
Figure 20: A comparison for the difference between the simulation results and Equation 31.
3.1.4 Piston and roller leakage
The piston and roller, as introduced in section 2.3.5, is, from the governing equations used in [4], known to be dependent on the slipping velocity and thus indirectly on the motor speed in revolutions per minute. The loss factor in Equation 12 which represents the leakage dependency on rotational speed was found to be a polynomial that matches the curve in Figure 12. The polynomial, a quintic function, was found by applying a set of points on the curve in the plot in Figure 12 and then using Matlabs curve fit function to find the representing curve for the applied points [31]. The resulting curve that is accepted as a replicate of the simulated result can be seen in the Figure 21 and the quintic function can be seen in Equation 32.
f (ω) = aω5 + bω4 + cω3 + dω2 + eω + f (32)
where a, b, c, d, e and f is constants that creates the replicated curve seen in Figure 21.
Figure 21: The result of the curve fitting for replicating the simulation results in Figure 12.
The simulation results was received from simulations at only one viscosity and pressure difference and to relate the results for different pressure differences and different viscosities two additional constants is added to Equation 32.
η0
where ΔP0 is the pressure difference and η0 is the viscosity used in the simulation. The leakage for
the roller-piston interface thus becomes
ΔP η0
Qr = aω5 + bω4 + cω3 + dω2 + eω + f (34)
η ΔP0
3.1.5 Flushing
For the systems modelled in this thesis affected by flushing, the flushing was dealt with in two different ways. The first case was to set the flushing value to a fixed value according to the specification of the system. And the other case was to let the estimation procedure estimate the flushing value with a initial value of the flushing specification for the system.
3.2 Matlabs non-linear grey box model object
With all the governing equations for the total leakage of the system it is time to set up the structure of the grey box model. This is done with the help of Matlabs non-linear grey box object, idnlgrey, and its corresponding option sets [32]. The idnlgrey object is constructed with a set of inputs. These inputs can be seen in table 3.
Inputs of idnlgrey construction Model structure filename Model structure orders Parameters of the model
Initial states of the state-space model Time-step size of the discrete model Optional inputs
Table 3: Description of the input parameters of the construction function for constructing an idnlgrey object.
The model structure filename points to the filename for the file holding the governing equations of the model and which specifies the inputs, states and outputs of the model. The states relates to the state-space model, which is a method of defining equations for a problem or a system that is represented by inputs, states and outputs for a first order differential, or difference, equation. The states in a state-space model is thus for example the derivative states at a given for the system.
The second input in the construction of the non-linear grey box object is the orders of the model. That is how many outputs, inputs and states the model contain. The model has three inputs, Pressure difference, Fluid temperature and rotational speed, one output, the fluid leakage after the motor and, as introduced in the paragraph above, no states.
The parameters of the model is a set describing the boundaries and initial values, if the values are to be estimated, of the model structure dependent parameters. The specifications of the parameters that can be set can be seen in Table 4
Specification of parameter set Fixed or free
Initial value Lower value bound Upper value bound Optional inputs
Table 4: Description of the specification inputs parameter input in the construction of an idnlgrey object.
where Fixed or free specifies if the parameter shall be estimated or not, Initial value is the set value of the parameter before estimation is proceeded, Lower value bound is the lowest value an estimated parameter can receive, Upper value bound is the highest value an estimated parameter can receive and optional inputs are the descriptive name and unit, respectively, of the parameters for increased readability when the model is presented.
Initial states of the state-space model is the value of the states in the beginning of the time-series. The specification inputs of the initial states input are the same as in Table 4.
Time-step size of the discrete model is a scalar value for the time difference between the input and output data that is fed to the model through an iddata object, the iddata object will be introduced in section 3.3.3.
The optional inputs of the construction of the idnlgrey object are different descriptive specifications as the name, time unit, input and output names of the object.
For the parameters in the governing equations there are only a few which are unknown and that have to be estimated. These are the different gap heights of the interfaces and the eccentricities of the balance pistons and the balance sleeves. Thus these parameters have been set to be free and all others set to be fixed. The fixed parameters are either from curve fitting regarding the roller-curve interface, Equation 34, and the deformation term constants XO and XS in Equation 29 and Equation 31 respectively. Or
3.3 Data preprocessing
The available data for this project is, as previously mentioned, data for the pressure difference, fluid temperature, rotational speed and leakage from the motor. The selection and correction of the sampled data available will in this section be presented.
The data for this project had, for different reasons, some anomalies of e.g. spikes, values out of physically reasonable bounds and missing values for some time periods.
The available datasets was fortunately very large for this type of project and the estimation problem that it poses. Several different sets of data was found that held low amounts of anomalies that was enough to use for estimation.
3.3.1 Temperature sample correction
The data that has been sampled has some different sample times. For the types of data used in this project it is the data for the fluid temperature that has a lower sampling rate than the other collected inputs and outputs. Temperature change in any process is a rather slow process and in the systems replicated in this project the temperature has been sampled every second while the other data in the dataset is sampled in a sub-second manner.
The data used in the model is necessary to have the same time-step size and thus the missing data points in the temperature data set will have to be interpolated in some way. An alternative to this would have been to neglect the sub-second samples in the other data sets and only use the second-wise data points, but from the slow characteristics of temperature change it seems to be acceptable to interpolate the temperature data and instead keep as much information about the faster dynamics of the system as possible. The interpolation of the temperature data was done with linear interpolation. 3.3.2 Outlier removal and noise smoothing
In the chosen datasets used for training and validation there are some apparent anomalies and flaws that had to be dealt with. These are the low sensor resolution for the leakage data. That is that the datapoints in the datasets have large instantaneous differences throughout the measurements. To solve this a MA filter was applied to better the possibility for efficient parameter estimation by revealing the underlying trend of the data and rid the continuous spikes in data values.
Another apparent anomaly of the data for the leakage is a known flaw, in the context of this thesis and the intended use of the sensor. That is when a back flow of fluid is passing the sensor, which happens when the motor is halted. This gives a spike in values of the sensor just before values of the flow rate go towards the minimum of the values. The value of the flow rate sensor, for when fluid is running in the backwards direction from the intended measuring direction, is set to give a maximum value. Thus this flaw i bypassed by choosing a dataset which does not contain any sudden spikes to the maximum value of the flow rate sensor.
