Design of a power loss model for vehicle drivetrains

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MASTER'S THESIS

Design of a power loss model for vehicle

drivetrains

Jesper Berglund

Master of Science in Engineering Technology

Mechanical Engineering

Luleå University of Technology

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I

Abstract

Lately the discussion on how big impact vehicles has on the environment has grown bigger and bigger. The vehicle manufacturers are building cars, trucks, wheel loaders etc. that are getting more and more fuel efficient for every year. But where do the engineers at the companies make the improvements? The drivetrain is one area of the vehicle where many different sources of losses are found. The gearbox is one example, where the losses are both dependent and independent on the torque transferred by the gear pairs. The gearbox consists of several components that all of them contribute to the total system power loss. It is therefore of great interest to understand the magnitude of the losses in different components for different operating conditions, and also to be able to determine how they change with, for instance, rotational speed and load.

There are currently some commercial software’s on the market that analyses how the losses in a drivetrain are distributed. These tools are useful for engineers when analyzing existing drivetrains, or when developing new products. The software’s shows how losses change depending on components, oils, engine speeds and load which helps the engineers concentrate on the right areas. Power loss calculations within these software’s are often based on simple and generalized methods. For instance could the user be needed to specify a contact friction coefficient of the system that will be held constant during the simulation. For starters a constant friction coefficient is a rather big simplification, and the task of assessing this value for the engineer is an intricate manner.

In this Master of Science thesis work a module based, user friendly software is developed. The model is taking the biggest loss sources, for example gearbox, differential and tires into account in a system with static conditions. No losses due to the dynamics in the system will be taken in consideration in this model. The final model is a good base containing most of the components of a person’s vehicle with rear wheel drive. All components from crankshaft to asphalt are modelled in simple modules built by Simulink blocks and Matlab m-code.

In the model created in this thesis all calculations are based on formulas from scientific experiments. This makes the loss calculations more advanced and accurate then the generalized formulas in most commercial software’s. It also gives the user bigger opportunities to for example use different methods to obtain the gear tooth contact friction coefficients. Two methods are currently implemented in the model. One method is to use an empirical formula, and the other is to use friction maps generated by experiments. As a result from the research program ProAct, lookup tables with friction coefficient depending on slide to roll ratio and entrainment speed can be implemented in the model. This makes it possible to show the difference in contact friction coefficients between the empirical formula and the friction maps. Ultimately, the purpose of the model is to be able to evaluate how a change in the system will affect the overall power loss in the drivetrain. A looped simulation between two mineral oils in this model shows that the gear frictional losses has the biggest influence on the total loss of the gearbox in a simulation of 31 investigated cases from 1000 rpm to 4000 rpm with an constant torque input of 100 Nm.

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III

Preface

The following report is the result of a Master of Science in mechanical engineering thesis work with construction as main direction. The thesis work is conducted in cooperation with the department of Engineering Sciences and Mathematics, at the Division of machine elements, as a subproject of the research program ProAct at Luleå University of Technology, LTU.

During the thesis work I have had a close contact with my supervisors at the division of machine elements. I especially want to thank Marcus Björling, Dr. Pär Marklund and Prof. Roland Larsson who had shared knowledge and guidance during this thesis work.

Through this thesis work I got extended knowledge in areas of literature surveys, analysis and simulation with advanced calculations and software’s.

Jesper Berglund

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V

1. Nomenclature ... 2. Introduction ... 1 2.1 Problem ... 1 2.2 Powertrain ... 2 2.2.1 Seals... 2 2.2.2 Bearings ... 2 2.2.3 Clutch ... 3 2.2.4 Gearbox ... 3 2.2.5 Differential ... 4 2.2.6 Tires ... 5 2.2.7 Friction maps ... 5 2.3 Delimitations ... 6 3. Method ... 7 3.1 Software survey ... 7 3.2 Components ... 8 3.2.1 Seals... 8 3.2.2 Bearings ... 8 3.2.3 Clutch ... 9 3.2.4 Gearbox ... 9 3.2.5 Differential ... 13 3.2.6 Tires ... 14 4. Model ... 15 4.1 Clutch ... 15

4.1.1 Make a clutch Variant ... 15

4.2 Gearbox and Differential ... 18

4.2.1 Gearbox ... 18

4.2.2 Differential ... 21

4.3 Bearing ... 21

4.3.1 Make a bearing variant ... 21

4.4 Seals ... 22

4.4.1 Make a sealing variant ... 22

4.5 How to run a simulation ... 23

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VI

5.1 Tooth friction loss ... 25

5.2 Churning loss ... 29

5.3 Total loss ... 30

6. Discussion and conclusion ... 31

7. References ... 33

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VII

1. Nomenclature

Symbol Description

Tcd Torque loss in clutch [Nm]

Ndisc Number of discs

R Average radius [m]

 Fluid density [kg/m3]

rc1 Outer radius of friction plate [m]

rc2 Inner radius of friction plate [m]  Angular velocity rad/s

hc Distance between discs [m]

Ts Torque loss of sealing [Nm]

ds Diameter of shaft, contact with seal [m]

ns Rotating speed of shaft [rpm]

Mbi Total power loss in SKF bearing [Nm]

Φish Inlet shear heating reduction factor

Φrs Kinematic replenishment/starvation reduction factor

Mrr Rolling frictional loss [Nm]

Msl Sliding frictional loss [Nm]

Mseal Frictional loss from sealing [Nm]

Mdrag Frictional loss from oil/air drag [Nm]

Grr Coefficient depending on axial-, radial load and mean diameter

ν Kinematic viscosity coefficient of fluid in bearing [mm/s2] nb Rotational speed of bearing [rpm]

Gsl Coefficient depending on axial-, radial load and mean diameter

fsl Sliding frictional coefficient

KS1 Constant depending on bearing type

KS2 Constant depending on bearing type

dsb Seal counter face diameter [mm]

βber Exponent depending on bearing and seal type

Vm Variable depending on oil level

Kball Constants depending on bearing

Kroll Constants depending on bearing

bb Bearing width [m]

dm Mean bearing diameter [m]

ιc Coefficient of viscosity in clutch [Ns/m2]

Ttfl Tooth friction loss [Nm]

Tin Incoming torque [Nm]

Hv Tooth geometry factor

fgf Gear friction coefficient

zi Number of teeth i=1,2,3…

εα Tooth contact ratio

AB Line of action, tooth contact length [m] pbt Base pitch diameter [m]

p Normal pitch diameter [m] bg Gear width [m]

C1 29.66 constant

Wn Normal load on tooth [N]

µgb Dynamic viscosity of gearbox oil [mPas]

Vs Sliding velocity [m/s]

Vr Rolling velocity [m/s]

2.14 Adjustment factor SRR Slide to Roll Ratio

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VIII V1 Velocity on gear [m/s]

V2 Velocity on pinion [m/s]

Tct Total churning loss torque [Nm]

Tpi Pocketing churning loss [Nm]

Tcdi Total drag churning loss torque [Nm]

Tdpi Power loss due to air/oil drag on the periphery [Nm]

Tdfi Power loss due to oil/air drag on the sides of the gear [Nm]

Trfi Power loss occurring while filling the cavity between teeth with oil [Nm]

Ppi Average powerloss due to pocketing [W]

M Number of increments vb,ij Backlash flow velocity [m/s]

ve,ij Flow velocity on end areas [m/s]

Fb,ij Force on backlash area [N]

Fe,ij Force on end area [N]

Pb,ij Pressure on backlash area [Pa]

Pe,ij Pressure on end area [Pa]

ξ Flow factor (Backlash area divided to total area)

Ab,ij Backlash area [m

2

]

Ae,ij End area [m2]

ϕ Increment change [rad]

Vij Volume depending on increment [m 3

] Height ratio of oil coverage

ra Tip radius of the gear [m]

ρgb Density of gearbox oil [kg/m3]

νgb Kinematic viscosity of gearbox oil [cSt]

Ncav Number of cavities below oil level

θ Angle spanning the tooth cavity

rva Tip radius of an equivalent spur gear [m]

rvb Base radius of an equivalent spur gear [m]

rv Back cone distance [m]

εγd Contact ratio for spiral bevel gear

Tt Torque loss from tire [Nm]

Mcar Mass of the car [kg]

g Gravitation [m/s2]

Cr Rolling resistance factor [kg/ton]

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1

2. Introduction

This master thesis is conducted in cooperation with a subproject to the research program ProAct, at Luleå University of Technology, focusing on gear tribology together with the industrial partners Scania, Volvo Construction Equipment and Vicura AB (former SAAB powertrain).

The automotive industry of today faces increased demands on lowering emissions due to legislations and customer preferences, which in turn drives development to lower fuel consumption. Decreasing fuel consumption can be accomplished in several ways, for instance reducing the weight of the vehicle, improving aerodynamics or increasing the efficiency of the powertrain. It is therefore of great interest to understand how decreases in frictional losses in for instance a gearbox will affect the total losses or efficiency of the complete powertrain.

2.1 Problem

The objective of this thesis work is to create a simulation model for a complete vehicle drivetrain from crankshaft to the driving wheels including gears, bearings, seals, wheels etc. The model should be modularized so that different parts of the drivetrain are described in separate modules. In this way the complexity of each model could be changed depending on priorities and input from other research projects. The final model should be able to predict friction losses and efficiency for a specific drivetrain under certain running conditions.

The work in this thesis was divided into sub-problems with the topics:

 Literature survey, to see what already has been done.

 Software survey, to find the most suitable software for this task.

 Modeling, creating the model.

 Report writing, to make a summary of the thesis work and an instruction on how to use and complement the model.

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2.2 Powertrain

In the definition of drivetrain there are two main areas. The first one is to generate power and is most commonly done by a combustion engine. The second area is to deliver this generated power in form of torque and rotation to the road surface. The composition of a vehicle drivetrain varies depending on the demands and area of usage of the vehicle. The one thing in common for all vehicle drivetrains is to transport the energy generated from fuel in form of torque and rotation to the road surface and carry the vehicle from point A to B. This may sound like a rather simple task but is in reality highly advanced, especially with increasing demands on for instance efficiency and long lifetime. The model created in this thesis focuses on the second part of the drivetrain. With cooperation of various machine elements deliver power from the engines outgoing shaft, crankshaft, to the road surface. The most commonly used components within a mechanical drivetrain are: clutch, gearbox differential and tires, Figure 1.

Figure 1. Drivetrain including the most common components.

In the drivetrain the losses can be divided into two subgroups, one dependent and one independent on the torque transferred by the components. Between these two subgroups there are no dependency, therefore it does not matter how much torque the gearbox are transferring for example at 4000 rpm, the churning losses (which are torque-independent) in the gearbox will stay identical. A more correct term of the drivetrains components in the model are subsystem. Each subsystem contains of a various amount of components like seals, bearings etc. In the following sections the components used within each different subsystem will be described.

2.2.1 Seals

One of the torque-independent losses comes from the seals. In drivetrains it is most common to use rubber seals. The seals purpose is to prevent oil exposure to the outside of the component and

contamination of the oil inside of the component. The seals have to be close to the rotating shaft and to fulfill its purpose a small contact will occur, thereby creating friction.

2.2.2 Bearings

The purpose of the bearing is to make the shafts rotate with as little friction as possible with external axial and radial forces acting on them. Since the efficiency of a bearing decreases with external load applied on it, the bearing is seen as a torque-dependent component. Depending on the bearings location and gearbox or differential design there are different kinds and amount of bearings in these subsystems.

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3 2.2.3 Clutch

The clutch’s function is to transmit rotation and torque in certain situations with help of rotating friction plates. For instance when a driver wants to change gear, he disengages the clutch by pressing the clutch pedal down. This makes the two friction plates separated from each other, breaking the transfer of torque and rotation, and the driver can change gear without any problem. When the driver then releases the clutch pedal, the clutch engages and the friction plates are brought together,

transferring torque and rotation further on the drivetrain. There are two main clutch systems in vehicle drivetrains. The first system is the so-called dry clutch. This is most common in vehicles with a manually shifted gearbox. The second one is the wet clutch system. It is built on the same theory as the dry clutch but it often has several friction plates rotating in oil, thereby called a wet clutch. This type of clutch is commonly used in automatic shifted gearboxes, four wheel drive differentials and limited slip differentials. There are also other types of clutches, and one of them is for example the dog clutch commonly used inside trucks gearboxes to connect axels between different gear pairs.

2.2.4 Gearbox

One of the biggest subsystems of the drivetrain is the gearbox. The gearbox itself consists of several components. The components modeled in this thesis are seals, gears and bearings. Depending on type of gearbox the quantity of components varies. In a gearbox there are two types of power losses generated by the different components. When gears transfers torque it emerges frictional losses in the contact between the gear teeth. These losses are called torque-dependent losses, since they vary with the amount of torque transferred. Another torque-dependent loss occurring when gears transfers torque, is the loss from gearbox bearings. When gears transfers torque it occurs axial and radial forces within the shaft the gears are mounted on. These forces will act like external forces on the bearing and thereby decreasing the bearings efficiency, as explained above. The torque-independent losses are occurring when the components for example rotates in oil. The torque-independent loss comes from oil churning and oil shearing which are identical whether the gears are transferring torque or not. In the gearbox there is not only just the active gear pairs that rotates, even the inactive gear pairs rotates and contributes to the total churning loss.

How the torque and rotation are transferred through the gearbox modeled in this thesis depending on active gear pair is shown in Figure 2.

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In general for gearboxes running at low speeds and transmitting high torques the losses from tooth friction is a substantial part of the total gearbox loss. But with higher speeds the percentage of the total loss coming from tooth friction decreases and instead the churning loss dominates. The tooth frictional loss depends both on rotational speed and torque.

The coefficient of tooth friction is continuously changing along the line that arises between two teeth in contact, the so-called Line of action (LOA). The gear tooth friction loss is among other depending on the geometry of the gears and the rotational speed. Different geometries make the ratio between rolling and sliding vary in the contact along LOA, which in turn influences the frictional behavior. The amount of rolling relative to sliding in a contact is usually explained as the slide to roll ratio, SRR. Both SRR and entrainment speed are varying along LOA as shown in Figure 3.

Figure 3. SRR and entrainment speed varying along line of action.

2.2.5 Differential

The differential’s or the final drive’s function is to divide and transfer torque and rotation from the gearbox further to the wheels. The first step of the differential is a spiral bevel gear commonly used in person’s vehicles with rear wheel drive, or a hypoid gear used in heavier vehicles. Both of them are built on the same theory. The second step is two straight bevel gears facing each other. The big difference between bevel gears and hypoid gears is the position of the hypoid’s pinion gear, where the center line of the pinion axle does not intersect with the centerline of the gear as shown in Figure 4. For front wheel drive cars with transverse engines the differential/final drive is built by ordinary helical gears.

Figure 4. The difference between a spiral bevel and a hypoid gear [7].

Since this kind of differential is in principle a gearbox, but with only one gear ratio, the losses occurring are of the same kind as in a gearbox.

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5 2.2.6 Tires

The most complex subsystem of the drivetrain is the tires. Tires are probably also the biggest source of losses in the whole second area of the drivetrain. The loss factors of tires vary with road conditions, tire composition and many more factors.

2.2.7 Friction maps

In the research program ProAct several investigations have been made that focuses on how different entrainment speeds and SRR are influencing the contact friction in gears. Friction maps for two mineral oils, one with high viscosity (w211) and one with low viscosity (w326), has been retrieved from experiments [1]. The oil data is specified in Table 1. These maps are implemented as lookup tables in Simulink, Figure 5 and Figure 6. The values from experiments in the maps are used as an alternative to the values calculated by the formula (15) explained further in section 3.2.4.1.

Figure 5. Friction coefficient map for w326.

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Table 1. Oil data.

TYPE w211 w326

Additives None None

Kinetic viscosity at 40°C (cSt) 30.8 109.3 Dynamic viscosity at 40°C (mPas) 27.1 94.9 Kinetic viscosity at 100°C (cSt) 5.3 11.98 Dynamic viscosity at 100°C (mPas) 4.46 9.97

Viscosity index 104 99

Type Mineral Mineral

2.3 Delimitations

The models developed in this master thesis work is only considering static conditions in the drivetrain. Any transient effects in different components that will arise due to changes in component speeds and loads etc. as seen in a dynamic system will therefore not be included.

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3. Method

To not make a work that is already been done and to secure the quality of the model a literature and software survey was carried out. By reading scientific articles, thesis reports and books regarding the area of drivetrains, a great understanding of the system and the systems losses was built.

During the literature survey the level of complexity on the model was set. As stated in the problem description the drivetrain power loss model should be made such that models of different complexities could be used for different parts in the drivetrain. For this reason a more advanced power loss model is desired for the gear teeth friction and churning losses, to be in line with current research projects, while the other parts of the drivetrain could be modeled with less complex formulas. So the survey was to find models that were suitable for the first level of complexity. The following section 3.2 describes the calculation model chosen for each area.

The software survey was carried out to be able to fulfill the criteria of the model regarding availability and the simplicity of use.

3.1 Software survey

One desirable feature of the model is that it is easy for the companies to use and to implement in their software library. One of the demands of the model was to take only static cases in consideration and to leave the dynamics due to high complexity. While doing the literature survey there was a couple of different software’s used in similar cases. Software allowing modelling with a graphic interface would make a good overview of the model. The most common software’s using a graphic interface was SimDriveline, Simulink and Modelica. The first two presented by MathWorks and the last by Modelica Association. SimDriveline and Modelica use modelling with physical signals where each “line” transfers multiple signals such as torques, velocities and other physical quantities. These signals are best when modelling dynamic systems but can also be used, with high complexity, with static problems. Simulink uses standard signals containing only one user defined constant signal. Simulink and SimDriveline can be combined to calculate both physical signals and standard signals but that makes the model more complex than it has to be. Simulink and Matlab can also be combined. In this way more complex calculations are made in Matlab, sending a variable to Simulink. As mentioned earlier Modelica, SimDriveline and Simulink uses a graphic interface to build models which gives a good understanding and thereby also easy to use. With the criteria that the model should be easy to implement to their software library it is convenient to use software that many companies has available. Another aspect of this criteria is that the software is backward-compatible so that the model created now can be used even in five years from now. The latest version of SimDriveline (SimDriveline 2.0) does not contain the same modules as the previous version, which makes it hard to use with a model, created in an old version or vice versa. Simulink and Matlab is backward compatible, which makes them qualified. Most companies do have Matlab already and often Simulink to. SimDriveline is kind of new and not all companies have that MathWorks toolbox.

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3.2 Components

The formula chosen for each component within the drivetrain during the literature survey will be described in the following sections.

3.2.1 Seals

The friction losses from seals in contact with rotating axles, Ts, are given by [2] as

(1)

where -5 is a loss factor of rubber, ds are the shaft diameter and ns is the shafts rotating speed in rpm.

3.2.2 Bearings

As mentioned earlier, depending on vehicle design and position of the bearing there are different kinds to apply. To all different kinds of bearings it is the same base method to calculate the total loss in bearings. The SKF method to calculate power loss in bearings is expressed as [3]

Φ Φ (2)

where Mbi is the total power loss of the bearing, i=1,2,3… depending on which bearing on order it is.

Mrr is the rolling frictional moment, Msl is the sliding frictional moment, Mseal is the frictional

moments from sealing and Mdrag is the friction moment of drag losses. Φish and Φrs are inlet shear

heating reduction factor and kinematic replenishment/starvation reduction factor respectively. Rolling frictional moment is generally given by

(3)

where Grr depends on the axial and radial loads and the mean diameter of the bearing,

Grr={Fa , Fr , dm}, ν is the kinematic coefficient of the lubricant and n the rotational speed in rpm.

Sliding frictional moment is given by

(4)

where Gsl={Fa , Fr , dm} and µsl is the sliding frictional coefficient. Sealing frictional moment formula

is for the bearings with seals built in it and is not the same as the seal-formula (1) explained earlier. Mseal is given by

(5)

where KS1 , KS2 and βber are constants depending on bearing and seal type, dsb is the seal counter face

diameter. The moment losses from drag in bearings are given by

(6)

or

(7)

depending if it is a ball or roller bearing. Vm is a variable as a function of the oil level, Kball and Kroll

are constants depending on bearing and Bb is the bearing width.

As mentioned in earlier formulas are varying depending on what type of bearing it is and the formulas can be found in [3].

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9 3.2.3 Clutch

The loss in a dry clutch in a static case is so small that they don’t need to be considered in the model. However, during a dynamic drive cycle the dry clutch would have a significant part of the torque peaks of the system. Since this thesis work don’t take dynamics in considerations the dry clutch would be seen as ideal.

In wet clutches there are multiple discs rotating in fluid both when active and inactive. This results in drag losses. Hisano Kitabayashi’s model of drag losses in wet clutches is expressed as [4]

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where Ndisc is number of discs, R is the effective radius, ιc is the dynamic viscosity of the fluid inside

the wet clutch, rc with index 1 and 2 are outer and inner radius respectively of the friction plate, ω is

the angular velocity of the shaft and hc is the distance between discs.

3.2.4 Gearbox

As mentioned earlier both losses who are dependent and independent of the transmitted torque contributes to the total loss of the gearbox. Methods for calculating both of these loss types will be described in following sections.

3.2.4.1 Torque-dependent

As explained in previous section one of the torque-dependent losses is the loss occurring in the contact between two contacting gear teeth. One of the contributing parts when calculating the friction

coefficient, SRR is defined as [5]

(9)

V1 and V2 is the velocity on the gear and pinion respectively and vary along the contact line. The

sliding, Vs, and rolling, Vr, velocities are given by

(10)

and

(11) where Ue is the entrainment speed along the line of action. The complete calculation of the gear

geometry and SRR can be found in the m-code in Appendix 8.2. These velocities are used to calculate the friction coefficient between two gears in contact, either by the formula explained below, or trough lookup tables as explained in section 2.2.7. The tooth frictional loss formula [6] is chosen for this thesis and given as

(12)

(19)

10 The tooth geometry factor Hv is given by

(13)

and εα is the tooth contact ratio given by [7]

(14) where AB is the tooth contact length (Line of action), pbt is the base pitch diameter, p is the normal

pitch diameter and bg is the width of the gear. The friction coefficient of the gear, fgf, can be calculated

as [8]

(15)

where C1=29.66 is a constant, Wn is the normal load on the gear, 2.14 is an adjustment factor, Vs and

Vr are sliding and rolling velocities respectively and µgb is the dynamic viscosity of the gearbox oil.

3.2.4.2 Torque-independent

The torque-independent losses generated from gears are the oil churning. It is the same theory as for the wet clutch, but instead of discs it is gears rotating in oil. According to [9] the total oil churning loss can be calculated with

(16)

where Tcdi is the churning loss due to drag losses and Tpi is the churning loss due to pocketing. The

total churning loss due to drag is defined by

(17)

the three included sub losses is Tdpi, power loss due to air/oil drag on the periphery, Tdfi, power loss

due to oil/air drag on the sides of the gear and Trfi is power loss occurring while filling the cavity

between teeth with oil. The sub losses is given by

(18) where ra is the outer radius of the gear and is the height of oil coverage. Tdfi is given by

(19) for laminar and

(20)

for turbulent working condition. Here νgb is the kinematic viscosity and ρgb is the density of the oil

(20)

11 Trfi is given by

(21) where Ncav is number of cavities below oil level and θ is the angle spanning the tooth cavity. D3 and D4

are determined by applying boundary conditions to the solution of the biharmonic equation [9]. The pocketing contribution to the total churning loss depends on change of volume between gear teeth. The volume change in four increments is approximated as shown in Figure 7. The detailed calculation of volume change can be found in [9]. The striped area, Atot, as shown in Figure 8 below is the

approximated area of a cavity.

Figure 7. How the volume change showed with four increments.

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The total area in Figure 8 above symbolizes the fully filled cavity. At m=0 the Volume V(0) is

decreased to approximately 50% of the total volume. At m=1 the volume V(1) is decreased into 2/3 of the last mentioned area. At m=2 it is just the bottom half circle remaining. m=3 is only half of the bottom circle remaining.

The pocketing loss is given by

(22) where Pp,ij is the power loss due to pocketing for the total volume decrease. M is the number of

increments used. Pp,ij is given by:

(23) where vb,ij is the oil backlash flow velocity, ve,ij is the velocity of the oil flow escaping through the

ends. Fb,ij and Fe,ij is the forces acting on the oil depending on which end it escapes. The forces are

given by

(24)

And

(25)

where Ae,ij is the areas described above and Ab,ij is approximated as shown in Table 2.

Table 2. Aproximated backlash area.

Increment m=0 m=1 m=2 m=3 Area A0b,ij = bg*1*10 -3 A1b,ij = Ab 0 b,ij/2 A 2 b,ij = Ab 1 b,ij/2 A 3 b,ij = 2*Ab 2 b,ij/3

The pressure at each of the areas are given by

(26) for the backlash pressure and

(27) for the pressure on the ends. The velocity on the escaping oil is given by

(28)

(22)

13 For the ends

(29)

where ϕ is the increment change in radians, ξ is a ratio between backlash flow area and the total flow area, given by

(30)

Due to approximated area values, (28) and (29) is calculated with the volume that occurs in the different increments. The percentage of the volume change is timed to the loss part of each step, for detailed calculations, see Appendix 8.5.

3.2.5 Differential

As for the helical gears in the gearbox the same losses occur here. The same formulas for calculating churning are used in this model as in the gearbox model. The model contains a differential with spiral bevel gear. In [10][11] describes a model of calculating gear geometry for spiral bevel gears. These geometry formulas may differ for a hypoid gear. Losses for spiral bevel and hypoid gears are assumed to be calculated the same way as losses in helical gears (12). The following calculations consider geometry of spiral bevel gears:

(31)

where the torque coming from gearbox is Tind and the friction coefficient fgf is calculated the same way

as before (15). Hvd is the geometry factor for the differential and is given by

(32)

The transverse contact ratio for spiral bevel gears is given by

(33)

and overlap ratio for spiral bevel gears is given by

(34)

where rv is the back cone distance given by

(35)

dd is the pitch diameter

(23)

14 and δ the pitch cone angle

and (37)

where is the shaft angle. rva is the tip radius of an equivalent spur gear given by

(38)

where hdai is the addendum given by

(39)

and

(40)

The base radius of an equivalent spur gear, rvb is given by

(41)

where αdt is the normal pressure angel of the differential gear. The total contact ratio is given by

(42)

3.2.6 Tires

Due to the complexity of this area a general calculation model was chosen depending on weight of the car, Mcar, the rolling resistance factor given by the tire manufacturer, Cr, the gravity, g, and the wheel

radius including rim and tire, rw [11]:

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4. Model

At first the structure of the model was sketched in Simulink without any loss calculations to verify the correct structure. Further on loss calculations were implemented to the Simulink sketch. To give the model a clean look the majority of calculations were made in Matlab which was coupled to the Simulink model. In the following sections the model will be closer described.

4.1 Clutch

The clutch is modeled in two variants, one ideal clutch that symbolizes the dry clutch and the second variant is symbolizing the wet clutch with losses as shown in Figure 9 below.

Figure 9. Simulink model of Wet and Dry clutch.

4.1.1 Make a clutch Variant

To implement the clutch type to the model the following steps were followed:

 Find a suitable equation for the model

 Make a new Subgroup in the Clutch Variant block by right clicking the Variant block, choose “Subsystem Parameters”, Click the “create and add a new subsystem choice”, Figure 10.

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 Rename the block by double clicking at the name and make it “Variant3”, and give it condition A==3 (or the next in order) by pressing the “Edit” button, Figure 11.

Figure 11. Press the Edit button to change condition

 A simple equation can be made directly in the new sub block in Simulink with suitable calculation blocks from the library.

 Constants can be typed in a Matlab m-file and be requested from Simulink or directly made in Simulink Constant block, Figure 12.

Figure 12. Simulink constant block.

 If more advanced equations are used, it is often more suitable to perform the calculations in an M-file called from the Simulink environment. Save the m-file with a suitable name like “wet_clutch.m” or “dog_clutch.m”. Save it to the same directory as the rest of the model. Right click on the newly created block and choose “Block Properties”, press callback and type the name of the m-file in StopFcn (or other suitable function parameter).

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 Right click on the Clutch variant block, go to variant, override variant and choose the new version of clutch, Figure 13.

Figure 13. Change variant by right clicking variant subsystem block.

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4.2 Gearbox and Differential

Next in order of the drivetrain is the gearbox and differential subsystem block with two variants, one ideal and one including losses, Figure 14. In the Losses variant there are many calculations, due to this m-files for the blocks was created, see Appendix 8.2 to Appendix 8.7.

Figure 14. Gearbox and Differential variants

4.2.1 Gearbox

As mentioned earlier the gearbox includes a lot of subcomponents itself. To make the model as close to real life an old Volvo gearbox was used to make measurements in. In the gearbox, Figure 15, the first sub component is in this case both sealing and bearing, Figure 16. These will be described in section 4.3 and 4.4.

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Figure 16. Bearing and sealing block described in section 5.3 and 5.4.

The second subcomponent is the gear steps including the tooth frictional losses and churnings losses. The gear step is modeled in three levels. Level 1 is shown in Figure 17, level two, Figure 18, enables to choose gear from the main block. Third level is represented by gear 1, the following gears steps are created in the same way, Figure 19.

Figure 17. Gears level 1.

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Figure 19. Gear 1 block's calculations.

The two last blocks of the gearbox are the Shearing block and a subcomponent block containing bearing and sealing calculations, Figure 20.

Figure 20. Shearing and bearing 2 + sealing 2 blocks.

4.2.1.1 Make a gearbox variant

To make a new gearbox model the same steps as for the clutch are used, see 4.1.1. The differences are:

 Write the gear parameters such as module, profile shift and number of teeth in an m-file which is named after the gearbox.

 The oil parameters are filled in the start_value.m file.

 Build the gearbox model from an exploded view of the gearbox. Place the components in the same order as they are inside the gearbox.

 Modify the bearing and sealing blocks to the same as in the gearbox and implement them in the model.

 Churning and teeth friction is calculated in the same way as described in Appendix 8.4 and 8.3.

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The differential model is a simple variant of the final gear based on a spiral bevel gear, see section 3.2.5. The model of the differential is based on the gearbox model. The differences are that the torques and angular velocities are divided into two sources, each of the driving wheels. The vehicle is

supposed to drive in a straight line which leads to a transfer of 50% power to each tire. There is also one bearing to each axle which means that the second subcomponent block containing bearings and seals are each multiplied with two.

4.2.2.1 Make a Differential variant

The making of a differential model is similar as for the gearbox, see section 4.2.1 and 4.1.1. The big differences are the names of the m-files and inside them the calculations of gear geometry, see m-file in Appendix 8.7.

4.3 Bearing

The bearing model in Simulink is built up on the four loss components of the bearing loss formula, Figure 21. The formula is calculated in the m-file and the results are requested from the Simulink model.

Figure 21. Bearing block containing the four loss components.

4.3.1 Make a bearing variant

To make a new bearing variant it is unnecessary to make any new block containing the calculations, make the new variant by following these steps.

 Write the variables of the bearing into the m-file.

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 Make a test simulation and verify to SKF homepage. - If the results is the same, the formula is correct

- If the results differ, change the exponents in the Grr and Gsl formulas to obtain the correct

results.

 If there are many bearings, just copy the component block and the formulas in the m-file and rename the result and the component blocks to suitable names.

 Implement the new calculations to the block that will obtain the bearing by typing the correct names to the blocks in the bearing block.

 Start the simulation and verify the result.

4.4 Seals

Depending on type and design of the gearbox or differential the seals may vary. In this case the sealing model is a general rotating shaft friction loss formula. The formula is modeled directly in Simulink due to the simplicity of the equation which makes use of a separate M-file unnecessary, Figure 22.

Figure 22. Sealing block with all calculations made directly in block.

4.4.1 Make a sealing variant

To make a new variant of seals, do the following steps:

 Copy the sealing constants from the consisting m-file.

 Rename the constants, with new ending numbers for example.

 Copy the formulas and rename the formulas with the new names from the constants.

 Write the new names of the constants in the new sealing block.

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4.5 How to run a simulation

To run a successful simulation the right callback functions has to be set, Figure 23. To set the callback functions, right click on the subsystem that is going to have the function, chose “block parameters”. Right click also in the “Main” window on an empty spot. Choose “Model Properties” and control that “start_value” is in InitFcn.

Figure 23. Gearbox block containing two callback stop functions.

When all the callbacks are right:

 Type in the value of engine speed and torque constants.

 Make a test simulation with “Ideal” variant on “Gearbox and Differential” block to verify the correct callbacks are used.

 To simulate losses, change “Gearbox and Differential” block to “Losses” variant.

 Run the simulation

 Show the losses by writing name of the loss or plot the losses by writing the plot commands in the Matlab prompt.

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To run a simulation with ramping engine speed complete the first two steps explained above, then:

 Type in the ramping value of engine speed in “Loop_Simulation.m”.

 Type the names of the parameters in the constant blocks of the Simulink model, Figure 24.

Figure 24. Type the name of the parameters in Simulink model.

 Type the names of the values that are wished to be saved, Figure 25.

Figure 25. Saves the requested values in a matrix.

 Type the plot parameters to make a plot when the loop is done.

 Run the “Loop_Simulation.m” file.

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5. Results

The results of this thesis are focused on the losses occurring in the gearbox. Two simulations are made to see how a change in oil viscosity is affecting tooth frictional losses and oil churning losses in the gearbox. The simulation was also carried out to see which of the loss factors having the biggest contribution to the total loss. The values used from oils are at 40°C, Table 1.

5.1 Tooth friction loss

To see how the friction coefficient for all gear steps differs between the two oils calculated by formula (15) and lookup table the following plots were made at 1000 rpm and 100 Nm, Figure 26 to Figure 29.

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Figure 27. How the friction coefficient varies for gear 1, step 2 depending on oil viscosity calculation method.

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Figure 29. How the friction coefficient varies for gear 3, step 2 depending on oil viscosity and calculation method.

To see how the losses changes with changing engine speed, a looped simulation were made with engine speeds from 1000 rpm to 4000 rpm with steps of 100 rpm and the same torque as earlier (100 Nm). The mean friction coefficient was used for each speed to calculate the friction loss for each gear and oil, Figure 30 to Figure 33.

Figure 30. How the tooth friction loss changes for gear 1, step 1 with changing engine speed for the different oils and calculation methods.

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5.2 Churning loss

The effect of oil viscosity and rotational speed on churning losses is shown in, Figure 34.

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5.3 Total loss

The total gearbox loss differs depending on what oil and calculation method that is used to calculate the friction coefficient, Figure 35. With this plot it is also obvious which loss component which has the biggest influence on the total loss of the gearbox. The total power loss of the gearbox is shown in Figure 36.

Figure 35. Total loss in gearbox depending on oil and calculation method.

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6. Discussion and conclusion

The model is created in MathWorks toolboxes Simulink and Matlab. This makes it is useable for almost anyone, because most companies already have MathWorks and these toolboxes which make the model easy to implement in their software library. It doesn’t matter what version of MathWorks the companies have to be able to use the model, because files created in these toolboxes are backward-compatible.

The model is module based and makes a good base for further work. Most of the components in a drive train are represented in the model. By choosing a modular design of the model it is easy to modify certain parts of the model, such as the power loss formulas, or simply add additional modules to include additional bearing types and gear geometries. It is just to add a new component as described in section 4 above and copy the existing m-code and type in the data that describes the components. In order to work, the m-file and the Simulink blocks has to be renamed and placed where they should be in the model. Because of the callback function in Simulink it easy to change what calculations that should be done by changing initial, start or stop function callbacks in Simulink. The model m-file library has no limitations but hard disc space, which allows the user to build a big component and subsystem library.

More advanced or other formulas are easy to implement because of the use of the m-code. More gear friction maps from testing can easily be implemented and simulated due to the same reason.

The gear geometry calculations examples in this thesis are built on measurable data from an existing gearbox from which all dimensions are measured. By doing this it is hard to know if the gears are different from gear standards, for example regarding profile shift or tip relief. As seen in for example Figure 26, the value of SRR is around an absolute value of 2.0. A SRR value over an absolute value of 2.0 is impossible to achieve. When the SRR absolute value reaches over 2.0, it means that one of the gear surfaces is moving backwards towards the other. The gear geometry errors occurring from incorrect tip relief or profile shift could lead to these unrealistic SRR values.

The pocketing part of the churning losses has the biggest influence on the total losses. The

approximations of the geometry to this part of the churning loss can lead into lowered losses. With more exact gear tooth geometry data a closer estimation can be calculated with the formulas presented in section 3, Method.

The friction maps shown in Figure 5 and Figure 6, has the lowest value of zero or close to zero, to extrapolate these maps results in negative friction coefficients, which is not possible. This will in the future be solved by more extensive testing so that no extrapolating is required. As seen in Figure 26 to Figure 29, the friction coefficient from the experimental values differs quite much from the values calculated with equation (15). Taking the mean values from these curves could lead to reasonably big difference between the tooth friction losses calculated with the two different methods. In this looped simulation case, is clearly shows a difference between the methods comparing the results of the total power loss of the gearbox.

In a looped simulation with 31 static cases between 1000 rpm and 4000 rpm and constant torque at 100 Nm, using both the low viscosity oil (w211) compared with a use of high viscosity oil (w326) in the modelled gearbox. The summation of the results of this looped simulation shows a decrease of the churning losses with almost 12 percent. Due to this decrease of power loss the total power loss is still about 17 percent lower for the high viscosity oil. To remember is that for a person’s vehicle the torque input to the gearbox in this case it is very high. Normally lower torques are dominant in a car’s driving

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cycles and that is probably the reason why the trend among today’s vehicle manufacturers is to use lower viscosity lubricants. Depending on the application of the vehicle I think that the domination of torque-dependent and torque-independent losses can vary. In for example a truck that is transmitting high torques at low engine speeds, probably the torque-dependent losses will be dominating the total loss of the gearbox. But for example a motorcycle with high engine speeds and low torque transfer, the torque-independent losses would be dominant.

In total this model shows that it is the torque-dependent losses that have the biggest influences on the total loss of this gearbox. These results would look the same for other similar systems like the

differential. Just like in the gearbox there are gears rotating in oil in the differential as well. The gears do not rotate as fast, but they are bigger and that could contribute to big losses too. Since this thesis has focused on gearbox losses there are great opportunities for further work to improve the differential subsystem and to implement a hub reduction subsystem where there are big gear ratios which

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7. References

[1] Björling, M., Larsson, R., Marklund, P., & Kassfeldt, E. (2011). Elastohydrodynamic

lubrication friction mapping- the influence of lubricant, roughness, speed and slide-to-roll ratio. Luleå: SAGE.

[2] Müller, H. K., & Nau, B. S. (1998). Fluid Sealing Technology. New York: Marcel Dekker Inc.

[3] SKF. (2003). General Catalogue. SKF.

[4] Kitabayashi, H., Li, C. Y., & Hiraki, H. (2003). Analysis of the Various Factors Affecting

Drag Torque in Multiple-Plate Wet Clutches. Yokohama: SAE.

[5] Amaro, R. I., Martins, R. C., Seabra, J. O., & Brito, A. T. (2004). MoST low friction

coating for gears application. Porto: Elsevier Ltd.

[6] Changenet, C., Oviedo-Marlot, X., & Velex, P. (2006). Power Loss Prediction in Geared

Transmission Using Thermal Networks-Applications to a Six-Speed Manual Gearbox.

ASME.

[7] Eriksson, E., Kassfeldt, E., Höglund, E., Isaksson, O., Lagerkrans, S., & Lundberg, J. (1990). Maskinelement, kompendium. Luleå.

[8] Heingartner, P., & Mba, D. (2005). Determing Power Losses in the Helical Gear Mesh. Gear Technology.

[9] Changenet, C., Leprince, G., Ville, F., & Velex, P. (2011). A Note on Flow Regimes and

Churning Loss Modeling. ASME.

[10] QTC. (2000). Handbook of Metric Gears. Quality Transmission Components. [11] KHK. (2010). Gear Technical Reference. KHK.

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8. Appendix

Appendix list

8.1 Initial conditions m-file.

8.2 4-speed Volvo gearbox

8.3 Gear losses m-file.

8.4 Gearbox churning m-file.

8.5 Gearbox pocketing m-file.

8.6 Gearbox bearing m-file.

8.7 Differential spiral bevel including Churning and bearing m-file.

8.8 Loop Simulation m-file.

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8.1 Initial conditions m-file.

%% start values %%% Oils

raw_w326=885; %Density kg/m^3 @40 degrees

ny_w326=109.3e-6; %Kinetic cSt @40 degrees 109.3

my_w326=94.9e-3;%Dynamic mPas 94.9e-3

raw_w211=872; %Density kg/m^3 @40 degrees

ny_w211=30.8e-6; %Kinetic cSt @40 degrees

my_w211=27.1e-3;%Dynamic mPas

raw80w90=887; ny80w90=15e-6; %15cSt 100grader my80w90=raw80w90*ny80w90; %%% g=9.81; %gravitation %Clutch Variables R=0.100; %Disc radius

jota=0.05; %Oil density

hc=0.002; %Distance between disc

NN=4; %Number of discs %Oil Shearing %my=47.6e-6*887.3; %48.6cSt 887.3kg/m3 my=my_w326; j=0.04; L=0.2; %axle length %Seal variables k=9; %seal koefficient ds=0.03; %diameter seal t=0.003; %seal thickness

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8.2 4-speed Volvo gearbox

%% Volvo m40 gearbox %%%%%%%%% Gear 1 step 1

alpha_deg=20; beta_deg=20;

le1=0.02; %face width

z(1)=19; %20 x(1)=0.38; %0.38 m(1)=0.0025; %0.0025 z(2)=27; %30 x(2)=0.7; %0.7 m(2)=m(1); %0.005 %%%%%%%%% Gear 1 step 2 alpha2_deg=20; %

beta2_deg=20; %Helical angel

z2(1)=15; %20 x2(1)=0.76; %0 m2(1)=0.002256; %0.005 z2(2)=36; %30 x2(2)=-0.2; %0 m2(2)= m2(1); %0.005 %%%%%%%%% Gear 2 step 2 alpha3_deg=20; beta3_deg=20;

z3(1)=20; %20 x3(1)=-0.26; %0 m3(1)=0.0024; %0.005 z3(2)=28; %30 x3(2)=0.56; %0 m3(2)= m3(1); %0.005 %%%%%%%%% Gear 3 step 2 alpha4_deg=20; beta4_deg=20;

z4(1)=23; %20 x4(1)=0.68; %0 m4(1)=0.00245; %0.005 z4(2)=24; %30 x4(2)=0.58; %0 m4(2)= m4(1); %0.005

u1=z(2)/z(1); %Gear ratio gear 1 step 1

u2=z2(2)/z2(1); %Gear ratio gear 1 step 2

u3=z3(2)/z3(1); %Gear ratio gear 2 step 2

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2(9) %%%%%%%%%%%%%% Calculations start nr_gear=2; alpha=alpha_deg*pi()/180; beta=beta_deg*pi()/180; beta_b=asin(cos(alpha)*sin(beta)); alpha_t=atan(tan(alpha)/cos(beta)); input_speed=rpm_in; torque=Tg_in; omega=input_speed*2*pi()/60; gear_pair_count=0;

%%%%%%%%%%%%%%%%%%%%%First gear step%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

S=0; for j=1:1:nr_gear-1 inva_t =tan(alpha_t)-alpha_t; invalphawt=inva_t+2*(x(2)+x(1))*tan(alpha)/(z(2)+z(1)); alpha_wt=alpha; for i=1:100,

alpha_wt = alpha_wt - (tan(alpha_wt)-alpha_wt-invalphawt)/(1/(cos(alpha)^2)-1);

end

a(j)=(m(j)*(z(j)+z(j+1)))/(2*cos(beta));

% Calculates the reference centrum distanse

a_w(j)=(a(j)*cos(alpha_t))/(cos(alpha_wt));

% Calculates the centrum distance (if addendum coefficient others than 0)

r(j)=((m(j)*z(j))/cos(beta))/2; % Calulates reference radius /delningsradie/rullcirkel

r(j+1)=((m(j+1)*z(j+1))/cos(beta))/2; d(j)=r(j)*2;

d(j+1)=r(j+1)*2;

rb(j)=r(j)*cos(alpha_t); % Calculates base/picth/ cirle, grundcirkel

rb(j+1)=r(j+1)*cos(alpha_t);

delta_haj(j,:)=m(j)*(((z(1)+z(2))/(2*cos(beta)))+x(1)+x(2))-a_w; ha(j)=m(j)*(1+x(j))-delta_haj(j);

da(j)=d(j)+(2*ha(j));

ra(j)=da(j)/2; % Calculates top cirkle, topcirkel

delta_haj(j+1,:)=m(j+1)*(((z(1)+z(2))/(2*cos(beta)))+x(1)+x(2))-a_w; ha(j+1)=m(j+1)*(1+x(j+1))-delta_haj(j+1); da(j+1)=d(j+1)+(2*ha(j+1)); ra(j+1)=da(j+1)/2;

AN(j)=sqrt(ra(j)*ra(j)-rb(j)*rb(j)); % calc distances along LOA

AN(j+1)=sqrt(ra(j+1)*ra(j+1)-rb(j+1)*rb(j+1)); NT(j)=sqrt(r(j)*r(j)-rb(j)*rb(j));

NT(j+1)=sqrt(r(j+1)*r(j+1)-rb(j+1)*rb(j+1));

N(j)=(rb(j)+rb(j+1))*tan(alpha_wt);

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AT(j+1)=AN(j)-NT(j); % calc dist från rullcent till "avslut"

l_a_length(j)=AN(j)+AN(j+1)-N(j); % Calculates length of line of action

prec=256; %

step=l_a_length/(prec-1); % sets precision of calculation

S=(-AT(j):step:AT(j+1));

omega(j+1,:)=z(j)*omega(j)/z(j+1); %Calculates the speed in second gear

v1(j,:)=omega(j,:)*(r(j)*sin(alpha)+S);

v1(j+1,:)=omega(j+1,:)*(r(j+1)*sin(alpha)-S); v_sj=(v1(j,:)-v1(j+1,:));

v_rj=(v1(j,:)+v1(j+1,:))/2;

u_snok(j,:)=(v1(j,:)+v1(j+1,:))/2; %Calculates the mean entrainment velocity STR1(j,:) = 2*(v1(j,:)-v1(j+1,:))./(v1(j,:)+v1(j+1,:)); %Calculates slide-to-roll ratio vr_max=max(v_rj); vr_min=min(v_rj); vs_max=max(v_sj); vs_min=min(v_sj); STR_max=max(STR1); STR_min=min(STR1); end

%%%%%%%%%%%%%%%%%%%%%%%Losses in first gear step%%%%%%%%%%%%%%%%%%%%%%

AB(1)=l_a_length(j); pb1=pi*m(1)*cos(alpha_t); p=pi*m(1); pbt=pb1/cos(beta); epett1=AB(1)/pbt; eptva1=(le1*sin(beta))/p; epalfa1=(AB(1)/pbt)+((le1*sin(beta))/p); Hv1=(pi/cos(beta_b))*((1/z(1))+(1/z(2)))*(1-epalfa1+epett1^2+eptva1^2); Wn=(torque(1)/0.02)/epalfa1; C1=29.66; myfrik1=(0.0127*log10((C1*(Wn/(l_a_length(j))))./(my_w326.*abs(v_sj).*v_rj) ))./2.14; myfrik21=(0.0127*log10((C1*(Wn/(l_a_length(j))))./(my_w211.*abs(v_sj).*v_rj )))./2.14; min_mf1=min(myfrik1); max_mf1=max(myfrik1); mean_mf1=mean(myfrik1); min_mf21=min(myfrik21); max_mf21=max(myfrik21); mean_mf21=mean(myfrik21); Pgfl1=mean_mf1*torque(1)*Hv1;

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%% First gear, step 2

nr_gear2=2; alpha2=alpha2_deg*pi()/180; beta2=beta2_deg*pi()/180; beta_b2=asin(cos(alpha2)*sin(beta2)); alpha_t2=atan(tan(alpha2)/cos(beta2)); omega2=omega(2);

%%%%%%%%%%%%%%%%%%%%% Second gear step, first gear %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S2=0; for i=1:1:nr_gear2-1 inva_t2 =tan(alpha_t2)-alpha_t2; invalpha_wt2=inva_t2+2*(abs(x2(2))+x2(1))*tan(alpha2)/(z2(2)+z2(1)); alpha_wt2=alpha2; for p=1:100,

alpha_wt2 = alpha_wt2 - (tan(alpha_wt2)-alpha_wt2-invalpha_wt2)/(1/(cos(alpha2)^2)-1);

end

a2(i)=(m2(i)*(z2(i)+z2(i+1)))/(2*cos(beta2));%Calculates the reference centrum distanse

aw2(i)=(a2(i)*cos(alpha_t2))/(cos(alpha_wt2));

r2(i)=((m2(i)*z2(i))/cos(beta2))/2;% Calulates reference radius /delningsradie/rullcirkel

r2(i+1)=((m2(i+1)*z2(i+1))/cos(beta2))/2; d2(i)=r2(i)*2;

d2(i+1)=r2(i+1)*2;

rb2(i)=r2(i)*cos(alpha_t2); %Calculates base/picth/ cirle, grundcirkel

rb2(i+1)=r2(i+1)*cos(alpha_t2);

delta_ha2(i,:)=m2(i)*(((z2(1)+z2(2))/(2*cos(beta2)))+x2(1)+abs(x2(2)))-aw2;

ha2(i)=m2(i)*(1+x2(i))-delta_ha2(i); da2(i)=d2(i)+(2*ha2(i));

ra2(i)=da2(i)/2; % Calculates top cirkle, topcirkel

delta_ha2(i+1,:)=m2(i+1)*(((z2(1)+z2(2))/(2*cos(beta2)))+x2(1)+abs(x2(2)))-aw2; ha2(i+1)=m2(i+1)*(1+x2(i+1))-delta_ha2(i+1); da2(i+1)=d2(i+1)+(2*ha2(i+1)); ra2(i+1)=da2(i+1)/2; da2=2*ra2;

AN2(i)=sqrt(ra2(i)*ra2(i)-rb2(i)*rb2(i)); % calc distances along LOA

AN2(i+1)=sqrt(ra2(i+1)*ra2(i+1)-rb2(i+1)*rb2(i+1)); NT2(i)=sqrt(r2(i)*r2(i)-rb2(i)*rb2(i));

NT2(i+1)=sqrt(r2(i+1)*r2(i+1)-rb2(i+1)*rb2(i+1));

N2(i)=(rb2(i)+rb2(i+1))*tan(alpha_wt2);

AT2(i)=AN2(i+1)-NT2(i+1); %calc dist från ingrepp till "pitch point"

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l_a_length2(i)=AN2(i)+AN2(i+1)-N2(i); % Calculates length of line of action

prec2=256;

step2=l_a_length2/(prec2-1); % sets precision of calculation

S2=(-AT2(i):step2:AT2(i+1)); omega2(i+1,:)=z2(i)*omega(2)/z2(i+1);%Calculates the speed in second

gear vi_2(i,:)=omega2(i,:)*(r2(i)*sin(alpha2)+S2); vi_2(i+1,:)=omega2(i+1,:)*(r2(i+1)*sin(alpha2)-S2); v_sj2=(vi_2(i,:)-vi_2(i+1,:)); v_rj2=(vi_2(i,:)+vi_2(i+1,:))/2; u_snok2(i,:)=(vi_2(i,:)+vi_2(i+1,:))/2;

% Calculates the mean entrainment velocity

STR2(i,:) = 2*(vi_2(i,:)-vi_2(i+1,:))./(vi_2(i,:)+vi_2(i+1,:));

% Calculates slide-to-roll ratio

vr2_max=max(v_rj2); vr2_min=min(v_rj2); vs2_max=max(v_sj2); vs2_min=min(v_sj2); STR2_max=max(STR2); STR2_min=min(STR2); end

%%%%%%%%%%%%%%%%%%%%%%%Losses in second gear step, first gear%%%%%%%%%%%%%%%%%%%%%% Wn2=((torque(1)-Pgfl1)*(z(2)/z(1)))/le2; myfrik2=(0.0127.*log10((C1.*(Wn2/(l_a_length2(i))))./(my_w326.*abs(v_sj2).* v_rj2)))/2.44; myfrik22=(0.0127.*log10((C1.*(Wn2/(l_a_length2(i))))./(my_w211.*abs(v_sj2). *v_rj2)))/2.44; min_mf2=min(myfrik2); max_mf2=max(myfrik2); mean_mf2=mean(myfrik2); min_mf22=min(myfrik22); max_mf22=max(myfrik22); mean_mf22=mean(myfrik22); AB2(1)=l_a_length2(i); pb2=pi*m2(1)*cos(alpha_t2); p2=pi*m2(1); pbt2=pb2/cos(beta2); epett2=AB2(1)/pbt2; eptva2=(le2*sin(beta2))/p2; epalfa2=(AB2(1)/pbt2)+((le2*sin(beta2))/p2); Hv2=(pi/cos(beta_b2))*((1/z2(1))+(1/z2(2)))*(1-epalfa2+epett2^2+eptva2^2);

%% step2, second gear

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%%%%%%%%%%%%%%%%%%%%% Second gear step, second gear %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S3=0; for i=1:1:nr_gear3-1 inva_t3 =tan(alpha_t3)-alpha_t3; invalpha_wt3=inva_t3+2*(abs(x3(2))+abs(x3(1)))*tan(alpha3)/(z3(2)+z3(1)); alpha_wt3=alpha3; for p=1:100,

end

a3(i)=(m3(i)*(z3(i)+z3(i+1)))/(2*cos(beta3));

% Calculates the reference centrum distanse

r3(i)=((m3(i)*z3(i))/cos(beta3))/2; % Calulates reference radius /delningsradie/rullcirkel

d3(i+1)=r3(i+1)*2;

rb3(i)=r3(i)*cos(alpha_t3); % Calculates base/picth/ cirle, grundcirkel

rb3(i+1)=r3(i+1)*cos(alpha_t3); delta_ha3(i,:)=m3(i)*(((z3(1)+z3(2))/(2*cos(beta3)))+abs(x3(1))+abs(x3(2))) -aw3; ha3(i)=m3(i)*(1+x3(i))-delta_ha3(i); da3(i)=d3(i)+(2*ha3(i));

ra3(i)=da3(i)/2; % Calculates top cirkle, topcirkel

delta_ha3(i+1,:)=m3(i+1)*(((z3(1)+z3(2))/(2*cos(beta3)))+abs(x3(1))+abs(x3( 2)))-aw3; ha3(i+1)=m3(i+1)*(1+x3(i+1))-delta_ha3(i+1); da3(i+1)=d3(i+1)+(2*ha3(i+1)); ra3(i+1)=da3(i+1)/2; da3=2*ra3;

AN3(i)=sqrt(ra3(i)*ra3(i)-rb3(i)*rb3(i)); % calc distances along LOA

AT3(i+1)=AN3(i)-NT3(i); % calc dist från rullcent till "avslut"

l_a_length3(i)=AN3(i)+AN3(i+1)-N3(i);%Calculates length of line of action

prec3=256;

S3=(-AT3(i):step3:AT3(i+1)); omega3(i+1,:)=z3(i)*omega(2)/z3(i+1);%Calculates the speed in second

gear

v3(i,:)=omega3(i,:)*(r3(i)*sin(alpha3)+S3);

v3(i+1,:)=omega3(i+1,:)*(r3(i+1)*sin(alpha3)-S3); v_sj3=(v3(i,:)-v3(i+1,:));

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u_snok3(i,:)=(v3(j,:)+v3(j+1,:))/2;% Calculates the mean entrainment velocity

STR3(i,:) = 2*(v3(i,:)-v3(i+1,:))./(v3(i,:)+v3(i+1,:));

% Calculates slide-to-roll ratio

vr3_max=max(v_rj3); vr3_min=min(v_rj3); vs3_max=max(v_sj3); vs3_min=min(v_sj3); STR3_max=max(STR3); STR3_min=min(STR3); end

%%%%%%%%%%%%%%%%%%%%%%% Losses in second gear step, second gear %%%%%%%%%%%%%%%%%%%%%% Wn3=((torque(1)-Pgfl1)*(z(2)/z(1)))/le3; myfrik3=(0.0127.*log10((C1.*(Wn3./(l_a_length3(i))))./(my_w326.*abs(v_sj3). *v_rj3)))/2.14; myfrik23=(0.0127.*log10((C1.*(Wn3./(l_a_length3(i))))./(my_w211.*abs(v_sj3) .*v_rj3)))/2.14; min_mf3=min(myfrik3); max_mf3=max(myfrik3); mean_mf3=mean(myfrik3); min_mf23=min(myfrik23); max_mf23=max(myfrik23); mean_mf23=mean(myfrik23); AB3(1)=l_a_length3(i); pb3=pi*m3(1)*cos(alpha_t3); p3=pi*m3(1); pbt3=pb3/cos(beta3); epett3=AB3(1)/pbt3; eptva3=(le3*sin(beta3))/p3; epalfa3=(AB3(1)/pbt3)+((le3*sin(beta3))/p3); Hv3=(pi/cos(beta_b3))*((1/z3(1))+(1/z3(2)))*(1-epalfa3+epett3^2+eptva3^2);

%% step2, third gear

%%%%%%%%%%%%%%%%%%%%% Second gear step, second gear %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% S4=0; for i=1:1:nr_gear4-1 inva_t4 =tan(alpha_t4)-alpha_t4; invalpha_wt4=inva_t4+2*(abs(x4(2))+abs(x4(1)))*tan(alpha4)/(z4(2)+z4(1)); alpha_wt4=alpha4; for p=1:100,

end

a4(i)=(m4(i)*(z4(i)+z4(i+1)))/(2*cos(beta4));

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r4(i)=((m4(i)*z4(i))/cos(beta4))/2;% Calulates reference radius /delningsradie/rullcirkel

d4(i+1)=r4(i+1)*2;

rb4(i)=r4(i)*cos(alpha_t4); % Calculates base/picth/ cirle, grundcirkel

rb4(i+1)=r4(i+1)*cos(alpha_t4); delta_ha4(i,:)=m4(i)*(((z4(1)+z4(2))/(2*cos(beta4)))+abs(x4(1))+abs(x4(2))) -aw4; ha4(i)=m4(i)*(1+x4(i))-delta_ha4(i); da4(i)=d4(i)+(2*ha4(i)); ra4(i)=da4(i)/2; delta_ha4(i+1,:)=m4(i+1)*(((z4(1)+z4(2))/(2*cos(beta4)))+abs(x4(1))+abs(x4( 2)))-aw4; ha4(i+1)=m4(i+1)*(1+x4(i+1))-delta_ha4(i+1); da4(i+1)=d4(i+1)+(2*ha4(i+1)); ra4(i+1)=da4(i+1)/2; da4=2*ra4; AN4(i)=sqrt(ra4(i)*ra4(i)-rb4(i)*rb4(i)); % calc distances along LOA

AT4(i+1)=AN4(i)-NT4(i); %calc dist från rullcent till "avslut"

l_a_length4(i)=AN4(i)+AN4(i+1)-N4(i); %Calculates length of line of action

prec4=256;

S4=(-AT4(i):step4:AT4(i+1));

%

omega4(i+1,:)=z4(i)*omega(2)/z4(i+1);% Calculates the speed in second gear

v_4(i,:)=omega4(i,:)*(r4(i)*sin(alpha4)+S4);

v_4(i+1,:)=omega4(i+1,:)*(r4(i+1)*sin(alpha4)-S4); v_sj4=(v_4(i,:)-v_4(i+1,:));

v_rj4=(v_4(i,:)+v_4(i+1,:))/2;

u_snok4(j,:)=(v_4(j,:)+v_4(j+1,:))/2; % Calculates the mean entrainment velocity

STR4(i,:) = 2*(v_4(i,:)-v_4(i+1,:))./(v_4(i,:)+v_4(i+1,:));% Calculates slide-to-roll ratio vr4_max=max(v_rj4); vr4_min=min(v_rj4); vs4_max=max(v_sj4); vs4_min=min(v_sj4); STR4_max=max(STR4); STR4_min=min(STR4); end

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%%%%%%%%%%%%%%%%%%%%%%% Losses in second gear step, second gear %%%%%%%%%%%%%%%%%%%%%% Wn4=((torque(1)-Pgfl1)*(z(2)/z(1)))/le4; myfrik4=(0.0127.*log10((C1.*(Wn4./(l_a_length4(i))))./(my_w326.*abs(v_sj4). *v_rj4)))/2.14; myfrik24=(0.0127.*log10((C1.*(Wn4./(l_a_length4(i))))./(my_w211.*abs(v_sj4) .*v_rj4)))/2.14; min_mf4=min(myfrik4); max_mf4=max(myfrik4); mean_mf4=mean(myfrik4); min_mf24=min(myfrik24); max_mf24=max(myfrik24); mean_mf24=mean(myfrik24); AB4(1)=l_a_length4(i); pb4=pi*m4(1)*cos(alpha_t4); p4=pi*m4(1); pbt4=pb4/cos(beta4); epett4=AB4(1)/pbt4; eptva4=(le4*sin(beta4))/p4; epalfa4=(AB4(1)/pbt4)+((le4*sin(beta4))/p4); Hv4=(pi/cos(beta_b4))*((1/z4(1))+(1/z4(2)))*(1-epalfa4+epett4^2+eptva4^2); %%% Pgfl2=mean_mf2*((torque(1)-Pgfl1).*(z(2)/z(1)))*Hv2; Pgfl3=mean_mf3*((torque(1)-Pgfl1).*(z(2)/z(1))).*Hv3; Pgfl4=mean_mf4*((torque(1)-Pgfl1).*(z(2)/z(1))).*Hv4;

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8.3 Gear losses m-file.

Pgfl1=mean_mf1*torque(1)*Hv1; Pgfl21=mean_mf21*torque(1)*Hv1; PGFL1=mean(fric_w1(1,:)).*torque(1)*Hv1; PGFL21=mean(fric_w21(1,:))*torque(1)*Hv1; Pgfl2=mean_mf2*((torque(1)-Pgfl1).*(z(2)/z(1)))*Hv2; Pgfl22=mean_mf22*((torque(1)-Pgfl1).*(z(2)/z(1)))*Hv2; PGFL2=mean(fric_w2(1,:)).*torque(1)*Hv2; PGFL22=mean(fric_w22(1,:))*torque(1)*Hv2; Pgfl3=mean_mf3*((torque(1)-Pgfl1).*(z(2)/z(1))).*Hv3; Pgfl23=mean_mf23*((torque(1)-Pgfl1).*(z(2)/z(1))).*Hv3; PGFL3=mean(fric_w3(1,:))*torque(1)*Hv3; PGFL23=mean(fric_w23(1,:))*torque(1)*Hv3; Pgfl4=mean_mf4*((torque(1)-Pgfl1).*(z(2)/z(1))).*Hv4; Pgfl24=mean_mf24*((torque(1)-Pgfl1).*(z(2)/z(1))).*Hv4; PGFL4=mean(fric_w4(1,:)).*torque(1)*Hv4; PGFL24=mean(fric_w24(1,:))*torque(1)*Hv4;

Design of a power loss model for vehicle drivetrains

MASTER'S THESIS