bricks theory

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Version 3.2

Theory

March 2003


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Theory BRICKS Version 3.2

Address comments concerning this document to:

AVL LIST GmbH

A-8020 Graz Hans-List-Platz 1

Phone: +43 316 787-1675Telefax: +43 316 787-1922E-Mail: [email protected]

Web Site: http://www.avl.com

Revision Date Description Document No.

A 30-Nov-1999 Theory v3.0 02.0201.5722B 14-Feb-2000 Theory v3.0 02.0201.5727C 29-Oct-2000 Theory v3.1 02.0201.5732

D 03-Mar-2003 Theory v3.2 02.0201.5738

Copyright 2003, AVL

All rights reserved. No part of this publication may be reproduced, transmitted, transcribed, storedin a retrieval system, or translated into any language or computer language in any form or by anymeans, electronic, mechanical, magnetic, optical, chemical, manual or otherwise, without priorwritten consent of AVL.

This document describes how to run the BRICKS software. It does not attempt to discuss all theconcepts of design analysis required to obtain successful solutions. It is the users responsibility todetermine if he/she has sufficient knowledge and understanding of fluid dynamics to apply thissoftware appropriately.

This software and document are distributed solely on an "as is" basis. The entire risk as to theirquality and performance is with the user. Should either the software or this document provedefective, the user assumes the entire cost of all necessary servicing, repair, or correction. AVLand its distributors will not be liable for direct, indirect, incidental or consequential damagesresulting from any defect in the software or this document, even if they have been advised of thepossibility of such damage.


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AST.02.0201.5738 - 03-Mar-2003 i

Table of Contents

1. Introduction ____________________________________________________ 1-1

1.1. Scope _______________________________________________________________________1-1

1.2. User Qualifications ___________________________________________________________1-1

1.3. Symbols _____________________________________________________________________1-1

1.4. Configurations _______________________________________________________________1-2

1.5. Documentation_______________________________________________________________1-2

2. Theoretic Fundamentals________________________________________ 2-1

2.1. Crank Train Load Analysis ____________________________________________________2-1

2.1.1. Kinematic Fundamentals of the Crank Mechanism ___________________________2-1

2.1.1.1. Piston Displacement___________________________________________________2-12.1.1.2. Piston Velocity________________________________________________________2-2

2.1.1.3. Piston Acceleration____________________________________________________2-3

2.1.2. Force Resolution at the Crank Train ________________________________________2-4

2.1.3. Gas Forces _______________________________________________________________2-6

2.1.4. Mass Forces ______________________________________________________________2-9

2.1.5. Mass Torque ___________________________________________________________ 2-11

2.2. Crankshaft Balancing _______________________________________________________ 2-12

2.2.1. Balancing of Mass Forces at One Crank Mechanism ________________________ 2-12

2.2.2. Balancing of Mass Forces at Multi Crank Mechanism _______________________ 2-15

2.2.3. Balancing of Free Couples _______________________________________________ 2-16

2.2.4. Internal Bending Moments_______________________________________________ 2-20

2.2.5. Influence from Rotating Counterweights on Cross Tipping Moment __________ 2-21

2.2.5.1. Alternating Torque Caused from Rotating Counterweights ______________ 2-21

2.2.5.2. Balancing of Gas and Mass Alternating Torque _________________________ 2-21

2.2.6. Table for Different Engine Types _________________________________________ 2-27

2.3. Crank Throw Optimization __________________________________________________ 2-28

2.3.1. Optimum SelectionEfficiency Value Analysis____________________________ 2-28

2.3.1.1. Partial Efficiency Value Determination for Strength ____________________ 2-29

2.3.1.2. Partial Efficiency Value Determination for Torsional Stiffness ___________ 2-30

2.3.1.3. Partial Efficiency Value Determination for Bearing Load ________________ 2-31

2.3.1.4. Partial Efficiency Value Determination for Crank Throw Mass ___________ 2-32

2.3.1.5. Partial Efficiency Value Determination for Crank Throw Unbalance ______ 2-33

2.4. Crank Bearing Calculation __________________________________________________ 2-34

2.4.1. Bearing Shell Calculations _______________________________________________ 2-34

2.4.1.1. Hoop Stress (Tangential Stress) in Shell _______________________________ 2-34

2.4.1.2. Radial Pressure between Shell and Housing ____________________________ 2-342.4.1.3. Bearing Clearances __________________________________________________ 2-34


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AST.02.0201.5738 - 03-Mar-2003 iii

2.5.3.2. Stepped Shaft_______________________________________________________ 2-81

2.5.3.3. Influence of Transition Radius________________________________________ 2-82

2.5.3.4. Moment of Resistance _______________________________________________ 2-82

2.5.4. System Excitation_______________________________________________________ 2-83

1.1.1.1. The Tangential Force at the Crankpin _________________________________ 2-832.5.4.2. Harmonic Analysis of Exciting Torque_________________________________ 2-83

2.5.4.3. Mass Force Harmonic________________________________________________ 2-84

2.5.4.4. Indicated Cylinder Power of the Oscillator _____________________________ 2-84

2.5.5. Determination of Rotation Angles and Torsional Torques ___________________ 2-85

2.5.5.1. Sine Component of Harmonic (j) of the total Exciting Torque ____________ 2-86

2.5.5.2. Extended Holzer Tabulation__________________________________________ 2-87

2.5.5.3. Boundary Conditions ________________________________________________ 2-87

2.5.6. Harmonic Synthesis_____________________________________________________ 2-88

2.5.7. Maxima/Minima-Determination __________________________________________ 2-89

2.5.7.1. The Maximum Alternating Amplitude _________________________________ 2-89

2.5.7.2. The Maximum Stress ________________________________________________ 2-89

2.5.8. Degree of Irregularity ___________________________________________________ 2-90

2.5.9. Maximum Rotational Acceleration of Single Masses_________________________ 2-91

2.5.10. Actual Damper Damping and Heat Load__________________________________ 2-91

2.6. Strength___________________________________________________________________ 2-92

2.6.1. Crankshaft Strength AVL Standard Method______________________________ 2-92

2.6.1.1. Fillet Fatigue Strength_______________________________________________ 2-92

2.6.1.2. Fillet Stresses_______________________________________________________ 2-96

2.6.1.3. Fatigue Strength Data for Typical Crankshaft Steets ___________________ 2-100

2.6.1.4. Fatigue Strength Data of Nodular Cast Iron Crankshafts _______________ 2-101

2.6.1.5. Fatigue Strength Data f0 (N/mm2) for the Test Crankshafts

acc. gf. Paper "Gegossene Kurbelwellen _____________________________________ 2-102

2.6.1.6. Formulas for the Evaluation of Stress Concentration Factorsacc. to MTZ-Publication____________________________________________________ 2-103

2.6.1.7. Torsion ___________________________________________________________ 2-104

2.6.2. UR-M 53 Theory_______________________________________________________ 2-105

2.6.2.1. Principles of Calculation ____________________________________________ 2-105

2.6.2.2. Calculation of Stresses ______________________________________________ 2-105

2.6.2.3. Calculation of Stress Concentration Factors ___________________________ 2-109

2.6.2.4. Additional Bending Stresses _________________________________________ 2-110

2.6.2.5. Calculation of Equivalent Alternating Stress __________________________ 2-111

2.6.3. Piston Pin Analysis ____________________________________________________ 2-112

2.6.3.1. Evaluation of Pin Deflections ________________________________________ 2-112

2.6.3.2. Unit Loads in the Pin Bearings ______________________________________ 2-113

2.6.3.3. Piston Pin Stresses according to SCHLFKE/KUHM __________________ 2-1132.6.3.4. Evaluation of Permissible Deflections ________________________________ 2-114


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BRICKS Version 3.2 Theory

iv AST.02.0201.5738 - 03-Mar-2003

List of Figures

Figure 2-1: Conventional Crank Train ....................................................................................................................2-1

Figure 2-2: Kinematic of Various Crank Trains .....................................................................................................2-3Figure 2-3: Forces and their Components on Conventional Crank Train ...........................................................2-4

Figure 2-4: Gas Pressure Diagrams of Different Engines .....................................................................................2-6

Figure 2-5: Cylinder Pressure Diagrams for Different Speeds of a Diesel Engine .............................................2-7

Figure 2-6: Scatter Range of Gas Pressure Curve on Gasoline and Diesel Engines...........................................2-8

Figure 2-7: Dissection of a Crankthrow into Simple Geometric Parts.................................................................2-9

Figure 2-8: Mass Forces at Crank Mechanism .....................................................................................................2-10

Figure 2-9: Forces and Moments at the Engine ...................................................................................................2-12

Figure 2-10: Mass Forces 1stand 2ndOrder on One Cylinder Crank Train .......................................................2-13

Figure 2-11: Counterweight for the Balancing of Rotating Masses in One Cylinder Crank Train.................2-13

Figure 2-12: Balancing of the Rotating Mass Forces ...........................................................................................2-13

Figure 2-13: Vector Diagram of the Oscillating Mass Forces..............................................................................2-14Figure 2-14: Total Balancing of Mass Forces 1stand 2ndOrder on One Cylinder Crank Train .......................2-14

Figure 2-15: Free Inertia Forces 1stOrder in y and z Directions for Various Counterweight Sizes ...............2-15

Figure 2-16: Crank Throw 1st and 2ndOrder for Inline Engines (four stroke) with 3 - 6 Cylinders...............2-16

Figure 2-17: Arrangement of a Simple Reference System to Determine the Free Couples .............................2-16

Figure 2-18: Free Couples 1stand 2ndOrder of a 3 Cylinder Crank Shaft .........................................................2-18

Figure 2-19: Counterweight Direction for Balancing of Mass Moment 1stOrder 90 to Middle

Throw of Shaft.................................................................................................................................................2-19

Figure 2-20: Schematic Overview for Total Balancing of 1stand 2ndOrder Couples at 3 Cylinder I

nline Engine.....................................................................................................................................................2-19

Figure 2-21: Inner Couples on Three Different Crankshafts for an 8 Cylinder Engine...................................2-20

Figure 2-22: Balancing of mass force 1. order and influencing of alternating torque 1. order by

using one in opposite direction with rotating intermediate shaft at a 1-cylinder engine

(1-cylinder diesel engine mo/Vn=2.4kg/l, =0.292 ) .....................................................................................2-23

Figure 2-23: Rangement of the Balancing Shafts for Balancing of Mass Torque .............................................2-26

Figure 2-24: Balancing of mass force 2. order and influencing of the alternating torque 2. order

using two reverse rotating intermediate shafts at a 4-cylinder inline engine...........................................2-26

Figure 2-25: Arrangement of crankshaft throw, firing order, ratio numbers of free couples, internal

bending moments and alternating torque of four stroke inline engines. .................................................2-27

Figure 2-26: Class Function of Partial Efficiency Values for Strength CFHOPT respectively

CFGOPT = 1.4 ................................................................................................................................................2-29

Figure 2-27: Class Function of Partial Efficiency Value for Torsional Stiffness ..............................................2-30Figure 2-28: Class Function of Partial Efficiency Values for Bearing Load ......................................................2-31

Figure 2-29: Class Function of Partial Efficiency Values for Crank Throw Mass ............................................2-33

Figure 2-30: Designations for Main Bearing.........................................................................................................2-39

Figure 2-31: Designations for Connecting Rod Bearing ......................................................................................2-40

Figure 2-32: Designations for Piston Pin Bearing with Pin Fixed in Piston.....................................................2-41

Figure 2-33: Designations for Piston Pin Bearing with Pin Fixed in Connecting Rod ....................................2-42

Figure 2-34: Equilibrium of Forces at the Journal Increasing Eccentricity......................................................2-49

Figure 2-35: Equilibrium of Forces at the Journal Decreasing Eccentricity.....................................................2-51

Figure 2-36: Crankshaft Torsion 1st kind according to Grammel.......................................................................2-75

Figure 2-37: Crankshaft Torsion 2nd kind according to Grammel......................................................................2-75

Figure 2-38: Crankshaft Bending as a Beam at Torsion 2nd kind.......................................................................2-76Figure 2-39: Dimensions of Crankshaft taken into consideration from Empirical Formulae.........................2-77


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AST.02.0201.5738 - 03-Mar-2003 v

Figure 2-40: Crank shaft Dimensions taken in consideration from B.I.C.E.R.A. Method ...............................2-78

Figure 2-41: Torsion of Shaft with Constant Cross Section ................................................................................2-80

Figure 2-42: Torsion of Stepped Shaft with Circular Cross Section...................................................................2-81

Figure 2-43: Stepped Shaft with Transition Radius ............................................................................................2-82

Figure 2-44: Fatigue Strength Diagram with Constant Stress Relations..........................................................2-92

Figure 2-45: Fatigue Strength Diagram with Constant Mean Stresses.............................................................2-93Figure 2-46: Construction Method of Fatigue Strength Diagram ......................................................................2-94

Figure 2-47: Crankthrow and Cross section A-A for Crankshaft Stress Analysis.............................................2-97

Figure 2-48: Notch Factor as a Function of the Form Factor...........................................................................2-102

Figure 2-49: Piston Pin Loading ..........................................................................................................................2-112


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03-Mar-2003 1-1

1. INTRODUCTIONThis document describes the functions and methods for the BRICKS Version 3.1 programfor the design analysis of the crank train. The BRICKS Theory Manual contains all

necessary information about the program kernel.

1.1. ScopeThe chapters of this manual describe the theory of the BRICKS software. They do notattempt to discuss all concepts of design analysis that are required to obtain successfulsolutions. It is the users responsibility to determine if he/she has sufficient knowledge andunderstanding of design analysis to apply this software appropriately.

1.2. User QualificationsUsers of this manual:

Must be qualified in basic UNIX or Windows NT

Must be qualified in basic Design Analysis

1.3. SymbolsThe following symbols are used throughout this manual. Safety warnings must be strictlyobserved during operation and service of the system or its components.

!Caution: Cautions describe conditions, practices or procedures whichcould result in damage to, or destruction of data if not strictly observed or

remedied.

Note: Notes provide important supplementary information.

Convention Meaning

ItalicsFor emphasis, to introduce a new term or for manual titles.

monospace To indicate a command, a program or a file name,

messages, or input / output on a screen or file contents.

SCREEN-KEYS A SCREENfont is used for the names of windows andkeyboard keys, e.g. to indicate that you should type a

command and press the ENTERkey.

MenuOpt A MenuOptfont is used for the names of menu options,

submenus and screen buttons.


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1-2 03-Mar-2003

1.4. ConfigurationsSoftware configurations described in this manual were in effect on the publication date ofthis manual. It is the users responsibility to verify the configuration of the equipmentbefore applying procedures in this manual.

1.5. DocumentationBRICKSdocumentation is available in PDF format and consists of the following:

Release Notes

User's Guide

Primer

AVL Workspace Installation Guide (Windows NT and UNIX)

AVL Workspace GUI Introduction

FLEXlm User's Guide


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03-Mar-2003 2-1

2. THEORETIC FUNDAMENTALSAll calculations are based on a statically determined system of the crank train, i.e. all partsare rigid and only the crankshaft has joints between each crank throw. The only difference

is the torsional vibration calculation. Here the torsional stiffness of the crankshaft partsare taken into consideration.

The theoretical principles of the following task are described in this manual:

Crank Train Load Analysis

Crankshaft Balancing

Hydrodynamic Bearing Analysis

Torsional Vibration Analysis

Strength calculation

2.1. Crank Train Load Analysis

2.1.1. Kinematic Fundamentals of the Crank Mechanism

The turning point of the piston pin and crankshaft lies on one line with the centerline ofthe cylinder.

Figure 2-1: Conventional Crank Train

2.1.1.1. Piston Displacement

With Figure 2-1 the relation for the piston displacement follows:

s r l r l0+ + = +cos cos [2.1.1]


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2-2 03-Mar-2003

The relation between the angle ,the conrod length land the angle is:

sin sin sin = = r

l[2.1.2]

and

cos sin = 1 2 2 [2.1.3]

With the upper relations the piston displacement becomes:

x s

r= = + 0 2 21

1 11cos sin

[2.1.4]

With the development into a FOURIER-series it follows:

== 6cos364cos162cos4cos 642

100 AAA

AAr

sx [2.1.5]

A03 51

1

4

3

64

5

256= + + + A1 1=

A23 51

4

15

128= + + + .. . A4

3 51

4

3

16=

A659

128= + ... A8

71

39=

In practice usual used approximation formula

s

r

0 14 4

2 +

cos cos [2.1.6]

2.1.1.2. Piston Velocity

For constant crankshaft speed

d

dt

const

= = =

The velocity of the piston follows from the first derivation of piston displacement:

=

= +

x

s

r

sin

sin cos

sin

0

2 21

[2.1.7]

or expressed in a series

= = + + + +x

s

r A

A A A

sin sin sin sin ...0

1

2 4 6

2 2 4 4 6 6 [2.1.8]


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03-Mar-2003 2-3

In practice usual used approximation formula for piston velocity

sin sin

s

r

0

22

+

[2.1.9]

2.1.1.3. Piston Acceleration

The piston acceleration follows from the second derivation of piston displacement:

( )=

= +

+

x

s

r

cos

cos sin sin

sin

0

2

2 2 3 4

2 23

1

[2.1.10]

or expressed in a series

=

= + + + x s

rA A A A

cos cos cos cos0

2 1 2 4 62 4 6

[2.1.11]

In practice usual used approximation formula for the piston acceleration

cos cos

s

r

0

2 2

+

[2.1.12]

In Figure 2-2 displacement, velocity and acceleration for various crank trains are shown

(only the first one is used in BRICKS).

Figure 2-2: Kinematic of Various Crank Trains


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2-4 03-Mar-2003

2.1.2. Force Resolution at the Crank Train

The load of the engine components depends on the transmitted forces. Therefore it isnecessary to determine the forces acting on the engine parts.

The forces acting on the crank train follows from the force resolution with the geometric

relations from Figure 2-3.

Figure 2-3: Forces and their Components on Conventional Crank Train

sin sin sin = = r

l[2.1.13]

cos sin sin = = 1 12 2 2 [2.1.14]

F F FS Z Z= =

1 1

1 2 2cos sin [2.1.15]

or transformed in a FOURIER series

( )F F A A A AS Z= + + + +0 2 4 62 4 6cos cos cos ..

A02 4 61

1

4

9

64

25

256= + + + + .. .

A22 4 61

4

3

16

75

512= . ..

A44 63

64

15

256= + A665

512=


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03-Mar-2003 2-5

Piston Side Force:

F F FN Z Z= =

tan

sin

sin

1 2 2[2.1.16]

or expressed as a FOURIER series

F F B B BN Z= + + +( sin sin sin . ..1 3 53 5 [2.1.17]

B13 53

8

15

64= + + + . . . B3

3 51

8

15

128= .. . B5

53

128= + ...

Radial Force at the Crank:

( )F F FR Z Z=

+=

cos

coscos

sin

sin

2

2 21[2.1.18]


( )...6cos4cos2coscos 64210 +++++= AAAAAFF ZR [2.1.19]

A03 51

2

3

16

15

128= . .. A1 1=

A23 51

2

1

4

45

256= + + + ... A4

3 51

16

9

128= .. .

A653

256= + .. .

Tangential Force at the Crank:

( )F F FT Z Z=

+= +

sin

cos

sin sin cos

sin

1 2 2

[2.1.20]


( )F F B B B BT Z= + + + +1 2 4 62 4 6sin sin sin sin ... [2.1.21]

B1 1= B23 51

2

1

8

15

256= + + + ...

B43 51

16

3

64

= ... B653

256

= + ...


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2-6 03-Mar-2003

2.1.3. Gas Forces

The gas forces in the cylinder are acting on the piston head, the cylinder head and on theside walls of the cylinder. The gas forces are transmitted to the crankshaft through thepiston and the connecting rod.

Figure 2-4 shows the gas pressure diagrams of different engines. On the left side thepressure-time (p-t) diagrams are depicted. The gas pressure in relation to the stroke isshown on the right side. This is the pressure-volume (p-v)diagram.

Figure 2-4: Gas Pressure Diagrams of Different Engines

a) 4-stroke gasoline engine at average speed (cylinder diameter 80mm)

b) 4-stroke diesel engine at maximum torque (cylinder diameter 125mm)c) 4-stroke diesel charged engine at rated speed (cylinder diameter 370mm)


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03-Mar-2003 2-7

Figure 2-5 shows cylinder pressure diagrams for full load and different engine speeds.

Figure 2-5: Cylinder Pressure Diagrams for Different Speeds of a Diesel Engine

The maximum cylinder pressure of gasoline engines lies in the range from 50 to 65 bar andin Diesel engines without turbo charger from 70 to 90 bar. The maximum gas pressure of

turbo charged Diesel engines reaches 130 to 150 bar.

The load of the engine components mainly depends on the maximum combustion pressure.This depends on the compression ratio, compression end pressure, combustion process,mixture and the load of the engine.

Different working principles (Diesel, Otto) require different compression end pressures. Ifthe compression end pressure is higher, the temperature in the cylinder is also higher.Diesel engines need a higher temperature at the end of the compression for self-ignition.Therefore Diesel engines need a higher compression ratio than Otto engines.

= +V VV

h c

c

[2.1.22]

... compression ratio

Vh ... displacement

Vc ... dead volume


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2-8 03-Mar-2003

Regarding the engine component stress, the scatter range of the gas pressure can be ofinterest.

In Figure 2-6 the scatter range of the gas pressure curve of gasoline and diesel engines areshown. The reasons for the scattering are differences in the ignition delay, spaceexpansion of the flame front and differences of the cylinder load through pressureoscillations and different residual gas portions.

Figure 2-6: Scatter Range of Gas Pressure Curve on Gasoline and Diesel Engines


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03-Mar-2003 2-9

2.1.4. Mass Forces

Mass forces are acting because of the acceleration and deceleration of moving engine parts.The mass forces are orientated in the opposite direction of the acceleration.

F m a= [2.1.23]

To determine mass forces, the masses and the center of gravity of the moving parts mustbe known.

Figure 2-7: Dissection of a Crankthrow into Simple Geometric Parts

To determine the mass and the center of gravity of complicated parts, they can be dissectedinto simple geometric parts as it is shown in the figures above.

The moving engine parts carry out the following motions:

1. Rotating motion around the crank shaft axis

Crank throw and the part of the conrod near the crank shaft.

2. Oscillating motion

Piston and the part of the conrod near the piston.

3. Revolution path similar to an ellipse

Middle part of the conrod with the center of gravity:

To calculate the effect of this system an equivalent system should be created with the samestatic and dynamic effects as the real system. The mass concentrated in one point is anecessary part of this equivalent system. From this comes the system of connected pointmasses.

The mass of the conrod is divided into a rotating portion and an oscillating portion.

The following mass forces are distinguished:

1. Rotating mass forces acting on the crank throw in radial direction

2. Oscillating mass force acting in the direction of the cylinder axis


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2-10 03-Mar-2003

Figure 2-8: Mass Forces at Crank Mechanism

Rotating mass force:

F m rr r= 2

[2.1.24]

Oscillating mass force:

( )F m r A Ao o= + + 2

1 2 2cos cos .. . [2.1.25]

A1 1= A2 3 514

15128

= + +

For the oscillating mass forces only the first and the second orders are important becausehigher orders have only low influence, and are therefore not considered in BRICKS. Also,for the calculation of A2, components with higher orders than the first in are omitted.

The following mass force components in z- and y-direction are acting on a one crankmechanism running with constant speed .

Inline Engine

[ ]F r m m A m Az r o o= + + + 2

1 2 2cos cos cos [2.1.26]

F r my r= 2 sin [2.1.27]

V-engine

( )F r m m A m Az r o o= + + + +

+

2 1 21

2

3

22cos cos cos cos cos cos .. .

[2.1.28]

( )F r m m A m Ay r o o= + +

+

2

1 212

3

22sin cos sin cos cos sin .. .

[2.1.29]


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03-Mar-2003 2-11

2.1.5. Mass Torque

From the inertia effect of the accelerated and decelerated piston and conrod movement, aperiodic changing torque follows at the crankshaft.

The mass torque is related to the piston areaAKand to the crank radius rwith regard to

the later use of this relation to be superimposed with the gas torque.

( ) ( )[ ]PA r

r m m x x m u u w wTK

K=

= + + + 2 2 3 [2.1.30]

= + u l

lx

l

l

e

l

1 2 sin cos

= + + u l

lx

l

l

e

l

1 2 cos sin

= + w l

l

e

l

e

l

x2 cos sin

= + w l

l

e

l

e

lx2 sin cos

x is the related piston speed, x is the related piston acceleration.

The following relation is obtained for the mass tangential pressure in the representation asFOURIER series

( ) ( ) ( )[ ]P r m m B k m B kT K k k= + + 2 2 3 3sin sin [2.1.31]

with the coefficients

B13 51

4

1

16

15

512= + + + .. . B

l

lB31

11=

B24 61

2

1

32

1

32= . .. B

l

lB

l

l

l

l32

1

2

21 2=

B33 53

4

9

32

81

512= . .. B

l

lB33

13=

B42 4 61

4

1

8

1

16= ... B

l

lB34

1

2

4=

B53 55

32

75

512= + + . . . B l

lB35

15=

B64 63

32

3

32= + + . . . B

l

lB36

1

2

6=

where:

mK... piston mass

m2 ... conrod mass at piston side

m3 ... conrod mass in the center of gravity of the conrod


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2-12 03-Mar-2003

2.2. Crankshaft BalancingThe following figure shows the fundamental forces and moments at an engine caused fromthe gas and mass forces.

Figure 2-9: Forces and Moments at the Engine

2.2.1. Balancing of Mass Forces at One Crank MechanismThe following mass force components in z- and y-direction are acting on a one crankmechanism running with constant speed .

Inline Engine

[ ]F r m m A m Az r o o= + + + 2

1 2 2cos cos cos [2.2.1]

F r my r= 2 sin [2.2.2]

V-engine

( )F r m m A m Az r o o= + + + +

+

2 1 21

2

3

22cos cos cos cos cos cos

[2.2.3]

( )F r m m A m Ay r o o= + +

+

2 1 21

2

3

22sin cos sin cos cos sin [2.2.4]


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03-Mar-2003 2-13

Figure 2-10: Mass Forces 1stand 2ndOrder

on One Cylinder Crank Train

Figure 2-11: Counterweight for the

Balancing of Rotating Masses in One

Cylinder Crank Train

The rotating masses only cause the centrifugal force F m rr r= 2 which acts in the

radial outside direction (rotating with the crank angle ). The centrifugal force is balanced

through a force in the opposite direction and the same value. To do this counterweights aremounted at the webs with the same force effect as the centrifugal force as schematicallyshown in Figure 2-12. The rotating mass forces can be balanced totally by counterweights.

Figure 2-12: Balancing of the Rotating Mass Forces

The effect caused from the oscillating masses can be shown through a vector diagram. Theoscillating mass force part is replaced through two vectors with the half length. One vectorrotates in the crankshaft direction the other in the opposite direction.


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2-14 03-Mar-2003

Figure 2-13: Vector Diagram of the Oscillating Mass Forces

Therefore there is a vector couple rotating with for the first order and another vectorcouple rotating with 2for the second order. The resultant of the two vectors gives theinstantaneous value of the oscillating mass forceFo.

A total balancing of the oscillating mass forces first and second order can be done bybalancing systems rotating in opposite direction with single and double crankshaft speed.The balancing forces must have the same value as the rotating mass force vectors.Balancing systems for the first and the second order are expensive therefore only seldommade.

Figure 2-14: Total Balancing of Mass Forces 1stand 2ndOrder on One Cylinder CrankTrain


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03-Mar-2003 2-15

A partial balancing of the mass forces first order can be done by putting the mass for theoscillating mass force in the counterweights of the crank web. With this method theoscillating mass force only changes the direction. The decreased part from the cylinderdirection (z) appears in the cross cylinder direction (y). The ratio of the decreased part in z-direction to the original value of the oscillating mass force first order is the "balancing

rate".

For general use a balancing rate of 50% is the best solution.

Figure 2-15: Free Inertia Forces 1stOrder in y and z Directions for VariousCounterweight Sizes

A total balancing of the mass forces first order can be reached by increasing the crankshaftcounterweight of mo 2 and with an additional balancing shaft rotating with crankshaft

speed in opposite direction with the balancing mass of mo 2 .

2.2.2. Balancing of Mass Forces at Multi Crank Mechanism

At multi crank mechanism there is the possibility to choose a construction with a goodbalancing of the mass forces. The mass forces first order are balanced if the crank throwsare shifted constant and if all crank throws have the same mass. Therefore the crankshaftfor the first order is drawn to see if the mass forces first order are balanced. To get the

crankshaft for the second order the angle between crank throw and cylinder axis has to bedoubled. In Figure 2-16 all crankshafts for the first order are balanced. The mass forces forthe second order are balanced for the 3, 5 and 6 cylinder crankshafts only. The crankthrows second order for the 4 cylinder crankshaft are all pointed in the same direction.This results in big mass forces for 4 cylinder inline engines. To balance these mass forces abalancing shaft rotating with double crankshaft speed would be necessary.


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2-16 03-Mar-2003

Figure 2-16: Crank Throw 1st and 2ndOrder for Inline Engines (four stroke) with 3 - 6Cylinders

2.2.3. Balancing of Free Couples

Engines with multi throw crankshafts with total balanced mass forces can have freecouples because the crankshaft mass forces are distanced each other at the cylinderdistance. Free mass couples occur if all crank throw mass forces result in a force couple atthe crankshaft.

They are called "outer moments" or "longitudinal tipping moments". The value of the freecouples is calculated as the sum of all products from mass force multiplied with thedistance to a reference point.

Figure 2-17: Arrangement of a Simple Reference System to Determine the FreeCouples

The sum of all moments around the reference point must be zero: M= 0


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03-Mar-2003 2-17

From this relation for the mass, moment 1storder can be written:

( ) ( ) ( )[ ]F a M m m r a k Mz k k y r o kk

n

y

k

n

1 1

2

1

1

1

0, cos + = + + + ===

[2.2.5]

( ) ( ) ( )[ ]F a M m m r a k My k k z r o kk

n

z

k

n

1 1

2

1

1

1

0, sin = + + ===

[2.2.6]

and for the 2ndorder:

( ) ( )[ ]F a M m r A a k Mz k k y o kk

n

y

k

n

2 2

2

2

1

2

1

2 0, cos + = + + ===

[2.2.7]

( ) ( )[ ]F a M m r A a k My k k z o kk

n

z

k

n

2 2

2

2

1

2

12 0, sin = + =

== [2.2.8]

or for the free couples 1stand 2ndorder with =0

( )M

M

m m r aky

y

r o

k

k

n

11

21

=+

= =

cos [2.2.9]

( )M

M

m m r akz z

r o

k

k

n

11

21

=+

= =

sin [2.2.10]

MM

m r A aky

y

o

k

k

n

22

2

2 1

2=

= =

cos [2.2.11]

M M

m r A akz z

o

k

k

n

22

2

2 1

2=

= =

sin [2.2.12]

From the vector calculation the free couples can be determined

M M My z1 12

1

2= + [2.2.13]

M M My z2 22

2

2= + [2.2.14]


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2-18 03-Mar-2003

Figure 2-18: Free Couples 1stand 2ndOrder of a 3 Cylinder Crank Shaft

The determination of the free couples 1stand 2ndorder is shown in Figure 2-18. Thefollowing values are obtained

( )

MM

m m r ay

y

r o

11

2 1 0 866 2 0 3 0 866 1 732=

+ = + + =

. . .

( )M

m m r az

z

r o

11

2 0 5 1 1 2 0 5 3 0=

+ = + =

. .

( )M

m m r aM M

r o

y z11

2 1

212

1732=+

= + =

.

MM

m r A ay

y

o

22

2

2

1 0 5 2 1 3 0 5 0=

= + =

. .

Mm r A a

zz

o

22

2

2

1 0 866 2 0 3 0 866 1 732=

= + + =

. . .

( )M

m m r aM M

r o

y z22

2 22

22

1732=+

= + =

.

The moment 1storder rotates with the crank shaft. The moment vector points in theopposite direction of crank throw number 2. The effect of the moment is like a force couple

rotating with the crankshaft.


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03-Mar-2003 2-19

The counterweights to eliminate the free moment have to be mounted in the same forcecouple plane as the acting force couple and in the opposite direction. An appropriatelocation of the counterweights is shown in Figure 2-19.

Figure 2-19: Counterweight Direction for Balancing of Mass Moment 1stOrder 90 toMiddle Throw of Shaft

The total balancing of the mass moment 1storder requires tuned counterweights at thecrank shaft for the balancing of the rotating and the half oscillating mass moment and arotating intermediate shaft balancing the half oscillating mass moment as it is shown inFigure 2-20.

Two additional balancing shafts are necessary for balancing of the mass moment 2 ndorderrotating with 2.

Counterweights on the crankshaft

( )m m mG r o= +1732

40 5

..

1. Order balancing shaft

m m a

bA r o1 1 732 0 5= . .

2. Order balancing shaft

m m A a

cA r o2 2

1732

8=

.

Figure 2-20: Schematic Overview for Total Balancing of 1stand 2ndOrder Couples at 3Cylinder Inline Engine

In practice the balancing of mass moment 1. and 2. order is hardly done.

The main influence on the mass moment has the longitudinal symmetry of the crankshaft.If the crankshaft is symmetric no free couples 1. order will occur.


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03-Mar-2003 2-21

2.2.5. Influence from Rotating Counterweights on CrossTipping Moment

In the opposite direction rotating counter weights can be used at engines to balance theoscillating mass forces. This rotating counter weights cause a periodic force in the opposite

direction to the mass forces of the oscillating engine parts. With this a total balance can bereached. If the counterweights are shifted in their height to each other and to thecrankshaft or if they are unsymmetrical to the cylinder axis, they additionally generate aperiodically changing moment on the crankcase. This moment is superimposed with thetipping moment caused from the gas and mass tangential pressure.

Therefore the additional rotating counterweights can be used to reduce the tippingmoments.

2.2.5.1. Alternating Torque Caused from Rotating Counterweights

To prevent an additional mass moment around the vertical (z) axis the resulting effectdirection of the counter weights has to coincide with the resulting cylinder forces.

2.2.5.2. Balancing of Gas and Mass Alternating Torque

The tangential gas forces of order numberqcause following periodic torque:

( )M V a q b q k kGq h q q z v= + 1

2cos sin [2.2.15]

The tangential mass forces of order numberqcause following torque:

M m r B q k kMq o q z v= 2 2 sin [2.2.16]

with B Bq = 1 and B Bq= 2 for 1. and 2. order

B13 51

4

1

16

15

512= + + B2

4 61

2

1

32

1

32=

If the three alternating torquesthe gas pressure alternating torqueMGq

the mass momentMMq

and the balancing torqueMAq

of the order numberqare summed up, the total alternating torqueMwqof the ordernumberqis obtained.

If the total alternating torque is related to the piston area and the force is acting on the

arm of the crank radius r ( )A r VK h = 1 2 there follows the dimension of a tangential

pressure. The angular velocity is replaced by the mean piston speed:


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2-22 03-Mar-2003

4 2 2 2 2 = r vm

PM

V

A q B qTwq

h

q q=

= + 1

2

cos sin [2.2.17]

( ) ( )A k k a m

Vr q y yq z v q

A

h

I II= + +2 2

[2.2.18]

( ) ( )B k k b m

VV

k k

qB

m

Vr q z z q z v q

o

h

mz v

qA

h

I II=

+

2

2

2

2

22 [2.2.19]

C A Bq q q= +2 2

2.2.5.2.1. One Cylinder Engine

At one cylinder engines the mass forces 1. order are often balanced by using a 50%balancing rating at the crankshaft with an additional balancing shaft rotating withcrankshaft speed.

The alternating torque is obtained as the sum of following moments 1. order:

Gas torque

( )M V a bG h1 1 11

2= + cos sin [2.2.20]

Mass torque:

M m r BM o12 2

1= sin [2.2.21]

B13 51

4

1

16

15

512= + +

Moment of the reverse rotating balancing mass

( ) ( )[ ]M r m z yA o1 2 0 5 180 180= . sin cos [ ]= + r m d d o

2 0 5. cos sin sin cos [2.2.22]

Finally for the total alternating moment 1. order related to A r VK h = 1 2 can be

written:

P

V

A BTw

h

=

= + 1

2

1 1cos sin [2.2.23]


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03-Mar-2003 2-23

A a m

VV

d

r

o

h

m1 1

20 52

= + . sin

B b m

VV B

d

r

o

h

m1 1

22

12

0 5=

. cos

Figure 2-22 shows the alternating torque 1. order in relation to the mean piston speed for a1 cylinder engine full load. For the distance between the middle of the crankshaft andcounter weight axis two different distance relations are chosen.

Figure 2-22: Balancing of mass force 1. order and influencing of alternating torque 1.

order by using one in opposite direction with rotating intermediate shaft at a 1-cylinder engine (1-cylinder diesel engine mo/Vn=2.4kg/l, =0.292 )


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2-24 03-Mar-2003

The figure shows:

1. To achieve a reduction of the alternating torque 1. order the axis of the counterweight shaft has to be near the crankshaft axis.

2. The angle between the connecting line of counter weight shaft axis and

crankshaft center and cylinder axis should be between 0 and 30 in the direction ofthe crankshaft.

The small distance to the crankshaft axis makes the use of a continuous counterweightshaft impossible. If one counterweight is transferred to the end of the crankshaft a gooddistance and angle can be realized. But a mass moment is generated because of thedistance between counterweight and cylinder axis. This can be avoided by using twocounterweights at the two ends of the crankshaft. But in series this is not done because ofthe high costs of the double gear drive and the double bearings.

2.2.5.2.2. Four Cylinder Inline Engine

At four cylinder inline engines there are no mass forces and no mass moments 1. order.But there are mass forces 2. order. Also the second order is dominating at the alternatingtorque generated from the tangential gas and mass forces.

The mass forces 2. order of a 4 cylinder engine are:

F F m r Az z o2 22

22 4 2=

= cos cos [2.2.24]

with A23 51

4

15

128= + +

The balancing of forces is given with two reverse rotating intermediate shafts with theforce effect of

( ) ( ) ( )[ ]F m rA z A22

2 180 2 180 2= + + cos cos

( )= m rA 2 2 22

cos [2.2.25]

( ) ( ) ( )[ ]F m rA y A22

2 180 2 180 2= + + sin sin

( ) [ ]= + =m rA 2 2 2 02

sin sin [2.2.26]

Out of equation 2.1 and 2.2 follows the value necessary for the balancing reduced on thecrank radius, to:

m m AA o= 1

2 2 [2.2.27]

With these two counterweight shafts rotating reverse to the crankshaft with double speedeven the alternating torques 2. order can be reduced.

From the tangential gas forces following torque is obtained

( )M V a bG h2 2 24 12

2 2= + cos sin [2.2.28]


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2-26 03-Mar-2003

Figure 2-23: Rangement of the Balancing Shafts for Balancing of Mass Torque

The layout of the torque balancing for a total elimination of the periodic mass torque onlyis not optimal as it is shown in Figure 2-24. The distance of the balancing shafts is shorter,also the gas force components of the alternating torque are reduced. So the alternatingtorque balance can be optimized by choosing the distance between the balancing shafts.

Figure 2-24: Balancing of mass force 2. order and influencing of the alternating torque2. order using two reverse rotating intermediate shafts at a 4-cylinder inline engine.


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03-Mar-2003 2-27

2.2.6. Table for Different Engine Types

Figure 2-25: Arrangement of crankshaft throw, firing order, ratio numbers of freecouples, internal bending moments and alternating torque of four stroke inline

engines.


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2-28 03-Mar-2003

2.3. Crank Throw Optimization

2.3.1. Optimum Selection Efficiency Value Analysis

The total efficiency value NUTZ shall be used in the evaluation of the quality of the shapeof the crank throw in a simple way. It combines the information of all evaluation criteriahaving influence on it. By it the selection problem of higher order is reduced to a lineardecision characterized through a single factor.

The total efficiency value determination is made by using a known scheme depicted in thefollowing table:

Table 2-1: Evaluation table for Total Efficiency Determination

Criterion PartialEfficiency Value

Valence Valenced PartialEfficiency Values *

Strength of the crank pin fillet TNFH BEWFH

Strength of the main journal fillet TNFG BEWFG

Torsional stiffness TNST BEWST

bearing load conrod bearing TNLH BEWLH

bearing load at main bearing TNLG BEWLG

Mass of crank throw TNMAS BEWMAS

Unbalance of throw TNUNW BEWUNW

total efficiency value 100 NUTZ

* valenced partial efficiency value = partial efficiency value x valence

Valence of the Single Evaluation Criteria:

The valence of a certain partial efficiency in relation to the other partial efficiencies is doneby valence factors. These depend on several criteria, for example the size of the engine,operating purpose, design, etc.

The standard valences are stated in the table below, with which the optimization programis working, if no other valences are entered.

Table 2-2: List of Standard Valences

Valence Small Size Engines Large Size Engines

BEWFH 20 18

BEWFG 20 18

BEWST 10 20

BEWLH 15 17

BEWLG 15 17

BEWMAS 20 10


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03-Mar-2003 2-29

2.3.1.1. Partial Efficiency Value Determination for Strength

Two partial efficiency values are used. The partial efficiency value TNFH serves theevaluation of the strength in the crank pin fillet radii and is a function of the belongingevaluation values CFH from the strength calculation. The second partial efficiency value

TNFG is valid for the strength valuation in the main journal fillet radii and is derived fromthe evaluation value CFG.

CFH(CFG)=

Strength

-----------------------------

occurring stress

1 2 3 4 5

CFH, CFG

Figure 2-26: Class Function of Partial Efficiency Values for Strength CFHOPT

respectively CFGOPT = 1.4

Class function for strength(in the example of TNFH)

from CFH=1 to CFH=3/2CFHOPT-1/2 :

( )( )

TNFH CFHOPT CFH

CFHOPT=

1

1

2

2

from CFH=3/2CFHOPT-1/2 to CFH=5CFHOPT-4

( )

( )( )TNFH

CFHOPT

CFH CFHOPT=

+

49 1

20 9 101

/

from CFH=5CFHOPT-4

TNFH CFHOPT CFH = 1 3 0, ,2


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03-Mar-2003 2-31

Class function for torsional stiffness:

from CT=0 to CT=CTOPT :

( )TNST

CT

CTOPT

= 1 12

2.3.1.3. Partial Efficiency Value Determination for Bearing Load

Two partial efficiency values are used. The partial efficiency value TNLH serves theevaluation of the connecting rod bearing load and is a function of the belonging evaluationvalues CLAH from the calculation of the nominal bearing load. The second partialefficiency value TNLG is valid for the evaluation of the main load and is derived from theevaluation value CLAG.

CLAH(CLAG)=

allowed nominal bearing press allowed deviation

---------------------------------------------------------- occurring nominal bearing press

CLAHOP(CLAGOP) = allowed deviation

TNLH, TNLG

-0,8

-0,6

-0,4

-0,2

0

0,2

0,4

0,6

0,8

1

0 0,5 1 1,5 2 2,5 3 3,5 4

CLAH, CLAG

TNLH,TNLG

Figure 2-28: Class Function of Partial Efficiency Values for Bearing Load


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2-32 03-Mar-2003

Class function for bearing load (in the example of TNLH)

from CLAH=1 to CLAH=3/2CLAHOP-1/2 :

( )

( )TNLH

CLAHOP CLAH

CLAHOP

=

1

1

2

2

from CLAH=3/2CLAHOP-1/2 to CLAH=5CLAHOP-4

( )

( )( )TNLH

CLAHOP

CLAH CLAHOP=

+

49 1

20 9 101

/

from CLAH=5CLAHOP-4

TNLH CLAHOP CLAH= 1 3 0 2, ,

2.3.1.4. Partial Efficiency Value Determination for Crank ThrowMass

A partial efficiency value for the evaluation of the throw mass is required.

A crank throw shape, which has the smallest mass and fulfills all other criteria ispreferred.

The partial efficiency value TNMAS is deviated from the respective throw mass for eachvariant.

MASOPT ... smallest throw mass under all calculated variants


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03-Mar-2003 2-33

TNMAS

-1,5

-1

-0,5

0

0,5

1

0 0,5 1 1,5 2 2,5

MASS/MASOPT

TNMAS

Figure 2-29: Class Function of Partial Efficiency Values for Crank Throw Mass

Class function for crank throw mass:

from MASS=MASOPT:

TNMAS MASS

MASOPT

MASS

MASOPT=

2

2.3.1.5. Partial Efficiency Value Determination for Crank ThrowUnbalance

Analogue to 2.3.1.4partial efficiency determination for crank throw mass.


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2-34 03-Mar-2003

2.4. Crank Bearing Calculation

2.4.1. Bearing Shell Calculations

Formulas for the evaluation of bearing shell data are as follows:

2.4.1.1. Hoop Stress (Tangential Stress) in Shell

( )

=

+2 w 1 w D

1

B Beff eff B H

[2.4.1]

2.4.1.2. Radial Pressure between Shell and Housing

Pr

D

1

B BB H

=

+

[2.4.2]

2.4.1.3. Bearing Clearances

2maxmaxminmin 2 DDwDC s += [2.4.3]

C D w D Dsmax max min min= +2 2 [2.4.4]

The elasticity characteristicsBBandBHas well as the increase of the saddle bore diameter

Ddue to the pressfit of the bearing shell which are used in the above formulas are furtherobtained from the following relations:

( ) ( )( )

( )B

D

E w w D DB

B B weff

B eff eff

= + +

1 1 1 2

4 1

2

[2.4.5]

( ) ( ) ( )

( )B

D D

E D DH

H H

H

= + +

1 1

1

0

2

0

2

[2.4.6]

D B

B B

H

B H

=+

[2.4.7]

where:

... interference between free O.D. of shell and saddle bore

D ... diameter of saddle bore

D0 ... estimated outer diameter of the bearing abutments


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03-Mar-2003 2-35

Average values forD0/Dare:

D0/D= 1.5 for conrod bearings

D0/D= 1.5 - 2 for main bearings in cast iron housing

D0/D= 2 for main bearings in aluminum housing

EB, EH ... modulus of elasticity of the bearing shell and housing respectively

B, H ... poisson's ratio of bearing shell and housing respectively

w ... total wall thickness of the shell

weff. ... effective wall thickness of the shell (= steel thickness + 1/2 lining thickness)

b ... total width of the shell

Ds ... shaft diameter

With the standard drawing specifications for bearing shells which in general specify a

protrusion of the bearing shell half over the centerline of an inspection block the free outerdiameter of the bearing shells is further obtained from the following formulas:

( )D p v DF Imin min= + +2 [2.4.8]

( )D p v DF Imax max= + +2 [2.4.9]

where:

pmin, pmax ... minimum and maximum protrusion of bearing shell half over thecenterline of the inspection block

DI ... diameter of the inspection block

v F D

w b

I

eff

=

6 10 6 0 .... amount by which the protrusion is reduced by the test loadFo(N).

With these free outer diameters of the bearing shell, the minimum and maximuminterference between the bearing shell and the saddle bore is further obtained from thefollowing formulas:

min= DFmin- Dmax [2.4.10]

max= DFmax- Dmin [2.4.11]

where:

DminandDmaxare the minimum and maximum saddle bore diameters

The relevant interferences for the evaluation of D1and D2in the bearing clearancecalculations are further obtained from the following consideration:


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2-36 03-Mar-2003

The minimum clearance is determined by the minimum saddle bore diameterDminand bythe smallest possible bore increase D1i.e. by the smallest interference which is possiblewith the smallest bore diameter

1= DFmin- Dmin. [2.4.12]

The maximum clearance is determined by the maximum saddle bore diameter and by themaximum possible bore increase D2i.e. by the biggest interference which is possible withthe maximum bore diameter

2= DFmax- Dmax. [2.4.13]

2.4.2. Abbreviations and Literal Notations

ai coefficients

B displacement angle (B= s- s)BR width of the bearing shell

BR/D width ratio: width/diameter

Ci coefficients

d0 diameter of the oil supply drilling

d diameter of the journal pin

D diameter of the bearing shell

e, E eccentricity of the journal's midpoint

EV one and a half step (Runge-Kutta calculations)

F force vector

FD supporting power due to rotation

FN frictional force

h height of the oil gap

i counter of supporting points in circumferential direction

Ki coefficients

L length of conrod

Mi constants

ni number of supporting points in circumferential direction

p pressure

pz supply pressure

P value of the load vector

P value of the resulting supporting power (P)

PR frictional power

PH for intermediate storage ofP(Runge-Kutta)

qB, qT diminution factor (factor of reduction)


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03-Mar-2003 2-37

Q oil flow

Q related oil flow

r radius of the journal

R radius of the bearing shell

RK radius of the crank

Rki tangens of Runge-Kutta calculation

So Sommerfeld-figure (dimensionless supporting power)

t time

v velocity vector

u, v, w components of the velocity in circumferential, radial and axial direction

u1 circumferential velocity of the bearing shell

U2 circumferential velocity of the journal pinUMRZ = /180 (to convert radiant to degree)

x, y, z coordinates in circumferential, radial and axial direction

Z related coordinate in axial direction z B z= 2

Z0 width of the oil well

crank angle

step width (Runge-Kutta)

displacement angle

, s direction of load vector with regard to the shell

max angle between direction of the load and location of the pressure maximum

, s angle of the smallest oil gap in the spatial coordinate system

angular velocity of the smallest gap (also: , d/dt)

related angular velocity of the smallest gap (= d/d)

relative eccentricity ( )= e R r

radial velocity of the journals midpoint (also: )

related relative displacement velocity (= d d ) dynamic viscosity

W width ratio: with/diameterBR/D

ratio: crank radius/length of conrodR/L

D, V dimensionless pressure figures

coefficient of friction

proportionality factor of oil flow

circle constant (3.141592654)

dimensionless pressure

density


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2-38 03-Mar-2003

math. notation for summation

angle starting at the widest gap

Sommerfeld transformation ( ) =tg 2

relative bearing clearance ( )= D d D

K angular velocity of the shaft ( )= = d dt n 30

S angular velocity of the bearing shell

Z angular velocity of the shaft

hydrodyn. effective angular velocity (also: )

v angular velocity ratio

vH for intermediate storage of v(Runge-Kutta)

Nabla-operator

Indices

D rotation

F frictional

G groove in circumferential direction

K crank shaft

m medium

P load, power

r relative (moving coordinate system)

S shell

S due to supply

ST conrod (on figures)

W oil well

V displacement

Z journal

Indices not listed above are explained when used.

Exponents

angle measured in degrees

angle measured in radians


51/127


03-Mar-2003 2-39

Figure 2-30: Designations for Main Bearing


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2-40 03-Mar-2003

Figure 2-31: Designations for Connecting Rod Bearing


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2-42 03-Mar-2003

Figure 2-33: Designations for Piston Pin Bearing with Pin Fixed in Connecting Rod


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03-Mar-2003 2-43

2.4.3. Hydrodynamic Oilfilm Theory

The hydrodynamic calculations in the existing program can be done following Holland, Eberhard and Lang or Butenschoen . Only the fundamentals anddifferences of these theories are discussed. Refer to the Literature for more detailed

information.

2.4.3.1. Reynold's Differential Equation

Taking the equation of Navier-Stokes,

= + dv

dtF gradp v

[2.4.14]

assuming that the mass forces are negligible compared to the frictional forces and the samealso applies to external forces (gravitation), i.e.

=dv

dt

0 ;

F=0 [2.4.15]

the equation of Navier-Stokes will take the following form:

gradp v=

[2.4.16]

The geometric relations within the slider bearing (the height of the gap is very lowcompared to the radii of curvature of journals and bearings) results that the bearing shellcan be developed in one plane. Furthermore, the change of the pressure in the direction ofthe gap's height is without significance compared to the change of the pressure in thedirections of the width and of the periphery. The second derivations of the speedcomponents in the directions of the gap's height are larger than in the other two directionsby powers of ten. Thus, the latter can be neglected, i.e.

p

x

u

y=

2

2[2.4.17]

p

z

w

y=

2

2[2.4.18]

Taking into consideration the no-slip condition, the equations of the motions can be foundfrom:


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( ) ( )u p

xy yh U U

y

hU= + +

1

2

2

2 1 1

[2.4.19]

( )w p

z

y yh= 1

2

2

[2.4.20]

By means of the continuity equation

divv u

x

v

y

w

z

= + + =

0 [2.4.21]

and the definitions of the hydrodynamically effective angular velocity

= + s z d dt2 [2.4.22]

the equations of the motions can be joined to form the well-known Differential Equation ofReynold's

( ) ( )

1 1

32

3+

+

+

=cos cos

D

BR z z

( ) ( ) ( )=

6

2 2

sin sin cos

d

dt

d

dt[2.4.23]

This is the basis of all these theories. If the positions and the motions of journal and shellare known, the pressure distribution in the oil gap can be found. Conversely, the motion ofthe journal can also be calculated. This calculation is based on the equilibrium condition ofthe oilfilm's supporting capacity and the external force, if a non-steady load is given, i.e. ifthe force curve is variable concerning the dimension and the direction.

2.4.4. Solution of the Differential Equation of Reynolds (RDEQ)

To solve this differential equation, the journals motion is divided into

1. Plain rotary motion with = + s z d dt2

2. Plain displacement with d dt

These two cases are then solved separately. The use of the hydrodynamically effectiveangular velocity * (Fraenkel ) reduces the non-steady load to a steady load with the

journal velocity * . The conclusion drawn from this is that every relative velocitydirected towards the gap will cause an increase in pressure. And a mere rotation of the gapwill generate a supporting capacity that is twice as large as in the case of mere journalrotation. Thus, the supporting capacity in constantly loaded bearing will break down if thegap rotates in the same direction with half the angular velocity of the shaft (in the case of

bearings without displacement, d /dt = 0).


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03-Mar-2003 2-45

By introducing dimensionless pressure values

D

Dp=

2

for rotation

V

Vp

d dt=

2 for displacement

the pressure distribution can be given for every combination between viscosity , angularvelocity * and relative bearing clearance , without having to solve the differential

equation anew each time.

2.4.4.1. Boundary Conditions

Unique solutions for the differential equations for rotation and displacement can be foundby boundary conditions only. These boundary conditions are different for the three ways ofsolving the equations.

The first boundary condition requests the pressure at the bearings edges to be 0:

( )p z BR, = =2 0

The second boundary condition for the rotation requests the pressure in the widest gap tobe 0:

( )p zD = =0 0,

At the end of the pressure distribution of the rotation (after the narrowest gap) thepressure must become 0 there, where the pressure-gradient in circumferential directionbecomes 0. This will take place on a bent line = 0(z)

( )( )p z zD = =0 0, with ( ) pD = =0

0

As this boundary condition, formulated by Reynolds, is difficult to handle mathematically,it was not considered by Holland and Lang. By means of measurements (e.g. by Carl) ofsteadily loaded bearings, it has been shown that a good correlation can be reached whenusing Reynolds boundary condition. The result of the measurements was that the pressuredistribution in the direction of the periphery ended only after the narrowest gap.

Thus Butenschoen chose the boundary conditions such that they met the physicaldemands: the maintenance of continuity and the incapability of the oilfilm to absorbtensions. It is easier to apply the boundary conditions named after Sommerfeld .These result from the periodicity of the pressure distribution. But they result in very highunderpressures which have been equalized to 0 by Gmbel . This fulfills the conditionthat no tensions are absorbed. The continuity, however, is not maintained.


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The pressure distribution in the axial direction is assumed to be parabolical. Hollandintroduced a factor of reduction for the supporting capacity (as in the case of a limitedwidth of the bearing, the pressure in the middle is less than in bearings with an unlimitedwidth), which results from the continuity condition.

2.4.5. Non-isothermal Consideration of the Slider Bearing

All reflections on the slider bearing considered so far apply to constant temperature andviscosity. Due to friction, there are spatial temperature distributions which result in avariable viscosity.

As shown by Motosh, the influence on the pressure distribution is low when it concerns theoil inlet temperature for slider bearings in combustion engines (at 80C), which is due tothe variable viscosity. Apart from the viscositys insignificant dependency at hightemperatures and the oppositely directed, and thus compensating, influence of the oilviscositys pressure dependency, the dependency is further reduced nowadays by using

multigrade oils. Thus, non-isothermal calculations are not necessary in the case of engineslider bearings. It is advisable to put in the viscosity for a suitable mean value of inlet andoutlet temperatures.

2.4.6. Calculation of the Journal Displacement

This chapter deals with the deduction of the differential equation system of the journaldisplacement and with its numerical evaluation in the AVL-Program.

2.4.6.1. The Steadily Loaded Bearing

In the steadily loaded cylindrical slider bearing, the load balances the resultant of apressure distribution in the oilfilm which is due to the fact that - by its rotation - thejournal pulls with it the viscous oil. In the state of equilibrium, there is no radial motion ofthe journal and the narrowest oilfilm gap is always found at the same place.

In this case, the supporting powerPDand - indirectly following from it - the eccentricityand the minimum oilfilm thickness only depend on:

1. the bearing's geometry:

diameter of the shell D

width BR

relative clearance of the bearings

= (D-d)/D

(d= diameter of the journal)

2. the dynamic oil viscosity

3. the absolute value of the journal's angular velocity


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03-Mar-2003 2-47

According to Sommerfeld, this dependency can be represented by the function:

P SoBR D

D D=

2[2.4.24]

the "Sommerfeld figure of the rotational supporting power SoD" being calculated accordingto the laws of hydromechanics based on Reynold's theory (, ).

2.4.6.2. The Non-steadily Loaded Bearing

In the non-steadily loaded cylindrical slider bearing, the value and direction of the loadchange according to any functions of time, and according to that, the state of equilibriumwill change, too.

This causes the following consequences:

1. There will be radial motions of the journal. These will cause a squeezing of theoilfilm (squeeze-effect), which in turn will result in a changing of the pressuresin the oilfilm, the resultant of which will act on the journal as an additionalsupporting power due to displacementPV.

If we assume that - in a bearing with a rotating journal - this displacementsupporting power is generated by the same physical procedure as in a bearingwith a static journal (an assumption that is not true, but is admitted accordingto Holland, Lang and Butenschoen),PVwill depend exclusively on thebearing's geometry and on the oil viscosity, and in addition to this, on thevelocity of the radial motion

=

d

dt

E

D d

2

E=E(t) being the journals eccentricity

And, once more, the functional correlation can be expressed as a factor ofproportionality by means of a Sommerfeld figure:

P So BR D

V V=

2

PVconstantly changes its value and its direction, which is due to the changingof the value and the direction of . It will even have a reducing effect on thetotal supporting power if the journal moves away from the shell, i.e. if

becomes negative.

2. The total supporting powerPwhich is to counter-balance the load is theresultant fromPDandPV.P's temporal change, however, is not compensatedexclusively byPV, asPDandPVlie in the same direction only seldomly formoments. Thus,PD, also has to temporarily change its value and direction and,in addition to this, the geometry of the flow-area will also change as a result ofthe eccentricitys change.

Both the change of the load component, which is compensated byPD, and thechange of the eccentricity cause a temporal change of the position of the


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2-48 03-Mar-2003

minimum oilfilm gap. i.e., the azimuth angle of the minimum oilfilm gap will

also change its value with the angular velocity .

The latter is of particular importance, as this angular velocity of the minimumoilfilm gap's motion on its part has a direct influence on the creation of the

rotational supporting power.If you look first at a steadily loaded slider bearing, the journal and shell rotatewith angular velocities Zand S, the substantial flow velocity in theminimum oilfilm gap will be the one of the medium oilfilm streamline

uD D D

mz s z s= +

= +

=2

2

with z s+ =

; [2.4.25]

and if s= 0;

uD D

mz= =

with z=

;

Otherwise, if the minimum oilfilm gap moves with an angular velocity . , then

uD D D D

mz s z s z s= +

= +

=

+ =

2

2

2

2

2

; [2.4.26]

with the "hydrodynamically effective velocity"

= + z s 2

In the case of a non-steadily loaded radial slider bearing, * is used instead ofZin the formula for the steadily loaded bearing with static shell.

The following is obtained:

P SoBR D

D D=

2 with = + z s 2 [2.4.27]

In addition, as we derivated beyond, exists:

P So BR D

V V=

2[2.4.28]

In these formulae the Sommerfeld figures themselves are functions of the relativeeccentricity and the width ratioBR/D.


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03-Mar-2003 2-49

To calculate them, authors have used different approximations, as a precise calculationfrom Reynold's differential equation is not possible. The formulae alternatively used in thissoftware originate from Lang , Holland and Butenschoen .

We will list these formulae at the end of this survey and state that we have corrected someobvious publishing errors in the formulae of Lang and Holland.

In concern to the explicit deductions of the formulae, the reader is referred to literature asthese theoretical fundamentals bear no direct reference to our program. In our program,only the final formulae which are relevant for the user are given.

2.4.7. Differential Equations for the Journal Displacement

This chapter deals with the set up of the differential equations for the journaldisplacement (eccentricity ratio).

This is done by generating the equilibrium of forces at the bearing journal:

2.4.7.1. Equilibrium of Forces at Increasing Eccentricity

Equilibrium of forces at the journal with positive rotation sense of * and increasingeccentricity.

Figure 2-34: Equilibrium of Forces at the Journal Increasing Eccentricity


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2-50 03-Mar-2003

With the angle designation according to Figure 2-34 the following applies:

Projection on the x-axis:

( ) ( )P B PD = cos cos90 90

P B PD = sin sin [2.4.29]

Projection on the y-axis:

( ) ( )P B P PD V = +sin sin90 90

P B P PD V = +cos cos [2.4.30]

Equation 2.4.30: P P B P D V = cos costherefore, with equation 2.4.29:

tan sin

cos=

P B

P B PV

P B P P BV = cos tan tan sin

( )P P B BV

= cos sin tan

P P BD = sin sin

As with 0 always S S , here always B0 . Further, always B90 . Therefore,also sinB0 .

The absolute value of the angle betweenPDand the eccentricity vector is always 90 andbmin=B. Therefore, b always is within the 1

stquadrant. For that reason always sin 0

and tan 0 , so that the following will apply, too:

( )P P B BV

= cos sin tan

P P BD = sin sin

Note: sinB could have been written instead of sinB.

Note: We could have written instead of sinBat this place, but didnot so deliberately.

As sinB or tanB can be computed as functions of the ratioBR/D(width/diameter) and

of the relative eccentricitye, finally:


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03-Mar-2003 2-51

( )P P B

B

BR DV=

cos

sin

tan , [2.4.31]

( )P P B

BR DD =

sin

sin , [2.4.32]

The equalization of equation 2.4.27 and equation 2.4.31 results in:

( ) cos

sin

tan ,

=

P

So BR DB

B

BR DV

2

[2.4.33]

The equalization of equation 2.4.28 and equation 2.4.32 results in:

( )

=

P

So BR DB

BR DD

2

sinsin ,

[2.4.34]

2.4.7.1.1. Equilibrium of Forces at Decreasing Eccentricity

Equilibrium of forces at the journal with positive rotation sense of and decreasingeccentricity.

Figure 2-35: Equilibrium of Forces at the Journal Decreasing Eccentricity


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2-52 03-Mar-2003

With the angle designation according to Figure 2-35 the following applies:

Projection on the x-axis:

( ) ( )P B PD = cos cos90 180 90

= P B PDsin sin [2.4.35]

Projection on the y-axis:

( ) ( )P B P PD V = +sin sin90 180 90

= +P B P PD Vcos cos [2.4.36]

Equation 2.4.36: P P B PV = cos costherefrom with equation 2.4.35:

tan sin

cos=

P B

P B PV

( )P P B BV= cos sin tan

P P BD = sin sin

With 0 is 0 90 B , sin ; 0

Under these circumstances, is always within the 3rdquadrant.

Therefore, sin ,< 0 tan .> 0

For that reason, both formulas can be written in the form:

( )P P B BV= cos sin tan

P P BD = sin sin

Or with ( )sin sin , = BR D

and ( )tan tan , = BR D :

( )P P B

B

BR DV=

cos

sin

tan , [2.4.37]

( )P P

B

BR DD =

sin

sin , [2.4.38]


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03-Mar-2003 2-55

v= +

1

1 2 2

cos

sin[2.4.42]

again

= K Sv 2 [2.4.43]

2.4.8.3. Calculation of for the Piston-pin Bearing

Pin is fixed in the Piston

The pin is fixed in the piston and rotates in the rod end. The hydrodynamic calculationswill be done in the reference system of the connecting-rod.

Note:Here, like in the previous and following chapter, Kisorientated against the positive sense of rotation of all the otherangles.

=

Z S r

2 (10)

Z S K S Zr= = + = +180

= = Z S Zr K

= K Sr2 ; ( ) S Sr

or = K Srv 2 (11)

with

v=

cossin1 2 2

(13)

Pin is fixed in the Conrods End

The pin is fixed in the conrods end and rotates in the piston-pin bearings of the piston. Thehydrodynamic calculations will be done in the absolute reference system.

= + Z S 2 (9)

as the shell doesnt move: S=0

Z K= 180


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2-56 03-Mar-2003

(because Kis orientated against the positive sense of rotation)

Z K=

= K S2

thus = K Sv 2 (11)

with

v=

cos

sin1 2 2(14)

2.4.9. Differences Equation System for Journal Displacement

Set up the differences equation system for the journal displacement in that form how it is

used in this program.

Form:

( ) cos

sin

tan ,

=

P

So BR DB

BR DV

2B

(7)

( )

=

P

So BR D BR DD

2 sin

sin ,

B(8)

with B S S= ; 0 90 B

= K Sv 2 (11)

the following will be obtained:

= = = d

dt

d

d

d

dt

d

d UMRZ

n

K

180 1

with UMRZ= 180

K= angular velocity of the crank shaft

and with change-over to finite differences , K :

( ) ( )

( )

K K V

P UMRZ

BR D SoB UMRZ

B UMRZ

BR D=

2

cossin

tan ,(15)

and with:


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The calculation of the time function of the relative eccentricity for sections withdecreasing journal (moving towards the shell) is done at any rate by integrating thedifference equation 15. For sections with increasing (moving away from the shell) journal,this calculation is done by integrating the difference equation 15, only then if theSommerfeld figures are calculated by the formulae of Butenschoen.

If calculating the Sommerfeld figures according to Holland or Lang also for increasingjournal, this method would provide rather unprecise values for . For that reason, thecomputation of for an increasing journal in these cases will be done by using a formulagiven by Lang, ignoring the displacement supporting power (which usually is very small incase of an increasing journal):

( ) ( ) = cos sinC UMRZ C C UMRZ 3 4 32 (18)

must be computed according to formula 17a. The coefficients C3and C4(QC3, QC4 in theprogram) are tabulated as functions of the width ratio BR/D.

In general, it must be stated that the computation by using the Sommerfeld figures ofButenschoen is the more modern and more reliable method. Generally, it providesconsiderably lower eccentricity values and thus higher values for the minimum oilfilmthickness than the two other methods.

The computation methods according to Holland and Lang have been used by AVL inBRICKS to make possible comparative considerations on previous computation results, forwhich only these two methods had been available.

2.4.10. Sommerfeld Figures

The computation equations for the Sommerfeld figures given by the authors mentioned arediscussed here.

Sommerfeld Figures of the Rotational Supporting Power

According to Holland and Lang:

( )( ) ( )So M M D M= + 0 1 0 65 12 2 . (19a)

the constantM0,M1,M2 (QM0, QM1, QM2 in the program) being tabulated as functions

of the width ratioBR/D.According to Butenschoen:

( ) ( )( ) ( ) ( )So D BR a aD= + 0 5 1 1 16 12 2 2 2 2 2

1 2. (19b)

a1and a2(BTS3 and BTS4 in the program) must be calculated as 4th-degree polynoms ofthe width ratioBR/D.


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03-Mar-2003 2-59

Sommerfeld Figure of the Displacements Supporting Power

According to Holland (we have corrected sign errors in the publication):

( )( )

( )So

D

BR

V=

+

+

+

+

6

2 1

1

11

1 3 353429

2 3

2

2

2

2

arctan

.

(20a)

According to Lang (sign errors in the publication have been corrected):

( )So

D

BR

V= +

+

+

++

+

3 6 2 2

12

1

1

2 1 16

2 2 4

3

2

2

2 4

2 2

arctan

.(20b)

According to Butenschoen:

( )( ) ( )So

D

BR aa

V=

+ +

4

1 20 5 1 2 15 1 1

2

2 2 5

2 2

1

2

. . arccos . (20c)

a1and a2(BTS1, BTS2 in the program) being calculated by means of cubic polynomials ofthe width ratioBR/D.

2.4.11. Solution of the Equation System for the Displacement

It is understandable that due to the complexity of the equation-system 16 to 20, it is notpossible to obtain an analytic solution and that only a numerical iteration method can besuccessful.

In case of such a numerical calculation, it is assumed that the terms of the equations,which are dependent on the variables and , remain constant over short intervals of anindependent parameter that is common to all variables (here, this parameter is thecrankangle ). Thus, the values of the target-variables can be computed at the end of eachsmall "step" from the values at the start of the step and from the known changes of theindependent variable load value P and the loads angle of direction as well as of theeventually changeable angular velocity of the journal K v .


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2-60 03-Mar-2003

A necessity for this methods success is that the procedure converges, i.e. that theconsequences of the initial values arbitrary assumption are reduced in the course of steps,and that the method is steady, i.e. that the errors arising from the approximation remainminimal and - in successive steps - alternatively deviate to higher or lower values. Thus, apolygon is generated that repr

bricks theory

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