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ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND APPLIED SCIENCES MSc THESIS CEM BOĞA NUMERICAL FREE VIBRATION ANALYSIS OF MULTI SHAFT-DISC SYSTEMS DEPARTMENT OF MECHANICAL ENGINEERING ADANA, 2006

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Page 1: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

ÇUKUROVA UNIVERSITY

INSTITUTE OF NATURAL AND APPLIED SCIENCES

MSc THESIS

CEM BOĞA

NUMERICAL FREE VIBRATION ANALYSIS OF

MULTI SHAFT-DISC SYSTEMS

DEPARTMENT OF MECHANICAL ENGINEERING

ADANA, 2006

Page 2: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

ÇUKUROVA ÜNİVERSİTESİ

FEN BİLİMLERİ ENSTİTÜSÜ

NUMERICAL FREE VIBRATION ANALYSIS OF

MULTI SHAFT-DISC SYSTEMS

CEM BOĞA

YÜKSEK LİSANS TEZİ

MAKİNE MÜHENDİSLİĞİ ANABİLİM DALI

Bu Tez 28/02/2006 Tarihinde Aşağıdaki Jüri Üyeleri Tarafından Oybirliği ile

Kabul Edilmiştir.

İmza: İmza: İmza:

Prof. Dr. Vebil YILDIRIM Prof. Dr. Orhan AKSOĞAN Prof. Dr. Naki TÜTÜNCÜ

DANIŞMAN ÜYE ÜYE

Bu Tez Enstitümüz Makine Mühendisliği Anabilim Dalında Hazırlanmıştır.

Kod No:

Prof. Dr. Aziz ERTUNÇ Enstitü Müdürü

Not: Bu tezde kullanılan özgün ve başka kaynaktan yapılan bildirişlerin, çizelge, sekil ve fotoğrafların kaynak gösterilmeden kullanımı, 5846 sayılı Fikir ve Sanat Eserleri Kanunundaki hükümlere tabidir.

Page 3: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

I

ABSTRACT

MSc THESIS

Cem BOĞA

DEPARTMENT OF MECHANICAL ENGINEERING INSTITUTE OF NATURAL AND APPLIED SCIENCES

UNIVERSITY OF ÇUKUROVA

Supervisor : Prof. Dr. Vebil YILDIRIM

Year: 2006, Pages: 88

Jury : Prof. Dr. Vebil YILDIRIM

: Prof. Dr. Orhan AKSOĞAN

: Prof. Dr. Naki TÜTÜNCÜ

As it is known, multi shaft-disc systems are subjected to the bending, axial and torsional effects under static/dynamic forces. Among those, the shearing stresses due to the torsional moment are known as the most dangerous stresses on rotating shafts. Therefore, free or forced torsional vibration analysis of rotating shafts is of utmost importance in the dynamic design stage. The torsional vibration analysis may be carried out by both numerical and analytical methods such as the transfer matrix method, the finite element method and Holzer method.

In this study, the numerical free vibration analysis of multi shaft-disc systems is studied. A Fortran code is developed for the undamped free vibration analysis of the simple/branched shaft-discs systems based on the lumped-parameter model with the help of the transfer matrix method. For the same purpose, software package ANSYS is also used to perform the finite element analysis. The results obtained from both the transfer matrix and finite element methods are compared with those available in the literature. After verification of the present results, a parametric study is performed to investigate the effects of variation of the number of discs, the torsional rigidity of shafts, inertia of the discs, the mass of the shaft and the boundary conditions on the first three natural frequencies. Parametric results are given in graphical forms. The mode shapes are also presented.

Keywords: Shaft-Disc, Free Vibration, Transfer Matrix, Natural Frequency, Torsional Vibration.

NUMERICAL FREE VIBRATION ANALYSIS OF MULTI SHAFT-DISC

SYSTEMS

Page 4: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

II

ÖZ

YÜKSEK LİSANS TEZİ

Cem BOĞA

ÇUKUROVA ÜNİVERSİTESİ FEN BİLİMLERİ ENSTİTÜSÜ

MAKİNE MÜHENDİSLİĞİ ANABİLİM DALI

Danışman : Prof. Dr. Vebil YILDIRIM

Yıl: 2006, Sayfa: 88

Jüri : Prof. Dr. Vebil YILDIRIM

: Prof. Dr. Orhan AKSOĞAN

: Prof. Dr. Naki TÜTÜNCÜ

Bilindiği üzere statik/dinamik koşullarda çoklu mil-disk sistemleri eğilme, eksenel ve burulma etkilerine maruzdur. Bunlar arasında, burulma momentlerinden doğan kayma gerilmeleri dönel millerde en tehlikeli gerilme tipleri olarak bilinir. Bu nedenle dönel mil-disk sistemlerinin serbest veya zorlanmış burulma titreşimlerinin analizi dinamik tasarım aşamasında büyük önem taşımaktadır. Burulma titreşim analizi transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir.

Bu çalışmada çoklu mil-disk sistemlerinin serbest titreşim analizi sayısal olarak çalışılmıştır. Ayrık kütle modeli ve taşıma matrisi yöntemi yardımı ile, basit/dallı mil-disk sistemlerinin sönümsüz serbest titreşim analizi için Fortran dilinde bir program geliştirilmiştir. Ayni amaç için, sonlu elemanlar analizi için ANSYS paket programı da kullanılmıştır. Taşıma matrisi ve sonlu elemanlar yönteminden elde edilen sonuçlar, literatürde bulunan sonuçlarla karşılaştırılmıştır. Mevcut sonuçların doğrulanmasından sonra, disk sayısının, millerin burulma rijitliğinin, disklerin ataletlerinin, mil kütlesinin ve sinir şartlarının değişiminin ilk üç serbest titreşim frekansına etkisini incelemek için parametrik bir çalışma gerçekleştirilmiştir. Parametrik sonuçlar grafik olarak verilmiştir. Mod şekilleri de sunulmuştur.

Anahtar Kelimeler: Mil-Disk, Serbest Titreşim, Taşıma Matrisi, Doğal Frekans, Burulma Titreşimi.

ÇOKLU ŞAFT-DİSK SİSTEMLERİNİN SERBEST TİTREŞİM ANALİZİ

Page 5: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

III

ACKNOWLEDGEMENT

I am very grateful to my supervisor Prof. Dr. Vebil YILDIRIM for her guidance,

inspiration and encouragement during all my thesis studies. Also I would like to thank

my special committee members Prof. Dr. Orhan AKSOĞAN and Prof. Dr. Naki

TÜTÜNCÜ.

I would like to thank my friend Research Assistant Uğur EŞME for his support

and encouragement.

Finally, special thanks to my family and my girl friend İnci BUYUR for their

endless supports.

Page 6: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

IV

CONTENTS

PAGE

ABSTRACT………………………………………………………………… I

ÖZ…………………………………………………………………………... II

ACKNOWLEDGEMENT..…………………………………………………

CONTENTS…………………………………………………………………

III

IV

NOMENCLATURE.……………………………………………………….. VI

LIST OF TABLES………………………………………………………….. VIII

LIST OF FIGURES.………………………………………………………... IX

1. INTRODUCTION……………………………………………………. 1

2. PREVIOUS STUDIES........................................................................... 3

3. MATERIAL AND METHOD.............................................................. 7

3.1. Transfer Matrix for a Torsional System………………………. 8

3.1.1. The Field Transfer Matrix for an Elastic Massless

Shaft………………………………………………….

9

3.1.2. The Point Mass Matrix for a Rigid Disc…………….. 12

3.1.3. The Overall Transfer Matrix for the Whole System… 13

3.1.4. Determination of the Natural Frequencies…………... 15

3.1.5. Determination of the Mode Shapes………………….. 16

3.2. Finite Element Method for a Torsional System………………. 17

3.2.1. Element Types……………………………………….. 19

3.2.2. Example for Finite Element Method of a Torsional

System………………………………………………..

23

3.2.3. Program Options…………………………………...... 23

3.2.4. Input Data Listing…………………………………… 24

3.3. Holzer’s Method for a Torsional System……………………... 25

4. RESULTS AND DISCUSSION............................................................ 32

4.1. Verification of the Present Results……………………………. 33

4.1.1. Torsional System for Free-Free Ends……………….. 33

4.1.2. Torsional System for Fixed-Free Ends……………… 35

4.1.3. Torsional System for Fixed-Fixed Ends…………….. 38

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V

4.1.4. Torsional Branched System…………………………. 40

4.2. Performing a Parametric Study……………………………….. 43

4.2.1. Effect of the Torsional Stiffness of the Shafts………. 44

4.2.2. Effect of the Inertia of the Discs…………………….. 48

4.2.3. Effect of the Mass of the Shafts……………………... 52

4.3. Discussion…………………………………………………….. 54

5. CONCLUSION……………………………………………………….. 61

REFERENCES……………………………………………………………... 63

CURRICULUM VITAE……………………………………………………. 66

APPENDIX A………………………………………………………………. 67

APPENDIX B………………………………………………………………. 71

APPENDIX C………………………………………………………………. 73

APPENDIX D………………………………………………………………. 82

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VI

NOMENCLATURE A Cross Sectional Area of Shaft

ia}{ Eigenvector Representing the Mode Shape of the thi Natural Frequency

vC Damping Constant

[ ]C Damping Matrix

f Natural Frequency (cycles per unit time)

[ ]F Field Transfer Matrix

G Shear Modulus of Shaft Material

I Torsional Mass Moment of Inertia

0I Polar Second Moment of Area of Shaft

xi Polar Radius of Gyration about x-Axis

TJ Torsional Second Moment of Area

k Spring Stiffness

][K System Stiffness Matrix

L Shaft Length

][M System Mass Matrix

dm Mass of Disc

n Number of Disc

][P Point Mass Matrix

dr Radius of Disc

sr Radius of Shaft

dt Thickness of Disc

t Time

T Torque

u Displacement

][U Overall Transfer Matrix

ω Natural Circular Frequency (radians per unit time)

Page 9: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

VII

{z} State Vector

φ Angle of Twist

µ Mass of Unit Length of Shaft

dρ Density of Disc Material

Page 10: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

VIII

LIST OF TABLES PAGE

Table 3.1. KEYOPT Conditions…………………………………………... 23

Table 3.2.

Holzer’s Tabulation of the First Natural Frequency and

Corresponding Mode Shapes of the System Shown in Figure

3.12. (Tse et al., 1978)………………………………………….

31

Table 4.1. Comparison of the Results of the First Example in Figure 4.1... 33

Table 4.2.

Comparison of the Results of the Second Example in

Figure 4.3……………………………………………………….

36

Table 4.3. Comparison of the Results of the Third Example in Figure 4.6.. 38

Table 4.4. Comparison of the Results of the Fourth Example in Figure 4.8 41

Table B.1. COMBIN14 Input Summary…………………………………... 71

Table B.2. MASS21 Input Summary……………………………………… 72

Page 11: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

IX

LIST OF FIGURES PAGE

Figure 3.1. General Torsional System…………………………………… 7

Figure 3.2. Massless Shaft with Discs…………………………................ 9

Figure 3.3. Free-body diagram of the shaft (Pestel and Leckie, 1963)….. 9

Figure 3.4. Free-Body Diagram of the Disc (Pestel and Leckie, 1963)…. 12

Figure 3.5. Idealized Representation of a Four-Cylinder Engine with a

Flywheel (Pestel and Leckie, 1963)………………………….

14

Figure 3.6. Plot for Determination of Natural Frequencies……………… 16

Figure 3.7. Modeling the System………………………………………... 19

Figure 3.8. COMBIN14 Spring-Damper.................................................... 20

Figure 3.9. MASS21 Structural Mass........................................................ 22

Figure 3.10. Residual Torque Versus (Tse et al, 1978)……………............ 26

Figure 3.11. A Torsional System with Three Discs………………………. 27

Figure 3.12. A Torsional System for Analytical Solution by Holzer’s

Method (Tse et al, 1978)……………………………………..

29

Figure 4.1. Torsional System for Free-Free Ends (SETO, 1964)………... 33

Figure 4.2. Mode Shapes of the First Example………………………….. 34

Figure 4.3. Multi Shaft-Disc System for Fixed-Free Ends (SETO, 1964). 35

Figure 4.4. Determinant – Frequency Curve for the Second Example….. 36

Figure 4.5. Mode Shapes of the Second Example……………………….. 37

Figure 4.6.

Multi Shaft-Disc System for Fixed-Fixed Ends (SETO,

1964)…………………………………………………………

38

Figure 4.7. Mode Shapes of the Third Example…………………………. 39

Figure 4.8. Torsional Branched System (SETO, 1964)…………………. 40

Figure 4.9. Mode Shapes of the Fourth Example………………………... 42

Figure 4.10. Illustration of the Boundary Conditions for the Torsional

System Having the Same Number of Discs (n=number of

disc)…………………………………………………………..

43

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X

Figure 4.11. The Effects of the Torsional Stiffness and the Number of

Discs on the First Three Natural Frequency for Free-Free

Ends…………………………………………………………..

45

Figure 4.12. The Effects of the Torsional Stiffness and the Number of

Discs on the First Three Natural Frequency for Fixed-Free

Ends……..................................................................................

46

Figure 4.13. The Effects of the Torsional Stiffness and the Number of

Discs on the First Three Natural Frequency for Fixed-Fixed

Ends…………………………………………………………..

47

Figure 4.14. The Effects of the Inertia of the Discs and the Number of

Discs on the First Three Natural Frequency for Free-Free

Ends…………………………………………………………..

49

Figure 4.15. The Effects of the Inertia of the Discs and the Number of

Discs on the First Three Natural Frequency for Fixed-Free

Ends……..................................................................................

50

Figure 4.16. The Effects of the Inertia of the Discs and the Number of

Discs on the First Three Natural Frequency for Fixed-Fixed

Ends…………………………………………………………..

51

Figure 4.17. Geometric Shape of Disc for Mass Moment of Inertia……… 53

Figure 4.18. Geometric Shape of Shaft…………………………………… 53

Figure 4.19. The Effects of Mass of the Shaft on the Variation of the First

and Third Frequencies for Free-Free Ends

(L=constant)………………………………………………....

55

Figure 4.20. The Effects of Mass of the Shaft on the Variation of the First

and Third Frequencies for Fixed-Free Ends

(L=constant)…………………………………………………

56

Figure 4.21. The Effects of Mass of the Shaft on the Variation of the First

and Third Frequencies for Fixed-Fixed Ends

(L=constant)………………………………………………....

57

Figure 4.22. The Effects of Mass of the Shaft on the Variation of the First

and Third Frequencies for Free-Free Ends (I=constant)…….

58

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XI

Figure 4.23. The Effects of Mass of the Shaft on the Variation of the First

and Third Frequencies for Fixed-Free Ends (I=constant)…...

59

Figure 4.24. The Effects of Mass of the Shaft on the Variation of the First

and Third Frequencies for Fixed-Fixed Ends (I=constant).....

60

Figure A.1. Shaft Element Under Torsion (Pestel and Leckie, 1963)…… 67

Figure A.2. Shaft Under Torsion (Pestel and Leckie, 1963)……………... 69

Page 14: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

1. INTRODUCTION Cem BOĞA

1

1. INTRODUCTION

Study of vibration is concerned with the oscillatory motions of bodies. All

bodies possessing mass and elasticity are capable of vibration. Thus, most engineering

machines and structures experience vibration to some degree, and their design generally

requires consideration of their oscillatory behavior.

Existence of vibration in a machine produces noise, high stresses, wear, fatigue

failure, etc., and therefore it is generally undesirable. However, there are numerous

machines which are basically based on vibration. That is, there are several useful

applications of vibrations in industry.

Unwanted vibration causes two main problems: fatigue failure and failure due to

excessive deformation. Excessive deformation may be caused if vibration occurs at a

resonance frequency. A famous failure caused by resonance was that of the Tacoma

Narrows Bridge. The general principle of the bridge collapse is straightforward: a

resonance effect. High winds set up vibrations in the bridge, causing it to oscillate at the

frequency near to one of the natural frequencies of the bridge structure. Once

established, this resonance condition led to the bridge’s collapse.

Every system has a natural fundamental vibration frequency. The natural

frequencies and the mode shapes of a system may give quite useful information about

the multi shaft-disc system. Resonances are critical speeds which a designer must

certainly avoid.

Resonance occurs when the frequency of the excitation is equal to the natural

frequency of the system. When this happens, the amplitude of vibration will increase

without bound and is governed only by the amount of damping present in the system.

Therefore, in order to avoid disastrous effects resulting from very large amplitude of

vibration at resonance, the natural frequency of a system must be known and properly

taken care of.

In this study, the numerical free vibration analysis of multi shaft-disc systems is

studied. A Fortran code is developed for the undamped free vibration analysis of the

simple/branched shaft-discs systems based on the lumped-parameter model with the

Page 15: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

1. INTRODUCTION Cem BOĞA

2

help of the transfer matrix method. For the same purpose, software package ANSYS is

also used to perform the finite element analysis. The results which are obtained from

both the transfer matrix and finite element methods are compared with the results

available in the literature. After verification of the present results, a parametric study is

performed to investigate the effects of variation of the number of discs, the torsional

rigidity of shafts, inertia of the discs, the mass of the shaft and the boundary conditions

on the first three natural frequencies. Parametric results are given in graphical forms.

The mode shapes are also presented.

Page 16: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

2. PREVIOUS STUDIES Cem BOĞA

3

2. PREVIOUS STUDIES

Although there are numerous works to study the static/dynamic behavior of

multi disc-shaft systems and their single components with the help of different

analytical/numerical methods in the literature, a complete and accurate bending,

torsional and axial analyses based on the continuous parameter model of the whole

cracked/uncracked rotor which considers the nonlinear, damped behavior of the

system with tapered shafts and different types of the discs are very limited. That is,

most of the existing studies are not sufficient to describe in full the response of the

whole rotor consisting of almost complex components. Here, just limited number of

works which are related to the present study will be cited.

Holzer (1921) presented a very popular analytical method for determination

of torsional vibration of shaft-disc systems based on the lumped parameter model and

the method of the trial and error. Although the Holzer method may be applied to both

free and forced vibration problems it takes considerable time to determine the higher

frequencies (Den Hartog and Li, 1946; Spaetgens and Vancouver, 1950). Application

of the method may be found in text books (Pasin, 1988).

Wu and Chen (2001) studied the torsional vibration analysis of ear-branched

systems by finite element method. The fundamentals of the finite elements method for

torsional systems may be available in many books.

The study of Pestel and Leckie (1963) is assumed to be the main reference for

the application of the transfer matrix method for lumped systems. They present the

field transfer matrix of the shafts with/without mass and point transfer matrix for discs

explicitly. As is known, this method gives an accurate and economical solution by

using matrices with smallest dimensions. So, this method is generally used for

complex rotor analysis by scientists and industrial companies (Belek, 1984; Doughty

and Vafaee, 1985; Yardımcıoğlu and Sabuncu, 1993; Tsai and Wang, 1997; Kang et

al., 1997; Aleyaasin et al., 2000a-b, 2001; Aleyaasin and Ebrahimi, 2000; Akdağ and

Sabuncu, 2001; Akdağ, 2002). The known advantage of this method is that it is very

difficult to improve a general-purpose program like finite element softwares. Some

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2. PREVIOUS STUDIES Cem BOĞA

4

strategies such as combining the transfer matrix-finite element methods may be

attempted as Bhutani and Loewl (1999) did for two dimensional structures.

Doughty and Vafaee (1985) worked out the damped torsional vibration.

Asçıoğlu (2002) have studied the damped free vibration analysis of industrial

rotor systems with bears by using both RotorInsa and ANSYS softwares. He used

BEAM3, PIPE16, MASS21 and COMBIN14 element types in ANSYS solution to

compare the results.

Belek (1984) has found the critical speeds of rotor-disc systems due to the

bending vibrations. The critical speeds have been calculated by using transfer matrix

and finite element methods for different boundary conditions.

Yardımcıoğlu and Sabuncu (1993) have worked the free vibration and static

stability analysis of discs having constant or variable thickness in the radial direction.

Free vibrations and static stability of shaft-disc systems under static radial forces are

carried out employing finite element and transfer matrix methods.

Tsai and Wang (1997) have used the Timoshenko beam theory and the

transfer matrix method to obtain the frequency equation of a continuous multi-crack

rotor system. The cracks are assumed to be in the first mode of fracture, i.e. the

opening mode. The effect of transverse cracks oriented in different directions is also

discussed. The steady of natural mode of cracked rotating shaft can be inspected to

estimate the crack position. It is shown that the transfer matrix method for

continuous systems is successfully applied to solving the frequency equation of

multi-crack rotors.

Aleyaasin et al. (2000a) have presented the flexural vibration of a rotor

mounted on fluid film bearings. The analysis is carried out via lumped modelling. It

is shown that by implementing the transfer matrix method the natural frequencies are

found with greater accuracy.

Kang et al. (1997) have utilized a modified transfer matrix method to analyze

the instability in unsymmetrical rotor-bearing systems. The modified transfer matrix

of a shaft segment was derived from a continuous-system point of view to decrease

the number of matrix-multiplication operations and to achieve a higher accuracy than

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2. PREVIOUS STUDIES Cem BOĞA

5

that found from a lumped-system point of view. Most commercial software packages

(for example, ANSYS, COSMOS for finite element method, and RAPIDD-RSR for

transfer matrix method) do not have any particular element for analyzing

unsymmetrical rotor-bearing systems and establishing the code for a parametric

instability. This study has provided these complementary contents by using the

approach of the modified transfer matrix method.

To make an improvement in the formulation Aleyaasin et al. (2000b, 2001)

and Aleyaasin and Ebrahimi (2000) considered the shaft as continuous system in

their further studies.

Akdağ (2002) worked out a parametric study of the undamped free vibration

analysis of multi shaft-disc systems with uniform/variable sections of shafts. Transfer

matrix procedure for lumped systems is used in the study.

There are some studies about the investigation of the branched torsional

systems (Sankar, 1979; Wu and Chen, 2001; Gilbert, 1972; Shaikh, 1974; Dawson

and Davies, 1981; Aleyaasin et al., 2000b, 2001) by using numerical and analytical

methods.

Sreeram (2005) presented a graph theory based parametric influences applied

to torsional vibration analysis.

Chen (2006) presented an exact solution for free torsional vibration of a

uniform circular shaft carrying multiple concentrated elements. Because classical

analytical method is lengthy and tedious, he adopted numerical assembly method to

tackle the problem. He has found out that, exact natural frequencies and

corresponding mode shape of twisting angles can be easily determined with the

numerical assembly method.

Recently, Gürgöze (2006) presented an analytical method to express the sums

of the squared reciprocal eigen frequencies of torsional vibrations of an elastic bar of

given length and torsional rigidity to which n discs are attached for both fixed-free

and fixed-fixed cases.

In this study, the free torsional vibration analysis of multi shaft-disc-systems

is studied by using both the Transfer Matrix and Finite Element methods with

ANSYS. The free vibration problem is modeled by using the lumped-parameter

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2. PREVIOUS STUDIES Cem BOĞA

6

system in which the masses of the shafts are assumed to be negligible with respect to

the masses of discs. After verifying the results obtained in this study by comparing

the analytical results in the literature, a parametric study is performed. In the

parametric study the effects of the number of discs, the boundary conditions, and

different shaft and disc rigidities are considered and presented by charts. The

applicability of the method to the branched system is also demonstrated.

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3. MATERIAL AND METHOD

The most dangerous stresses for rotating shafts are known as shearing stresses

due to torsional moments. Therefore in the design stage of shaft-disc systems, the

torsional vibration analysis is inevitable together with the determination of critical

speeds of shafts, which are obtained by the bending vibration of the system.

The natural frequencies and the mode shapes of a system may give quite

useful information about the system.

A torsional system consists of a shaft with discs located at different points

along its length and bearings (Figure 3.1). As shown in Figure (3.1) φ represents the

angle of twist and T represents the corresponding torque.

Figure 3.1. General torsional system

In general, the shaft has a mass and its material may exhibit different

properties such as elastic/viscoelastic/plastic, isotropic/anisotropic, homogeneous/

nonhomogeneous, linear/nonlinear behaviors. Moreover the geometry of the cross-

section may be in different forms such as uniform/variable along the axis, solid

circle/hollow circle etc. The disc may have similar material and section properties.

For the continuous parameter model, all the elements of the system (disc,

shaft and bearing) are assumed to have mass, elastic and damping characteristics at

each section in the mathematical formulation of the problem.

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For simplicity of the analysis, the lumped parameter model is generally

preferred. In this model, the shaft is considered as elastic and massless. The disc is

assumed to be rigid.

In this study, the free undamped torsional vibration analysis of multi shaft-

disc system is studied numerically. The free vibration problem is modeled by using

the lumped-parameter system in which the masses of the shafts are assumed to be

negligible with respect to the masses of discs and elastic properties of the disc are

assumed to be negligible with respect to the elastic properties of the shaft. The

effects of the bearing on the natural frequencies of the system and damping are not

included in the present study.

The torsional vibration analysis of multi shaft-disc systems may be studied by

both numerical and analytical methods. In this chapter Holzer’s analytical technique,

Transfer Matrix and Finite Element numerical methods will be outlined for lumped

parameter model.

Obviously the matrix method has no special advantage in simple cases, but

when the system is more complicated, the advantages of matrix method will become

obvious.

3.1. Transfer Matrix for a Torsional System

Denoting the twist angle by φ and the torque applied to the shaft by T, the

state vector at ith section of the shaft is given by (Pestel ad Leckie, 1963);

ii T

z

=

φ (3.1)

We shall now consider the torsional vibrations of an elastic massless shaft of

uniform circular cross section, with discs attached at discrete points along its axis

(Figure 3.2). As stated before, the shaft is assumed to be elastic and massless, that is

without rotational inertia. The disc is rigid (not elastic) and has a mass, that is a

rotational mass moment of inertia I.

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Figure 3.2. Massless shaft with discs

3.1.1. The Field Transfer Matrix for an Elastic Massless Shaft

The shaft between sections i-1 and i is isolated, the end rotations and torques

being indicated in Figure (3.3). In view of the fact that only small deformations are

considered throughout, the end rotations φ are depicted as axial vectors.

Figure 3.3. Free-body diagram of the shaft (Pestel and Leckie, 1963)

Since there is no discontinuity in the torque along the portion of the shaft,

from the equilibrium condition we have;

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R1i

Li TT −= (3.2)

Where the subscript i represents locations of the discs attached to the shaft,

and superscripts L and R indicate the left and right sections of ith disc, respectively.

This means that, the left section of the disc shows, in fact, the right section of the

shaft of ith disc.

We know the following from the basic concepts of strength of materials;

GJT

dxd

T

=φ (3.3)

Where x is any section over the shaft portion, TJ is the torsional second

moment of area of the shaft and G is the shear modulus of the shaft material.

Integration of equation (3.3) over the shaft length L gives;

GJTL

T

=φ∆ (3.4)

The following may be written considering Figure (3.3);

iT

iR

iRi

Li GJ

LT)(

11

−− =−φφ (3.5)

Defining the rigidity of the shaft by;

LGJk T .

= (3.6)

Equation (3.5) becomes;

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R1i

R1i

Li T

k1

−− += φφ (3.7)

Combining equations (3.2) and (3.7) we have;

R

ii

T

L

i TGJL

T 1

.10

1

=

φφ (3.8)

or;

R

1ii

L

i T.

10k11

T−

=

φφ (3.9)

Using the definition of the state vector in equation (3.1), the above equation is

rewritten in the following compact form;

R

1iiLi −= zFz (3.10)

where F is the field transfer matrix for the massless shaft.

F

=

=

10k11

10GJ

L1T

(3.11)

If the mass of the shaft is not neglected, equation (3.8) becomes (Pestel and

Leckie, 1963);

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1

.cossin

sincos

−=

i

R

T

T

L

i TL

GJGJ

L

λλλ

λλ

λφ (3.12)

where;

G.J.L.i

Tx

µωλ = (3.13)

The radius of gyration about the x-axis (shaft axis) is;

AI

i 0x = (3.14)

A is the cross sectional area of the shaft, I0 is the polar second moment of area

of shaft, µ is the mass of the unit length of the shaft. For circular sections, torsional

moment of inertia is equal to the polar moment of inertia, that is JT =I0.

From equation (3.12), we know that as λ approaches zero ( 0→λ ), λsin

goes to zero, λcos goes to unity and finally λ

λsin goes to unity ( 0sin →λ ,

1cos →λ , 1/sin →λλ ). So equation (3.11) given for massless shaft is obtained.

Derivation of equation (3.12) is given in Appendix A.

3.1.2 The Point Mass Matrix for a Rigid Disc

Figure 3.4. Free-body diagram of the disc (Pestel and Leckie, 1963)

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Consider the free body diagram of the disc (Figure 3.4). As a result of the

inertia torque of the disc there is a discontinuity in the torque.

0ITT i2

iL

iR

i =+− φω (3.15)

Where, ω (rad/s) is the free vibration frequency. We note that the angle of

twist remains unchanged, so that;

R

iL

i φφ = (3.16)

When relating the state vectors Riz and L

iz on either side of the disc i, we have;

L

ii2

R

i T.

1I01

T

=

φω

φ (3.17)

or;

Lii

Ri zPz = (3.18)

where P is referred to the point mass matrix of the rigid disc.

3.1.3 The Overall Transfer Matrix for the Whole System

To explain the derivation of the overall transfer matrix from both the point

mass matrix and field transfer matrix, consider the idealized representation of a four-

cylinder engine with a flywheel attached shown in Figure (3.5). This system consists

of five rigid discs and four elastic shafts. The boundary conditions of the ends of the

shaft are free-free.

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Figure 3.5. Idealized representation of a four-cylinder engine with a

flywheel (Pestel and Leckie, 1963)

The overall transfer matrix U, is obtained by appropriate successive

multiplications of point mass and field transfer matrices from one end to the other

end. For instance, the relationship between the state vectors at sections numbered by

0 and 4, L0z and R

4z , is written in the following form.

L0

L0011223344

R4 UzzPFPFPFPFPz == (3.19)

where;

LR444 zPz = for the disc 4 RL344 zFz = for the shaft 4

LR333 zPz = for the disc 3

RL233 zFz = for the shaft 3

LR222 zPz = for the disc 2 RL122 zFz = for the shaft 2 LR111 zPz = for the disc 1 RL011 zFz = for the shaft 1

LR000 zPz = for the disc 0 (3.20)

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The explicit form of equation (3.19) is;

=

2221

1211R

4

4

UUUU

. L

0

0

T

φ (3.21)

The portioned overall transfer matrix is;

U

=

2221

1211

UUUU

(3.22)

3.1.4. Determination of the Natural Frequencies

Imposing the appropriate boundary conditions, overall transfer matrix can be

reduced to a matrix, from which the characteristic equation of the system is obtained.

The roots of the characteristic equation will be the natural frequencies. Then the

associated mode shapes can be evaluated by finding the response at each station for

each natural frequency.

For example, since the torque at free end is zero,

0TT L0

R4 == (3.23)

The characteristics equation becomes 0=U21 0φ , that is;

U21=0 (3.24)

The eigenvalues (roots) that make the determinant of the characteristic matrix zero

are called as the natural frequencies of the system.

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0U 21 = (3.25)

In this study, the roots of the characteristic equation are determined by

attributing numerical values to the natural frequencies as in Figure (3.6).

ω (rad/sn)

0 100 200 300 400 500 600 700

U21

-6.0e+6

-4.0e+6

-2.0e+6

0.0

2.0e+6

4.0e+6

6.0e+6

8.0e+6

1.0e+7

1.2e+7

ω1

ω2 ω3

ω4

Figure 3.6. Plot for determination of natural frequencies

3.1.5. Determination of the Mode Shapes

Considering Figure (3.5), from the state vector at the beginning;

L00

R0 zPz = (3.26)

we get L0

R0 φφ = . Taking the numerical value of the twist angle at the

beginning as unit;

L

0R

0 φφ = =1 (3.27)

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The twist angles at just right sections of the shafts (that is at left sections of

the discs) are determined from the equations given below.

L001

L1 zPFz =

L00112

L2 zPFPFz =

L0011223

L3 zPFPFPFz =

L0011223

L4 zPFPFPFPFz 34= (3.28)

These angles may be written as multipliers of 0φ ;

10 =φ

011 c φφ =

022 c φφ =

033 c φφ =

044 c φφ = (3.29)

By the method explained above, the mode shapes of corresponding

frequencies are obtained. The condition for the orthogonality of the first and the

second modes is given by Pestel and Leckie (1963) as follows;

∑=

=5

0i2i1ii 0..I φφ (3.30)

3.2. Finite Element Method for a Torsional System

The modal analysis is used to determine the natural frequencies and mode

shapes of the system. The system has constant stiffness and mass effect. The

equation of motion for an undamped system is:

}0{}]{[}]{[..

=+ uKuM (3.31)

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Note that ][K , the structure stiffness matrix, may include prestress effects. For

a linear system, free vibrations will be harmonic of the form:

tau ii ωcos}{}{ = (3.32)

where, ia}{ is eigenvector representing the mode shape of the thi natural

frequency, iω is thi natural circular frequency (radians per unit time), t is time. Thus

equation (3.31) becomes:

}0{}]){[][( 2 =+− ii aKMω (3.33)

This equality is satisfied if either }0{}{ =ia or if the determinant of

])[]([ 2 MK ω− is zero. The first option is trivial and, therefore, is not of interest.

Thus, the second one gives the solution:

0][][ 2 =− MK ω (3.34)

This is an eigenvalue problem which may be solved for up to n values of 2ω

and n eigenvectors ia}{ which satisfy equation (3.33) where n is the number of

degrees of freedom.

Rather than outputting the natural circular frequencies ),(ω the natural

frequencies )( f are output; where;

πω2

iif = (3.35)

Where if is thi natural frequency (cycles per unit time). Finite element

representation of a torsional system is illustrated in Figure (3.7).

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a) Real system

b) Finite element model

Figure 3.7. Modeling the system

3.2.1. Element Types

COMBIN14 is chosen for spring elements and MASS21 for mass elements.

COMBIN14 has longitudinal or torsional capability in one, two, or three

dimensional applications (Figure 3.8). The longitudinal spring-damper option is a

uniaxial tension-compression element with up to three degrees of freedom at each

node: translation in the nodal x , y and z directions. No bending or torsion is

considered. The torsional spring-damper option is a purely rotational element with

three degrees of freedom at each node: rotations about the nodal x , y and z axes.

No bending or axial loads are considered. The spring-damper element has no mass.

Masses can be added by using the appropriate mass element.

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The geometry, node location and the coordinate system for this element are

shown in Figure (3.8). The element is defined by two nodes, a spring constant )(k

and damping coefficients 1)( VC and 2)( VC . The damping capability is not used for

static or undamped modal analyses.

Figure 3.8. COMBIN14 Spring-Damper

Single degree of freedom per node (KEYOPT(2)>0). The orientation is

defined with the KEYOPT command and the two nodes are usually coincident.

KEYOPT(2) = 1 through 6 is used for defining the element as a one-

dimensional element. With these options, the element operates in the nodal

coordinate system. The KEYOPT(2) = 7 and 8 options allow the element to be used

in a thermal or pressure analysis. The KEYOPT(3) = 1 is used for defining the

element as a three-dimensional torsional spring-damper. If the KEYOPT(3) =1, the

degrees of freedom is ROTX, ROTY, ROTZ. Also, if KEYOPT(3) =1 (torsion) is

used with large deflection, the coordinates will not be updated. Consider the case of a

single degree of freedom per node first. The form of the element stiffness and

damping matrices are:

[ ]

−=

1111

kKe (3.36)

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[ ]

−=

1111

ve CC (3.37)

Consider the case of multiple degrees of freedom per node. The stiffness and

damping matrices in element coordinates are developed as:

[ ]

=

000000000000001001000000000000001001

kK l (3.38)

[ ]

=

000000000000001001000000000000001001

vCCl (3.39)

where subscript l refers to element coordinates. An input summary for

COMBIN14 is given in Table B.1 in Appendix B.

MASS21 is a point element having up to six degrees of freedom: translations

in the nodal x , y and z directions and rotations about the nodal x , y and z axes

(Figure 3.9). A different mass and rotary inertia may be assigned to each coordinate

direction.

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Figure 3.9. MASS21 Structural mass

The element coordinate system may be initially parallel to the global

Cartesian coordinate system or to the nodal coordinate system (KEYOPT(2)). The

element coordinate system rotates with the nodal coordinate rotations during a large

deflection analysis. Options are available to exclude the rotary inertia effects and to

reduce the element to a two-dimensional capability (KEYOPT(3)). If the element

requires only one mass input, it is assumed to act in all appropriate coordinate

directions.

The element mass matrix is:

=

fe

dc

ba

M e

000000000000000000000000000000

][ (3.40)

Table (3.1) shows the KEYOPT conditions for MASS21.

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Table 3.1. KEYOPT conditions

KEYOPT(3)=0 KEYOPT(3)=2 KEYOPT(3)=3 KEYOPT(3)=4

a 1 1 1 1

b 2 1 1 1

c 3 1 - -

d 4 - - -

e 5 - - -

f 6 - 2 -

For the mass summary, only the first real constant is used, regardless of

which option of KEYOPT(3) is used. Analyses with inertial relief use the complete

matrix. A summary of the element input is given in Table B.2 in Appendix B.

3.2.2. Example for the Finite Element Method of a Torsional System

The finite element method is used to determine the natural frequencies for

torsional vibration of the system shown in Figure (3.5).

3.2.3. Program Options

Mode-frequency analysis is used as analysis type. As an element type three-

dimensional torsional spring COMBIN14 is used for the shaft. MASS21 is used for

disc.

Material properties of shaft-disc system are: 103210 ==== IIII

2sec. −− inlb ; 204 =I 2sec. −− inlb ; 6321 10.5,1=== kkk radinlb /.− ;

64 10.2=k radinlb /.−

All shafts have the same length. The natural frequencies are found as; o=0ω

sec/rad , 358.2141 =ω sec/rad , 672.4442 =ω sec/rad , 244.6343 =ω sec/rad ,

345.7444 =ω sec/rad .

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3.2.4. Input Data Listing

/prep7

et,1,combin14,,,1

et,2,mass21,,,3

r,1,1500000

r,2,2000000

r,3,1,10

r,4,1,20

n,1,

n,2,,,1

n,3,,,2

n,4,,,3

n,5,,,4

type,1

real,1

e,1,2

e,2,3

e,3,4

type,1

real,2

e,4,5

type,2

real,3

e,1

e,2

e,3

e,4

type,2

real,4

e,5

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/view,1,1,1,1

finish

/solu

antype,modal

modopt,redu,,,,5

m,1,rotz

m,2,rotz

m,3,rotz

m,4,rotz

m,5,rotz

outpr,basic,1

solve

finish

*get,f1,mode,1,freq

*get,f2,mode,2,freq

*get,f3,mode,3,freq

*get,f4,mode,4,freq

*get,f5,mode,5,freq

FINISH

3.3. Holzer’s Method for a Torsional System

Holzer’s method uses these tables for the analytical determination of the

frequencies. The method assumes a trial frequency. A solution is found when the

assumed frequency satisfies the constraints of the problem. Usually, this requires

several trials. The tabulation also gives the mode shape of the system.

For the successive discs numbered by i and i+1, the following equation is

considered.

P

i

1PP

1i,i

2n

i1i Jk

φω

φφ ∑=+

+ −= (3.41)

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Where JP is the mass moment inertia of the disc with respect to the shaft axis,

φ is the amplitude of the torsional vibration, k is the torsional spring constant of the

shaft.

For ,nωω = the residual torque of equation (3.41) becomes zero;

0J P

i

1PP =∑

=

φ (3.42)

Assuming either an arbitrary numerical value or a numerical value with some

approximation for natural frequency, ,ω and taking ,11 =φ the amplitudes of all the

discs are computed. For chosen frequencies, the residual which is different from zero

is calculated from the above equation. After plotting the curve for (residual-ω ), it is

checked whether the desired number of nodes are achieved (for nth frequency, it is

necessary to obtain n nodes). The operations will continue to get any satisfactory

frequency (Holzer, 1921; Pasin, 1988). A typical residual torque versus 2ω plot is

shown in Figure (3.10).

Figure 3.10. Residual torque versus 2ω (Tse et al., 1978)

Let’s explain the method for the undamped system shown in Figure (3.11).

The motion is harmonic at a principal mode of vibration.

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Figure 3.11. A torsional system with three discs

The scalar equations of motion from Newton`s second law are (Tse et al.,

1978);

)( 211112 φφφω −−=− kI

)()( 322121222 φφφφφω −−−−=− kkI

)( 232332 φφφω −−=− kI (3.43)

Summing the equations gives;

03

1

2 =∑=i

iiI ωφ (3.44)

Correspondingly, for an n disc system;

01

2 =∑=

n

iiiI ωφ (3.45)

The equation states that the sum of the inertia torque of a semi definite system

must be zero. The trial frequency ω must satisfy this constraint. Hence equation

(3.45) is another form of the frequency equation.

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To begin the tabulation, assume a trial frequency ω arbitrarily and let ,11 =φ

calculate 2φ from the first equation in equation (3.43), and 3φ from the second

equation, that is;

11 =φ

1112

12 / kI φωφφ −=

222112

23 /)( kII φφωφφ +−= (3.46)

The values of 1φ , 2φ and 3φ are substituted in equation (3.44) to check

whether the constraint is satisfied. If not, a new value of ω is assumed and the

process repeated. Note that the equations for 2φ and 3φ can be generalized for an n

disc system as;

∑−

=−− −=

1

11

2

1

j

iii

jjj J

kφωφφ j = 2,3,…,n (3.47)

In summary, the method consists of the repeated application of equations

(3.45) and (3.47) for different trial frequencies. If the trial frequency is not a natural

frequency of the system, equation (3.45) will not be satisfied. The residual torque in

equation (3.45) represents a torque applied at the last disc. This is equivalent to a

condition of steady-state forced vibration. The amplitudes iφ (i = 1, 2, 3,…,n) also

give the mode shape for the given natural frequency.

As explained above, Holzer method is essentially a systematic tabulation of

the frequency equation of the system. The method has general applications, including

systems with rectilinear and angular motions, damped or undamped, semidefinite or

branched systems. Holzer’s method may also be applied to the forced vibration

problems (Den Hartog and Li, 1946; Spaetgens and Vancouver, 1950). The

procedure can be programmed for computer applications. Its main disadvantage is

that it is time consuming due to the trial and error.

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The method will be explained below thorough a numerical example. The

torsional system and its properties are given in Figure (3.12). We wish to find just

natural frequencies.

Figure 3.12. A torsional system for analytical solution by Holzer’s method

(Tse et al., 1978)

As stated before, there is no standard procedure for estimating a trial

frequency. As an initial trial, assume an equivalent system, consisting of 1I and

( 43 II + ) at the two ends connected by a shaft with 1k and 2k in series. The spring

constant tk of this equivalent system is;

=+

=21 /1/1

1kk

kt radmkN /.320

20/110/1103

=+

(3.48)

From this equation we get the initial frequency as follows;

)]/(1/1[ 3212 IIIkt ++=ω (3.49)

( 173=ω rad/s)

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Now we proceed with the calculation as shown in Table (3.2). First, the

values of I and k are entered in columns 1 and 5, respectively. The remaining

values are determined as follows;

Row 1 Column 2: Assume 11 =φ rad

Column 3: Compute 211 ωφI = 0.4(1)(173)2 = 11.97 kNm

Column 4: Sum values in Column 3 = 11.97 kNm

Column 6: Divide Column 4 by Column 5 = 11.97/10 =1.197 rad

Row 2 Column 2: Compute −= 12 φφ (twist in k1) = 1-1.197 = -0.197 rad

Column 3: Compute 222 ωφI = 0.1(-0.197)(173)2 = -0.59 kNm

Column 4: Sum values in Column 3 = 11.97-0.59 = 11.38 kNm

Column 6: Divide Column 4 by Column 5 = 11.38/20 = 0.569 rad

(twist in k2)

The values in rows 3 and 4 are determined in a like manner. The residual

torque is found as (-0.92) kNm, which is not zero. The negative sign indicates that the

trial frequency is too high for the first mode.

Similarly, a frequency 165=ω rad/s is used for the second trial and the

residual torque is 1.3 kNm. A linear interpolation between the first and the second

trial gives 170=ω rad/s. The calculated values for this frequency are shown in Table

(3.2). The residual torque is -0.01 kNm, which may be sufficiently accurate for the

purpose.

The natural frequencies for the second and third are found in like manner

( 1701 =ω rad/s, 3542 =ω rad/s, 5883 =ω rad/s).

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Table 3.2. Holzer’s tabulation of the first natural frequency and corresponding mode

shapes of the system shown in Figure (3.12). (Tse et al., 1978)

1 2 3 4 5 6 Column

Row

I

(m2kg)

φ

(rad)

2φωI

(kNm)

∑ 2φωI

(kNm)

k

(kNm/rad) ∑ 21 φωI

k

(rad)

(a) First trial for the first mode: 173=ω rad/s

1 0.4 1 11.97 11.97 10 1.197

2 0.1 -0.197 -0.59 11.38 20 0.569

3 0.4 -0.766 -9.17 2.21 10 0.218

4 0.1 -1.043 -3.12 -0.92 Residual torque

(b) Second trial for the first mode: 165=ω rad/s

1 0.4 1 10.89 10.89 10 1.089

2 0.1 -0.089 -0.24 10.65 20 0.532

3 0.4 -0.621 -6.77 3.88 10 0.388

4 0.1 -1.009 -2.74 1.13 Residual torque

(c) Third trial for the first mode: 170=ω rad/s

1 0.4 1 11.56 11.56 10 1.156

2 0.1 -0.156 -0.45 11.11 20 0.555

3 0.4 -0.711 -8.22 2.88 10 0.288

4 0.1 -1 -2.89 -0.01 Residual torque

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4. RESULTS AND DISCUSSION

A Fortran code is developed for the application of the Transfer Matrix

Method to the undamped free vibration of multiple shaft-disc systems based on the

lumped parameter model. That computer program for both shafts with/without mass,

input and output files are given in Appendices C-D.

Elastic shafts with/without mass are assumed to have a uniform circular cross

section. Material of the shaft is chosen as isotropic.

In this section, various examples are solved numerically based on both the

transfer matrix and finite element methods. Different boundary conditions such as

free-free ends, fixed-free ends, fixed-fixed ends are studied for the first three natural

frequencies which are different from rigid body modes. Their mode shapes are also

presented. Applicability of the method to the branched torsional system is also

demonstrated. It is shown that, present study which is performed with the help of the

transfer matrix method gives very satisfactory results for the design of such torsional

systems.

After verifying the present numerical results, a parametric study is performed

to investigate the effect of the mass of the shaft, the number of discs, the torsional

rigidity of shafts, inertia of the discs, and the boundary conditions on the natural

frequencies based on the transfer matrix method. Parametric results are presented in

graphical forms.

It is expected that the present study will be very helpful to the engineers and

designers.

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4.1. Verification of the Present Results

4.1.1. Torsional System for Free-Free Ends

Figure 4.1. Torsional system for free-free ends (Seto, 1964)

In this example the torsional system has three discs and two massless shafts

(Figure 4.1). The boundary condition for the shaft is free-free. Material and

geometrical properties of the shaft-disc system are: k1 = k2 =1 in-lb/rad=0.1129792

Nm/rad and I1 = I2 = I3 = 1 in-lb-sec2/rad= 0.1129792 kg.m. The results are given in

Table (4.1) in a comparative manner. Seto (1964), get those results by Holzer’s

method. As seen from the table, natural frequencies are very close to each other.

Mode shapes of the first example are shown in Figure (4.2).

Table 4.1. Comparison of the results of the first example in Figure (4.1)

ω (rad/s) Present Study

(Transfer Matrix)

ANSYS

(Finite Element)

Seto (1964)

(Holzer’s Method)

1ω 0 0 0

2ω 1 0.99994 1

3ω 1.73205 1.73205 1.7

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0.5 1 1.5 2

0.5

1

1.5

2

1ω = 0 rad/s

φ0=1; φ1=1; φ2=1

0.5 1 1.5 2

-1

-0.5

0.5

1

2ω = 1 rad/s

φ0=1; φ1=0; φ2= -1

0.5 1 1.5 2

-2

-1.5

-1

-0.5

0.5

1

3ω = 1.73205 rad/s

φ0=1; φ1= -2; φ2=1

Figure 4.2. Mode shapes of the first example

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4.1.2. Torsional System for Fixed-Free Ends

Figure 4.3. Multi shaft-disc system for fixed-free ends (Seto, 1964)

Material and geometrical properties of the shaft-disc system shown in Figure

(4.5) are: k1 =30.106 in-lb/rad=3389376 Nm/rad, k2 =10.106 in-lb/rad=1129792

Nm/rad, k3 =10.106 in-lb/rad=1129792 Nm/rad, I1 = 4000 in-lb-sec2/rad=451.9168

kg.m2, I2 = 2000 in-lb-sec2/rad=225.9584 kg.m2, I3 = 1000 in-lb-sec2/rad=112.9792

kg.m2.

The first three natural frequencies are presented in Table (4.2). As seen from

the table all results obtained by the transfer matrix method, the finite element method

and Holzer’s method show a good agreement with each other.

The variation of the determinant of the characteristic equation with respect to

the natural frequencies is illustrated in Figure (4.4).

The corresponding mode shapes are also given in Figure (4.5).

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ω (rad/sn)0 20 40 60 80 100 120 140

u 22

-0 6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2

ω1 ω2 ω3

Figure 4.4. Determinant – Frequency curve for the second example

Table 4.2. Comparison of the results of the second example in Figure (4.3)

ω (rad/s) Present Study

(Transfer Matrix)

ANSYS

(Finite Element)

Seto (1964)

(Holzer’s Method)

1ω 45.7636 45.76358 46

2ω 100 99.99994 100

3ω 133.812 133.81215 134

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0.5 1 1.5 2 2.5 3

2´ 10-8

4´ 10-8

6´ 10-8

8´ 10-8

1´ 10-7

1.2 ´ 10-7

1ω = 45.7636 rad/s

φ0=0; φ1=3.33.10-8; φ2=1.05.10-7; φ3=1.33.10-7

0.5 1 1.5 2 2.5 3

-3´ 10-8

-2´ 10-8

-1´ 10-8

1´ 10-8

2´ 10-8

3´ 10-8

2ω = 100 rad/s

φ0=0; φ1=3.33.10-8; φ2=0; φ3= -3.33.10-8

0.5 1 1.5 2 2.5 3

-1´ 10-7

-5´ 10-8

5´ 10-8

1´ 10-7

3ω = 133.812 rad/s

φ0=0; φ1=3.33.10-8; φ2=-1.05.10-7; φ3=1.33.10-7

Figure 4.5. Mode shapes of the second example

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4.1.3. Torsional System for Fixed-Fixed Ends

Figure 4.6. Multi shaft-disc system for fixed-fixed ends (Seto, 1964)

The necessary properties of the system shown in Figure (4.6) are as follows:

k1 = k2 = k3 = k4 =1000 in-lb/rad=112.9792 Nm/rad, I1 = I2 = I3 = 10 in-lb-

sec2/rad=1.129792 kg.m2. Results are presented in Table (4.3). A very good

agreement in frequencies is observed.

Mode shapes are illustrated in Figure (4.7).

Table 4.3. Comparison of the results of the third example in Figure (4.6)

ω (rad/s) Present Study

(Transfer Matrix)

ANSYS

(Finite Element)

Seto (1964)

(Holzer’s Method)

1ω 7.65367 7.65366 7.66

2ω 14.1421 14.14213 14.12

3ω 18.4776 18.47759 18.57

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3

3

3

1 2 3 4

0.0002

0.0004

0.0006

0.0008

0.001

0.0012

0.0014

1ω = 7.65367 rad/s

φ0=0; φ1=0.001; φ2=0.001414; φ3=0.001; φ4=0

1 2 3 4

-0.001

-0.0005

0.0005

0.001

2ω = 14.1421 rad/s

φ0=0; φ1=0.001; φ2=0 ; φ3= -0.001; φ4=0

1 2 3 4

-0.001

-0.0005

0.0005

0.001

3ω = 18.4776 rad/s

φ0=0; φ1=0.001; φ2= -0.001414; φ3=0.001; φ4=0

Figure 4.7. Mode shapes of the third example

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4.1.4. Torsional Branched System

(a) Real system (b) Equivalent system

Figure 4.8. Torsional branched system (Seto, 1964)

Material properties of the branched system shown in Figure (4.8) are:

I1=2I2=10 lb-in-sec2/rad=1.129792 kg.m2, k1=k2=k=1 in-lb/rad=0.1129792 Nm/rad,

gear reduction ratio R = 3 (Seto, 1964).

If the moment of inertia of the gears is negligible the following may be

written for equivalent stiffness of the equivalent shaft;

21

21eq kk

k.kk+

= (4.1)

This is done by replacing 2I with 22´

2 IRI = and 2k with kRk 22 = .

kRkk.R.kk 2

2

eq += k9.0= (4.2)

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The natural frequency of the equivalent system is given by (Seto, 1964);

´21

´21eq

I.I

)II(k +=ω (4.3)

where;

112

22´

2 5.42

IIRIRI === (4.4)

and;

kRk 22 = (4.5)

The natural frequencies of the equivalent system in Figure (4.8.-b) are

presented in Table (4.4). A very good harmony of the results is seen in the table.

Table 4.4. Comparison of the results of the fourth example in Figure (4.8)

ω (rad/s) Present Study

(Transfer Matrix)

ANSYS

(Finite Element)

1ω 0 0

2ω 0.331662 0.331662479

Mode shapes of the branched system are given in Figure (4.9).

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R

Disc number1 2

φ

0,0

0,5

1,0

1,5

2,0

1ω = 0 rad/s

φ0=1; φ1=1 (rigid body mode)

Disc number

1 2

φ

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

2ω = 0.331662 rad/s

φ0=1; φ1= -0.222222

Figure 4.9. Mode shapes of the fourth example

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4.2. Performing a Parametric Study

In this section, all discs and shafts are assumed to have corresponding

identical properties. That is, while all the discs have the same inertias, the shafts have

the same rigidities.

a) Free-free (n-1 shafts)

b) Fixed-Free (n shafts)

c) Fixed-Fixed (n+1 shafts)

Figure 4.10. Illustration of the boundary conditions for the torsional system

having the same number of discs (n=number of disc).

As seen from Figure (4.10), for the same number of discs, torsional systems

have different number of shafts. In general, increase in the number of the shafts

decreases the rigidity of the system. The intermediate supports are not included in the

present study.

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4.2.1. Effect of the torsional stiffness of the shafts

First, taking the inertia of the discs as constant, I=30 in-lb-s2/rad=3.389376

kg-m2-s2/rad=constant, shaft rigidities are changed between k=1000000-10000000

in-lb/rad to investigate the effect of shaft rigidities on the first three free vibration

frequencies. In these examples, mass of the shafts is neglected as Akdağ and

Sabuncu (2001) and Akdağ (2002) do. These investigators studied just the first

fundamental frequency based on the same procedure presented in this study and just

consider the free-free boundary conditions.

Results are presented in Figures (4.11-4.13) for three boundary conditions

(for free-free ends, fixed-free ends, and for fixed-fixed ends). For the fundamental

frequencies and free-free ends, a good accordance is observed between the present

graphs and graphs of Akdağ and Sabuncu (2001).

As seen from these figures, when the shaft rigidity increases the natural

frequencies also increase for all boundary conditions. The variation of the increase in

the natural frequencies is clear for small number of discs and frequencies of higher

modes. That is the curves of small number of disc are steeper than higher number of

discs. This is also valid for higher modes.

However, increasing the number of discs reduces the frequencies. This kind

of behavior is very clear for especially the first frequency.

The order of the numerical values of the frequencies with respect to the

boundary conditions may be given as free-free, fixed-fixed, fixed-free in descending

order. This is due to the number of shafts for the same number of discs (see Figure

4.10). As stated before, increase in the number of shafts for the same number of discs

makes the system less rigid.

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Free-Free End (Ι = 30 lb-in.-sec

2 = constant)

(n=number of disc)

k (lb-in./rad)

0,0 2,0e+6 4,0e+6 6,0e+6 8,0e+6 1,0e+7 1,2e+7

ω2(

rad/

sec)

0

200

400

600

800

1000

n

n=2

n=3

n=4

n=5n=6n=7n=8n=10

Free-Free End (Ι = 30 lb-in.-sec

2 = constant)

(n=number of disc)

k (lb-in./rad)0,0 2,0e+6 4,0e+6 6,0e+6 8,0e+6 1,0e+7 1,2e+7

ω3

(rad/

sec)

0

200

400

600

800

1000

1200

n

n=3

n=4

n=5

n=6n=7n=8

n=10

Free-Free End (Ι = 30 lb-in.-sec

2 = constant)

(n=number of disc)

k (lb-in./rad)0,0 2,0e+6 4,0e+6 6,0e+6 8,0e+6 1,0e+7 1,2e+7

ω4(

rad/

sec)

0

200

400

600

800

1000

1200

n

n=4

n=5

n=6

n=7

n=8

n=10

Figure 4.11. The effects of the torsional stiffness and the number of discs on the first three natural frequency for free-free ends

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Fixed-Free End(Ι = 30 lb-in.-sec

2 = constant)

(n=number of disc)

k (lb-in./rad)0,0 2,0e+6 4,0e+6 6,0e+6 8,0e+6 1,0e+7 1,2e+7

ω1(ra

d/se

c)

0

100

200

300

400

n

n=2

n=3

n=4

n=5n=6n=7n=8n=10

Fixed-Free End(Ι = 30 lb-in.-sec

2 = constant)

(n=number of disc)

k (lb-in./rad)

0,0 2,0e+6 4,0e+6 6,0e+6 8,0e+6 1,0e+7 1,2e+7

ω2(ra

d/se

c)

0

200

400

600

800

1000

n

n=2

n=3

n=4

n=5

n=6n=7n=8n=10

Fixed-Free End(Ι = 30 lb-in.-sec

2 = constant)

(n=number of disc)

k (lb-in./rad)0,0 2,0e+6 4,0e+6 6,0e+6 8,0e+6 1,0e+7 1,2e+7

ω3(r

ad/s

ec)

0

200

400

600

800

1000

1200

n

n=3

n=4

n=5

n=6

n=7n=8

n=10

Figure 4.12. The effects of the torsional stiffness and the number of discs on the first three natural frequency for fixed-free ends

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Fixed-Fixed End(Ι = 30 lb-in.-sec

2 = constant)

(n=number of disc)

k (lb-in./rad)

0,0 2,0e+6 4,0e+6 6,0e+6 8,0e+6 1,0e+7 1,2e+7

ω1(r

ad/s

ec)

0

100

200

300

400

500

600

700

n=2

n=3

n=4

n=5

n=6n=7n=8n=10

n

Fixed-Fixed End(Ι = 30 lb-in.-sec

2 = constant)

(n=number of disc)

k (lb-in./rad)

0,0 2,0e+6 4,0e+6 6,0e+6 8,0e+6 1,0e+7 1,2e+7

ω2( ra

d/se

c)

0

200

400

600

800

1000

1200

n=2

n=3

n=4

n=5n=6n=7n=8n=10

n

Fixed-Fixed End(Ι = 30 lb-in.-sec

2 = constant)

(n=number of disc)

k (lb-in./rad)

0,0 2,0e+6 4,0e+6 6,0e+6 8,0e+6 1,0e+7 1,2e+7

ω3(

rad/

sec)

0

200

400

600

800

1000

1200

n=3

n=4

n=5

n=6n=7n=8

n=10

n

Figure 4.13. The effects of the torsional stiffness and the number of discs on the first three natural frequency for fixed-fixed ends

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4.2.2. Effect of the inertia of the discs

Second, as the torsional rigidity of the shafts are being constant, k=1000000

in-lb/rad=1129792 Nm/rad=constant, the inertia of the discs is changed from I=1 lb-

in-s2/rad to I=50 lb-in-s2/rad.

Akdağ and Sabuncu (2001) performed the same parametric study for

fundamental frequency and free-free boundary condition. In this study, the second

and the third frequency are also considered with additional two boundary conditions

(fixed-free and fixed-fixed).

Variation of the first three free vibration frequencies with respect to the

inertia of the discs is presented in Figures (4.14-4.16). The graphs of both the

presents study and Akdağ and Sabuncu’s study (2001) are in a very good agreement.

While the torsional rigidity of the shaft is kept constant, variation of the mass

moment of inertia of the discs changes the natural frequencies. Increasing the inertias

of the discs, in general, decreases the natural frequencies. This is very apparent for

smaller inertias.

Apart from this, increasing the number of discs reduces free vibration

frequencies for all boundary conditions.

The order of the numerical values of the frequencies with respect to the

boundary conditions may be given as, fixed-free, fixed-fixed, free-free, in ascending

order. This is due to the number of shafts for the same number of discs (see Figure

4.10). As stated before, although the increase in constraints increases the rigidity of

the system, increase in the number of shafts for the same number of discs makes the

system less rigid.

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4. RESULTS AND DISCUSSION Cem BOĞA

49

Free-Free End(k=1000000 lb-in./rad=constant)(n=number of disc)

Ι (lb-in./sec2) 0 10 20 30 40 50 60

ω2(ra

d/se

c)

0

200

400

600

800

1000

1200

1400

1600

n=2

n=10

n

Free-Free End(k=1000000 lb-in./rad=constant)(n=number of disc)

Ι (lb-in./sec2)0 10 20 30 40 50 60

ω3(

rad/

sec)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

n=3

n=10

n

Free-Free End(k=1000000 lb-in./rad=constant)(n=number of disc)

Ι(lb-in./sec2)

0 10 20 30 40 50 60

ω4(

rad/

sec)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

n=4

n=10

n

Figure 4.14. The effects of the inertia of the discs and the number of discs on the first three natural frequency for free-free ends

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4. RESULTS AND DISCUSSION Cem BOĞA

50

Fixed-Free End(k=1000000 lb-in./rad=constant)(n=number of disc)

Ι (lb-in./sec2) 0 10 20 30 40 50 60

ω1(r

ad/s

ec)

0

100

200

300

400

500

600

700

n=2

n=10

n

Fixed-Free End(k=1000000 lb-in./rad=constant)(n=number of disc)

Ι (lb-in./sec2) 0 10 20 30 40 50 60

ω2(ra

d/se

c )

0

200

400

600

800

1000

1200

1400

1600

1800

n=2

n=10

n

Fixed-Free End(k=1000000 lb-in./rad=constant)(n=number of disc)

Ι (lb-in./sec2) 0 10 20 30 40 50 60

ω3(r

ad/s

ec)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

n=3

n=10

n

Figure 4.15. The effects of the inertia of the discs and the number of discs on the first three natural frequency for fixed-free ends

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4. RESULTS AND DISCUSSION Cem BOĞA

51

Fixed-Fixed End(k=1000000 lb-in./rad=constant)(n=number of disc)

Ι (lb-in./sec2)

0 10 20 30 40 50 60

ω1(

rad/

sec)

0

200

400

600

800

1000

1200

n=2

n=10

n

Fixed-Fixed End(k=1000000 lb-in./rad=constant)(n=number of disc)

Ι (lb-in./sec2)0 10 20 30 40 50 60

ω2(

rad/

sec)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

n=2

n=10

n

Fixed-Fixed End(k=1000000 lb-in./rad=constant)(n=number of disc)

Ι (lb-in./sec2)

0 10 20 30 40 50 60

ω3(

rad/

sec)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

n=3n=10

n

Figure 4.16. The effects of the inertia of the discs and the number of discs on the first three natural frequency for fixed-fixed ends

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4. RESULTS AND DISCUSSION Cem BOĞA

52

4.2.3. Effect of the mass of the shafts

As stated in the previous chapter, for the field transfer matrix for shafts

with/without mass the following is used in the vibration analysis (Pestel and Leckie,

1963).

Fwithout mass

=

=

10k11

10GJ

L1T

(4.6)

Fwith mass

−=

λλλ

λλ

λ

cossinL

GJ

sinGJ

Lcos

T

T (4.7)

where;

G.J.L.i

Tx

µωλ = (4.8)

For 0→λ , two matrices given above are identical. The same parametric

study given in (4.2.1-2) is performed here for the torsional system considering the

mass of the shaft in equation (4.7).

To change the inertia of the discs, just the radius of the disc (rd) is varied.

And, in order to alter the rigidity of the shafts, just the length of the shafts (L) is

modified. Other geometrical and material properties are kept constant in the analysis.

The mass of the disc is determined by (Figure 4.17)

dddd trm ρπ ××= 2)( (4.9)

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4. RESULTS AND DISCUSSION Cem BOĞA

53

Figure 4.17. Geometric shape of disc for mass moment of inertia

Where ρd is the density of disc material and td is the thickness. For the mass

moment of inertia for disc the following is used.

2

21

dd rmI = (4.10)

Denoting sr as the radius of the shaft, the torsional moment of inertia of the

shaft for uniform closed circular section is computed by (Figure 4.18)

4sT r.

21J π= A.iI 2

x0 == (4.11)

Figure 4.18. Geometric shape of shaft

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4. RESULTS AND DISCUSSION Cem BOĞA

54

Where ix is the radius of gyration about the x axis (shaft axis), and A is the

area of the cross section.

For ix we may have the following;

AI

ix0= =

AJT = 2

4

.2.

s

s

r

r

π

π

=2sr

(4.12)

Graphs are presented in Figures (4.19-4.24) for just the first and third

frequencies to see well the variation of the quantities in the same page. The similar

behavior in case that the effect of torsional rigidities and inertias of the discs are

changed is observed.

As expected, including the mass of the shaft in the mathematical model

reduces all the frequencies. The difference between the results for shafts with and

without mass is obvious for small torsional rigidities and small inertias of the discs.

4.3. Discussion

The method presented in this study may be used for the first step of the design

since it gives effective and economical results. However for more accurate results, it

is necessary to use the distributed parameter model for the analysis. It will be very

useful to include the bearing effects in the problem.

Apart from those, the bending and torsional vibrations may be studied in the

same analysis. That is coupled bending-torsional vibration analysis may be carried

out for both free and forced vibration problems instead uncoupled modes.

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4. RESULTS AND DISCUSSION Cem BOĞA

55

Free-Free EndL=26 in=constantn=number of disc

rd (in)8 10 12 14 16 18 20 22

ω2( r

ad/s

ec)

0

20

40

60

80

100

120

140

160

180

200

Shaft with massMassless shaft

n

n=2

n=10

Free-Free EndL=26 in=constantn=number of disc

rd (in)8 10 12 14 16 18 20 22

ω4(

rad/

sec)

0

50

100

150

200

250

300

Shaft with massMassless shaft

n

n=4

n=10

Figure 4.19. The effects of mass of the shaft on the variation of the first and third frequencies for free-free ends (L=constant)

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4. RESULTS AND DISCUSSION Cem BOĞA

56

Fixed-Free EndL=26 in=constantn=number of disc

rd (in)8 10 12 14 16 18 20 22

ω1

(rad

/sec

)

0

20

40

60

80

100

Shaft with massMassless shaft

n

n=2

n=10

Fixed-Free EndL=26 in=constantn=number of disc

rd (in)8 10 12 14 16 18 20 22

ω3 (

rad/

sec)

0

50

100

150

200

250

300

Shaft with massMassless shaft

n

n=3

n=10

Figure 4.20. The effects of mass of the shaft on the variation of the first and third frequencies for fixed-free ends (L=constant)

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4. RESULTS AND DISCUSSION Cem BOĞA

57

Fixed-Fixed EndL=26 in=constantn=number of disc

rd (in)8 10 12 14 16 18 20 22

ω1 (

rad/

sec)

0

20

40

60

80

100

120

140

Shaft with massMassless shaft

n

n=2

n=10

Fixed-Fixed EndL=26 in=constantn=number of disc

rd (in)8 10 12 14 16 18 20 22

ω3( r

ad/s

ec)

0

50

100

150

200

250

300

Shaft with massMassless shaft

n

n=3

n=10

Figure 4.21. The effects of mass of the shaft on the variation of the first and third frequencies for fixed-fixed ends (L=constant)

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4. RESULTS AND DISCUSSION Cem BOĞA

58

Free-Free Endrd=10 in=constantn=number of disc

L (in)15 20 25 30 35

ω2(

rad/

sec)

0

50

100

150

200

250

Shaft with massMassless shaft

n

n=2

n=10

Free-Free Endrd=10 in=constantn=number of disc

L(in)15 20 25 30 35

ω4

(rad

/sec

)

50

100

150

200

250

300

350

Shaft with massMassless shaft

n

n=4

n=10

Figure 4.22. The effects of mass of the shaft on the variation of the first and third frequencies for free-free ends (I=constant)

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4. RESULTS AND DISCUSSION Cem BOĞA

59

Fixed-Free Endrd=10 in=constantn=number of disc

L (in)15 20 25 30 35

ω1 (

rad/

sec)

0

20

40

60

80

100

120Shaft with massMassless shaft

n

Fixed-Free Endrd=10 in=constantn=number of disc

L (in)15 20 25 30 35

ω3 (

rad/

sec)

50

100

150

200

250

300

350

Shaft with massMassless shaft

n

n=3

n=10

Figure 4.23. The effects of mass of the shaft on the variation of the first and third frequencies for fixed-free ends (I=constant)

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4. RESULTS AND DISCUSSION Cem BOĞA

60

Fixed-Fixed Endrd=10 in=constantn=number of disc

L (in)15 20 25 30 35

ω1

(rad

/sec

)

20

40

60

80

100

120

140

160

180

Shaft with massMassless shaft

n

n=2

n=10

Fixed-Fixed Endrd=10 in=constantn=number of disc

L (in)15 20 25 30 35

ω3 (

rad/

sec)

50

100

150

200

250

300

350

Shaft with massMassless shaft

n

n=3

n=10

Figure 4.24. The effects of mass of the shaft on the variation of the first and third frequencies for fixed-fixed ends (I=constant)

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5. CONCLUSION Cem BOĞA

61

5. CONCLUSION

In this study, the free undamped torsional vibration analysis of multi shaft-

disc-systems is studied by using both the Transfer Matrix and Finite Element

methods with ANSYS. A computer program in Fortran code is developed for

torsional systems without bearings.

The free vibration problem is modeled by using the lumped-parameter system

in which the masses of the shafts are assumed to be negligible with respect to the

masses of discs and discs have no elastic properties. The cross section of the shafts is

taken as uniform and circular section. Materials of the shafts and discs are isotropic

and linear. Small deformation is considered in the formulation.

After verifying the results obtained in this study by comparing the analytical

results in the literature, a parametric study is performed. In the parametric study the

effects of the number of discs, the boundary conditions, and different shaft and disc

rigidities are considered and presented by charts. The effect of the mass of shafts on

the free vibration behavior is also studied. The applicability of the method to the

branched system is also demonstrated.

The results obtained from the present study are outlined below:

Insertion of the mass of the shafts in the analysis, decrease the natural

frequencies. This gives more accurate results.

While the torsional rigidity of the shafts is kept constant, increasing inertias

of the discs decreases the frequencies.

Increase in the torsional rigidity of the shaft increases the frequencies in cases

that the inertia of the discs is kept constant.

When the number of the discs increases, the frequencies decrease.

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5. CONCLUSION Cem BOĞA

62

The transfer matrix method is found as the practical, effective and economical

for also complicated systems. This method is also used the natural frequencies for

coupled modes. That is finding the critical speeds from the bending vibration may be

added to the problem together with torsional and axial vibrations.

The present study may be developed by using distributed-parameter model in

the analysis. The shafts with non-uniform or tapered cross sections may be

considered to expand the study. The type of the material may be changed such as

composites. And finally a nonlinear analysis may be performed.

As is well known, for a complete analysis of the whole system, damping

effects due to bearings should also be included in the problem.

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63

REFERENCES AKDAĞ, M., and SABUNCU, M., 2001. Çoklu Disk-Mil Sistemlerinin Serbest

Titreşimlerinin Transfer Matrisi Yöntemi ile Analizi. 10. Ulusal Makine

Teorisi Sempozyumu, Konya, 785-793.

AKDAĞ, M., 2002. The Free Vibration of Multi Disc-Shaft Systems. Master Thesis,

Dokuz Eylül Üniversitesi, FBE, 82p.

ALEYAASIN M., and EBRAHIMI, M., 2000. Hybrid Modelling for Analysis

and Identification of Rotors. Computer Methods in Applied Mechanics and

Engineering, 182/1-2, 163-176.

ALEYAASIN, M., EBRAHIMI M., and WHALLEY, R., 2000a. Multivariable

Hybrid Models for Rotor-Bearing Systems. Journal of Sound and Vibration,

233/5, 835-856.

ALEYAASIN, M., EBRAHIMI M., and WHALLEY, R., 2000b. Vibration Analysis

of Distributed-Lumped Rotor Systems. Computer Methods in Applied

Mechanics and Engineering, 89/2, 545-558.

ALEYAASIN, M., EBRAHIMI M., and WHALLEY, R., 2001. Flexural Vibration of

Rotating Shafts by Frequency Domain Hybrid Modelling. Computers &

Structures, 79/3, 319-331.

AŞÇIOĞLU, B., 2002. Rotor Sistemlerinin Sonlu Elemanlar Yöntemi

ile İncelenmesi. Master Tezi, KTÜ, FBE, 95s.

BELEK, H. T., 1984. Kritik Hızların Sonlu Elemanlar-Transfer Matris Yöntemiyle

Hesaplanması. 1. Ulusal Makine Teorisi Sempozyumu, 349-358.

BHUTANI, N., and LOEWY, R., G., 1999. Combined Finite Element-Transfer

Matrix Method. Journal of Sound and Vibration, 226/5, 1048-1052.

CHEN, W. D., 2006. An Exact Solution for Free Torsional Vibration of a Uniform

Circular Shaft Carrying Multiple Concentrated Elements. Journal of Sound

Vibration, 291, 627-643.

DAWSON, B., and DAVIES, M., 1981. An Improvement to Shaikh's Method for

Torsional Vibration Analysis of Branched System. The Shock and Vibration

Bulletin, 51, 1–10.

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64

DEN HARTOG, J. P., and LI, J. P., 1946. Forced Torsional Vibrations with

Damping:An Extension of Holzer's Method. Journal of Applied Mechanics,

Transactions of ASME, 68, 276–280.

DOUGHTY, S., and VAFAEE, G., 1985. Transfer Matrix Eigen Solutions for

Damped Torsional Systems. Transactions of ASME, Journal of Vibration,

Acoustics, Stress and Reliability in Design, 107, 128–132.

GÜRGÖZE, M., 2006. On Some Relationships between the Eigenfrequencies of

Torsional Vibrational Systems Containing Lumped Elements. Journal

of Sound and Vibration, 290, 1322 – 1332.

GILBERT, A. C., 1972. A Note on the Calculation of Torsional Natural Frequencies

of Branch Systems. Journal of Engineering and Industrial Transactions of

ASME, 94, 279–.

HOLZER, H., 1921. Analysis of Torsional Vibration. Springer, Berlin.

KANG, Y., LEE, Y.G., and CHEN, S.C., 1997. Instability Analyis of Unsymmetrical

Rotor-Bearing Systems Using Transfer Matrix Method. Journal of Sound and

Vibration, 199/3, 381-400.

PASİN, F., 1988. Mekanik Titreşimler Ders Notları. İ.T.Ü. Rektörlük Matbaası,

İstanbul, No 21, 151p.

PESTEL, E. C., and LECKIE, F. A., 1963. Matrix Methods in Elastomechanics.

McGraw-Hill, USA, 435p.

SANKAR, S., 1979. On the Torsional Vibration of Branched Systems Using

Extended Transfer Matrix Method. Journal of Mechanical Design,

Transactions of ASME, 101, 546–53.

SETO, W. W., 1964. Theory and Problems of Mechanical Vibrations. McGraw-Hill,

USA, 199p.

SHAIKH, N., 1974. A Direct Method for Analysis of Branched Torsional System.

Journal of Engineering and Industrial Transactions of ASME, 96,

1006–1009. SPAETGENS, T. W., and VANCOUVER, B. C., 1950. Holzer Method for Forced-

Damped Torsional Vibrations. Journal of Applied Mechanics, 17, 59–63.

SREERAM, T. R., 2005. Graph Theory Based Parametric Influences Applied to

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65

Torsional Vibration Analysis. Advances in Engineering Software, 36/4,

209-224.

TSAI, T. C., and WANG, Y. Z., 1997. The Vibration of a Multi-Crack Rotor. Ins. J.

Mech. Sci., 39/9, 1037-1053.

TSE, F. S., MORSE, I. E., and HINKLE, R. T., 1978. Mechanical Vibrations Theory

and Applications. Allyn and Bacon, USA, 449p.

WU, J. S., and CHEN, C. H., 2001. Torsional Vibration Analysis of Ear-Branched

Systems by Finite Element Method. Journal of Sound and Vibration,

240/1, 159-182.

YARDIMCIOĞLU, B., and SABUNCU, M., 1993. Şaft-Disk Sistemlerinin Serbest

Titreşim ve Statik Stabilite Analizi. VIII. Ulusal Mekanik Kongresi, 639-648.

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66

CURRICULUM VITAE

Cem BOĞA was born in ADANA in 1978. He graduated from Adana Erkek

High School in 1994. He enrolled in Mechanical Engineering Department of

Çukurova University in 1996. He graduated from the same university as a

mechanical engineer in 2000. He has been working at Çukurova University

Vocational School of Karaisali since 2003.

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APPENDIX A Cem BOĞA

67

Appendix A. Transfer Matrix for the Torsional Vibration of an Elastic Shaft (Pestel and Leckie, 1963)

Figure A.1. Shaft element under torsion (Pestel and Leckie, 1963)

Figure (A.1) illustrates the force system acting on the shaft and the

displacements of an element of length dx of the shaft. The inertia couple is

φωµ 2)2( dxxi , where xi is the radius of gyration about the x axis and µ is the mass per

unit length. From the equilibrium condition we obtain the equation;

0idxdT 22

x =+ φµω (A.1)

and from the elastic properties;

GJT

dxd

T

=φ (A.2)

where GJT is the torsional stiffness of the shaft. The elimination of T gives the

second-order differential equation in φ :

0GJ

idxd

T

22x

2

2

=+ φωµφ (A.3)

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APPENDIX A Cem BOĞA

68

the solution of which is, for a shaft portion of length L (Figure A.2);

)cos()sin(LxB

LxA λλφ += (A.4)

where;

GJLi

T

222x2 ωµ

λ = (A.5)

G.J.L.i

Tx

µωλ = (A.6)

At 0=x , which coincides with point 1−i on the shaft, the boundary

conditions are 1−= iφφ and 1−= iTT , from which we obtain;

1iB −= φ (A.7)

and;

GJLT

AT

i

λ1−= (A.8)

The solution of Equation (A.3) is then;

)sin()cos( 11 Lx

GJLT

Lx

Tii λ

λλφφ −− += (A.9)

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APPENDIX A Cem BOĞA

69

Figure A.2. Shaft under torsion (Pestel and Leckie, 1963)

and the expression for T is;

)cos()sin( 11 LxT

Lx

LGJ

T iT

i λλλ

φ −− +−= (A.10)

at Lx = , coinciding with point i of the shaft,

λλ

λφφφ sincos 11 GJLTT

iii −− +== (A.11)

λλλ

φ cossin 11 −− +−== iT

ii TL

GJTT (A.12)

Expressed in matrix form, these two equations become;

1

.cossin

sincos

−=

iT

T

i TL

GJGJ

L

λλλ

λλ

λφ (A.13)

or;

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APPENDIX A Cem BOĞA

70

1. −= iii zUz (A.14)

If the shaft is massless (that is, 0→λ ), Equation (A.14) reduces to the form;

1

.10

1

=

iT

i TGJL

Tφφ

(A.15)

which is identical to Equation (A.15). In equation (A.15), JT is the torsional second

moment of area of the shaft.

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APPENDIX B Cem BOĞA

71

Appendix B.1. A summary of element input for COMBIN14.

Table B.1. COMBIN14 Input summary

Element Name COMBIN14 Nodes JI ,

Degrees of freedom

UZUYUX ,, if KEYOPT(3) = 0 ROTZROTYROTX ., if KEYOPT(3) = 1

UYUX , if KEYOPT(3) =2 see list below if KEYOPT(2)>0

Real Constants 2,1, CVCVK Material Properties None

Surface Loads None Body Loads None

Special Features

Nonlinear (if 2CV is not zero), Stress stiffening, Large deflection, Birth and death

KEYOPT(1)

0 Linear Solution (default) 1 Nonlinear solution (required if 2CV is non-zero)

KEYOPT(2)

0 Use KEYOPT(3) options 1 1-D longitudinal spring-damper (UX degree of freedom) 2 1-D longitudinal spring-damper (UY degree of freedom) 3 1-D longitudinal spring-damper (UZ degree of freedom) 4 1-D torsional spring-damper ( ROTX degree of freedom) 5 1-D torsional spring-damper ( ROTY degree of freedom) 6 1-D torsional spring-damper ( ROTZ degree of freedom) 7 Pressure degree of freedom element 8 Temperature degree of freedom element

KEYOPT(3)

0 3-D longitudinal spring-damper 1 3-D torsional spring-damper 2 2-D longitudinal spring-damper(2-D elements must lie in a X-Y plane)

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Appendix B.2. A summary of element input for MASS21.

Table B.2. MASS21 Input summary

Element Name MASS21 Nodes I

Degrees of freedom

ROTZROTYROTXUZUYUX .,,,, if KEYOPT(3) = 0

UZUYUX ,, if KEYOPT(3) = 2 ROTZUYUX ,, if KEYOPT(3) =3

UYUX , if KEYOPT(3) =4 (degrees of freedom are in the nodal coordinate system)

Real Constants

IZZIYYIXXMASSZMASSYMASSX ,,,,,if KEYOPT(3)=0 MASS if KEYOPT(3)=2 MASS , IZZ if KEYOPT(3)=3 MASS if KEYOPT(3)=4 (Mass and moment of inertia directions are in the element coordinate system, see KEYOPT(2))

Material Properties None Surface Loads None

Body Loads None Special Features Large deflection, Birth and death

KEYOPT(2)

0 Element coordinate system is initially parallel to the global Cartesian coordinate system

1 Element coordinate system is initially parallel to the global nodal coordinate system

KEYOPT(3)

0 3-D mass with rotary inertia 2 3-D mass without rotary inertia 3 2-D mass with rotary inertia 4 2-D mass without rotary inertia

Note: all 2-D elements are assumed to be in the global Cartesian X – Y plane

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73

Appendix C.1. The program written in Fortran for massless shaft. PROGRAM SHAFT3 PARAMETER(N=2) IMPLICIT REAL*8 (A-H,O-Z) INTEGER*2 TOPMIL,TOPDISK CHARACTER*8 VERI,out1 DIMENSION FRE(50),FR(50) write(*,*) 'VERI DOSYASI ISMI=? (MAX 8 KARAKTER OLACAK)' READ(*,*) VERI write(*,*) 'CIKTI DOSYASI ISMI=? (MAX 8 KARAKTER OLACAK)' READ(*,*) out1 OPEN(5,FILE=VERI,FORM='FORMATTED',STATUS='OLD') OPEN(6,FILE=out1,FORM='FORMATTED',STATUS='NEW') OPEN(7,FILE='OUT',FORM='FORMATTED',STATUS='NEW') write(*,*) 'disk sayisi=? , mil sayisi=?' READ(*,*) TOPDISK,TOPMIL WRITE(7,*) ' w(rad/s)',' w(hertz)',' *determinant' WRITE(7,*) ' ' write(*,*) 'sinir sarti=?' write(*,*) '1==>bos-bos, 2==>ankastre-bos, 3=ankastre-ankastre' read(*,*) nb if(nb.eq.1) then ifr=topdisk-1 else ifr=topdisk endif IF(NB.EQ.1) THEN WRITE(6,*) 'BOS-BOS TL=0 T0=0' ELSE IF(NB.EQ.2) THEN WRITE(6,*) 'ANKASTRE-BOS TL=0 FI0=0' ELSE WRITE(6,*) 'ANKASTRE-ANKASTRE FI0=0 FIL=0' ENDIF PI=3.14159265898D0 ZHER0=0.0000001D0 ZHER=ZHER0 DELZH1=0.1d0 DO 4000 II=1,10 4000 FRE(II)=0.D0 ISAY=0 do 950 III=1,100000000 CALL DET(TOPDISK,TOPMIL,NB,ZHER,ZET) zz=zher/(2.*pi) WRITE(7,*) ZHER, zz, ZET

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74

IF(ZHER.EQ.ZHER0) THEN ZHER1=ZHER ZHER2=ZHER ZET1=ZET ZET2=ZET GOTO 9998 ELSE ZHER1=ZHER2 ZHER2=ZHER ZET1=ZET2 ZET2=ZET ENDIF WRITE(*,*) III,ZET,ZHER IF(ZET1.LT.0)THEN NZET1=1 ELSE NZET1=2 ENDIF IF(ZET2.LT.0)THEN NZET2=1 ELSE NZET2=2 ENDIF IF(NZET1.EQ.NZET2) THEN GOTO 9998 ELSE ZILK=ZHER1 ZHER=ZILK ZSON=ZHER2 DELARA=(ZSON-ZILK)/9999. ENDIF DO 9501 IIII=1,10000 CALL DET(TOPDISK,TOPMIL,NB,ZHER,ZET) IF(ZHER.EQ.ZILK) THEN ZHER3=ZHER ZHER4=ZHER ZET3=ZET ZET4=ZET GOTO 9000 ELSE ZHER3=ZHER4 ZHER4=ZHER ZET3=ZET4 ZET4=ZET ENDIF

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IF(ZET3.LT.0)THEN NZET3=1 ELSE NZET3=2 ENDIF IF(ZET4.LT.0)THEN NZET4=1 ELSE NZET4=2 ENDIF IF(NZET3.EQ.NZET4) THEN GOTO 9000 ELSE ISAY=ISAY+1 WRITE(*,*)'ISAY= ',ISAY ENDIF FRE(ISAY)=ZHER IF(ISAY.EQ.IFR) GOTO 2000 9000 ZHER=ZHER+DELARA 9501 CONTINUE 9998 ZHER=ZHER+DELZH1 950 CONTINUE 2000 write(6,*) ' rad/s',' Hz' write(6,*) ' ' DO 898 II=1,ifr FR(II)=FRE(II)/(2*PI) 898 write(6,1258) fre(ii),fr(ii) 1258 FORMAT(1X,E20.9,1X,E20.9) do 33 ii=1,ifr wilk=fre(ii) 33 CALL EIGENVECTOR(WILK,TOPMIL,TOPDISK,nb) CLOSE (7, STATUS = 'DELETE') STOP END SUBROUTINE MATMUL (A,B,C,IC,JC,KC) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION A(IC,JC),B(JC,KC),C(IC,KC) DO 11 I=1,IC DO 11 K=1,KC C(I,K)=0.0D0 DO 11 J=1,JC 11 C(I,K)=C(I,K)+A(I,J)*B(J,K) RETURN END

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SUBROUTINE DET(TOPDISK,TOPMIL,NB,ZHER,ZET) IMPLICIT REAL*8 (A-H,O-Z) INTEGER*2 TOPMIL,TOPDISK REAL*8 IDISK,L,J,KRIJ DIMENSION U(2,2),P(2,2),F(2,2),C(2,2),ARA(2,2),ARA1(2,2) REWIND 5 ITOP=TOPDISK IF(NB.EQ.3) ITOP=TOPMIL DO 111 I=1,ITOP N=2 READ(5,*) IDISK READ(5,*) KRIJ DO 14 II=1,N DO 14 JJ=1,N P(II,JJ)=0.D0 ara(ii,jj)=0.d0 14 F(II,JJ)=0.D0 P(1,1)=1.D0 P(2,1)=-ZHER*ZHER*IDISK p(2,2)=1.d0 F(1,1)=1.D0 F(2,2)=1.D0 IF ((I.EQ.ITOP).AND.(NB.EQ.1)) GOTO 11 F(1,2)=1.D0/KRIJ GO TO 12 11 F(1,2)=0.D0 12 IF ((I.EQ.1).AND.(NB.EQ.3)) p(2,1)=0.d0 CALL MATMUL(P,F,C,2,2,2) if ((i.eq.1).AND.(TOPDISK.EQ.1)) goto 98 DO 18 II=1,N DO 18 JJ=1,N 18 ARA(II,JJ)=C(II,JJ) if (i.eq.1) then nc=1 else if (i.eq.2) then nc=2 else if (i.gt.2) then nc=3 endif goto (101,102,103),nc 101 DO 19 II=1,N DO 19 JJ=1,N 19 ARA1(II,JJ)=C(II,JJ) goto 111 102 CALL MATMUL(ara1,ara,c,2,2,2) DO 166 II=1,N

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DO 166 JJ=1,N 166 U(II,JJ)=C(II,JJ) goto 111 103 CALL MATMUL(U,ARA,C,2,2,2) DO 16 II=1,N DO 16 JJ=1,N 16 U(II,JJ)=C(II,JJ) goto 111 98 DO 20 II=1,N DO 20 JJ=1,N 20 U(II,JJ)=C(II,JJ) 111 CONTINUE IF (NB.EQ.1) THEN ZET=U(2,1) ELSE IF (NB.EQ.2) THEN ZET=U(2,2) ELSE ZET=U(1,2) ENDIF RETURN END SUBROUTINE EIGENVECTOR(W,TOPMIL,TOPDISK,nb) IMPLICIT REAL*8 (A-H,O-Z) INTEGER*2 TOPMIL,TOPDISK REAL*8 IDISK,L,J,KRIJ DIMENSION U(2,2),P(2,2),F(2,2),C(2,2),ARA(2,2) DIMENSION PP(50,2,2),FF(50,2,2),FI(50) WRITE(6,*) ' ' WRITE(6,*) 'OZEL VEKTOR ==> w=' , w , ' rad/s icin' WRITE(6,*) ' ' REWIND 5 n=2 jjj=topdisk ITOP=TOPDISK IF (NB.EQ.3) ITOP=TOPMIL do 999 III=1,ITOP READ(5,*) IDISK READ(5,*) KRIJ DO 14 II=1,N DO 14 JJ=1,N P(II,JJ)=0.D0 14 F(II,JJ)=0.D0 P(1,1)=1.D0 P(2,1)=-W*W*IDISK p(2,2)=1.d0 F(1,1)=1.D0

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F(2,2)=1.D0 IF ((III.EQ.ITOP).AND.(NB.EQ.1)) GO TO 11 F(1,2)=1.D0/KRIJ GO TO 12 11 F(1,2)=0.D0 12 IF ((III.EQ.1).AND.(NB.EQ.3)) P(2,1)=0.d0 do 88 ii=1,2 do 88 jj=1,2 PP(jjj,ii,jj)=P(ii,jj) 88 FF(jjj,ii,jj)=F(ii,jj) 999 jjj=topdisk-iii DO 31 II=1,2 DO 31 JJ=1,2 P(II,JJ)=PP(1,II,JJ) 31 F(II,JJ)=FF(1,II,JJ) CALL MATMUL(P,F,U,2,2,2) IF (NB.EQ.1) then FI(1)=1.d0 else fi(1)=u(1,2) endif DO 111 I=2,itop DO 1 II=1,2 DO 1 JJ=1,2 P(II,JJ)=PP(I,II,JJ) 1 F(II,JJ)=FF(I,II,JJ) CALL MATMUL(P,F,C,2,2,2) DO 2 II=1,2 DO 2 JJ=1,2 2 ARA(II,JJ)=C(II,JJ) CALL MATMUL(ARA,U,C,2,2,2) DO 25 II=1,2 DO 25 JJ=1,2 25 U(II,JJ)=C(II,JJ) if (nb.eq.1) then FI(I)=u(1,1) ELSE FI(I)=U(1,2) endif 111 CONTINUE DO 77 KLM=1,itop 77 WRITE(6,*) 'FI(',KLM,')=',FI(KLM) WRITE(6,*) ' ' RETURN END

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79

Appendix C.2. Input files

1. Input file for Figure 4.1

1 1 1 1 1 1

2. Input file for Figure 4.3

1000 10000000 2000 10000000 4000 30000000

3. Input file for Figure 4.6

10 1000 10 1000 10 1000 10 1000

4. Input file for Figure 4.8.-b

45 0.9 10 0.9

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Appendix C.3. Output files

1. Output file for Figure 4.1

BOS-BOS TL=0 T0=0 rad/s Hz 0.100000010E+01 0.159154959E+00 0.173205331E+01 0.275664845E+00 OZEL VEKTOR ==> w= 1.00000009999947 rad/s icin FI( 1)= 1.00000000000000 FI( 2)= -1.999989558409965E-007 FI( 3)= -1.00000019999892 OZEL VEKTOR ==> w= 1.73205330532019 rad/s icin FI( 1)= 1.00000000000000 FI( 2)= -2.00000865247060 FI( 3)= 1.00002595748668

2. Output file for Figure 4.3

ANKASTRE-BOS TL=0 FI0=0 rad/s Hz 0.457635965E+02 0.728350258E+01 0.100000000E+03 0.159154943E+02 0.133812161E+03 0.212968669E+02 OZEL VEKTOR ==> w= 45.7635964596311 rad/s icin FI( 1)= 3.333333333333333E-008 FI( 2)= 1.054092431877339E-007 FI( 3)= 1.333332949078291E-007 OZEL VEKTOR ==> w= 100.000000100053 rad/s icin FI( 1)= 3.333333333333333E-008 FI( 2)= -2.668080991957141E-016 FI( 3)= -3.333333333333333E-008 OZEL VEKTOR ==> w= 133.812161316180 rad/s icin FI( 1)= 3.333333333333333E-008 FI( 2)= -1.054092602147664E-007 FI( 3)= 1.333333487520317E-007

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3. Output file for Figure 4.6 ANKASTRE-ANKASTRE FI0=0 FIL=0 rad/s Hz 0.765367547E+01 0.121812028E+01 0.141421443E+02 0.225079217E+01 0.184775979E+02 0.294080103E+01 OZEL VEKTOR ==> w= 7.65367547 rad/s icin FI( 1)= 0.001 FI( 2)= 0.001414 FI( 3)= 0.001 OZEL VEKTOR ==> w= 14.1421443 rad/s icin FI( 1)= 0.001 FI( 2)= -2.668080991957141E-016 FI( 3)= -0.001 OZEL VEKTOR ==> w= 18.47759 rad/s icin FI( 1)= 0.001 FI( 2)= -0.001414 FI( 3)= -0.001

4. Output file for Figure 4.8.-b

BOS-BOS TL=0 T0=0 rad/s Hz 0.331663266E+00 0.527858482E-01 OZEL VEKTOR ==> w= 0.331663266316641 rad/s icin FI( 1)= 1.00000000000000 FI( 2)= -0.222228024709143

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APPENDIX D Cem BOĞA

82

Appendix D. The program written in Fortran for shaft with mass. PROGRAM SHAFT3 PARAMETER(N=2) IMPLICIT REAL*8 (A-H,O-Z) INTEGER*2 TOPMIL,TOPDISK CHARACTER*8 VERI,out1 DIMENSION FRE(50),FR(50) write(*,*) 'VERI DOSYASI ISMI=? (MAX 8 KARAKTER OLACAK)' READ(*,*) VERI write(*,*) 'CIKTI DOSYASI ISMI=? (MAX 8 KARAKTER OLACAK)' READ(*,*) out1 OPEN(5,FILE=VERI,FORM='FORMATTED',STATUS='OLD') OPEN(6,FILE=out1,FORM='FORMATTED',STATUS='NEW') OPEN(7,FILE='OUT',FORM='FORMATTED',STATUS='NEW') c out dosyasi determinant grafigi cizmek icin kullanilacak C TOPDISK=TOPLAM DISK SAYISI C TOPMIL=TOPLAM MIL SAYISI write(*,*) 'disk sayisi=? , mil sayisi=?' READ(*,*) TOPDISK,TOPMIL WRITE(7,*) ' w(rad/s)',' w(hertz)',' *determinant' WRITE(7,*) ' ' write(*,*) 'sinir sarti=?' write(*,*) '1==>bos-bos, 2==>ankastre-bos, 3=ankastre-ankastre' read(*,*) nb c ifr=istenilen frekans sayisi if(nb.eq.1) then ifr=topdisk-1 else ifr=topdisk endif IF(NB.EQ.1) THEN WRITE(6,*) 'BOS-BOS TL=0 T0=0' ELSE IF(NB.EQ.2) THEN WRITE(6,*) 'ANKASTRE-BOS TL=0 FI0=0' ELSE WRITE(6,*) 'ANKASTRE-ANKASTRE FI0=0 FIL=0' ENDIF c zher=w (rad/s) PI=3.14159265898D0 ZHER0=0.0000001D0 ZHER=ZHER0 DELZH1=0.1d0 DO 4000 II=1,10 4000 FRE(II)=0.D0

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ISAY=0 do 950 III=1,100000000 CALL DET(TOPDISK,TOPMIL,NB,ZHER,ZET) zz=zher/(2.*pi) WRITE(7,*) ZHER, zz, ZET IF(ZHER.EQ.ZHER0) THEN ZHER1=ZHER ZHER2=ZHER ZET1=ZET ZET2=ZET GOTO 9998 ELSE ZHER1=ZHER2 ZHER2=ZHER ZET1=ZET2 ZET2=ZET ENDIF WRITE(*,*) III,ZET,ZHER C NZET=1 ===> NEGATIF NZET=2 ====> POZITIF IF(ZET1.LT.0)THEN NZET1=1 ELSE NZET1=2 ENDIF IF(ZET2.LT.0)THEN NZET2=1 ELSE NZET2=2 ENDIF IF(NZET1.EQ.NZET2) THEN GOTO 9998 ELSE ZILK=ZHER1 ZHER=ZILK ZSON=ZHER2 DELARA=(ZSON-ZILK)/9999. ENDIF DO 9501 IIII=1,10000 CALL DET(TOPDISK,TOPMIL,NB,ZHER,ZET) C.....TEKRAR ISARET DEGISIMI OLUP OLMADIGI KONTROL EDILECEK.... IF(ZHER.EQ.ZILK) THEN ZHER3=ZHER ZHER4=ZHER ZET3=ZET ZET4=ZET GOTO 9000

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ELSE ZHER3=ZHER4 ZHER4=ZHER ZET3=ZET4 ZET4=ZET ENDIF IF(ZET3.LT.0)THEN NZET3=1 ELSE NZET3=2 ENDIF IF(ZET4.LT.0)THEN NZET4=1 ELSE NZET4=2 ENDIF IF(NZET3.EQ.NZET4) THEN GOTO 9000 ELSE ISAY=ISAY+1 WRITE(*,*)'ISAY= ',ISAY ENDIF FRE(ISAY)=ZHER IF(ISAY.EQ.IFR) GOTO 2000 9000 ZHER=ZHER+DELARA 9501 CONTINUE 9998 ZHER=ZHER+DELZH1 950 CONTINUE 2000 write(6,*) ' rad/s',' Hz' write(6,*) ' ' DO 898 II=1,ifr FR(II)=FRE(II)/(2*PI) 898 write(6,1258) fre(ii),fr(ii) 1258 FORMAT(1X,E20.9,1X,E20.9) do 33 ii=1,ifr wilk=fre(ii) 33 CALL EIGENVECTOR(WILK,TOPMIL,TOPDISK,nb) CLOSE (7, STATUS = 'DELETE') STOP END SUBROUTINE MATMUL (A,B,C,IC,JC,KC) IMPLICIT REAL*8 (A-H,O-Z) DIMENSION A(IC,JC),B(JC,KC),C(IC,KC)

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DO 11 I=1,IC DO 11 K=1,KC C(I,K)=0.0D0 DO 11 J=1,JC 11 C(I,K)=C(I,K)+A(I,J)*B(J,K) RETURN END SUBROUTINE DET(TOPDISK,TOPMIL,NB,ZHER,ZET) IMPLICIT REAL*8 (A-H,O-Z) INTEGER*2 TOPMIL,TOPDISK REAL*8 IDISK,L,J,G,KRIJ,LAMDA,MU,GYR DIMENSION U(2,2),P(2,2),F(2,2),C(2,2),ARA(2,2),ARA1(2,2) REWIND 5 ITOP=TOPDISK IF(NB.EQ.3) ITOP=TOPMIL DO 111 I=1,ITOP N=2 READ(5,*) IDISK READ(5,*) L,J,G,MU,GYR LAMDA=GYR*L*ZHER*SQRT(MU/(J*G)) DO 14 II=1,N DO 14 JJ=1,N P(II,JJ)=0.D0 ara(ii,jj)=0.d0 14 F(II,JJ)=0.D0 P(1,1)=1.D0 P(2,1)=-ZHER*ZHER*IDISK p(2,2)=1.d0 F(1,1)=COS(LAMDA) F(2,2)=COS(LAMDA) IF ((I.EQ.ITOP).AND.(NB.EQ.1)) GOTO 11 F(1,2)=L*SIN(LAMDA)/(LAMDA*J*G) F(2,1)=-LAMDA*J*G*SIN(LAMDA)/L GO TO 12 11 F(1,2)=0.D0 12 IF ((I.EQ.1).AND.(NB.EQ.3)) p(2,1)=0.d0 CALL MATMUL(P,F,C,2,2,2) if ((i.eq.1).AND.(TOPDISK.EQ.1)) goto 98 DO 18 II=1,N DO 18 JJ=1,N 18 ARA(II,JJ)=C(II,JJ) if (i.eq.1) then nc=1 else if (i.eq.2) then

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nc=2 else if (i.gt.2) then nc=3 endif goto (101,102,103),nc 101 DO 19 II=1,N DO 19 JJ=1,N 19 ARA1(II,JJ)=C(II,JJ) goto 111 102 CALL MATMUL(ara1,ara,c,2,2,2) DO 166 II=1,N DO 166 JJ=1,N 166 U(II,JJ)=C(II,JJ) goto 111 103 CALL MATMUL(U,ARA,C,2,2,2) DO 16 II=1,N DO 16 JJ=1,N 16 U(II,JJ)=C(II,JJ) goto 111 98 DO 20 II=1,N DO 20 JJ=1,N 20 U(II,JJ)=C(II,JJ) 111 CONTINUE c NB=1 ........... BOS-BOS c NB=2 ........... ANKASTRE-B0S c NB=3 ........... ANKASTRE-ANKASTRE IF (NB.EQ.1) THEN ZET=U(2,1) ELSE IF (NB.EQ.2) THEN ZET=U(2,2) ELSE ZET=U(1,2) ENDIF RETURN END SUBROUTINE EIGENVECTOR(W,TOPMIL,TOPDISK,nb) IMPLICIT REAL*8 (A-H,O-Z) INTEGER*2 TOPMIL,TOPDISK REAL*8 IDISK,L,J,G,MU,GYR,LAMDA,KRIJ DIMENSION U(2,2),P(2,2),F(2,2),C(2,2),ARA(2,2) DIMENSION PP(50,2,2),FF(50,2,2),FI(50) WRITE(6,*) ' ' WRITE(6,*) 'OZEL VEKTOR ==> w=' , w , ' rad/s icin' WRITE(6,*) ' '

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REWIND 5 n=2 jjj=topdisk ITOP=TOPDISK IF (NB.EQ.3) ITOP=TOPMIL do 999 III=1,ITOP READ(5,*) IDISK READ(5,*) L,J,G,MU,GYR C READ(5,*) KRIJ LAMDA=GYR*L*W*SQRT(MU/(J*G)) DO 14 II=1,N DO 14 JJ=1,N P(II,JJ)=0.D0 14 F(II,JJ)=0.D0 P(1,1)=1.D0 P(2,1)=-W*W*IDISK p(2,2)=1.d0 F(1,1)=COS(LAMDA) F(2,2)=COS(LAMDA) IF ((III.EQ.ITOP).AND.(NB.EQ.1)) GO TO 11 F(1,2)=L*SIN(LAMDA)/(LAMDA*J*G) F(2,1)=-LAMDA*J*G*SIN(LAMDA)/L GO TO 12 11 F(1,2)=0.D0 12 IF ((III.EQ.1).AND.(NB.EQ.3)) P(2,1)=0.d0 do 88 ii=1,2 do 88 jj=1,2 PP(jjj,ii,jj)=P(ii,jj) 88 FF(jjj,ii,jj)=F(ii,jj) 999 jjj=topdisk-iii c sinir sarti bos-bos FI=U11 c.....P1*F1 OLUSTURULUYOR DO 31 II=1,2 DO 31 JJ=1,2 P(II,JJ)=PP(1,II,JJ) 31 F(II,JJ)=FF(1,II,JJ) CALL MATMUL(P,F,U,2,2,2) IF (NB.EQ.1) then FI(1)=1.d0 else fi(1)=u(1,2) endif DO 111 I=2,itop DO 1 II=1,2 DO 1 JJ=1,2

Page 101: ÇUKUROVA UNIVERSITY INSTITUTE OF NATURAL AND …transfer matrisi yöntemi, sonlu elemanlar yöntemi ve Holzer yöntemi gibi sayısal ve analitik yöntemlerle çalışılabilir. Bu

APPENDIX D Cem BOĞA

88

P(II,JJ)=PP(I,II,JJ) 1 F(II,JJ)=FF(I,II,JJ) CALL MATMUL(P,F,C,2,2,2) DO 2 II=1,2 DO 2 JJ=1,2 2 ARA(II,JJ)=C(II,JJ) CALL MATMUL(ARA,U,C,2,2,2) DO 25 II=1,2 DO 25 JJ=1,2 25 U(II,JJ)=C(II,JJ) if (nb.eq.1) then c FIL=U11.FI0+U12.MO=u11.FI0, FI0=1 FI(I)=u(1,1) ELSE c FIL=U11.FI0+U12.MO=u12.M0 FI(I)=U(1,2) endif 111 CONTINUE DO 77 KLM=1,itop 77 WRITE(6,*) 'FI(',KLM,')=',FI(KLM) WRITE(6,*) ' ' RETURN END