Çukurova university institute of natural and …transfer matrisi yöntemi, sonlu elemanlar yöntemi...
TRANSCRIPT
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
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.
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
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
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.
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
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
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)
VII
{z} State Vector
φ Angle of Twist
µ Mass of Unit Length of Shaft
dρ Density of Disc Material
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
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
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
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
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
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.
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
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
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
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.
3. MATERIAL AND METHOD Cem BOĞA
7
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.
3. MATERIAL AND METHOD Cem BOĞA
8
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.
3. MATERIAL AND METHOD Cem BOĞA
9
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;
3. MATERIAL AND METHOD Cem BOĞA
10
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;
3. MATERIAL AND METHOD Cem BOĞA
11
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);
3. MATERIAL AND METHOD Cem BOĞA
12
1
.cossin
sincos
−
−=
i
R
T
T
L
i TL
GJGJ
L
Tφ
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)
3. MATERIAL AND METHOD Cem BOĞA
13
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.
3. MATERIAL AND METHOD Cem BOĞA
14
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)
3. MATERIAL AND METHOD Cem BOĞA
15
The explicit form of equation (3.19) is;
=
2221
1211R
4
4
UUUU
Tφ
. 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.
3. MATERIAL AND METHOD Cem BOĞA
16
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)
3. MATERIAL AND METHOD Cem BOĞA
17
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)
3. MATERIAL AND METHOD Cem BOĞA
18
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).
3. MATERIAL AND METHOD Cem BOĞA
19
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.
3. MATERIAL AND METHOD Cem BOĞA
20
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)
3. MATERIAL AND METHOD Cem BOĞA
21
[ ]
−
−=
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.
3. MATERIAL AND METHOD Cem BOĞA
22
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.
3. MATERIAL AND METHOD Cem BOĞA
23
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 .
3. MATERIAL AND METHOD Cem BOĞA
24
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
3. MATERIAL AND METHOD Cem BOĞA
25
/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)
3. MATERIAL AND METHOD Cem BOĞA
26
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.
3. MATERIAL AND METHOD Cem BOĞA
27
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.
3. MATERIAL AND METHOD Cem BOĞA
28
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.
3. MATERIAL AND METHOD Cem BOĞA
29
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)
3. MATERIAL AND METHOD Cem BOĞA
30
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).
3. MATERIAL AND METHOD Cem BOĞA
31
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
4. RESULTS AND DISCUSSION Cem BOĞA
32
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.
4. RESULTS AND DISCUSSION Cem BOĞA
33
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
4. RESULTS AND DISCUSSION Cem BOĞA
34
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
4. RESULTS AND DISCUSSION Cem BOĞA
35
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).
4. RESULTS AND DISCUSSION Cem BOĞA
36
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
4. RESULTS AND DISCUSSION Cem BOĞA
37
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
4. RESULTS AND DISCUSSION Cem BOĞA
38
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
4. RESULTS AND DISCUSSION Cem BOĞA
39
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
4. RESULTS AND DISCUSSION Cem BOĞA
40
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)
4. RESULTS AND DISCUSSION Cem BOĞA
41
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).
4. RESULTS AND DISCUSSION Cem BOĞA
42
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
4. RESULTS AND DISCUSSION Cem BOĞA
43
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.
4. RESULTS AND DISCUSSION Cem BOĞA
44
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.
4. RESULTS AND DISCUSSION Cem BOĞA
45
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
4. RESULTS AND DISCUSSION Cem BOĞA
46
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
4. RESULTS AND DISCUSSION Cem BOĞA
47
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
4. RESULTS AND DISCUSSION Cem BOĞA
48
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.
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
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
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
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)
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
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.
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)
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)
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)
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)
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)
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)
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.
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.
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.
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
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.
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.
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)
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)
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
Tφ
λλλ
λλ
λφ (A.13)
or;
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.
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)
APPENDIX B Cem BOĞA
72
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
APPENDIX C Cem BOĞA
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
APPENDIX C Cem BOĞA
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
APPENDIX C Cem BOĞA
75
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
APPENDIX C Cem BOĞA
76
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
APPENDIX C Cem BOĞA
77
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
APPENDIX C Cem BOĞA
78
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
APPENDIX C Cem BOĞA
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
APPENDIX C Cem BOĞA
80
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
APPENDIX C Cem BOĞA
81
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
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
APPENDIX D Cem BOĞA
83
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
APPENDIX D Cem BOĞA
84
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)
APPENDIX D Cem BOĞA
85
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
APPENDIX D Cem BOĞA
86
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,*) ' '
APPENDIX D Cem BOĞA
87
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
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