A BRIEF STUDY ON DYNAMICS OF VISCOELASTIC

A BRIEF STUDY ON DYNAMICS OF VISCOELASTIC
A BRIEF STUDY ON DYNAMICS OF VISCOELASTIC
ROTORS – AN OPERATOR BASED APPROACH
A THESIS SUBMITTED IN THE PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology
In
MECHANICAL ENGINEERING
[Specialization: Machine Design and Analysis]
By
VIVEK SINGH
208ME102
Department of Mechanical Engineering
National Institute of Technology
[email protected]
June -2011
A BRIEF STUDY ON DYNAMICS OF VISCOELASTIC
ROTORS – AN OPERATOR BASED APPROACH
A THESIS SUBMITTED IN THE PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
Master of Technology
In
MECHANICAL ENGINEERING
[Specialization: Machine Design and Analysis]
By
VIVEK SINGH
208ME102
Under the supervision of
Dr. HARAPRASAD ROY
Department of Mechanical Engineering
National Institute of Technology
[email protected]
National Institute Of Technology
Rourkela
CERTIFICATE
This is to certify that the thesis entitled, ― A BRIEF STUDY ON DYNAMICS OF
VISCOELASTIC ROTORS – AN OPERATOR BASED APPROACH ‖ submitted by Mr. VIVEK
SINGH in partial fulfillment of
TECHNOLOGY
Degree
the
requirements
for
the
award
of
MASTER
OF
in ―MECHANICAL ENGINEERING‖ with specialization in
―MACHINE DESIGN AND ANALYSIS‖ at the National Institute of Technology, Rourkela (India)
is an authentic Work carried out by him under my supervision and guidance.
To the best of my knowledge, the matter embodied in the thesis has not been
submitted to any other University / Institute for the award of any Degree or Diploma.
Date:
Dr. HARAPRASAD ROY
Department of Mechanical Engineering
National Institute of Technology
Rourkela-769008
ACKNOWLEDGEMENT
First and foremost I offer my sincerest gratitude and respect to my supervisor and guide Dr.
HARAPRASAD ROY, Department of Mechanical Engineering, for his invaluable guidance and
suggestions to me during my study. I consider myself extremely fortunate to have had the opportunity
of associating myself with him for one year. This thesis was made possible by his patience and
persistence.
After the completion of this Thesis, I experience feeling of achievement and satisfaction. Looking
into the past I realize how impossible it was for me to succeed on my own. I wish to express my deep
gratitude to all those who extended their helping hands towards me in various ways during my tenure
at NIT Rourkela. I greatly appreciate & convey my heartfelt thanks to my colleagues‘ flow of ideas,
dear ones & all those who helped me in completion of this work.
I am especially indebted to my parents for their love, sacrifice, and support. They are my first
teachers after I came to this world and have set great examples for me about how to live, study
and work.
VIVEK SINGH
CONTENTS
ABSTRACT
1). INTRODUCTION
1.1 Background and importance ...............................................................................2
1.2 Linear viscoelasticity..................………………………………………...…......4
1.3 Material damping and modelling techniques ..………………………..….......7
1.4 Damping modelling in finite element analysis…………………………...…....9
1.5 Important researches on rotor dynamics………………………………….....12
1.6 Internal damping and its effect on rotor dynamics...........................................14
1.7 Layout of the present work.................................................................................15
2). VISCOELASTIC ROTOR AND ITS MODELLING
2.1 Constitutive relationships ……………………………………………….........18
2.2 Equations of motion…………………………………………………….......…22
2.3 Finite element formulation………………………………………………..…..26
3). RESULT AND DISCUSSION
3.1 The Rotor-Shaft System………………………………………………..……...29
3.2 Stability limit of spin speed and Unbalance response amplitude…………......30
3.3 Time response of the disc…………………………………………….................33
4). CONCLUSIONS AND SCOPE FOR FUTURE WORK
4.1 Conclusions……………………………………………………………………......38
4.2 Scope for future work………………………………………………………….....39
REFERENCES.....................................................................…39-44
LIST OF FIGURES
1). Figure 1.1. Stress strain curve………………………………………........…...5
2). Figure 1.2. Various spring-dashpot model………………………….......……6
3). Figure 2.1. Different Viscoelastic Models……………………………….......21
4). Figure 2.2. 3-Element model……………………………………………...…22
5). Figure 2.3. Displaced position of the shaft cross section………………...…23
6). Figure.2.4. Planer beam bending element……………………………......…26
7). Figure 3.1. Schematic diagram of Rotor………………………………..…..30
8). Figure: 3.2. Variation of maximum real parts vs. spin-speed………….......31
9). Figure: 3.3. Mass UBR and corresponding stability for disc at the middle.32
10). Figure 3.4. Unbalance Response Amplitude at different disc position…...33
11). Figure: 3.5. Time response of rotor in Stable Zone (Y- Direction)………34
12). Figure: 3.6 Time response of rotor in Stable Zone (Z- Direction)…......…35
13). Figure 3.7. Time response of rotor in Unstable Zone (Y- Direction)….......36
14). Figure3.8. Time response of rotor in Unstable Zone (Z- Direction)……....37
A Brief Study on Dynamics of Viscoelastic
Rotors - An Operator Based Approach
ABSTRACT
Viscoelasticity, as the name implies, is a property that combines elasticity and viscosity
or in other words such materials store energy as well as dissipates it to the thermal domain when
subjected to dynamic loading and most interesting the storage and loss of energy depends upon
the frequency of excitation. Modeling of viscoelastic materials is always difficult whereas
modeling the elastic behavior is easy, modelling the energy dissipation mechanism possess
difficulty. This work attempts to study the dynamics of a viscoelastic rotor-shaft system
considering the effect of internal material damping in the rotor. The rotation of rotors introduces
a rotary damping force due to internal material damping, which is well known to cause instability
in rotor-shaft systems. Therefore, a reliable model is necessary to represent the rotor internal
damping for correct prediction of stability limit of spin speed and unbalance response amplitude
of a rotor-shaft system. An efficient modelling technique for viscoelastic material, augmenting
thermodynamic field (ATF) has been found in literature.
Here the material constitutive relationship has been represented by a differential time
operator. Use of operators enables to consider general linear viscoelastic behaviours, represented
in the time domain, for which, in general, instantaneous stress and its derivatives are proportional
to instantaneous strain and also its derivatives. The operator may be suitably chosen according to
the material model. The constitutive relationships for ATF approach is represented in differential
time operator to obtain the equations of motion of a rotor-shaft system after discretizing the
system using beam finite element method. The equations thus developed may easily be used to
find the stability limit of spin speed of a rotor-shaft system as well as the time response as a
result of unbalance when the rotor-shaft system is subjected to any kind of dynamic forcing
function.
In this work dynamic behavior of an aluminium rotor is predicted through viscoelastic
modelling of the continuum to take into account the effect of internal material damping. To study
the dynamics of an aluminium rotor-shaft system stability limit of spin speed, unbalance
response amplitude and time response are used as three indices. It is observed that, the operator
based approach is more suitable for finding the equation of motion of a viscoelastic rotor which
is used to predicts the dynamic behaviour of that continuum.
CHAPTER ONE
Introduction
1.1 Background and Importance
Rotordynamics is a specialized branch of applied mechanics concerned with the
behavior and diagnosis of rotating structures. It is commonly used to analyze the behavior of
structures ranging from jet engines and steam turbines to auto engines and computer disk
storage. At its most basic level rotordynamics is concerned with one or more mechanical
structures (rotors) supported by bearings and influenced by internal phenomena that rotate
around a single axis. The supporting structure is called a stator. As the speed of rotation increases
the amplitude of vibration often passes through a maximum that is called a critical speed. This
amplitude is commonly excited by unbalance of the rotating structure; everyday examples
include engine balance and tire balance. If the amplitude of vibration at these critical speeds is
excessive catastrophic failure occurs. In addition to this, turbomachinery often develops
instabilities which are related to the internal makeup of turbomachinery, and which must be
corrected. This is the chief concern of engineers who design large rotors. Texts by Goodwin
(1989), Kramer (1993), Lalanne and Ferraris (1998) and Vance (1988) are valuable sources of
information on dynamic behaviour of rotor-shaft systems and their analysis.
Unlike the viscoelastic structures (which do not spin) viscoelastic rotors are acted upon
by rotating damping force generated by the internal material damping, that tends to destabilize
the rotor shaft system by generating a tangential force proportional to the rotor spin speed. Thus
a reliable model is necessary to represent the constitutive relationship of a rotor-material by
taking into account the internal material damping for understanding the dynamic behaviour of a
viscoelastic rotor. Such a model is useful for getting an idea about safe speed ranges of rotation,
where the rotor is stable. Expression of the constitutive relationships in the frequency domain,
i.e. the stress-strain relationship under a sinusoidal excitation at a fixed frequency is well known
from literature, but such constitutive relationships are not useful for analysis of transient
response for investigating stability.
This work presents the development of equations of motion of a rotor-shaft-system with a
viscoelastic rotor after discretizing the system into finite elements. Subsequently these equations
are used to study the dynamics of the rotor-shaft system in terms of stability limit of spin speed
and time response of a disc as a result of unbalance. For this, the material constitutive
relationship has been represented by a differential time operator. Use of operators enables one to
consider general linear viscoelastic behaviours, represented in the time domain by multi-element
(3, 4 or higher elements) spring-dashpot models or internal variable models (ATF), for which, in
general, instantaneous stress and its derivatives are proportional to instantaneous strain and its
derivatives. Again such representation is fairly generic, in a sense that the operator may be
suitably chosen according to the material model to obtain the equations of motion of a rotor-shaft
system.
1.2 Linear viscoelasticity
Viscoelasticity, as the name implies, is a property that combines elasticity and viscosity.
A material, which is viscoelastic in nature, thus stores and also dissipates energies and therefore
the stress in such materials is not in phase with the strain. For this reason, it is extensively used
in various engineering applications for controlling the amplitude of resonant vibrations and
modifying wave attenuation and sound transmission properties, increasing structural life through
reduction in structural fatigue. Nakra (1998) has reported many such applications.
The classical theory of elasticity states that for sufficiently small strains, the stress in an
elastic solid is proportional to the instantaneous strain and is independent of the strain rate. In a
viscous fluid, according to the theory of hydrodynamics, the stress is proportional to the
instantaneous strain rate and is independent of the strain. Viscoelastic materials exhibit solid and
fluid behavior. Such materials include plastics, amorphous polymers, glasses, ceramics, and
biomaterials (muscle). Viscoelastic materials are characterized by constant-stress creep and
constant-strain relaxation. Their deformation response is determined by both current and past
stress states, and conversely, the current stress state is determined by both current and past
deformation states. The Stress-Strain Curves for a purely elastic and a viscoelastic material are
shown in figure 1.1. Due to loss of energy during loading and unloading time, the stress strain
curve for viscoelastic material is elliptic in nature (Thompson and Dahleh (1998)). The area
enclosed by the ellipse is a hysteresis loop and shows the amount of energy lost (as heat) in a
loading and unloading cycle
Figure 1.1 The stress strain curve
Some phenomena in viscoelastic materials are:
I. If the stress is held constant, the strain increases with time (creep);
II. If the strain is held constant, the stress decreases with time (relaxation);
III. The effective stiffness depends on the rate of application of the load;
IV. If cyclic loading is applied, hysteresis (a phase lag) occurs, leading to dissipation of
mechanical energy;
V. Acoustic waves experience attenuation;
VI. Rebound of an object following an impact is less than 100%;
VII. During rolling, frictional resistance occurs.
Viscoelastic behaviour has elastic and viscous components modeled as linear
combinations of springs and dashpots, respectively. These models, which include the Maxwell
model, the Kelvin-Voigt model, and the Standard Linear Solid Model, are used to predict a
material's response under different loading conditions. The Maxwell model of a one dimensional
viscoelastic material consists of a linear spring and a linear dashpot connected in series as shown
in Figure 1.2a. The Kelvin-Voigt model is consists of a linear spring and a linear damper
connected in parallel (Figure 1.2b). More than one spring or linear dampers are included in the
model to better approximate material behavior over a broad frequency range. The three-element
model of a standard viscoelastic solid is shown in Figure 1.2c. It consists of a linear spring K1 in
series with a linear Kelvin element (spring K and dashpot C in parallel). The four-element model
is shown in Figure 1.2d. 2-element Voigt model simulates only the creep and 2-element
Maxwell‘s model simulates only the stress relaxation behaviours. Generally 3 or higher element
models are capable of depicting both constant-stress creep and constant-strain relaxation
behaviours as shown by viscoelastic solids. Bland (1960), Christensen (1982) and Shames and
Cozzareli (1992) and many others have given different network comprising linear springs and
dashpots to model linear viscoelastic solids.
k2
k
d
k
x
(a)
x
x
(b)
k1
k1
d
x
d1
d2
d1
x
x
(c)
k2
x
x
(d)
Figure 1.2 Various spring-dashpot model
Each model differs in the arrangement of these elements, and all of these viscoelastic
models can be equivalently modeled as electrical circuits. In an equivalent electrical circuit,
stress is represented by voltage, and the derivative of strain (velocity) by current. The elastic
modulus of a spring is analogous to a circuit's capacitance (it stores energy) and the viscosity of a
dashpot to a circuit's resistance (it dissipates energy). All materials exhibit some viscoelastic
response. In common metals such as steel or aluminium, as well as in quartz, at room
temperature and at small strain, the behaviour does not deviate much from linear elasticity.
Which may be classified as weakly viscoelastic as the amount of energy dissipated is much less
than the energy stored whereas synthetic polymers, wood, and human tissue as well as metals at
high temperature display significant viscoelastic effects. Those are generally strongly
viscoelastic as energy dissipated is substantially more. In some applications, even a small
viscoelastic response can be significant. To be complete, an analysis or design involving such
materials must incorporate their viscoelastic behaviour.
1.3 Material damping and modelling techniques
Vibration damping is essential to the attainment of performance goals for a variety of
advanced engineering systems. In terms of performance, higher damping reduces steady state
vibration level at resonance and time needed for vibration to settle. The dynamic analysis of
structural components with viscoelastic damping treatment has been subject for the treatment of
many years. In common built-up structures that operate in the atmosphere, air damping and joint
damping typically dominate system damping. However material damping can also be an
important contributor to overall damping in many applications, such as aerospace vehicles, large
space structures, etc. Viscoelastic material damping is generally a complex function of
frequency, temperature, type of deformation, amplitude and structural geometry.
Damping is the dissipation of mechanical energy and is produced by the some
nonconservative forces acting on a given structure. Damping may be classified into external and
internal, depending upon the nature of the non-conservative forces acting on the structure.
External damping is caused by forces acting on the object, such as damping due to air resistance
or Coulomb damping due to friction. Internal damping is caused by physical phenomena
intimately linked to the structure of the material. There are various types of models for predicting
the material damping. Viscous damping law where the damping force is a function of excitation
frequency. For structural materials, under harmonic excitation dissipate energy proportional
primarily to the square of excitation amplitude. Lazan (1968) has reported system damping
values for different structural materials under different types of deformations. A very good
survey has been observer in Rusovici (1999).
The material moduli for linear viscoelastic materials depend on frequency and
temperature, but not on strain and stress levels. If harmonic stress and strain states are
considered, then the material moduli can be expressed in term of complex number. Where the
real part is the material storage modulus and the ratio between imaginary and real part is the loss
modulus. The material loss factor is equal to the tangent of the phase angle between the
harmonic stress input and the corresponding harmonic strain. Both the loss and storage moduli
are frequency and temperature dependent. The frequency and temperature dependence of the
complex modulus components must be captured accurately; simple models such as Maxwell or
Kelvin model are not able to do that. The Maxwell model behaves like a fluid at low frequencies,
while the Kelvin model becomes infinitely stiff at high frequencies. Prediction accuracy
increases by adding more springs and dampers. Though the frequency-domain description is
simple to use, but faces difficulties when the forcing function contains more than one frequency
components (as in the case of periodic excitation) and also for obtaining the transient response
(Bert (1973)).
1.4 Damping modelling in finite element analysis
The structural response of a system is determined in computational structural dynamics
by solving a set of n simultaneous, time-dependent equations. One of the first methods to model
damping is to consider proportional damping (Thompson and Dahleh (1998)). In this method, the
damping matrix is considered to be proportional to the mass and stiffness matrices through two
coefficients (Rayleigh‘s coefficients).which are calculated directly from the modal damping
ratios. The damping ratios are the ratio of actual damping to critical damping for a particular
mode of vibration. The drawback of this method is that the damping matrix depends on the
arbitrarily determined parameters, and thus has no physical motivation.
Segalman (1987) uses a perturbation approach to model linear viscoelastic structures.
Hereditary integrals are used in the equation of motion. Henwood (2002) tried to represent a
hysteretic damping matrix by a viscous damping matrix (for values of loss factor 0.4) and
extended the work of Crandall (1970) for single-degree-of-freedom to a general structure. The
actual energy dissipation mechanism is, however, very complicated. In the internal variable
approach the effect of energy dissipation is taken care of by including a dissipation coordinate.
Under this approach Bagley and Torvik (1983, 1985) use differential operators of fractional
order to model linear frequency dependence of viscoelastic materials using finite elements
approach with four empirical model parameters. Time-dependent stresses and strains are related
by derivatives of fractional orders. This model connected the molecular theories for uncross
linked polymer solids and linear viscoelastic models through fractional calculus. The stiffness
matrix of a viscoelastic element was built and the ensuing finite element equations were obtained
in the frequency domain. The solutions of the system of equations had to be transformed from
the frequency domain back into the time domain. Only loads that have a Laplace transform may
be included in subsequent analyses. The Bagley-Torvik approach is only applicable for uncross
linked polymer solids of linear viscoelastic materials.
Padovan (1987) based his computational algorithms for finite element analysis of
viscoelastic structures on fractional integral-differential operators. Viscoelastic, displacement
based, time domain finite element model results, which may be solved for any type of loading.
The Padovan model is valid only for viscoelastic materials that may be modeled through
fractional calculus.
The model developed by Golla and Hughes (1985) incorporated the hereditary integral
form of the viscoelastic constitutive law in a finite element model. The finite element equations
are derived in the Laplace domain through the Ritz technique. The time domain equations are
obtained from the frequency domain equations by the linear theory of realizations. The method
yields a system of second-order matrix differential equations. Internal dissipation coordinates
augment the stiffness, damping, and mass matrices. This technique is applicable only for linear
viscoelastic materials and may be applied only when the system is initially at equilibrium.
McTavish and Hughes (1992, 1993) extended the Golla-Hughes model and formulated the GHM
(Golla-Hughes-McTavish) model for linear viscoelastic structures. This new formulation the
material modulus is modeled as the sum of mini-oscillators are characterized by a viscous
damper, spring and mass. The motion of the mass represents the internal dissipation coordinate.
Again, this method leads to a system of second order differential equations of motion in the
frequency domain, where the mass, stiffness, and damping matrices are augmented by the
internal dissipation coordinates.
Initially, the methodology for modeling frequency-dependent material damping in
structural dynamics was motivated by the material science results. Such observation led to the
inclusion of augmenting thermodynamic fields (ATF) that could interact with the usual
mechanical displacement fields. This method, developed by Lesieutre (1989), had its foundation
on the irreversible thermodynamics. ATF method to model frequency dependent material
damping of linear viscoelastic structures in a finite element context.
In the first ATF paper introduced by Lesieutre and Mingori (1990), one-dimensional
formulation of the ATF model was developed. Lesieutre (1992) further developed the ATF
model to include the behavior of high-damping materials and also to demonstrate the use of
multiple augmenting fields. The material constitutive equations are derived from the Helmholtz
free energy function. The ATF evolution equation is determined from the irreversible
thermodynamics assumption that the rate of change ATF is proportional to its deviation from an
equilibrium value. The coupled equations of motion are discretized using the method of weighted
residuals. The ATF parameters are obtained by iteratively curve fitting or by minimization of the
error between the complex modulus to frequency dependent experimental values of storage
modulus and loss factor (Roy (2008)). The descriptions of frequency dependent storage modulus
and loss factor for different polymeric materials are given by Ferry (1980) and Nashif et. al.
(1985) and others. Roy et al. (2008) used the ATF approach to model a viscoelastic continuum of
a rotor shaft, to obtain the equations of motion and studied the dynamic behaviour in terms of
stability limit of spin speed and unbalance response amplitude.
Lesieutre and Bianchini (1995), Lesieutre et. al. (1996) (approach was developed to
extend the ATF method to three-dimensional state) developed a time-domain model of linear
Viscoelasticity based on a decomposition of the total displacement into elastic and anelastic
parts. They used the motion of anelastic displacement field (ADF) to describe the part of the
strain that was not instantaneously proportional to stress. The total and anelastic displacement
fields were used to build differential equations for the mass particles and the relaxation of the
ADFs respectively, resulting in coupled governing equations that were not explicitly time
dependent. The ADF approach is the extension of ATF approach, is more straightforward
development in finite element context. The displacement field was made of an elastic component
and an anelastic component, where anelastic field is introduced to take in to account the
dissipation.
Just like in the ATF model, the ADF parameters are obtained by curve fitting or
minimization of errors between ADF complex moduli to the corresponding experimental data
(Roy (2008)). The equations of motion are discretized by the weighted residuals method to
obtain a finite element model. Multiple ADF may be used to better approximate experimental
frequency dependent modulus data for a broad range of frequency.
In the recent paper, Roy et. al. (2009) used the both ATF and ADF approaches to model
the viscoelastic beam. This method also applied to study the dynamic behaviour of composite
beam comprising of several viscoelastic layers. All the models, ATF, ADF and GHM employ
additional coordinates to model damping more accurately, whereas the ‗dissipation coordinate‘
of GHM is internal to individual elements, while these are continuous in ATF or ADF from
element to element. For this advantage ATF/ ADF approaches are used in this work to represent
the viscoelastic material behavior.
1.5 Important researches on rotor dynamics
The history of rotordynamics is replete with the interplay of theory and practice. W. J. M.
Rankine first performed an analysis of a spinning shaft in 1869, but his model was not adequate
and he predicted that supercritical speeds could not be attained. In 1895 Dunkerley published an
experimental paper describing supercritical speeds. Gustaf de Laval, a Swedish engineer, ran a
steam turbine to supercritical speeds in 1889, and Kerr published a paper showing experimental
evidence of a second critical speed in 1916.
The Jeffcott rotor (named after Henry Homan Jeffcott), also known as the de Laval rotor
in Europe, is a simplified lumped parameter model used to solve these equations. The Jeffcott
rotor is a mathematical idealization that may not reflect actual rotor mechanics. Henry Jeffcott
was commissioned by the Royal Society of London to resolve the conflict between theory and
practice. He published a paper now considered classic in the Philosophical Magazine in 1919 in
which he confirmed the existence of stable supercritical speeds. August Föppl published much
the same conclusions in 1895, but history largely ignored his work.
Between the work of Jeffcott and the start of World War II there was much work in the
area of instabilities and modeling techniques culminating in the work of Prohl and Myklestad
which led to the Transfer Matrix Method (TMM) for analyzing rotors. The most prevalent
method used today for rotordynamics analysis is the Finite Element Method. Nelson (2003) has
written extensively on the history of rotordynamics and most of this section is based on his work.
There are many software packages that are capable of solving the rotordynamic system of
equations. Rotordynamic specific codes are more versatile for design purposes. These codes
make it easy to add bearing coefficients, side loads, and many other items only a rotordynamicist
would need. The non-rotordynamic specific codes are full featured FEA solvers, and have many
years of development in their solving techniques. The non-rotordynamic specific codes can also
be used to calibrate a code designed for rotordynamics.
1.6 Internal damping and its effect on rotor dynamics
All types of damping associated to the non-rotating parts of the structure have a usual
stabilizing effect. On the other hand damping associated to rotating parts can trigger instability in
supercritical ranges. Rotation of rotors introduces a rotary damping force due to internal material
damping, which is well known to cause instability in rotor-shaft systems. Thus a reliable model
is necessary to represent the rotor internal damping for correct prediction of stability limit of spin
speed (SLS) of a rotor-shaft system. Rotary machines, such as motors, compressors and turbines
are very common and widely used. Recently the designers of these machines have been required
to meet very severe specifications from the demands of high speed operating power or
improvements in efficiency and reliability for the design. In such situations, finding some robust
and reliable mathematical models, in conjunction with special numerical solution procedures,
which enable designers to make an accurate assessment of the relevant parameters, the critical
speeds and the dynamic behavior of the system, especially the response of the system to
unbalance excitation, is of great importance in order to design for increased speeds of rotation, to
optimize weight, to improve reliability, and to reduce maintenance costs.
Modelling the rotor internal damping using the viscous and hysteretic model has been
attempted by many researchers [Dimentberg (1961), Tondl (1965), Genta (2005)]. Most of the
authors have considered, in general, viscous form of internal damping and used 2-element Voigt
model for representing the material behaviour to study the dynamics of rotor-shaft systems.
Again Zorzi and Nelson (1977), Ozguven and Ozkan (1984), Ku (1998) developed a finite
element model of the rotor material damping by representing its constitutive relationship with a
Voigt model (2-element model) where internal material damping force was considered as a
superposition of viscous and hysteretic damping forces to take into account the frequency
dependent and frequency independent components of energy dissipation per cycle for properly
representing the properties of structural materials like steel. In this regard Genta (2004) pointed
out the correct interpretation and use of the hysteretic damping model.
However viscous and hysteretic damping models are unsuitable for proper representation
of viscoelastic material behaviour, which shows considerable dependence on wide range of
excitation frequencies. Not many papers are found to report dynamic simulation of viscoelastic
rotors. Grybos (1991) used 3-element material model and studied the dynamics of a viscoelastic
rotor. Roy et al. (2008) reported a finite-element approach, where viscoelastic behaviour of a
rotor-continuum was represented by ATF (Augmenting Thermodynamic Field). Recently Dutt
and Roy (2010) obtained the equation of motion of viscoelastic rotor-shaft system after
discretizing the continuum by finite beam element method. The rotor-shaft material was assumed
to behave as a linear viscoelastic solid for which the instantaneous stress was obtained by
operating the instantaneous strain by a generic linear differential time operator. The advantage of
using a generic operator approach is that it may be suitably tailored according to the material
constitutive relationship to obtain the equations of motion for a particular material model.
1.7 Layout of the present work
As aluminium is normally taken as an elastic material but a careful modelling as done by
Lesieutre (1989) and Lesieutre and Mingori (1990) considered aluminium as a viscoelastic
material. Viscoelastic modeling of aluminium rotor-shaft is also important from the point of view
of predicting the SLS of the rotor-shaft system. Concepts of finite element of rotor shaft-system
as given by Zorzi and Nelson (1977) , Nelson and McVaugh (1976) and Rao JS (1996) are
extended to take into account the internal material damping using ATF parameters.
In this present reporting, an attempt has been made to study theoretically the stability
limit of the spin speed, unbalance vibration response and subsequently time response within the
stable zone of operation of a simply supported aluminium rotor-shaft having a central disc made
of aluminium. The rotor-shaft material is assumed to behave as a linear viscoelastic solid for
which the instantaneous stress is obtained by operating the instantaneous strain by a linear
differential time operator. The internal variable approach i.e. ATF is used for modelling the
viscoelastic material. The constitutive relationships for ATF approach is represented in
differential time operator, where the coefficients of the operator are formed by ATF parameters.
The equations of motion of a rotor-shaft system are obtained after discretizing the continuum
using finite beam element. So this work is useful for dynamic analysis of viscoelastic rotors
under any type of dynamic forcing function.
Based on that review, the objective and scope proposed in this work are as follows.
(a) Finding out the equations of motion and Development Finite Element formulation of the
viscoelastic rotor . Operator based constitutive relationship is obtained by using augmenting
thermodynamic fields (ATF) approach. The equations of motion is developed by using that
constitutive relationship. Finite element method is used to discretised the rotor continuum.
(b) Dynamic behaviour of an aluminium rotor is predicted through viscoelastic modelling of
the continuum to take into account the effect of internal material damping. Stability limit of spin
speed, unbalance response amplitude and time response are found to study the dynamics.
CHAPTER TWO
Viscoelastic Rotor and Its Modelling
CHAPTER TWO
Viscoelastic Rotor and Its Modelling
This chapter forms the basis of the entire work as it presents the derivation of the
equations of motion of a viscoelastic rotor and Development Finite Element analysis procedure.
The material constitutive relationship has been represented by a differential time operator, where
the instantaneous stress and its derivatives are proportional to instantaneous strain and also its
derivatives. The operator based constitutive relationship plays an important role in developing
the equation of motion in the time domain. The finite element method is used to discretised the
rotor continuum. The dynamic behaviour of the rotor shaft system includes the stability limit of
spin speed, unbalance response amplitude and time response within the stable zone of operation.
2.1 Constitutive relationships
The constitutive relationships are obtained from the Helmholtz free energy density
function, 'H' representing a thermodynamic potential, where strain (ε) is an independent variable.
The function, 'H' is defined as [Lesieutre and Mingori (1990)]
H
1 2
1
E     2
2
2

H
 E  

A 
H

   
(1)
(2a)
2b)
In the preceding equation E is the un-relaxed modulus, ζ is the mechanical stress, ξ is the
ATF, A is the affinity, α is a material property relating the changes in A to ξ and δ is the strength
of coupling between the mechanical displacement field and the thermodynamic field.
With the assumption [Lesieutre and Mingori(1990)] that  , the rate of change of ξ with
time, varies proportionally to its deviation from an equilibrium value for irreversibility of the
thermodynamic action, a first order differential equation or relaxation equation is given as
  SA   B(   )
(3)
In the preceding equation B is the inverse of relaxation time, S is a constant of
proportionality, and  denotes the values of  at equilibrium where A = 0. Putting A = 0 in
equations (2b) the values of  , is obtained as
    .
Substituting the values of  in equation (3), the first order relaxation equation is given as
  B 
B

x
(4)
Equation (4) is a first-order linear non-homogeneous differential equation of ξ with
respect to time ‗t‘. Equation (4) can be rewritten as
 B  1
  
x

B

D
 
(5)
d
Where D= dt , is the first order differential time Operator. Putting values of  from
equation (5) in equation (2a), the constitutive relationship is rewritten as:

2 
  ED
B E 
 

x 
x
BD
(6)
Any standard linear viscoelastic solid can be represented as a combination of multiple
linear springs and linear dashpots. The material modulus is a function of operator of time. The
general constitutive relation for a linear viscoelastic solid can be expressed as
σx=E( )εx
……… (7)
The instantaneous normal stress x is obtained from the equation (7) i.e. by operating the
expression of εx by the operator E ( ), called the modulus operator. The generic form of modulus
operator is expressed in equation (8) below, where num(D) and den(D) are the numerator and
denominator polynomials of differential time operator, D ≡ d/dt. For the special case of linear
elastic behaviour, E is a constant called the Young‘s modulus.
The generalized expression of modulus operator for any viscoelastic solid is written as
n
E () 
num(D)

den(D)
a
j 0
m
j
Dj
 bj D j
(8)
j 0
Mechanical models for physical representation of viscoelastic behaviour are given by
Bland (1960) among others. Expressions of E ( ), for 2, 3, and 4-element models for example, as
shown in Figure 2.1, are given below.
Figure 2.1: Different Viscoelastic Models
E2 ()  a0  a1D, a0  k1 , a1  d1
E3 () 
E4 () 
(9a)
a0  a1D
kd
d
,a0  k1 , a1  d1  1 1 , b0  1, b1  1
(b0  b1D)
k2
k2
a0  a1D  a2 D 2
kd
dd
d
, a0  k1 , a1  d1  d 2  1 2 , a2  1 2 , b0  1, b1  1
(b0  b2 D)
k2
k2
k2
(9b)
(9c)
For the nomenclature, all springs and dampers directly connected to the ground are called
‗primary‘ and those connected in series are called ‗secondary‘. Following this, all springs and
dashpots with subscript ‗1‘ are primary and the ones with subscript ‗2‘ are secondary.
The modulus operator for single ATF model

2 

  ED
B E 


E   
BD
(10)
The equation (10) represents a three element model. On comparing equation (9b) and
equation (10), the polynomial coefficient for numerator and denomerator are given as

  2 
a0  B  E     ,
  

a1  E ,
b0  B ,
b1  1
Using the mechanical analogy (Lesieutre et. al. (1996)) as shown in figure 2.2, which
represents the model used in a single ATF approach.
Figure 2.2: 3-Element model
 E2
E 
Where, K2 = E, k1 =  2  1 E , d1 =
B 2


2.2 Equations of motion
The equations of motion of a rotor-shaft system were found out by Zorzi and Nelson
(1977) by considering viscous and hysteretic components of internal material damping. A Voigt
element (a 2-element spring-dashpot representation) was used to represent the stress is the sum
of two parts one is proportional to strain and another to the strain rate. Thus the representation
forms a special case of operator function. For general viscoelastic materials, however, the stress
strain relationship is governed by operating an operator, which is a function of D, of which the
viscous damping law is a special case (as the modulus operator E( ) for a 2-element Voigt model
is obtained by putting b1 = 0 in equation (9b). Hence, in this work the procedure of Zorzi &
Nelson (1977) for the viscous damping case has been extended for a general viscoelastic solid by
substituting the modulus operate for the 2-element model by modulus operator function for 3-
element models as define above. This process is quite general in a sense that any journal
modulus operator may be substituted once the configuration of the spring- dashpot network is
decided.
A rotor shaft system has been considered. Figure 2.3 shows the position of a shaft cross
section defined by the coordinates of its centre (v, w) at a distance u from any suitable reference
and an element of differential radial thickness dr and at a distance r (where r varies from 0 to r0)
subtending an angle d(Ωt) (where Ωt varies from 0 to 2π) at any instant of time ‗t‘. The crosssection of rotor undergoes two simultaneous rotations due to whirl (Ω) and the spin (ω)
respectively radian per second.
Figure 2.3 Displaced position of the shaft cross section
ζx, εx denote, respectively, the mechanical stress and strain induced in the element at the
instant of time. Zorzi and Nelson (1977) express the mechanical strain in the ‗x‘ direction as
 2 R( x, t )
 x  r cos(t  t )
x 2
(11)
Where, R is the displacement of the rotor centre line, ω is the whirl speed.
Zorzi and Nelson (1977) obtained the bending moment expressions after considering a 2element material model (Voigt model) to represent the constitutive-relationship of the material.
The bending moment expressions at any instant of time about the y and z-axes, Myy and Mzz
respectively, are expressed as given below.
M yy 
2 r0
  (w  r sin(t ))
x
rdrd (t )
0 0
M zz 
............ (12)
2 r0
   (v  r cos(t ))
x
rdrd (t )
0 0
The instantaneous bending moments are written next by extending the work by Zorzi and
Nelson (1977). Substituting x from equation (6) in the bending moment expressions (equations
(12)) and utilizing the expressions of x given in equation (11), the bending moment expressions
are rewritten as
 2 R ( x, t ) 
a0 a1D 
M zz      v  r cos  t 
r cos  t  t 
rdrd (t )
b0 b1D 
0 0
x 2 
2 ro
2 ro

0 0
M yy    w  r sin  t 

a0  a1D
b0 b1D

 2 R ( x, t ) 
 r cos  t  t  x 2  rdrd (t )


…….... (13)
It may be noted in equation (13) that the operator E ( ) is operated exclusively on the
terms inside the bracket [ ] containing the expression of strain to give the stress. The operator
does not work on other terms, (v + r cos (t) and w + r sin (t)) forming the momentums in the
respective planes at any instant of time,‗t‘. Hence the expressions of momentums are perceived
as constants as far as the operator E ( ) is concerned. Following this logic the equation (13) may
be rewritten as
M zz 
1
b0 b1D
2 ro

2
 r  v  r cos  t  [a0 cos  t   t 
0 0
 a1      sin  t   t 
M yy  
1
b0 b1D
2 ro

2 R
3 R

a
cos

t


t

 2
1
x 2
x t
2 R
]drd   t 
x 2
2
 r  w  r sin  t  [a0 cos  t   t 
0 0
2 R
3 R

a
cos

t


t

 2
1
x 2
x t
2 R
 a1      sin  t   t  2 ]drd   t 
x
After performing the integration
M zz 
I [a v  a v  a w]
1
1
b0 b1D 0
M yy  
I [a w  a w  a v]
1
1
b0 b1D 0
In matrix form
  a0 a1   v   a1
 M zz 
I




  w   0
M
a


a
b

b
D
 
0
 yy  0 1   1
0   v 
 
a1   w
........... (14)
2.3 Finite element formulation
For developing the equations of motion, the rotor-shaft continuum is discretized by using
beam finite elements. Due to the motion in x-y and z-x plane of the rotor, the finite element
formulation approximates the mechanical coordinates in two planes simultaneously. Each
element contains two nodes at the ends and 4 degrees of freedom, which are the displacements
and slopes in x-y and z-x planes, as shown in Figure 2.5.
Figure.2.4. Planer beam bending element
The expression of translations and relations with rotations are given as
vx, t  
T

   x  qt 
wx, t 
,

w
x

,
v
x
………. (15)
Where v and w denote the deformations along y and z axes and Φ, Ґ are the rotations about the y
and z axes respectively. The Hermite shape function is given as
 x 
 ( x)  

xy
0
0 
zx x  ,
with subscripts in the elements showing the respective planes.
Assuming that rotor is rotating at a uniform speed (Ω). Using the expressions for
differential bending energy as shown in equation (16), along with the expressions of v(x, t) and
w(x, t) and its spatial derivatives from preceding equation (where {q}(8x1) are nodal displacement
vector (as the rotor continuum is discretized using beam finite elements).
1   M zz 
   

2  M yy 
T
dPB
e
(16)
The equations of motion may easily be written using complex coordinates. Expressions
of the stiffness, damping and circulatory matrices due to bending are obtained from the strain
energy and dissipation function calculated from the expression of bending moments given in the
equation (14). Diagonal elements of each coefficient matrix in this equation give rise to a direct
matrix (e.g. direct stiffness, direct damping matrix) whereas the cross-diagonal elements give
rise cross coupled matrices; reference (Ozguven and Ozkan (1984)) may be seen for details. The
expressions of generalized force vectors comprising of forces and moments acting in the ‗x-y‘
and ‗z-x‘ planes i.e.
F 
xy (1 x 4 )
F 

T
xy (1 x 4 )
. Composition of stiffness, circulatory as well as
damping matrices are also shown below.
Fxy  
I

(1x4) 
a 0  K b  q  a1  K c  q  a1  K b  q 


(1x8)
(1x8)
(1x8) 
(8x8)
(8x8)
(8x8)

b

b
D


F


0
1
 zx (1x4) 
The expression of [Kb] and [Kc] are given as
l
 Kb    I    x    x 
0
T
 Kc   0 I  ( x) 1
l
dx,
0

1
 ( x)
0

T
dx
…….. (17)
Operating by the operator den(D) = (b0 + b1D) throughout and arranging terms with
same orders of differentiation together, the equations of motion of one shaft element are given
below. Assuming the rotor is rotating at a uniform speed ().
b1  M  q    b0  M   b1 G  q   b0 G   a1  K b  q
  a0  Kb   a1  K c  q   b0 b1D   P
……… (18)
Where, [M](8x8) = [MT](8x8) + [MR](8x8) is the inertia matrix, [MT](8x8) is the translational mass
matrix, [MR](8x8) is the rotary inertia matrix, [GT](8x8) is the gyroscopic matrix. Effects due to
simultaneous action of spin and vibratory motion the rotary inertia and gyroscopic matrix are
taken into account. The expressions of translational mass matrix, rotary inertia matrix and
gyroscopic matrix are given below after following Rao (1996).
 M R   0  I   x     x T dx G   0 2 I   x  01
l
l
 M T     A  x    x T dx,
0
,
l
,
1
T
    x  dx
0
Where, ρ is the density of the material, A is the cross-sectional area. The symbols (•) and (´)
stand for single partial differentiation with respect to time,


and space,
respectively.
t
x
It may be noted that the 3rd order differential equation result in this process. Dutt and
Nakra (1992) also obtained a third order differential equation as they modelled the rotor supports
of an elastic rotor-shaft using a 4-element spring-dashpot model.
Again equation (18) may be further combined and rewritten as
 A X   B X   P 
(19)
 A3 
 A    0
  0

  0   A3   0 
q
0 
 0  0 



 
 A3  0  , B    0 0   A3  , X   q , P   b b D  0 
q
P
 A0   A1 
0  A3 
 A2  

 
 
0
1
CHAPTER THREE
Results and Discussion
3.1. The Rotor-Shaft System
A rotor shaft system as shown in figure 3.1, is made of aluminium (E = 7.13e10 Pa, ρ =
2750 Kg/m3), has been considered. The rotor shaft (Length = 0.75 m, Diameter = 0.05 m) is
mounted at the ends in bearings considered as simply supported ends. The aluminium disc (Disc
diameter = 0.15 m, Disc thickness = 0.03 m) is put centrally and has an unbalance, U = 10 gmmm. Following Lesieutre and Mingori (1990) the ATF parameters of aluminium are B = 8000
sec-1,  = 8000 Pa,  = 4.7766e6 Pa.
Figure 3.1 Schematic diagram of Rotor
3.2. Stability limit of spin speed and Unbalance response amplitude
Spin-speed for stability of the aluminium rotor–shaft system (Figure 3.2) has been found
out from the sign of the maximum real part of the eigenvalues. The figure shows a plot of the
maximum real parts for various spin speed. SLS corresponds to the spin speed when the
maximum value of the real part of all the eigenvalues touches the zero line. Steady state
synchronous Unbalance Response Amplitude (UBR) of the disc is plotted in figure 3.3 within the
respective stable speed zones of operation. In certain speed the response has a maximum
amplitude, which indicate the resonance of the system.
Figure: 3.2 Variation of maximum real parts vs. spin-speed
Figure: 3.3 Mass UBR and corresponding stability for disc at the middle.
Figure 3.4 shows the unbalance response amplitude at different disc positions when the
rotor is divided into 10 elements having 11 nodes. It is observed at 6th node (when disc is at the
middle) the amplitude of unbalance response is maximum and further shifting disc towards the
left and right from the middle the UBR amplitude decreases despite of changing the unstable
zone (beyond spin speed of 1.05e+004 rpm). So, figure 3.4 shows that by changing disc position
only UBR amplitude changes without changing unstable zone.
Figure 3.4 Unbalance Response Amplitude at different disc position.
3.3 Time response of the disc
In this section the theoretical results of time response both for transient and steady state
excitation have been analyzed. In transient analysis the rotor is perturbed by an initial velocity of
0.38 m/s at the disc node, whereas in the case of steady state analysis, the rotor is excited due to
unbalance.
Figure 3.5 and figure 3.6 shows the UBR amplitude for different time span in stable zone,
both in Y and Z direction when the rotor is perturbed by initial velocity at the centre of the rotor.
As it is a case of transient excitation, after certain instant both the transient response in Y and Z
direction decreases and reaches at zero. It is due to the nature of the viscoelastic material of the
rotor, because after certain instant the energy dissipates and vibration reduces.
Also, figure 3.4 and figure 3.5 shows the, the UBR amplitude for different time span for
steady state in stable zone, both in Y and Z direction when the rotor is excited, due to unbalance.
As it is a case of steady state excitation, the response amplitude is constant throughout the time
span.
Figure: 3.5 Time response of rotor in Stable Zone (Y- Direction).
Figure: 3.6 Time response of rotor in Stable Zone (Z- Direction).
Figure 3.7 and figure 3.8 shows time response of rotor in unstable zone. When the rotor is
allowed to rotate above SLS, the response amplitude monotonically increases with time. So
beyond this speed the system becomes unstable and does not reach any steady state. It is the
uncontrolled state of vibration, after certain time the failure of rotor takes place. So it is advisable
to rotate much below that speed.
Figure 3.7: Time response of rotor in Unstable Zone (Y- Direction)
Figure3.8: Time response of rotor in Unstable Zone (Z- Direction).
CHAPTER FOUR
Conclusions and scope for future work
4.1 Conclusions
This project work has given the equations of motion of a rotor-shaft system having a
viscoelastic rotor. The linear viscoelastic rotor-material behaviour is represented in the time
domain where the instantaneous stress is obtained by operating the instantaneous strain. The
mechanical analogy i.e. rheological model is sometime difficult to represent for all viscoelastic
materials. The operator may be suitably chosen according to the material model. The formulation
has been found very useful to generate equations motion by discretizing the rotor continuum into
finite beam elements and study the dynamic behaviour of rotor-shaft systems in terms of stability
limit of the spin speed as well as unbalance response of the disc. Temporal variation of disc
response has also been plotted as a further verification of stability of the rotor-shaft system. So
this work is useful for dynamic analysis of viscoelastic rotors under any type of dynamic forcing
function.
4.2 Scope for future work
This study has given birth to different other possibilities which may be taken up as future
research activities in this area.
1) In the present work ATF approach is represented in differential time operator based
approach to obtain the equations of motion of a rotor-shaft system is more suitable for
finding the equation of motion of a viscoelastic rotor which is used to predicts the
dynamic behaviour of the continuum.
2) In this work 3-element mechanical model is used for physical representation of
viscoelastic behaviour for simplicity. Here, 4-element model can also be used for
better prediction of stability limit speed (SLS) and unbalance response (UBR) of the
rotor-shaft system.
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