Report of Rotor

1. 1
Report of Rotor
Winter 2021

2. Abstract
In this project, we try to consider mechanical properties of rotor system as
dynamically and statistically in order to save big archive will use to solve problem
in rotor system at Oil Turbo Compressor Equipment (OTCE). In this report shows
first definitions, literature reviews, mathematical models to how know of rotor
system as well. Structure rotor is including, shaft, impeller, bearing are main
components and these parts which are important duty in rotordyanmic. Also, every
part of rotor system might be dramatically faced faults, so condition monitoring
and simulation as mechanically, can help us to reduce future problems, certainly.
Some fundamental phonemes are happening on rotor systems that are containing
vibration, shock, misalignment, unbalancing, instability, unhealthy bearing and
fractures as randomly as it can negatively effect on rotor performance.

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Aim and Scope
In this report, we are presenting different concepts of rotor system. These concepts
are containing as:
1. Preliminary concepts and explanations
2. Literature review
3. Math Modeling
4. Conclusion
5. References
Structure of chapters
 Preliminary concepts and explanations
Certainly, for every system have to been defined so many variables, parameters, physics and
finally definitions that are helping for everybody can initially understand how know in which
rotor systems, for instance. So, in this section, we would prefer to define various aspects of rotor
system, rotor dynamic and so on for example rotor response, stability and other concepts will
gradually add in this report.
 Literature review
Surely, literature review can show old researches during many years, which were presented from
researchers and scientists in order to extension and modifications of system, especially rotor
systems into industries and universities, the main aim and scope of this part to illustrate short
definitions and model as experimentally and theoretically as well. We are looking forward to
read some research papers and reports to reach just below target:
To know and implement old version modeling for practical usage into industry.
 Math Modeling
In math modeling, we want to model rotor systems from basic model to advanced model by
using literature reviews in previous section as practically as we consider experimenting in our
activities. Therefore, math modeling is most significant part in this report, because it can be
helped us to develop and modify our system in higher quality performance and output. Also we
will show results either graphs or tables.

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 Conclusion
In this part, obviously we must explain all results of which models as details in the other hand, in
this section, conclusions have to help us to solve our challenges’ industry very good.
 Reference
In final report, we must refer all references which were used during parts of report. These
references are also technical reports, research papers, books and other related notifications.

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Chapter 1
Preliminary concepts and explanations

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 Rotor Response
In rotor response, there are some concepts that model mechanical behavior of rotor
performance such as lateral analysis, torsional vibration and stability analysis.
Engineers develop mass-elastic models for rotating components (e.g., compressors,
turbines, pumps, motors, and gearbox shafts), accounting where appropriate for the
lateral stiffening effect of interference fits. The resultant model enables prediction
of rotor system dynamic characteristics, such as:
 Lateral critical speeds
 Torsional critical speeds
 Response to unbalance excitation
 Stability
 LATERAL ANALYSIS
To predict the lateral critical speeds of rotors and to determine sensitivity to
unbalance.
 Undamped critical speeds
 Bearing performance
 Stiffness and damping coefficients
 Damped unbalance response amplitudes and frequencies
 Rotor stability
 Mode shapes
 TORSIONAL VIBRATION
To predict the torsional critical speeds of the entire train, including the effects of
gear boxes, couplings, etc.
 Undamped critical speeds
 Transient critical speeds, including synchronous motor start-ups
 Mode shapes
 Cumulative fatigue criteria, such as maximum starts
 STABILITY ANALYSIS

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To determine system stability and sensitivity to instability mechanisms. These
analyses involve:
 Aerodynamic cross coupling
 Logarithmic decrement predictions
 BEARINGS
Fluid film, tilting pad bearings influence the dynamics of turbomachinery rotor
systems. Plain fluid film bearings act as highly loaded dynamic elements in
reciprocating engines. Rolling element bearings carry the high-speed rotors of
modern aircraft gas turbine engines and their derivatives in power generating and
mechanical drive service. Squeeze film dampers help moderate resonant vibration
levels in gas turbine engines, and some manufacturers use them to stabilize high-
performance centrifugal compressors.
 Campbell Diagram
The Campbell diagram is one of the most important tools for understanding the
dynamic behavior of the rotating machines. It basically consists of a plot of the
natural frequencies of the system as functions of the spin speed. Although being
based on complete linearity, the Campbell diagram of the linearized model can
yield much important information concerning a nonlinear rotating system. A
critical speed of order k of a single – shaft rotor system is defined as spin speed for
which a multiple of that speed coincides with one of the system’s natural
frequencies of precession.
Aspects of rotating machine behavior
Lateral vibration
Rotor lateral vibration (sometimes called transverse or flexural vibration) is
perpendicular to the axis of the rotor and is the largest vibration component in most
high-speed machinery. Understanding and controlling this lateral vibration is

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important because excessive lateral vibration leads to bearing wear and, ultimately,
failure. In extreme cases, lateral vibration also can cause the rotating parts of a
machine to come into contact with stationary parts, with potentially disastrous
consequences.
Lateral vibration is generally caused by lateral forces, the most common of which
are unbalance forces that are present in all rotating machines, despite efforts to
minimize or eliminate them. In subsequent chapters, we discuss the effects or
rotate unbalance and the methods for balancing real machines, but the balance will
never be perfect.
As in all elastic systems, a machine has natural frequencies of lateral vibration
determined by the lateral stiffness and mass distribution of the rotor-bearing
foundation system. When the rotational speed and, hence, the frequency of the out-
of-balance forces is equal to any of these natural frequencies the vibration
determined by the lateral stiffness and mass distribution of the rotor-bearing-
foundation system. When the rotational speed- and, hence, the frequencies, the
vibration response becomes large and the rotor is considered to be rotating at
critical speed. When a machine is accelerated from rest to its operating speed, it
might have to pass through one or more of these critical speeds. For most classes
of machine, it is important that it is not permitted to operate at or close to critical
speed for any length of time.
Because the rotor can vibrate laterally in two mutually perpendicular directions,
the vibration combines to create an orbit for the rotor motion. If the supporting
structure of the bearings of a horizontal rotor has identical stiffness and damping
properties in both the horizontal and vertical directions, then this orbit is circular
and the bending stresses in the rotor are constant. In many instances, however, the
structure supporting the bearings is stiffer in the vertical than in the horizontal
direction. In such a situation, the rotor orbit is elliptical and the bending stress in
the rotor varies at twice the rotational speed.
In the discussion thus far, the role of dissipative or damping forces on the motion
has not been mentioned. As in structural dynamic, damping has a major influence
close to the resonant frequencies. Although it might be anticipated that damping

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always tends to reduce vibration, this is not always the case. If the damping forces
arise in the supporting structure, then the effects are invariably beneficial and may
be treated in much the same way as damping in any structural system. Problems
arise, however, when there is damping in the rotor itself. Far from being beneficial,
this type of damping can be destabilizing.
Axial Vibration
The ultimate function of a jet engine is to produce thrust in the axial direction. A
thrust bearing must be fitted to transmit this thrust to the housing and, hence, the
aircraft to which it is attached. Without this thrust bearing, the rotor would simply
be propelled away from the engine housing and, therefore, would be ineffective. Of
course, there is some time-varying fluctuation about the mean level of thrust,
which gives rise to axial vibrations of the rotor, with this motion having its own set
of resonance frequencies. In contrast to the lateral motion of the rotor, stresses
arising from axial vibration are uniform across a complete cross-section of the
rotor. There may be cross-coupling between axial and lateral vibrations-for
example, in helical and verbal gear meshes.
Torsional Vibration
The third type of vibration is torsional vibration, or a twisting motion of the rotor
about its own axis in some respects, this is relatively straight-forward to model
because bearings and supporting structures have little influence on the natural
frequencies. There is also a practical problem: lateral and, to a lesser extent, axial
vibrations become obvious by their effects on the machine and its surroundings,
enabling the deployment of appropriate effort to resolve developing problems. In
complete contrast, torsional problems can go unnoticed without special
instrumentation. Furthermore, because little motion is transmitted to components
other than the rotor torsional modes often have low damping. During this
undetected phase, however, considerable damage may be caused to a machine.
Introduction Rotor Dynamic

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Rotor-Dynamics is a specialized branch of applied mechanics concerned with the
behavior and diagnosis of rotating structures. It is commonly used to analysis the
behavior of structures ranging from jet engines and steam turbines auto engines
and computer disk storage.
Rotor-dynamic Failure
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 rotating structure; everyday examples include engine balance and tire
balance.
If the amplitude of vibration at these critical speeds is excessive, then catastrophic
failure occurs (loss of the equipment, excessive wear and tear on the machinery,
catastrophic breakage beyond repair or even human injury and loss of lives). In
addition to this, turbomachinery often develop instabilities which are related to the
internal makeup of turbomachinery, and which must be corrected. This is the major
concern of engineers who design large rotors.
Critical speed and approach to solve Rotor dynamics
1. The critical speed of a rotating machine occurs when the rotational speed
matches its natural frequency. The lowest speed at which the natural
frequency is first encountered is called the first critical speed, but as the
speed increases, additional critical speeds are seen. Hence, minimizing
rotational unbalance and unnecessary external forces are very important to
reducing the overall forces which initiate resonance
2.
3. Finite element method (FEM), which is one of the approach for modeling
and analysis of the machine for natural frequencies.
4. Practical approach of balancing the components is by using balancing
machine.
Applications of rotating machines

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1. Marine propulsion/turbomachines
2. Power stations/automobiles
3. Machine tools/household machines
4. Aerospace applications
Campbell diagram for a simple rotor
The Campbell diagram, also known as “Whirl speed map” or a “Frequency
interference Diagram”, of a simple rotor system is shown on the below.
Figure 1: A sample of Campbell diagram
The pink and blue curves show the backward whirl (BW) and forward whirl (FW)
modes. Respectively, which diverge as the spin speed increases. When the BW
frequency or the FW frequency equal the spin speed Ω, indicated by the
intersections A and B with the synchronous spin speed line, the response of the
rotor may show a peak. This is called a critical speed.
The equation of motion, in generalized matrix form, for an axially symmetric rotor
rotating at a constant spin speed Ω is
𝑀𝑞̈(𝑡) + (𝐶 + 𝐺)𝑞̇(𝑡) + (𝐾 + 𝑁)𝑞(𝑡) = 𝑓(𝑡)

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Where:
M is the symmetric Mass matrix
C is the symmetric damping matrix
G is the skew-symmetric gyroscope matrix
K is the symmetric bearing or seal stiffness matrix
N is the gyroscopic matrix of deflection for inclusion of e.g., centrifugal elements.
Q is the generalized coordinates of the rotor in inertial coordinates and f is a
forcing function, usually including the unbalance.
The gyroscopic matrix G is proportional to spin speed Ω. The general solution to
the above equation involves complex eigenvectors which are spin speed dependent.
Engineering specialists in this field rely on the Campbell Diagram to explore these
solutions.
Reference Frames
Rotor dynamics simulation can be performed
1. Stationary reference frame
2. Applies to a rotating structure (rotor) along with a stationary support
structure
3. Rotating part of the structure to be modelled must be axisymmetric
4. A non-axisymmetric part can be transformed into equivalent axisymmetric
mass
5. Axisymmetric is the basic assumption for rotor dynamic theory in the
stationary reference frame but deviations from axisymmetric are supported
as small deviations will introduce small accuracy loss.

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Rotating reference frame
The structure has no stationary parts and the entire structure is rotating
Consider only the Coriolis force.
Jeffcott Rotor
 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.
 It is a rotating machinery equivalent to the single spring mass damper
system with a lumped mass on a massless elastic shaft.
 It is a simple system that is generally used to introduce rotor dynamics
characteristics.
Types of Jeffcott
1. Flexible Rotors
 The rotor shaft is much flexible compared to bearing and foundation
support
 Rotor mounted on very stiff bearings resulting in shaft modes
2. Rigid rotors
 The rotor shaft is much stiffer compared to bearing and foundation
support
 Rotor mounted on very flexible bearings resulting in bearing modes
Rotor dynamic response analysis types
Following types of analysis can be performed in ANSYS
 Modal analysis

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It gives the different modes shapes along with natural frequencies. Also obtain the
critical speeds from the Campbell diagram which is useful to review the stability
and resonance occurrence possibilities.
 Harmonic
It allows you to calculate the response to balance and unbalance excitations.
 Transient analysis
It allows you to study the response of the structure under transient loads (for
example, a 1G shock) on the related components.
 Instability
Self-excited vibrations in a rotating structure cause an increase of the vibration
amplitude overtime such as shown below, such instabilities, if unchecked, can
result in equipment damage.
The most common sources of instability are:
 Bearing characteristics (in particular when non-symmetric cross-terms are
present)
 Internal rotating damping (material damping)
 Contact between rotating and static parts
Rotor unbalance
The major cause of excessive vibrations in rotating shafts is the residual unbalance
and it might be due to material in homogeneities, manufacturing processes,
keyways, slots, etc.
During its life cycles, rotor may subjected to wear between rotating parts, thermal
loads(bending) and dust/dust accumulation, which results in rotor deterioration in
its balance condition. So it is important to determine the rotor response due to
specified unbalance.

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Excitations that depend on rotation velocity are:
 Mass unbalance
 Coupling misalignment
 Aerodynamic excitations as in centrifugal compressors
To perform a harmonic analysis of an unbalanced excitation, the effect of the
unbalanced mass is represented by forces in the two directions perpendicular to the
spinning axis:
Figure 2: the effect of the unbalanced mass on rotor dynamic
Table 1: Parameters effect on unbalanced rotor
symbol definition
x Spin axis
m Unbalance mass
Ω Rotational velocity
r Radius of the eccentric mass
w Angular velocity in rad/sec
x
m
z
𝐹𝑧
𝐹𝑏 = 𝑚𝑟Ω2
= 𝐹0Ω
2
y
r
𝐹𝑦

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Transient Analysis
3. Transient structural analysis is used to find out the dynamic response of a
structure under the action of any general time-dependent loads. It is dynamic
analysis in time domain.
4. Loading will be time history and response will get as time history of response.
It means, load vs. time will be the input and response (displacement, stress,
velocities, acceleration) vs. time will be the result.
5. It can use it to determine the time-varying displacements, strains, stresses, and
forces in a structure as it responds to any transient loads.
6. It can includes, inertia, damping and nonlinear effects.
7. It is also known as time-history analysis or transient structural analysis.
Shrink-Fitting
Shrink fitting is a procedure in which heat is used to produce a very strong joint
between two pieces of metal, one of which is inserted into the other. Heating
causes one piece of metal to contract or expand on to the other, producing
interference and pressure which holds the two pieces together mechanically.
Most applications involve a shaft with a given outside diameter and another part
such as a gear, steering knuckle, rollers or washer which has a bore hole. There are
several forms of this type of joint, characterized by the amount of tolerance (space)
between the two parts. The shrink fitting process is used for the interference fit
type joint.
By heating the mass around the bore hole uniformly, it is possible to significantly
expand the size of the hole. The shaft is then easily inserted into the expanded hole.
Upon cooling, the mass around the hole shrinks back to its original size and
frictional forces create a highly effective joint. Industries and applications of
shirink-fitting
•Railway - gearboxes, wheels, transmissions
•Machine tools - lathe gearboxes, mills
•Steel works - roll bearings, roll neck rings
•Power generation - various generator components
•Cement Industries rollers

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•Rerolling mill rollers
Due to the need to insert a core and also that to be effective, the core has to be in
relatively close proximity to the bore of the part to be heated, there are many
application in which the above bearing heater type approach is not feasible.
There are a huge number of industries and applications which benefit from shrink
fitting or removal. In practice, the methodology employed can vary from a simple
manual approach where an operator assembles or disassembles the parts to fully
automatic pneumatic or hydraulic press arrangements.
•Automotive starter rings onto flywheels
•Timing gears to crankshafts
•Motor stators into motor bodies
•Motor shafts into stators
•Removal and re-fitting of a gas turbine impeller
•Removal and re-fitting of hollow bolts in electrical generators
•Assembly of high precision roller bearings
•Shrink-fitting of 2-stroke crankshafts for ship engines
Figure 3: Experimental Shrink-fitting

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Chapter 2
Literature review

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Introduction
As you know, in this chapter, we want to survey research findings of rotor systems
in some subjects such as vibration rotors, vibration control rotors, rotor response,
finite element modeling in which rotors and condition monitoring rotors. So, every
title are describing in every sub-section.
1. Vibration Rotors
 1960
The stability of motion of a rotor with unsymmetrical shaft on an elastically
supported mass foundation [1].
 1965
Large wound-rotor motor with liquid rheostat for refinery compressor drives [2].
Figure 4: Compressor and motor drive installation
 1966
Some intake flow maldistribution effects on compressor rotor blade vibration [3].
 1972
A finite element model for distributed parameter turbo rotor systems [4]. Effect of
an adjustable non-uniform pitch in the distributor on the alternating stresses in
compressor rotor blades [5].

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 1974
Influence of the material of the rotor blades of an axial-flow compressor under
flutter initiating conditions [6].
 1976
The dynamics of rotor-bearing systems using finite elements [7].
Figure 5: Typical system configuration
 1977
Finite element simulation if rotor-bearing systems with internal damping [8].
Figure 6: Rotor system configuration
The influence of non-Newtonian oil film short journal bearings on the stability of a
rigid rotor [9].

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 1979
Study of the flow field behind a transonic axial compressor rotor using laser-
anemometry and unsteady pressure measurements [10]. Turbulence characteristics
in the near wake of a compressor rotor blade [11].
 1980
Prediction of dynamic properties of rotor supported by hydrodynamic bearings
using the finite element method [12].
Figure 7: schematic shaft-disk- bearing system
Figure 8: Showing complete dynamic analysis
The dynamic of rotor-bearing systems with axial torque- a finite element approach
[13]. Transient, three-dimensional, finite element analysis of heat flow in turbine-
generator rotors [14]. Dynamic reduction in rotor dynamics by the finite element
method [15]. Comparison between optical measurements and a numerical solution
of the flow field within a transonic axial flow compressor rotor [16].

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 1981
Laser anemometer measurements in a transonic axial flow compressor rotor [17].
 1982
Dynamic stability of a rotor blade using finite element analysis [18]. Finite element
analysis of the dynamic of flexible disk rotor systems [19]. Casing wall boundary-
layer development through an isolated compressor rotor [20]. Three dimensional
flow field inside the passage of a low speed axial flow compressor rotor [21].
 1983
An experimental study of the unsteady response of the rotor blades of an axial flow
compressor operating in the rotating stall regime [22]. Vibration control of multi-
mode rotor bearing systems [23].
Figure 9: Symmetrically supported flexible rotor carrying 3 rigid discs
 1984
Whirl speeds and unbalance response of multi-bearing rotors using finite elements
[24].

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Figure 10: Three bearing model, dual squeeze film damper
Nonlinear transient finite element analysis of rotor-bearing-stator systems [25]. A
general method of analysis for dynamic problems of rotor systems [26]. Laser
Doppler velocimetry measurements in the tip region of a compressor rotor [27].
Thermal analysis of a high-pressure compressor rotor of an aero-engine-venting as
a means for life improvement [28].
 1985
Finite element approach to rotor blade modeling [29]. Numerical simulation of
unsteady flow in a compressor rotor cascade [30]. An experimental investigation of
stator/rotor interaction influence on multistage compressor rotor flow [31].
 1986
Finite element analysis of rotor bearing systems using a modal transformation
matrix [32]. Whirl and whip rotor/bearing stability problems [33].
 1987
Rotor wake segment influence on stator-surface boundary layer development in an
axial-flow compressor stage [34].
Figure 11: Surface hot-film sensor data acquisition system

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Laser anemometry study of the flow field in a transonic compressor rotor [35].
 1988
Modal analysis of continuous rotor-bearing systems [36]. On the transient analysis
of rotor-bearing systems [37].
Figure 12: Model rotor system
On the stability of a spinning, fluid filled and sectored rotor [38]. Some comments
on the stability of spinning-rotor gauges [39]. The principle of general energy
conservation and application to the stability analysis of a rotor-bearing system
[40]. Stability of whirl and whip in rotor/bearing systems [41]. Self-tuning adaptive
control of forced vibration in rotor systems using an active journal bearing [42].
Figure 13: Schematic of the active journal bearing

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Figure 14: A test rig for a three-bearing rotor system
Experimental vibration damping characteristics of the third-stage rotor of a three-
stage transonic axial-flow compressor [43].
Figure 15: Multistage compressor test facility
 1989
Structure of tip clearance flow in an isolated axial compressor rotor [44].
 1990
Tapered Timoshenko finite elements for rotor dynamic analysis [45].
 1991
Finite element analysis of natural whirl speeds of rotating shafts [46]. Modeling of
complex rotor systems by combining rotor and substructure models [47].
Application of higher harmonic control to hinge less rotor systems [48].

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 1992
Unbalance response of rotor systems by the dynamic stiffness method [49].
Figure 16: Considering 18 nodes for finite element method of rotor
Finite element analysis of the rotor/stator contact in a ring-type ultrasonic motor
[50].
 1993
Stochastic finite element analysis for high speed rotors [51]. Effects of rotor
eccentricity and parallel windings on induction machine behavior: A Study using
finite element analysis [52]. Generalized polynomial expansion method for the
dynamic analysis of rotor-bearing systems [53].
 1994
Influence of non-classical friction on the rubbing and impact behavior of rotor
dynamic systems [54]. A finite element model for a flexible non-symmetric rotor
on distributed bearing, a stability study [55]. Engineering feasibility of induced
strain actuators for rotor blade active vibration control [56]. Indexing a dual-rotor
compressor with nonlinear rotor synchronization [57]. The improvement of screw
compressor performance using a newly developed rotor profile [58]. Active
vibration control of rotor systems [59].
 1995
The effect of damping on the stability of a finite element model of a flexible non-
axisymmetric rotor on tilting pad bearings [60]. Characterization of the first-stage
rotor in a two-stage transonic compressor [61]. The effect of adding roughness and

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thickness to transonic axial compressor rotor [62]. Tip clearance flow-shock
interaction in a transonic compressor rotor [63]. Experimental investigation of the
rotor blade vibration in the three compressors of the 11-by 11-foot transonic wind
tunnel [64]. Unsteady flow and shock motion in a transonic compressor rotor [65].
 1996
Dynamic response and stability of a rotor-support system with non-symmetric
bearing clearances [66].
Figure 17: The Jeffcott rotor system
Coupled torsional-flexural vibration of shaft systems in mechanical engineering-
finite element model [67]. The dynamic stiffness method for linear rotor-bearing
systems [68]. Movement simulation in finite element analysis of electric machine
dynamic [69]. Experimental and computational investigation of the tip clearance
flow in a transonic axial compressor rotor [70].
 1997
Modeling, synthesis and dynamic analysis of complex flexible rotor systems [71].

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Figure 18: A hypothetic complex rotor system C1-coupling of nodes 4 and 5 by the inter-shaft bearing; C2-
coupling of nodes 7 and 9 by the gear pair
The non-linear analysis of the effect of support construction properties on the
dynamic properties of multi-support rotor systems [72].
Figure 19: The diagram of viscous-elastic characteristics of the oil film (a) and the model of heat exchange in the
bearing (b).

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Figure 20: The flow chart of the KINWIR and NLDW programs
Dynamic analysis of a rotor system considering a slant crack in the shaft [73].
Figure 21: A rotor-bearing system model with a slant crack element

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Figure 22: Vibration response of rotor
The non-linear analysis of the effect of support construction properties on the
dynamic properties of multi-support rotor systems [74].
Figure 23: The diagram of viscous-elastic characteristics of the oil film (a) and the model of heat exchange in the
bearing (b)
On the dynamic analysis of rotors using modal reduction [75].

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Figure 24: The configuration of multi-stepped rotor bearing system
Dynamic analysis of a rotor system considering a slant crack in the shaft [76].
 1997
Numerical investigation of the rotor-stator investigation in a transonic compressor
stage [77]. Vibration control of rotor systems with non-collocated sensor/actuator
by experimental design [78]. Large eddy simulation of unsteady rotor-stator
interaction in a centrifugal compressor [79]. Near-wake measurements in a
rotor/stator axial compressor using slanted hot-wire technique [80]. Rotor blade
pressure measurement in a high speed axial compressor using pressure and
temperature sensitive paints [81]. Numerical investigation of intercoupling
vibration of a mistuned turbomachinery compressor rotor wheel [82].
 1998
Dynamic analysis and reduced order modelling of flexible rotor-bearing systems
[83]. Design, fabrication, and testing of 10 MJ composite flywheel energy storage
rotors [84].
Figure 25: Rotor on lid of spin pit

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Finite element analysis of whirl speeds for rotor-bearing systems with internal
damping [85]. An algorithm for response and stability of large order non-linear
systems application to rotor systems [86]. Experimental and numerical
investigation of unsteady rotor-stator interaction on axial compressor stage (with
IGV) performance [87]. An experimental investigation of IGV-rotor interactions in
a transonic axial-flow compressor [88]. Turbulence energy and spectra
downstream an axial compressor rotor [89]. Unsteady aerodynamics in transonic
compressor rotor blade passage [90]. Mistuning characteristics of a bladed rotor
from a two-stage transonic compressor [91]. Three- dimensional flow
measurements down-stream of an axial compressor rotor using a four-hole pressure
probe [92].
 1999
Dynamic analysis of multi-stepped, distributed parameter rotor-bearing systems
[93]. Statical and dynamical behavior of rotors with transversal crack, comparison
between different models [94].
Figure 26: Forces and moments acting on the cracked element in the vertical plane
Vibration characteristics of cracked rotors with two open cracks [95]. Coupled
bending torsional vibration of rotors using finite element [96]. Rotor dynamic
response at blade loss [97].

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Figure 27: FE model of the turbine, the supports and the skid structure
Design of optimum support parameters for minimum rotor response and maximum
stability limit [98].
Figure 28: Rotor system diagram
A PIV investigation of rotor-IGV interactions in a transonic axial-flow
compressor[99]. 3D-shock visualization in a transonic compressor rotor [100].
Three-Dimensional viscous analysis of rotor-stator interaction in a transonic
compressor [101]. Effect of upstream rotor vertical disturbances on the time-
averaged performance of axial compressor stators, framework of technical
approach and wake-stator blade interactions [102].

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 2000
Dynamic analysis of rotor-shaft system with viscoelastically supported bearings
[103].
Figure 29: The rotor shaft system with viscoelastically supported bearings.
Identification of the oil-film dynamic coefficients in a rotor-bearing system with a
hydrodynamic thrust bearing [104]. Rigid rotor dynamic stability using Floquet
theory [105]. Dynamic response of an unbalanced rotor supported on ball bearings
[106]. Identification of transverse crack position and depth in rotor systems [107].
Figure 30: 4-bearing 2-shaft MODIAROT test-rig on flexible foundation with fully instrumented bearing housing
Design and finite element analysis of an outer-rotor permanent-magnet generator
for directly coupled wind turbines [108]. Improved parameter modeling of interior
permanent magnet synchronous motor based on finite element analysis [109]. On
the discrete-continuous modeling of rotor systems for the analysis of coupled
lateral torsional vibrations [110].

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Figure 31: (a) Finite element and (b) discrete-continuous models of the rotating machine
Stability of complicated rotor-bearing system by overlapping decomposition-
aggregation method [111]. Stability of a rotor supported on journal bearings with
piezoelectric elements [112]. Navier-stockes simulation of IGV-rotor-stator
interactions in a transonic compressor [113]. Adaptive-Q control of vibration due
to unknown disturbances in rotor/magnetic bearing systems [114].
Figure 32: Schematic of vibration control problem
 2001
Determination of the rubbing location in a multi-disk rotor system by means of
dynamic stiffness identification [115]. Modeling the dynamic behavior of a
supercritical rotor on a flexible foundation using the mechanical impedance
technique [116].

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Figure 33: Model adopted for a multi-span coupled rotor-bearing-foundation system. (a) Projection on yz plane (b)
general projection on xy plane
Using genetic algorithms and finite element methods to detect shaft crack for rotor-
bearing system [117]. Experimental and torsional modal analysis of three support
rotor test rig using LMS CADA-X and ABAQUS [118].
Figure 34: Rig for investigations of rotor and bearings dynamics (1)shaft(2)bearings(3)bearings loading
disc(4)coupling(5)resistance bearing(6)frame(7)frame supports(8)foundation block(9)pneumatic absorbers(10)
propelling engine(11)gear box

37. 37
Figure 35: Three stages of numerical simulation of test rig
Experimental-calculation determination of dynamic stability of blade assemblies of
GTE compressor rotor wheels [119]. Characterizing unsteady periodic disturbances
in the tip leakage vortex of an idealized axial compressor rotor blade [120].
 2002
Non-linear modelling of rotor dynamic systems with squeeze film dampers- an
efficient integrated approach [121]. Effects of rotor misalignment in airgap on
dynamic response of an eccentric rotor in BLDC motor [122]. Dynamic
characteristics of indeterminate rotor systems with angular contact ball bearing
subject to axial and radial loads [123]. DPIV study of near-stall wake-rotor
interactions in a transonic compressor [124].
Figure 36: Flow path of 200-hp compressor aerodynamic research laboratory (CARL) facility
Experimental analysis of the rotor stator interaction within a high pressure
centrifugal compressor [125].

38. 38
 2003
Dynamic response and stability analysis of an automatic ball balancer for a flexible
rotor [126]. Modeling of solid conductors in two-dimensional transient finite
element analysis and its application to electric machines [127]. Transverse crack
modeling and validation in rotor systems, including thermal effects [128].
Combining a nonlinear static analysis and complex eigenvalue extraction in brake
squeal simulation [129]. Rotor design and analysis, a technique using
computational fluid dynamics (CFD) and heat transfer analysis [130].
Figure 37: Frequency response of rotor
Dynamics and control of a rotor using an integrated SMA/composite active bearing
actuator[131]. Stability and bifurcation of unbalance rotor labyrinth seal
system[132]. Numerical simulation of rotor clocking effect in a low-speed
compressor[133]. Robust control of multiple discrete frequency vibration
components in rotor- magnetic bearing systems [134].
 2004
Dynamic characteristic of a flexible rotor system supported by a viscoelastic foil
bearing (VEFB) [135]. Non-linear dynamic interactions of a Jeffcott rotor with

39. 39
preloaded snubber ring [136].On the non-linear dynamic behavior of a rotor-
bearing system [137]. Dynamic characteristic of a flexible bladed-rotor with
coulomb damping due to tip-rub [138]. Analysis of whirl speeds for rotor-bearing
systems supported on fluid film bearing [139]. Rotor bearing analysis for
turbomachinery single and dual rotor systems [140].
Figure 38: Dual rotor system model
Nonlinear stability analysis of a complex rotor/stator contact system [141]. Impact
of wake on downstream adjacent rotor in low-speed axial compressor [142] .A
CFD study of the flow through a transonic compressor rotor with large tip
clearance [143] .Fracture of a compressor made from grey cast iron [144].
 2005
Dynamic analysis of hydrodynamic bearing- rotor system based on neural network
[145]. Finite element analysis of coupled lateral and torsional and torsional
vibrations of a rotor with multiple cracks [146] .Hybrid model based identification
of local rubbing fault in rotor systems [147].
Figure 39: Sketch of a rotor-bearing-support system. Node 1 and 11: bearing location e: unbalanced mass

40. 40
Figure 40: The mechanical model of oil film bearing and support
AI vibration control of high-speed rotor systems using electrorheological
fluid[148].
Figure 41: Schematic diagram of the test rig
An efficient scheme for stability analysis of finite element asymmetric rotor
models in a rotating frames [149]. Stability and response analysis of symmetrical
single-disk flexible rotor-bearing system [150]. Unsteady tip clearance flow in an
isolated axial compressor rotor [151]. Unsteady rotor-stator interaction in high
speed compressor and turbine stages [152].
 2006
Non-linear dynamic behavior and bifurcation analysis of a rigid – rotor supported
by a relatively short externally pressurized porous gas journal bearing system[153].
Simultaneous identification of residual unbalances and bearing dynamic

41. 41
parameters from impulse response of rotor-bearing systems [154]. Rotor dynamic
instability analysis on hybrid air journal bearings [155]. Nonlinear dynamic
behaviors of a rotor-labyrinth seal system [156]. Nonlinear dynamic analysis of a
flexible rotor supported by micro polar fluid film journal bearings [157]. Influence
of the mechanical seals on the dynamic performance of rotor-bearing systems
[158]. A finite element transient response analysis method of a rotor-bearing
system to base shock excitations using the state-space Newmark scheme and
comparison with experiments [159].
Figure 42: Set-up of the rotor test-rig and electro-magnetic shaker
Stability, bifurcation and chaos of a high-speed rub-impact rotor system in MEMS
[160]. Optimum design of rotor-bearing system stability performance comparing
an evolutionary algorithm versus a conventional method [161]. The effect of
cavitation on the vibration behavior of nonlinear rotor bearing systems [162].
Figure 43: Schematic drawing of rotor bearing test rig

42. 42
Figure 44: Journal bearing
A design approach to reduce rotor losses in high-speed permanent magnet machine
for turbo-compressor [163].
 2007
Spinning up rotor dynamics [164]. Rotor dynamic capabilities in ANSYS
mechanical [165].
Figure 45: Beam model of a two-spool rotor with symmetric bearings (left) and displacement plot (right)
Dynamic simulation of a flexible rotor during drop on retainer bearings [166].
Effect of axial preload of ball bearing on the nonlinear dynamic characteristics of a
rotor-bearing system [167]. Dynamic stability of rotor-bearing systems subjected
to random axial forces [168]. Simulation investigations of the effect of a

43. 43
supporting structure defect on the dynamic state of the rotor supported on slide
bearing [169].
Figure 46: Rotor of the research
Identification of crack in a rotor system based on wavelet finite element method
[170]. Dynamic stability of rotor-bearing systems subjected to random axial forces
[171]. Effects of a crack on the stability of a nonlinear rotor system [172]. An air
crash to failure of compressor rotor [173]. Unsteady tip clearance flow pattern in
an isolated axial compressor rotor with micro tip injection [174]. Coating failure in
compressor rotor blades of an aero engine [175]. A statistical model for the effect
of casing treatment recesses on compressor rotor performance [176]. Blade
parameterization and aerodynamic design optimization for a 3d transonic
compressor rotor [177].
 2008
Stability and coupling dynamic behavior of nonlinear journal active
electromagnetic bearing rotor system [178]. Dynamic behavior analysis of cracked
rotor [179]. Non-linear dynamic analysis of rub-impact rotor supported by
turbulent journal bearings with non-linear suspension [180]. Dynamic response of
a rub-impact rotor system under axial thrust [181]. Three-dimensional finite
element simulation of nonlinear dynamic rotor systems of a turbocharger [182].

44. 44
Figure 47: Synchronous mode
Vibration based operational model analysis of rotor systems [183]. Control of
flexible rotor systems with active magnetic bearings [184]. Analysis of motion
stability of the flexible rotor-bearing system with two unbalanced disks [185].
Stability and vibration analysis of a complex flexible rotor bearing system [186].
Influence of orifices on stability of rotor-aerostatic bearing system [187].
Numerical simulation of sand erosion phenomena in rotor/stator interaction [188].
Investigation of laser shock peening on aero-engine compressor rotor blade [189].
Rotor-blades profile influence on a gas-turbine’s compressor effectiveness [190].
Numerical investigation of inlet distortion on an axial flow compressor rotor with
circumferential groove casing treatment [191].
Figure 48: Cross-sectional view of tested axial flow compressor

45. 45
Control algorithm design for passing through resonance of two-rotor vibration unit
[192].
 2009
Finite element analysis for gate rotor shaft of single screw compressor [193].
Figure 49: The finite element model for gate rotor shaft
Figure 50: The displacement distribution of gate rotor shaft
Figure 51: The Von-Mises stress distribution of gate rotor shaft

46. 46
Figure 52: The top six vibration shapes of gate rotor shaft
Analytical study of nonlinear synchronous full annular rub motion of flexible
rotor-stator system and its dynamic stability [194]. Simultaneous estimation of the
residual unbalance and bearing dynamic parameters from the experimental data in
a rotor-bearing system [195].
Figure 53: The rotor-bearing test rig
Preload effect on nonlinear dynamic behavior of a rigid rotor supported by
noncircular gas-lubricated journal bearing systems [196]. Crack identification by

47. 47
multifractal analysis of a dynamic rotor response [197]. Dynamic behavior of
hydrodynamic journal bearings in presence of rotor spatial angular misalignment
[198]. Vibration analysis of misaligned shaft-ball bearing systems [199].Rotor
dynamic analysis of 3D-modeled gas turbine rotor in ANSYS [200]. Simulation
and characterization of rotor dynamic properties for hydropower units [201]. Aero
elastic analysis of rotor systems using trailing edge flaps [202]. Methods to
incorporate foundation elasticities in rotor dynamic calculations [203].
Figure 54: Low pressure rotor and balance shop
Simulation of wear and contact pressure distribution at the pad-to-rotor interface in
a disc brake using general purpose finite element analysis software[204].
Stability boundaries of a spinning rotor with parametrically excited gyroscopic
system [205]. Flutter synchronization for turbo-compressor rotor blades [206].
Modelling of turbulent flows in transonic axial flow compressor NASA rotor
37[207].
 2010
Bifurcation and nonlinear dynamic analysis of a rigid rotor supported by two-lobe
noncircular gas-lubricated journal bearing system [208]. Robust design of a HDD
spindle system supported by fluid dynamic bearing utilizing the stability analysis
of five degrees of freedom of a general rotor-bearing system [209]. The dynamic
behavior of a rotor system with a slant crack on the shaft [210]. Couple stress fluid
improves rub-impact rotor-bearing system – nonlinear dynamic analysis [211].

48. 48
Effects of flexible support stiffness on the nonlinear dynamic characteristics and
stability of a turbo pump rotor system [212].
Figure 55: A liquid fuel turbo-pump rotating assembly
Figure 56: Rotor dynamic finite element model
Theoretical rotor dynamic analysis of two-pole induction motors regarding
excitation due to static rotor eccentricity [213]. Dynamic hydro elastic scaling of
self-adaptive composite marine rotors [214]. Modal reduction of geared rotor
systems with general damping and gyroscopic effects [215]. Aerodynamic
sweeping study and design for transonic compressor rotor blades [216]. Active
vibration control of flexible rotors on maneuvering vehicles [217].

49. 49
Figure 57: Rotor bearing system on moving base with different coordinate system
Limit cycle of the shelf-excited oscillations of the rotor blades of a centrifugal
compressor [218]. Numerical investigation of a high-subsonic axial-flow
compressor rotor with non-axisymmetric Hub Endwall [219]. Unbalance
compensation in a rotor-bearing system by dynamic stiffness control and
acceleration scheduling [220].
Figure 58: Rotor wake variability in a multistage compressor

50. 50
Figure 59: The cross section of the research compressor
Vibration absorption in a rotor-bearing system using a cantilever beam absorber
[221].
Figure 60: Schematic diagram of the discretized rotor-bearing
 2011
Complete determination of the dynamic coefficients of coupled journal and thrust
bearings considering five degrees of freedom for a general rotor-bearing system
[222].Effectiveness of impact-synchronous time averaging in determination of
dynamic characteristics of a rotor dynamic system [223]. A time-domain
methodology for rotor dynamics: analysis and force identification [224]. Nonlinear
dynamic analysis of fractional order rub-impact rotor system [225]. Balancing of
flexible rotors at low speed [226]. Dynamic behaviors of a full floating ring
bearing supported turbocharger rotor with engine excitation [227]. A multiple
whirls phenomenon and heuristic problems in rotor-bearing systems [228].

51. 51
Figure 61: Photograph of the testing stand and the FEM discretization of rotor and supporting structure
Modelling of parametric excitation of a flexible coupling- rotor system due to
misalignment [229].
Figure 62: Showing the experimental setup
High-speed stability of a rigid rotor supported by aerostatic journal bearings with
compound restrictors [230]. Advanced comparison solutions for CCS, EOR and
offshore CO2[231].
Figure 63: RG 56-10 compressor in Russia

52. 52
Figure 64: Impellers of RG 80-8 in North Dakota
Rotor natural frequency in high-speed permanent-magnet synchronous motor for
turbo-compressor application [232].
Figure 65: Mode shape of shaft
Figure 66: Mode shape of completed rotor
Application of active magnetic force actuator for control of flexible rotor system
vibrations [233].

53. 53
Figure 67: Bently Nevada rotor kit RK4 with active magnetic actuator
Wet compression performance of a transonic compressor rotor at its near stall point
[234]. Knowledge extraction from aerodynamic simulation data of compressor
rotor [235]. The challenge of stereo PIV measurements in the tip gap of a transonic
compressor rotor with casing treatment [236]. Effect of circumferential grooves
and tip recess on stall characteristic of transonic axial compressor rotor [237].
Efficiency enhancement in transonic compressor rotor blades using synthetic jets, a
numerical investigation [238].
 2012
Thermal effects for shaft-pre-stress on rotor dynamic system [239].
Figure 68: Experimental setup for thermal gyroscopic mode testing

54. 54
Figure 69: A finite element model
Evaluation of critical speed of generator rotor with external load [240].
Figure 70: Generated model for undamped critical speed with external load
The experimental analysis of vibration monitoring in system rotor dynamic with
validate result using simulation data [241].
Figure 71: Finite element simulation

55. 55
Evaluation of gas turbine rotor dynamic analysis using the finite element method
stochastic modelling of flexible rotors [242]. Analyzing the dynamic response of a
rotor system under uncertain parameters by polynomial chaos expansion [243].
Finite element analysis of turbine generator rotor winding shorted turns [244].
Figure 72: Steps taken in modeling a two – pole 266-MVA generator
Computational tradeoff in modal characteristics of complex rotor systems using
finite element method [245].
Figure 73: Actual rotor and its solid model
Automatic balancing of rotor bearing systems [246]. Study on numerical
simulation of flouting in compressor rotor [247]. Levitation and vibration
suppression of an elastic rotor by using active magnetic bearings [248]. An
improvement on the efficiency of a single rotor transonic compressor by reducing
the shock wave strength on the blade suction surface [249]. Inspection of rotor

56. 56
disks of HPT and LPT of ITK-10-4 Gas-compressor units by the ultrasonic flaw
detection method [250]. Stochastic modeling of flexible rotors [251].
 2013
Introduction of rotor dynamics using implicit method in LS-DYNA [252].
Prediction life of horizontal rotors by natural frequency evolution [253]. On the
finite element modeling of the asymmetric cracked rotor [254].
Figure 74: The MFS-RDS spectra-quest rotor dynamic simulator used for experimental analysis
Aero-elastic bearing effects on rotor dynamics, a numerical analysis [255].
Dynamic analysis of three-dimensional helical geared rotor system with geometric
eccentricity [256].
Figure 75: Typical mode shapes of the helical geared rotor system
Stability analysis for transverse breathing cracks in rotor systems [257]. Transient
small wind turbine tower structural analysis with coupled rotor dynamic interaction

57. 57
[258]. Flow structure in the tip region for a transonic compressor rotor [259].
Controlled passage through resonance for two-rotor vibration unit [260]. Design of
rotor and magnetic bearings for 200RT class turbo refrigerant compressor [261].
 2014
Analysis of rotor Dynamics acceptance criteria in large industrial rotors [262].
Estimation of natural frequencies and mode shapes of a shaft supported by more
than three bearings [263].
Figure 76: ANSYS output for natural frequencies and mode shapes
Figure 77: Result from ANSYS workbench for Max shear stress=2.0759 MPa
Study of different stresses induced in rotor shaft of electric motor [264].

58. 58
Figure 78: Shaft Deflection
Rotor dynamic analysis of steam turbine rotor using ANSYS [265].
Figure 79: Full body Rotor structure

59. 59
Figure 80: Mode Shape Bending
Rotor dynamics analysis of a multistage centrifugal pump [266].
Figure 81: Mode shapes and rotor orbits on ANSYS
Unbalanced response of rotor using ANSYS parametric design for different
bearings [267].

60. 60
Figure 82: Model of Nelson rotor with various sections
Figure 83: Variation of amplitude of vibration
Utilizing a general purpose finite element approach for assessing the rotordynaimc
response of a flexible disk/shaft system [268].

61. 61
Figure 84: Mode shapes calculated at 0 rpm with all effects included
Three-dimensional structural evaluation of a gas turbine engine rotor [269].
Figure 85: Discretized 3-D FEA models of the two engine rotors, including the boundary conditions
Figure 86: Normalized equivalent stress distribution in SW501F and GE-7FA rotors at steady state condition

62. 62
Figure 87: Transient response at three time points/steps in GE-7FA engine rotor
Dynamic analysis of a high speed rotor-bearing system [270]. Composite shaft
rotor dynamics, an overview [271]. Isothermal boundary condition at casing
applied to the rotor 37 transonic axial flow compressors [272]. A new multi-
objective evolutionary algorithm for optimizing the aerodynamic design of HAWT
rotor [273]. Active vibration control in a rotor system by an active suspension with
linear actuators [274]. Modeling and analysis of flexible multistage rotor systems
with water-lubricated rubber bearings [275]. Vibration control of multi-mode rotor
bearing systems [276].
 2015
Model identification and dynamic analysis of ship propulsion shaft lines [277].

63. 63
Figure 88: FE model of the outer shaft line
Stability analysis and backward whirl investigation of cracked rotors with time-
varying stiffness [278]. Combined explicit finite and discrete element methods for
rotor bearing dynamic modeling [279]. Efficient modelling of rotor-blade
interaction using sub-structuring [280]. Dynamic performance of turbocharger
rotor bearing systems [281]. The statement of design and application questions for
the gyroscope with a gas-dynamic suspension of ball rotor in the navigation
support drilling system [282]. Unified approach for accurate and efficient modeling
of composite rotor blade dynamics [283].Modal component mode synthesis in
torsional vibration analysis rotor-blade interaction [284]. In-process, non-
destructive, dynamic testing of high-speed polymer composite rotors [285]. Crack
fault diagnosis of rotor systems using wavelet transforms [286]. Field balancing
and harmonic vibration suppression in rigid AMR-rotor systems with rotor
imbalances and sensor runout [287].
 2016
Vibration modes of the rotor system of turbocharger with floating-ring bearing
[288].
Figure 89: Finite element model of rotor

64. 64
Figure 90: Mode shapes
Vibration analysis of a shaft in rotor bearing system by changing dimensional
parameters [289].

65. 65
Figure 91: Vibration mode shape
Campbell diagram analysis of open cracked rotor [290].Static analysis of shaft
(EN24) of foot mounting motor using FEA [291].
Figure 92: CAD model of shaft

66. 66
Figure 93: Deformation change
Finite element modeling of rotor using ANSYS [292].
Figure 94: Meshed model
Figure 95: Mode shapes of system
Rotor dynamic validation of a twin rotor-bearing system considering gyroscopic
forces and bearing dynamic with a multibody formulation: application to a geared

67. 67
UHBR gas turbine [293]. Critical speed analysis of the turbocharger rotor system
based on ANSYS workbench [294].
Figure 96: Vibration mode shape
Rotor dynamic analysis of centrifugal compressor due to liquid carries over a new
dynamic model of ball-bearing rotor systems based on rigid body element [295].
Figure 97: The bearing rotor test rig
An algorithm for response and stability of large order non-linear systems
application to rotor systems [296].
 2017
Rotor bearing system FEA analysis for misalignment [297].

68. 68
Figure 98: A perfect aligned meshed model of rotor bearing system (nodes 18,400)
Figure 99: Maximum deflection values of parallel misalignments at bearing 1
Figure 100: Maximum deflection values of parallel misalignments at bearing 2

69. 69
Dynamic behavior of high-speed rotor [298]. On the dynamic analysis of rotating
shafts using nonlinear super element and absolute nodal coordinate formulation
[299].
Figure 101: Finite element model of a complex-shaped component
Rotor dynamic design analysis of a squeeze film damper test rig [300]. Modelling
and simulation of single rotor system [301].
Figure 102: Mode shape of shaft-disk

70. 70
Figure 103: Variation of amplitudes of vibration
The dynamic analysis of rotors mounted on composite shafts with internal damping
[302].The experimental identification of the dynamic coefficients of two
hydrodynamic journal bearings operating at constant rotational speed and under
nonlinear conditions [303].

71. 71
Figure 104: Test rig
Identification of speed-dependent active magnetic bearing parameters and rotor
balancing in high speed rotor systems [304]. A combined EFEM-DEM dynamic
model of rotor-bearing-housing system [305].
Figure 105: The combined rotor-bearing model with radial load applied at rotor center
Figure 106: Deformed configurations of EFEM rotor and housings at first critical speed

72. 72
Investigations on rotor dynamic characteristics of a floating ring seal considering
structural elasticity [306]. Active vibration control of a rotor bearing system using
piezoelectric patch actuators and an LQR controller [307].
 2018
Rotor dynamic analysis of the AM600 turbine-generator shaft line [308].
Figure 107: Meshed shaft line mode
Figure 108: Concept 1 mode 3 torsional natural frequency and deformation
Figure 109: Concept 1 Mode 4 torsional natural frequency and deformation

73. 73
Figure 110: Concept 2 Mode 4 torsional natural frequency and deformation
Figure 111: Concept 2 Mode 5 torsional natural frequency and deformation
Determination of the critical bending speeds of a multi-rotor shaft from the
vibration signal analysis [309].

74. 74
Figure 112: Bending mode shapes in ANSYS
Figure 113: Experimental models
An enhanced axisymmetric solid element for rotor dynamic model improvement
[310]. Design and modal analysis to calculate critical speed of shaft [311].
Vibration analysis of shaft in SOLIDWORKS and ANSYS [312]. Effects of
unbalance on the nonlinear dynamics of rotors with transverse cracks [313].
Figure 114: 3D finite element model of a cracked rotor

75. 75
Figure 115: Beam-based finite element model of the cracked rotor
Dynamic response of a cracked rotor with an unbalance influenced breathing
mechanism [314]. Interactions in the rotor-bearings- support structure system of
the multi-stage ORC micro turbine [315].
Figure 116: Diagram of the rotor of the seven-stage ORC micro turbine
Figure 117: Diagram of the micro turbine and generator rotors set coupled using a belt gear
Dynamic behavior of the composite rotor blade using an adaptive damper [316].
Mechanical model development of rolling bearing-rotor systems, a review [317].

76. 76
Figure 118: High-speed rotor models based on (a) 1D beam and (b) 3D solid element
Figure 119: Flowchart of the combined model
Influence of gradual damage on the structural dynamic behavior of composite
rotors, simulation assessment [318]. The effect of time-periodic base angular
motions upon dynamic response of asymmetric rotor systems [319].

77. 77
Figure 120: Coordinate system of the composite shaft
 2019
Transverse vibration modal analysis on offset rotor shaft of large centrifugal fan
[320].
Figure 121: Transverse vibration with various angular velocities

78. 78
Natural frequency analysis of a functionally graded rotor system using three-
dimensional finite element method [321].
Figure 122: Meshed FE model of rotor bearing system
An enhanced axisymmetric solid element for rotor dynamic model improvement
[322]. The effects of coupling mechanism on the dynamic analysis of composite
shaft [323]. Influence of manufacturing errors on the unbalance response of
aerodynamic foil bearings [324]. New backward whirl phenomena in intact and
cracked rotor systems [325].
Figure 123: MFS-RDS spectra Quest rotor dynamics simulator
Vibration signature of a rotor-coupling-bearing system under angular misalignment
[326]. A general dynamic model coupled with EFEM and DBM of rolling bearing
rotor system [327].

79. 79
Figure 124: The experimental test rig
Dynamic study of composite material shaft in high-speed rotor-bearing systems
[328]. Dynamic behavior of three-dimensional planetary geared rotor systems
[329]. Simulation of deep leaning control systems to reduce energy loses due to
vibration and friction in rotor bearings [330]. Active vibration control of rotor-
bearing systems by virtual dynamic absorber [331].
 2020
Critical speed analysis of rotor shaft using Campbell diagram [332]. FEA and
modal analysis of a damped flywheel with unbalanced masses [333].
Figure 125: Von Mises stress over the cross section of the flywheel

80. 80
Figure 126: Several mode shapes
Investigation of bending stiffness of gas turbine engine rotor flanged connection
[334].
Figure 127: Model experiments

81. 81
Figure 128: The FE model of the HP rotor
 2021
Effect of transmission ratio on the nonlinear vibration characteristics of a gear-
driven high-speed centrifugal pump [335]. Sensitivity analysis and vibration
control of asymmetric nonlinear rotating shaft system utilizing 4-pole AMBs as an
actuator [336]. Simulation of torsional vibration of driven railway wheelsets
respecting the drive control response on the vibration excitation in the wheel-rail
contact point [337]. Systems of vibration parameters automated control for
diagnostics of equipment technical state [338]. Control of a nonlinear flexure-
jointed X-Y positioning stage using LTV-FIR command pre-filtering for finite-
time error cancellation [339].

82. 82
Chapter 3
Math Modeling

83. 83
2. Aim and Scope
In chapter 3, we will show some various models of rotor-shaft based on different
method. Mechanical behavior of rotor system is main important aim for showing
rotor response. This chapter mathematical modeling is designing from simple to
complicated methods either theoretically or numerically. The first section, we want
to simulation modal analysis either numerically or analytically via ANSYS and
MATLAB in order to valid and verify natural frequencies and critical speeds with
practical model. As we considered, this chapter was shown industrial rotor
functionally to solve our problems or more information. There are different types
of dynamic displacement into segment which is vibrated in coordination’s.
Therefore, vibrations are including transitions and rotations that are happen in 3
coordination’s are X, Y,Z. Transition is related transitional displacement are axial
and lateral( other name is so-called bending, transverse and flexural) and rotation
is also named rotational or angular displacement too.
Figure 1: Shaft-disc and degree of freedom
𝜏𝑦
𝐹𝑦
Y
𝐹𝑧
𝜏𝑧
Z
X
𝐹𝑥
𝜏𝑥

84. 84
Simulation of Modal Analysis
Based on second’s law of Newton describes as:
𝑀𝑞̈ + 𝐶𝑞̇ + 𝐾𝑞 = 𝐹 (1)
In eq.1, F can be defined as vector as follows:
𝐹 = [
𝑓
𝜏
] = [
𝑓𝑜𝑟𝑐𝑒
𝑡𝑜𝑟𝑞𝑢𝑒
] (2)
Also, q is vector of displacement with 6 degrees of freedom are including
transitional and rotational displacements, respectively:
𝑞 =
[
𝑥
𝑦
𝑧
𝜃𝑥
𝜃𝑦
𝜃𝑧]
(3)
Based on eq.1 if 𝐹 = 0 than it is explained free vibration and we can calculate
natural frequency and mode shape. And also, if 𝐹 ≠ 0 than it is defined forced-
vibration as well, so in this stage, we can consider disturbance such as external
forces, torques and noise on the system(rotor). Now, we are asking two practical
questions of which free vibration and after that, extracting eigenvalues and
eigenvectors.
What is the benefit of Eigenvalue?
x
𝜃𝑥
𝜃𝑦
𝜃𝑧
y
z
Bending
Bending
Lateral

85. 85
 Frequency
 Critical speed
 Transitional and rotational vibration (rotary machine)
What is the benefit of Eigenvector?
 Mode shape (mechanical behavior)
 Maximum and minimum displacement
 Effects of degree of freedom
 Effects of boundary conditions
Eigenvalue and Eigenvector
Based on eq.1, we should calculate eq.4 in order to evaluate eigenvalues and
eigenvectors.
|𝐾 − 𝑀𝑤2|𝜆 = 0 (4)
In eq.4, eigenvalue is determined:
𝑑𝑒𝑡(𝐾 − 𝑀𝑤2) = 0
Eigenvalue is 𝑤2
and also we knew frequency formulation was shown as:
𝑤 = √
𝐾
𝑀
(5)
How to calculate K and M matrices?
In dynamical system, there are 3 types of parameters are describing mechanical
behavior of system as well, such as potential energy, kinetic energy and virtual
work, but in free vibration, just potential and kinetic energies are more important to
describe system to compute stiffness (K) and M(mass) matrices, therefore, we
have:
Potential Energy

86. 86
U =
1
2
∫ σεdV
(6)
or
U =
1
2
qT
Kq
(7)
Kinetic Energy
V =
1
2
∫ σε̇dV
(8)
or
𝑈 =
1
2
𝑞𝑇
𝐾𝑞
(9)
Node and Element Concepts
Node and element are two characters are described finite element methods and
other numerical methods too. Because node and element are coupled concepts to
show geometry structure and also mechanical behaviors of structure. For example,
main model of rotor is designed in SOLIDWORKS is based on
Figure 2: Solidworks model with boundary conditions

87. 87
Figure 3: SOLIDWORKS model of rotor with schematically boundary conditions
To simplify Fig.3 and show simple model of rotor such shaft-disc (Fiq.4), now we
want to import nodes and elements in shaft-disc structure as correct and also every
element has 6 degree of freedom(DoF) in generally as showing:
Figure 4: Finite element model of shaft-disc
But in our future model (ANSYS model) we will determine 3 DoFs for each
element because of special boundary conditions (Fiq.6) or there are 3 Dofs of
rotational displacements in X,Y and Z axis.
How to know types of direction angular velocity?
Generally, every element can move in 3 axis with 6 various displacements of
which rotational and transitional displacements, so K or M matrices must be 6*6
degree (36 arrays) as we show:
𝑁6
𝑁5
𝑁4
𝑁3
𝑁2
𝑁1
x
y
z
𝜃𝑧
𝜃𝑦
𝜃𝑥

88. 88
But in fig.4 is showing one sample element of rotor with 3 Dofs that are just
maintaining rotational displacements, hence, K or M matrices are reduced as 3*3
elements.
Industrial model of rotor is shown in below figure. In this real model, boundary
conditions are one of the types of conditions (figure 4). Schematically, in our
model of rotor, based on figure 5, the rotor (figure 2) is considered with 2 simply
supports and one bearing as spring and damper.
𝑲 =
[
∎ ∎ ∎ ∎ ∎ ∎
∎ ∎ ∎ ∎ ∎ ∎
∎
∎
∎
∎
∎
∎
∎
∎
∎
∎
∎
∎
∎
∎
∎
∎
∎ ∎
∎ ∎
∎
∎
∎
∎]
x
y
z
𝜽𝒚
𝜽𝒛
𝜽𝒙
x y z 𝜽𝒙 𝜽𝒛
𝜽𝒚
𝐾 =
∎ ∎ ∎
∎ ∎ ∎
∎ ∎ ∎
𝜃𝑥 𝜃𝑦 𝜃𝑧
𝜃𝑥
𝜃𝑦
𝜃𝑧

89. 89
Figure 5: Real model of rotor with fixture
Figure 6: schematic model of rotor with boundary conditions
ANSYS Models
Although ANSYS is user friendly software, it can completely use into industries
around the world. The capabilities of ANSYS are so much in many fields such as

90. 90
engineering and physic sciences as well. Nowadays, ANSYS can easily solve
multi-physic models, so in this section, firstly, we simulate modal analysis of rotor
for understanding natural frequencies (eigenvalues) and mode shapes
(eigenvectors).
Rotor
In this section, we are showing results of modal analysis of rotor in ANSYS. These
results are including contour of mode shape (eigenvector), natural frequencies
(eigenvalue), critical speed and Campbell diagram too. In the mode shape results,
there are 10 mode shapes which are presented
Figure 7: 1st
of mode shape

91. 91
Figure 8: 2nd
of mode shape
Figure 9: 3rd
of mode shape

92. 92
Figure 10: 4th
of mode shape
Figure 11: 5th
of mode shape

93. 93
Figure 12: 6th
of mode shape
Figure 13: 7th
of mode shape

94. 94
Figure 14: 8th
of mode shape
Figure 15: 9th
of mode shape

95. 95
Figure 16: 10th
of mode shape
Now, there are illustrated natural frequencies based on ANSYS simulation of rotor
in below table and also critical speed as rotational velocity is calculated with :
𝐶𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑆𝑝𝑒𝑒𝑑 (𝑟𝑝𝑚) = 60 × 𝑁𝑎𝑡𝑢𝑟𝑎𝑙 𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦(𝐻𝑧)
According to every mode of natural frequency, obviously, we are having critical
speed.
Figure 17: frequencies and rotational velocities of analysis modal

96. 96
Natural frequencies and critical rotational velocities.
Table 2: Natural frequency and critical speed of rotor
Rotor (Hz) Critical Speed (rpm)
41.74 2504.4 𝜃̇𝑥
42.971 2578.26 𝜃̇𝑦
43.239 2594.34 𝜃̇𝑧
187.86 11271.6 𝜃̇𝑥
190.72 11443.2 𝜃̇𝑦
512.61 30756.6 𝜃̇𝑧
847.37 50842.2 𝜃̇𝑥
854.48 51268.8 𝜃̇𝑦
1116 66960 𝜃̇𝑧
1119 67188 𝜃̇𝑥
Figure 18: Campbell diagram of modal analysis of rotor

97. 97
Numerical Free Vibration
Figure 19: Simply Supported Beam
Figure 20: Simply-Supported Beam with Overhanging
Figure 21: Simply-Supported Shaft - Disc with Overhanging
Vibration Simulation
 Model- 1
The model 1 is simple view that shows rotor-shaft-bearing (fig.1). In this section,
we are showing the one of the simplest modeling in rotor dynamic as flexural
vibration based on below schematic figure:
Figure 22: Analogy of Rotor system, real model (right) and lumped model (left)
Based in Fig.1, math model is defined that:
𝑚𝑑
Bearing
Impeller
Shaft
𝑘1 𝑐1 𝑘2
𝑐2
𝑚𝑑
𝑚𝑠

98. 98
𝑚𝑠𝑦̈ + (𝑐1 + 𝑐2)𝑦̇ + (𝑘1 + 𝑘2)𝑦 = 𝐹 + 𝑚𝑑𝑔 + 𝑚𝑠𝑔 (10)
𝐼𝜃̈ + (𝑘2𝑦 + 𝑐2𝑦̇)𝐿 + (𝑚𝑑𝑔 + 𝑚𝑠𝑔)
𝐿
2
= 𝐹𝑥0 + 𝜏
(11)
And moment inertia of shaft is:
𝐼𝑠 =
𝑀𝑅2
4
+
𝑀𝐿2
3
(12)
Figure 23: Flexural vibration of Shaft
Model- 2
The second model is more complicated model than primer model. Based on second
Newton’s law, we always explain dynamic model as:
𝑀𝑞̈ + (𝐶 + Ω𝐺)𝑞̇ + 𝐾𝑞 = 𝐹 + 𝜏 (13)

99. 99
Parameters are in (4), which can define for 2
𝑀 = [
𝑚 0
0 𝑚
] 𝐹 = [
𝑓𝑦
𝑓𝑧
]
(14)
𝐾 = [
𝑘𝑦𝑦 𝑘𝑦𝑧
𝑘𝑧𝑦 𝑘𝑧𝑧
] 𝜏 = [
𝜏𝑦
𝜏𝑧
]
𝐶 = [
𝑐𝑦𝑦 𝑐𝑦𝑧
𝑐𝑧𝑦 𝑐𝑧𝑧
] 𝑞 = [
𝑞𝑦
𝑞𝑧
]
𝐺 = [
0 𝑔
𝑔 0
]
x
y
x
y

100. 100
Figure 24: 2DOF vibrations of shaft and limit cycles
Model- 3
Figure 25: Rotor Bearing System (a) schematic (b) end of shaft
𝑚𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 𝑚(𝑢𝜑̈ sin 𝜑 + 𝑢𝜑̇ 2
cos 𝜑) (15)
𝑚𝑦̈ + 𝑐𝑦̇ + 𝑘𝑦 = 𝑚(𝑢𝜑̇ 2
sin 𝜑 − 𝑢𝜑̈ cos 𝜑) (16)
𝑚𝜑̈ + 𝑐𝜑̇ + 𝑘𝜑 = 𝜏 − 𝑝 = 𝑚(𝑥̈𝑢 sin 𝜑 − 𝑦̈𝑢 cos 𝜑) (17)

101. 101
𝑚𝑏𝑏̈ + 𝑐𝑏𝑏̇ = 𝐹 (18)
Figure 26: 3-DOF vibrations of shaft and limit cycles

102. 102
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