LIST OF FIGURES
LIST OF TABLES
1.1 Gas Turbine
1.1.1 Theory of Operation
1.2 Lateral and Torsional Analysis of Rotor
2. LITERATURE REVIEW
2.1 Lateral Vibration Analysis
2.2 Torsional Vibration Analysis
3. MATHEMATICAL BACK GROUND
3.1.1 Single Degree Freedom Systems
3.1.2 Equation of motion for SDOF
3.1.3 Multi Degree of Freedom (MDOF) System
3.1.4 Vibration Model
3.1.5 Equations of Motion
3.1.6 Viscously Damped of Free Vibration
3.2 Rotor Dynamics
3.2.1 Definition of terms
3.2.2 Damped Unbalance Response Analysis
3.3 Finite Element Method in Rotor Dynamics
3.3.1 Division of Rotor into Discrete Systems
3.3.2 Gyroscopic Effects
3.3.4 Shear Deformation
3.3.5 Damped Natural Frequency/Stability
3.4 Imbalance Response of Rotor
3.5 Torsional Analysis (Method of Calculation)
3.5.1 Transfer Matrix Analysis
3.5.2 Generation of Campbell Diagrams
3.5.3 Electrical Fault Torque Calculations
4. RESULTS AND DISCUSSIONS
4.1 Lateral vibration analysis of Frame-GT Train
4.1.1 GT Rotor modeling
4.1.2 Damped Critical Analysis considering minimum stiffness
4.1.3 Unbalance Response Analysis
4.1.4 Logarthmic Decrement and Stability Analysis
4.2 Torsional Vibration Analysis of Rotor Train
4.2.1 Natural Frequency
4.2.2 Torsional Mode Shape
4.2.3 Campbell Diagram
4.2.4 Torque Magnification Factor
4.2.5 Alternating Shaft Stresses
The purpose of this report is to determine the lateral and torsional dynamic characteristics of the complete system under synchronous conditions of excitation and response. A damped natural response study was made in order to investigate the combined effect of oil film stiffness and damping coefficients on system damping and stability characteristics at all damped natural resonance speeds. An unbalance response analysis is also performed to study the system sensitivity.
This study was performed to investigate the lateral vibration characteristics of the subject system in order to avoid vibration problems that might interfere with the smooth and reliable operation of the system. Total system studies are important in that often the coupling effects of marrying driver and driven equipment result in resonant speeds that are not calculable when investigating the response of the separate components. Oil film stiffness and damping for all bearings must be properly considered in the system calculations along with the effective stiffness and damping of pedestal supports as required. The above effects are in the following calculation to ensure the proper calculation of resonant speeds.The following study concerns itself with the lateral analysis of gas turbine, load coupling, and 50 Hz/15.75Kva generator. This study reports the lateral natural frequencies and mode shapes calculated from the mass and stiffness distribution of the beam elements modeled using the DYROBES software. An unbalanced response analysis is also performed to study the system sensitivity.
The significance of torsional vibration in high speed rotating machinery is well established. It is desirable to keep all torsional natural frequencies away from operating speed as well as twice the electrical frequency of the system. However , this is not always feasible and, therefore torsional criticals can be tolerated within these regions provided the response to excitation levels are low enough to keep the alternating shear stress within acceptable levels The following study concerns itself with the complete torsional analysis of gas turbine rotor including load coupling, gear box and 50Hz/15.75KVA generator rotor. This study reports the torsional natural frequencies, mode shapes and Campbell diagram by using transfer matrix method. The transient response shear stresses were also calculated for fault condition.
1.1 Gas Turbine
A gas turbine is an engine where fuel is continuously burnt with compressed air to produce a stream of hot, fast moving gas. This gas stream is used to power the compressor that supplies the air to the engine as well as providing excess energy that may be used to do other work. The engine consists of three main parts viz., compressor, combustor and turbine.
The compressor usually sits at the front of the engine. There are two main types of compressor, the centrifugal compressor and the axial compressor. The compressor will draw in air and compress it before it is fed into the combustion chamber. In both types the compressor rotates and it is driven by a shaft that passes through the middle of the engine and is attached to the turbine.
The combustor is where fuel is added to the compressed air and burnt to produce high velocity exhaust gas. Down the middle of the combustor runs the flame tube. The flame tube has a series of holes in it to allow in the compressed air. It is inside the flame tube that fuel is injected and burnt. There will be one or more igniters that project into the flame tube to start the mixture burning. Air and fuel are continually being added into the combustor once the engine is running. Combustion will continue without the use of the igniters once the engine has been started. The combustor and flame tube must be very carefully designed to allow combustion to take place efficiently and reliably. This is especially difficult given the large amount of fast moving air being supplied by the compressor. The holes in the flame tube must be carefully sized and positioned. Smaller holes around where the fuel is added provide the correct mixture to burn. This is called the primary zone. Holes further down the flame tube allow in extra air to complete the combustion. This is the secondary zone. A final set of hole just before the entry to the turbine allow the remainder of the air to mix with the hot gases to cool them before they hit the turbine. This final zone is known as the dilution zone. The exhaust gas is fed from the end of the flame tube into the turbine.
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Fig. 1.1: Simplified gas turbine diagram.
The turbine extracts energy from the exhaust gas. The turbine can, like the compressor, be centrifugal or axial. In each type the fast moving exhaust gas is used to spin the turbine. Since the turbine is attached to the same shaft as the compressor at the front of the engine the turbine and compressor will turn together. The turbine may extract just enough energy to turn the compressor. The rest of the exhaust gas is left to exit the rear of the engine to provide thrust as in a pure jet engine. Or extra turbine stages may be used to turn other shafts to power other machinery such as the rotors of a helicopter, the propellers of a ship or electrical generators in power stations the end of the flame tube into the turbine.
Cold air is drawn in from the left into the compressor (blue). The compressed air (light blue) then goes into the combustor. From the outside of the combustor the air goes through holes (purple) into the flame tube (yellow). Fuel is injected (green) into the flame tube and ignited. The igniters are not show here. The hot exhaust gas flows from the end of the flame tube past the turbine (red) rotating it as it passes. From there the exhaust exits the engine. The turbine is connected via a shaft (black) to the compressor. Hence as the turbine rotates the compressor rotates with it drawing in more air to continue the cycle.
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Fig.1.2: Gas turbine
Energy is released when air is mixed with fuel and ignited in the combustor. The resulting gases are directed over the turbine's blades, spinning the turbine and, cyclically, powering the compressor. Finally, the gases are passed through a nozzle, generating additional thrust by accelerating the hot exhaust gases by expansion back to atmospheric pressure.
Energy is extracted in the form of shaft power, compressed air and thrust, in any combination, and used to power aircraft, trains, ships, electrical generators, and even tanks.
1.1.1 Theory of operation
Gas turbines are described thermodynamically by the Brayton cycle, in which air is compressed isentropically, combustion occurs at constant pressure, and expansion over the turbine occurs isentropically back to the starting pressure.
In practice, friction and turbulence cause:
a) non-isentropic compression – for a given overall pressure ratio, the compressor delivery temperature is higher than ideal.
b) non-isentropic expansion – although the turbine temperature drop necessary to drive the compressor is unaffected, the associated pressure ratio is greater, which decreases the expansion available to provide useful work.
c) pressure losses in the air intake, combustor and exhaust – reduces the expansion available to provide useful work.
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Fig. 1.3 Idealized Brayton Cycle
As with all cyclic heat engines, higher combustion temperature means greater efficiency. The limiting factor is the ability of the steel, nickel, ceramic, or other materials that make up the engine to withstand heat and pressure. Considerable engineering goes into keeping the turbine parts cool. Most turbines also try to recover exhaust heat, which otherwise is wasted energy. Recuperators are heat exchangers that pass exhaust heat to the compressed air, prior to combustion. Combined cycle designs pass waste heat to steam turbine systems and combined heat and power (co-generation) uses waste heat for hot water production.
Since neither the compression nor the expansion can be truly isentropic, losses through the compressor and the expander represent sources of inescapable working inefficiencies. In general, increasing the compression ratio is the most direct way to increase the overall power output of a Brayton system.
The compressor portion of the gas turbine rotor is an assembly of wheels, a speed ring, ties bolts, the compressor rotor blades, and a forward stub shaft .Each wheel has slots broached around its periphery. The rotor blades and spacers are inserted into these slots and held in axial position by staking at each end of the slot. The wheels are assembled to each other with mating rabbets for concentricity control and are held together with tie bolts. Selective positioning of the wheels is made during assembly to reduce balance correction.
After assembly, the rotor is dynamically balanced. The forward stub shaft is machined to provide the thrust collar which carries the forward and aft thrust loads. The stub shaft also provides the journal for the No. 1 bearing, the sealing surface for the No. 1 bearing oil seals and the compressor low-pressure air seal Of the many factors affecting the efficient working of a simple gas turbine, including unbalanced forces, vibrations are the prominent that lead to development of cyclic stresses which inturn results in fatigue failure.
Machines in the best of operating condition will have some vibration because of small, minor defects. Therefore, each machine will have a level of vibration that may be regarded as normal or inherent. However, when machinery vibration increases or becomes excessive, some mechanical trouble is usually the reason. Vibration does not increase or become excessive for no reason at all. Something causes it - unbalance, misalignment, worn gears or bearings, looseness, etc.
1.2 Lateral and Torsional Analysis of rotor
The primary purpose of the lateral analysis is to calculate the frequency location of a unit’s response sensitivity at the critical speeds to anticipated levels of unbalance. If the critical speeds are adequately separated from the units operating speeds or are heavily/critically damped, then the possibility of the unit encountering problematic vibrations from any excitation mechanism (including rubs) during normal operation is greatly reduced
The objective of this study is to determine the torsional critical frequencies, mode shapes and prediction of the torque and resonant shear stress values reached in the shafts of the system rotor, in order to judge the acceptability of the system and appropriate design modification from a torsional vibration viewpoint
Many single-drive-line rotors are stiff enough in torsion so that torsional natural frequencies are sufficiently high to avoid forced resonance by the time-varying torque components transmitted in the rotor. When single rotors are coupled together, the possibility is greater for excitation of coupled-system torsional natural frequency modes. In most coupled drive trains, it is the characteristics of the couplings, gear trains, and electric motors or generators that instigate torsional rotor vibration problems.
If the frequency of a machine's torque variation matches one of the resonant torsional frequencies of the drive train system, large torsional oscillations and high shear stresses can occur within the vibrating components. If a machine experiencing such torsional vibration is continuously operated, an unwarranted fatigue failure of weak system components is imminent. One of the major obstacles in the measurement and subsequent detection of torsional vibration in a machine is that torsional oscillations cannot be detected without special equipment. However, prediction of torsional natural frequencies of a system and consequent design changes that avoid the torsional natural frequencies from occurring in the operating speed range of a machine is necessary. This necessitates the analysis of the torsional characteristics of the system components.
2. Literature Review
The science of rotor dynamics has been extensively developed as a consequence of the realization by industry that the lateral dynamics characteristics and behavior of rotating equipment profoundly impacts plant operation This study was performed to investigate the lateral vibration characteristics of the subject system in order to avoid vibration problems that might interfere with the smooth and reliable operation of the system.
One of the major obstacles in the measurement and subsequent detection of torsional vibration in a machine is that torsional oscillations cannot be detected without special equipment. However, prediction of torsional natural frequencies of a system and consequent design changes that avoid the torsional natural frequencies from occurring in the operating speed range of a machine is necessary. This necessitates the analysis of the torsional characteristics of the system components Rotating machinery such as turbines, pumps and fans are very important components in many machines and systems. Some examples are aircraft engines, power stations, large flywheels in a hybrid transmission of motorcars, etc. Therefore, the behaviour of these rotor-dynamic components can influence the performance of the whole system. Namely, for certain ranges of rotational speed, such systems can exhibit various types of vibration which can be so violent that they can cause significant damage.
There are many possible causes for such behaviour. Some examples are friction or fluid forces in the bearings in which a shaft is borne, mass-unbalance in the rotor which can lead to whirling motions, flexibilities present in the system, etc.
Lateral vibrations in rotor systems have been analyzed extensively by Fritz  Lee ; Muszynska ; Tondl . Lee , Tondl  considered different types of rotor systems; but in all those systems, lateral vibrations are induced by the mass-unbalance in a rotor.On the other hand, Fritz [3,4] and Muszynska  derived expressions for fluid forces which can also induce lateral vibrations in systems with a long vibrating rotor which rotates in a stator.
Interaction between torsional and lateral vibrations in different rotor systems is studied in [Gunter ;Lee ; Tondl. In various mechanical systems it is noticed that increase of mass- unbalance can have both stabilizing and destabilizing effects in the considered system. For example, Tondl  and Lee  consider a simple disc with a mass-unbalance connected to a shaft which is elastic in both torsional and lateral direction. They noticed that in such systems, under certain conditions, instabilities can appear if the unbalance increases. On the other hand, in [Gunter  the opposite effect has been noticed. Namely, the behaviour of flexible rotor-bearing systems is analyzed and it is concluded that the mass-unbalance can stabilize some rotor systems.
S SARKAR,[ 9 ] deals with with a two-step nonlinear finite element analysis for misaligned multi-disk rotors on short oil-film bearings of various types (cylindrical, pocket, symmetrical three-lobed, unsymmetrical three-lobed). . he proposed a method for computing the displacement-dependent stiffness terms from the experimental static loaddisplacement data the orbit of the rotor around the static equilibrium is determined using a time-integration scheme
Songtao Li · Qingyu Xu Xiaolong Zhang[ 10 ] deal with dynamic behaviors of the unbalanced rotor-seal system with sliding bearing based on the adopted model and Newmark integration method. The influence of the labyrinth seal on the nonlinear characteristics of the rotor system is analyzed by the bifurcation diagrams and Poincare’ maps
A. R. GUIDO and G. ADILETTA  analysed dynamic behaviour of a rigid rotor with nonlinear elastic restoring forces was carried out, an experimental confirmation of the theoretical data from that analysis was sought .Comparison of the collected data with the corresponding theoretical results made it possible to infer that system nonlinearity in the presence of small damping can give rise to motions that are periodic, whether synchronous or not, or quasi-periodic, but never chaotic
Guangchi Ying · Guang Meng · Jianping Jing[ 12 ] To investigate the effect of foundation excitation on the dynamical behavior of a turbocharger, a dynamic model of a turbocharger rotor-bearing system is established, results obtained by numerical simulation show that the differences of dynamic behavior between the turbocharger rotor systems with/without foundation excitation are obviously. With the foundation excitation, the dynamic behavior of rotor becomes more complicated, and develops into chaos state at a very low rotational speed
Ulrich Werner[ 13 ] in his paper shows that dynamic air gap torques in converter-fed inductionmotorsmay not only cause torsional vibrations in the drive train, butmay also cause lateral vibrations of the motor itself—including bending vibrations of the rotor shaft—, if the motor ismounted on a soft foundation. Based on a simplified analytical model, themathematical correlation between rotor-stator interaction, oil film characteristics of sleeve bearings, the influence of a soft foundation—coupling angular and lateral displacement of the stator—and excitation from dynamic air gap torques is shown.
E. V. Ur’ev, A. V. Kistoichev,and A. V. Oleinikov1[ 14 ] studied the possible causes of residual deflections in seamless forged rotors are examined. It is shown that measures to eliminate a residual deflection must take its cause into account. A method for correction of rotors proposed by the Leningrad Metal Factory (LMZ), which involves mounting special systems of balanced and “antibalancing” loads on a bent rotor, is analyzed
XU Hua, ZHU Jun[ 15 ] In his paper considered the effects of mechanical seals, a lumped-mass model and the transfer matrix method are used to establish the equations for the dynamics performance of rotor–bearing system.The analysis results show that the mechanical seal more or less has effects on the rotor–bearing system. The mechanical seal has much more effects on the flexible rotor–bearing system than on the rigid one.
A. L. Stel’makh, A. D. Len[ 16 ] presented the results of investigations of the aerodynamic stability of flexural-torsional vibrations of compressor blades under conditions of attached and separated flow with regard for the cross and mutual aerodynamic links of the blades in a broad range of variations of the phase shift, the ratio of the amplitudes of translational and angular components of their displacements, and the angle of attack
M. Attia Hili, T. Fakhfakh, and M. Haddar[ 17 ] studied Shaft misalignment and rotor unbalance in rotating machinery. In order to understand the dynamic characteristics of these machinery faults, a model of a complete motor flexible-coupling rotor system capable of describing these failures was developed. Generalized system equations of motion for a rotor under misalignment and unbalance conditions were derived using the finite element method
R. Bumby and J. M. Wilson [ 18 ]In this paper a finite element model and simple lumped mass and spring models of the rotor, for the calculation of the undamped torsional natural frequencies, are described and compared. A method by which equivalent spring stiffnesses for both the inner and outer rotors can be derived is described, allowing one to use a rotor model with one lumped mass and equivalent spring stiffness for each of the inner and outer rotors
A Campbell diagram plot represents a system's response spectrum as a function of its oscillation regime. It is named for Wilfred Campbell (1884-1924).
viggiano and Schmied has studied turborotor trains with gears, the torsional and lateral motion of gear wheels are coupled, torsional vibration problem was solved by exchange pinion gear bearings. Due to lateral vibration component the radial bearings can influence the damping of the torsional vibration modes. Normally they add damping, but they can also destabilize
Corbo, M. A. and Malanoski, S. B Performed multi-level lateral rotordynamic analysis and torsional vibration analysis for two vertical long-shaft , Performed design audit on GE gas turbine for a major utility
W.J.Chen & E.J.Gunter authored DyRoBeS is a powerful engineering design/analysis software tools,for complete rotordynamic analysis and comprehensive bearing performance calculations. The FE based codes combine a user interface, with a powerful set of modeling and analysis capabilities for the most demanding requirements used for gas turbine modeling
Z. Luo, X. Sun and J. N. Fawcett has proposed a general model of multistage and multimesh geared shaft systems, which takes into account the coupling effects between torsional, lateral and axial vibrations. The system can be divided into several substructures which are dynamically analysed using a finite element method.
Pankaj Kumar and S.Narayanan,  has solved Fokker Planck equation numerically by employing PI solution procedure to examine some features to examine some features of chaotic and stochastic responses of nonlinear rotor systems.
3. Mathematical Background
Vibration is considered with the oscillating motions of the bodies and the forces associated with them. All bodies possessing mass and elasticity are capable of vibration. The mass is inherent of the body and the elasticity is due to relative motion of the parts of the body. The objective of the designer is to control the vibration when it is objectionable. Objectionable vibrations in a machine may cause the loosening of the parts, its malfunctioning or its eventual failure. The ultimate goal in the study of the vibration is to determine its effect on the performance and safety of the system under consideration. The performance of many instruments depends on the proper control of the vibrational characteristics of the devices.
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The elements that constitute a vibratory system are shown in figure. They are idealized and called mass, spring, damper and the excitation force. The first three elements describe the physical system.
Energy may be stored in the mass and spring, and dissipated in the damper in the form of heat. Energy enters the system through the application of an excitation. The mass may gain or lose kinetic energy in accordance with the velocity change of the body. The spring possesses elasticity and is capable of storing the potential energy under deformation. The damper has neither mass nor elasticity and is capable of dissipating the energy. Viscous damping in which the damping force is proportional to the velocity is generally assumed in engineering. There are two general classes of vibrations viz., free vibrations and Forced vibrations.
Free vibrations takes place when a system oscillates under the action of forces inherent in the system itself, and when external impressed forces are absent. The system under free vibration will vibrate at one or more of its natural frequencies, which are properties of the dynamic system established by its mass and stiffness distribution.
Vibration that takes place under the excitation of the external forces is called forced vibration. When the excitation is oscillatory, the system is forced to vibrate at the excitation frequency. If the frequency of excitation coincides with one of the natural frequencies of the system, a condition of resonance is encountered, and dangerously large oscillations may result. The failure of the major structures such as bridges, buildings or airplane wings is an awesome possibility under resonance. Thus, the calculation of the natural frequencies is of major importance in the study of the vibration.
Vibrating systems are all subject to damping to some degree because friction and other resistances dissipate energy. If the damping is small, it has very little influence on the natural frequencies of the system, and hence the calculations for the natural frequencies are generally made on the basis of no damping. On the other hand, damping is of great importance in limiting the amplitude of oscillation at resonance.
3.1.1 Single Degree Freedom (SDOF) Systems
The number of independent coordinates required to describe the motion of a system is called degrees of freedom of the system. The system shown in figure has one degree of freedom and hence called SDOF system. The SDOF system is the keystone for more advanced studies in vibrations of multi-degree (MDOF) systems or practical structures. The idealized elements in fig-1 form a model of a vibrating system, which in reality can be quite complex. The spring shown in fig-1 may possess mass and elasticity. In order to consider it as an idealized spring, either its mass is assumed negligible or an appropriate portion of its mass is lumped together with the other masses of a system. For example, a beam has its mass and elasticity inseparably distributed along its length. The vibrational characteristics of a beam or more generally of an elastic body or a continuous system are approximated by a finite number of lumped parameters. This method is a practical approach to the study of some very complicated structures, such as an aircraft or missile.