Abstract— Smooth functioning of power plant equipment is very

important for uninterrupted power supply. The vibration signatures

generated by the machinery is the most important consideration to

understand the dynamic behavior of rotating machine for its smooth

operation. To meet the existing demand for power, several machines

of different design and integration of equipment is taking place both

new ones as well as used machines. These turbo machines are

available for 60 Hz and 50 Hz application. The 60 Hz machine needs

to be converted into 50 Hz for application in our country.

The process involves clear understanding blade dynamics, rotor

dynamics, electrical aspects of generators and control systems. This

paper describes rotor dynamic aspects of the system like Proper

rotor dynamic analysis of turbine configuration, bearings, couplings,

gear boxes, generators and supporting structure. This work is

reviewed to study the theoretical aspects and software available and

various aspects of design and condition monitoring of the above set.

A case study of a typical refurbished 5O MW, 60Hz Turbo

Generator system set to 50 MW, 50Hz is presented.

Keywords—Transverse analysis, Torsional Analysis, Vibrations,

mode shapes, Critical Speeds.

I. INTRODUCTION

UE to rapid industrialization there is a huge growing

demand for power in the country. Though new power

generating equipments are produced by Indian industries

and imports are also made, it is impossible to meet the demand

in a stipulated time. To meet the ever growing demand, it is

also important to import used machinery from world over

various equipment and integrate them properly from various

aspects to generate trouble free power. This involves,

importing 60 Hz direct drive machinery, turbines from one

location, generators from some other make, gear boxes can be

new or old along with coupling. Here rotor dynamics [1] plays

vital role to make them smoothly free from vibrations and

noise and also avoid catastrophe.

The rotor dynamic investigations involves complete

Transverse [2]and Torsional analysis [3] of the rotating

system, study of previous vibration data like vibration

condition monitoring , type of bearings used, structural

Narisimha Murthy1 is with the Adama Science and Technology University,

Adama, Ethiopia. Perumalla Janaki Ramulu2 is with the Adama Science and Technology

University, Adama, Ethiopia.

Tariku Desta3 is with the Adama Science and Technology University, Adama, Ethiopia.

NRR Anbusagar4 is with the Adama Science and Technology University,

Adama, Ethiopia.

integrity quality of balancing adopted, blade Campbell

diagrams [4], Torsional oscillations and short circuit analysis.

The main objective of this research is that due to severe

shortage of power, countries like India is buying old sets of

30years usage with 60HZ operation to generate power at a

shorter time. The adaptability of the 60HZ to 50HZ power,

rotor dynamic challenges like the excitation and resonances

are explained when a gear box is presented.

1. Procedure for Analysis

In this research, the procedure consists of two analysis such

as Transverse analysis and Torsional analysis respectively. In

this work the authors described about the best bearing that can

be suited along with a case study of 50MW Turbo generator

set.

1.1 Transverse Analysis

The transverse analysis is very important auditing for

Turbine, Generator as well. Higher power ratings operating at

3600 rpm are normally connected directly. Once if the

generator need to be operated at 3000 rpm, it is essential to

have a gear box with speed reduction from 3600 rpm to 3000

rpm with a idler in-between and two couplings of flexible

nature are to be connected. Here the turbine operates at 3600

rpm; hence it does not need any modification from bending

point of view. Whereas the generator need to be checked for

transverse analysis. The transverse analysis involves critical

speed calculations [5, 6], undamped mode shapes, and damped

mode shapes unbalance response as per API 612 or any other

acceptable international standards. The independent stability

analysis [7] needs to be carried out. Sufficient care need to be

taken while fixing up coupling design and it need to be

worked out jointly with coupling manufacturer.

For the Transverse analysis it can be done by Transfer

Matrix Methods [8] and many other techniques are developed.

For Transverse analysis here we deals with such systems

having a continuous element which is incorporated into

transfer matrix methods. The solution is limited to steady state

whirl orbits for unbalance response. Case study on a 50MW

turbine set in Fig.1 is presented.

Rotor Dynamic Considerations in Refurbishing

Turbo Machinery

T. Narisimha Murthy1, P. Janaki Ramulu

2, Tariku Desta

3, and NRR Anbusagar

4

D

7th International Conference on Latest Trends in Engineering & Technology (ICLTET'2015) Nov. 26-27, 2015 Irene, Pretoria (South Africa)

http://dx.doi.org/10.15242/IIE.E1115049 159

Fig. 1 50 MW Turbine Rotor Model

1.2 Torsional Analysis

The Torsional response characteristics of rotating and

reciprocating equipment should be analyzed and evaluated to

ensure the system's reliability. Severe Torsional vibrations

often occur with the only indication of a problem being gear

noise or coupling wear. Excessive Torsional vibrations can

result in gear wear, gear tooth failures, key failures, shrink fit

slippage and broken shafts in severe cases. Specifications such

as those published by The American Petroleum Institute (API)

provide guide lines and criteria for evaluating system

Torsional response characteristics. The two major analyses

techniques used are the Holzer Method [9] and the

Eigenvector- Eigen value procedure (Modal Superposition

Method).

In performing a Torsional analysis, the first step is to

calculate the Torsional natural frequencies and mode shapes

utilizing an Eigenvector – Eigen value Method. The calculated

natural frequencies are then compared to the excitation

frequencies such as running speed and its multiples to

determine if the system satisfies the appropriate specifications.

For example, API 617 for centrifugal compressors specifies

that the Torsional mode shapes of the complete system should

be at least 10% below any operating speed or at least 10%

above the trip speed.

After the Torsional natural frequencies are calculated,

forced vibration response calculations are performed to

calculate the shaft Torsional stresses. If the system does not

meet the proposed criteria, a parametric analysis can identify

sensitive elements which may be adjusted to modify the

Torsional response and make the system acceptable.

If resonance cannot be avoided, coupling selection can be

optimized based on Torsional shaft stress calculations, as well

as location of critical speeds. When changes in coupling

specifications result in a coupling that has a different weight

than the originally specified, the transverse critical speeds may

be affected. Heavier couplings will lower the transverse

critical speeds while lighter couplings will raise them. The

transverse critical speeds should be at least 20% from the

operating speed. In the Torsional Analysis we can develop

Modeling of the system, Torsional Inertias Calculations,

Torsional Stiffnesses Calculations which are included as

shafts, Couplings, Gear Teeth and Geared Systems also.

2. Journal Bearings

In this case study of 50MW Turbo generator Set we

introduced an Elliptical Bearing Instead of a Cylindrical

bearing [10].

3.1 Case Study

The bearing presently used is elliptical bearing of diameter

Φ = 300 mm, length= 240 mm with a preload of 0.3. Using

XL Rotor software, the bearing is sized for load carrying

capacity and stability. Fig. 2 shows the typical elliptical

bearing layout of 50MW Generator Rotor.

Fig. 2 Typical Elliptical Bearing Layout

Fig. 3 Speed Vs Stiffness

Fig. 4 Speed Vs Damping

Fig. 5 Speed Vs Eccentricity

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http://dx.doi.org/10.15242/IIE.E1115049 160

Table 1 shows the bearing pad parameters.

Table 2 shows the Lubricant properties, bearing length,

diameter, oil holes etc.,

Table 3 shows Speed Vs Eccentricities and load carrying

capacity values.

Table 4 shows Speed Vs Stability, power losses in bearings,

temperature rise in bearings.

Table 5 shows Speed Vs Bearing dynamic properties for rotor

dynamic calculations and two charts i.e., Speed Vs Stiffness

shown in Fig.3 and Speed Vs Damping in Fig. 4 and

eccentricities plot are shown in Fig. 5.

TABLE I

BEARING PAD PARAMETERS

TABLE II

PROPERTIES OF A LUBRICANT

XLPocket Spreadsheet for Fixed Lobe Bearing Coefficients

Title: Conmec IR MGGB-633 OEM Pressure Dam Bearing **pdb.xls**

jcn/rmt 30August2005

Click on Cell then

Output File Names

TGSAT

Viscosity Option

0

Press Alt-F1 for Help

Selected Lubricant

Operating Film T 65.6

oC

Operating Viscosity 16.74

cp

Lubricant Ref Ta 40.0 oC Operating Density 0.85 gm/cm

3

Lubricant Ref Tb 100.0 oC Number of Visc Iter 2 --

Lubricant Ref Va 204.0 SSU Number of Holes 2 --

Lubricant Ref Vb 47.0 SSU Inlet Hole Type

0.00E+00

Lubricant Ref SGa 0.859 -- Inlet Hole Length 12.7 mm

Lubricant Ref SGb 0.826 -- Inlet Hole Width 6.35 mm Lubricant API Grav 30.6 -- Inlet Hole Diameter 2.54 mm

Journal Diameter 300.00 mm Weep Hole Type

1.00E+00

Bearing Length 240.00 mm Weep Hole Depth 0.762 mm

Diametral Clearance 0.3500 mm

Weep Hole Chamfer 45

degrees

Gravity Load 90,742 N Weep Hole Radius 0 mm First Critical Speed 1862 rpm Oil Inlet T

48.9

oC

Ecc X Initial Guess 0 -- Supply Pressure 0.138 MPa

Ecc Y Initial Guess 0 -- Horz or Vert Flag

0

Hot Oil Carry Over 60

percent

Hot Shaft T

0

oC

Pressure Field Flag

0

Brg Housing P 0 MPa

K and C File Format 1

TABLE III

SPEED & LOAD CARRYING CAPACITIES VALUES

Speed Somm

Ecc X Ecc Y

Epp Psi W Resultant W Res

Angle Loading P Max

P Max

Angle

rpm -- -- -- -- degrees Newtons degrees MPa MPa Degrees

4,000 0.185 0.3160039 0.0392684 0.31843441 97.0835794 90742 270 1.26029884 4.00565441 268.4

3,500 0.182 0.3292312 0.0346524 0.3310498 96.0084008 90742 270 1.26029884 3.92016368 270

3,000 0.1757 0.3444204 0.0282091 0.34557368 94.6822512 90742 270 1.26029884 3.84501457 271.6

2,000 0.1519 0.3830 0.0057 0.38301789 90.8454106 90742 270 1.26029884 3.74504557 278

1,000 0.1049 0.4329 -0.0604 0.43710028 82.057802 90742 270 1.26029884 3.92154256 281.2

500 0.06573 0.4443 -0.1587 0.47178104 70.3397197 90742 270 1.26029884 4.46620125 281.2

TABLE IV

BEARING STABILITY VALUES

Speed Stab Thres Stable? SurfaceV PwrLoss Drain Flow T Max Min H Min H Angle

rpm rpm -- m/s kW lpm oC mm Degrees

4,000 3816.8 0 62.81928 67.9671569 4.11E+01 128.88889 0.077216 303.51

3,500 3848.1 0 54.98592 53.3148845 3.77E+01 121.66667 0.075184 306.7

3,000 3906.1 0 47.12208 40.5267688 3.39E+01 114.44444 0.072898 306.7

2,000 4086 0 31.42488 20.2596561 26.9148265 100 0.065532 306.7

1,000 4100.6 0 15.6972 6.5245488 17.5646688 83.333333 0.051054 306.7

500 3612.5 0 7.86384 2.17733514 11.8864353 72.777778 0.036068 303.51

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TABLE V

BEARING DYNAMIC PROPERTIES

Kxx Kxy Kyx Kyy Cxx Cxy Cyx Cyy

rpm N/m N/m N/m N/m N-s/m N-s/m N-s/m N-s/m

4,000 316.71E+6 1.64E+9 -2.43E+9 2.66E+9 5.46E+6 2.99E+6 843.32E+3 15.86E+6

3,500 392.36E+6 1.47E+9 -2.30E+9 2.51E+9 5.80E+6 2.57E+6 297.89E+3 16.85E+6

3,000 442.37E+6 1.33E+9 -2.24E+9 2.39E+9 6.53E+6 2.14E+6 -245.74E+3 18.61E+6

2,000 556.71E+6 892.44E+6 -2.06E+9 2.14E+9 8.07E+6 56.70E+3 -2.74E+6 23.76E+6

1,000 722.98E+6 306.35E+6 -2.01E+9 2.10E+9 10.82E+6 -7.93E+6 -11.89E+6 41.29E+6

500 798.16E+6 -30.34E+6 -2.20E+9 2.80E+9 15.05E+6 -21.61E+6 -29.47E+6 88.22E+6

4.2 Transverse Analysis

The rotor is divided into number of elements. Table 6 and

Table 7 shown the Section number, Length, and Outer

diameter, Inner diameter, Bearing location, unbalance plane

location etc.,. A model is shown in Fig.1 obtained.

TABLE VI SECTIONAL DATA OF 50MW GENERATOR ROTOR MODEL

INPUT TABLE OF BEAM AND STATION DEFINITIONS, MORE THAN ONE BEAM PER STATION IS OK

Station Length OD ID Density Elastic Modulus Shear Modulus Added

Weight Added Ip

Added

It

Speed

Factor

# meters meters meters kg/m3 N/m2 N/m2 kg kg-m2 kg-m2

1 1.0E+00 6.7E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 3.7E+02 0 0 1.0

2 1.1E+02 6.7E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

3 4.5E+02 3.0E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

4 4.5E+02 3.0E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

5 6.2E+01 3.0E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

6 1.0E-01 3.0E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

7 1.8E+02 3.0E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

8 2.9E+02 4.1E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

9 6.9E+02 4.5E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

10 7.6E+02 8.6E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

11 7.6E+02 8.6E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

12 1.0E-01 8.6E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

13 7.6E+02 8.6E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

14 7.6E+02 8.6E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

15 6.9E+02 4.5E+02 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

16 2.1E+02 3.8E+02 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

17 2.1E+02 3.8E+02 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

18 1.8E+02 3.0E+02 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

19 2.0E-01 3.0E+02 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

20 1.8E+02 3.0E+02 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

21 1.7E+02 3.3E+02 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

22 1.0E+02 2.3E+02 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

23 1.8E+02 1.7E+02 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

24 1.1E+02 1.7E+02 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

25 1.1E+02 2.5E+02 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

26 1.0E+02 9.9E+01 6.4E+01 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

27 1.3E+01 9.9E+01 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

28 1.0E+00 9.9E+01 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

29 1.2E+01 1.8E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

30 3.1E+01 1.0E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

31 6.2E+01 9.5E+01 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

32 5.0E+01 8.0E+01 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

33 1.0E-01 8.0E+01 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

34 5.0E+01 8.0E+01 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

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35 5.5E+01 9.5E+01 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

36 7.9E+01 8.0E+01 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

37 1.9E+01 1.4E+02 0.0E+00 7833.545 2.068E+11 8.273E+10 0.0E+00 0 0 1.0

38 0 0 0 0 0 0 0 0 0 1.0

Figure 10 shows the static bending deflection curve (Sag

curve) along with the coupling hub. The maximum

displacement is occurring at the overhang portion of the

coupling with a deflection value of 0.25 mm. This translates to

a rigid critical speed of 1890 rpm.

Figure.11 shows the undamped critical speed and mode

shapes of the system in vertical direction. The stiffness is

taken as vertical bearing stiffness at the operating speeds. The

first critical speed is 1956 rpm and second critical speed is

3007 rpm which is close to operating speed. This is the reason

when the system is free from resonance at 3600 rpm (60Hz)

operation.

As per the standards, the rotors of this design are normally

balanced as per API 612. Hence the residual unbalance is

computed as

U =

gr mm

Where U is the residual unbalance

W is the weight of the rotor

N is the Operating speed.

This unbalance will be kept at the center of the rotor to

excite the first mode and a couple will be created by

placed

at second mode anti mode points so that the second mode will

be get excited. Using the above, the unbalance response plots

are generated for both the casing by adopting 'UN BAL'

programme which considering the hydro dynamic bearing

properties of elliptical bearing. The elliptical bearing

properties are generated by using XL-ROTOR.

The response curves shown in Figure 12 to 17 indicate a

well damped critical speed at 1500 rpm and critical speed

close to 3500 rpm. The amplitudes of vibration are less than

20 μm pp at operating speed; which is the allowed vibration

limits as per API 612 are 50 μm pp. The effect of coupling is

also considered and the system is free from coupling

unbalance at 3000 rpm.

Using XL-ROTOR soft ware the overall system stabilities

by including the bearing properties are computed. Figure.17

shows a graph that Speed Vs Logarithmic Decrement chart.

The logarithmic decrement value at 3000 rpm is 0.26 indicate

that the system is safe from stability point of view.

TABLE VII BEARING DETAILS

Bearing Position

Length (mm)

Diameter (mm)

Location Section

Type

Front Bearing 240 300 6

2 Lobe sleeve bearing

Rear Bearing 240 300 33 2 Lobe sleeve bearing

Fig. 10 SAG Curve of Generator

Fig. 11 Undamped Mode Shape of the Generator Rotor

Fig 12: Unbalance Response of Generator Rotor for Central

Unbalance

Fig 13 Speed Vs Phase for Coupled Unbalance

7th International Conference on Latest Trends in Engineering & Technology (ICLTET'2015) Nov. 26-27, 2015 Irene, Pretoria (South Africa)

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Fig 14: Dynamic force due to API Central Unbalance

Fig 15: Response due to Coupled API Unbalance

Using ' TORS', the Torsional natural frequencies are

computed and the values are 1250 rpm and the second is 5075

rpm. Fig.19 shows the Torsional mode shapes of the system.

A steady state torque is applied on generator to steady the

Torsional response of the system. Fig. 20 shows the response

in time domain at coupling location. The maximum induced

torques is less than the actual torques applied and hence the

system is safe from electric resonance effect.

Fig 16: Response due to Coupling Unbalance

Fig 17: Rotor Bearing Stability Chart

6.3 Torsional Analysis

Figure 18 shows the Torsional model of the system, which

comprises of steam turbine, coupling, gear box and Generator.

Fig 18: Torsional Model of Generator Rotor

Fig 19: Torsional Mode Shapes of the system

CONCLUSIONS

The following are the final conclusions arrived after carrying

out detailed analysis.

1. The unbalance response specifies a critical speed at

1800 rpm and above 3400 rpm for API 612 standard

configuration. With unbalance at coupling, the rotor

shows a significant variation and puts another critical

and a phase change at 3000 rpm.

2. The existing bearing is very close to instability but the

new bearing due to his high preloading, the instability

pushes beyond 4000 rpm and the 3000 rpm mode

under consideration is fully damped.

3. The vibrations with API standard are well in limits.

Though the dynamic forces are well in limit, the

peaking is shown belong 3000 rpm due to dynamic

forces. Hence it may be require light weight coupling

and balancing on the overhang after assembly to

control the forces.

4. Machines with 50Hz operation and overhung can have

a rigid coupling instead of flexible to shift the critical

when they are assembled. But the distance piece needs

balancing along with one of the rotor.

5. In the present case, the damping is high and hence

decrease in coupling weight may be helpful and it may

get moment hinge support from flexible member thus

may improve system.

6. The Torsional critical speeds with proposed and old

couplings are quite far away from operating speeds.

The system is free from Torsional resonance and is

7th International Conference on Latest Trends in Engineering & Technology (ICLTET'2015) Nov. 26-27, 2015 Irene, Pretoria (South Africa)

http://dx.doi.org/10.15242/IIE.E1115049 164

safe with both couplings. The Torsional critical speeds

are 1250 cpm and 5075 cpm.

7. The rigid and damped critical speeds are at 2000 rpm

and at 3000 rpm along with coupling. There is slight

variation in critical speed with the weight of the

coupling. With 560 Kgs weight the critical speed is

1890 rpm and 3000 rpm and with modified coupling

there is increase in critical speed of 200 to 300 rpm.

8. The mode shapes are sensitive to bearing stiffness and

coupling moment. With lower bearing stiffness the

modes are under control and in all aspects the coupling

plays dominant rule. With higher stiffness the exciter

vibrations will be significant.

REFERENCES

[1] Rao, J.S., "Rotor Dynamics", Wiley Eastern Limited, New Delhi, 1983. [2] Chree, C., "The Whirling and Transverse Vibrations of Rotating Shafts",

Phil. Mag. 7, 1904, p.504.

http://dx.doi.org/10.1080/14786440409463146 [3] Lowey, R.G. and Piarulli, V.J., "The Dynamics of Rotating Shafts",

SUM No.4, The Shock and Vibration Information Center, Washington,

D.C., 1970. [4] Campbell Wilfred. “Protection of Steam Turbine Disk Wheels from

Axial Vibration", Trans. ASME., Vol. 46, 1924, pp. 31-160.

[5] Bansal, P.N. and Kirk, R.G., "Stability and Damped Critical Speeds of Rotor Bearing Systems", ASME Paper No. 75-DET-57.

[6] Darlow, M.S., Smalley, A.J. and Ogg.J.,"Critical Speeds and Response

of Large Vertical Pump", ASME, Paper No. 78-PUP-34. [7] Gunter, E.J. and DeChowdhury, P.,"Dynamic Stability of Flexible Rotor

Systems", ME-4040-104-700, University of Virginia, Charlottes Ville

1970. [8] Prohl. M.A., "a General Method for Calculating Critical Speeds of

Flexible Rotors", Trans. ASME, J. Appl. Mechanics, Vol.12, No.3,

Sept. 1945, pp. A142-A148. [9] Holzer, H.,"Die Berechnung der Drehswingunen", Springer Verlag

OHG, Berlin, Published by J.Edwards, inc.Ann ABN Mich. 1921. [10] W.J.Chen & E.J.Gunter., “DyRoBeS ( Dynamics of Rotor Bearing

Systems) is a powerful and complete Rotor Dynamics and Bearing

Analysis program based on Finite Element Analysis", A Comprehensive

Rotor Dynamics book, 1985.

7th International Conference on Latest Trends in Engineering & Technology (ICLTET'2015) Nov. 26-27, 2015 Irene, Pretoria (South Africa)

http://dx.doi.org/10.15242/IIE.E1115049 165