Optimizing Gas Turbine Bearing Vibration Using 2D D’Alembert’s Equation

OPTIMIZING GAS TURBINE BEARING VIBRATION USING 2-D D’ALEMBERT’S

EQUATION

By

ABSTRACT

The ability to identify faulty equipment under severe

conditions allows the manufacturers and operators to take

appropriate proactive measures to rectify the faults and

provide an assessment for early detection of the engine defects

that could lead to catastrophic failure. This work therefore

shows that such counter measures can be carried out by

optimization, modeling, simulation or using a reliable analysis

of gas turbine engine, rotor shaft and bearing vibration data

through 2-Dimensional D‟Alembert‟s equation. Gas turbine

plants on industrial duty for electricity generation were thus

used to actualize the research. The data for vibration

amplitude of rotor bearings from the reference engine which

varied between 0.19 and 3.81 mm/s were compared with those

obtained from a failed engine of similar characteristics just

before failure and used for the simulation and modeling. The

given engine speed and active load were also determined as

falling between 7264 rpm to 7436 rpm and 10 MW to 20 MW

respectively. A well packaged computer program code-named

“MELBF” written in C++ programming language was

developed. The results show that the machine should not be

run beyond 3.81 mm/s vibration amplitude in order to avoid

resonance and downtime of the engine.

Keywords: 2-D D’Alembert’s Equation, Bearing Vibration,

Catastrophic failure, Modeling and Simulation,

Optimization, Proactive Vibration Monitoring,

Rotor Shaft Faults.

___________________________________

NOMENCLATURE

k = Shaft stiffness constant (kN.m/rad)

ω = Engine rotational speed (rad/sec)

ωn = Natural frequency (rpm)

M = Lumped mass of shaft (kg)

.x = Vibration velocity amplitude

( mm/s)

C = Damping coefficient/constant of bearing

in translation (kN. s/m)

Fo =Forcing function at initial frequency (N)

..x = Vibration acceleration displacement (mm/s2)

X= Vibration displacement amplitude (mm)

N = Cycle per minute (rpm)

Cc = Critical damping coefficient constant

ξ = Damping ratio

m = Infinitesimal mass (g) causing Eccentricity (e)

e = Eccentricity (mm)

ψ , h (v) = Arbitrary function

= Phase angle

T = frequency (rpm)

ρ = mass per unit length (kg/m)

F(t)= Inertia force

1. INTRODUCTION

Gas turbine (GT) engines when in operation are mainly

affected by rotor shaft faults such as unbalance, misalignment,

looseness, distortion, bearing vibration and eccentric journal

etc (Ogbonnaya, 2009). Ability to identify faulty components

under severe condition allows the users and operators to take

appropriate proactive measures to rectify the faults and

Ezenwa A. Ogbonnaya*

Department of Mechanical Engineering,

Micheal Okpara University of Agriculture,

Umudike-Umuahaia, Nigeria.

Email: [email protected]

Kombo Theophilus-Johnson

Department of Marine Engineering,

Rivers State Univesity of Science and

Technology, Port Harcourt, Nigeria.


Hyginus U. Ugwu

Department of Mechanical Engineering,

Micheal Okpara University of Agriculture,

Umudike-Umuahaia, Nigeria.


Peter B. Forsman

Department of Welding, Oil and Gas

Engineering, Petroleum Training Institute,

Effurun, Delta State, Nigeria.


Charles U. Orji

Department of Marine Engineering,

Rivers State Univesity of Science and

Technology, Port Harcourt, Nigeria.


*Corresponding Author: E.A Ogbonnaya, 0803-777-8406,

[email protected]

Proceedings of ASME Turbo Expo 2012 GT2012

June 11-15, 2012, Copenhagen, Denmark

GT2012-68213

1 Copyright © 2012 by ASME

provide an assessment for early detection of the engine faults

that could lead to catastrophic failure (Carter, 1993).

However, vibration analysis has been used in fault diagnosis

and in the view of Mobley (1999), it is suitable for application

in rotating machines with rolling bearing for decades.

Recently, evidence has placed vibration condition monitoring

technique as one of the best amongst non- destructive

techniques as it provides greater and more effective

maintenance programme with large cost benefits to industry

(Luo et al., 2000; Ahmadi and Mollazade, 2009). Vibration

analysis in particular has for some time been used as a

predictive maintenance procedure and as a support for

scheduling machinery maintenance periodicity (Ogbonnaya

2004; Ahmadi and Mollazade 2009).

According to Azovtsev et al (1997), the wear mechanisms

primarily made up of sliding and rubbing for element of

bearing such as inner race, outer race or balls wear can be

represented by an increase in vibration amplitude and a narrow

band region of increasing energy content in the frequency

spectrum. An evident offset of the spectrum from the baseline

of zero amplitude may also provide an indication of wear of

the bearing elements. Figure 1 shows a sectional/pictorial view

of a defective and a normal journal bearing.

1.1 Other Approaches/Techniques

1) Hyunchual and Shen (2009) presented an approach on

mode evolution of asymmetric rotor assembled to

flexible bearing and housing. Two mathematical criteria

were used to identify balance modes.

2) Tachung and Kai-Shang (2009) dealt specifically on

identification of speed-dependent bearing coefficients

from unbalanced responses in one test run. In that work,

second order polynomials were assumed to fit the

bearing coefficients from where unbalance

measurements were made at several rotating speeds in

that speed range to solve for the polynomial

coefficients.

3) A diagnozing and prognozing approach of GT rotor

shaft faults was carried out in Ogbonnaya (2009). In

that work, a method of computer application in visual-

basic programming language was suggested. A

software named MICE was also used for the analysis.

Fig. 1: A sectional view of defective and normal journa l bearing (Meier-Peter, 2009): Indian Exporters’ Guide, (2012)

1.2 Approach Used in this Work

This paper provides a comprehensive optimization of GT rotor

bearing vibration using D‟Alembert‟s solution. A program

code-named MELBF written in C++ programming language

was used to bring it to fruition. MELBF stands for Misaligned,

Eccentricity, Loose Bearing, Failure. It further provides a

detailed description of these phenomenon/threats and their


consequences. Also this present work looked at general

bearing faults that could be detected with vibration monitoring

(Barkov and Barkova, 1995).

2. MATERIALS AND METHODS

The method adopted in this work is through the use of

D‟Alembert‟s solution as to further express the problems of

vibration. Generally, D‟Alembert‟s principle is a method to

convert a dynamic system into an equivalent static system by

adding the inertia force, taken in the reverse direction to the

restoring forces (Sadhu, 2006). For the type of system shown

in figure 2, the principle states that

..xM + C + kx = F(t) 1

Fig. 2: Damped vibration system Thomson, (1972)

2.1 Derivation of D’Alembert’s Solution for partial

differential equation

According to Kreyzig (2010) and Saeed (2010),

2x

u 2 2C

2t

u 2

2

where ρ

T 2C

The wave equation can be immediately obtained by

transferring equation 2 in a suitable way namely, by

introducing the new independent variables. Hence,

ct x z ,ct' x v 3

Then u becomes a function of v and z. The derivatives in

equation 2 can now be expressed in terms of derivations with

respect to v and z.

Denoting partial derivations by subscripts such that 1 vx

and 1 z x For simplicity t x,u becomes

zvxzxvx u u zu v u u 4

zzvzvv

xzzvxvzvzvxx

u 2u u

z u u v u u xu u u

5

Similarly equation 2 can be transformed in the same procedure

finding

u u 2 u c u zzvzvvtt2 6

By inserting these into results, we get

0 v z

u2 vzU

7

Obviously, the point of the present method is that the resulting

equation 7 can be readily solved by two successive

integrations.

Hence,

v h v

u

v z

u 2

8

where v h is an arbitrary function of v, and so

z Ψ dν v h u 9

where z is also an arbitrary function of z. Since the

integral is a function of v, say, ϕ(v)

The solution u is of the form z v u relating to

equation 3, we have

ct - x Ψ ct x φ t x,u 10

Equation 10 is called second order D‟ALembert‟s solution of

the wave equation 2.

GT rotor bearing when in operation is subjected to forced

harmonic motion as is usually modeled in the spring-mass-

damper system. From equation 3, we can assume the

following in equation (10) and advance to what is stated in

(Sadhu, 2006). See equation (1).

Thus from the elementary modeled form, the reversed inertia

force is..xM , the restoring forces acting on the object are

.xC

and spring force Kx in which equation 10 can be translated and

assumed as shown below

W

f (t)

W

K

M M

C K

x

C

M


f(t) ct - x Ψ ct xφ t x,u 11

Also recall from equation (1)

..xM + C + kx = F(t)

Thus, the steady state solution of equation 11 yields

M

e m

nω

ω 2ξ

22

nω

ω1

2

nω

ω

X

12

Equation 12 was used to develop a program named MEBLF

written in C++ programming language. A flow chart drawn

from equation 12 is shown in figure 3. It is a single loop flow

chart with two subroutines to convert N to and X to The

vibration readings from the test engines were taken in units of

displacement amplitude. The actual and theoretical data

collected were statistically analyzed as shown in table 1. The

link between D‟Alembert‟s solution and equation 1 is thus

shown in equation 11. Hence why it was possible to use

equation 12 for the simulation in this work.

2.2 Instrumentations and Test Equipment

The data used in the various aspects of this work were

collected from an operational H25 GT (Delta Unit 5) plant on

industrial duty for electricity generation. The plant is located

in Ughelli, Delta State of Nigeria. The various locations of the

bearings along the shaft line are shown in Appendix A while a

typical print-out/screen capture from the equipment is shown

in Appendix B. The most important operating data/parameters

considered were those of vibration amplitude and turbine

speed.

It is note-worthy that the „historian‟ was used to get the last

vibration amplitude reading just before failure and matched

with turbine speed from a defective Delta Unit 3 H-25 engine

in the same location. This helped to actuallize the simulation

carried out in this work when compared with the readings of

the normal running condition obtained from the Delta Unit 5

H-25 engine. Eccentricity of the shaft through deformed

bearing was not possible to be established in the process of the

work. Hence lumped vibration equations were used in the

calculations. Due to the above reason, wave equations as

introduced in D‟Alembert‟s principles were also used in the

research.

A secondary source of data collection technique which

involved a case study approach was used in the work. This

involved data collection on a large number of variables and

enhanced validation of findings.

Fig. 3: Program code in C++ programming language

Speeds were taken at various loads with a view to finding the

effect of load on turbine speed and vibration amplitude. These

vibration amplitudes were matched with the corresponding

turbine speeds at which they occurred.

The measuring instrumentations used were those already fitted

on the GT plants. These were the model CV-87 high

temperature velocity transducer, a self-generating electro-

.xC

..xM Kx

Start

Input total Number

of iterations NN

Input K, C, M, e,

Input N(I), Time (I)

I=0

No

Yes

I=I+1

I ˂NN

Stop

Declare and define

variables used

= 52.36*X*106*ω

ω(I)=2πN(I), Time (I)

Print (I), ω(I), N(I), Time (I)


dynamic moving coil instrument which required no external

power supply and a module VT-355P electromagnetic type

speed detector (Model PA-150).

Table 1: Reading taken from GT-3 and GT-5

Turbine Speed

(rpm)

reference Vibration

Amplitude (mm/s)

load (MW)

Bearing 1 Bearing 2 Bearing 4 Bearing 5

GT-5

normal

GT-3

before

failure

GT-5

normal

GT-3

before

failure

GT-5

normal

GT-3

before

failure

GT-5

normal

GT-3

before

failure

7264 2.41 - 2.31 3.1 2.12 2.72 0.97 1.56 0.19 0.58

7326 2.92 15 2.88 3.29 2.06 2.62 0.91 1.49 0.21 0.62

7326 2.82 20 2.83 3.14 2.06 2.62 0.91 1.51 0.25 0.61

7328 2.71 - 2.67 2.99 2.59 3.07 0.76 1.66 0.25 0.59

7331 3.81 10 3.73 4.45 3.47 4.04 1.24 1.81 0.45 0.82

7341 2.5 20 2.38 2.99 2.08 2.64 0.83 1.41 0.25 0.62

7341 2.92 - 2.86 3.2 2.37 2.91 1.02 1.64 0.23 0.65

7342 3.39 20 3.39 4.08 3.09 3.63 0.76 1.38 0.36 0.69

7348 2.92 20 2.86 3.08 2.22 2.8 1.08 1.67 0.23 0.61

7349 2.92 20 2.71 3.18 2.5 3.08 0.79 1.34 0.25 0.66

7351 2.63 10 2.59 2.93 2.61 3.14 1.11 1.7 0.32 0.71

7352 2.64 10 2.73 3.09 2.25 2.85 1.09 1.62 0.33 0.74

7352 2.84 20 2.82 3.19 2.65 3.22 0.77 1.39 0.32 0.72

7355 2.92 20 2.87 3.24 2.76 3.31 0.81 1.34 0.32 0.73

7358 3.09 20 2.92 3.35 2.65 3.2 0.85 1.44 0.32 0.72

7359 2.92 20 2.76 3.11 2.65 3.2 0.85 1.49 0.32 0.74

7359 2.71 20 2.63 3.02 2.63 3.19 0.71 1.36 0.34 0.69

7360 3.55 20 3.19 3.71 2.13 2.71 0.51 1.06 0.28 0.7

7369 2.88 20 2.77 3.13 2.61 3.15 1.11 1.68 0.32 0.74

7371 3.21 20 3.09 3.49 2.96 2.52 0.78 1.39 0.36 0.74

7379 2.96 20 2.9 3.34 2.65 3.25 0.83 1.47 0.38 0.8

7388 3.17 10 3.1 3.4 1.7 2.26 0.81 1.39 0.43 0.88

7392 3.04 20 3.05 3.51 2.63 3.34 0.81 1.43 0.36 0.62

7395 3.26 20 3.11 3.62 2.69 3.99 0.89 1.5 0.36 0.73

7404 3.26 20 3.09 3.46 2.5 3.08 1.04 1.61 0.32 0.73

7407 3.34 20 3.33 3.81 2.98 3.58 0.81 1.52 0.72 0.99

7436 3.26 20 3.26 3.7 2.7 3.35 0.74 1.34 0.38 0.77

The vibration and speed measurement module are installed in

the turbine supervisory system cabinet outside the engine

room while the output is indicated on the indicator or recorder

on the computer monitor (GT Manuals 1985, 2001 and

Ogbonnaya, 2004). Recordings and data could always easily

be obtained from a computer monitor while permanently

stored data are left in a historian attached to each engine. Data

from the historian are mostly retrievable only after an

emergency. The graphs shown in figures 4 to 7 are combined

plots of amplitude of vibration for bearings 1 to 4 of the

defective and reference GT engines obtained from

measurements. It was thus possible to carry out a simulation

while correlating a threshold operating limit of the failed GT

engine (Delta Unit 3) and that of the reference H-25 (Delta

Unit 5). The readings obtained were then fed into the

analogous equation 12 to that of D‟Alembert‟s equation.


3.0 ANALYSIS AND DISCUSSION OF RESULTS

Values shown in table 1 represent parameters measured from

the engines used to demonstrate the authenticity of D‟

Alembert‟s solution to model and simulate bearing vibration

failure. No matter the form in which vibration is represented,

this work confirms that oscillation is always present whenever

a mechanical system is in operation.

Furthermore, the graphs shown in figures 4 to 7 helped to

show how trajectories generated from D‟Alembert‟s equation

can be used to make maintenance decision from vibration

spectrum. The load associated with each turbine speed is also

shown in table 1. The spectrum above in figures 4 to 7 are

those from a failed engine, GT-3 with similar characteristics

as GT-5 used for the simulation. Bearing 3 is not shown on the

shaft line of the plant because it exists as a thrust bearing on

which no transducer is fitted. The analysis of the recorded

vibration amplitudes versus turbine speeds is discussed below.

The vibration readings are tabulated bearing by bearing since

the transducers are fitted to the bearing housing. The effect of

vibration amplitude on bearing 1 increased rapidly at about

7326 rpm and attained an unusual waveform up to 7440 rpm.

The speed from 7264 to 7326 rpm is the run-up speed when

there is no load on the turbine. From the vibration spectrum of

bearing 1, it is noticed that the critical speed of the engine is

7331 rpm where vibration is highest. This speed

approximately corresponds to the specific critical speed of

7330 rpm in the manufacturer‟s manuals.

Fig. 4: Vibration amplitude for bearing 1 against turbine speed.

It is interesting to note that the vibration spectrum for bearing

2 generally has the same pattern with those of bearing 1 except

during the period of run-up. This is due to the fact that much

of the factors that lead to the pattern of slope during the run-up

period in bearing 1 have reduced.

Bearing 3 is the thrust bearing. No transducer is fitted to it.

Hence no vibration measurement is taken from it. It is thus

equally omitted in the labeling of the bearings in table 1.

Fig. 5: Vibration amplitude of bearing2 against turbine speed

Bearing 4 has the run-up vibration spectrum further sloping

downwards more than that of bearing 2 due to the same reason

proffered for bearing 2. However the peak-peak vibration

levels are of much more reduced levels than those of bearings

1 and 2. This is because bearing 4 is further away from the

reference point than bearings 1 and 2.

Fig. 6: Vibration amplitude of bearing 4 against rotational speed

Bearing 5 has almost a flat trajectory during run-up but very

slightly sloping upwards before the set of conspicuous

vibration. Its peak-peak values are equally much less than

those of other bearings. It could be noticed that the spectra for

GT 3 bearing just before failure have remarkable deviation

from these for GT 5 under normal operation. These deviations

would have been used to assertain the onset of failure even

before GT 3 finally grounded to a halt.

Reference velocity vibration amplitude is normally generally

imposed on all the bearings by either the engine manufacturers

or the engine operator/watch keeper. This is the limit of

0

1

2

3

4

5

7250 7300 7350 7400 7450 Vib

rati

on

am

plit

ud

e fo

r b

eari

ng

1 (m

m/s

)

Speed (rpm)

GT-5 GT-3

0

1

2

3

4

5

7250 7300 7350 7400 7450

Vib

rati

on

am

pli

tud

e f

or

bearin

g 2

(m

m/s

)

Speed (rpm) GT-5 GT-3

0

0.5

1

1.5

2

7250 7300 7350 7400 7450

Vib

rati

on

am

pli

tud

e fo

r

bea

rin

g 4

(m

m/s

)



vibration level as shown in table 1 that no bearing vibration is

to exceed.

Fig. 7: Vibration amplitude of bearing 5 against speed

The reference vibration amplitude is also shown in fig 8 which

contains a general trajectory of all the bearings on one graph

to give a quick comparison of the various vibration amplitude

of the bearings for GT-5.

4 CONCLUSION AND RECOMMENDATIONS

4.1 Conclusion

A work on optimization of GT bearing vibration using 2-D

D‟Alembert‟s equation has been carried out. An already

existing equation with bearing vibration amplitude, speed and

other terms as constants was infused into D‟ Alembert

equation to come up with the very useful innovation to handle

GT bearing defects. Results thus show that monitoring the

health of the bearings can help in the prevention of downtime

of a GT plant. The bearing at the compressor end of the GT

plant carries much load and thus high vibration amplitude

occurs there. So a step-by-step analysis of plant parameters

obtained was used to write a program on C++ programming

language to determine the vibration displacement amplitude of

the plant.

It was also found out that the amount of defect meted to the

GT depends on amplitude of vibration and friction as

measured at the bearing housings. The work also shows that vibration diagnostics is most efficient for detection and

identification of incipient defects in bearings. Therefore

vibration monitoring and analysis of GT bearings are very

important to reduce catastrophic downtime of the plant.

Fig. 8: Combined actual vibration amplitudes against turbine speeds

4.2 Recommendations

From the results obtained in this work, the following recommendations are made;

The performance of GT bearings should be constantly

monitored during the engine operation.

The software developed should be incorporated to help in

monitoring the vibration amplitude of the bearings.

The bearing at the turbine side should be rigidly

constructed.

0

0.2

0.4

0.6

0.8

1

1.2

7250 7300 7350 7400 7450

Vib

rati

on

am

pli

tud

e f

or

bearin

g 5

(m

m/s

)



Acknowledgments

The authors wish to acknowledge the roles played by Mr John

N. Woji and Mr Kuvie A. Ejabefio in collecting, collating the

data used for this work. They also helped in the typesetting of

the manuscript. We are also indebted to the Management and

Staff of Delta Thermal Station, Ughelli, Delta State, Nigeria

for allowing the use of their equipment. God bless you all.

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Marine Engineering, Rivers State University of Science

and Technology, Port Harcourt, Rivers State, Nigeria.

Ogbonnaya, E.A. (2009) Diagnozing and Prognozing Gas

Turbine Rotor Shaft Faults Using the “MICE”, ASME, Turbo-Expo, Paper No GT2009-59450, Orlando, Florida,

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23


http://www.indianexpoters.com/whitemetal-babbitt-metal-reconditioning-and%20-relining-sell99558.html

http://www.indianexpoters.com/whitemetal-babbitt-metal-reconditioning-and%20-relining-sell99558.html

http://www.dvvmedia.com/

APPENDIX A:

Schematic of a 25MW GT rotor shafting system showing the locations of the bearings.

APPENDIX B:

A typical printout from the screen of Unit Ughelli H-25 GT system


Optimizing Gas Turbine Bearing Vibration Using 2D D’Alembert’s Equation

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