Condition Monitoring Lectures

7/29/2019 Condition Monitoring Lectures

1/249

CONTENTS

Module A

1. Introduction

2. Failure

3. Single-Degree-of-Freedom Vibration

4. Multi-Degree-of-Freedom Vibration

5. Balancing

6. Steam Turbine7. Gas Turbine

8. Generator

9. Pumps

Module B

10. Hydraulic Turbines

11. Fans

12. Wind Turbines

13. Gearboxes and Rolling Element Bearings

14. Condition Monitoring

15. Sensors and Instrumentation

16. Analysis Techniques

17. Fault Diagnosis

MECH7350 is delivered in two intensive modules over one semester. This volume contains the

course notes for Module A. Supplementary material will be provided through lectures from

industry experts.


2/249

MECH7350 Rotating Machinery 1. Introduction

1-1

1. INTRODUCTION

(This section is based largely on Black and Veatch)

Fig. 1.1 is a diagram of a typical pulverised coal-fuelled electrical generating facility. In this

section the main components are identified and an overview of the course is presented.

Throughout the course typical rotating machinery is explained. It is left to students to

identify features of particular plant in their situations.

1.1 Coal Handling and Pulverising

Coal is usually delivered to the facility by trains or long conveyor belts. The coal handling

system unloads the coal, then stacks, reclaims, crushes and conveys it to a storage silo. Coal

is fed from the storage silos, pulverised to a powder, and blown into the steam generator.

Fig. 1.1 Typical pulverised coal fuelled electrical generating facility (from Black and Veatch)


3/249


1-2

There is rotating machinery in the coal handling and pulverising systems but it will not be

considered explicitly in this course. There are electric motors, gearboxes and mills that have,

for example, generic bearing and vibration issues but such issues will be addressed when

other parts of the overall facility are addressed.

1.2 Steam Generator

Pulverised coal is mixed with air in the steam generator and combusted, and the combustion

energy is used to produce, to superheat and to reheat steam. The only parts of the steam

generator that will be addressed are the forced draft fans.

1.3 Turbine

The steam turbine converts the thermal energy of the steam to rotating mechanical energy,

and the generator which is coupled to the turbine, converts the mechanical energy to

electrical energy. Aspects that are addressed include:

Configurations

Speed of rotation

Design features

Casings

Couplings

Alignment

Rotors

Balancing

Blading

o Types of blade; impulse and reaction stages

o Materials

o Losses

o Blade attachment

o Blade vibration

o Blade erosion

Bearings

Bearing vibration

Instrumentation

Condition monitoring


4/249


1-3

1.4 Generator

Aspects addressed include:

Basics of synchronous generator theory

Three-phase windings

Rotor design features

Bearings

Rotor winding

Sliprings, brushgear and shaft earthing

Stator design features

Stator winding Cooling

Excitation systems

Vibration

Balancing

Instrumentation


1.5 Condenser and Cooling Towers

Steam exhausted from the low-pressure section of the steam turbine is condensed to liquid in

the condenser. The condensed water is moved from the condenser by condensate pumps

through low-pressure regenerative feedwater heaters to a deaerator. Boiler feed pumps move

the deaerated water through high pressure regenerative heaters to the steam generator. Other

pumps are used for circulating water through the cooling towers, and for oil lubrication.

Aspects of pumps that will be addressed include: Design features

Performance

Configurations

Operating characteristics

Cavitation


5/249


1-4

1.6 Forced Draft, Primary Air and Induced Draft Fans

Combustion air is supplied to the steam generator by forced draft fans. Primary air fans

transport pulverised coal into the steam generator. Induced draft fans remove the flue gases

from the steam generator and exhaust them to the stack. Aspects of fans to be addressed

include:

Design features

Performance

Configurations

Operating characteristics

1.7 Electric Motors

The generator is the largest rotating electrical machine in a power generation plant.

However, many electric motors, large and small, are used. Issues to be addressed include:

Rolling element bearings

Balancing

1.8 Gearboxes

There are a number of large gearboxes in a modern power station, e.g. on the coal pulveriser.

Issues to be addressed include:

Design of gear trains

Rolling element bearings

Fatigue

Lubrication


1.9 Other Types of Rotating Plant

Other types of rotating plant to be addressed are:

Gas turbines, current and future

Wind turbines

Hydraulic turbines


6/249


1-5

1.10 Fundamentals

Rotating machinery can fail catastrophically or wear to such an extent that performance is

degraded or there is a risk of catastrophic failure. To enable an appreciation of failure modes,

the course includes an introduction to the phenomena of wear and fracture mechanics of

materials.

Component failures are often caused by the cyclic forces that can develop in rotating

machinery. A fundamental understanding of the following is presented:

Newtonian dynamics in rotating machinery

Vibration theory

Balancing

Vibration measurement


Signal processing

Fault diagnosis using vibration analysis

Basic understanding of the design, performance and behaviour of turbines, fans, pumps and

hydrodynamic lubrication requires some knowledge of fundamentals of fluid mechanics.

This is presented.


7/249


8/249

MECH7350 Rotating Machinery 2. Failure

2-2

2.2 Fracture Mechanics

2.2.1 Types of Failure

Failure of a loaded member can be regarded as any behaviour that renders it unsuitable for its

intended function. Static loading can result in objectionable deflection and elastic instability,

as well as plastic distortion and fracture. Distortion, or plastic strain, is associated with shear

stresses and involves slipping along natural slip planes. Failure is defined as having occurred

when the plastic deformation reaches an arbitrary limit. Fracture, on the other hand, is clearly

defined as the separation or fragmentation of a member into two or more pieces. It normally

constitutes a pulling apart, associated with tensile stress.

In general, materials prone to distortion failure are classed as ductile, and those prone to

fracture without significant prior distortion as brittle. Unfortunately, there is an intermediate

grey area wherein a given material can fail in either a ductile or a brittle manner, depending

on circumstances. Normally ductile materials can fracture in a brittle manner at sufficiently

low temperatures. Other factors promoting brittle fracture are sharp notches and impact

loading.

2.2.2 Basic Concepts

The fracture mechanics approach begins with the assumption that all real materials contain

cracks of some size even if only submicroscopic. If brittle failure occurs, it is because the

conditions of loading and environment (primarily temperature) are such that they cause an

almost instantaneous propagation to failure of one or more of the original cracks. If there is

fatigue (cyclic) loading, the initial cracks may grow very slowly until one of them reaches a

critical size, at which time total fracture occurs

2.2.3 Stress Concentration

g = gross-section tensile stress

2

P

wt=

PP

t

Stress concentration

2w


9/249


2-3

However, stress is much higher at the base of a crack (stress concentration factor). If radius

of crack root approaches zero, stress concentration factor approaches infinity. This means

that ductile yielding will occur in a small volume of material at the crack root, and the stress

will be redistributed. Thus the stress concentration factor is considerably less than infinity.

In the fracture mechanics approach, a stress intensity factorKis evaluated theoretically (more

soon) and is compared with a limiting value of K that is found from standard tests to be

necessary for crack propagation in that material. The limiting value is called fracture

toughness or critical stress intensity factorc

K . Failure occurs when Kexceedsc

K .

Values of cK are substantially lower for thick members (plane strain) than for thin members(plane stress), so it is conservative to assume thick members. Thick members offer less

opportunity for redistributing high crack root stresses by shear yielding.

Table 2.1 contains typical mechanical properties of 25.4 mm thick plates made of common

aircraft structural materials. Note:

The relatively high fracture toughness of the titanium alloy in comparison to its

ultimate strength uS ;

The room temperature comparison ofc

K for the two steels of nearly equivalent

ultimate strengths; and

The reduction inc

K with temperature for the high-toughness D6AC steel.

Yield strengthy

S is the tensile stress at which plastic yielding first occurs in a specimen

tensile test. Ultimate stressu

S is the stress in the tensile stress specimen when it is carrying

the maximum possible load before failure. SyandSu are defined below.


10/249


2-4

Table 2.1 Strength properties of 25.4 mm thick plates values of Ultimate Stressu

S , Yield Stress yS and

Critical Stress Intensity Factorc

K .

Material Temperatureu

S

MPay

S

MPac

K

MPa(m)1/2

7075-T651 aluminium alloy Room 538 483 29.67

Ti-6A1-4V (annealed) titanium alloy Room 896 827 71.44

D6AC high toughness steel Room 1517 1310 76.90

D6AC high toughness steel -40C 1565 1358 49.46

4340 steel Room 1793 1496 57.15

Cracks generally begin in thick members at the surface, and have a somewhat elliptical form,

as shown adjacent. Research has established that if:

62

>t

w

2

a

c= about 0.26

3>c

w

0.5a

t<

0.8g

yS

<

Then at the edge of the crack, Kis given approximately by:

( )2

0.39 0.053 /

g

g y

a

S

Fracture would be predicted forK>c

K .

2.3 Fatigue

Fatigue failure might better be described as progressive fracture under fluctuating or repeatedloading. Fatigue fractures begin with a minute (usually microscopic) crack at a critical area

t


11/249


2-5

of high local stress. This is almost always at a geometric stress raiser. Fatigue failure results

from repeated plastic deformation, such as breaking of a wire by bending it back and forth.

Whereas a wire can be broken after a few cycles of gross plastic yielding, fatigue failures

typically occur after thousands or even millions of cycles of minute yielding. Fatigue failure

can occur at stress levels far below the conventionally determined yield point or elastic limit.

The initial fatigue crack usually results in an increase in local stress concentration. As the

crack progresses, the material at the crack root at any particular time is subjected to the

destructive localised reversed yielding. As the crack deepens, thereby reducing the section

and causing increased stresses, the rate of crack propagation increases until the remaining

section is no longer able to support a single load application and final fracture occurs.

Engineering practice relies on empirical fatigue data from the standardised R.R. Moore

fatigue test rotating beam, shown diagrammatically below.

S-N curves are generated for materials. The figure below is typical. (S = stress, N = number

of cycles to failure at the amplitude S of oscillatory stress.)

Notch

Small region behavesplastically

Main body behaveselastically


12/249


2-6

The adjacent figure is a typical S-N curve for

steel and shows an endurance limit. This is the

stress below which fatigue failure does not

occur, even for an indefinitely large number of

loading cycles. For a low carbon steel the

endurance limit is about one-half of the ultimate

strength.

2.4 Surface Damage

More machine parts fail through surface damage than breakage. Various mechanisms for

surface damage are described briefly.

2.4.1 Corrosion

Corrosion is the degradation of a material (normally a metal) by chemical or electrochemical

reaction with its environment. It can combine with static or fatigue stresses to produce a

more destructive action than would be expected by considering the actions of corrosion and

stress separately.

2.4.2 Cavitation Damage

Cavitation damage is the formation of bubbles in a liquid that is moving with respect to a

nearby solid surface. Bubbles are formed when the liquid pressure drops below its vapour

pressure. When these bubbles subsequently collapse at or near the solid surface, pressure

waves impinge upon the surface causing local stresses that can be great enough to cause

plastic deformation of many metals. A surface damaged by cavitation appears roughened,

with closely spaced pits. In severe cases, enough material is removed to give the surface a

spongy texture. Cavitation commonly occurs in centrifugal pumps and turbine blades.

2.4.3 Adhesive Wear

When two surfaces slide across each other, the contact pressure and frictional heat of sliding

are concentrated at the small local areas of contact (asperities). Local temperatures and

pressures are extremely high and conditions are favourable for welding at these points. These

welds fail in shear, and new welds form, and so on. This is called adhesive wear. Loose

particles resulting from the wear can cause further damage.


13/249


2-7

2.4.4 Surface Fatigue

When curved elastic bodies, such as parts of a rolling-element bearing, are pressed together,

finite contact areas are developed because of deflections. These contact areas are so small

that very high compressive stresses can result in a cyclic manner. Fatigue failures can be

initiated by minute cracks that propagate to permit small pieces of material to separate from

the surfaces. This is pitting or spalling.


14/249

MECH7350 Rotating Machinery 3. Single Degree-of-Freedom Vibration

3-1

3. SINGLE DEGREE-OF-FREEDOM VIBRATION

Understanding of complex vibration problems begins with understanding of the vibration of a

system with a single degree-of-freedom, i.e. a system for which the motion can be described

by the time variation of a single coordinate. In many situations the important features of the

behaviour of complicated multi-degree-of-freedom systems can be described adequately with

a single degree-of-freedom system.

3.1 Notation

Vectors are used in this section. A vector quantity is represented by an underscore. For

example, F

can be a force which has a scalar magnitude Fand a direction of action.

The notation F indicates the sum of vectors, i.e.,1 2 ..... nF F F F = + + +

This is shown diagrammatically in Fig. 3.1.

Displacementx

, velocity v

and acceleration a

can also be regarded as vector quantities,

although a scalar representation is adequate in systems with a single degree-of-freedom. The

following notation is used for time-derivatives.

dxv x

dt= =

dva v x

dt= = =

Fig. 3.1 vector notation

1F

2F

3F

nF

F


15/249


3-2

3.2 Units

mass kilograms, kg

time seconds, s

displacement metres, m

velocity m/s

acceleration m/s2

force kg m/s2

(from Second Law), called newtons, N

angle radians, rad

/s r =

For a full circle,02

2 360r

radr

= = =

d

dt

= angular velocity, rad/s

2/60 rad/s = 1 rpm

3.3 Some Fundamentals of Dynamics

Here some important dynamics results are summarised in the context of rotating machines.

Full detail is available in Bedford and Fowler.

3.3.1 Newtons Laws

Newtons laws of motion were first enunciated in Sir Isaac Newton (1687) Philosophiae

Naturalis Principia Mathematica. They are written in modern language as follows (from

Bedford and Fowler).

First Law: When the sum of the forces acting on a particle is zero, its velocity is constant.

In particular, if the particle is initially stationary, it will remain stationary.

Second Law: When the sum of the forces acting on a particle is not zero, the sum of the

forces is equal to the rate of change of momentum of the particle. If the mass is constant, the

sum of the forces is equal to the product of the mass of the particle and its acceleration.

mv =

momentum

( )d mv dvF m

dt dt = =

if m is constant

2

2

d rm ma

dt= =

r

s


16/249


3-3

Third Law: The forces exerted by two particles on each other are equal in magnitude and

opposite in direction; e.g. gravitational attraction, swinging a mass on a string.

3.3.2 Rotational Equivalent

Consider holding a shaft with two hands and twist each in opposite directions. This is

equivalent to applying a moment (or torque) to the shaft with zero resulting force.

T = rF + rF = dF

Strictly, T is a vector. It acts about an axis which has direction. Consider again the Second

Law for translation;

F = ma

Mass might well have been called inertia. The rotational equivalent is:

T = I

Where = angular acceleration

=d

dt

2

2

d

dt

=

I= moment of inertia, kgm2

For a particle of mass m rotating in a circular path of radius r,I = mr2about the axis.

For a uniform rigid disc of mass m and radius r,I = mr2

Translational (linear) momentum of mass m is mv

Angular momentum of a rigid body about a fixed axis is I

Kinetic energy of translation of mass m is mv2.

Kinetic energy of rotation is I2. (Detail is in Bedford and Fowler.)

T

F

F

is equivalent to

radius r

mkm

jkf kjf jk kjf f=


17/249


3-4

3.3.3 Centre of Mass

Consider a system of two particles.

Define centre of mass as;

1 2

1 2

m r m r r

m m

+=

+

1r=

ifm2 is small.

Differentiate twice with respect to time.

1 1 2 2

1 2

m r m r r

m m

+=

+

Consider both external and internal forces acting on each particle.

For 1m , Second Law gives 1 21 1 1F f m r + =

For m2, 2 12 2 2F f m r + =

Add:

( )1 2 21 12 1 20 Total massof systemSum of external forces on system

F F f f m m r

=

+ + + = +

So;

Sum of external forces on a system of particles = total mass of system acceleration of

mass centre

F ma=

3.3.4 Vibration of Single Degree-of-Freedom Systems

Consider, for example, the simplest model of vibration of

an unbalanced turbine (rotor + casing) on elastic

mountings (Fig. 3.2). For simplicity, turbine is

constrained to move only vertically. A similar model

could be set up for horizontal vibration. Assume:

M= total mass of turbine plus casing

m = equivalent unbalance point mass

k= spring stiffness of support; (spring force = kx)

c = viscous damping coefficient of support;

(damping force = c x )

Viscous damping is a convenient approximation to more realistic non-linear damping because

it leads to analytical solutions that approximate actual systems well.

y

x

m2

m2

1r

1r

Fig. 3.2 (from Thompson).


18/249


3-5

In the equilibrium position (rotor stationary), 0x x= =

And spring force balances weight. Only changes from equilibrium

are considered, so weight (gravity) can be ignored.

Treat as a system of two particles, m andM-m, in displaced position shown.

Position of centre of mass is given by:

( ) ( sin )

( )

M m x m x e tx

M m m

+ +=

+

Apply Second Law to motion of mass centre, recognising thatx is positive upwards.

2

2

( ) ( sin )d M m x m x e t kx cx M

dt M

+ + =

[ ]

2

2 sin

d

Mx me tdt = +

2sinMx me t =

Rearrange to get differential equation of motion of turbine casing.

2

0sin sinMx cx kx me t F t + + = = (3.1)

where F0 = me2

= centripetal force (constant for constant ).

me = unbalance.

Same unbalance can be caused by larger m at smaller e.

Some special cases are now considered.

3.3.5 Free Vibrations

No rotation of turbine rotor; = 0.

Equation of motion (3.1) becomes:

0Mx cx kx+ + =

0c k

x x xM M

+ + =

Rewrite as:22 0

n nx x x + + = (3.2)

where nk

M = and

2 n

c

M

=

The physical significance of n and will become apparent.

System motion is caused by initial conditions,

Mg

kxo


19/249


3-6

(0)x = initial displacement at t = 0, and

(0)x = initial velocity.

Solution of (3.2) is:

2 2

2

( ) ( )

sin 1 ( )cos 11

nt n

n n

n

x o x o

x e t x o t

+

= +

(3.3)

This can be obtained from the theory of

differential equations, Laplace transforms, or by

substituting (3.3) into (3.2). Solution is the sum

of oscillatory sine and cosine terms, multiplied by

a time-decaying exponential (Fig. 3.3).

For no damping, c = 0 and = 0. The

solution of (3.2) is:

( )sin ( ) cosn n

n

x ox t x o t

= +

The system vibrates at undamped natural frequency nk

M = .

For < 1, the frequency of damped oscillation is 21d n = .

For > 1, the solution of (3.2) is:

2 2( 1) ) ( 1) )n nt tx Ae Be +

= + where:

2

2

(0) ( 1) (0)

2 1

n

n

x xA

+ + =

2

2

(0) ( 1) (0)

2 1

n

n

x xB

=

The motion is an exponentially decreasing function

of time, as shown in Fig. 3.4.

Fig. 3.3

Fig. 3.4


20/249


3-7

For 21, 1 0 = = and the solution of (3.2) is:

[ ]{ }(0) (0) (0) ntnx x x x t e = + +

Three types of responses are shown in Fig. 3.5 for

different initial velocities.This is called critical damping. If damping is any

higher, exponential decay is slower.

For 1, 12 n

c

M

= = so 2critical nc M=

Then is the ratio of actual damping present, c, to damping, 2 nM needed to achieve

critical damping. is called the damping ratio.

Summary = 0 undamped free vibrations

< 1 underdamped oscillations

1 = critical damping

> 1 overdamped

The frequency of damped vibrations is2

1d n = rad/s. This differs by only a small

amount from n for small . Hence n can be estimated well from the observed frequency

in transient bump tests.

3.3.6 Complex Frequency Response

In this section the system is forced to vibrate at frequency . It is convenient to use the

concept of phasors. Complex variable theory gives:

( )0 0 cos sini tX e X t i t = + where 1i =

So ( )0 0sin Imi t

X t X e =

Recall that

( )1 21 21 2 1 2.

ii iA e A e A A e +

=

and( )

1

1 2

2

1 1

2 2

ii

i

A e Ae

A e A

=

Return to (3.1).

2

0sin sinMx cx kx me t F t + + = = where2

0F me=

Fig. 3.5


21/249


3-8

Rewrite as

2 02 sinn nF

x x x tM

+ + = (3.4)

Solution is the sum of a constant amplitude forced vibration (particular integral) and a

transient response (complementary function) which decays to zero with time. We areinterested in the forced response after transients have died away.

Write [ ]0 0sin Im cos sinF F

t t i t M M

= +

0Imi tF

eM

=

We could have equally validly assumed a cosine forcing function in (3.5);

0 0cos Rei tF F

t e

M M

=

So put 0 0sin i tF F

t eM M

= in (3.4).

Now vibration x will also be a sine (or cosine) function but out-of-phase with 0 sinF

tM

.

So let( )

0

i tx x e

= (x lags 0F by )

i txe

= where 0

ix x e

= = complex amplitude of vibration

Then (3.4) becomes

2 2 02i t i t i t i t n n

Fxe i xe xe e

M

+ + =

( )2 2 02 n nF

i xM

+ + =

2

2 2 2

0

1/1

/ 2 1 2

n

n n

x

F M i i

= =

+where

n

=

Take amplitudes (moduli).

( ) ( )

2

0

120 0 2 22

1/

/ /1 2

nxx

F M F M

= =

+

(3.5)

Using2

/n k M = gives,

( ) ( )1

20 2 22

1

/1 2

static

x x

F k x

= =

+

and2

2tan

1

=


22/249


3-9

Case A Rotating unbalance

2

0F me=

So

( ) ( )

2

0

12 2 22

1 2

Mx

me

=

+

(3.6)

0Mx

meand are plotted on Fig. 3.6.

Case B 0F = constant, independent of. It follows that

( ) ( )

0

120 2 22

1

/1 2

x

F k

=

+

and (3.7)

2

2tan

1

=

(3.8)

(3.7) and (3.8) are plotted in Fig. 3.7.

Fig. 3.6


23/249


3-10

Case C Vibration Isolation

Spring and damper transmit forces to foundation that are 90o

out-of-phase.

So magnitude of force transmitted is

( ) ( ) ( )1 1

2 2 22 2

0 0 0 1 2TF kx c x kx = + = + (3.9)

Combine (3.7) and (3.9) to get force transmissibility,

0

TFTRF

= =( )

( ) ( )

1

22

2 22

1 2

1 2

+

+

(3.10)

This is plotted in Fig. 3.8.

Fig. 3.7


24/249


3-11

3.3.7 Vibration Measuring Instruments

Consider a vibrating mass m suspended inside the casing of a vibration measuring instrument,

which in turn is attached to a vertically vibrating surface (Fig. 3.9).

Fig. 3.8

Fig. 3.9.


25/249


3-12

Displace m in positivey-direction with positive velocity and draw a free-body diagram.

Assume weight of m is balanced by initial spring

force.

Equation of motion is

( ) ( )c x y k x y mx =

Assume instrument gives an output signal

proportional to relative motion z = x-y.

If 0 siny y t=

then2

0 sinmz cz kz m y t + + =

This is identical to the case of rotating unbalance, withz and m2y replacingx and me

2.

So, if ( )0 sinz z t = , then

( ) ( )

0

1220 2 22

1

1 2

z

y

=

+

(3.11)

When the natural frequency n of the instrument is high compared to that of the vibration

0 siny y t= to be measured, / 1n = and ( ) ( )1

2 2 221 2 +

approaches unity.

Then2

2 0 00 0 2 2 2

n n n

y y accelerationz y

= = = =

Hencez is proportional to the acceleration to be measured. Fig. 3.11 is a plot of 02

0/ n

z

y for

various damping ratios.

m

( )k x y

( )c x y

+vey

Fig. 3.10.


26/249


3-13

If we can choose 0.7 = , we can get a useful frequency range 0 / 0.20n with a

maximum error of 0.01 percent.

But the widely used piezoelectric crystal accelerometers (barium titanate) have 0 and can

operate up to 0.06 n = .

/ 2n nf = can be as high as 50,000 Hz, so instruments can operate to 3000 Hz. Fig. 3.11

shows the rugged construction of a piezoelectric accelerometer.

The crystal produces a charge q

proportional to z and so to

acceleration 2 .y But crystal has a

very small capacitance C, so voltage

V=q/C is greatly reduced if output

cable has a high capacitance. This is

overcome by using a charge

amplifier.

3.4 Whirling

(This section is based largely on Thompson)

Rotating shafts, with or without rotors, tend to bow out atcertain critical speeds and whirl. Whirling is the rotation of the

plane made by the bent shaft and the line of centres of the

bearings. The shaft and rotor (and SG in the figure below)

rotate with angular velocity . Whirling (and rotation of OS)

occurs at angular velocity which may or may not be equal to

and can be in the same or the opposite direction to .

Whirling can be caused by mass unbalance, eccentricity, cyclic

fluid friction in bearings and gyroscopic forces. Synchronous

whirl is when = .

Fig. 3.11 Construction of a piezoelectric accelerometer.


27/249


3-14

Ifk

m= , the critical speed, the amplitude of whirl is limited only by the damping which is

usually small. Severe damage can occur. If rotor operation is above critical speed, run-up

must pass quickly through that speed.

Long shafts without a rotor can whirl. This is a complicated problem to analyse because of

its distributed-parameter nature.

3.5 Spring Stiffness

Only linear springs are considered in this course. For small displacements, most materials

behave approximately linearly. This is an adequate assumption when calculating or

measuring natural frequencies.

Force, f kx=

k = spring stiffness

x = displacement from unstretched position

Moment, T=tk

tk = torsional spring stiffness

= rotational displacement (rad)from unstrained position

Springs in series

1k 2k

f

x

x x

time time

Fast run-up Slow run-up

f

x

T

1 2

1

1/ 1/ k

k k=

+


28/249


29/249


30/249

MECH7350 Rotating Machinery 4. Multi-Degree-of-Freedom Systems

4-1

4. MULTI-DEGREE-OF-FREEDOM SYSTEMS

(This section is based largely on Thompson)

4.1 Translational Systems

In the following example (Fig. 4.1) damping is assumed to be negligible.

Draw free-body diagrams for each mass displaced in the positive direction.

Apply Second Law to each mass.

Look for modes of vibration where each mass vibrates harmonically at the same frequency

and passes through equilibrium at the same time (in-phase or 180 degrees out-of-phase).

Put;

1 1 1

2 2 2

sin

sin

i t

i t

x A t or A e

x A t or A e

=

=

Then the two equations of motion become;

( )

( )

2

1 2

2

1 2

2 0

2 2 0

k m A kA

kA k m A

=

+ =(4.1)

Or, in matrix notation;

21

22

02

02 2

Ak m k

Ak k m

=

For non-trivial solutions (i.e. not 1 2 0A A= = ),

( )

( )

1 1 2 1 1

1 2 2 22

kx k x x m x

k x x kx mx

=

+ =

Fig. 4.1 (from Thompson).

m 2m

1kx ( )1 2k x x 2kx


31/249


4-2

2

2

20

2 2

k m k

k k m

=

Put 2 = and expand;

2

2

3 302

k k

m m

+ =

This is called the characteristic equation of the system. The two roots, 1 and 2 , are the

eigenvalues of the system.

Here1

2

0.634

2.366

k

m

k

m

=

=

and the natural frequencies are;

1 1

2 2

0.634

2.366

k

m

k

m

= =

= =

From (4.1) the ratio of the amplitudes of vibration can be found;

1

2

2

0.7312

A k

A k m= =

for

1 =

= -2.73 for 2 =

These are plotted in Fig. 4.2.

These are the natural modes of vibration.

At 1 , masses vibrate naturally in phase.

At 2 , masses vibrate naturally out of phase (or in opposition).

The actual amplitudes of vibration at natural frequencies depend on the magnitude, a, of

initial conditions.

1x

a

= 2

Initialdisplacement

= 1

a 0.731a

1x 2x

2x

2.73a

Fig. 4.2


32/249


33/249


34/249


4-5

Table 4.1.

Here:2

4

EI

= = density = frequency, rad/s


35/249


36/249


37/249

MECH7350 Rotating Machinery 5. Balancing

5-3

angle between reference marks on disc and support.

Step 3: Add a known trial masst

m at known radiust

r and known angle to reference

mark. Run disc and measure:

magnitude of vibrationu w

A+

angle between reference marks on disc and support.

Step 4: Construct a vector diagram (Fig. 5.5).

Calculate from to know where to add balancing mass, as follows.

Cosine rule gives:

( )1

2 2 22 cosw w u u w u u w

A A A A A A + +

= = +

2 2 2

1cos2

u w u w

u w

A A A

A A +

+ =

The magnitude of the original unbalance mass is

u

o t

w

Am m

A= . It is located at the same radial distance

tr as the trial mass. The procedure can

be repeated to get a finer (more accurate) balance.

5.3 Dynamic or Two-Plane Balancing

Any unbalance mass in a long rotor is equivalent to two unbalance masses in any two planes.

For equivalence of forces,

2 2 21 2m R m R m R = +

1 2m m m= +

For equivalence of moments, sum moment about axis into plane of diagram through O.

2 2

1 133

lm R m Rl m m = =

Fig. 5.5 (from Rao).

is equivalent to

O


38/249


5-4

Hence,

1 2

2

3 3

m mm and m= =

This argument can be extended to show that a distribution of unbalance masses (distributed

along rotor and at different angular positions) is equivalent to unbalance masses in any twoplanes and at generally different angular positions.

Now consider use of transducers on each bearing A and B, a vibration analyzer and a strobe

light. While stationary, put reference marks on each end of rotor and on stator.

Step 1: At operating speed, measure amplitude and phase due to original unbalance at

each bearing. Measurement of amplitude and phase at bearing A is in some way due to

equivalent unbalance at both left and right plane.

A AL L AR R

All unknown

V A U A U = +

(5.1)

Similarly

B BL L BR R

All unknown

V A U A U = + (5.2)

Step 2: Add a known trial weightL

W

in left plane at a known angular position and

measure displacement and phase at the two bearings.

( )A AL L L AR RV A U W A U = + +

(5.3)

( )B BL L L BR RV A U W A U = + +

(5.4)

(5.3) (5.1) A AAL

L

V VA

W

=

(5.5)

(5.4) 5.2) B BBL

L

V VA

W

=

(5.6)

Step 3: RemoveL

W

and add knownR

W

.

Measure

( )A AR R R AL LV A U W A U = + +

(5.7)


39/249


5-5

( )B BR R R BL LV A U W A U = + +

(5.8)

(5.7) (5.1) A AAR

R

V VA

W

=

(5.9)

(5.8) (5.2) B BBR

R

V VA

W

=

(5.10)

Having calculatedAL

A

,BL

A

,AR

A

andBR

A

we can substitute into (5.1) and (5.2) to find the

original unbalances;

BR A AR B

L

BR AL AR BL

A V A VU

A A A A

=

(5.11)

BL A AL B

R

BL AR AL BR

A V A VU

A A A A

=

(5.12)

The rotor can be balanced by adding equal and opposite balancing weights in each plane.

Useful Vector Algebra:

If;

AA a =

andB

B b =

1 2a ia= + 1 2b ib= + 1i =

where;

1 cos Aa a = 1 cos Bb b =

2 sin Aa a = 2 sin Bb b =

Then;

( ) ( )1 1 2 2A B a b i a b = +

( ) ( )

( )1 1 2 2 2 1 1 2

2 2

1 2

a b a b i a b a bA

B b b

+ + =

+

( ) ( )1 1 2 2 2 1 1 2A B a b a b i a b a b= + +i

5.4 Balancing of Flexible Rotors

If the flexibility of a rotor is significant, 2-plane balancing can be ineffective and balancing

must be applied to three or more planes. This is a specialised procedure. It is visited in

Section 8 on generators and addressed in detail in Harris and Piersol (2002).


40/249

MECH7350 Rotating Machinery 6. Steam Turbines

6-1

6. STEAM TURBINES

6.1 Turbine Types

Large steam turbines are all of the axial-flow type

(Fig. 6.1). They may use single flow, double flow or

reversed flow (Fig. 6.2, where blades are not shown).

Double flow avoids excessively long blades and can

reduce axial thrust. Steam enters and leaves cylinder

radially, so design must leave space for flow to turn to

axial direction with minimum losses.

The limit of a single-cylinder turbine is about 100

MW. Multi-cylinder designs are used in large plant,

e.g. one high pressure (HP) turbine, one intermediate

pressure (IP) turbine and two low pressure (LP)

turbines (Figs 6.3 and 6.4 show various multi-cylinder

turbine arrangements). The IP and LP turbines are

usually double flow.

Cross compound machines avoid long shafts and can

enable fewer LP turbines if LP turbine shafts are run atdifferent speeds. Mainly used with 60Hz grid

frequency.

6.2 Speed of Rotation

Speed of shaft rotation isf = pn

f= grid frequency (Hz)

p = number of generator pole pairs

n = rotational speed (Hz)

Machine type Rotational speed (rpm)

Two-pole (full-speed) 3000 (50Hz)

Four-pole (half-speed) 1500 (25Hz)

Turbines to drive boiler feed pumps operate at variable speeds, as high as 8500 rpm, to

accommodate the operational range of the driven machine.

Fig.6.1 Axial-flow turbine (from MPSP).

Fig.6.2 Direction of flow in turbines(from MPSP).


41/249


6-2

Fig. 6.3 Multi-cylinder turbine arrangements (from MPSP).


42/249


43/249


6-4

6.3 Turbine Stages

An impulse stage consists of stationary blades forming nozzles through which the steam

expands, increasing velocity as a result of decreasing pressure. The steam then strikes the

rotating blades and performs work on them, which in turn decreases the velocity (kinetic

energy) of the steam. The stream then passes through another set of stationary blades which

turn it back to the original direction and increases the velocity again though nozzle action.

Ideal reaction stages would consist of rotating nozzles with stationary blades (buckets) to

redirect the flow for the next set of rotating nozzles. The expansion in the rotating blades

causes a pressure force (reaction) on them that drives them. However, it is impractical to

admit steam to rotating nozzles. The expansion of steam in the stationary nozzles of a

practical reaction turbine is an impulse action. Therefore, the reaction stage in actual turbine

actions is a combination if impulse and reaction principles.

A reaction stage has a higher blade aerodynamic efficiency than an impulse stage, but tip

leakage losses are higher because of the pressure drop across the rotating stage. This is

significant for short blades (HP) but becomes insignificant for long blades (LP).

Modern turbines are neither purely impulse nor purely

reaction. They are a combination of both, with a

highly twisted profile so that the inlet and outlet angles

conform to the three-dimensional flow characteristics

at all blade heights, e.g. Fig. 6.5.

Blade efficiencies are not ideal. Profile loss is due to formation of a boundary layer on the

blade surface. Secondary loss is due to friction on the casing wall and on the blade root. It is

a boundary layer effect. Tip leakage is due to steam passing through the necessary small

clearance between the moving blade tip and the casing, or between the end of the fixed blades

and the rotating shaft.

A shroud band extends around the entire circumference of the moving blades, joining the tips.

The shroud is sealed against the casing by several knife edges (Fig. 6.6).

Fig. 6.5 LP last stage moving blade(from MPSP).


44/249


45/249


46/249


6-7

Low Pressure Stages

Blades of up to one metre long can be used. A coverband or lacing wire must behave as a

beam spanning the blade pitch in resisting centrifugal loading, and must accommodate the

substantial circumferential strains due to elastic extension of the blades and the tendency of

the blades to untwist at speed. When lacing wires are used, they are usually of the loose

type with circumferential restraint on only one blade in each group, and are free to move

circumferentially in adjacent blades, centrifugal forces providing the necessary damping

through friction.

A coverband of conventional design is not feasible for slim sections and where the peripheral

speed might be approaching Mach 2, but a continuous ring of stiffening devices of sufficient

elasticity may be used to accommodate the circumferential strains. The elastic arch banding

shown in Fig. 6.10 braces the blade tips and provides some resistance to blade untwist as well

as permitting circumferential strain.

Zigzag spool rods are sometimes used at the tips of last-stage LP blades (Fig. 6.11). They

provide no restraint against circumferential expansion or centrifugal untwist, but the reduced

sections at the ends of the rods are forced against the holes in the blades by centrifugal action

Fig. 6.10 Arch coverbands (from MPSP).


47/249


6-8

and the sliding friction provides effective damping, minimising blade vibration and high

frequency flutter at the blade tip.

Fig. 6.11 Zigzag spool rod tip-ties (from MPSP).


48/249


6-9

6.4.4 Moving Blade Root Attachments

Last stage blades develop centrifugal forces of hundreds of

tonnes when running. Strong methods of attachment are

needed. Fir-tree roots are widely used (Fig. 6.12). There is

some looseness in fir-tree roots for assembly but this

becomes rigid when the blades rotate. However, it is not

possible to measure the zero-speed vibration characteristics

of the blades. Pinned roots overcome this but blade

replacement is not easy.

6.4.5 Blade Materials

Blade material must have some or all of the following properties, depending on the position

and role.

corrosion resistance (especially in the wet LP stage)

tensile strength (to resist centrifugal and bending stresses)

ductility (to accommodate stress peaks and stress concentrations)

impact strength (to resist water slugs)

material damping (to reduce vibration stresses)

creep resistance

12% Cr stainless steels are a widely used material. Their weakness is at very high

temperatures (> 480C). A typical high temperature steel is 12% Cr alloyed with

molybdenum and vanadium (to 650C).

Titanium has some attractions but it is expensive and material damping is low. It has poor

vibration characteristics. Because of its high strength/weight ratio, titanium is used in lacing

wire and for coverbands and shrouding.

6.4.6 Blade Vibration Control

Blade vibration characteristics under operating conditions are very complex and difficult to

predict by calculation such as finite element analysis because:

an individual blade has a very complex geometry

Fig. 6.12 Types of fir-tree roots (fromMPSP).


49/249


6-10

there are vibration interactions among the blades through the blade disc,

diaphragms, coverbands and lacing wires.

The vibration of a fully-bladed disc is much more complicated than is suggested by the

characteristics of a single cantilever blade. There is a multiplicity of modes of vibration in

the turbine working frequency range. For a single blade, there are only two or three.

Sources of vibration excitation are:

Non-uniform flow caused by:

o steam entering over only a portion of the circumference

o complex axial to radial flow behaviour (which is minimised with good design)

o flow distortion caused by steam extraction passages for feedheater tappings

Periodic effects due to manufacturing constraints, e.g.

o Inexact matching at fir-tree roots

o Eccentricity of diaphragms

o Ellipticity of stationary parts

o Non-uniformity of manufacturing thicknesses

o Moisture removal slots

All of the above sources cause excitation at the

rotation frequency or low multiples (harmonics) of

that frequency. Recall the Fourier content of a

non-sinusoidal periodic wave.

Nozzle wake excitation as a rotating blade passes a stationary blade.

Excitation frequency = rotational frequency number of stationary blades and its

multiples.

Some sources can cause excitation at frequencies that are unrelated to rotational frequency,

acoustic resonances in inlet passages, extraction lines and other cavities

vortex shedding from bluff bodies

time

Contains fundamental and harmonics

flow

Vortex street


50/249


51/249


6-12

For the larger LP blades, natural frequencies are lower and may coincide with harmonics of

the rotation frequency below the 8th order. Testing must be at running speed because

centrifugal effects can change the stiffness of long blades. Testing is conducted at speed in a

vacuum wheel chamber. The presence of air would necessitate a huge amount of power to a

large electric motor to drive the disc. Windage near blade tips would cause overheating and

make results difficult to interpret.

The disc is run up to 115% of synchronous speed and blade vibration is detected with strain

gauges of piezoelectric crystals. A Campbell Diagram can be developed (Fig. 6.13).

Where lines cross there is a prospect of resonance in service. It is usual to confine attention

to 6% of synchronous speed (2820 to 3180 rpm). A common specification is that in this

range there be no resonances up to 8th order.

Problem modes can be tuned out by adjusting the blade mass near the tip, or by adding or

removing mass to or from the shrouding.

Fig. 6.13 Campbell diagram (from MPSP).


52/249


53/249


54/249


6-15

Fig. 6.15 Cross-section of HP cylinder (from MPSP).

Fig. 6.16 Axial section of IP turbine casing (from MPSP).


55/249


6-16

Fig. 6.17 HP and IP casings (from MPSP).


56/249


57/249


6-18

Ensure axial location or allow relative axial movement

Provide torsional resilience

Flexible or semi-flexible couplings can provide this but they are impracticable on large

turbines because of the high torque to be transmitted.

Rigid couplings are used in large turbines so that the joined shafts can behave as one

continuous rotor. They are either integral with the shaft forging (Fig 6.19) or shrunk on to

the shaft (Fig. 6.20). In the latter case, high pressure oil can be injected into annular grooves

to ensure correct seating during assembly, or to aid removal.

Couplings are designed to withstand a three-phase fault or out-of-phase synchronising

without damage (4-5 times full load torque).

6.19 Rigid forged coupling (from MPSP).


58/249


6-19

6.6.1 Rotor Alignment

Excessive misalignment of a multi-bearing shaft line can affect the vibration behaviour.

It causes bending moments at couplings which act like a rotating out-of-balance.

It can cause bearing unloading which alters shaft vibration behaviour.

A long shaft bends naturally under its own weight to form a catenary (Fig. 6.21) and revolve

around a curved centreline. The shape of the catenary depends on the masses and stiffnesses

of the rotors.

The aim of alignment is to ensure insignificant bending moments and shear at the couplings.

Bearing heights are adjusted so that coupling faces are square to each other, with centrelines

coincident and with the same slope where the faces meet. This is done by slightly separating

Fig. 6.20 Shrunk-on coupling (from MPSP).


59/249


6-20

the coupling and turning the rotor to different positions. Bearings are adjusted to get uniform

gap and concentricity (measured with a dial gauge).

Bending moment cyclic variations can be measured with strain gauges and optical

techniques. Lasers are used to set the catenary up initially, prior to adjusting it.

Outer bearings may be 25 mm above the level of central bearings. Changes in service to

pedestal bearings are monitored on-line with a manometric system.

6.7 Journal Bearings

Bearings on the shaft line of a large turbine/generator set are invariably white-metalled

journal bearings because of their:

high load capacity

reliability

absence of wear through use of hydrodynamically generated films of lubricating

oil (no metal-to-metal contact)

Turbine bearings have diameters up to 550 mm, with length/diameter (L/D) ratios of 0.5 to

0.7. Generator bearings have L/D ratios of 0.6 to 1.0 because of the weight of the generator

rotor. They are split in halves for assembly of the rotor, with bolts and local dowels (Fig.

Fig. 6.21 Shaft catenary for a large turbine-generator (from MPSP).


60/249


61/249


6-22

White metal (or babbit) is usually composed of 80 to 90 % tin to which is added about 3 to

8% copper and 4 to 14% antimony. These alloys have very little tendency to cause wear to

their steel journals because of their ability to embed dirt. They are easily bonded, cast and

shaped, and can have good load-carrying and fatigue properties.

The bores of journal bearings are usually elliptical to provide the geometry for hydrodynamic

lubrication. A circular bore is machined with shims in the horizontal split. The shims are

removed in assembly to give typical diametrical clearance/diameter ratios of 0.001 vertically

and 0.00015 horizontally. Oil is fed into the bearing via lead-in ports at two diametrically

opposite points on the horizontal centreline. This is to cool and lubricate the bearings and

comes from the main turbine lubricating-oil pump.

Each bearing also has a high pressure jacking oil supply at the bottom. This lifts the shaft

when starting from rest, until speed is high enough for hydrodynamic lubrication to start-up.

Instrumentation at each bearing normally gives:

white metal temperature

lubricating oil outlet temperature and inlet pressure

jacking oil pressure

vertical and horizontal vibration

6.7.1 Hydrodynamic Lubrication

(This section is taken from Williams)

Hydrodynamic bearings depend on the presence of a converging, wedge-shaped gap into

which a viscous fluid is dragged by the relative motion of two surfaces. A pressure is

generated which tends to push the surfaces apart. This balances the load on the bearing.

Large rotating machinery utilises hydrodynamic journal and thrust bearings. An analytical

solution of their behaviour is complicated but the elements of behaviour can be understood by

studying the simple two-dimensional pad bearing in Fig. 6.23.


62/249


6-23

The bearing is long in the y-direction, so that there is no fluid flow normal to the plane of the

paper. The upper, inclined fixed member is of lengthB, while the lower flat slider moves

from left to right with velocity U. Fig. 6.23 also shows the pressure distribution in the

viscous fluid. The integral of this pressure distribution supports W/L, the load per unit length

into the page.

The angle of the wedge is greatly exaggerated in Fig. 6.23. It is typically only a quarter of a

degree.

Consider the equilibrium of the small element of fluid within the gap in Fig. 6.23. The local

film thickness is h and it varies in a known way from ih at the entry to oh at the exit. Gravity

and inertia (acceleration) forces can be neglected.

Then

px z z x

x z

=

where

Fig. 6.23 Two-dimensional bearing pad (from Williams).


63/249


64/249


6-25

geometry of a journal bearing and the exaggerated wedge of fluid. Analysis of this

geometry is complicated by the finite length of a journal bearing and the flow out of its ends.

Fig. 6.25 also shows the configuration of

an ideal steady state hydrodynamic film.

If a vibratory load (e.g. due to rotating

unbalance) is superimposed on the

steady load (weight) the oil thickness

can change and move around the

circumference. This can lead to whirling

of the journal (shaft) and affect the

vibratory behaviour of the whole rotor

line.

Fig. 6.24 Geometry of journal bearing (from Williams).

Fig. 6.25 Possible oil filconfigurations (from MPSP).


65/249


6-26

6.7.2 Hydrodynamic Thrust Bearings

By pivoting the fixed part of the bearing in Fig. 6.23, the angle of tilt will vary with load so

that it is at the optimum for load-carrying capacity. This was the discovery of the Australian,

A.G.M. Michell, and led to his famous thrust bearing design which is used in

turbine/generator shaft lines. This is illustrated in Fig. 6.26.

A turbine thrust bearing is used to provide axial location for the turbine rotors relative to the

cylinders. Because solid couplings are used, only one thrust bearing is used in the shaft line.

It is usually located near areas where blade/cylinder clearances are a minimum. It is in two

halves for ease of assembly.

Thrust bearings are of the Michell tilting pad design (Fig 6.27).

Fig. 2.26 Thrust bearing configuration (from Stachowiak and Batchelor).


66/249


6-27

Fig. 6.27 Mitchell thrust bearing (from MPSP).


67/249


68/249


6-29

Alloy steels are chosen to have good creep resistance and high temperature and high fracture

toughness.

6.10.1 Overspeed Testing

A 20% proof overspeed test is specified on all large turbine/generator rotors at the time of

manufacture. This tests the forging against spontaneous fast fracture and confirms its

balance.

6.10.2 Rotor Balancing

With the blade discs assembled, the rotor is balanced both statically and dynamically. Each

blade disc is balanced individually before assembly. Rotors are dynamically balanced at low

speed (400 rpm) with weight adjustments made in two planes, one at each end of the rotor.

Provision is made to vary screwed plugs in tapped holes, or to add weights. The aim is to get

25 m< amplitude of vibration at the bearing pedestals.

Modal behaviour must be understood for long rotors. These rotors are often balanced at

running speeds and critical speeds in a vacuum chamber. When rotor flexibility is important,

balancing is done at three or more planes.

On-site vibration testing can be done but it is affected by variations in the stiffness of the

bearings, possible shaft misalignment and the coupling of the individually balanced rotors to

form the complete shaft system. Access holes are provided in the casing.

6.10.3 Critical Speeds

A stationary shaft and rotor between bearings has a

natural frequency of vibration with the shaft in

bending. If the speed of rotation coincides with this

natural frequency, any small unbalance can cause

dangerous vibrations. This is a critical speed.

If the critical speed is below the running speed, the shaft is regarded as flexible. Care is

needed to run up through this critical speed quickly. Modern units have rigid shafts, with

Rotor

Shaft Bearing


69/249


6-30

critical speed above operating speed. With the long shafts in large units, large shaft

diameters are needed.

However, each turbine does not act independently of others. There might be up to seven

individual rotors in a shaft line. The bearings are hydrodynamic and so have flexibility which

might increase with wear. As a result, there are then a number of critical speeds, two or three

of which can be below the operating speed. A typical speed-vibration curve from an

instrumented bearing housing is shown in Fig. 6.28.

Fig. 6.28 Typical speed-vibration curve at a pedestal (from MPSP).


70/249

MECH7350 Rotating Machinery 7. Gas Turbines

7-1

7. GAS TURBINES

(This section is based largely on Black and Veatch)

Gas turbines are only briefly covered in these notes because (a) many of their mechanical

fundamentals are similar to those for steam turbines, and (b) they are the topic of a guestlecturer.

Gas turbine technology is used in a

variety of configurations for electric

power generation. Conventional

applications in power stations are

simple cycle and combined cycle.

Simple cycle operation is used

primarily for peaking power

generation. Smaller units (about 15

MW) are used for black starts.

Fig 7.1 is a schematic of simple

cycle operation.

Combined cycles combine the gas turbine and steam turbine cycles into more efficient power

plants by utilising the gas turbine exhaust gas heat. This is shown schematically in Fig. 7.2.

Fig. 7.1 Simple gas turbine cycle (from Black and Veatch).

Fig. 7.2 Combined cycle gas turbine (from Black and Veatch).


71/249


7-2

Gas turbine applications generally rely on natural gas, with its environmental benefits, or fuel

oil for fuel. However, future power plants can be expected to use the integrated gasification

combined cycle (IGCC) in which coal is partially combusted in oxygen to produce syngas,

which in turn is burned in a combined cycle process to produce electricity.

7.1 Gas Turbine Components

The main components of a gas turbine (Fig. 7.3) are:

Inlet air system

Compressor

Combustion systems

Turbine

Exhaust system

generator

When the gas turbine is started, ambient air is drawn through the inlet air system, where it is

filtered and then directed to the inlet of the compressor where it is compressed and directed to

the combustion system. Inside the combustion system the air is mixed with fuel and the

mixture is ignited by a spark plug ignition system. The compressed and heated combustion

gases then flow to the turbine, expanding and causing it to rotate. The rotating turbine drives

Fig. 7.3 Major sections of a gas turbine assembly (from Black and Veatch).


72/249


7-3

the compressor and accessory equipment, such as the main lube oil pump. The number of

stages within the compressor and turbine may vary.

After leaving the last stage of the turbine, the exhaust gases are either released to the

atmosphere or directed through an exhaust system to heat recovery equipment.

7.2 Multi- and Single-Shaft Plants

In a multi-shaft combined-cycle plant, there are generally several gas turbines with heat

recovery generators producing steam for a single steam turbine. The steam and gas turbines

use different shafts and generators. With the largest gas turbines on the market, one steam

turbine per gas turbine or one steam turbine for two gas turbines is common.

If one steam turbine per gas turbine is installed, the single-shaft application is most common

gas turbine and steam turbine driving the same generator. A plant with two gas turbines

can be built either in a multi-shaft or a single-shaft configuration.

7.3 Sequential Combustion

In a gas turbine with sequential combustion, air enters the first combustion chamber after the

compressor. There, fuel is burned, raising the gas temperature to the inlet temperature for the

first turbine. The hot gas expands through this turbine stage, generating power before

entering the second combustion chamber where additional fuel is burned to reach the inlet

temperature for the second part of the turbine. There, the hot gas is expanded to atmospheric

pressure. Sequential combustion increases efficiency.

7.4 Materials

Gas and steam turbines experience similar problems. However, the magnitude of these

problems is different, leading to more demands on the materials in gas turbines. In modern,

high performance, long-life gas turbines, the critical components are the combustor liner and

the turbine blades.

Creep and corrosion are the primary failure mechanisms in a gas turbine blade, followed by

thermal fatigue. The first-stage blades must withstand the most severe conditions of

temperature, stress and environment. Temperatures can be as high as 1500C. Nickel-base

alloys are widely used with coatings to protect against hot corrosion.


73/249


7-4

7.5 An Example

An example of a large gas turbine used in power generation is the Alstom Model GT26

shown in Fig. 7.4. It has the following features:

Sequential combustion

Turbine speed of 3,000 rpm (50 Hz)

Gross electrical output of 288 MW

Natural gas primary fuel, with fuel oil as a backup

Blade cooling with air extracted from the compressor

Hydrodynamic journal and tilting-pad thrust bearings

Turbine outer casing and compressor casing split horizontally

Fig. 7.4 Alston GT26 gas turbine.


74/249

MECH7350 Rotating Machinery 8. Generators

8-1

8. GENERATORS

Fig. 8.1 is a reminder of the components of a turbine/generator system. Here it includes the

exciter which provides DC current to the rotor of the generator.

8.1 Synchronous Generator Theory

Synchronous means that the generators rotor runs at the constant mains frequency as load

varies.

8.1.1 Electromagnetic Induction

The instantaneous voltage induced in a stator conductor is given by;

dBe kl

dt=

where;

k= constant

l = length of conductor

dB

dt= rate of change of magnetic flux density

Ife is to be a sinusoid, dB/dtmust also be sinusoidal. Excitation coils are wound on the rotor

to produce a flux with a density which varies approximately sinusoidally around the

circumference. A two-pole arrangement (one pole pair) is shown in Fig. 8.2. The magnitude

Fig. 8.1 Components of a turbine-generator system (from Black and Veatch).


75/249


8-2

of the flux density can be changed by varying the direct current to the excitation coils on the

rotor.

Speed, Frequency and Pole Pairs

If f= frequency, Hz

n = rotational speed, r/s

p = number of pole pairs

Then f = pn

In Australia, generators are 2-pole (i.e.p = 1). Hence generators run at 3000 rpm.

Three-phase Windings

Fig. 8.3 shows how three phases are generated by having three winding spaces on the stator.

This is still a 2-pole generator. It is economical to have many stator conductors in parallel so

the individual conductor voltages are additive. Each go conductor is connected to a return

conductor, acted on by the pole of opposite polarity, and thence to a third conductor adjacent

to the first, and so on through the phase. The return conductors are disposed in a layer

displaced radially from the go conductor, both in the slots and in the end region. The

discrete nature of the windings gives rise to generation of harmonics.

Fig. 8.2 Production of sinusoidalvoltage (from MPSP).


76/249


8-3

Torque

The mechanical torque provided by the turbine is balanced by an electromagnetic torque

caused by the interaction of the magnetic flux and the current flowing in the stator windings.

8.2 The Rotor

(This section is mainly from MPSP)

The rotor must:

carry the excitation windings

provide a low reluctance path for the magnetic flux

transfer the rated torque from the turbine to the electromagnetic reaction at the air

gap

resist large centrifugal forces.

Steel is the most suitable material. A steel

forging is used with machined slots (Fig. 8.4).

Fig. 8.3 Arrangement of statorconductors (from MPSP).

Fig. 8.4 Rotor section (from MPSP)


77/249


78/249


8-5

Thick end rings are used to restrain the rotor end winding under the action of centrifugal

force. The ends of the windings are connected to flexible leads and there are radial copper

studs to connect to the sliprings and thence to the exciter. The contents of the winding slots

are retrained by an aluminium wedge (Figs 8.4 and 8.8).

Fig. 8.6 Rotor winding (from MPSP).

Fig. 8.7 Axial flow fans on rotor (from MPSP).


79/249


8-6

8.2.2 Sliprings, Brushgear and Shaft Earthing

For a 660 MW generator the excitation current is about 5000 A. This must flow through

sliprings with a large area (Figs 8.9 and 8.10). Brushes only last about six months but they

can be changed while running on-load.

In normal operation there is 10-50 volts between the two shaft ends of a generator, due

mainly to magnetic dissymmetry. To stop an axial current from flowing and damaging

bearings, an insulation barrier is provided at the exciter end.

Fig. 8.8 Rotor slot (from MPSP).


80/249


81/249


8-8

It is important that the shaft at the turbine end of the generator is maintained at earth

potential, using a pair of shaft-riding brushes connected to earth through a resistor. The

voltage at the exciter end is monitored as an indication that insulation is intact (Fig. 8.11).

8.3 The Stator

(This section is mainly from Klempner and Kerszenbaum)

The stator core with windings are assembled into a skeletal core frame which is inserted into

a strong outer casing.

The stator core provides paths for the magnetic flux from one rotor pole around the outside of

the stator winding and back into the other pole. It is made up from tens of thousands of

electrical grade steel laminates, each about 0.4 mm thick. This prevents large circulating

eddy currents with their associated losses. Each lamination is insulated on both sides with a

very thin layer of an organic or inorganic compound. Winding slots, location notches and

holes for ventilation are cut in one pressing operation. The laminate segments are fitted onto

key bars in a stator frame structure and clamped axially (Fig. 8.12).

The core is cooled with hydrogen which passes axially through ducts cut radially in each

laminate.

Fig. 8.11 Shaft earthing and monitoring (from MPSP).


82/249


8-9

The stator outer casing provides support for the stator core, which can be 500 tonnes, and acts

as a pressure vessel in case of an explosion of the hydrogen cooling gas. Casings are

fabricated steel cylinders of up to 25 mm thickness and reinforced externally (Fig. 8.13).

Fig. 8.12 Core frame (from MPSP).

Fig. 8.13 Outer stator casing (from MPSP).


83/249


84/249


8-11

Stator cooling is achieved indirectly if the strands within the conductor bars are all solid and

the heat generated (I2R) is removed by conduction to the stator core.

In directly gas-cooled bars, hydrogen passes from end to end in cooling ducts.

In direct water-cooled bars, the copper strands are made hollow to carry demineralised water.

These cooling designs are shown in Fig. 8.16.

Hydrogen is used (rather than air) because:

its low density minimises fan and rotor windage losses

its heat transfer coefficient is 50% more effective than that of air at the same

pressure.

Hydrogen is contained between the casing and the rotor shaft with journal type seals which

use a flow of seal oil which is supplied at a pressure slightly higher than the hydrogen

pressure.

Fig. 8.15 Bracing of end-windings (from Klempner and Kerszenbaum).


85/249


8-12

8.3.2 Generator Auxiliary Systems

There are five main auxiliary systems:

Lubricating oil system

Hydrogen cooling system

Seal oil system

Stator cooling water system

Excitation system

8.3.3 Exciter Systems

The exciter system provides the DC current to the generator rotor windings. The system is

designed to control the applied voltage, and thus the field current to the rotor, which in turn

gives control of the generator output. DC currents can be as high as 8000 amps. There are

three types of excitation system;

Rotating

Static

Fig. 8.16 Stator conductor bar cross sections. (a) Indirectly cooled statorconductor bar; (b) Directly gas-cooled stator conductor bar; (c) directly water-cooled stator conductor bar (from Klempner and Kerszenbaum).


86/249


87/249


8-14

Bearing and shaft vibration on both ends of the generator may be monitored in terms of

magnitude, phase and frequency at variable load conditions. Accelerometers and proximity

probes are used in two sets, set 90 degrees apart. Sophisticated vibration analysers are

available. Each manufacturer gives its own recommendation for alarm and trip. A typical

maximum allowable amplitude is 1 mm at 3000 rpm.

Generator rotors are relatively flexible and pass through two main critical speeds during run-

up to rated speed of 3000 rpm. Two-plane balancing is inadequate. Facilities for balancing

are provided along the length of the rotor in the form of tapered holes in the cylindrical

surface.

Imperfect equalisation of the stiffness of the rotor in two orthogonal directions (associated

with the creation of poles) will cause 100 Hz vibration to occur, superimposed on the normal

50Hz. It is important to distinguish between these two components. A significant crack in

the rotor will have a comparatively greater effect on the double frequency vibration

component; run-down traces are recorded and analysed to provide assurance that no

significant change has occurred since the previous run-down.

Oil whirl in bearings can cause vibration at 25 Hz.

The torsional resonance of the generator rotor coupled to the turbine rotors is at about 13 Hz.

It is important that this is significantly different from the frequency of torsional exciting

influences such as the steam governor control (1-2 Hz).

Transient oscillations in torque occur during electrical disturbances (switching operations,

lightning strikes, imperfect synchronising events). Some of the torque cycles may be large

enough to cause plastic deformation of the turbine-end shaft and the generator/excitor

coupling (if there is one).

8.4.2 Stator End-Winding Vibration

Vibration of the stator end-windings must be minimised. It can cause fatigue cracking in the

winding copper. This is a serious problem if cooling hydrogen can leak into the cooling

water circuit. Resonances close to 100 Hz must be avoided, since both the core ovalising and

the winding exciting force occur at this frequency. Accelerometers in the end-winding

structure allow any increase in vibration due to support slackening to be monitored.


88/249


8-15

Looseness can be corrected by tightening bolts or by inserting tightening wedges. If

permanent mounting of accelerometers is not possible, a bump test, or impact frequency

spectrum analysis is done with temporary vibration transducers and a calibrated impact

hammer. Vibration amplitude is highly dependent on current.

The general aim is to keep the maximum amplitude of vibration of stator end-windings to less

than 50 m peak-to-peak, with no natural frequencies within the ranges 40-65 Hz and 80-120

Hz.

8.4.3 Stator Core and Frame

The plus/minus magnetic field revolves. This alternating effect causes vibration of the core

at the rotational frequency and with harmonics, due to the nature of the flux patterns. In a

two-pole generator, the driving frequency is 50 Hz and there is a 100 Hz (twice per

revolution) component due to the four-node pattern of the flux. This can be seen in Fig. 8.17

where there are two areas of high flux density and two of minimum density at any given point

in time, as the flux patterns rotate at the rated speed. This causes the core to be distorted

minutely by the electromagnetic pull into an oval shape, in and out, in the radial direction.

The result is vibration of the core and subsequently the frame.

Fig. 8.17 Rotor flux pattern (from Kempner and Kerszenbaum).


89/249


8-16

Because of the inherent vibration and the large forces involved, the core must be held solidly

together such that there are no natural frequencies near the once and twice per revolution

forcing frequencies. Designers take care to ensure that the natural frequencies of the core are

not near 50 or 100 Hz. It is desirable to keep the natural frequencies at least 20% away

from the once and twice per revolution frequencies. Damped spring mounting of the whole

generator on its foundation might be needed, or spring mounting of the core in the casing.

In addition to the vibration due to the alternating flux, there is a large rotational torque

created by the electromagnetic coupling of the rotor and stator, across the airgap. This is in

the direction of rotor rotation. The torque due to the magnetic field in the stator iron is

transmitted to the core frame via the keybar structure at the core back. Therefore the stator

frame and foundation must be capable of withstanding this torque and large changes in torque

when there are transient upsets in the system or the machine.

Vibration in the stator core is naturally produced by the unbalanced magnetic pull in the

airgap, origination from the unequal magnetic field distribution of the rotor. The core must

be maintained tight or fretting will occur between the laminates. If the core becomes too

loose, the laminates and/or the space blocks might even fatigue, with loose core material

breaking off. Monitoring of core vibrations can be done with accelerometers mounted on the

core back in strategic locations.

Frame vibration is also excited by unbalanced magnetic pull and by any vibration produced in

the core. There are known results of vibration resonance occurring on the frame as a result of

the frame having a resonant frequency near line or twice line frequency. Severe damage to

the frame can occur by the initiation of cracks in the frame welds. Good core-to-frame

coupling is required to ensure that the core and the frame move together. Such vibrations

have been corrected by spring mounting of the core to the frame or installing a damping

arrangement to de-tune the vibration modes. Monitoring of frame vibration can be done with

accelerometers mounted on the keybars, frame ribs or casing structure.


90/249

MECH7350 Rotating Machinery 9. Pumps

9-1

9. PUMPS

9.1 Fundamentals of Fluid Mechanics

9.1.1 Bernoulli Equation

(This section is mainly from White)

We consider only two-dimensional, incompressible, frictionless (inviscid) flow.

Consider an elemental fixed streamtube control volume of variable area A(s) and length ds

(Fig. 9.1), where:

s = streamline direction

= fluid density (constant)

p = pressure

v = streamtube velocity

A(s) = streamtube cross-sectional area at s.

Conservation of mass gives:

0out inm m = because there can be no accumulation of mass in the control volume if

density is constant. Hence at any s, m Av= .

Now consider Newtons Second Law applied to fluid in the control volume. Sum elemental

forces in the streamwise direction.

( ) ( ) ( )s inoutF mv mv d mv= = (9.1)

Neglect shear forces on the walls (inviscid flow) so the forces are due to pressure and gravity.

, sins gravdF dW = (z positive up)

Fig. 9.1 (from White).


91/249


9-2

= sing Ads

= gAdz

= ( )Adz g =

To get pressure force, imagine pressurep subtracted from all faces of the control volume.

Then:

( ) ( ),1

02

s pressdF dp A dA A A dpdA= + + +

= Adp to first order.

Substitute into (9.1).

( )gAdz Ad d mv Avdv = =

Divide by A.

0dp vdv gdz

+ + =

This is Bernoullis equation for steady, frictionless flow along a streamline. Beware of its

limitations.

We can integrate between any two points 1 and 2 to get:

2 21 21 1 2 2

1 1

2 2

p pv gz v gz

+ + = + + (9.2)

or

2 21 1 2 21 2

2 2p v p vz z

g g + + = + + (9.3)

Summary of assumptions

1. Steady flow

2. Incompressible flow

3. Frictionless flow

4. Flow along a single streamline: different streamlines may have different

Bernoulli constants2

02

p vh z

g= + +

5. No shaft work between 1 and 2: no pumps or turbines on the streamline

6. No heat transfer between 1 and 2.


92/249


93/249


94/249


9-5

Advantages Disadvantages

Generate high pressures Low flow rates

Can handle high viscosities Pulsating flow

Self-priming

9.3 Rotodynamic Pumps

(This section is mainly from White)

Fig. 9.4 is a schematic of a typical centrifugal rotodynamic pum

Condition Monitoring Lectures

Documents

Transcript of Condition Monitoring Lectures