cyclic symmetric modal analysis

Full model of bladed disk assembly

Harish N. Dave Lalit Kr. Mishra

2012, Quality Engineering and Software Technologies, Inc. All rights reserved. UNCONTROLLED WHEN PRINTED

2

Technical Report

on

Cyclic Symmetric Modal Analysis

Sourabh Sawant

Raghavendra S S

Biju Kumar Girish Kulkar ni

Suresh S

Bharath B N

Ansil George

Gururaj S Nayak

With valuable Contributions from

Harish N. Dave

Lalit Kr. Mishra

By


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Contents

Chapter 1 Introduction

1.1 Concepts of symmetry

1.2 General Concepts of Modal Analysis

Chapter 2 Cyclic Symmetric Modal Analysis

2.0 Introduction

2.1 Cyclic Symmetric Modal Analysis ...10

2.1.1 Eigenvalue Problem

2.1.2 Concept of Nodal Diameters

Chapter 3 Results and Discussion

3.0 Introduction

3.2 Definition of Real Resonance

3.3 Results Interpretation

3.3.1 Why do we get DOUBLETS?

3.4 Interference Diagram

3.4.1 Plotting an Interference Diagram

3.4.2 Interference Diagram beyond NB/2 Limit

3.4.3 Studying an Interference Diagram

3.4.4 Interference Diagram vs. Campbell Diagram

3.5 Some Thoughts on Fatigue Life Estimation

Conclusions

Scope of future study ...36

References


4

Chapter 1 Introduction

1.0 Introduction:

Many engineering problems have identical substructures in the circumferential

direction. It is the field of mechanical engineering where one finds a variety of such

applications. To highlight a few turbomachinery blades and discs, centrifugal fans and

impellers, rotary pumps, gear wheels, pressure vessels, milling cutters, reamers, gear

hobbers etc. These systems can be analyzed in full 3600 but at a cost that is prohibitive.

Moreover, it also becomes cumbersome to analyze the full 3600 FE model. It is good to

exploit the cyclically symmetric nature of the bladed assembly and reduce the core, time

and labor involved in analyzing these systems. Using the cyclic symmetric principle both

static and modal analysis can be done. These problems can be broadly classified under

three heads,

1. Static or steady state analysis of cyclic symmetric objects subjected to cyclic symmetric loading.

2. Static or steady state analysis of cyclic symmetric objects subjected to general loading (asymmetric loading).

3. Eigenvalue problems (modal analysis).

Any structure that is circumferentially symmetric can be analyzed with the concept of

cyclic symmetry.

According to the principle of cyclic symmetry, it is enough to model only one sector

and behavior of the other sectors can be accurately predicted from the modeled sector.

These results in enormous reduction in computer core and labor, which means reduced

cost and a economical solution to the problem. Further by utilizing this principle, the

analyst can have the privilege of using more number of elements in single sector model.

This report is an effort in making this concept of cyclic symmetry more clear, to the

reader and to explain cyclic symmetric modal analysis in particular. In first chapter, the

concept of symmetry will be discussed with its certain advantages in general. Later, some

of the concepts of modal analysis will also be discussed. In second chapter a focus on

cyclic symmetric modal analysis of a bladed disc assembly of a steam turbine is

presented. Next mathematical formulation to the Eigenvalue problem for cyclic

symmetric structures will be carried out. The concept of nodal diameters and their

physical interpretation will also be discussed at a great length in this chapter. Third

chapter discusses the means of interpreting the results obtained from cyclic symmetric

modal analysis. Results interpreted are those of a bladed disc assembly of a steam

turbine. Interference diagram is most important tool used by engineers to evaluate the risk

of resonance. This will be discussed in detail and compared with conventionally used

Campbell diagram. Finally the conclusions of this study on cyclic symmetric modal

analysis will be presented.


5

1.1 Concepts of symmetry

Symmetry is a law of nature. Any symmetric properties in an analysis should be

exploited to the full if at all possible, as it can drastically reduce solution times and hence

facilitates more effective use of resources. In general, it is important that loads and

boundary conditions are symmetric in nature to allow for symmetry to be used in the

model. There are four major ways that symmetry can be exploited in the model, which

are discussed below.

Type of Symmetries: -

1) Planar or Reflective Symmetry: -

This is the most common type of symmetry found in finite element models.

Reflective symmetry is a condition where the same pattern is mirrored in a plane. The

image shown here indicates that the model is doubly symmetric, one plane in the

horizontal direction and another in the vertical direction. As this is a 2D model the

symmetric planes are comprised of lines. Planar symmetry can occur in 3D also, the

plane of symmetry would be defined by a surface. It is important to apply appropriate

symmetric boundary conditions. In the given case, the vertical constraint is to prevent any

horizontal movement, while the horizontal constraint would be to prevent any vertical

movement.

2) Repetitive Symmetry: -

Repetitive symmetry is a symmetry condition that is seen to repeat along the

length of a body. A typical example of this would be circular cooling fins equally spaced

along the axis of heat exchanger tubing. An example of this type of symmetry is given in

figure below.

1 Full Model Model utilizing reflective symmetry

2. Repetitive Symmetry


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3) Axisymmetry or Rotational Symmetry: -

If a shape can be defined by rotating a cross-section about an axis, then it is said

to be axisymmetric. If the loads and boundary conditions are also axisymmetric in nature,

then an axisymmetric analysis may be carried out. Axisymmetric elements are 2D planar

in nature, and are used to model a revolved 3D part in 2D space. No special boundary

conditions are applied to exploit axisymmetry. An example showing axisymmetry is

given in figure below.

4) Cyclic Symmetry: -

Cyclic symmetry is the geometric repetition in the form of cyclic sectors. The

structure is composed of a series of identical sectors that are arranged circumferentially to

form a ring. In order to utilize cyclic symmetry, appropriate cyclic symmetric boundary

conditions must be applied. Cyclic symmetry is further discussed and explained in detail

in this report. An example of the symmetry is shown in figure below.

Additional Considerations

3. Full Model Model utilizing Axi-Symmetry

S

4. Full Model consisting of 75 sectors One sector modeled using cyclic symmetry principle


7

Additional Considerations:

It is important to note that taking advantages of symmetric properties may have a

detrimental effect in frequency (modal) or eigenvalue buckling problems. The reason for

this is that the symmetric model will not be able to predict non-symmetric mode shapes.

Therefore, if a symmetric analysis is carried out, only the mode shapes that also have the

same symmetric properties will be resolved. In cases where only minor details disrupt a

structure's symmetry, one can often ignore them or treat them as being symmetric, in

order to gain the benefits of using a smaller analysis model.

1.2 General Concepts of Modal Analysis

Modal analysis is a vibration analysis. Objective of any modal analysis is to

determine the natural frequencies and its associated mode shapes. Any physical system

possesses its own natural frequency. Further any physical system has 3 unique properties

i.e. stiffness, mass and damping. Damping can be expressed as a function of both stiffness

and mass. Hence the stiffness and mass, govern the behavior of the system subjected to

any loading.

Natural Frequency and mode shapes:

Frequency Ratio

n/

U

Natural frequency is the frequency at

which stiffness force becomes equal and opposite

of inertia force and the structure will vibrate in the

absence of any external loading. At natural

frequency, damping of the system governs the

amplitude of response, and since material

damping is usually very small, the amplitude of

response becomes very high. When excitation

frequency is approximately equal to natural

frequency, the energy build up is maximum and

stresses induced in the system are very large and

the system may fail due to excessive vibrations.

Mode shape is the vibration pattern of the system

excited at its natural frequency. Mathematically

mode shape is nothing but the eigenvector. Each

natural frequency has its own unique mode shape.

Mode shape does not represent the absolute

displacement of the system but only represents the

relative displacement.


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Undamped Free Vibration Problem:

Consider an undamped free vibration problem,

Note: The figure represents the simplest idealization of any physical system, whose

governing parameters are stiffness and mass. i.e. it should not be thought as merely a

spring connected with a mass.

Thus higher the mass lower is its natural frequency. Stiffer the structure, higher its natural

frequency.

The eigenvalue problem as formulated above can be solved by any of the following

techniques,

Simultaneous iteration scheme

Subspace iteration scheme

Lanczos algorithm

Power dynamics scheme

0kxxm

Fundamental equation of motion:

Where, k is stiffness of the system

m is the mass of the system

x is the displacement about the mean position

)cos( tux

The solution of the above can be assumed to be,

Where u is the stationary amplitude

Thus we can formulate the eigenvalue problem as under,

muku n2 is the natural frequency

is the eigenvector or mode shape n

u

umuK n2

On similar lines, the eigenvalue problem of a multi DOF system system can be formulated as,

m

knFor a single DOF system, the natural frequency is given by

Where, is the stiffness matrix

n

u

m

K

is the mass matrix

is

the eigenvector

is

the natural frequency

is the eigenvector or mode shape

is the natural frequency


9

Fig-1 A typical industrial Steam Turbine Rotor

having 22 blades One sector analyzed using concept of cyclic symmetry

W

h

e

e

l

Bucket

V

a

n

e

Shroud

Wheel

Vane

Chapter 2 Cyclic Symmetric Modal

Analysis 2.0 Introduction

Turbomachinery bladed disk assembly is the most ideal example of cyclic

symmetric modal analysis. In this chapter cyclic symmetric modal analysis of bladed disk

assembly of a steam turbine rotor will be discussed in detail.

A rotating turbine blade is the component, which converts the energy of the

flowing fluid into mechanical energy. The reliability of these blades is very important for

the successful operation of a turbine. Researchers have ascertained the major cause of

failure to fatigue of metal. Usually the turbine blades are fabricated by single crystal

solidification techniques to avoid grain boundaries and hence regions of fatigue crack

initiation. Fluctuating forces cause fatigue failure. Turbine blades experience fluctuating

forces when they pass through non-uniform fluid flow from stationary vanes (nozzles).

Thus the basic design consideration is to minimize the dynamic stresses produced by

these fluctuating forces. Based on the theory of vibration of mechanical structures, the

dynamic behavior of the bladed disc assembly can be predicted. Combining the dynamic

behavior with the nature of the fluctuating forces helps a design engineer in evaluating a

blade design. Dynamic stresses reach a maximum at resonance. Thus a complete risk

evaluation due to resonance is of utmost importance. The turbine rotor disk with the

blades mounted evenly around its circumference possesses rotational periodicity in

geometry. It can be analyzed with the help of cyclic symmetric principle.


10

Modeling Considerations:

FE Model is generated for one sector as shown in the above figure. A typical

turbomachinery rotor disk assembly consists of a wheel, bucket (or dovetail), vane or

blade and shroud (or cover plate). These various parts are shown clearly in the above

figure. First we need to generate the meshed model and then generate a full FE Model,

which is to be used in the subsequent modal analysis. Modeling procedure will not be

discussed here as it changes from situation to situation. But however some modeling and

meshing considerations are highlighted here,

1. The sole objective of a cyclic symmetric modal analysis is to extract natural frequencies; thus the mesh need not be too fine, unless there is a special need to

maintain a finer mesh.

2. It is not required to capture all the minute geometric details, like very small fillets. 3. A flow through mesh between bucket and vane is very difficult to realize, thus

constraint equations need to be applied between vane bottom and bucket top. In

the same way, it needs to be done between vane top and shroud bottom.

4. Depending on the modeling requirements, couplings need to be generated at force transmitting points between bucket and wheel.

5. Displacement constraints must be applied at the appropriate locations. In the model shown it was applied at the bore region of the wheel.

6. It should be taken care that the displacement constraint, couplings, constraint equation should not contradict each other i.e. they should not appear on the same

node.

7. Cyclic symmetric B.C`s needs to be defined along the two extreme faces along the hoop direction. Use CECYC or CPCYC command in ANSYS to define the

same.

Thus a FE Model can be built taking into account the above points. Now we proceed

to carry out a cyclic symmetric modal analysis.

2.1 Cyclic Symmetric Modal Analysis

This method takes advantage of harmonic variation of modal displacement around

the circumference. Referring to figure 2 the cyclic symmetric object shown has N = 8

repeating sectors. Only 1 sector is modeled in FE Analysis. The forces acting on the

whole object can be expanded in Fourier series as below,

... (2.1)

Here kF is the force on the kth

sector, Nfff ,......, 21 are the Fourier coefficients and the

angle N/2 is the angle between any two repeated sectors. Thus if we have a force

vector we can formulate a matrix equation in between the forces acting and the Fourier

coefficients and subsequently solve it to obtain the Fourier coefficients. We may thus

interpret the force on the structure to act in N harmonics, with the complete behavior of

the structure being a superposition of its responses in all the N harmonics.

)1)(1()2)(1(

1

)1(

21 ......Nki

N

Nki

N

ki

k efefeffF


11

All calculations in cyclic symmetric

analysis consist of both real and imaginary

parts. For harmonic 0n , the uniform

force of 1f is considered as acting in all the

sectors. Thus displacement and stresses in

all the sectors would be identical for the

harmonic 0n . For the harmonic 1n , a

force of 2f acts in the first sector and ief2 acts on the second sector,

2

2

ief acts

in the third and so on. The displacement

also follows the same rule. Similarly for

the harmonic 2n , force 3f acts in the 1st

sector 2

3

ief in the 2nd

sector, 4

3

ief in the

3rd

sector and so on. Thus all the harmonics

can be easily analyzed for the behavior of

the structure, since the displacement and

force in one sector determine the

displacement and forces in the

remaining sectors for each harmonic. Fig-2 Cyclic symmetric object with N = 8 repeating sectors

To summarize:

1. If the forces in the repeating sectors are related as in above equation then the corresponding displacements in the substructures are also related.

2. The structure behavior can then be obtained from the analysis of a single

repeating sector, stipulating the above variation (in the form ie ) as a

precondition.

Thus, if a generalized external force acting on a cyclic symmetric structure (e.g. load

carried be a pair of meshing gears) is represented as a sum Fourier harmonics, then for

each such harmonic, analysis of a single repeated sector is possible. It is worthwhile to

mention here that cyclic symmetric analysis, in fact is an extension of the standard

practice for axisymmetric structures under arbitrary external loading. The forces and

corresponding displacements are expressed in terms of Fourier series and analysis is

carried out for each harmonic individually. The total response is obtained as a sum of

individual harmonic responses.

In general for a problem having N repeated structures, for values of Np /2

and NpN /)(2 the imaginary parts of the displacements values will be equal and

opposite i.e. they will be complex conjucate. Hence cumulative sum of displacements will

always be real. This is the reason why stress values are always real in a cyclic symmetric

static analysis. For a complete problem, it is enough to compute only (N/2) + 1

harmonics when N is even and (N+1/2 harmonics when N is odd.


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2.1.1 Eigenvalue Problem: In this section eigenvalue problem for cyclic symmetric structures will be

formulated. General definition of eigenvalue problem and various solvers was discussed

in 1st chapter. For a cyclic symmetric model, the eigenvector iu if the

thi repeated

sector will be connected to the thi )1( by the relationship

Where, N Number of repeated structures, m number of harmonics to be computed

The nodes in a one sector modeled using cyclic symmetric principle can be classified as:

Master / Slave boundary nodes: Slave boundary nodes are the nodes that also occur on

the next consecutive sector. They are common to thi and thi )1( sector. Hence the slave

boundary nodes are connected to the master boundary nodes by a complex constraint,

namely

i

i

i ueu 1

pN

2

Slave Nodes

Master Nodes

Interior Nodes

Fig 3Concept of Master, Slave, Adjacent and Interior nodes.

Adjacent nodes


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pN

i

e

2

where p is the harmonic.

Adjacent nodes: These are the nodes that are adjacent to the slave boundary

nodes.

Interior Nodes: These are the nodes that are neither master/slave nor adjacent nodes.

Thus it is clear that master and slave boundary nodes have a complex

relationship in between them.

The Eigenvalue problem of cyclic symmetric modal analysis in matrix form can

be arrange in the following form,

=

Where iz are the eigenvectors, iM is the mass matrix, 2

n is the eigenvalue and the

stiffness matrix is given as,

n

T

n

nn

i

n

i

n

T

T

T

BA

AB

eA

eA

BA

ABA

ABA

AB

1

11

43

332

221

11

............

............

............

............

nz

z

z

z

z

...

...

...

4

3

2

1

2

n

n

n

M

M

M

M

M

M

1

4

3

2

1

.........................

nz

z

z

z

z

...

...

...

4

3

2

1

n

T

n

nn

i

n

i

n

T

T

T

BA

AB

eA

eA

BA

ABA

ABA

AB

1

11

43

332

221

11

............

............

............

............

Master Nodes

Interior/Adjacent

Nodes

Slave Nodes


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Summarizing the above form of stiffness matrix:

1. The interaction stiffness terms of master and slave nodes are complex in nature.

2. The stiffness matrix has both real and imaginary parts. Comparatively, real part occupies more core than imaginary part. Imaginary stiffness terms occur

only in the interaction terms between master and slave nodes.

3. Mathematically, the stiffness matrix in a cyclic symmetric modal analysis is Hermitian in nature, because real parts are symmetric and imaginary parts are

skew-symmetric. Hence it is an eigenvalue problem of a Hermitian matrix. If

the transpose of a complex matrix is its conjucate, then a matrix is said to be

Hermitian. Its eigenvalues are real and eigenvectors are imaginary.

4. Cyclic symmetric modal analysis gives rise to a complex eigenvalue problem. However, this eigenvalue problem becomes real for 0 and .

5. The harmonic, at which it is excited, has a considerable influence on the stiffness of the structure.

Further if consistent mass is used then the mass matrix is also complex. But for a

lumped mass formulation it remains real. The Eigenvector and natural frequency can be

obtained by solving this complex Eigenvalue problem. Normally, Lanczos algorithm is

used. The Eigenvector thus obtained are that of a single sector. Eigenvector of other

sectors can be determined from the mathematical relation given by equation 2.2 and

hence the mode shape of the entire disk can be easily obtained.

Thus both natural frequency and mode shape are also dependent on the harmonic

and this harmonic is popularly referred to as Nodal Diameter in literature.

2.1.2 Concept of Nodal Diameters:

In previous sections, the Eigenvalue problem for cyclic symmetric structures was

explained. Nodal diameters play vital role in cyclic symmetric modal analysis and helps

to understand the deflection pattern of the bladed disk assemblies, thus become an

essential feature of the cyclic symmetry analysis. The influence of ND on stiffness of the

structure is evident. It should be noted here that the word nodal used here is in vibration

sense and not in finite element sense. Node here means a point of zero displacement, a

virtual hinge. Nodal diameter can be understood by the following,

Nodal diameter defines the number of complete sine waves that pass through the circumference. If we plot the displacement of the tip of the blade with angular

position, it follows a sinusoidal characteristic which is designated as a diametrical

pattern

Nodal diameter indicates number of zero displacement planes that pass around the circumference.

Mathematically, nodal diameter represents the harmonic content.


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Nodal diameter is the number of distinct harmonics that are required to completely define the system. The amplitude of vibration varies harmonically around the disk

and ND is nothing but the no. of distinct harmonics that are present in the system.

Nodal diameter will govern the cyclic symmetric B.C`s that are to be applied while carrying out cyclic symmetric modal analysis.

Physically, Nodal diameter is nothing but the disk effect. It simulates the disk mode shape.

Nodal diameter has a direct influence on natural frequency.

The following figure illustrates the harmonic variation of amplitude along the

circumference. At ND = 0 there exists no zero displacement planes. At ND = 1 +ve

indicates a positive displacement and ve indicates a negative displacement and hence

one zero displacement plane exists. Similarly at ND = 2, we have 2 zero displacement

planes. The variation of amplitude along circumferential position is also depicted below.

Similarly nth ND will have N complete sine waves passing around its circumference and

N zero displacement lines in the disc. In a cyclic symmetric structure, the number of

harmonics depends on the number of repeating sectors. If there are N repeating sectors in

the structure, then the number of distinct harmonics i.e. (nodal diameters) can be

calculated as,

(a) (N/2) + 1 nodal diameters, n

(b) (N+1)/2 nodal diameters,

It should be noted that this also includes the zeroth ND, and hence maximum count of ND

will be one less than the above quantity.


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Let us workout a simple example of a 3600 disc constrained at its bore and

conduct a cyclic symmetric modal analysis and study the nodal diameter pattern. One 600

model was analyzed in ANSYS and we have ND = 0,1,2,3. The nodal diameter patterns

are as shown.

Full 3600 disk having 6 60

0 sectors One 60

0 sector with constraint at bore.


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Nodal Diameter 0

Nodal Diameter 1

The disc mode shapes are shown in the given

figure. In ND = 0 disc mode, all the sectors move in

the same direction, either up or down. They all are in

phase i.e. there is no relative motion between the

sectors. The displacement pattern is similar in all the

sectors. All the sectors move together in this ND. In

other words, no interaction exists between different

sectors of the disc. The entire disc behaves like a

cantilever beam constrained at its bore. No sine waves

pass through the circumference. All the sectors are in

phase.

ND 0

The disc mode shapes are shown in the

given figure. In ND = 1 disc mode, the sectors

belonging to the 1800 moves in one direction

and the other 1800 moves in another direction

i.e. in a flexure mode, half of the disc moves up

while the other half moves down. The

displacement pattern of the disc is as shown.

One zero displacement plane occurs at this ND

i.e.1-sine wave pass through the circumference.

ND 1


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Nodal Diameter 2

Nodal Diameter 3

The disc mode shapes are shown in

given figure. In ND = 3 i.e. at the max. ND

all the sectors move in directions opposite to

their neighboring sectors. Three zero

displacement plane occurs at this ND. 3-sine

waves pass through the circumference. All

the sectors are out of phase.

ND 3

The disc mode shapes are shown in

the given figure. In ND = 2 disc mode, first

900 moves in one direction, second 90

0

moves in another, third 900

moves in same

direction as first and the last one moves in

same direction as second. Two zero

displacement plane occurs at this ND. The

two planes are perpendicular to each other. 2-

sine waves pass through the circumference.

ND 2


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Thus the main conclusions from the above study on nodal diameters can be

summarized as,

1. For the ND = 0 and ND = NDMAX nodal diameters, the sectors of the bladed disk

2. The mode shapes shown in the above figures were flexure modes. Similarly we can have tangential, axial or a complex modes for each nodal diameter.

3. ND is the parameter that depicts the interaction of each sector in the whole disc. In bladed disc assemblies, it depicts the interaction in between different blades.

In addition to nodal diameter there also exists a nodal circle mode in disc vibrations.

Nodal circle mode is similar to nodal diameter mode except that, in nodal circle mode a

zero displacement circle exists in contrast to that of a zero displacement line in nodal

diameter modes. Depending on the nature there also can exist combination modes of

nodal diameter and nodal circle. We limit our scope of discussion to nodal diameter

modes only. Thus the principle of cyclic symmetry can be successfully applied to modal

analysis. The next chapter is dedicated to results that we get from a cyclic symmetric

modal analysis and tools to analyze the results of a cyclic symmetric modal analysis

namely Interference Diagram and Campbell Diagram.

Chapter 3 Results and Discussion

3.0 Introduction:

Evaluation of the turbine bladed disc assembly design is the sole purpose of any

FE Analysis. Untimely shutdown disturbs the economics of both the user and the

manufacturer. And hence there is an urge for reliable turbine operation. High Cycle

Fatigue (HCF) plays a significant role in causing turbine blade failures. HCF failures can

result from resonant vibratory stresses sustained over a relatively shorter time. During

operation, the periodic fluctuation in the steam forces occurs at frequencies

corresponding to running speed and its harmonics and thus causes the disc to vibrate. The

amplitude of these vibrations depends on the proximity of the forcing frequencies to the

natural frequency. Large amplitude vibration can occur when the forcing frequency


20

approaches or becomes resonant with a natural frequency of the bladed disk. Vibratory

stresses at or near resonant condition aggravate HCF damage.

Steam turbine manufacturers typically design and manufacture bladed disks with

adequate margins between the forcing frequencies and the fundamental natural

frequencies to avoid resonance. However it is not possible to completely avoid resonance

under normal operating conditions of a steam turbine. This is because of manufacturing

variations, assembly variations, routine wear, varying operating speeds or any other

factor that modifies the mass or stiffness of the structure. A practical case of resonance in

operating conditions was due to solid build up at the root of blade because of corrosion,

which shifted the natural frequency, and the excitation frequency coincided with natural

frequency and produced resonance. Thus blade design must be reviewed continuously for

possible interference of natural frequency with excitation frequency.

3.1 Sources of Excitation in a Steam Turbine

Assembly: The root cause of vibration in a steam turbine is due to flow induced vibrations.

In a steam turbine, most common sources of excitation are nozzle-passing frequency

(NPF), running speed and harmonics of both NPF and running speed. The main source of

turbine blade vibration problems arising in turbomachines is the fact that the distribution

of the steam load on the blade is not constant with respect to the rotational angle but

varies during one rotation of blade. This angular dependence of blade load is caused by

various circumstances. The main reason for that is the fact that the rotor blades run

behind a row of still standing stator blades, which are fixed in the casing of the machine.

This causes a circumferential distortion in the flow, which may be due to inlet or exhaust

ducts. The pressure variation in the fluid flow causes the forces to vary circumferentially.

These forces depend on the relative position of blading with respect to nozzles and any

other interruptions like struts, diaphragms etc. Another reason for non-uniform blade load

in one revolution can occur by partial admission e.g. the loading of the rotor blades by

fluid forces in one or more sectors around the circumference while other sectors remain

unloaded. In both the cases, the load on the rotating blade is periodic and can be

described in the form of Fourier series as,

)cos()(0

nn

n

n tFtF

Where )(tF is the forcing function, nF are the Fourier coefficients which define the

forcing function )(tF , n is the phase angle, t is the time, n are the excitation

frequencies given by,

nn (or) NNnn

Where is the running speed of the turbine, NN is the number of nozzles (i.e. the

number of periodics of the blade load during one revolution). Here it should be noted that


21

excitation frequency arises either from running speed harmonics or nozzle passing

frequency.

Nozzle passing frequency excitations are caused by steam flowing through a set of

nozzles or diaphragms, which are used to turn and direct steam flow onto the blades.

Because of their design, nozzles have flow interruptions at regular or cyclic intervals,

which cause the force imparted on the blades to be cyclic. These excitations are caused

by nozzle vane trailing edge wakes as the flow leaves the vanes. A blade passes through

nozzle vane wakes a number of times equal to the number of nozzle vanes in one

revolution. An example of this wake formation zone is shown in the given figure below.

Due to formation of wakes in the airfoil region of nozzles, velocity of steam remains no

more constant and a variation in pressure takes place. Net effect of this variation in

velocity and pressure is to produce unbalance forces.

WAKES (REGIONS OF REDUCED

VELOCITY)

P2 V2

P1 V1

SHOCKS (LINES OF

VELOCITY AND

DENSITY CHANGE)

P2 > P1 V2 < V1


22

Fig: Wake formation in Stator vanes

Running speed harmonics occur due to interruptions in the fluid flow path.

Harmonics of running speed are multiples of rotor operating speed. For example, the fifth

harmonic of running speed would be a force that occurs five times for every revolution of

the wheel. For example, a turbine rotor running at a speed of 3600 RPM (60 cycles/sec or

Hz) would have running speed harmonics occurring at 120Hz, 180Hz, 240Hz and so on.

3.2 Definition of Real Resonance:

A good design of turbomachinery blading consists of following steps,

1. The investigation of natural frequency and the mode shapes of the rotor blades 2. The determination of amplitude, frequency, and shape of aerodynamic forces

acting on the rotor blades.

3. The determination of damping values and generating appropriate damping models.

4. Determination of dynamic stresses. 5. Life estimation based on cumulative damage fatigue theories.

Thus determination of true resonance condition is critical for turbomachinery design.

A turbine bladed disc can get into a state or resonance when the energy build up is

maximum. There are two simultaneous conditions for the energy build up per cycle of

vibration to be maximum. These conditions are:

The frequency of exciting force equals the natural frequency of the structure,

The exciting force profile has the same shape, as the associated mode shape of vibration i.e. the harmonic number must represent the nodal diameter.

Both of these must exist together for a real resonance to occur. If both the

conditions are satisfied, then work done by the system will be maximum and hence

energy build up will be maximal, thus causing maximum damage to the system.

3.3 Results Interpretation:

Results from cyclic symmetric modal analysis with objective of investigation of

natural frequency and mode shape pattern are discussed in this section. Basically it

should be noted that natural frequency would vary with nodal diameter. Consider a case

of 44 blades around the circumference and thus we have nodal diameter range from


23

Natural frequencies were extracted for first 12 modes using Lanczos

algorithm in ANSYS. A typical list of natural frequencies thus obtained from a cyclic

symmetric modal analysis is given below:

The modes as indicated in first column are the mode shapes of the individual

buckets/sectors and nodal diameter is the one that indicates the vibration mode of disc.

Observations:

1. Mode of disc vibration can be completely defined if and only if both ND and the

individual bucket (or) sector mode shape of one sector are specified i.e. 1F (3)

indicates 1st Flexure mode with 3 nodal diameters. If a combination of nodal

circle modes and nodal diameter modes occur, then we specify it as 1F (2C3D) 1st

flexure mode occurring in conjunction with a combination of 2nd

nodal circle

mode and 3rd

nodal diameter mode.

2. It is clearly evident from the above results that natural frequency depends to a considerable extent on nodal diameter.

3. Doublets in natural frequencies are observed for all nodal diameters except

ND =

0 and ND = NB/2.

4. As the number of nodal diameter increases, the disk assembly becomes more and

more stiffer in nature.

5. It may also happen that a natural frequency value can appear in more than one

nodal diameters, that means a particular natural frequency can excite many nodal

diameters.

Natural Frequencies at each Nodal Diameter

ND=0 ND=1 ND=2 ND=3 ND=4 ND=20 ND=21 ND=22

Mode # 1 1018.551 1051.936 1102.742 1169.668 1264.132 3156.285 3198.778 3212.413

Mode # 2 1296.730 1051.936 1102.742 1169.668 1264.132 3156.285 3198.778 3548.476

Mode # 3 2918.921 1390.389 1657.615 2020.396 2406.908 3535.594 3544.492 4504.928

Mode # 4 3549.565 1390.389 1657.615 2020.396 2406.908 3535.594 3544.492 5405.417

Mode # 5 4288.775 2939.808 2982.150 3081.082 3272.481 4499.048 4503.265 5955.164

Mode # 6 4838.825 2939.808 2982.150 3081.082 3272.481 4499.048 4503.265 6352.461

Mode # 7 5669.021 3861.894 4225.162 4260.962 4304.176 5400.065 5404.021 7297.392

Mode # 8 5998.684 3861.894 4225.162 4260.962 4304.176 5400.065 5404.021 7521.234

Mode # 9 6625.923 4300.977 4540.467 4766.320 4886.864 5981.648 5962.567 8351.090

Mode # 10 7116.876 4300.977 4540.467 4766.320 4886.864 5981.648 5962.567 8723.034

Mode # 11 8135.735 4857.238 4985.760 5294.683 5491.265 6371.552 6357.133 9671.878

Mode # 12 8213.375 4857.238 4985.760 5294.683 5491.265 6371.552 6357.133 10201.245

ND = 0 NDMAX


24

3.3.1 Why do we get DOUBLETS?

In general the mode shape of the bladed disc assembly can be given as,

Where MnlL , ...........3,2,1,0l , n is the number of buckets and if assembly is

packeted then it is the number of packets and l is the phase lag between the force and

response, which occurs due to damping. The basic reason for repetition of natural

ept at ND = 0 and ND = NB/2 or (NB-1)/2 depending on

whether the no. of buckets are even or odd respectively , is that they represent orthogonal

mode shapes.

Consider the case of 40 bladed disk assembly, thus we have no. buckets 40n ,

no. of nodal diameters (ND) from 0 to 20 and phase angle = 0 . Except at ND = 0 and

ND = 20 we expect doublets. Now let us expand the above equation for the following

four cases by substituting the various values of l, n and M.

Case 1: - ND = 0 Case 3: - ND = 15

0L 15L 40040L 55,251540L

80080L 95,651580L

Case 2: - ND = 20 Case 4: - ND =17

20L 17L 60,202040L 57,231740L

100,602080L 97,631780L

The values of L represent the harmonic content in the mode shape of the disc.

The 3600

model of the turbine rotor consists of 40 blades. Thus 0 to 40 blades represents

00 to 360

0 in a cyclic manner. For understanding purpose one can assume the angular

)sin()(0

l

l

L LAMX


25

position )ang. = (3600

/ no. of buckets n)

*{harmonic no. L})] starting from 00

to 3600 clockwise direction if it has been assumed

00 angular position at the top of the disk. This angular position will be the starting point

of the sinusoidal wave to that deflection pattern of bladed disk will follow. For better

understanding consider case 3 mentioned above in the table, this nodal diameter has

mode shapes at 15th

and 25th

harmonic, means angular position of the nodal lines i.e.

sinusoidal waves will start either at 1350 or 225

0 respectively. Similarly considering the

case 4 we have only 17th

and 23rd

harmonic, representing 1530 and 207

0 respectively

within first 3600. Thus for the case, 3 and 4 we have only two harmonics that

encompasses within full 3600

and other higher harmonic nodal lines will start exactly at

any of the two angular positions. Consider case 2, this nodal diameter has only one

starting angular position of the sinusoidal wave for each and every harmonics and which

is unique. For each harmonics for case 2, there will be 20 sinusoidal waves and same

number of nodal lines, which will be at 180

interval. Finally considering case 1 it has

mode shapes occurring at 0th

and 40th

harmonics thus it represents 00 and 360

0 degrees

respectively. Mathematically we can say 3600 in nothing but 0

0 (as the function is

harmonic in nature )360cos()0cos( and )360sin()0sin( ). For the zeroth nodal

diameter there is no sine wave which means no nodal line to be shown along the

circumference. Again for the zeroth nodal diameter all the system is moving all together

i.e. there is no relative motion among the components. Thus in the cases 1 and 2 we have

only one harmonic that encompasses within full 3600.

The above study reveals an interesting observation that, except at ND = 0 and

ND = NB/2 we have two different harmonic contents within 3600.

This leads to the natural modes to occur in spatially orthogonal pairs.

Orthogonal mode shapes are the ones in which nodal radii of one mode in the pair

become antinodes in the other and vice-versa.


26

The above figure represents an ND = 3 nodal diametric mode showing nodes

and antinodes. Solid lines show one mode and dotted the other of the two orthogonal

mode shapes i.e. nodes of solid lines are the antinodes of dotted line and vice-versa.

These two orthogonal mode shapes have same frequency as they represent nothing but

and 360 angular position. The basic nature of function assumed in cyclic symmetric -

analysis is harmonic in nature i.e. cos , )sin( and from mathematics we know that

)360cos()cos( and )360sin(sin . Thus there is no change in mode

shape as such, except an orientation change. Since mode shape remain unaffected, the

natural frequencies also occur in pairs for a pair of orthogonal mode shapes. This is true

only if the disc is perfectly cyclic symmetric.

Following observations can be made from the study :

The reason for repetition of frequencies is because of occurrence of orthogonal

The mode designation does not change in a pair of orthogonal mode shapes i.e.

The only change in a pair of orthogonal mode shape is the orientation i.e. the location of nodal line is indeterminate and depends usually on the position of

applied excitation.

Excitation at any ND can have harmonics i.e. it is not necessary that only 15th harmonic excites the system at ND = 15 and the system can get excited at ND =

15 even at 25, 55, 65,

The next obvious question is - how are orthogonal mode shapes implemented in

ANSYS?

In ANSYS 5.6 for cyclic symmetric modal analysis we use two macros

CYCGEN and CYCSOL. For generating cyclic symmetric B. C`s ANSYS offers

CECYC and CPCYC commands. The main purpose of CYCGEN is to generate a

duplicate sector. This duplicate sector is generated to take care of orthogonal mode shape

and leads to occurrence of natural frequencies in pairs. Thus duplicate sector and the

original sector put together ensure an orthogonal mode shape pair. CYCGEN macro has

to be executed before CYCSOL. The main purpose of CYCSOL macro is to generate

cyclic symmetric B.C corresponding to the nodal diameter and solve it. To generate

cyclic symmetric B.C, CYCSOL uses CECYC command. Thus we need to give in the

nodal diameter number or number of buckets as an input to CYCSOL macro. But we

also know that at ND = 0 and ND = NB/2 we do not have an occurrence of orthogonal

mode shapes and hence natural frequencies do not occur in pairs. ANSYS ensures that at

CYCGEN is not

included in the FE Analysis. This is taken care in CYCSOL macro by constraining all

DOF of the duplicate sector, for ND = 0 and ND = NB/2 respectively.

At Ansys 10 and above, the command CYCGEN is replaced by CYCLIC

command. This command also instructs Ansys to automatically identify the sector angle


27

and creates the Constraint equations between the sector end faces. The CYCOPT

command should be used in conjunction to the CYCLIC command to specify the number

of nodal diameters to be extracted, tolerance of variation in co-ordinates of nodes on the

end faces etc. The CYCSOL is no more used in the present version and a simple SOLVE

command suffices. However the total number of modes to be extracted & the mode

extraction algorithm is to be specified using the MODOPT command as in earlier

versions.

The reason for repetition of natural frequencies is explained and we now proceed

to interpret the results from a cyclic symmetric modal analysis using an Interference

diagram.

3.4 Interference Diagram: In cyclic symmetric structures, natural frequency also depends on nodal diameter apart from rotating speed, material properties, geometry and applied constraints. For a

given geometry, material property and applied constraints natural frequency varies with

rotating speed and nodal diameter. Thus we really need a three dimensional surface

),,( NDn to conduct a full-fledged investigation on risk due to resonance.

Engineers do not always prefer a three-dimensional surface. This is compounded

by the fact that, it is difficult to visualize a 3D plot and it is also a time consuming


28

process to create a 3D plot of ),,( NDn . So we need to simplify a 3D plot into 2D

plot for better and quicker analysis. Thus we can either create a ),( NDn plot or a

),( n plot. Plot of ),( NDn is nothing but Interference Diagram and plot of

),( n is a Campbell Diagram.

3.4.1 Plotting an Interference Diagram:

The interference diagram is a plot of nodal diameter along its absisca and natural frequency along its ordinate. Since interference diagram is a tool for risk evaluation

arising due to resonance, we also need to express all sources of excitations in the

diagram. A typical interference diagram is as shown below:

3.4.2 Interference Diagram beyond NB/2 Limit:


29

In chapter 2 we defined nodal diameters as the number of distinct harmonics

that are required to define the complete system. Again consider the case of a rotor with

44 blades around the circumference and let us plot interference diagram beyond NB/2

limit:

Full Range Interference Diagram

0

1000

2000

3000

4000

5000

6000

7000

0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88

Harmonic Number

Na

tura

l F

req

uen

cy

Mode 1

Mode 2

Mode 3

Mode 4

Mode 5

Mode 6

NPF LinesCritical Nodal Diameters

Running Speed Harmonics Line

Harmonic no(NDMAX)

after which natural frequency

values periodically repeats.

No of Blades = 44


30

Points to be noted:

1. ND = NB/2 for even no of blades and ND = (NB-1)/2 fully defines the system. Behavior at all harmonics higher than ND = NB/2 are just a replica of behavior

defined until ND = NB/2. This is possible due to the beauty of harmonic

functions, which was the basic assumption in the case of cyclic symmetric

structures.

2. If we map the running speed harmonic line in 0 to NB/2 range then it would be enough to completely interpret the risk due to resonance and hence carry out a full

vibration analysis. The plot thus obtained after mapping is nothing but

Interference Diagram.

3. It should also be noted that interference diagram is drawn keeping the speed as constant. In the present case running speed was assumed to be 60Hz.

3.4.3 Studying an Interference Diagram:

This section is devoted to physical understanding of interference diagram and

carrying out a risk evaluation due to resonance. The following important points are to be

noted while studying an interference diagram:

Resonance will occur at all points where running speed harmonics and/or NPF line cut the natural frequency line. But however the level of damage caused may

be different.

At any nodal diameter, the harmonics that can excite the system are given by the equation (3.2) i.e. it is not possible for any harmonic to excite any ND. There

exists a relationship between the nodal diameter to be excited and the possible

harmonics that can excite the system at that nodal diameter.

Experimentally or by experience the max harmonic NPF is decided. In the present case it was assumed till NPF2 .

There exists no general rule for change in natural frequency with respect to nodal diameters. In the present case we observe an increase in natural frequency with

increase in nodal diameter. This may be due to the fact that the system becomes

stiffer at higher modes and nodal diameter is nothing but disc mode. Some

researchers have observed that shrouded assembly is more sensitive to nodal

diameters than unshrouded assemblies. But strictly speaking there is no defined

pattern of natural frequency variation with respect to nodal diameters.

Among all possible nodal diameters it is possible to single out a few which will cause maximal damage to the system i.e. pump in max energy into the system.

These nodal diameters are referred to as critical nodal diameters.


31

Why are critical nodal diameters really critical?

First what are critical nodal diameters? They are nothing but the nodal

diameter at which both NPF, its harmonics have same forcing frequency as that of

running speed and its harmonics i.e. they are the nodal diameters corresponding to the

points of intersection of running speed harmonics line and NPF line. Thus the no. of

critical nodal diameters that can occur are usually equal to the no of harmonics of NPF

that are expected to excite the system i.e. maximum no. of harmonics of the NPF for

which the risk evaluation due to resonance is to be carried out. In present case study upto

NPF2 was carried out and hence we had only 2 critical nodal diameters. Sometimes

both the intersections can occur at same nodal diameter.

The main edge interference diagram possesses over other tools like Campbell

diagram are due to the critical nodal diameters. Critical nodal diameters represent the

interference of two waves i.e. running speed harmonic wave and NPF harmonic wave.

From elementary physics, interference occurs when two waves with same frequency are

in phase, the waves add up and produce a net wave, which has amplitude equal to the

summation of amplitudes of both the waves. Thus in the present case at critical nodal

diameters, the amplitude of force due to running speed harmonic and NPF harmonic add

up and inflict maximum damage to the system, if the forcing frequency equals natural

frequency. It should be taken care while designing that; the natural frequency of any

mode should not cut the point of interference. In actual practice resonance condition

prevails in a certain frequency zone. Thus we also need to evaluate the % of frequency

margin that exists between natural frequency and forcing frequency at critical nodal

diameters and compare it with design criteria.

Necessary conditions, for real resonance was stated earlier. The number of nodal

diameter must coincide with the number of harmonic. This is taken care in an

interference diagram by the concept of interference at critical nodal diameter and hence

interference diagram also helps to design the turbine rotor against the possibility of

occurrence of real resonance.

Thus capabilities of interference diagram are studied and an appreciation of its

suitability towards better turbine blade design is presented.


32

3.4.4 Interference Diagram Vs. Campbell Diagram:

In this section we make a comparative study of an interference diagram with a

Campbell diagram. Campbell Diagram is a ),( n plot with natural frequency, n

along its ordinate and running speed, along its absisca. This diagram predicts where the

have plotted running speed harmonics). This condition of frequency coincidence will be

termed a PROBABLE RESONANCE.

A further look at these coincident points with the knowledge of mode shape(s) and the

shape(s) of exciting forces allows the POSSIBLE RESONANCE points to be identified.

Figure below is a representation of such an analysis where the solid circles indicate the

few POSSIBLE RESONANCES among the many PROBABLE RESONANCES of

the blade frequency and force frequency intersections.

From these PROBABLE RESONANCES we need to focus our energies in avoiding

REAL RESONANCE. In interference diagram it occurs at the point of intersection (or

within the frequency margin) of NPF line, running speed harmonic line and natural

frequency line i.e. at critical nodal diameter. Condition of REAL RESONANCE in both

Interference diagram and Campbell diagram is illustrated below:

Operating range of speed

Steady speed


33

cyclic symmetric modal analysis

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