hassan final report

7/31/2019 Hassan Final Report

1/39

i

Vibratory Analysis of Turbomachinery Blades

by

Mohamed Hassan

A Project Submitted to the Graduate

Faculty of Rensselaer Polytechnic Institute

in Partial Fulfillment of the

Requirements for the degree of

MASTER OF ENGINEERING - MECHANICAL ENGINEERING

Approved:

___________________________________________Professor Ernesto Gutierrez-Miravete, Project Adviser

Rensselaer Polytechnic InstituteHartford, Connecticut

December, 2008


2/39

ii

Copyright 2008

by

Mohamed Hassan

All Rights Reserved


3/39

iii

CONTENTS

Vibratory Analysis of Turbomachinery Blades ............................................................... i

LIST OF TABLES ........................................................................................................iv

LIST OF SYMBOLS......................................................................................................v

LIST OF FIGURES..................................................................................................... vii

ACKNOWLEDGMENT ............................................................................................ viii

ABSTRACT..................................................................................................................ix

1. Introduction..............................................................................................................1

2. Modeling Methodology ............................................................................................6

3. Discussion and Results ...........................................................................................14

4. Conclusion .............................................................................................................23

5. References..............................................................................................................24

6. Appendix: ANSYS Macros.....................................................................................25


4/39

iv

LIST OF TABLES

Table 1: Results Summary Sequence ............................................................................15

Table 2: Full Model vs. Cyclic Symmetry Frequency Results .......................................15


5/39

v

LIST OF SYMBOLS

w = Driver Frequency (HZ)

mw = Natural Frequency (HZ)

I= Number of Nozzles around the Wheel

N= Speed (RPM)

nf = Forcing Function of the nth Harmonic

nF = Amplitude of the nth Harmonic

= Angular Position on the Disk

t = Time

W= Work

my = Periodic Distance Function

mY = Distance - Constant

m = Nodal Diameter

n = Harmonic Index

M= Mass Matrix

C= Damping Matrix

K= Stiffness matrix

x = displacement

x = velocity

x = acceleration

m}{ = Mode Shape

a = Sector Angle

mf = Coefficient corresponding to thejth Sector

1=i Complex Notation

Jj ,.....2,1= ,Jis the number of substructures,j sector number

P =J/2 for evenJand (J-1)/2 for oddJ

jU = Response of the full structure for sector number j

AU = Basic sector solution

BU = Duplicate sector solution


6/39

vi

A

HighU = Basic sector deflection on the high side

B

HighU = Duplicate sector deflection on the high side

A

LowU = Basic sector deflection on the low side

B

LowU = Duplicate sector deflection on the low side

= Phase angle


7/39

vii

LIST OF FIGURES

Figure 1: Jet Engine Schematic [7] .................................................................................1

Figure 2: Example of Nodal Diameter m [1] ...................................................................2

Figure 3: Cyclic Symmetry Layout [1] ...........................................................................9Figure 4: Cyclic Symmetry Process [1].........................................................................10

Figure 5: SOLID45 Element [1] ...................................................................................11

Figure 6: Cyclic Symmetry Sector Edges [1] ................................................................11

Figure 7: Cyclic Symmetry and Full Model ..................................................................12

Figure 8: Sector Model Boundary Conditions ...............................................................13

Figure 9: Full Model Boundary Conditions...................................................................13

Figure 10: Mode1 Easy Wise Bending ND1 .................................................................16



Figure 13: Mode1 Easy Wise Bending ND0 vs. ND12..................................................20

Figure 14: Torsion Mode variation with nodal diameters ..............................................20

Figure 15: Campbell Diagram.......................................................................................22


8/39

viii

ACKNOWLEDGMENT

I would like to thank Professor Ernesto Gutierrez-Miravete for his patience,

understanding and guidance. I also like to thank all my family and friends for their

motivation and continuous support to my education.


9/39

ix

ABSTRACT

The purpose of this paper is to introduce the theory and process of vibration analysis for

gas turbine blades. Turbomachinary are critical for aircraft propulsion and energy

production so they must be designed to operate safely for extended periods of time overvarious speed ranges. The engineers task is to be able to design blades that can operate

safely in harsh environments and be able to avoid vibratory resonances that can cause

failure. This requires in-depth structural analysis and finite element modeling is a very

powerful tool to address complex designs. In this paper a modal analysis was performed

on a bladed disk turbine wheel to obtain its dynamic characteristics by examining

different modeling techniques. Two methods were employed; the first was to model a

full wheel bladed-disk (24 blades) structure. The other method was to use ANSYS

modal cyclic symmetry capability by only modeling a single sector of the entire

structure. Results showed that both natural frequencies and mode shapes were almost

identical between the employed methods. However, cyclic symmetry was found to be

more superior since it accomplished the task at a fraction of time and computer

resources.


10/39

1

1. IntroductionBladed disks exist in modern jet engines and are very important to its operation. In a

basic jet engine, air enters the front intake and is compressed. The compressor is made

up of many blades that are attached to a disk and a shaft. The blades compress the airraising its pressure. The compressed air is sprayed with fuel and an electric spark lights

the mixture. The burning gases expand and pass through another fan-like set of blades

(turbine) which rotates the turbine shaft. This shaft, in turn, rotates the compressor,

thereby bringing in a fresh supply of air through the intake. Air rushes through the

nozzle at the back of the engine. As the gas shoots backwards, the engine and the aircraft

are thrust forward. See Figure 1 for a schematic of a jet engine. Jet engines are self

sustaining machines as long as fuel is provided they will keep operating.

Figure 1: Jet Engine Schematic [7]

It is very important that turbomachinery be designed in a way to assure trouble-free

operation. Jet Engine failures or in-flight shutdowns put lives at risk and can be very

costly for both the user and the original equipment manufacturer. Turbomachinary

components, especially blades, are exposed to loads that can cause failure, designing

reliable components requires in-depth vibration and stress analysis.

One of the main causes of turbine blade failure is high cycle fatigue or HCF. Fatigue

failure is caused by repeated cyclic loads on a structural member. The fatigue life of a

part is defined by the number of load cycles it can survive. The fatigue life depends on

the stress cycles magnitude and the parts material properties. In most cases the higher

the stress the shorter the fatigue life.


11/39

2

Fatigue failure occurs as follows. A crack initiates after a number of stress cycles.

This happens at a location of relatively high stress concentration. Cooling holes, sharp

fillets or internal core features of a turbine blade are typical high stress concentration

spots. By applying stress cycles the initiated crack grows. Final failure occurs very

rapidly after the crack reaches some critical length. HCF failure corresponds to fracture

due to a relatively large number of stress cycles caused by vibrations. [5]

Natural frequency is the frequency at which an object vibrates when excited by

force. At this frequency, the structure offers the least resistance to a force and if left

uncontrolled, failure can occur. Mode shape is deflection of object at a given natural

frequency. A guitar string is a good example of natural frequency and mode shapes.

When struck, the string vibrates at a certain frequency and attains a deflected shape. The

eigenvalue (natural frequency) and the accompanying eigenvector (mode shape) are

calculated to define the dynamics of a structure. A turbine bladed disk has many natural

frequencies and associated mode shapes. In the case of a bladed disk, the mode shapes

have been described as nodal diameters. The term nodal diameter is derived from the

appearance of a circular geometry, like a disk, vibrating in a certain mode. Mode shapes

contain lines of zero out-of-plane displacement which cross the entire disk as shown in

Figure 2. In other words, a node line is a line of zero displacement and the displacement

is out of phase on the sides of the line represented by white and gray shades in Figure 2.

These are commonly called nodal diameters. Hence the natural frequency and nodal

diameter are required to describe a bladed disk mode.

Figure 2: Example of Nodal Diameterm [1]

After establishing natural frequency and mode shapes, alternating forces must exist

to excite a structure and make it vibrate. These forces have inherent frequencies and

shapes just as bladed disks do. In a gas turbine, the most common sources of excitation

are running speed harmonics and vane passing frequencies [6]. Running speed

harmonics are multiple frequencies of the rotor operating speed. A turbine rotor running


12/39

3

at 7200 RPM (120 cycles/sec or HZ) would be having running speed harmonics

occurring at 240 HZ, 360 HZ, 480 HZ, and so on.

Vane passing frequency excitation is caused by air flowing through a nozzle or

vane. Vanes are static structures that are used to direct and control airflow onto the

blades. Because of their design, vanes have flow interruptions at regular or cyclic

intervals that cause a cyclic force on the blades. Other interruptions include struts or

combustor nozzles. For example, forty five symmetrically located vanes will create a

force that occurs forty five times per revolution or a ( )45sin excitation to the blades.

Blades will experience the same force pattern making it a periodic force. Moving a stick

along a picket fence is a similar situation.

Understanding natural frequency and periodic forces helps explain resonance.

Resonance is a condition where response or amplitude of vibration is a maximum and

resistance to an oscillating force is at minimum. At this condition, the shape and

frequency of a force must match the natural frequency and mode shape of the structure.

An example of resonance is the Tacoma-Narrows bridge failure [4]. The bridge, as any

structure, had inherent natural frequencies and associated mode shapes. When the wind

blew at a certain speed, it created a forcing function that matched one of the bridges

natural frequencies and mode shapes, the bridge started to oscillate. Vibration

amplitudes became so large that ultimately lead to bridge failure. In Turbomachinery, itis very difficult to have resonance free operation throughout the entire speed range.

Resonance is best avoided or at least controlled through the use of dampers.

A mathematical discussion of the condition of resonance is provided. In each

revolution of a turbine wheel, blades pass through a field of pressure fluctuation due to

nozzle or any other interruptions in the flow field. This fluctuation in pressure imposes a

time varying force on the blades. In general such forces can be broken into harmonic

components using Fourier analysis as follows:

....)sin(...)sin( 1110 ++++++= nnn twFtwFFF

The frequency of the harmonics depends on the speed of the turbine and the number

of interruptions in the annulus like the number of nozzles, and is expressed as:

60

NIw =


13/39

4

Where,

w = Driver frequency (HZ)

I= Number of nozzles around the wheel

N= Speed (RPM)

The frequency of any harmonic is an integer multiple of rotational speed and the number

of interruptions. The nth harmonic of the force can be expressed as:

)(sin),( += wtnFtf nn

Where,

nF = amplitude of the nth harmonic

= angular position on the disk

t = time

Resonance is achieved when the forces imposed on the blade do positive work. The

work is defined as:

=1

0

FdxW

Where Wis the work done by force Fto move the body to a distance 1.

The work done by the nth harmonic of the force on the mth nodal diameter of the bladed

disk in one period can expressed as

dtdN

tyt

tfW m

T

n2

),(),(

2

0 0

=

Finally,

===

ww

wwNFW

n

m

m

andmnfor0

andmnfor

Where,

nF = Amplitude of the nth Harmonic

mY = Distance - constant

mw = Natural frequency

= Angular position on disk

m = Nodal diameter

n = Harmonic index


14/39

5

The natural frequency mw of the bladed disk must equal to the frequency of the

driving force ww =m and the number of nodal diameters m must coincide with the

harmonic of the force n i.e. m = n. The second result suggests that the work done will be

zero when neither of the above conditions are satisfied which means no resonance will

occur. This explains why the natural frequencies and driver frequency must match and

also the force harmonic and structure mode shape or nodal diameter must match to

achieve resonance.


15/39

6

2. Modeling MethodologyModal analysis can be a powerful tool to assist in the identification and elimination

of high cycle fatigue problems. Advances in computing power lead to the use of finite

element models or FEM to investigate turbine blade responses under running conditions.FEM can be used to predict steady stresses and vibratory natural frequencies and mode

shapes. Knowledge of these frequencies is useful in avoiding excessive excitations such

as unbalance and vane passing thereby reducing the risk of fatigue failure. [2]

A continuous structure has an infinite number of degrees of freedom (DOF). The

finite element method approximates the real structures with a finite number of DOFs. Z

mode shapes can be found for a FEM having ZDOFs. Modal analysis is the process of

determining the Znatural frequencies and mode shapes. Based on the initial conditions

the structure will vibrate at one its natural frequencies and mode shapes. The dynamic

equation of a spring mass damper system is [4]

}{}]{[}']{[}'']{[ FxKxCxM =++

This is a second order non homogenous ordinary differential equation where the

mass M, damping Cand stiffness Kmatrices are constant with time and the unknown

displacements x vary with time. To perform a modal analysis and obtain natural

frequencies and mode shapes the forcing function must equal to zero so the roots of the

characteristic equation can be obtained. For an undamped system not excited by external

forces the governing dynamic equation reduces to

}0{}]{[}'']{[ =+ xKxM

The system vibrates at some particular frequency and mode shape

)cos(}{}{ twxmm

=

Where,

m}{ is the mode shape

mw is the natural frequency

We get the first derivative which is velocity

)sin(}{}'{ twwxmmm

=

Then obtain the second derivative which is acceleration


16/39

7

)cos(}{}''{ 2 twwx mmm =

Replacing the displacement and acceleration into the undamped equation of motion we

get

( ) }0{}{][][2

=+mm KwM

The equation has two solutions. The first solution is trivial and undesired

}0{}{ =m

The other solution is

0][][ 2 =+ KwM m

This presents an Eigen Problem

}]{[}]{[ uIuA =

Where is the eigenvalue and }{u is the eigenvector

Going back to our solution and rewriting it as an eigen problem

( ) }0{}{][][ 2 =+mm

KwM

mmm wMK }{][}]{[2 =

mmm IwKM }]{[}]{[][21 =

This is similar to

}]{[}]{[ xIxA =

Therefore the natural frequencies are eigenvalues 2mw and mode shapes are eigenvectors

m}{ for given mass and stiffness matrices.

A possible engineering challenge is posed when a structure with symmetric

geometry appears. Cyclic symmetry is present in many mechanical and civil engineering

structures such as domes, cooling towers, milling cutters, gears, and gas turbine engines

to name a few. Such structures can be considered a domain composed of identical sub-

domains that have symmetry with respect to an axis. The sub-domain or sector createsthe whole domain by rotating the sub-domain by 2/J. J is the number of identical

sectors. Analysis of one of the sub-domains and its high degree of repetition is the key to

obtaining major savings in computation time.

Commercial software such as NASTRAN and ANSYS have been developed to

handle linear and nonlinear static and dynamic analyses. ANSYS has the capability to


17/39

8

perform modal analysis on any type of structure. It can extract natural frequencies and

mode shapes. It is also capable of performing modal analysis on a pre-stressed structure

to include stiffening and thermal effects from static loads. ANSYS modal cyclic

symmetry obtains natural frequency and mode shapes of a cyclically symmetric structure

by modeling just one of its sectors. ANSYS modal analysis is a linear analysis. Any non-

linearities, such as plasticity and contact (gap) elements are ignored even if they were

defined. ANSYS has several mode extraction methods: Block Lanczos, subspace, PCG

Lanczos, reduced, unsymmetric, damped, and QR damped [1]. Those damped methods

allow for damping to be included in the structure. Block Lanczos will be used to analyze

the models in this paper. The Block Lanczos method is used for large symmetric

eigenvalue problems. It achieves higher convergence rate than the subspace method and

it uses the sparse matrix solver.

The general process for modal analysis consists of five steps: Build the FE model,

apply the loads and obtain the solution, expand the modes and review the results. In this

paper we will investigate two methods of modeling a rotating gas turbine stage. The first

will be referred to as the full method and will include a full wheel with 24 blades

representing the actual structure. The second method will be the modal cyclic symmetry

which will utilize one sector i.e. one blade with a disk sector and use that reduced model

for modal analysis to predict the natural frequencies and mode shapes of the entire

wheel. We can define the structure in terms of a primary segment which is repeated at

equally spaced intervals about the symmetry axis. If the displacement boundary

conditions of all segments are identical with respect to the axis of symmetry, we can

analyze the entire structure in terms of the mass and stiffness characteristics of a single

segment. As mentioned earlier the primary advantage of cyclic symmetry is the large

savings in CPU/elapsed time and computer resources. The two methods are shown in

Figure 7.

By using the ANSYS modal cyclic symmetry capability we can obtain the natural

frequencies and mode shapes of the entire structure for a prescribed range of nodal

diameters using a single sector model. Cyclic symmetry is implemented in ANSYS by

defining constraint relationships between the high and low edges of the basic sector as


18/39

9

shown in Figure 3. The definition of the constraint equations depends on the harmonic

index specified.

Figure 3: Cyclic Symmetry Layout [1]

The relationship between harmonic index, n, and nodal diameter, m, for a model

consisting ofJsectors is given by the following equation: [1]

0,1,2,3...i; == nJim

For example, if there are 24 sectors (J= 24) and we specify k= 2, ANSYS will obtain

the solution for nodal diameters 2, 22, 26, 46, 50, 70, 74 and so on. The harmonic index

range is from 0 toJ/2 ([J-1]/2 ifj is odd). The full structure dynamic equation is

}{}]{[}']{[}'']{[ FxKxCxM =++

The Finite Fourier Series force expansion in complex exponential form is

=

=

P

m

maji

mj etftF0

)1()()(

And the deflection is

=

=

P

m

maji

mj etutU0

)1()()(


19/39

10

Where,

a: Sector AngleJ

a2

=

mf : Coefficient corresponding to thejth sector

1=i complex notation

Jj ,...2,1= ,Jis the number of substructures,j is sector number

Pm ,.....2,1,0= , m is the nodal diameter

P =J/2 for evenJand (J-1)/2 for oddJ

The cyclically symmetric problem is solved on a single substructure enforcing the

compatibility boundary conditions between the cyclic substructures. Two most

commonly used solution methods are the Complex Hermitian and the Duplicate sector

which is used by ANSYS for its fast performance. During the solution stage, ANSYS

automatically generates a duplicate sector of elements at the same geometric location of

the basic sector and applies all boundary conditions, loads, coupling and constraint

equations present on the basic sector to the duplicate sector. Modal analysis stores

information in complex form with real and imaginary components. This is done since

cyclic symmetry structures require phase information to be stored to determine

displacement, stress and strain fields when the sector model is expanded to the full 360.

Figure 4: Cyclic Symmetry Process [1]

The procedure for creating cyclic symmetry models is shown in Figure 4. First we

need to define the basic sector, to do so we obtain bladed-disk sector geometry and mesh


20/39

11

it. The model uses solid45 elements for simplicity. SOLID45 elements are used for 3-D

modeling of solid structures. The element is defined by eight nodes having three

translation degrees of freedom at each node as shown in Figure 5.

Figure 5: SOLID45 Element [1]

The mesh on the cyclic faces of the disk section must match, that can be

accomplished by using the sweep mesh function, if thats not possible the MSHCOPY can

be used to copy the mesh from one sector side to the other. ANSYS imposes cyclic

symmetry compatibility conditions for each nodal-diameter solution by the use of

couples or constraint equations connecting the nodes on the low and high-edge

components on the basic and duplicate sectors as shown in Figure 6.

Figure 6: Cyclic Symmetry Sector Edges [1]

The nodes at the edge components are related by the following equation.

=

B

Low

A

Low

B

High

A

High

U

U

ma

ma

ma

ma

U

U

cos

sin

sin

cos


21/39

12

Using the automated procedure, ANSYS detects the coordinate system that the

geometry was built in, it also detects the symmetry planes and creates components from

them. It configures the symmetry planes so that the nodes match up with one another,

this is necessary in generating the constraint equations that enforce cyclic symmetry.

ANSYS detects the sector angle, and calculates the number of sectors necessary to

complete the full 360. Once the CYCLIC command is executed, ANSYS will echo the

number of sectors needed to complete the full 360, the components created in defining

the symmetry plane, and whether the component pairs are matched or unmatched. In our

case, cyclic symmetry boundary conditions were created successfully as ANSYS output

indicated that sectors have matched as shown in Figure 7. Boundary conditions included

ALL DOF constraints (displacement constrained in x, y and z directions) at the inner

disk diameter to both models as shown in Figure 8 and Figure 9. The full model was

basically a sector model physically duplicated 24 times to create a full wheel.

Figure 7: Cyclic Symmetry and Full Model


22/39

13

Figure 8: Sector Model Boundary Conditions

Figure 9: Full Model Boundary Conditions


23/39

14

3. Discussion and ResultsANSYS is capable of post processing cyclic symmetry results. The ANSYS command

/CYCEXPAND controls the number of sectors to expand and the phase angle shift. The

response of any sector is represented by

( ){ } ( ){ } ++= majUmajUU BAj 1sin1cos

Where,

jU = Response of the full structure for sector numberj

AU = Basic sector solution

BU = Duplicate sector solution

j = Sector number

m = Nodal diameter

a = sector angle

= Phase angle

Result can be expanded to any angle and up to the full 360. This option usually

requires extra time and memory for large models. The results may be plotted at any

phase angle. Cyclic symmetry post-processing is somewhat complicated yet it can be

coded. A macro is provided in the appendix section that automates the cyclic symmetry

post-processing. The macro basically loops through every frequency substep and nodal

diameter loadstep, determines the max phase angle of deflection and plots the deflection

results.Looking at the results, it was important to note the sequence difference between the

cyclic symmetry model and the full model. As shown in Table 1, the cyclic symmetry

model shows all the frequencies as substeps and nodal diameters as loadsteps. The full

model solution shows its predicted frequencies as loadsteps. The full model loadsteps

basically vary by nodal diameter but this is not shown in the results summary because

its inherent in the structure. In other words, the load steps are arranged by a blade mode

family (same type of deflection) over different disk nodal diameters. In cyclic symmetry,

ANSYS expands the solution for a specific nodal diameter and mode shape. This is very

useful since cyclic symmetry arranges results so there are multiple blade modes (natural

frequencies) for every nodal diameter solved for. The full model results arrangement is


24/39

15

not versatile since it only shows one blade mode shape type over all the possible nodal

diameters making it complicated as far as result tracking and post processing. It also

does not allow phase angle plotting since the phase angle is a part of that mode shape

(loadstep) and its not a parameter than can be expanded upon like in cyclic symmetry.

Table 1: Results Summary Sequence

Figure 10, Figure 11 and Figure 12 compare the frequency and mode shape (EWB

Easy Wise Bending) between the two models for ND1, ND2 and ND3, respectively.

Cyclic symmetry model mode shapes were expanded over the full wheel structure. All

the cyclic symmetry plots show the harmonic index (Nodal Diameter) value. Overall,

both cyclic symmetry and the full model frequencies and mode shapes were very similar.

Table 2 compares a sample of the frequency results. The highest difference was about

1.94% at ND0 while the difference diminished to 0% beyond ND4. Mode shapes for

ND1, ND2 and ND3 are shown in Figure 10, Figure 11 and Figure 12, respectively.

Table 2: Full Model vs. Cyclic Symmetry Frequency Results


25/39

16

Figure 10: Mode1 Easy Wise Bending ND1


26/39

17



27/39

18



28/39

19

The blue colored contours across the disk in the mode shapes plots in Figure 10,

Figure 11 and Figure 12 represent nodal diameters. As explained earlier, nodal diameters

are lines of zero out-of-plane displacement which cross the entire disk. The

displacement, for example, of the tip of each blade when plotted with angular position,

shows a sinusoidal characteristic over the circumference where the peaks are the out of

plane blade motion. There can be as few as zero nodal diameters where the entire blade

set is in phase with one another. The maximum number of nodal diameters is onehalf

the number of blades, in this situation every blade moves out of phase with its neighbor.

For our case, the disk contains 24 blades, so the maximum number of nodal diameters is

twelve. The nodal diameter mode shape ND1 is illustrated in Figure 10. There are two

phase changes in the bladed disk which are noted by the radial lines forming the one

nodal diameter mode shape. The same observation can be applied to the 2nd and 3rd

nodal diameter plots shown in Figure 11 and Figure 12, respectively. Figure 13 shows

both a zero and twelve nodal diameter mode shapes. Sometimes the disk is not

participating and the vibration is only at the blade this is typically known as a blade

alone mode mainly due to a stiff disk. Still blades are affected by the presence of nodal

diameters. As shown in Figure 13 and Figure 14 the disk has almost zero deflection and

the blades are out of phase. In Figure 14 multiple nodal diameters are shown for the

Torsion mode, it is clear that nodal diameters affect the blade deflection. At ND1 and

ND3 plots in Figure 14 there is almost stagnant or very small blade vibration where a

node line is crossing.


29/39

20

Figure 13: Mode1 Easy Wise Bending ND0 vs. ND12

Figure 14: Torsion Mode variation with nodal diameters


30/39

21

More than 50 years ago, Dr. Campbell working in the Rotor Dynamics Laboratory

of the General Electric Steam Turbine facility in Schenectady, New York, was looking

for a way to present information about turbine blade resonances during startups of large

steam turbines. He needed to evaluate the performance of a rotating turbine blade during

accelerations and decelerations. Campbell diagrams [3] are used to illustrate the

interference between natural frequencies and common exciting forces described earlier.

In the case of a gas turbine bladed disk natural frequencies are plotted against rotor

running speed. Diagonal lines represent the sources of excitation or engine orders such

as vane or strut counts. When the diagonal line crosses a bladed disk natural frequency

within the turbine speed range a resonance is said to occur.

Figure 15 shows a Campbell diagram of actual modes and possible engine order

drivers. The mode frequency drop is typically due to thermal affects, as speed increases

the temperature increase and elastic modulus decrease thereby dropping natural

frequency of the part. Depending on the speed, the frequency drop is usually within 15%

of the value at zero speed or room temperature. As stated earlier, drivers can be actual

static structures such as struts or vane passes or they can be running speed harmonics.

Whenever there is a crossing it represents possible resonance as shown by the blue

circles in Figure 15. During startups and shutdowns these modes would get excited but

the time spent at resonance is key. If it is passing through, then it has a small effect and

does not pose any concern but if there is a long time to be spent at a specific speed (like

during cruising of aircraft) then, care must be taken to avoid resonance at long dwells.


31/39

22

Figure 15: Campbell Diagram


32/39

23

4. ConclusionIn this paper we introduced the theory and the general process of analyzing the dynamic

characteristics of gas turbine rotating components. It is very important to design

turbomachinery to assure trouble free operation since failure can expose lives to danger.One of the engineer/designer main tasks is to design components that avoid deadly

resonance at operational speeds. In depth structural analysis using finite elements is

required to address the complex nature of the designs and to troubleshoot field problems.

In this study we looked at two methods on analyzing a turbine bladed disk using ANSYS

Finite element code. The first method was the full turbine wheel of 24 blades and the

other method used one sector by utilizing cyclic symmetry. Cyclic symmetry has many

advantages and its primary advantage is the large savings in CPU time and computer

resources. Its important to note that our cyclic symmetry model was only 1572 nodes,

while the full model was about 17,280 nodes, thats about 11x model size and yet cyclic

symmetry provided results with the same accuracy. Cyclic symmetry can be more

powerful when used to model actual hardware like a turbine airfoil or an integrally

bladed rotor that can potentially reach and exceed one million elements for a single

sector. In that case modeling a full wheel is not an option due to limited resources and

cyclic symmetry is the method of choice since it will obtain results with the same

accuracy at a fraction of time and resources. An important condition for the successful

application of finite element models is the validation and calibration. Although

validation details are not covered in this paper it typically includes lab testing, spin rigs

and full engine testing to validate designs at actual running condition. FEA can be most

valuable in situations where there is an extensive history of modeling, testing, and field

experience. Having a valid analysis approach available will likely save several design

iterations and produce a more robust blade design in the end.


33/39

24

5. References[1] ANSYS 11.0 Theory Reference, ANSYS Corporation, 2007

[2] T. Tomioka, Y. Kobayashi and G. Yamada Analysis of free vibration of rotating

disk-blade coupled systems by using artificial springs and orthogonal polynomialsJournal of Sound and Vibration " 191(1), 53-73, 1996

[3] J. Hou and B. Wicks Root Flexibility and Untwist Effects on Vibration

Characteristics of a Gas Turbine Blade Air Vehicles Division Platforms SciencesLaboratory DSTO-RR-0250, 2002

[4] William J. Palm Mechanical Vibration Wiley ISBN 0-471-34555-5, 2004

[5] Jerry H. Griffin "Unstable Resonant Response of Shrouded Bladed Disks,

Proceedings of the Third National Turbine Engine High Cycle Fatigue Conference,1998

[6] M. Singh, J. Vargo, D. Schiffer and J. Dello, Safe Diagram A Design andReliability Tool for Turbine Blading, Dresser-Rand Company, 2002

[7] John M. Vance Rotordynamics of Turbomachinary Wiley ISBN-13:9780471802587, 1988


34/39

25

6. Appendix: ANSYS Macros!**********************************************************************

!This Macro is for meshing the model and creating the cyclic boundary

!conditions setting up the solution parameter and the number of nodal

!diameters to solve for

!This macro requires the geometry file cyc_sym_model.iges to be in the

working directory

!**********************************************************************

FINISH

/CLEAR,START

!Allows the user to use ansys automated process or define it manually

(for more complicated structures)

auto_process = 0 !Automated cyc symmetry- 1, Manual cyc symmetry- 0

!Read IGES Geometry

/AUX15

!*

IOPTN,IGES,NODEFEAT

IOPTN,MERGE,YES

IOPTN,SOLID,YES

IOPTN,SMALL,YES

IOPTN,GTOLER, DEFA

IGESIN,'cyc_sym_model','iges',' '

VPLOT

!*

!Define Solid45 Element

/prep7

ET,1,SOLID45

!Define Material Properties

MP,DENS,1,0.000777

MP,EX,1,3e7

MP,PRXY,1,.3

!Meshing

ESIZE,0.1,0,

FLST,5,4,6,ORDE,2

FITEM,5,1

FITEM,5,-4


35/39

26

CM,_Y,VOLU

VSEL, , , ,P51X

CM,_Y1,VOLU

CHKMSH,'VOLU'

CMSEL,S,_Y

!*!*

VCLEAR,_Y1

MSHAPE,0,3d

VMESH,_Y1

!*

CMDELE,_Y

CMDELE,_Y1

CMDELE,_Y2

!*

!Apply Boundary Conditionsallsel

ASEL,S, , , 14

nsla,s,1

csys,0

NSEL,r,LOC,Y,-.32,.32

d,all,all

allsel

!Couple blade and disk

asel,s,,,7

asel,a,,,16

nsla,s,1

CPINTF,ALL,0.0001,

allsel

*if,auto_process,eq,1,then

CYCLIC, , , ,'CYCLIC'

!CPCYC,ALL,,1,,360/24

*else

!cyclic symmetry components

asel,s,,,22

asel,a,,,6

asel,a,,,17

CM,cyclic_m01h,AREA

asel,s,,,20

asel,a,,,4


36/39

27

asel,a,,,15

CM,cyclic_m01l,AREA

!asel,s,,,6

!asel,r,,,17

!CM,cyclic_m02h,AREA!asel,s,,,4

!asel,s,,,15

!CM,cyclic_m-2l,AREA

!CYCLIC, , , ,'CYCLIC'

csys,1

cyclic,24,360/24,1,cyclic,1

!CPCYC,ALL,,1,,360/24

*endif!Solution Controls

CYCOPT,HINDEX,0,12,,, !Defines 12 Nodal Diamters to solve for

/SOL

ANTYPE,2

MODOPT,LANB,10 !Will obtain the first 10 modes of the structures

EQSLV,SPAR

MXPAND,10, , ,1

LUMPM,0

PSTRES,0

!*

SAVE

Solve


37/39

28

!**********************************************************************

! This Macro to Automate Post-Processing it works for both full and

! cyclic symmetry models

!**********************************************************************

/post1

/WIND,ALL,OFF/WIND,1,LEFT

/WIND,2,RIGHT

GPLOT

EPLOT

/VIEW,1,,,-1

/ANG,1

/AUTO,1

/REP,FAST

!/VIEW,2,-1

!/ANG,2!/ANG,2,90,ZS,1

/VIEW,2,-1

/ANG,2

/AUTO,2

/REP,FAST

/FOC, 2, 1.02420727007 , -0.131829818902 ,

0.177885996090

/VIEW, 2, -0.706832267489 , -0.201670178251E-01, -

0.707093655061

/ANG, 2, 0.940571321353

set,last

*get,nlstp,active,,set,lstp

*get,nsub,active,,set,sbst

/cycexpand,1,amount,nrepeat,360


*do,k,1,nlstp,1

set,k,1

*get,freq_1,active,0,set,freq

set,k,2

*get,freq_2,active,0,set,freq

cycno2_key=0

test_freq=abs(freq_1-freq_2)

*if,freq_1,eq,freq_2,or,test_freq,le,0.1,then

cycno2_key=1

*endif

*if,cycno2_key,eq,1,then

*do,i,1,nsub,2


38/39

29

set,k,i

/graph,power

/PLOPTS,INFO,auto

/PLOPTS,LEG1,1

/PLOPTS,LEG2,0/PLOPTS,LEG3,1

/PLOPTS,FRAME,1

/PLOPTS,TITLE,1

/PLOPTS,MINM,1

/PLOPTS,LOGO,0

/PLOPTS,WINS,1

/PLOPTS,WP,0

/TRIAD,LBOT

/RGB,INDEX,100,100,100, 0

/RGB,INDEX, 80, 80, 80,13/RGB,INDEX, 60, 60, 60,14

/RGB,INDEX, 0, 0, 0,15

/SHOW,JPEG


cycphase,disp,1

cycphase,get,u,sum,max

/cycexpand,,phaseang,_cycphase

PLNSOL, U,SUM, 2,1.0

*enddo

*else

*do,i,1,nsub,1

set,k,i

/graph,power

/PLOPTS,INFO,auto

/PLOPTS,LEG1,1

/PLOPTS,LEG2,0

/PLOPTS,LEG3,1

/PLOPTS,FRAME,1

/PLOPTS,TITLE,1

/PLOPTS,MINM,1

/PLOPTS,LOGO,0

/PLOPTS,WINS,1

/PLOPTS,WP,0

/TRIAD,LBOT

/RGB,INDEX,100,100,100, 0

/RGB,INDEX, 80, 80, 80,13

/RGB,INDEX, 60, 60, 60,14


39/39

/RGB,INDEX, 0, 0, 0,15

/SHOW,JPEG


cycphase,disp,1

cycphase,get,u,sum,max

/cycexpand,,phaseang,_cycphasePLNSOL, U,SUM, 2,1.0

*enddo

*endif

*enddo

/show,close

/show,term

hassan final report

Documents