Turbomachinery Blade and Stage Vibrations
by
Patrick McComb
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING
Major Subject: MECHANICAL ENGINEERING
Approved:
_________________________________________
Professor Ernesto Guitierrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, CT
December, 2013
i
© Copyright 2013
by
Patrick McComb
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
LIST OF SYMBOLS ...................................................................................................... viii
ACKNOWLEDGMENT .................................................................................................. ix
ABSTRACT ...................................................................................................................... x
1. Introduction and Background ...................................................................................... 1
1.1
Gas turbine Engines ........................................................................................... 1
1.2
Blading ............................................................................................................... 2
1.3
Free Vibration .................................................................................................... 3
1.4
Forced Vibration ................................................................................................ 5
1.5
High Cycle Fatigue (HCF) ................................................................................. 8
1.6
Clipping .............................................................................................................. 9
2. Modeling and Methodology ...................................................................................... 10
2.1
Finite Element Analysis ................................................................................... 10
2.2
Finite Element Model ....................................................................................... 11
2.3
Boundary Conditions ....................................................................................... 13
2.4
2.3.1
Blade Alone.......................................................................................... 13
2.3.2
Fixed Sector Boundaries ...................................................................... 14
2.3.3
Cyclic Symmetry.................................................................................. 15
Solution ............................................................................................................ 17
2.4.1
Free vibration ....................................................................................... 17
2.4.2
Forced Vibration .................................................................................. 17
3. Results and Discussion .............................................................................................. 21
3.1
Free Vibration .................................................................................................. 21
3.1.1
Blade Alone.......................................................................................... 21
3.1.2
Fixed Sector ......................................................................................... 22
iii
3.2
3.1.3
Cyclic Symmetry.................................................................................. 23
3.1.4
Frequency comparison ......................................................................... 27
Forced Vibration .............................................................................................. 29
3.2.1
Frequency and Modeshape comparison ............................................... 29
3.2.2
Post processing ..................................................................................... 31
3.2.3
Mode 1 ................................................................................................. 33
3.2.4
Mode 2 ................................................................................................. 37
3.2.5
Mode 4 ................................................................................................. 41
4. Conclusions................................................................................................................ 46
5. References.................................................................................................................. 48
6. Appendix A: Matlab .................................................................................................. 49
7. Appendix B: ANSYS files ......................................................................................... 50
8. Appendix C: Cyclic Symmetry Modeshapes ............................................................. 53
iv
LIST OF TABLES
Table 1: Varying Boundary Condition Frequency Comparison ...................................... 27
Table 2. Clipped Blades Frequency Comparison ............................................................ 30
Table 3. Frequency, Deflection and Phase Angle ............................................................ 32
Table 4. Blade Amplitude Summary ............................................................................... 47
v
LIST OF FIGURES
Figure 1. Simple Brayton Cycle ........................................................................................ 2
Figure 2. Gas turbine engine compressor and turbine ....................................................... 3
Figure 3. Magnitude of Receptance ................................................................................... 7
Figure 4. Phase Shift of Receptance .................................................................................. 7
Figure 5. Blade Geomtry and Break-up ........................................................................... 11
Figure 6. ANSYS Solid45 elements ................................................................................ 12
Figure 7. Rotor and Blade mesh ...................................................................................... 13
Figure 8. Blade Alone Boundary Conditions .................................................................. 14
Figure 9. Fixed Sector Boundary Condititons ................................................................. 15
Figure 10. Cyclic Symmetry Sector Nodes ..................................................................... 16
Figure 11. Cyclic Symmetry Boundary Conditions ........................................................ 16
Figure 12. Forced Response Loading .............................................................................. 18
Figure 13. Baseline Blade ................................................................................................ 19
Figure 14. Clipped Blade 1 .............................................................................................. 19
Figure 15. Clipped Blade 2 .............................................................................................. 20
Figure 16. Blade Alone Modeshapes 1-9 ........................................................................ 21
Figure 17. Fixed Sector Modeshapes 1-9 ........................................................................ 23
Figure 18. 1st Bending and 1st Torsion @ ND 0 .............................................................. 24
Figure 19. 1st Bending and 1st Torsion @ ND 12 ............................................................ 24
Figure 20. Mode 2 @ ND 2 and ND 12........................................................................... 25
Figure 21. Mode 1-9 Nodal Diamter Map ....................................................................... 26
Figure 22. Cyclic Symmetry Modeshapes 1-9 ................................................................ 27
Figure 23. Modes 1-9 Campbell Diagram ....................................................................... 29
Figure 24. Baseline and clipped Modes 1, 2, 4 ................................................................ 31
Figure 25. Resonant Response Amplitudes ..................................................................... 33
Figure 26. Baseline Blade Mode 1 Forced Responses .................................................... 34
Figure 27. Clipped Blade 1 Mode 1 Forced Response .................................................... 35
Figure 28. Clipped Blade 2 Mode 1 Forced Response .................................................... 36
Figure 29. Mode 1 Deflection Comparison ..................................................................... 37
Figure 30. Clipped Blade 1 Mode 2 Forced Responses................................................... 38
vi
Figure 31. Clipped Blade 1 Mode 2 Forced Response .................................................... 39
Figure 32. Clipped Blade 1 Mode 2 Forced Response .................................................... 40
Figure 33. Mode 2 Deflection Comparison ..................................................................... 41
Figure 34. Clipped Blade 2 Mode 4 Forced Responses................................................... 42
Figure 35. Clipped Blade 2 Mode 4 Forced Response .................................................... 43
Figure 36. Clipped Blade 2 Mode 4 Forced Response .................................................... 44
Figure 37. Mode 4 Deflection Comparison ..................................................................... 45
vii
LIST OF SYMBOLS
m = mass
c = damping coefficient
k = stiffness coefficient
ẍ = acceleration
ẋ = velocity
x = displacement
F = force
ω = forcing frequency
ωn = natural frequency
ωd = damped natural frequency
ζ = damping ratio
α(ω) = receptance
r = frequency ratio
As = real coefficient
Bs = imaginary coefficient
Cs = magnitude
φs = phase angle
xs(t) = steady state response
[M] = mass matrix
[C] = damping matrix
[K] = stiffness matrix
{ẍ} = acceleration vector
{ẋ} = velocity vector
{x} = displacement vector
{F} = force vector
ND = Nodal Diameter
viii
ACKNOWLEDGMENT
I would like to thank my family and friends, especially my wife Kate for her motivation
and support through my degree and this paper. I would also like to thank Professor
Gutierrez-Miravete for his knowledge and guidance completing this paper.
ix
ABSTRACT
Blades or integrally bladed rotors in gas turbine engines are subject to dynamic
loading and vibrations. Vibration analyses provide a set of information that is valuable in
the design of blades, and this data is used to determine overall durability of blade
designs. Free vibration analysis of blades is important in identifying frequencies of
vibration and critical speed ranges. The free vibration analysis can be performed with
different levels of complexity, and simplified loading can simulate more complex
situations. Forced vibration analysis builds upon free vibrations by analyzing the
displacement or corresponding stress in a blade under a given cyclic load. The simple
forced response is important in trying to understand the dynamic response due to
complex pressure loading within a gas turbine machine.
x
1. Introduction and Background
1.1 Gas turbine Engines
The jet age began in the late 1930’s when Hans von Ohain and Frank Whittle
developed their first ideas for what has now become the modern jet engine. Since then,
the design of jet engines has evolved in size and complexity; however, one aspect that
has remained consistent from the onset is the use of rotating blades or turbomachinery.
In most modern day jet engines as well as other land based gas turbine engines,
understanding the complex mechanical and aerodynamic interaction of turbomachine
blading has become paramount to the success of engine design and durability.
Modern day axial flow engines typically consist of an inlet, a fan/compressor
section, a combustion section, a turbine section and an exit nozzle. Both the compressor
section and the turbine section typically consist of several rows, or stages, of blades that
vary in size and count from front to back and the two sections are connected by a shaft.
Each blade stage is followed by a row of vanes to make up the full stage. The stages of
compressor blades suck air into the engine and compress the air, increasing its pressure
and temperature for high energy combustion. After the combustion the high energy gas
expands through the stages of turbine blades, which turns the shaft and provides the
power for operability of the upstream compressor. This process is known as a Brayton
cycle engine, seen in Figure 1, and would not be possible for large axial flow machines
without the use of many compressor and turbine blades.
1
Figure 1. Simple Brayton Cycle [1]
1.2 Blading
Both compressors and turbines typically consist of several stages of blades that
are preliminarily designed for optimal aerodynamics, ignoring design and structural
constraints. Depending on the application optimal aerodynamics can imply certain levels
of efficiency for given thrust and operability requirements. Compressor blades decrease
in size from the entry stage to the exit to provide constant mass flow as the air density
increases due to compression. Turbine blades increase in size from entry to exit for the
same reason as the gas expands. In both cases, each stage consists of a set number of
blades that are distributed around a rotor disk which is connected to the rotating shaft.
Each blade is initially designed to input or extract a set amount of work at an ideal
aerodynamic efficiency for optimal operation. Figure 2 shows a large multi-stage
compressor connected by a shaft to a smaller multi-stage turbine section.
2
Figure 2. Gas turbine engine compressor and turbine [2]
However, just because a blade is ideal for aerodynamic operation does not mean
it is appropriate for the use in a gas turbine engine, especially in a jet engine. The blades
in jet engines are subject to extreme loads, many of which can cause vibrations that must
be tolerated throughout the operation of the engine. For jet engines, operational safety is
extremely important as in many cases many human lives are at stake. It is very important
to understand both the cause of the vibrations and the response of the blades to different
dynamic loadings.
1.3 Free Vibration
One of the most important aspects in the design of turbomachine blades is
understanding the dynamics of blade vibration. In reality a blade is a complex system
and its dynamics can be represented as a multi-degree of freedom system, where [M] is
the mass, [C] is the damping, and [K] is the stiffness.
[𝑀]{𝑥̈ } + [𝐶]{𝑥̇ } + [𝐾]{𝑥} = 𝐹
3
When used in the Finite Element Method, [M] represents the mass matrix created for
each discrete element. Similarly, [C] represents the damping matrix and [K] represents
the stiffness matrix. To understand the basics of free vibration a single degree of
freedom system is considered first. In this case the differential equation of motion is
𝑚𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 𝐹
Here m is mass, c is the damping coefficient and k is the stiffness (coefficient of
elasticity). The first step in understanding the vibration of a system is to understand the
free undamped vibration, or vibration due to initial disturbance of the system. The
equation of motion simplifies to
[𝑀]{𝑥̈ } + [𝐾]{𝑥} = 0
And the corresponding single degree of freedom equation simplifies to:
𝑚𝑥̈ + 𝑘𝑥 = 0
The equation of motion can be solved as an eigenvalue problem where the
eigenvectors are the modeshapes or displacements of vibration and the eigenvalues are
the natural frequencies of the blade. In many cases it is important to evaluate the free
vibrations unloaded, and loaded with certain forces experienced in turbomachinery such
as centrifugal loading and the steady pressure loading of the gas passing over the blades.
This free vibration analysis is used with other design aspects of a machine to determine
potential areas where rotational speed and forcing drivers can create vibrations which the
blade material cannot tolerate.
4
1.4 Forced Vibration
A free vibration analysis can identify potential regions of vibrational concern;
however, it only produces an eigenvalue solution that does not let one predict the
amplitude of a vibration due to cyclic loading, like that experienced in a jet engine. In jet
engines there are complex disturbances that can be approximately simulated by applying
a simple load cyclically as described by the equation below for a single degree of
freedom system.
𝑚𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 𝐹𝑐𝑜𝑠𝜔𝑡
In this equation F is the magnitude of the force and ω is the frequency of the force being
applied to the system. This equation can then be rearranged by dividing through by the
mass to:
𝑥̈ + 2𝜁𝜔𝑛 𝑥̇ + 𝜔𝑛 2 𝑥 =
𝐹
𝑐𝑜𝑠𝜔𝑡
𝑚
Where:
𝑐
= 2𝜁𝜔𝑛
𝑚
𝑘
= 𝜔𝑛 2
𝑚
𝜁 = √1 − (
𝜔𝑑 2
)
𝜔𝑛
The transient response of this system can be described with the solution:
𝑥𝑠 (𝑡) = 𝛼(𝜔)𝐹𝑒 𝚥𝜔𝑡 = 𝐹(𝐴𝑠 𝑐𝑜𝑠𝜔𝑡 + 𝐵𝑠 𝑠𝑖𝑛𝚥𝜔𝑡)
The response is harmonic with the frequency of the force and described in real and
imaginary terms. In the above equation α(ω) is known as the receptance, which is a
function of the frequency response. The receptance can be calculated using the equation:
5
1
1
2
𝑚𝜔
𝑚
𝑛
𝛼(𝜔) =
=
−𝜔 2 + 2𝜁𝜔𝑛 𝚥𝜔 + 𝜔𝑛 2
1 − 𝑟 2 + 2𝜁𝚥𝑟
Where r is the ratio of the forcing frequency to the natural frequency.
𝑟=
𝜔
𝜔𝑛
More practically the receptance can be described in terms of the coefficients As and Bs
as:
1 − 𝑟2
𝑚𝜔𝑛 2
𝐴𝑠 =
(1 − 𝑟 2 + (2𝜁𝑟)2
2𝜁𝑟
𝑚𝜔𝑛 2
𝐵𝑠 =
(1 − 𝑟 2 + (2𝜁𝑟)2
The real and imaginary solution can be described in the form:
𝑥𝑠 (𝑡) = 𝐹𝐶𝑠 cos⁡(𝜔𝑡 − 𝜑𝑠 )
In this description Cs is the magnitude and φs is the phase shift of the solution and are
calculated as:
𝐶𝑠 = √𝐴𝑠 2 + 𝐵𝑠 2 =
1
𝑚𝜔𝑛 2
√(1 − 𝑟 2 + (2𝜁𝑟)2
2𝜁𝑟
1 − 𝑟2
The magnitude and phase shift of the solution can be plotted versus the frequency ratio r.
𝜑𝑠 = 𝑡𝑎𝑛−1
The shape of the response is dependent on the damping ratio as seen in Figures 3 and 4
plotted in Matlab (Appendix A).
6
Figure 3. Magnitude of Receptance
Figure 4. Phase Shift of Receptance
The forced response solution has a peak amplitude when:
𝜔 = 𝜔𝑛 √1 − 2𝜁 2
The forced response solution at peak amplitude is known as resonance. Blades in jet
engines are subject to resonant responses resulting from cyclic loading due to pressure
loading of upstream and downstream disturbances. These disturbances include speed
harmonics, vane and rotor pass frequencies, and other aerodynamic and acoustic
disturbances. For example, if an upstream stage has 40 vanes, as a blade rotates 1 time
around the rotor it is subject to a similar pressure disturbance 40 times, every time it
passes by the wake of each vane. The rotating blade will vibrate at a natural frequency
7
harmonic relative to rotor speed and vane count. It is extremely important in the design
of turbomachine blading to understand the frequency and amplitude of responses due to
different cyclic loadings.
1.5 High Cycle Fatigue (HCF)
It is important to understand the cyclic forced vibrations in jet engines in order to
avoid failures due to high cycle fatigue. Fatigue is the repeated loading of a structure that
over time can cause a failure. As a blade vibrates, the fatigue cycles on a blade add up
and eventually cross a threshold dependent on blade material properties that cause a
blade to crack and eventually fail. The best way to avoid high cycle fatigue is to avoid
resonances; however, in application of real life machines, it is impossible to avoid all
resonances in the range that the machine must operate. Typically, resonances are
avoided in speed ranges where an engine may operate most often. As a result, forced
vibration analysis becomes very important in the design of blades at off design
conditions, where the engine may not operate most of the time. This analysis helps to
understand the frequencies and amplitudes of vibration when these resonances occur.
The blade predicted displacements correspond to a relative stress, which can be
compared to material capability to understand the robustness of a design. The robustness
of the design can vary, but typically blades are designed to stay below a certain stress
value as a result of resonance to assure that a fatigue failure can never occur.
8
1.6 Clipping
Although forced vibration analysis is used in the design, the complex nature of the
loading can produce a prediction of resonant amplitude that does not match that
measured in a test environment. In this situation, forced vibration is valuable as a tool to
produce relative magnitude results dependent on geometry changes with the same load.
In some cases, blades can be altered in minor ways, sometimes known as clipping, to
change the natural frequency or the magnitude of the blade response (or both), to make a
design less susceptible to a fatigue failure.
9
2. Modeling and Methodology
2.1 Finite Element Analysis
Although simple hand calculations can be very effective in the preliminary
understanding of a blade design, with the modern computing power today, the best way
to perform vibration analyses is with the Finite Element Method. Finite element models
(FEMs) can be made to approximate blade geometries, and can then be loaded to
simulate real life operational conditions. Static and dynamic analyses are then used to
help understand the failure mechanism such as HCF.
Finite element analysis takes a real structure and approximates it by breaking it
up into a number of discrete elements with a finite number of degrees of freedom. In a
free vibration analysis the number degrees of freedom of a given model corresponds to
the number of natural frequency modes that can be extracted, which in most cases is
more than is necessary. The equations for the vibrating system with multiple degrees of
freedom neglecting damping as discussed earlier is:
[𝑀]{𝑥̈ } + [𝐾]{𝑥} = 0
In this scenario [M] and [K] are the mass and stiffness matrices respectively, which are
generated from the model elements and nodes. The {x} is a vector of displacements of
each degree of freedom of the system. If the solution is assumed to be harmonic then the
resulting solution is an eigenvalue problem in the form:
det([𝐾] − 𝜔2 [𝑀]) = 0
The solution of the eigenvalue problem includes both the eigenvalue (natural
frequencies) and the eigenvectors (mode shapes). The equation for the forced vibration
analysis with damping is:
10
[𝑀]{𝑥̈ } + [𝐶]{𝑥̇ } + [𝐾]{𝑥} = {𝐹}
Finite element analysis is used to determine the corresponding magnitudes of deflection
at a given frequency and phase angle.
2.2 Finite Element Model
Commercial modeling software, Unigraphics NX6 and ANSYS version 12.1,
were used to produce the geometry and finite element model for this analysis. The blade
model represents a cyclic symmetric sector of an integrally bladed rotor (IBR) typically
used in modern compressors. To produce the geometry a simple full rotor disk was
revolved 360 degrees. Then a simple blade was place onto the rotor disk and merged
together. Twenty-four blades were placed symmetrically around the disk. A 15 degree
sector was used to produce a symmetric sector. Finally, the rotor sector was broken up
strategically using the curvature of the blade to produce clean sweep meshable geometry.
The resulting geometry for this analysis can be seen in Figure 5.
Figure 5. Blade Geomtry and Break-up
11
The model was then meshed using ANSYS solid45 elements and default key options.
These elements are 3-D 8 noded brick/hexagonal elements, with three translational
degrees of freedom at each node seen in Figure 6.
Figure 6. ANSYS Solid45 elements
The best way to mesh cyclic symmetric geometry is to sweep the mesh through the
rotation. This will assure that the mesh on each cyclic edge of the rotor will have the
same mesh, which is important when running cyclic symmetry. The final meshing step is
to sweep mesh the blade. It is important to specify an appropriate mesh size for the
model to produce good frequency results, while minimizing run time. The model was
meshed using a general element edge length of .2”. The blade itself was meshed with 4
elements through the thickness in order to properly capture the bending stiffness using
solid elements. The fidelity of the mesh can be verified by refining the mesh until
frequencies remain constant and converged. The final blade mesh has 5256 elements and
6602 nodes and is shown in Figure 7.
12
Figure 7. Rotor and Blade mesh
The blade and rotor were assumed to be made from a titanium alloy (Ti 6-4) with a
modulus of elasticity of 16.5 MSI, Poisson’s ratio of .35, and a density of .160 lb/in3.
2.3 Boundary Conditions
2.3.1
Blade Alone
The simplest way to model a blade and rotor is to model just the blade. This
method is used in blade design optimization, when solution accuracy may not be as
critical as run time. This model only has 924 elements and 1345 nodes. Physically, the
blade alone solution tends to be stiffer and have higher frequencies since it essentially
assumes that the rotor is infinitely stiff. In this model, the blade nodes on the bottom of
the blade are constrained in all 3 translational degrees of freedom as shown in Figure 8.
13
Figure 8. Blade Alone Boundary Conditions
2.3.2
Fixed Sector Boundaries
Another simplified way to model the boundary conditions of the model is to fix
the sector boundary nodes in all degrees of freedom. In this scenario more of the
elasticity of the rotor structure is considered in the solution, however, the model will still
be ignoring some of elastic properties of the rotor. In the case where the rotor is very
stiff, the solution can typically be very similar to an N/2 cyclic symmetry solution
(where N is the number of blades in the rotor). The model with these BCs can be seen in
Figure 9.
14
Figure 9. Fixed Sector Boundary Condititons
2.3.3
Cyclic Symmetry
ANSYS has the capability to model a full 360 degree rotor by using cyclic symmetry
such that the full rotor does not have to be modeled. This reduces model complexity and
run time while, allowing for full interaction between the rotor and blade. To set up the
model for cyclic symmetry the two sector edges of the rotor have to have matching
meshes by either sweep meshing or using a mesh copy. ANSYS has an automated
procedure that detects the coordinate system of symmetry and the proper sector edges to
produce the appropriate constraint equations seen in Figure 10. This produces a full 360
degree rotor model in which all sectors are identical.
15
Figure 10. Cyclic Symmetry Sector Nodes
Several rows of nodes on the inside diameter of the rotor are fixed in all degrees of
freedom to simulate the rotor tied to an arbitrary shaft. It is important not to select nodes
on the cyclic symmetry boundary. These boundary conditions are shown in Figure 11.
Figure 11. Cyclic Symmetry Boundary Conditions
Unlike the first use of the simpler boundary conditions, the cyclic symmetry
solution provides analytical results for the interaction between the rotor vibration and the
blade vibration. The blade frequencies are solved corresponding to a particular disk
mode or nodal diameter (ND). Different rotor nodal diameters cause the blade mode
shapes and frequencies to change. Depending on the purpose of the analysis it is
16
important to track the blade frequencies and modes for each nodal diameter from ND 0
to ND N/2, N being the number of blades. For example, a stiff wise bending blade mode
may couple more with the vibration of the disk, and may be a resonance to avoid.
2.4 Solution
2.4.1
Free vibration
ANSYS has a variety of methods to solve modal analysis. The model in this project was
solved using the Block Lanczos mode extraction method, which is efficient at solving
large eigenvalue problems using the sparse matrix solver. Linear or non-linear prestressed static analysis can be solved before the modal solution to capture the effects of
stress stiffening, spin softening, and large displacements. Also, thermal affects can be
included in the model using temperature dependent material properties and thermal
mapping. Each step of increased complexity increases run time. As a result, the solutions
presented here ignore both thermal and pre-stress effects on the rotor and blade system.
2.4.2
Forced Vibration
The forced vibration solution uses the frequency and displacement results from the linear
free vibration analysis. The fixed sector boundary conditions were selected above to
perform this analysis for simplicity. It will be demonstrated that the rotor modeled is
significantly stiff enough to assume similar modal results to the cyclic symmetry N/2
analysis. An arbitrary 10 lb force is applied at the leading edge tip of the blade normal to
the blade surface, which is used as a simple harmonic force in the analysis. This is a
simple loading case; a more complex multi-point load, or pressure load could be used in
17
a real application to more accurately simulate the loading. Figure 12 shows the forced
response model with the applied force.
Figure 12. Forced Response Loading
ANSYS uses a harmonic analysis to solve the forced vibration problem. The mode
superposition solution method was used because it is the fastest of the methods and is
recommended by ANSYS. The sparse solver was also used in the solution and the output
format of the solution is real and imaginary. The damping ratio for the harmonic solution
was assumed to be .01, which is a standard value for metals, but could be varied. The
harmonic solution can be solved for a set number of sub-steps over a range of
frequencies. The frequency range can sweep and capture a large number of resonances,
or the range can be narrowed to focus on a known resonant frequency. A control file for
performing this analysis is found in Appendix B.
Three models were used to complete the forced response investigation. All three
used the same rotor model, but each had a different blade. The second and third blade
models are the same as the baseline blade except with small clips of the trailing edge
blade tips. Clipped blade 1 has material removed .058” chord wise from the trailing edge
18
and 1” down the span. Clipped blade 2 has material removed .116” chord wise from the
trailing edge and 1” down the span. The blade models can be seen in Figured 13-15. The
blade response of these three models will be compared using the forced response
analysis.
Figure 13. Baseline Blade
.058” x 1” clip
Figure 14. Clipped Blade 1
19
.116” x 1” clip
Figure 15. Clipped Blade 2
20
3. Results and Discussion
3.1 Free Vibration
The free vibration mode shapes and frequencies will be presented below for each set of
boundary conditions. All of the results will be compared and summarized.
3.1.1
Blade Alone
The simplest solution results are from the model with blade alone boundary
conditions. The first nine modes were extracted within the first 20,000 hertz. The nodal
solution combined X, Y, and Z deflections (USUM) were used to plot the modeshapes
seen in Figure 16. Corresponding frequencies are summarized in Table 1 on page 24.
Figure 16. Blade Alone Modeshapes 1-9
21
The first two modes (M1 and M2) are first bending and first torsion respectively, with
mode shape complexity typically increasing with each mode. Identify blade modes can
be more difficult as a blade shape gets more complex; however, for this simple model
M5 is second torsion and M9 is third torsion, while M3 is second bending and M6 is
third bending. M4 and M8 have a stiff wise bending component.
3.1.2
Fixed Sector
The first nine modes within the first 20,000 hertz were also extracted for the fixed
rotor sector boundary conditions. The nodal solution USUM (X,Y,Z) deflections were
used to plot the modeshapes seen in Figure 17. Corresponding frequencies are
summarized in Table 1 on page 24.
22
Figure 17. Fixed Sector Modeshapes 1-9
The first 9 mode shapes with the fixed sector boundary conditions are the same as the
blade alone, except modes 7 and 8 have flipped in order. Also, as expected, the addition
of the rotor sector has decreased the natural frequencies of each mode. For example,
blade alone M1 is ~1374 Hz, while the fixed sector M1 is ~1264 Hz.
3.1.3
Cyclic Symmetry
Cyclic symmetry provides rotor and blade vibration combined solutions. The
/CYCEXPAND command can be issued to expand the number of sectors around the
rotor from 1 to a full 360 degree rotor. If the rotor is expanded to a full 360 degrees the
full rotor modeshape can be viewed, as well as the interaction with the blades. The
23
analysis was run for every nodal diameter from 0 to N/2, which is ND 12 for this system.
Figure 18 shows the blade first bending mode and first torsion mode respectively at ND
0 with all of the blades vibrating in phase.
Figure 18. 1st Bending and 1st Torsion @ ND 0
Figure 19 shows the blade first bending (1B) mode and first torsion (1T) mode
respectively at ND 12 with all of the blades vibrating out of phase.
Figure 19. 1st Bending and 1st Torsion @ ND 12
Even though the blade mode shapes are similar for ND 0 and ND 12 the system
vibration and frequency can be different. In some cases the disk is stiff enough and blade
frequencies only have small variations from the blade alone simulation, but certain blade
24
and rotor modes can interact to dramatically change the system vibration. Figure 20
shows the mode shape for mode 2 at nodal diameters 2 and 12.
Figure 20. Mode 2 @ ND 2 and ND 12
The modeshape at ND 12 is 1T, but at ND 2 the mode shape has changed to a stiffwise
bending mode driven by the rotor, indicated by the displacement seen in the rotor. The
M2 ND2 image shows the lighter blue shades into the rotor section, implying a larger
amplitude of displacement and more vibration interaction. The blade and disk interaction
has changed the modeshape. Also, not only is the mode shape affected, but the frequency
will vary for each nodal diameter. A good way to track mode sensitivity to nodal
diameter is to plot nodal diameter versus frequency for each mode, sometimes referred
to as a nodal diameter map. Figure 21 shows a nodal diameter map for ND0 to N/2 for
the first 9 modes, each colored symbol representing a different mode.
25
Figure 21. Mode 1-9 Nodal Diamter Map
This chart shows how the lower ND can have a dramatic effect on the system
fundamental frequencies. This can be extremely important in understanding certain
resonant crossing in the engine. One can also observe that as the higher nodal diameters
approaching ND 12 are less influenced by the rotor vibration and the ND 12 mode
shapes and frequencies can be approximated by the fixed sector boundary conditions. To
plot mode shapes for a single sector, a phase sweep is performed to find the phase angle
with the peak vibration amplitude. Then each modal deflection is plotted at that phase
angle. Figure 22 shows the first 9 modes up to 20,000 Hz for nodal diameter 12. These
mode shapes are comparable to those for the fixed sector in Figure 17. The mode shapes
for the other nodal diameters can be found in Appendix C.
26
Figure 22. Cyclic Symmetry Modeshapes 1-9
3.1.4
Frequency comparison
Each set of model boundary conditions produces different frequency predictions
for the first 9 modes, summarized in Table 1.
Table 1: Varying Boundary Condition Frequency Comparison
Mode
1
2
3
4
5
6
7
8
9
Delta (Fixed
Sector to
Blade Alone (Hz) Fixed Sector (Hz) Cyclic ND 12 (Hz) Cyclic ND12)
1373.898809
1263.495429
1259.04
0.35%
3853.311373
3652.017458
3636.17
0.44%
5017.390281
4556.490203
4525.04
0.70%
6012.422854
5251.624095
5197.94
1.03%
9011.647358
8637.582689
8617.83
0.23%
12005.7108
11256.90835
11192.8
0.57%
15589.20011
13139.89331
12908.9
1.79%
15985.13839
13715.12062
13497.9
1.61%
16238.25959
15448.72193
15408.3
0.26%
27
Table 1 shows that the blade alone model produces higher frequencies than the
other methods as a result of the assumption that the rotor is infinitely stiff. However,
even though the fixed sector model makes some simplified assumptions, the frequencies
are very similar to the nodal diameter 12 frequencies, with the largest difference being
about 1.8% for mode 7. All other modes except mode 8 had less than a 1% delta in
frequency. Looking at the mode shapes it can also been observed that the fixed sector
and ND 12 model have nearly identical mode shapes. Since the fixed sector result
correlate so well to the cyclic symmetry model, for simplicity the following forced
response analysis is performed using the fixed sector model.
In the jet engine application it is extremely important to plot the natural
frequencies versus rotor speed, which is known as a Campbell or interference diagram.
The Campbell diagram was introduced by Wilford Campbell who used this tool to
understand the interaction between modal frequencies and excitation forces. The
diagonal lines beginning at the origin are called driver lines or engine orders and
represent the potential disturbances that could affect the rotating blades. Figure 23 is an
example of a Campbell diagram where every 10 engine orders are called out as potential
drivers of concern. The horizontal lines corresponds to the modes of concern shown in
Table 1.
28
Figure 23. Modes 1-9 Campbell Diagram
Every mode and driver line crossing represents a potential resonant response; however,
known drivers like vane and blade counts of concern can be identified in a design. In this
simulation frequencies are the same for each speed range, however, thermal and prestress affects can influence the blade frequencies in a true engine application. The
conditions will affect the speed at which particular resonances can occur.
3.2 Forced Vibration
3.2.1
Frequency and Modeshape comparison
The first step in running the forced vibration analysis was to run a free vibration
analysis with fixed sector boundary conditions for each of the three blades being
investigated (Figures 13-15). Table 2 shows the frequencies and deltas for the first 9
modes; Modes 1, 2, and 4 are highlighted because these modes were investigated using
the forced vibration analysis. Delta 1 is the frequency delta between the baseline and
clipped blade 1. Delta 2 is the frequency delta between the baseline and clipped blade 2.
29
Table 2. Clipped Blades Frequency Comparison
Fixed Sector (Hz) Clip1 (Hz) Clip2 (Hz)
1263.495429
1269.8
1278.6
3652.017458
3672.1
3695.4
4556.490203
4606.9
4661.1
5251.624095
5257.6
5273.7
8637.582689
8722.4
8814.8
11256.90835
11259
11286
13139.89331
13155
13200
13715.12062
13767
13835
15448.72193
15513
15592
Delta 1
0.50%
0.55%
1.11%
0.11%
0.98%
0.02%
0.11%
0.38%
0.42%
Delta 2
1.20%
1.19%
2.30%
0.42%
2.05%
0.26%
0.46%
0.87%
0.93%
Clipped blade 1 has a frequency delta of .5% to 1% when compared to the baseline
blade. Clipped blade 2 has a frequency delta of 1% to 2% when compared to the baseline
blade. Since material was clipped from the trailing edge tip, these results confirm the
natural frequencies should increase. Similarly, clipped blade 2 has a larger frequency
delta since it has a larger clip.
Although clipping can be done to intentionally change the modeshape, the intent
of this study was to understand the change in response to a particular mode due the
material removal. Figure 24 shows that Modes 1, 2, and 4 remain the same for the 3
blade geometries. Mode 1 is first bending, Mode 2 is first torsion and Mode 4 is second
bending. These modes were selected because they all have considerable deflection at the
leading edge tip; mode 3 has a node line through the leading edge tip so it was not
investigated in this manner.
30
Figure 24. Baseline and clipped Modes 1, 2, 4
3.2.2
Post processing
After each harmonic analysis run is complete it needs to be post processed by
both the time history post processor and the general post processor. The time history post
processor is used to define the frequency and phase angle to expand the peak amplitude
results with the expansion pass solution. The frequency (.rfrq) results file needs to be
opened in the time history post processor. The leading edge tip node tangential
deflection is added as a variable to track phase angle versus deflection and select a
frequency and angle to expand the solution. The tangential deflection was chosen
because it had the largest amplitude for these modes. Table 3 shows the tangential
deflections and corresponding phase angles for the baseline blade swept from 1250 to
1300 Hz.
31
Table 3. Frequency, Deflection and Phase Angle
Mode1 - Baseline
Frequency (Hz)
1250
1256
1260.8
1262.5
1263.1
1263.4
1263.5
1263.6
1263.8
1264.5
1266.2
1271.1
1284.6
1300
UY (in.)
Phase angle ( ˚ )
0.036168
-42.6445
0.045217
-58.8116
0.051192
-77.5298
0.052096
-85.2486
0.052191
-88.0802
0.052194
-89.1008
0.052188
-89.6737
0.052177
-90.2465
0.052145
-91.2671
0.051969
-94.0987
0.050848
-101.817
0.044377
-120.536
0.026298
-148.51
0.016520
-160.306
An expansion pass solution can then be run to produce the full solution at the
frequency and phase angle with the largest corresponding deflection, in this case, 1263.4
Hz and -89.1degrees. Once the expansion pass is complete the overall deflections can be
plotted to represent peak magnitude of the forced response.
Figure 25 is a plot of tangential displacement (UY) of the LE tip node versus
frequency for the baseline blade. Each spike on the chart is a resonance which
corresponds to modal frequencies M1, M2, M4 and their amplitude. This analysis was
run from 0 to 8000 Hz to capture the first 4 modes. Mode 3 again does not show up here
because there is no displacement of the LE tip node.
32
Figure 25. Resonant Response Amplitudes
3.2.3
Mode 1
Figures 26 through 28 show the amplitude versus frequency plots for each blade
from 1250 to 1300 Hz and corresponding peak amplitude. The plots show the resonance
increasing in frequency and show a small varying peak tip node amplitude.
33
Figure 26. Baseline Blade Mode 1 Forced Responses
34
Figure 27. Clipped Blade 1 Mode 1 Forced Response
35
Figure 28. Clipped Blade 2 Mode 1 Forced Response
Figure 29 compares the peak amplitude of deflection for the 3 blades analyzed.
For first bending, the peak tip deflection increases as the clip size is increased.
36
Figure 29. Mode 1 Deflection Comparison
3.2.4
Mode 2
Figures 30 through 32 show the amplitude versus frequency plots for each blade
from 3600 to 3700 Hz and corresponding peak amplitude. The plots show the resonance
increasing in frequency and show a small varying peak tip node amplitude.
37
Figure 30. Clipped Blade 1 Mode 2 Forced Responses
38
Figure 31. Clipped Blade 1 Mode 2 Forced Response
39
Figure 32. Clipped Blade 1 Mode 2 Forced Response
Figure 33 compares the peak amplitude of deflection for the 3 blades analyzed.
For first torsion, the peak tip deflection decreases as the clip size is increased opposite of
the first mode.
40
Figure 33. Mode 2 Deflection Comparison
3.2.5
Mode 4
Figures 34 through 36 show the amplitude versus frequency plots for each blade
from 5200 to 5300 Hz and corresponding peak amplitude. The plots show the resonance
increasing in frequency and show a small varying peak tip node amplitude.
41
Figure 34. Clipped Blade 2 Mode 4 Forced Responses
42
Figure 35. Clipped Blade 2 Mode 4 Forced Response
43
Figure 36. Clipped Blade 2 Mode 4 Forced Response
Figure 37 compares the peak amplitude of deflection for the 3 blades analyzed.
For second bending mode, the peak tip deflection for the baseline blade and smaller clip
are about the same. For the larger clipped blade the peak deflection than decreases.
44
Figure 37. Mode 4 Deflection Comparison
45
4. Conclusions
Understanding the characteristics of turbomachinery blade vibration is essential
in the design of gas turbine engines. It is even more important in jet engines when the
livelihood of possibly hundreds of passengers is on the line. Free vibration analysis can
be run using a variety of methods and boundary conditions ranging from the simple
blade alone analysis to the more complex cyclic symmetry model with non-linear prestress. Each type of model can have value in the blade design. While the cyclic
symmetry model may capture the most representative physics, it may be too large or
complicated for optimization work. It was shown that in some situations it is necessary
to understand the complex interactions between the individual blade vibrations coupled
with the rotor. However, in the case of the simple rotor and disk presented here,
simplified boundary conditions were sufficient to capture modal frequency results within
1-2% of the more costly cyclic symmetry analysis.
The forced vibration analysis uses the results of the free vibration analysis and
predicts blade responses to different harmonic loadings. The true unsteady aerodynamic
loading may be difficult to model, but can be simplified to excite blades in certain
modes. One way this analysis can be valuable is by evaluating the response of blades
with slight geometric variations. By loading the blades in a consistent manner,
deflections (or corresponding stresses) can be compared to understand and improve a
blade design
Table 4 summarizes the deflection amplitudes for each of the three blades
analyzed, with Delta 1 being the change from baseline to clipped blade 1 and Delta 2
from the baseline to clipped blade 2.
46
Table 4. Blade Amplitude Summary
Mode
1
2
3
Baseline (in.)
0.052988
0.010053
0.00397
Clip 1 (in.)
0.05327
0.009913
0.003973
Clip 2 (in.)
0.053422
0.009789
0.003947
Delta 1
0.53%
-1.39%
0.08%
Delta 2
0.82%
-2.63%
-0.58%
First bending increases deflection with clip size from about .5% to .8% with the
larger clip. The torsion is more sensitive than either first bending mode and deflection
decreases from about 1.4% with the smaller clip to 2.6% with the larger clip. The second
bending mode shows an interesting trend, with the smaller clip having very little affect at
all, while the larger clip shows a decrease in deflection of about .6%. Overall, these
deflections only track one node on the blade, but do show a trend for how a small change
in geometry can cause a change in predicted deflection.
The results show how much detail must be put into understanding blade
vibration. A small clip may help to improve one mode deflection, but could increase the
response on a different mode. At this point in the blade design it is usually very
important to gather data from true running conditions to correlate the amplitudes of
blade response for different modes at different resonant crossing speeds. The more
information a blade designer can gather from vibrations analysis and machine testing the
more robust a design can be, and the safer each engine can perform.
47
5. References
Images
[1] http://web.mit.edu/16.unified/www/SPRING/propulsion/notes/node27.html
[2] http://www.powergeneration.siemens.com/products-solutions-services/
[3] ANSYS 12.1 Theory Reference, ANSYS Corporation, 2009.
[4] Friswell, M.J., Penny, J. E. T., Garvey, A. D., Lees, A. W. (2010). Dynamics of
Rotating Machines. Cambridge University Press, New York, New York.
[5] Hassan, Mohammed (2008). Vibratory Analysis of Turbomachinery Blades.
Rensselaer Polytechnic Institute Master’s Report, Hartford, CT.
[6] Hartog, J.P Den (1985). Mechanical Vibrations. Dover Publications, Inc. , New
York, New York.
[7] Hill, Philip G, Peterson, Carl R. (1992). Mechanics and Thermodynamics of
Propulsion. Addison-Wesley Publishing Company, Inc., Reading, Massachusetts.
[8] Singh, M., Vargo, J., Schiffer, D., Dello, J. (2002). Safe Diagram – A Design and
Reliability Tool for Turbine Blading. Dresser-Rand Company, Wellsville, NY.
[9] Synder, Daniel (2011). A Modeling Study of the Sensitivity of Natural Frequency of
Vibration to Geometric Variations in a Turbine Blade. Rensselaer Polytechnic Institute
Master’s Report, Hartford, CT.
[10] Timoshenko, S., Young, D.H., Weaver Jr., W. (1974). Vibration Problems in
Engineering, Fourth Edition. John Wiley & Sons, New York, New York.
48
6. Appendix A: Matlab
%Matlab Script to plot receptance for various damping ratios%
clear;
Xo=1;
Co=1;
m=1;
On=1;
xsi=.1;
xsi2=.2;
xsi3=.5;
O=On*sqrt(1-(xsi^2));
r = 0:.01:2;
Cs=((1/(m*On^2))./sqrt((1-r.*r).*(1-r.*r)+(2*xsi*r).*(2*xsi*r)));
Cs2=((1/(m*On^2))./sqrt((1-r.*r).*(1-r.*r)+(2*xsi2*r).*(2*xsi2*r)));
Cs3=((1/(m*On^2))./sqrt((1-r.*r).*(1-r.*r)+(2*xsi3*r).*(2*xsi3*r)));
plot(r,Cs,r,Cs2,r,Cs3)
%Matlab Script to plot phase angle for various damping ratios%
clear;
Xo=1;
Co=1;
m=1;
On=1;
xsi=.1;
xsi2=.2;
xsi3=.5;
eta=.2;
O=On*sqrt(1-(xsi^2));
r = 0:.01:2;
n=2*xsi*r;
d=1-r.*r;
t=n./d;
phi=atan2(n,d);
n2=eta;
d2=1-r.*r;
t=n2./d2;
phi2=atan2(n2,d2);
plot(r,phi,r,phi2)
49
7. Appendix B: ANSYS files
!This file can run both a modal blade alone analysis or the fixed sector boundary analysis
!with meshed geometry and defined boundary conditions.
/filname,ibr
resume,ibr,db
/prep7
!om
= 837.758
nmodes = 9
fqlow = 0.0
fqhi = 20000.0
!Enter Preprocessor
!Define angular velocity (omega)
!Run for 9 modes
!Low end of frequency range 0 Hz
!High end of frequency range 20,000 Hz
MP,DENS,1,.000414
MP,EX,1,1.65e6
MP,PRXY,1,.35
!Define Material Density
!Define Material Modulus
!Define Material Poisson
!Modal Solution!
/solu
antype,2
eqslv,spar
!Enter Solution
!Modal Solution
!Use Sparse Solver
!Define frequencies & modes!
modopt,lanb,nmodes,fqlow,fqhi,,off
mxpand,nmodes,,,yes
!omega,om,,,
!Define Frequency Range
!Number of Modes and element results
!Apply Angular velocity if necessary
save
solve
finish
/exit,nosa
50
!This file can run modal analysis cyclic symmetry IBR sector model
!with meshed geometry and defined boundary conditions.
/clear
/filname,ibr
resume,ibr,db
/title, ibr,
!om
= 837.758
nmodes = 18
fqlow = 0.0
fqhi = 20000.0
ndlow = 0
ndhi = 12
ndinc = 1
!Define angular velocity (omega)
!Run for 18 modes (2X for first 9)
!Low end of frequency range 0 Hz
!High end of frequency range 20,000 Hz
!Low end of Nodal Diameter Range
!High end of Nodal Diameter Range
!Run each Nodal Diameter increment
/prep7
!Enter Preprocessor
mp,dens,1,.000414
mp,ex,1,1.65e6
mp,prxy,1,.35
!Define Material Density
!Define Material Modulus
!Define Material Poisson
allsel,all
cycopt,toler,3e-3
csys,6
cyclic,24,360/24,6,cyc,1
!Select the whole model
!3 thousandths of a inch cyclic tolerance
!Cylindrical coordinate system-rotation about X
!24 blade sector, 15 degrees each
!Modal Solution!
/solu
antype,2
eqslv,spar
!Enter Solution
!Modal Solution
!Use Sparse Solver
!Define frequencies & modes!
modopt,lanb,nmodes,fqlow,fqhi,,off
pstress,0
mxpand,nmodes,,,0
!omega,om,,,
cycopt,hindex,ndlow,ndhi,ndinc,0,
!Define Frequency Range
!No Pre-stress
!Number of Modes and element results
!Apply Angular (x) velocity if necessary
!Define Nodal Diameter range
solve
finish
save
51
!This file runs a modal analysis with fixed sector boundary conditions
!Defines a point load at the tip LE of the blade
!Runs a harmonic modal superposition analysis based on frequency inputs, and modal solutions.
/clear
/filname,ibr
resume,ibr,db
/output,ibr,output
/prep7
forcen=6511
appf=10
!Modal Solution!
/solu
antype,2
eqslv,spar
dmprat,0.01
modopt,lanb,9,,,,OFF
mxpand,9,,,1
solve
save,,,,,,model
!Application of the Harmonic Force!
/prep7
nwpave,6511
nwplan,-1,6511,6523,6577
cswpla,15,0,1,1,
csys,15
nrotat,6511
F,forcen,FY,appf
CSYS,0,
finish
!Harmonic Solution!
/solu
antype,harmic
hropt,msup,4,1,yes
hrout,on,on,on
harfrq,1250,1300
nsubst,25
kbc,1
save
solve
finish
/eof
!Enter Preprocessor
!Node for applied force
!Magnitude of point force at above node
!Enter Solution
!Modal Solution
!Use Sparse Solver
!Damping ratio for forced response
!Define Frequency Range
!Number of Modes and element results
!Enter Preprocessor
!Node of work plane origin
!Nodes defining work plane normal to blade surface
!Define local csys (coordinate system)
!Work in local csys
!Rotating nodal coordinate to csys 15
!Apply harmonic force
!Returning to global csys
!Enter Solution
!Run Harmonic solution
!Run modal superposition, 4 modes, viscous damping
!Output Real/Imaginary, Cluster, Mode Contributions
!Frequency range to explore
!Number of substeps per freq. range
!Step changed loads
52
8. Appendix C: Cyclic Symmetry Modeshapes
53
54
55
56