Free and Forced Blade Vibrations using Finite Element Analysis
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-Miravette, Thesis 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 ....................................................................................................... vii
ACKNOWLEDGMENT ................................................................................................ viii
ABSTRACT ..................................................................................................................... ix
1. Introduction and Background ...................................................................................... 1
1.1
Gas turbine Engines ........................................................................................... 1
1.2
Blading ............................................................................................................... 2
1.3
Free Vibration .................................................................................................... 3
1.4
Forced Vibration ................................................................................................ 3
1.5
High Cycle Fatigue (HCF) ................................................................................. 6
1.6
Clipping .............................................................................................................. 7
2. Modeling and Methodology ........................................................................................ 8
2.1
Finite Element Analysis ..................................................................................... 8
2.2
Finite Element Model ......................................................................................... 9
2.3
Boundary Conditions ....................................................................................... 11
2.4
2.3.1
Blade Alone.......................................................................................... 11
2.3.2
Fixed Sector Boundaries ...................................................................... 12
2.3.3
Cyclic Symmetry.................................................................................. 13
Solution ............................................................................................................ 14
2.4.1
Free vibration ....................................................................................... 14
2.4.2
Forced Vibration .................................................................................. 14
3. Discussion and Results .............................................................................................. 17
3.1
Free Vibration .................................................................................................. 17
3.1.1
Blade Alone.......................................................................................... 17
3.1.2
Fixed Sector ......................................................................................... 18
iii
3.2
3.1.3
Cyclic Symmetry.................................................................................. 19
3.1.4
Frequency comparison ......................................................................... 22
Forced Vibration .............................................................................................. 24
3.2.1
Frequency and Modeshape comparison ............................................... 24
3.2.2
Post processing ..................................................................................... 26
3.2.3
Mode 1 ................................................................................................. 27
3.2.4
Mode 2 ................................................................................................. 28
3.2.5
Mode 4 ................................................................................................. 29
3.2.6
Summary .............................................................................................. 30
4. Conclusions................................................................................................................ 31
5. References.................................................................................................................. 32
6. Appendix A: Cyclic Symmetry Modeshapes ............................................................ 33
7. Appendix B: Matlab .................................................................................................. 37
8. Appendix C: ANSYS files ......................................................................................... 38
iv
LIST OF TABLES
Table 1: Varying Boundary Condition Frequency Comparison ...................................... 22
Table 2. Clipped Blades Frequency Comparison ............................................................ 24
Table 3. Frequency, Deflection and Phase Angle ............................................................ 26
Table 4. Blade Amplitude Summary ............................................................................... 30
v
LIST OF FIGURES
Figure 1. Simple Brayton Cycle ........................................................................................ 2
Figure 2. Magnitude of Receptance ................................................................................... 5
Figure 3. Phase Shift of Receptance .................................................................................. 6
Figure 4. Blade Geomtry and Break-up ............................................................................. 9
Figure 5. ANSYS Solid45 elements ................................................................................ 10
Figure 6. Rotor and Blade mesh ...................................................................................... 11
Figure 7. Blade Alone Boundary Conditions .................................................................. 12
Figure 8. Fixed Sector Boundary Condititons ................................................................. 12
Figure 9. Cyclic Symmetry Boundary Conditions and Sector Nodes ............................. 13
Figure 10. Forced Response Loading .............................................................................. 15
Figure 11. Baseline Blade, Clip 1, and Clip 2 ................................................................. 16
Figure 12. Blade Alone Modeshapes 1-9 ........................................................................ 17
Figure 13. Fixed Sector Modeshapes 1-9 ........................................................................ 18
Figure 14. 1st Bending and 1st Torsion @ ND 0 .............................................................. 19
Figure 15. 1st Bending and 1st Torsion @ ND 12 ............................................................ 19
Figure 16. Mode 2 @ ND 2 and ND 12........................................................................... 20
Figure 17. Mode 1-9 Nodal Diamter Map ....................................................................... 21
Figure 18. Cyclic Symmetry Modeshapes 1-9 ................................................................ 22
Figure 19. Modes 1-9 Campbell Diagram ....................................................................... 23
Figure 20. Baseline and clipped Modes 1, 2, 4 ................................................................ 25
Figure 21. Resonant Response Amplitudes ..................................................................... 27
Figure 22. Mode 1 Forced Responses.............................................................................. 27
Figure 23. Mode 1 Deflection Comparison ..................................................................... 28
Figure 24. Mode 2 Forced Responses.............................................................................. 28
Figure 25. Mode 2 Deflection Comparison ..................................................................... 29
Figure 26. Mode 4 Forced Responses.............................................................................. 29
Figure 27. Mode 4 Deflection Comparison ..................................................................... 30
vi
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
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
vii
ACKNOWLEDGMENT
Type the text of your acknowledgment here.
viii
ABSTRACT
Type the text of your abstract here.
ix
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 was the use of rotating blades or turbo
machinery. In most modern day jet engines as well as other land based gas turbine
engines understanding the complex interaction of turbo machine 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 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 it pressure
and temperature for high energy combustion. After the combustion the high energy gas
is expanded 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 and would not be possible without for large axial flow machines without
the use of many compressor and turbine blades.
1
Figure 1. Simple Brayton Cycle
1.2 Blading
Both compressors and turbines typically consist of several stages of blades that
are preliminarily designed for ideal aerodynamics. Compressor blades decrease in size
from the entry stage to the exit to provide constant mass flow as the air density is
increased due to compression. Turbine blades increase in size increase in size from entry
to exit for the same reason as the gas expands. In both cases the, 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.
ADD PICTURE OF COMPRESSOR AND TURBINE
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 on the line. It is very
important to understand both the cause of the vibrations and the response of the blades to
different dynamic loading.
2
1.3 Free Vibration
One of the most important aspects in the design of turbo machine blades is
understanding the dynamics of blade vibration. The basic equation of motion for the
vibration of a blade is:
𝑚𝑥̈ + 𝑐𝑥̇ + 𝑘𝑥 = 𝐹
Here m is mass, c is the damping coefficient and k is the stiffness (coefficient of
elasticity). The above equation is simplified for a single degree of freedom system, but
will be applied with many degrees of freedom. The first step in understanding the
vibration of a blade is to understand the free vibration, or vibration due to initial
disturbance of the system. If damping is ignored then the equation simplifies to:
𝑚𝑥̈ + 𝑘𝑥 = 0
The equation of motion can be solved to provide the natural frequencies of the
blade at many different conditions. In many cases it is important to evaluate the free
vibrations unloaded, and loaded with certain forces experienced in turbo machinery 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.
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
3
engines there are complex disturbances that can be simulated by applying a simple load
cyclically described by the equation below.
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
The steady-state 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:
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:
4
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 XX and
XX.
Figure 2. Magnitude of Receptance
5
Figure 3. Phase Shift of Receptance
The forced response solution has a peak amplitude when:
𝜔 = 𝜔𝑛 √1 − 2𝜁 2
This phenomena described by the above forced response solution 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
harmonic relative to rotor speed and vane count. It is extremely important in the design
of turbo machine blading to understand the frequency and amplitude of responses due to
different cyclic loading.
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
6
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 off the engine design
conditions to understand the frequencies and amplitudes of vibration. 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.
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 is 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.
7
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 Finite Element Analysis (FEA). Finite element
models (FEMs) can be made to approximate blade geometries, and can be loaded to
simulate real life operational conditions. Static and dynamic analyses are then used to
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 multiple degrees of freedom vibration are
similar to the single degree of freedom equations discussed earlier:
[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 is:
8
[𝑀]{𝑥̈ } + [𝐶]{𝑥̇ } + [𝐾]{𝑥} = {𝐹}
The 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 XX.
Figure 4. Blade Geomtry and Break-up
9
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 XX.
Figure 5. 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 final blade mesh has 5256 elements and 6602 nodes and
seen in figure XX.
10
can be
Figure 6. Rotor and Blade mesh
The blade and rotor were assumed to be made from titanium 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 is essential
assumes that the rotor is infinitely stiff and has not elasticity. In this model, the blade
nodes on the bottom of the blade are constrained in all 3 translational degrees of freedom
as seen in figure XX.
11
Figure 7. Blade Alone Boundary Conditions
2.3.2
Fixed Sector
Boundaries
Another simplified way to model the boundary conditions of the model is the 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. The
model with these BCs can be seen in figure XX.
Figure 8. Fixed Sector Boundary Condititons
12
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 saves a lot of 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.
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. These boundary conditions can be seen in figure XX.
Figure 9. Cyclic Symmetry Boundary Conditions and Sector Nodes
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
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.
13
2.4 Solution
2.4.1
Free vibration
ANSYS has a variety of methods to solve modal analysis. This model 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 pre-stressed
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 solution
demonstrated here ignores both thermal and pre-stress effects on the rotor and blade
system.
2.4.2
Forced Vibration
The forced vibration solution uses the 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 a real application to more
accurately simulate the loading. Figure XX shows the forced response model with the
applied force.
14
Figure 10. 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 this solution the output
format 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 C.
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. The blade response of these three models will be compared using the forced
response analysis.
15
Figure 11. Baseline Blade, Clip 1, and Clip 2
16
3. Discussion and Results
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 to summarize the results.
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 USUM (X,Y,Z) deflections were used to plot the mode shapes and
corresponding frequencies and can be seen in figure XX.
Figure 12. Blade Alone Modeshapes 1-9
17
The first two modes are first bending and first torsion respectively, with mode shape
complexity typically increasing with each mode. Modes 4 and 8 have a stiff wise
bending component.
3.1.2
Fixed Sector
The first nine models 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 mode shapes and corresponding frequencies and can be seen in figure
XX.
Figure 13. Fixed Sector Modeshapes 1-9
18
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.
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 blade. The
analysis was run for every nodal diameter from 0 to N/2, which is ND 12 for this system.
Figure XX shows the blade first bending mode and first torsion mode respectively at ND
0 with all of the blades vibrating in phase.
Figure 14. 1st Bending and 1st Torsion @ ND 0
Figure XX 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 15. 1st Bending and 1st Torsion @ ND 12
19
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
and rotor modes can interact to dramatically change the system vibration. Figure XX
shows the mode shape for mode 2 at nodal diameters 2 and 12.
Figure 16. Mode 2 @ ND 2 and ND 12
At nodal diameter 2 the mode shape has changed from 1T at nodal diameter 12 to a
stiffwise bending mode driven by the rotor, indicated by the displacement seen in the
rotor. 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 XX shows a nodal diameter map for ND0 to N/2 for the first 9 modes.
20
Figure 17. 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 XX shows the first 9 modes up to 20,000 Hz for nodal diameter 12. The
mode shapes for the other nodal diameters can be found in Appendix A.
21
Figure 18. Cyclic Symmetry Modeshapes 1-9
3.1.4
Frequency comparison
Table 1: Varying Boundary Condition Frequency Comparison
Mode
1
2
3
4
5
6
7
8
9
Blade Alone (Hz) Cyclic ND 12 (Hz) Fixed Sector (Hz)
1373.898809
1259.04
1263.495429
3853.311373
3636.17
3652.017458
5017.390281
4525.04
4556.490203
6012.422854
5197.94
5251.624095
9011.647358
8617.83
8637.582689
12005.7108
11192.8
11256.90835
15589.20011
12908.9
13139.89331
15985.13839
13497.9
13715.12062
16238.25959
15408.3
15448.72193
Delta
0.35%
0.44%
0.70%
1.03%
0.23%
0.57%
1.79%
1.61%
0.26%
Each set of model boundary conditions produces different frequency predictions
for the first 9 modes. It is clear from Table 1 that the blade alone model has higher
frequencies than the other methods as the result the assumption that the rotor is infinitely
stiff. However, even though the fixed sector model makes some simplified assumptions,
22
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 force
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 XX is an
example of a Campbell diagram where every 10 engine orders are called out as potential
drivers of concern.
Figure 19. Modes 1-9 Campbell Diagram
23
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.
3.2 Forced Vibration
3.2.1
Frequency and Modeshape comparison
A free vibration analysis with fixed sector boundary conditions was run for each
of the three blades being investigated. 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.
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
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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 XX 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.
Figure 20. Baseline and clipped Modes 1, 2, 4
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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. 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.
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 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 XX 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
26
corresponds to a modal frequency and 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.
Figure 21. Resonant Response Amplitudes
3.2.3
Mode 1
Figure XX shows 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.
Figure 22. Mode 1 Forced Responses
Figure XX 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.
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Figure 23. Mode 1 Deflection Comparison
3.2.4
Mode 2
Figure XX shows 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.
Figure 24. Mode 2 Forced Responses
Figure XX 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.
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Figure 25. Mode 2 Deflection Comparison
3.2.5
Mode 4
Figure XX shows 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.
Figure 26. Mode 4 Forced Responses
29
Figure 27. Mode 4 Deflection Comparison
Figure XX 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.
3.2.6
Summary
Table 4 summarizes the deflections amplitudes for each blade, with Delta 1 being
the change from baseline to clipped blade 1 and Delta 2 from the baseline to clipped
blade 2.
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 bending mode and deflection
decreases from about 1.4% with the smaller clip to 2.6% with the larger clip. Second
bending mode shows an interesting trend, with the smaller clip have 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 blade, but do show a trend for how a small change in
geometry can cause a change in predicted deflection.
30
4. Conclusions
31
5. References
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.
Hartog, J.P Den (1985). Mechanical Vibrations. Dover Publications, Inc. , New York,
New York.
Timoshenko, S., Young, D.H., Weaver Jr., W. (1974). Vibration Problems in
Engineering, Fourth Edition. John Wiley & Sons, New York, New York.
Hill, Philip G, Peterson, Carl R. (1992). Mechanics and Thermodynamics of Propulsion.
Addison-Wesley Publishing Company, Inc., Reading, Massachusetts.
Hassan, Mohammed (2008). Vibratory Analysis of Turbomachinery Blades.
Synder, Daniel (2011). A Modeling Study of the Sensitivity of Natural Frequency of
Vibration to Geometric Variations in a Turbine Blade.
ANSYS 12.1 Theory Reference, ANSYS Corporation, 2009
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6. Appendix A: Cyclic Symmetry Modeshapes
33
34
35
36
7. Appendix B: Matlab
37
8. Appendix C: ANSYS files
38