Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads
The results of modal analysis of the turbine blade were presented. The turbine blade during the operation of the engine is subjected to complex thermomechanical loads induced by centrifugal forces of the rotating blade and a nonuniform thermal field. These loads have a great influence on the natural...
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| Опубліковано в: : | Проблемы прочности |
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
2016
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| Цитувати: | Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads / L. Witek, F. Stachowicz // Проблемы прочности. — 2016. — № 4. — С. 6-13. — Бібліогр.: 21 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860246202312818688 |
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| author | Witek, L. Stachowicz, F. |
| author_facet | Witek, L. Stachowicz, F. |
| citation_txt | Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads / L. Witek, F. Stachowicz // Проблемы прочности. — 2016. — № 4. — С. 6-13. — Бібліогр.: 21 назв. — англ. |
| collection | DSpace DC |
| container_title | Проблемы прочности |
| description | The results of modal analysis of the turbine blade were presented. The turbine blade during the operation of the engine is subjected to complex thermomechanical loads induced by centrifugal forces of the rotating blade and a nonuniform thermal field. These loads have a great influence on the natural frequencies of the blade. In the first section of the study, modal analysis of the blade was performed using the vibration system. As a result, the resonant frequencies of the real blade were obtained. In order to check the effect of the rotational engine speed and the thermal field on the natural frequencies of the blade, the finite element method was employed. At the first stage of computations static analysis was conducted for the blade subjected to mechanical and thermal loads. Then modal analysis was used to isolate the natural frequencies and vibration modes of the blade. In modal analysis the stress state from the first stage was considered as preliminary conditions. Several results of numerical calculations and experimental analysis were compared to detect the relative error of natural frequency estimates
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| first_indexed | 2025-12-07T18:36:59Z |
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SCIENTIFIC AND TECHNICAL
SECTION
UDC 539.4
Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads
L. Witek
1
and F. Stachowicz
Rzeszów University of Technology, Faculty of Mechanical Engineering and Aeronautics, Rzeszów,
Poland
1 lwitek@prz.edu.pl
The results of modal analysis of the turbine blade were presented. The turbine blade during the
operation of the engine is subjected to complex thermomechanical loads induced by centrifugal
forces of the rotating blade and a nonuniform thermal field. These loads have a great influence on
the natural frequencies of the blade. In the first section of the study, modal analysis of the blade was
performed using the vibration system. As a result, the resonant frequencies of the real blade were
obtained. In order to check the effect of the rotational engine speed and the thermal field on the
natural frequencies of the blade, the finite element method was employed. At the first stage of
computations static analysis was conducted for the blade subjected to mechanical and thermal loads.
Then modal analysis was used to isolate the natural frequencies and vibration modes of the blade. In
modal analysis the stress state from the first stage was considered as preliminary conditions. Several
results of numerical calculations and experimental analysis were compared to detect the relative
error of natural frequency estimates.
Keywords: turbine blade, stress analysis, finite element method, vibration, modal analysis.
Introduction. The turbine blades have small bending stiffness and are particularly
susceptible to resonant vibrations. In operation of the engine, the blades are excited by an
unbalanced rotor. In the worst case, the excitation frequency overlaps the resonant
frequency of the blade. During the resonance, a high stress amplitude results in high-cycle
fatigue, thus the blade can be damaged after relatively short time. If a problem arises in the
turbine section, it will significantly affect the whole engine function and safety of the
aircraft. Failures of any high-speed rotating components can be dangerous to passengers
and personnel.
The problem of turbine failure was the subject of several investigations [1–4]. The
results of stress, fracture and modal analysis of the turbine and compressor blades were
described in [5–11]. The results [12] show that the vibrations have a negative influence on
the fatigue life of the aircraft engine blades. The fatigue process of aircraft components can
be also accelerated with corrosion [13, 14]. The investigations of vibration analysis of the
blades are described in [15–18].
The main objective of this study is to determine the resonant frequencies and free
vibration modes of the turbine blade subjected to complex thermomechanical loads. The
effect of the rotational turbine speed on the natural frequencies of the aircraft engine blade
was additionally analyzed.
1. Experimental Modal Analysis of the Blade. Experimental modal analysis of the
blade made use of an Unholtz-Dickie UDCO TA-250 vibration system presented in Fig. 1a.
The blade was horizontally mounted on the movable head of the vibrator (Fig. 1b). In
© L. WITEK, F. STACHOWICZ, 2016
6 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4
modal analysis the excitation frequency was increased from 500 to 4000 Hz with the sweep
rate of 1 Hz/s. The intensity of excitation was defined as 2g (where g � 9.81 m/s 2).
In experimental analysis two measurement channels were activated. The first channel
was used to measure the acceleration amplitude of the vibrator head (signal A1). The
acceleration in this channel was measured by a large sensor (10g). The signal ( )A2 in the
second channel was delivered from the miniature piezoelectric acceleration sensor (0.2g)
(Fig. 1b). This sensor was located 15 mm above the bottom of the blade airfoil section. This
position is more advantageous than the location of the sensor on the top of the blade. The
sensor position closer to the blade fir-tree slot (blade constraint) causes smaller disturbances
in dynamics of an investigated object. The acceleration sensors were connected to the blade
with special wax.
During modal analysis the excitation frequency was gradually increased. At this time
the signals from the two measurement channels were recorded. The main result of modal
analysis is the amplitude-frequency characteristic shown in Fig. 2. On the horizontal axis
the actual excitation frequency is plotted. The relative amplitude of the blade is laid off on
the vertical axis. The relative amplitude is the ratio of the acceleration amplitude from the
second channel (A2) to the signal from the first channel (acceleration amplitude of the
vibrator head, A1).
As is seen in Fig. 2, a high peak typical of a resonance phenomenon is observed at
969.7 Hz. This value is the resonant frequency (vibration mode I). The relative amplitude
of vibration I mode equals 119.11. The second peak is visible on the amplitude-frequency
characteristic at 2585.2 Hz. This value is the resonant frequency for vibration mode II. The
third resonant frequency equals 3727.4 Hz (Fig. 2).
2. Numerical Model of the Blade, Loads and Boundary Conditions. The numerical
model of the turbine blade was constructed using an MSC-PATRAN program [19]. The
model reproduces the geometry of the real blade (Fig. 3a). The discrete model of the turbine
blade presented in Fig. 3b consists of 353478 TET-4 finite elements (with linear function
[20]) and 70754 nodes. The average size of elements equals 0.54 mm. The material of the
blade (EI-437-y nickel alloy) was defined as linear elastic with the following properties:
modulus of elasticity = 200 GPa, Poisson’s ratio = 0.3, and density = 8200 kg/m3 [21]. An
a b
Fig. 1. Vibration system for experimental modal analysis (a) and the blade fixed to the head of the
vibrator (b).
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4 7
Modal Analysis of the Turbine Blade ...
EI-437-y alloy has a good creep resistance and was used for hot rotational components of
turbine engines.
In modal analysis the surfaces of the fir-tree slots (blade lock) were totally constrained
( ).T T Tx y z� � � 0 The thermal field on the turbine (considered in Sect. 4) was shown in
Fig. 3c. The modulus of elasticity and the thermal expansion coefficient for an EI-437-y
alloy were assumed to be temperature-dependent.
3. Results of Numerical Modal Analysis. Numerical modal analysis was performed
using an ABAQUS solver [20]. As a result of computations, the modes and natural
frequencies of the blade were obtained. At the first stage of the analysis, the rotational
speed of the engine rotor was defined as n� 0 whereas the constant temperature was equal
to T � �0 C (only the blade was fixed). These conditions were employed in experimental
modal analysis of the blade.
Fig. 2. Amplitude–frequency characteristic for an examined turbine blade.
a b c
Fig. 3. Object of investigations (a), numerical model of the blade (b), and the temperature field in
the blade during the operation of the engine [height h � 2500 m and engine power of 900 HP
(662 kW)] (c).
8 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4
L. Witek and F. Stachowicz
The results of computations demonstrated that the resonant frequency (for vibration
mode I) equalled 992 Hz (Table 1). The experimental result is equal to 969.7 Hz.
To estimate the validity of the numerical solution, the relative error E was defined by
the formula:
E
F F
F
res nf
res
�
�
�100%, (1)
where Fres is the resonant frequency (experimental result) and Fnf is the natural
frequency (numerical result).
The relative error E (mode I) equals 2.3% (Table 1). For vibration mode II the error is
3%. A greater relative error (10.2%) is observed for free vibration mode III. The results
from the next modes cannot be compared because of limitations of a UDCO TA-250
vibration system used in experimental investigations (5–4000 Hz).
4. Effect of the Thermal Field and Rotational Turbine Speed on the Natural
Frequency of the Blade. The rotation of the engine rotor induces large centrifugal stresses
in the blade. Tensile stresses cause an increase in the natural frequencies of the blade during
the acceleration of the turbine. High temperatures on the turbine blade also exert influence
on its natural frequency. This complex phenomenon is associated with a decrease in the
modulus of elasticity of the blade material at high temperatures. Additionally, the hot blade
has a larger size (because of thermal expansion), which influences a decrease in the natural
frequencies as well. In this section the effect of the rotational turbine speed and thermal
field on the natural frequencies of the blade was investigated.
For solving the problem, the procedure of numerical analysis was divided into the
three stages. At the first stage of static analysis, the thermomechanical loads were applied
to the blade. As a result, the stress distribution for the blade subjected to rotational speeds
and nonuniform thermal fields was obtained. At the second slage modal analysis of the
blade was performed using an ABAQUS command: Frequency [20]. At the third stage
Complex frequency analysis [20] was used to isolate the natural frequencies of the blade. In
this procedure, the stress state reached at the first stage (for the hot blade subjected to
rotation) is considered a preliminary condition for modal analysis.
The equivalent stress distributions (according to the Huber–Mises–Hencky criterion)
for free vibration modes I, II, and III of the blade are presented in Fig. 4. The results
correspond to the maximum vibration amplitude of the blade A � 1 mm. The maximum
stress area for mode I (296 MPa, Fig. 4a) is located in the zone above the blade lock. Mode
I can be classified as pure flexural (Table 2).
For free vibration mode II the maximum stress area (229 MPa) is located in the blade
airfoil section. Mode II can be classified as torsional-flexural (combination of torsion and
bending).
Free vibration mode III is pure torsional. The large stress area in this case is located
on the leading edge of the blade (743 MPa). Maximum stresses obtained in modal analysis
T a b l e 1
Comparison of Experimental and Numerical Results (n � 0, cold blade T � �0 C)
Natural frequency/resonant frequency
Mode I Mode II Mode III
Fnf ,
Hz
Fres ,
Hz
E,
%
Fnf ,
Hz
Fres ,
Hz
E,
%
Fnf ,
Hz
Fres ,
Hz
E,
%
992 969.7 2.3 2663 2585.2 3.0 4108 3727.4 10.2
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4 9
Modal Analysis of the Turbine Blade ...
should be treated as reliable results. The stress at the blade resonance depends on the real
vibration amplitude of the blade.
The results of modal analysis for the blade subjected to rotational speeds and thermal
fields (Fig. 3c) are summarized in Table 3. The maximum rotational speed of the engine
rotor (22,490 rpm) should be related to a 100% speed of the turbine. In this study the
following speeds: 0, 50, 80, and 90% were used. Moreover, the overspeed condition of the
turbine (110%) was also considered.
The results of the two analyses are depicted in Fig. 5. Curve 1 is related to the rotating
blade in the nonuniform thermal field (Fig. 3c). Curve 2 presents the natural frequencies of
the cold rotating blade (without specifying the temperature).
As is seen in Fig. 5, the natural frequency of the cold blade for the rotor speed of 0%
equals 992 Hz (Table 1). After acceleration of the turbine to the speed of 100%, the natural
frequency of the cold blade is equal to 1178 Hz (increase by about 19%).
After heating the blade to the operation temperature (Fig. 3c), the natural frequency
(for the rotor speed of 0%, mode I) equals 908 Hz (Fig. 5, Table 3). It means that the
thermal field (for n� 0) causes a decrease in the natural frequency by about 8.4%. After
acceleration of the hot blade to the speed of 100%, the natural frequency increases to
1112 Hz (increase by about 21%).
For free vibration mode II the natural frequency of the hot blade increases from 2419
Hz to 2586 Hz (increase by 7%) (Fig. 6, Table 3).
In the torsional vibration mode (mode III) the natural frequency of the blade (increase
in frequency by about 1.7%) is slightly influenced by the rotational speed of the turbine
(Fig. 6, Table 3). In this case the tensile stress of the rotating blade exerts little influence on
the shear stress arising during the torsional vibration of the blade.
T a b l e 2
Classification of Free Vibration Modes of the Blade
Mode number I II III IV V VI
Mode shape
(classification)
flexural torsional-
flexural
torsional complex complex complex
a b c
Fig. 4. Equivalent stress distributions (according to the Huber–Mises–Hencky criterion) in the turbine
blade for free vibration modes I (a), II (b), and III (c) (blade amplitude A �1 mm, n � 22,490 rpm,
thermal field).
10 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4
L. Witek and F. Stachowicz
Conclusions. In this study the effect of the rotational speed of the engine rotor and the
thermal field in the turbine on the natural frequencies of the blade was investigated. In the
first section of the study experimental analysis was performed to determine the real
resonant frequencies of the turbine blade. In the next section numerical modal analysis of
the blade (with consideration of the rotational turbine speed and the nonuniform thermal
field on the blade) was conducted. As a result, the free vibration modes and frequencies of
the cold and hot rotating blades were determined.
This study give rise to the following findings.
1. The relative error of the numerical solution (natural frequency estimates) makes up
2.3% (mode I), 3.0% (mode II), and 10.2% (mode III).
2. A considerable error (10.2%) for vibration mode III can be caused by large-sized
finite elements of the numerical model of the blade. Convergence of numerical solutions
obtained with different types and number of finite elements in the model (with parabolic
function) was corroborated.
T a b l e 3
Natural Frequencies of the Blade for Different Rotational Speeds of the Turbine
[Thermal Fields of the Blade for the Engine Operation at h � 2500 m,
Engine Power of 900 HP (662 kW)]
Turbine
speed [%]
Turbine
speed [rpm]
Natural frequency [Hz]
Mode I Mode II Mode III Mode IV Mode V Mode VI
0 0 908 2419 3741 4855 6031 9366
50 11,245 954 2456 3758 4893 6083 9398
80 17,992 1021 2511 3776 4951 6165 9446
90 20,241 1050 2534 3785 4977 6202 9466
100 22,490 1079 2559 3798 5005 6239 9487
110 24,739 1112 2586 3804 5036 6281 9509
Fig. 5. Effect of the rotational turbine speed on the natural frequencies of the cold and hot blades
(vibration mode I).
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Modal Analysis of the Turbine Blade ...
3. The operation temperature on the turbine causes a decrease in the natural frequency
of the blade by about 8%.
4. The natural frequencies of the hot blade for modes I and II (after turbine
acceleration from n� 0 to 22,490 rpm) increase by 21% and 7%, respectively.
5. The natural frequency (for mode III) of the rotating blade increases only by 1.7%.
The centrifugal forces of the rotating blade exert little influence on the shear stress arising
during pure torsional vibration.
Acknowledgments. The research received funding from the People Programme (Marie
Curie International Research Staff Exchange) of the European Union’s Seventh Framework
Programme FP7/2007-2013/ under REA Grant PIRSES-GA-2013-610547.
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12 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2016, ¹ 4
L. Witek and F. Stachowicz
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Received 10. 08. 2016
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Modal Analysis of the Turbine Blade ...
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/HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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| id | nasplib_isofts_kiev_ua-123456789-173501 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 0556-171X |
| language | English |
| last_indexed | 2025-12-07T18:36:59Z |
| publishDate | 2016 |
| publisher | Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| record_format | dspace |
| spelling | Witek, L. Stachowicz, F. 2020-12-07T20:04:57Z 2020-12-07T20:04:57Z 2016 Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads / L. Witek, F. Stachowicz // Проблемы прочности. — 2016. — № 4. — С. 6-13. — Бібліогр.: 21 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/173501 539.4 The results of modal analysis of the turbine blade were presented. The turbine blade during the operation of the engine is subjected to complex thermomechanical loads induced by centrifugal forces of the rotating blade and a nonuniform thermal field. These loads have a great influence on the natural frequencies of the blade. In the first section of the study, modal analysis of the blade was performed using the vibration system. As a result, the resonant frequencies of the real blade were obtained. In order to check the effect of the rotational engine speed and the thermal field on the natural frequencies of the blade, the finite element method was employed. At the first stage of computations static analysis was conducted for the blade subjected to mechanical and thermal loads. Then modal analysis was used to isolate the natural frequencies and vibration modes of the blade. In modal analysis the stress state from the first stage was considered as preliminary conditions. Several results of numerical calculations and experimental analysis were compared to detect the relative error of natural frequency estimates The research received funding from the People Programme (Marie Curie International Research Staff Exchange) of the European Union’s Seventh Framework Programme FP7/2007-2013/ under REA Grant PIRSES-GA-2013-610547. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads Модальный анализ лопатки турбины при сложных термомеханических нагрузках Article published earlier |
| spellingShingle | Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads Witek, L. Stachowicz, F. Научно-технический раздел |
| title | Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads |
| title_alt | Модальный анализ лопатки турбины при сложных термомеханических нагрузках |
| title_full | Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads |
| title_fullStr | Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads |
| title_full_unstemmed | Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads |
| title_short | Modal Analysis of the Turbine Blade at Complex Thermomechanical Loads |
| title_sort | modal analysis of the turbine blade at complex thermomechanical loads |
| topic | Научно-технический раздел |
| topic_facet | Научно-технический раздел |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/173501 |
| work_keys_str_mv | AT witekl modalanalysisoftheturbinebladeatcomplexthermomechanicalloads AT stachowiczf modalanalysisoftheturbinebladeatcomplexthermomechanicalloads AT witekl modalʹnyianalizlopatkiturbinyprisložnyhtermomehaničeskihnagruzkah AT stachowiczf modalʹnyianalizlopatkiturbinyprisložnyhtermomehaničeskihnagruzkah |