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Original article / Оригинальная статья DOI: http://dx.doi.org/10.21285/1814-3520-2020-4-756-767
Evaluation of spring-mass-damping models of mistuned turbomachine impellers to analyse vibration and fatigue life
Igor N. Ryzhikov*, Tien Quyet Nguyen**
*Irkutsk National Research Technical University, Irkutsk, Russia **Viettel aerospace Institute, Hanoi, Vietnam
Abstract: This study is aimed at evaluating the feasibility of a spring-mass-damping model which reduces both the amount of RAM (random access memory) usage and the time taken to calculate results when analysing the effect of turbomachine impeller mistuning on both vibrations and fatigue life. The mistuning is the result of asymmetries in the assembled turbomachine impeller. The use of a spring-mass-damping model of a mistuned impeller tailored to calculate forced vibrations by introducing the variable of exciting load, as well as the finite element method and methods for calculating the fatigue life, allowed us to assess the fatigue life of a turbomachine impeller. To assess the accuracy of the model, calculated results for vibration frequencies and fatigue life were compared with the results of experiments with a turbomachine impeller at the Brandenburg University of Technology. The calculated natural vibrations demonstrated good accuracy (3-4% for lowest vibration modes) compared with the experimental results. The fatigue life was calculated for five different mistuning schemes simulated experimentally by attaching additional masses to the impeller rim. The calculated results were again compared with the results of experiments on the modified impeller. Recommendations on the assembly of such impellers were given. The error between the calculated results and experimental results of the natural vibrations is sufficiently small, which permits the use of the proposed model in the analysis of the effect of the vibrations of turbomachine impellers caused by mistuning on their fatigue life. It is shown that the amount of RAM required for the calculation is reduced by orders of magnitude, since the proposed model contains only two degrees of freedom per sector. As a result, the proposed model is of great utility in modelling modern structures with a complex shape. Using the results of this study, the fatigue life of impellers can be maximised by following the "sawtooth" law of the locating of the mistuned blades on the impeller rim when assembling turbomachine impellers.
Keywords: rotor blades, vibrations, mistuning, spring-mass model, finite element method, fatigue life
Information about the article: Received June 08, 2020; accepted for publication July 14, 2020; available online August 31, 2020.
For citation: Ryzhikov IN, Nguyen Tien Quyet. Evaluation of spring-mass-damping models of mistuned turbomachine impellers to analyse vibration and fatigue life. Vestnik Irkutskogo gosudarstvennogo tehnicheskogo universiteta = Proceedings of Irkutsk State Technical University. 2020;24(4):756-767. https://doi.org/10.21285/1814-3520-2020-4-756-767
УДК 534.1:539.3
Использование пружинно-массово-демпферных моделей при анализе колебаний и долговечности рабочих колес турбомашин с расстройкой параметров
© И.Н. Рыжиков*, Тьен Кует Нгуен**
*Иркутский национальный исследовательский технический университет, г. Иркутск, Россия **Аэрокосмический институт Viettel, г. Ханой, Вьетнам
Резюме: Цель - исследовать возможность применения эффективной пружинно-массово-демпферной модели, позволяющей уменьшить требуемую оперативную память и увеличить производительность расчетов при проведении анализа влияния расстройки параметров рабочих колес турбомашин, вводимой путем добавления к лопаткам дополнительных масс, на их колебания и долговечность. С использованием пружинно-массово-демпферной модели рабочего колеса с расстройкой параметров, доработанной для расчета вынужденных колебаний путем учета переменной возбуждающей нагрузки, а также метода конечных элементов и методов расчета долговечности был проведен анализ долговечности реального рабочего колеса турбомашины. Сравнение расчетных данных с использованием уточненной пружинно-массово-демпферной модели с данными эксперимента,
ВЕСТНИК ИРКУТСКОГО ГОСУДАРСТВЕННОГО ТЕХНИЧЕСКОГО УНИВЕРСИТЕТА 2020;24(4): 756-767
проведенного в Бранденбургском техническом университете, показало хорошую точность (3-4% для низших форм колебаний) расчетов свободных колебаний с использованием данной пружинно-массово-демпферной модели. Также были проведены расчеты долговечности рабочего колеса и сравнение результатов пяти вариантов распределения массы лопаток по ободу рабочего колеса. Разработаны рекомендации по сборке рабочих колес. Приведенные результаты расчетов свободных колебаний реального рабочего колеса в сравнении с результатами эксперимента, демонстрируют достаточно малую погрешность, что позволило применить разработанную модель для решения задач анализа влияния расстройки параметров колебаний рабочих колес турбомашин на долговечность. При этом показано, что поскольку данная модель содержит лишь 2 степени свободы, ее использование позволит на порядки уменьшить объем требуемой оперативной памяти, что имеет большое значение при моделировании современных конструкций, отличающихся сложной формой. На основании анализа результатов расчета долговечности различных вариантов распределения дополнительных масс лопаток разработана рекомендация соблюдать при сборке рабочих колес турбомашин «пилообразный» закон расположения расстроенных лопаток на ободе колеса.
Ключевые слова: рабочие лопатки ротора, колебания, расстройка параметров, пружинно-массовая модель, метод конечных элементов, долговечность
Информация о статье: Дата поступления 08 июня 2020 г.; дата принятия к печати 14 июля 2020 г.; дата он-лайн-размещения 31 августа 2020 г.
Для цитирования: Рыжиков И.Н., Нгуен Тьен Кует. Использование пружинно-массово-демпферных моделей при анализе колебаний и долговечности рабочих колес турбомашин с расстройкой параметров. Вестник Иркутского государственного технического университета. 2020. Т. 24. № 4. С. 756-767. https://doi.org/10.21285/1814-3520-2020-4-756-767
1. INTRODUCTION
The rotor impeller of a turbomachine has a cyclically symmetric design. Computer models designed to study impeller vibrations often represent the impeller rotor blades as a set of idealised symmetrical sectors, in which case, no deviations in symmetry can be accommodated. An actual impeller will always have very small deviations in shape, size, characteristics of materials, fixing and loading as a result of imperfect manufacturing technologies (the so-called mistuning). The model described by J. Nipkau1 was selected following a review of spring-mass models designed to represent these small imperfections. Many authors assert [1-12] that mistuning often has a negative effect on the stress-strain state of the impeller due to vibrations significantly reducing impeller service life. Representing this mistuning in a finite element method (FEM) model of an impeller is a serious complication requiring a significant increase in computer memory usage and processing time. Development of an effective model to analyse the mistuning effect on the service life of impellers, which nevertheless uses computer re-
sources more efficiently, is an urgent requirement.
One of such potential approaches is modelling the vibrations of complex mechanical systems using similar spring-mass-damping models. The application of this model in conjunction with the FEM significantly reduces the computer resources required for the analysis of mistuned impellers as a result of the simplicity of the spring-mass-damping model, which has a significantly smaller number of freedom degrees compared with FEM.
2. REVIEW OF RESEARCH LITERATURE
The application of spring-mass models in studies of impeller vibrations was first recorded around the middle of the twentieth century. An example of these applications [13] describes a model of an impeller with a flexible disk having no mass (fig. 1) to represent the effect of disk flexibility on the vibration frequency of the blades. Fig. 1: m,- blade mass; k - blade stiffness; d - damping in the blade material; ks - impeller stiffness.
1
Nipkau J. Analysis of mistuned blisk vibrations using a surrogate lumped mass model with aerodynamic influences. 2010, PhD thesis, Brandenburg University of Technology Cottbus, Cottbus, Germany. 180 p.
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Fig. 1. Flexible disk model [13] Рис. 1. Модель с упругим диском [ 13]
In the model described in [14, 15] (fig. 2), spring elements are used to represent inter-blade connections. Fig. 2: x, - movements; kc - stiffness of the interblade connection; mb - blade mass; kb - blade stiffness; db - damping in the blade material.
An alternative approach to the previous model [16] (fig. 3) represents the connection between the blade masses as torsion spring elements. Fig. 3: mi - blade mass; kc - stiffness of interblade connection; r - radius of the blade mass centre; kt - torsional stiffness of the blade.
In a paper [17] devoted to the study of forced disk vibrations, the model again incorporates interblade dampers.
A model with springs and dampers between the blade masses [18] was also applied to a gas turbine engine to determine the relative vibration amplitudes of the fan blades while taking into account aerodynamic forces.
The authors developed a finite element model of a disc composed of flat elements connected to single-mass blades, each having one degree of freedom. The disc studied was of a polymer composite material reinforced by carbon fibre. The aeroelastic behaviour of the rotor was simulated to investigate the effect of flux density on the distribution of vibration amplitudes.
More complex models with two or three degrees of freedom for each sector were elaborated in [19].
Thus, as many researchers note, the application of spring-mass-damping models to the finite element method is a successful approach to obtaining adequate results with significant savings in computer resources compared to a standalone finite element method. This consideration determined the choice of an approach for the analysis of the effect of mistuning on impeller vibrations.
-AVW—I
Fig. 2. Model with interblade spring elements [14, 15] Рис. 2. Модель с межлопаточными пружинными элементами [14, 15]
ВЕСТНИК ИРКУТСКОГО ГОСУДАРСТВЕННОГО ТЕХНИЧЕСКОГО УНИВЕРСИТЕТА 2020;24(4): 756-767
Fig. 3. Model with torsion spring elements [16] Рис. 3. Модель с торсионными пружинными элементами [16]
3. DESCRIPTION OF THE SPRING-MASS-DAMPING MODEL FOR MISTUNED TURBOMACHINE IMPELLERS FOR THE ANALYSIS OF VIBRATIONS AND FATIGUE LIFE
In the present study, the authors applied the impeller spring-mass-damping model described in detail by J. Nipkau1 in which each sector contains only two degrees of freedom.
Fig. 4 represents the model of the impeller sector with and without mistuning. In this approach, the mistuning was represented by introducing the Ama additional mass to the mass of the blade. Fig. 4: ksec - sector stiffness; dsec -damping in the sector material; msec - sector mass; Si - sector movements; kc - stiffness of the intersector connection; kb - blade stiffness; db - damping in the blade material; mb - the mass of the blade; Ama - additional weight; Sn, x - blade movements; ka - stiffness of the inter-blade connection; da - damping in the interblade connection.
The equation of motion for natural vibrations of the impeller is as follows:
MS + KS = 0,
matrix; 5 is the acceleration at nodal points; <5"is the vector of nodal point movement.
Blade vibrations are excited by many factors when the turbomachine is operating. The matrix form of the equation of motion for forced vibrations is:
MS + DS + KS = P(t\
where D is the damping matrix; 8 is the velocity of nodal point movements; P(t) is the vector of variable excitation loads.
The mass matrix is defined as
М =
m
0
0 0
0m
0 0
0
m sec 0 ...
0 mb 0
; 0 mb
0 0
0 0
0
0 m,
where M is the mass matrix; K is the stiffness
Damping matrix:
ВЕСТНИК ИРКУТСКОГО ГОСУДАРСТВЕННОГО ТЕХНИЧЕСКОГО УНИВЕРСИТЕТА 2020;24(4):756-767
dsec+db ... 0 0 ~dbx 0 • 0 0
0 dsec+db 2 0 0 0 —d-bo. • 0
0 0 0 ~db,N-1 0
D = 0 0 0 ^sec+dbN 0 0 0
~db ,1 0 0 0 dbx 0 0 0
0 0 0 0
0 1 0 0 db,N-l 0
0 0 0 0 0 0
Stiffness matrix:
"k* -k 0 c 0 -k с -hi 0 0 0 0
-k c k* -k 1 с 0 0 ~K,2 0
0 -к '■• j 0 ;
• -k с 0 • 0
0 • • -к kN-1 -K 0 ~kb,N-1 0
K = -k c 0 0 ~k С kN 0 0 ... 0 0 -K,N
0 0 0 0 Kx 0 0 0 0
0 0 0 K.2 0
0 j 0
* • • • 0 * • 0
0 • • ~kb,N-1 0 0 kb,N-1 0
0 0 0 0 -KM 0 0 ... 0 0 K.N
where k* = ksec + 2 kc + kbi.
In this study, the excitation of blade vibrations was modelled by 29 nozzle blades. The load exciting vibrations were taken to be constant along the entire length of the airfoil blade. The force determined by the Fourier series was considered as the variable load:
P(t) = L0 (1 + 0.5 cos (p + 0.025 cos 2$),
where L0 is the static part of the load; <p is the impeller rotation angle.
In order to analyse forced vibrations the authors extended the model described in the paper1, by developing a spring-mass-damping model of the impeller with a P(t) exciting load (fig. 5).
4. RESULTS
The authors compared the results calculated by their model with the results of an experiment carried out at the Brandenburg University of Technology (BTU) [20]. This latter experiment was undertaken using a 29-blade compressor
ВЕСТНИК ИРКУТСКОГО ГОСУДАРСТВЕННОГО ТЕХНИЧЕСКОГО УНИВЕРСИТЕТА 2020;24(4): 766-767
b
Fig. 4. Model of one sector: a - without additional mass; b - with additional mass Рис. 4. Модель одного сектора: а - без дополнительной массы; b - с дополнительной массой
а
| Ш)
Fig. 5. Spring-mass-damping model with an excitation load Рис. 5. Пружинно-массово-демпферная модель с возбуждающей нагрузкой
Fig. 6. Real impeller with additional masses Рис. 6. Реальное рабочее колесо с дополнительными массами
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impeller as shown in fig. 6. When required, the mistuning was simulated by the addition of masses to the periphery, outer circumference, of the blades.
In the first set of experiments no masses were added in order to calculate the baseline natural vibrations of the impeller using the standard FEM, i.e. without spring-mass-damping, and assuming an idealised cyclically symmetrical mass distribution. Fig. 7 represents the first three natural vibration modes of the impeller without mistuning.
In the second set of experiments, three ex-
amples of mistuning were considered each characterised by a different distribution of additional masses on the outer circumference of the impeller:
1) additional masses of 0.00211 kg were added to 28 of 29 blades;
2) additional masses of 0.005 kg were added to 28 of 29 blades;
3) additional masses were added to all blades as listed in tab.1.
Tab. 2 compares the results of the spring-mass-damping FEM calculation and the results of the experiment.
Mode 1
Mode 2
Mode 3
Fig. 7. Natural vibration modes of the impeller without mistuning Рис. 7. Формы собственных колебаний рабочего колеса без расстройки
Table 1. Order of the additional mass correction
Таблица 1. Порядок присоединения дополнительных масс_
Blade No. Am, kg Blade No. Am, kg Blade No. Am, kg
1 0.001251440 11 0.001451640 21 0.001356856
2 0.001302147 12 0.001302100 22 0.001125158
3 0.000025486 13 0.000065488 23 0.000085489
4 0.001102156 14 0.001502634 24 0.001402132
5 0.001305234 15 0.001600680 25 0.001202183
6 0.001205214 16 0.001166480 26 0.001102121
7 0.000802923 17 0.000735542 27 0.001302524
8 0.001100598 18 0.001553548 28 0.000356542
9 0.001302003 19 0.001135789 29 0.001205558
10 0.001202100 20 0.001257365
Table 2. Calculation of mistuned impeller vibrations
Таблица 2. Результаты расчета колебаний колеса с расстройкой параметров
Vibration mode Example 1 | Example 2 | Example 3
Natural frequencies, Hz
Finite element method Experiment (Brandenburg University of Technology) Finite element method Experiment (Brandenburg University of Technology) Finite element method Experiment (Brandenburg University of Technology)
1 403.3854 397.8125 410.3864 402.9375 416.3864 403.8574
2 1296.5417 1261.0000 1300.2214 1252.6875 1304.5584 1255.0780
3 1803.3784 1766.3125 1826.6854 1765.6750 1843.3258 1766.0640
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The third set of experiments assessed the impeller fatigue life, calculated as operating hours and loading cycles, with different examples of mistuning simulated by the addition of masses on the outer circumference of the impeller. The method used for the calculation of service life is described in detail by Nguyen Tien Quyet2.
Five different examples of mistuning were considered, each simulated by the attachment of additional masses to the outer circumference of the impeller:
1) the additional masses are identical (0.00211 kg) and were attached to 28 blades, leaving one unloaded blade;
2) the additional masses are identical (0.005 kg) and were attached to 28 blades, leaving one unloaded blade;
3) various additional masses were attached to all blades as listed previously in tab. 1;
4) the additional masses attached to the blades were gradually increased radially, ie.from one blade to the next, from a minimum (0.000025486 kg) to a maximum (0.001600680
kg), (tab. 3); thus the first and last (29th) blade in this scheme have the maximum possible mass difference between adjacent blades;
5) in contrast to the previous example, the additional masses on adjacent blades were not changed gradually, but were changed by following the "sawtooth" law in order to achieve the minimum difference in mass between all adjacent blades (tab. 4).
The fatigue life was calculated and compared with the experimental results for each example of additional mass distribution described above. The results are shown in tab. 5 and fig. 8.
5. CONCLUSION
The calculated results for natural vibrations using the spring-mass-damping model developed in this study compare favourably and with good accuracy to the experimental results with a compressor impeller. No attempt was made to evaluate the RAM usage when executing the spring-mass-damping model. However, it is known that memory usage is dependent on the
Table 3. Distribution of additional masses by blades (option 4)
Таблица 3. Распределение дополнительных масс по лопаткам (вариант 4)
Blade No. Am, kg Blade No. Am, kg Blade No. Am, kg Blade No. Am, kg
1 0.000025486 9 0.001102156 17 0.00125144 25 0.001402132
2 0.000065488 10 0.001125158 18 0.001257365 26 0.001451640
3 0.000085489 11 0.001135789 19 0.001302003 27 0.001502634
4 0.000356542 12 0.001166480 20 0.001302100 28 0.001553548
5 0.000735542 13 0.001202100 21 0.001302147 29 0.001600680
6 0.000802923 14 0.001202183 22 0.001302524
7 0.001100598 15 0.001205214 23 0.001305234
8 0.001102121 16 0.001205558 24 0.001356856
Table 4. Distribution of additional masses by blades (option 5)
Таблица 4. Распределение дополнительных масс по лопаткам (вариант 5)
Blade No. Am, kg Blade No. Am, kg Blade No. Am, kg Blade No. Am, kg
1 0.000025486 9 0.001402132 17 0.001302100 25 0.001102156
2 0.001302003 10 0.000802923 18 0.001257365 26 0.001502634
3 0.000085489 11 0.001135789 19 0.000065488 27 0.001125158
4 0.001302524 12 0.001451640 20 0.00125144 28 0.001553548
5 0.000735542 13 0.001202100 21 0.001302147 29 0.001100598
6 0.001166480 14 0.001305234 22 0.000356542
7 0.001600680 15 0.001205214 23 0.001356856
8 0.001102121 16 0.001205558 24 0.001202183
2Nguyen Tien Quyet. Mathematical models and a software package for assessing the effect of mistuning in power turbomachine impellers on their service life 05.13.18: Dissertation Cand. Sci. (Tech.). Irkutsk, 2018. 153 p. / Нгуен Тьен Кует. Математические модели и программный комплекс для оценки влияния расстройки параметров рабочих колес энергетических турбомашин на их долговечность: дис. ... канд. техн. наук: 05.13.18. Иркутск, 2018. 153 с.
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Table 5. Fatigue life calculation with the mistuning caused by different masses of blades
Таблица 5. Результаты расчета долговечности с расстройкой параметров, вызванной разной массой лопаток
Vibration mode Option 1 | Option 2 | Option 3
Natural frequencies, Hz
Finite element method Experiment (Brandenburg University of Technology) Finite element method Experiment (Brandenburg University of Technology) Finite element method Experiment (Brandenburg University of Technology)
1 403.3854 397.8125 410.3864 402.9375 416.3864 403.8574
2 1296.5417 1261.0000 1300.2214 1252.6875 1304.5584 1255.0780
3 1803.3784 1766.3125 1826.6854 1765.6750 1843.3258 1766.0640
Fatigue life 1.6603 E + 6 h 1.2402 E + 6 h 1.7626 E + 6 h
Vibration mode Option 4 Option 5
Natural frequencies, Hz
Finite element method Experiment (Brandenburg University of Technology) Finite element method Experiment (Brandenburg University of Technology)
1 420.5684 - 391.2154 -
2 1360.2235 - 1125.5652 -
3 1903.2365 - 1793.3654 -
Fatigue life 1.3603 E + 6 h 1.7903 E + 6 h
Option 1 Option 2
Option 3 Option 4 Option 5
Fig. 8. Mistuned impeller fatigue life for various options Рис. 8. Долговечность рабочего колеса с расстройкой параметров
number of degrees of freedom in the model; the standard FEM model has hundreds of degrees of freedom (depending on the density of the finite element mesh) for each sector, whereas the spring-mass-damping impeller model has only two degrees of freedom for each sector. Therefore, the memory usage for the spring-mass-damped model can be orders of magnitude less than that used for the standard FEM.
The calculated fatigue life using the adjusted
spring-mass-damping model indicated that assembling the impeller by following the "sawtooth" law resulted in the longest fatigue life when compared to the alternative schemes for which calculated results are presented in this study. A gradual blade mass increase radially from blade to blade which results in a maximum possible difference between the first and last adjacent blades results in a decrease in fatigue life.
References
1. Irretier H. Spectral Analysis of Mistuned Bladed Disk Assemblies by Component Mode Synthesis. In: Vibrations of Bladed Disk Assemblies: Proceedings of the ASME 9th Biennial Conference on Mechanical Vibration and Noise. 11-14 September 1983, Dearborn. New York: American Society of Mechanical Engineers; 1983, p. 115-125.
2. Zainchkovskij KS, Repeckij OV, Lopatin AB, Malikov VF, Ol'shevskij AG, Persiyanov VV, et al. Automation of Turbomachinery Strength Calculations /eds. OV Repeckij. Irkutsk: Irkutskii soyuz NIO; 1990, 100 p. (In Russ.)
3. Ryzhikov IN. To Gas Turbine Engine Rotor Durability Assessment. In: Aviamashinostroenie i transport Sibiri: sbornik statej VI Vserossijskoj nauchno-prakticheskoj kon-ferencii = Aircraft Engineering and Transport in Siberia: Collected articles of VI All-Russian scientific and practical conference. 13-16 April 2016, Irkutsk. Irkutsk: Irkutsk National Research Technical University; 2016, p. 288-294. (In Russ.)
4. Nguyen Dinh Duong, Ryzhikov VI. Study of the Influence of Turbomachine Impellers Parameters Detuning on their Free Oscillations. Vestnik Irkutskogo gosudarstven-nogo tehnicheskogo universiteta = Proceedings of Irkutsk State Technical University. 2010;4(44):22-26. (In Russ.)
5. Repetskiy O, Ryjikov I. Modeling and Simulation of Dynamic Processes with Help of Program Package BLADIS+. Innovations and Advanced Techniques in Systems, Computing Sciences and Software Engineering. 2008:219-220. https://doi.org/10.1007/978-1-4020-8735-6_41
6. Repetski O, Rygikov I, Springer H. Numerical Analysis of Rotating Flexible Blade-Disk-Shaft Systems. In: ASME 1999: International Gas Turbine and Aeroengine Congress and Exhibition. 1999. Available from: https://asmedigitalcollection.asme.org/GT/proceedings/GT 1999/78613/V004T03A034/248248 [Accessed 14th April 2020]. https://doi.org/10.1115/99-GT-317
7. Bladh R, Castanier MP, Pierre C. Component-Mode-Based Reduced Order Modeling Techniques for Mistuned Bladed Disks - Part I: Theoretical Models. Journal of Engineering for Gas Turbines and Power. 2001;123(1):89-99. https://doi.org/10.1115/1.1338947
8. Ewins DJ, Han ZS. Resonant Vibration Levels of a Mistuned Bladed Disk. Journal of Vibration, Acoustics, Stress, and Reliability in Design. 1984;106(2):211-217. https://doi.org/10.1115/1.3269171
9. Beirow B, Figaschewsky F, Kuhhom A, Bornhorn A. Modal Analyses of an Axial Turbine Blisk with Intentional Mistuning. Journal of Engineering for Gas Turbines and Power. 2018;140(1):012503. Available from: https://asmedigitalcollection.asme.org/gasturbinespower/a rticle-abstract/140/1/012503/374565/Modal-Analyses-of-an-Axial-Turbine-Blisk-With?redirectedFrom=fulltext [Accessed 21st April 2020].
https://doi.org/10.1115/1.4037588
10. Figaschewsky F, Kuhhom A. Analysis of Mistuned Blade Vibrations Based on Normally Distributed Blade Individual Natural Frequencies. In: ASME Turbo Expo 2015: Turbine Technical Conference and Exposition. 2015. Available from:
https://asmedigitalcollection.asme.org/GT/proceedings-abstract/GT2015/56772/V07BT32A020/237792 [Accessed 14th April 2020]. https://doi.org/10.1115/GT2015-43121
11. Wagner JT. Coupling of Turbomachine Blade Vibrations through the Rotor. Journal of Engineering for Power. 1967;89(4):502-513. https://doi.org/10.1115/1.3616718
12. Repetckii O, Nguyen Tien Quyet, Ryzhikov I. Investigation of Vibration and Fatigue Life of Mistuned Bladed Disks. In: Actual Issues of Mechanical Engineering (AIME 2017): Proceedings of the International Conference. 2729 July 2017, Tomsk. Tomsk; 2017, vol. 133, p. 702-707. https://doi.org/10.2991/aime-17.2017.114
13. Wagner MB, Younan A, Allaire P, Cogill R. Model Reduction Methods for Rotor Dynamic Analysis: A Survey and Review. International Journal of Rotating Machinery. 2010. Available from:
http://downloads.hindawi.com/journals/ijrm/2010/273716.p df [Accessed 25th April 2020]. https://doi.org/10.1155/2010/273716
14. Sinha A. Calculating the Statistics of Forced Response of a Mistuned Bladed Disk Assembly. AIAA Journal. 1986;24(11):1797-1801.
https://doi.org/10.2514/3.9526
15. Repetsky OV, Do Manh Tung. Analyzing Parameter Detuning Effect on Turbomachine Rotor Wheel Vibrations based on Spring-Mass Model. Vestnik Irkutskogo gosu-darstvennogo tehnicheskogo universiteta = Proceedings of Irkutsk State Technical University. 2013;10(81):56-63.
16. Happawana GS, Nwokah ODI, Bajaj AK, Azene M. Free and Forced Response of Mistuned Linear Cyclic
ВЕСТНИК ИРКУТСКОГО ГОСУДАРСТВЕННОГО ТЕХНИЧЕСКОГО УНИВЕРСИТЕТА 2020;24(4):766-767
Systems: a Singular Perturbation Approach. Journal of Sound and Vibration. 1998;211(5):761-789. https://d0i.0rg/l 0.1006/jsvi.1997.1349
17. Griffin JH, Hoosac TM. Model Development and Statistical Investigation of Turbine Blade Mistuning. Journal of Vibration, Acoustics, Stress and Reliability in Design. 1984;106(2):204-210. https://doi.org/10.1115/1.3269170
18. Griffin JH, Sinha A. The Interaction between Mistuning and Friction in the Forced Response of Bladed Disk Assemblies. Journal of Engineering for Gas Turbines and Power.
1985;107(1):205—211. https://doi.Org/10.1115/1.3239684
19. Pierre C, Murthy DV. Aeroelastic Modal Characteristics of Mistuned Bladed Assemblies: Mode Localization and Loss of Eigenstructure. National Aeronautics and Space Administration. 1991. Available from: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/1991 0018277.pdf [Accessed 14th April 2020].
20. Beirow B. Grundlegende Untersuchungen zum Schwingungsverhalten von Verdichterlaufrädern in Integralbauweise. Maastricht: Shaker Verlag; 2009, 172 p.
Библиографический список
1. Irretier H. Spectral analysis of mistuned bladed disk assemblies by component mode synthesis // Vibrations of Bladed Disk Assemblies: Proceedings of the ASME 9th Biennial Conference on Mechanical Vibration and Noise (Dearborn, 11-14 September 1983). New York: American Society of Mechanical Engineers, 1983. Р. 115-125.
2. Заинчковский К.С., Репецкий О.В., Лопатин А.Б., Маликов В.Ф., Ольшевский А.Г., Персиянов В.В. [и др.] Автоматизация прочностных расчетов турбомашин / под ред. О.В. Репецкого. Иркутск: Изд-во Иркутского союза НИО, 1990. 100 с.
3. Рыжиков И.Н. К оценке долговечности роторов газотурбинных двигателей // Авиамашиностроение и транспорт Сибири: сборник статей VI Всерос. науч.-практ. конф. (г. Иркутск, 13-16 апреля 2016 г.). Иркутск: Изд-во ИРНИТУ, 2016. С. 288-294.
4. Нгуен Динь Дыонг, Рыжиков В.И. Исследования влияния расстройки параметров рабочих колес турбо-машин на их свободные колебания // Вестник Иркутского государственного технического университета. 2010. № 4(44). С. 22-26.
5. Repetskiy O., Ryjikov I. Modeling and simulation of dynamic processes with help of program package BLADIS+ // Innovations and Advanced Techniques in Systems, Computing Sciences and Software Engineering. 2008. P. 219220. https://doi.org/10.1007/978-1-4020-8735-6_41
6. Repetski O., Rygikov I., Springer H. Numerical analysis of rotating flexible blade-disk-shaft systems // ASME 1999: International Gas Turbine and Aeroengine Congress and Exhibition. 1999. [Электронный ресурс]. URL: https://asmedigitalcollection.asme.org/GT/proceedings/GT 1999/78613/V004T03A034/248248 (14.04.2020). https://doi.org/10.1115/99-GT-317
7. Bladh R., Castanier M.P., Pierre C. Component-mode-based reduced order modeling techniques for mistuned bladed disks - Part I: theoretical models // Journal of Engineering for Gas turbines and Power. 2001. Vol. 123. Issue 1. P. 89-99. https://doi.org/10.1115/1.1338947
8. Ewins D.J., Han Z.S. Resonant vibration levels of a mistuned bladed disk // Journal of Vibration, Acoustics, Stress, and Reliability in Design. 1984. Vol. 106. Issue 2. P. 211-217. https://doi.org/10.1 115/1.3269171
9. Beirow B., Figaschewsky F., KQhhorn A., Bornhorn A. Modal analyses of an axial turbine blisk with intentional mistuning // Journal of Engineering for Gas Turbines and Power. 2018. Vol. 140. Issue 1. Р. 012503. [Электронный ресурс]. URL:
https://asmedigitalcollection.asme.org/gasturbinespower/a rticle-abstract/140/1/012503/374565/Modal-Analyses-of-an-Axial-Turbine-Blisk-With?redirectedFrom=fulltext (21.04.2020). https://doi.org/10.1115/1.4037588
10. Figaschewsky F., KQhhom A. Analysis of mistuned blade vibrations based on normally distributed blade individual natural Frequencies // ASME Turbo Expo 2015: Turbine Technical Conference and Exposition. 2015. [Электронный ресурс]. URL: https://asmedigitalcollection.asme.org/GT/proceedings-abstract/GT2015/56772/V07BT32A020/237792 (14.04.2020). https://doi.org/10.1115/GT2015-43121
11. Wagner J.T. Coupling of turbomachine blade vibrations through the rotor // Journal of Engineering for Power. 1967. Vol. 89. Issue 4. P. 502-513. https://doi.org/10.1115/1.3616718
12. Repetckii O., Nguyen Tien Quyet, Ryzhikov I. Investigation of vibration and fatigue life of mistuned bladed disks // Actual Issues of Mechanical Engineering (AIME 2017): Proceedings of the International Conference (Tomsk, 27-29 July 2017). Tomsk, 2017. Vol. 133. P. 702-707. https://doi.org/10.2991/aime-17.2017.114
13. Wagner M.B., Younan A., Allaire P., Cogill R. Model reduction methods for rotor dynamic analysis: a survey and review // International Journal of Rotating Machinery. 2010. [Электронный ресурс]. URL: http://downloads.hindawi.com/journals/ijrm/2010/273716.p df (25.04.2020). https://doi.org/10.1155/2010/273716
14. Sinha A. Calculating the statistics of forced response of a mistuned bladed disk assembly // AIAA Journal. 1986. Vol. 24. No. 11. P. 1797-1801.
https://doi.org/10.2514/3.9526
15. Репецкий О.В., До Мань Тунг. Анализ влияния расстройки параметров на колебания рабочих колес тур-бомашин на основе пружинно-массовой модели // Вестник Иркутского государственного технического университета. 2013. № 10(81). С. 56-63.
16. Happawana G.S., Nwokah O.D.I., Bajaj A.K., Azene M. Free and forced response of mistuned linear cyclic systems: a singular perturbation approach // Journal of Sound and Vibration. 1998. Vol. 211. Issue 5. P. 761 -789. https://doi.org/10.1006/jsvi.1997.1349
17. Griffin J.H., Hoosac T.M. Model development and statistical investigation of turbine blade mistuning // Journal of Vibration, Acoustics, Stress and Reliability in Design. 1984. Vol. 106. Issue 2. P. 204-210. https://doi.org/10.1115/1.3269170
ВЕСТНИК ИРКУТСКОГО ГОСУДАРСТВЕННОГО ТЕХНИЧЕСКОГО УНИВЕРСИТЕТА 2020;24(4):7Б6-767
18. Griffin J.H., Sinha A. The interaction between mistun-ing and friction in the forced response of bladed disk assemblies // Journal of Engineering for Gas Turbines and Power. 1985. Vol. 107. Issue 1. P. 205-211. https://doi.org/10.1115/1.3239684
19. Pierre C., Murthy D.V. Aeroelastic modal characteristics of mistuned bladed assemblies: mode localization and
Authorship criteria
Ryzhikov I.N., Nguyen Tien Quyet declare equal participation in obtaining and formalization of scientific results and bear equal responsibility for plagiarism.
Conflict of interests
The authors declare that there is no conflict of interests regarding the publication of this article.
The final manuscript has been read and approved by all the co-authors.
INFORMATION ABOUT THE AUTHORS
Igor N. Ryzhikov,
Cand. Sci. (Eng.), Associate Professor, Associate Professor of the Department of Engineering Technologies and Materials, Irkutsk National Research Technical University, 83, Lermontov St., Irkutsk 664074, Russia; L < e-mail: rin111@list.ru
Tien Q. Nguyen,
Cand. Sci. (Eng.),
Senior Researcher,
Viettel Aerospace Institute,
Hanoi, Toa nha Viettel 20 Tang,
Khu CNC Hoa Lac, Thach That, Ha Noi, Vietnam;
e-mail: cavoixanh@mail.ru
loss of eigenstructure // National Aeronautics and Space Administration. 1991. [Электронный ресурс]. URL: https://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/1991 0018277.pdf (14.04.2020).
20. Beirow B. Grundlegende Untersuchungen zum schwingungsverhalten von verdichterlaufrädern in integralbauweise. Maastricht: Shaker Verlag, 2009. 172 p.
Критерии авторства
Рыжиков И.Н., Нгуен Тьен Кует заявляют о равном участии в получении и оформлении научных результатов и в равной мере несут ответственность за плагиат.
Конфликт интересов
Авторы заявляют об отсутствии конфликта интересов.
Все авторы прочитали и одобрили окончательный вариант рукописи.
СВЕДЕНИЯ ОБ АВТОРАХ
Рыжиков Игорь Николаевич,
кандидат технических наук, доцент,
доцент кафедры машиностроительных
технологий и материалов,
Иркутский национальный исследовательский
технический университет,
664074, г. Иркутск, ул. Лермонтова, 83, Россия;
IX e-mail: rin111@list.ru
Нгуен Тьен Кует,
кандидат технических наук,
старший научный сотрудник,
Аэрокосмический институт Viettel,
г. Ханой, Toa nha Viettel 20 Tang, Khu CNC Hoa Lac,
Thach That, Ханой, Вьетнам;
e-mail: cavoixanh@mail.ru
ВЕСТНИК ИРКУТСКОГО ГОСУДАРСТВЕННОГО ТЕХНИЧЕСКОГО УНИВЕРСИТЕТА 2020;24(4):766-767
