Figure 7-10 - uploaded by Philipp Schmiechen
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shows the evolution of the spectrum of the signal with time of the rotor and Figure 7-11 that of the stator. Each vertical line in these spectrograms is a short-time, discrete, Fourier transform of part of the signal. The abscissa indicates the starting time of the particular time sample while the record length follows from the frequency resolution. By presenting the transforms in order of increasing starting time of the signal sample, the time variation of the spectra with time is visualised. In order to obtain visually smooth diagrams along the time axis, the analysed time records overlap.
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A linear theory is used to study the beam-wave coupling in a double-beam gyrotron traveling wave amplifier (Gyro-TWA). Numerical results show that the beam-wave coupling in this type of amplifier excites six waves, which include one constant-amplitude backward wave, and five forward waves consisting of two growing waves, two decaying waves and one...
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... Both graphs (a and b) are plotted in the rotating coordinate system attached to the gear, which rotates in the clockwise direction. It can be observed that the displacement distribution moves in the counter-clockwise direction with respect to the rotating system, shifting the nodal diameter of roughly 1/71 of the gear circumference over one mesh period [46,47]. Moreover, the travelling waves superimpose in such a way that their maximum and minimum amplitudes depend on the nodal diameter orientation. ...
The main objective of this paper is to analyse the dynamic behaviour of thin -rimmed/-webbed gears. A hybrid model is presented, which can capture the instant contact conditions between the teeth as well as the global dynamic behaviour of transmissions including the internal displacements and stresses in gear bodies. Finite elements are combined with condensed sub-structures, lumped parameter elements along with mortar-based mesh interfaces to connect non-compatible models. The results show that gear body dynamic displacements and stresses in thin-rimmed/-webbed gears are stationary in the fixed coordinate system attached to the meshing and relatively decoupled from the dynamic mesh forces. From a numerical viewpoint, it is found that numerical convergence on gear body stresses at high speeds requires significant computational times, thus highlighting the interest of lighter hybrid gear models as opposed to full finite element simulations.
... 与转静子碰摩作用下的转静子 耦合振动 [8][9][10][11] 。前者认为流体激励引起叶片的颤 振、强迫振动，叶片振动与流场相耦合，进而导 致流场出现周向压力脉动，作用于机匣即会激起 机匣的周向振动，属于流致振动范畴，这类耦合 振动发生的条件是转子叶片/叶鼓与机匣的某阶固 有频率 [12] 或行波波速 [13][14] 相近；后者是由转子不 对中、质量不平衡导致叶尖与机匣间的碰摩，而 碰摩产生的接触力进一步诱发机匣振动，其源头 为转子的不平衡响应，这类耦合振动发生的条件 是转子临界转速对应的频率或倍频与机匣的某阶 固有频率相近，因此一般发生在临界转速附近。 在《航空发动机设计手册》中通过转、静子 的行波频率与转动圆频率给出安全工作裕度 [15] 的 定义，以在设计过程中避免转、静子发生耦合振 动。此时可通过调整二者的固有频率使得发动机 在工作过程中避开耦合区。在工程应用中，通常 采用在结构板面或壳面上加筋的方式来改变结构 固有频率 [16] 。增加加强筋在增强结构刚度的同时 不会增加过多的质量，且在一定程度上可以抑制 结构的振动响应，是一种较为经济的减振调频方 法。加强筋不同的几何形状 [17][18] 、尺寸大小 [19] 、 铺设方式 [20] 以及位置分布 [21] 等均会对加筋结构固 有频率产生不同的影响，因此在实际应用中，需 要根据不同的工程需求对加强筋及机械结构进行 优化。在当前针对航空发动机等较为复杂的结构 进行优化时，通常根据目标函数对设定的参数进 行逆向优化 [22][23][24] [14] 与 Mathias Legrand [8] 等对发动机风扇部件转静子耦合振动的研究中中 合振动的危险模态进行判断。其共振裕度 [15] 定义 [2] 漆文凯, 王向辉. 基于转静子耦合的组合压气机动力 ...
Flow-induced coupling vibration between rotor blades/bladed drum and stator may occur when the modal frequencies with the same nodal diameter of the rotor and stator are close to each other. Structural optimization of the rotor and stator is commonly adopted to adjust their resonant frequencies, but the state-of-the-art inverse optimization method usually needs iterative calculation and lacks theoretical basis. An ad-hoc forward design method based on displacement and strain energy density distribution is proposed to tailor the frequency margin. This method provides a solid theoretical basis for designing the gap of the frequency, and the stiffened area can be preliminarily determined through a single modal analysis, which significantly shortens design cycles. The method is then applied to an industrial rotor/stator finite element model to improve the resonance margin. The influence of structural parameters of stiffeners on the modal characteristics of rotor and stator is investigated. The resonance margin of the dangerous mode is increased from 3.47% to 10.56% by stiffening. The generality of this method is good, and it is suitable for other types of engines.
... Note that this may not always be guaranteed due to pollution error -for more, the interested reader is referred to [17,18]. When quadratic elements are used, as is the case in all simulations for the present work, the rule of thumb recommends for mesh size h to resolve at least 3 to 5 elements per wavelength [19,20]. COMSOL's Free Tetrahedral feature is used to create an unstructured mesh with a maximum element size h max = 6.86 mm, corresponding to n res = 5 for a frequency of 10 kHz. ...
Developing high-fidelity digital twins of head and torso simulators (HATS) provides us with reliable simulations that could replace tedious measurements and facilitate the rapid and efficient design of AR/VR products. In this paper, we aim to develop and validate a digital twin of the Brüel and Kjær high-frequency HATS Type 5128. The digital twin uses an accurate scan of the HATS and captures the behavior of the ear simulator (Type 4620), including the full average human ear canal geometry and a termination coupler emulating an average eardrum impedance response. As a natural first step, finite-element acoustic simulations in COMSOL are set up to validate the far-field head-related transfer function (HRTF) with measurements for three spatial directions in the horizontal plane. To increase the confidence in the validation results, a rough convergence study is conducted for the simulations, and measurements are compared against the manufacturer’s reference measurements. Finally, we show how validation studies may be improved by investigating some of the commonly-used modeling assumptions.
... Consequently, significant savings in terms of computational costs can be achieved by applying SAFE to suitable classes of problems. Note that, to ensure accurate numerical results, researchers have proposed to use at least 6 to 10 nodes [4][5][6] per wavelength if a piecewise linear approximation of the displacement field is employed. This is a rather rough estimate and according to the findings discussed in Refs. ...
In this research, high-order shape functions commonly used in different finite element implementations are investigated with a special focus on their applicability in the semi-analytical finite element (SAFE) method being applied to wave propagation problems. Hierarchical shape functions (p-version of the finite element method), Lagrange polynomials defined over non-equidistant nodes (spectral element method), and non-uniform rational B-splines (isogeometric analysis) are implemented in an in-house SAFE code, along with different refinement strategies such as h-, p-, and k-refinement. Since the numerical analysis of wave propagation is computationally quite challenging, high-order shape functions and local mesh refinement techniques are required to increase the accuracy of the solution, while at the same time decreasing the computational costs. The obtained results reveal that employing a suitable high-order basis in combination with one of the mentioned mesh refinement techniques has a notable effect on the performance of the SAFE method. This point becomes especially beneficial when dealing with applications in the areas of structural health monitoring or material property identification, where a model problem has to be solved repeatedly.
... Initially considered rigid with a pre-distorted geometry, the casing's vibrations are in fact non-negligible, just as for the bladed-disk. These vibrations are propagated in both structures, according to the direction of rotor rotation, in a forward direction for the bladed-disk and in the opposite direction for the casing [Schmiechen, 1997]. Resulting from this consideration is the existence of a critical condition of interaction between the two structures, occurring at a given rotation velocity, for which both traveling wave speeds would be coincident. ...
Aeronautical compressor fans currently operate with minimal blade-casing clearance. This therefore makes the occurrence of rub events very likely. Under specific circumstances, the blade undergoes excessive amplification of contact-induced oscillations, called hereafter divergence, which can be critical for the structural integrity of the engine. This paper proposes an investigation of the mechanisms responsible for the blade divergence. Experiments are conducted on a fully instrumented laboratory set-up, consisting of a single flat blade being moved towards a rotating cylinder to initiate interactions, while monitoring the vibrations and the evolution of wear on the abradable coating. Two synchronization mechanisms have been identified as facilitating the divergence: (i) the inherent set-up synchronization between the vibration modes related to the horizontal and vertical motion of the blade; (ii) the preferential blade-coating interactions in the vicinity of periodically distributed irregularities of the abradable coating which act as a source of excitation of the vibrations.
... However, this is not sufficient to approximate the function. Schmiechen [46] recommends 6-10 points 565 per element for an accurate reconstruction. This threshold should never be crossed on the coarse grid. ...
A sequential estimator based on the Ensemble Kalman Filter for Data Assimilation of fluid flows is presented in this research work. The main feature of this estimator is that the Kalman filter update, which relies on the determination of the Kalman gain, is performed exploiting the algorithmic features of the numerical solver employed as a model. More precisely, the multilevel resolution associated with the multigrid iterative approach for time advancement is used to generate several low-resolution numerical simulations. These results are used as ensemble members to determine the correction via Kalman filter, which is then projected on the high-resolution grid to correct a single simulation which corresponds to the numerical model. The assessment of the method is performed via the analysis of one-dimensional and two-dimensional test cases, using different dynamical equations. The results show an efficient trade-off in terms of accuracy and computational costs required. In addition, a physical regularization of the flow, which is not granted by classical KF approaches, is naturally obtained owing to the multigrid iterative calculations. The algorithm is also well suited for the analysis of unsteady phenomena and, in particular, for potential application to in-streaming Data Assimilation techniques.
... Specific experimental benches reproducing the interactions between a geometrically simplified blade and an abradable coating [2][3][4][5][6][7] are generally used to conduct these types of investigations. To be the most representative of complex aircraft engines environment, few full-scale facilities were also designed and implemented [8][9][10]. Investigations of abradable seal systems led to a better understanding of wear mechanisms occurring during interactions [6,11], taking into account operating parameters such as the tangential velocity, the blade incursion rate, and temperature. ...
Abradable coatings are part of dynamic seals used in aircraft engines to reduce inter-stage leakages. However, during running, severe interactions may occur between the rotating blades and the coating of the casing leading to these parts getting damaged. Numerical approaches simulating these interactions could be a way to prevent these contact risks but they require both a characterization of the thermomechanical behavior of the abradable material and of the blade. This paper presents a quasi-static compression behavior study of a heterogeneous and porous AlSi-PE abradable used in the low-pressure compressor of aircraft engines. Combined with the results obtained for tests conducted at high compression strain-rates and extreme temperatures recently published in a previous paper (Skiba et al., in Dyn Behav Mater 6:213–223, 2020. 10.1007/s40870-020-00242-y), a thermoelastoviscoplastic bilinear constitutive law is proposed. This formulation allows the prediction of the macroscopic behavior of a heterogeneous and porous AlSi-PE abradable material from room temperature to 360 °C and with strain-rates ranging from 10–3 to 103 s−1.
... According to the Nyquist-Shannon sampling theorem, the sampling frequency of a signal must be at least twice as high as the highest frequency of interest. Building on this, Schmiechen (1997) recommends an additional factor between three and five in order to be able to consider the relevant effects using an approximation. This results in the typical number of six to ten nodes per wavelength (Marburg 2002). ...
The Noise, Vibration and Harshness (NVH) behaviour of vehicles is nowadays a key purchase criterion. Validated simulation methods are required to evaluate the NVH behaviour in the early phases of the product development process. This paper presents a methodology for model-based assessment of the NVH behaviour of a tractor with hydrostatic-mechanical power split transmission (PST). The aim of the simulation methodology is the assessment of structure-borne and airborne noise caused by transmission excitations. A multi-domain overall system model is proposed to accommodate numerous interactions and couplings within the system. The model combines the aspects of hydrostatic force excitation, structural dynamics and subsequent acoustic response. This paper introduces the main components of the simulation model and demonstrates its application for a given standard tractor with PST. Simulation results of structure-borne and airborne noise for a wide operation range of the tractor are presented and compared with measurement results. The comparisons of simulation and measurement results show a high degree of conformity. The modelling method therefore allows identifying and optimising acoustic anomalies in the NVH behaviour even before prototypes and measurements at system level are available.
... Zu der Frage, wie viele Knoten pro Wellenlänge erforderlich sind, ist eine umfangreiche Literaturbasis vorhanden. In [115] beispielsweise wird, basierend auf dem Abtasttheorem von Shannon, festgestellt, dass zwei Knoten pro Wellenlänge mathematisch prinzipiell ausreichend sind. Dennoch führt eine derartige Diskretisierung zu einer ungenügenden Auflösung der akustischen Wellen. ...
Die Arbeit im Bereich der Akustik ist, wie in vielen anderen Fachdisziplinen, vom Einsatz numerischer Berechnungsmethoden geprägt. Das wahrscheinlich am weitesten verbreitete Berechnungsverfahren ist die zu den Diskretisierungsverfahren gehörende Finite-Elemente-Methode (FEM). Trotz stetig steigender Leistung der verwendeten Computer sind derartige Berechnungen speicher- und zeitintensiv. Daher haben sich Modellreduktionsverfahren etabliert. Dennoch stellt die Akustik besondere Herausforderungen, die durch existierende Modellreduktionsverfahren nur unzureichend adressiert werden.
Diese Arbeit entwickelt ein Multi-Modell-Verfahren speziell für Berechnungen in der Akustik im Frequenzbereich. Das Verfahren ermöglicht den Einsatz vereinfachter Modelle gemeinsam mit einem zu lösenden Referenzmodell zur Lösung einer gegebenen Berechnungsaufgabe. Das Referenzmodell ist dasjenige Modell, von dem im gesamten Frequenzbereich eine ausreichende Ergebnisqualität erwartet wird. Die zur Verfügung stehenden vereinfachten Modelle werden gegen das Referenzmodell validiert. Die Kernaspekte der Arbeit sind die Entwicklung des erforderlichen Validierungsprozesses sowie einer die Rechenzeitersparnis absichernden Budgetplanung.
Das entwickelte Verfahren wird für einzelne Berechnungen verifiziert und auf stochastische Untersuchungen angewendet. Es wird eine Rechenzeitersparnis bei einem akzeptablen Fehler der approximierten Systemantwort erreicht, dabei ermöglicht das Verfahren die Abwägung zwischen einer genauen, langsamen Lösung und einer schnellen Näherungslösung. Insgesamt steht mit dem entwickelten Multi-Modell-Verfahren ein Ansatz zur Verfügung, welches die Anforderungen der Berechnung von Wellenausbreitung mit numerischer Berechnung berücksichtigt und dadurch beschleunigt.
... While a reduction of operating clearances provides a very significant improvement of the overall efficiency [19,20,21], it also unavoidably favors structural contacts between rotating and static components of the engine [22,23]. Following such contact events, a variety of complex phenomena including blade failures [24], abnormal increases of temperature [25], and even modal interactions [26,27] have been reported in the literature. This is the reason why the analysis and prediction of blade-tip/casing contact events have been a very active field of research over the past decade [28,29,30,31,32,33,34,35,36]. ...
This work focuses on the experimental and numerical characterization of stress levels within a shroud segment made of ceramic matrix composite (CMC) material undergoing repeated blade contacts. The dedicated experimental set-up consists in a rotating disk with three notched mock blades that impact the shroud segment as they rotate. The underplatform on which the shroud is fixed progressively gets closer to the blades, so that the abradable layer deposited onto the shroud is progressively worn out from a blade revolution to another. Four sensors, located under the shroud segment, are used to record forces during the experiments. Different blade/shroud relative positions are tested, in such a way that impacts may occur at distinct locations. In addition to the experimental tests, a numerical model is built based on both reduced order models of the blade and the shroud segment, in order to numerically predict the forces at the four sensors. This numerical model is first calibrated based on a single reference test case to retrieve the same force magnitude at each sensor. Then, the other tests are simulated using the calibrated model. Predicted contact forces are in good agreement with experimental data, which validates the numerical model. Finally, simulations are carried out considering engine-like conditions, which could not be reproduced using the experimental test bench. The influence of several parameters (radial velocity, impact location, number of blades in contact and angular speed) is analyzed in detail.
