Cyclic Symmetric Structures Research Articles

Physics-informed machine learning approach for reduced-order modeling of integrally bladed rotors: Theory and application

Integrally bladed rotors are commonly used in aircraft and rocket turbomachinery and known to exhibit complex dynamics when subject to operational loading conditions. Though nominally cyclic symmetric structures, in practice, cyclic symmetry is destroyed due to mistuning caused by random sector-to-sector imperfections in material properties and geometry. Simulating mistuned blisk dynamics using high-fidelity models can be computationally expensive, thus, a variety of physics-based reduced-order models have been previously developed. However, these models cannot easily incorporate experimental data nor leverage potential benefits of data-driven and machine-learning-based approaches. Here, we present a novel first-of-its-kind physics-informed machine learning modeling approach that incorporates physical laws directly into a novel network architecture while maintaining a sector-level viewpoint. The approach is combined with an assembly procedure resulting in a significantly smaller linear system based on blade-alone response data, and can directly incorporate physical response data like that measured with blade tip timing and/or traveling-wave excitation. Validation is shown using a large-scale finite-element model, with multiple traveling-wave forced-response predictions and response selection cases considered. Using only as little as a single degree of freedom per sector from the blade tip, this approach shows high accuracy relative to high-fidelity simulations.

Topology optimization for infinite fatigue life of cyclic symmetric structures subjected to non-proportional loading

This paper presents a density based topology optimization method for infinite fatigue life constraints of non-proportional load cases, with a specific focus on parts with cyclic symmetry. Considering non-proportional loads in topology optimization significantly broadens the types of design problems that can be handled. The method estimates the local variation in Signed von Mises stress using a smooth min/max function and constrains the resulting stress amplitude using established stress based topology optimization methods. Accounting for non-proportionality of loading significantly increases the computation cost with respect to existing proportional methods, as the time-varying stress field needs to be computed. Inertia effects are neglected in the structural analysis. Therefore, a quasi-static analysis is used to obtain the stress history. To reduce the computational cost, advantage is taken of cyclic symmetric properties to reduce the number of necessary time steps to evaluate. This reduces the computational cost roughly proportional to the number of unique load time steps present in the repeated segments as opposed to a standard implementation. The method is tested on numerical examples in 2D and 3D for both proportional and non-proportional loads and was found to be locally accurate up to the accuracy of the constraint aggregation.

A scaled boundary finite element partitioning based reduced order algorithm for the elastic analysis of cyclically symmetric structures

AbstractAn efficient reduced order algorithm is proposed for the elastic SBFE analysis for 2‐D cyclic symmetric structures with or without a common node. The general stiffness matrices of scaled boundary finite element (SBFE) is proved to be block‐circulant, and can be constructed via the basic region, instead of the whole computing domain. Thus, the expense on eigenvalue analysis required in generating stiffness matrix can be significantly reduced, and the solution scale can be reduced by partitioning the system equation into a series of small independent subproblems. Furthermore, the presented algorithm is combined with the Woodbury formula to reduce the computational cost on the analysis of incomplete cyclically symmetric structures, the original system equation is transformed into the equations with block‐circulant coefficient matrices, which is efficiently solved by partitioning algorithm. Four numerical examples are provided to demonstrate the effectiveness and advantages of the proposed approach.

Model reduction of a cyclic symmetric structure exhibiting geometric nonlinearity with a normal form approach

This papers deals with the computationally costly simulation of the frequency response function of geometrically nonlinear structures exhibiting cyclic symmetry. Its objective is to propose a reduction method that extends the model of a single sector reduced via the direct normal form to a full cyclically symmetric system. After validation on simple structures, the methodology is applied on a 3D meshed blade, representative component of the new generation of high bypass ratio turbofans. Based on both the proposed reduction methodology and theoretical conditions that predict the possible appearance of internal resonances as a function of the number of sectors and the engine order of the excitation, the complex nonlinear dynamics of a finely meshed full fan cyclic system is then investigated. This could not have been achieved without an appropriate reduction method. Branch switching algorithm, stability analysis, and bifurcation tracking compose the numerical tools employed along the harmonic balance method to simulate the forced response of the structure.

Impact of Mistuned Underplatform Dampers on the Nonlinear Vibration of Bladed Disks

Abstract Before the final experimental validation and certification of a turbo-engine, designers perform a numerical simulation of its vibratory properties, among other things, in order to estimate its lifespan and adjust the design in an optimization process. One possible practical solution to decrease the vibratory response is to add underplatform dampers to the system. These components dissipate energy by friction and are widely employed in turbomachinery. However, a specific underplatform damper is usually efficient only for a specific mode. The purpose of this work is to investigate the possibility of adding different kinds of underplatform dampers to the cyclic structure in order to decrease the vibratory energy over a larger panel of modes. Different methods exist to determine the vibrations of nonlinear cyclic symmetric systems, but creating a robust methodology to account for the additional effect of mistuning remains a big challenge in the community. In this paper, the structure is mistuned through the friction coefficient of the dampers and not by altering its geometry, as is usually done in the literature. First, assuming a cyclic symmetric structure, the performance of the dampers is assessed for specific modes. Then, employing a method recently developed, the efficiency of an intentional mistuning pattern of underplatform dampers is studied and an optimal pattern proposed.

Optimal analysis for optimal design of cyclic symmetric structures subject to frequency constraints

In general, the design optimization of large-scale structural systems is an extremely computationally expensive process. However, optimal structural analysis provides efficient and practical methods of structural analysis which can be employed as powerful tools to reduce significantly the computational time of structural analyses performed in the structural optimization process. In this paper, an efficient eigensolution method for free vibration analysis of rotationally repetitive structures is developed for the optimal design of cyclic symmetric structures subject to frequency constraints. Through the present eigensolution method, the initial free-vibration eigenproblem is decomposed into some sub-eigenproblems with much smaller dimensions using an efficient block-diagonalization technique. The main advantage of the present method is that it requires significantly less computational time and memory than the existing classical eigensolution method. Finally, two large-scale numerical examples are given to demonstrate the efficiency and accuracy of the present eigensolution method and compare it with the existing classical method in both terms of computational time and memory requirements. Numerical results indicate that the present eigensolution method not only guarantees the accuracy of the analysis results but also significantly reduces the computational time and memory required compared to the existing classical method. More importantly, the computational time of the optimization process is also significantly reduced. Optimization is carried out by a set-theoretical-based Jaya algorithm, named ST-JA. The proposed ST-JA aims to enhance the diversification and intensification capabilities of the original JA and strike a fine balance between them. Optimization results confirm that the proposed ST-JA outperforms the original JA and has superior or comparable performance to other state-of-the-art optimization algorithms.

Rub-Impact Dynamics of Shrouded Blades under Bending-Torsion Coupling Vibration

Shroud devices which are typical cyclic symmetric structures are widely used to reduce the vibration of turbine blades in aero engines. Asymmetric rub-impact of adjacent shrouds or aerodynamic excitation forces can excite the bending-torsion coupling vibration of shrouded blades, which will lead to complex contact motions. The aim of this paper is to study the rub-impact dynamic characteristics of bending-torsion coupling vibration of shrouded blades using a numerical method. The contact-separation transition mechanism under complex motions is studied, the corresponding boundary conditions are set up, and the influence of moments of contact forces and aerodynamic excitation forces on the motion of the blade is considered. A three-degree-of-freedom mass-spring model including two mass blocks with the same size and shape is established to simulate the bending-torsion coupling vibration, and the dynamic equations of shrouded blades under different contact conditions are derived. An algorithm based on the fourth-order Runge–Kutta method is presented for simulations. Variation laws of the forced response characteristics of shrouded blades under different parameters are studied, on the basis of which the method to evaluate the vibration reduction characteristics of the shrouded blade system when the motion of the blade is chaotic is discussed. Then, the vibration reduction law of shrouded blades under bending-torsion coupling vibration is obtained.

Interaction Between Coriolis Forces and Mistuning on a Cyclic Symmetric Structure With Geometrical Nonlinearity

Abstract An investigation of the interaction between Coriolis forces and mistuning on a cyclic symmetric structure is presented in this paper. The sensitivity of the eigenvalues and eigenvectors to mistuning is first studied with the perturbation method. A lumped parameter model is used to perform a modal analysis using a numerical approach after which geometrical nonlinearity is added to compare behavior with the linear case. Two different modes are thoroughly investigated for different rotational speeds, the first with an eigenvalue isolated from the others and the second presenting a frequency veering zone. The evolution from a standing wave domination at low speeds to a traveling wave domination at high speeds is observed for the isolated mode, whereas a standing wave domination remains around the veering zone for the second mode studied. It is also shown that the geometrical nonlinearity reinforces the mistuning effect versus the Coriolis forces.

A Prediction Method with Altering Equivalent Stiffness for Damping Evaluation of Shrouded Bladed Disk Dynamic Systems

In turbomachinery bladed disks are typical cyclic symmetric structures where high-cycle fatigue of the blades can easily occur. Increasing the shroud structure is an effective blade vibration reduction method using the dry friction of the shroud contact surface to dissipate vibration energy. During one vibration cycle of the blade, the contact surface of the shroud may have one or several states of stick, slip, and separation due to the different amplitude of blade, which affects not only the damping, but also the stiffness of the system. This paper proposes a method to analyze the damping characteristics of blades with shroud considering the change of equivalent contact stiffness of the shroud by using the finite element method. With a 2D contact model, this method calculates the equivalent damping and obtains the damping characteristic curve to evaluate the damping of shroud. During the design phase, the objective is that when resonance occurs, the shroud can provide the optimal damping ratio under the allowable vibration stress of the blade. The parameter sensitivity of the damping characteristic for one bladed disk with parallel shroud is investigated. It is shown that vibration phase angle, contact stiffness, contact force, friction coefficient significantly influence the damping characteristics.

Crack identification in cyclic symmetric structures based on relative indicators of frequency separation

Abstract. Cyclic symmetric structures are an important class of structures in the fields of civil and mechanical engineering. In order to avoid accidents due to cracks in such structures, an effective method for crack identification is presented in this paper. First, the dynamic model of cyclic symmetric structures with gapless cracks is developed using a structure's sector model and rotation transformation. Then, the effects of cracks on the free vibration characteristics of a cracked cyclic symmetric structure are addressed, with particular interests in the distortion of mode shapes and the shift and split of natural frequencies. On the basis of crack-induced phenomena, an effective method based on relative indicators of frequency separation is developed for quantitative crack identification. Numerical results illustrate that the relative indicators are sensitive to small cracks and insensitive to the predicting model used during analysis. Finally, the method is validated by experiments conducted on an impeller-shaft assembly. The results show the effectiveness of the frequency separation indicators in crack identification in cyclically symmetric structures.

Design of dry friction and piezoelectric hybrid ring dampers for integrally bladed disks based on complex nonlinear modes

This paper investigates a new passive damper coupling the energy dissipative mechanisms of dry friction and piezoelectric shunting circuit for Integrally Bladed Disks (blisks). The idea is to distribute piezoelectric material to the dry friction ring so that the elastic deformation of the dry friction ring is utilized to generate additional damping. Mounting piezoelectric material on the additional structure instead of the host structure is a more practical alternative to install piezoelectric damping devices to realistic mechanical systems. This improves the reliability and maintainability of both dampers and host structures. Based on the concept of Complex Nonlinear Modes (CNMs), the dry friction damping effect of the proposed damper is measured through the modal damping ratio for target modes. Modal Electromechanical Coupling Factor (MEMCF) is extended to nonlinear electromechanical coupling systems in order to quantify the additional piezoelectric damping ratio. The damping effect of the hybrid damper is also evaluated by the maximal response amplitude in the frequency domain by the forced response analysis. On one hand, it validates the results from the nonlinear modal analysis, on the other hand, it serves as a reference to choose the optimum design parameters of the hybrid damper. Steady-state response of the cyclic symmetric structure under engine-order excitation is calculated by the Multi-Harmonic Balance Method (MHBM). A phenomenological lumped parameter model representing a blisk with a hybrid ring damper is firstly studied to demonstrate the methodology. Then a blisk finite element model is used as a demonstrator to quantitatively verify the effectiveness of the idea. It is shown that for those modes that can be damped by the dry friction damper, the addition of piezoelectric material further enhances the damping effect.

Energy transfer between nodal diameters of cyclic symmetric structures exhibiting polynomial nonlinearities: Cyclic condition and analysis

The recent trend of new design for large slender blades in turboengines facilitates structural large deformation and geometrical nonlinear vibratory effects. The cyclic symmetry properties of such structures, combined with these nonlinearities give rise to specific complex phenomena such as internal resonances or energy localization. Robust and efficient methods have been developed to recover the solutions of such problems but they currently lack proper tools to analyze the results. In this paper, a formula is provided to compute polynomial nonlinear forces directly in the cyclic (spectral) domain of a cyclic symmetric structure. This new approach enables to express the equation of motion directly in the spectral domain, and offers two main advantages. It first provides a reduction of the system by determining a priori which nodal diameters will be coupled and by giving a closed-form expression for a direct evaluation of the cyclic nonlinear force. The proposed approach also facilitates results interpretations. The method is applied to a simplified bladed disk with a cubic nonlinearity that models symmetric large deformation, although the analytical development is valid for any polynomial nonlinearity. A detailed analysis of the different phenomena occurring in the cyclic structure excited along a particular nodal diameter is provided. More precisely, a multiple scales analysis is performed and yields interesting insights of internal resonances between different nodal diameters. This analytical method is completed with several numerical simulations involving the harmonic balance method and bifurcation algorithms. Both of these analyses recover coherent results on the prediction of different internal resonances for traveling or standing wave excitations.

Modal Analysis of Mistuned Turbine Blade Packet Due to Combined Blade and Lacing Wire Damage

The turbine disk blade system is a cyclic symmetric structure, initially tuned with all its blades perfectly identical in geometry and material properties; similarly interconnecting lacing wires are of equal stiffness. The cyclic symmetry of the bladed disks gets destroyed due to small differences in material properties or geometric variation between individual blades or lacing wires causing mistuning. Although mistuning is typically small, it can have a drastic effect on the dynamic response of the system. In particular, mistuning can also cause vibration localization for a few blades and the associated concentration of vibration energy can lead to an increase in blade amplitude and stress levels. Numerical simulations are performed with the characteristic equations of the simplified continuum model. Two different damage severity indices are included in the model to study the combined effect of cracked blades and damaged lacing wires on the natural frequencies of grouped blades. This study highlights the characteristic changes in the sub modal frequencies under combined damage in a stand still position. Although the major cause of mistuning is blade damage, lacing wire damage is more frequent and often acts as a precursor to blade damage and thus the present study focuses on mistuning due to combined damage.

Self-Adaptive Fault Feature Extraction of Rolling Bearings Based on Enhancing Mode Characteristic of Complete Ensemble Empirical Mode Decomposition with Adaptive Noise

Originally, a rolling bearing, as a key part in rotating machinery, is a cyclic symmetric structure. When a fault occurs, it disrupts the symmetry and influences the normal operation of the rolling bearing. To accurately identify faults of rolling bearing, a novel method is proposed, which is based enhancing the mode characteristics of complete ensemble empirical mode decomposition with adaptive noise (CEEMDAN). It includes two parts: the first is the enhancing decomposition of CEEMDAN algorithm, and the second is the identified method of intrinsic information mode (IIM) of vibration signal. For the first part, the new mode functions (CIMFs) are obtained by combing the adjacent intrinsic mode functions (IMFs) and performing the corresponding Fast Fourier Transform (FFT) to strengthen difference feature among IMFs. Then, probability density function (PDF) is used to estimate FFT of each CIMF to obtain overall information of frequency component. Finally, the final intrinsic mode functions (FIMFs) are obtained by proposing identified method of adjacent PDF based on geometrical similarity (modified Hausdorff distance (MHD)). FIMFs indicate the minimum amount of mode information with physical meanings and avoid interference of spurious mode in original CEEMDAN decomposing. Subsequently, comprehensive evaluate index (Kurtosis and de-trended fluctuation analysis (DFA)) is proposed to identify IIM in FIMFs. Experiment results indicate that the proposed method demonstrates superior performance and can accurately extract characteristic frequencies of rolling bearing.

Dynamic model order reduction of blisks with nonlinear damping coatings using amplitude dependent mistuning

In this paper, a reduced order model is developed to simulate the dynamics of a bladed disk or blisk with nonlinear damping coatings adhered to its blades. The nonlinear forces exerted by these coatings on the underlying linear blisk structure are a function of the local strain. It is known that coatings modify the stiffness and damping of each blade depending on its amplitude. Blisks, which are designed as perfectly cyclic symmetric structures with identical blades, never behave as such in practice due to various uncertainties encountered during their manufacturing. This asymmetry in the structure is also referred to as mistuning. Mistuning in the linear blisk structure, which causes different blades to respond with non-identical amplitudes, interacts with the coating nonlinearity to yield a mistuning pattern which depends on the blade amplitudes. Additional stiffness and damping parameters that are dependent on the blade amplitude are introduced into a reduced linear model to formulate the nonlinear reduced order model. It is found that this model captures the nonlinear amplitude dependent mistuning effect and predicts the nonlinear coated blisk responses accurately near isolated blisk mode families in blade-dominated frequency regions where these coating effects are likely to be dominant. Significant reductions in the computational effort are achieved through this reduction.

Periodic Layout Optimization of Cyclic Symmetric Structure

This paper proposes a new optimization method to solve periodic layout optimization of a cyclic symmetric structure by means of the guide-weight method. The mathematical model for periodic layout optimization under multiple constraints is built on a single working condition. By constructing a virtual fan-shaped subdomain, periodic layout optimization of the cyclic symmetric structure is transformed into the conventional topology optimization of the virtual fan-shape subdomain. Starting from the Lagrange equation, the general iterative criterion of the virtual fan-shaped subdomain is deduced by the guide-weight method. Taking the problem of minimum compliance as an example, the iterative criterion between relative density and strain energy density of elements in the virtual fan-shaped subdomain is obtained under the condition of mass constraint. The physical significance of iterative criterion is explained and the optimization flow of periodic layout optimization is provided. Two examples of the cyclic symmetric structure show that the proposed method is effective and robust to solve the problem of periodic layout optimization.

Rolling Bearing Fault Diagnosis Based on EWT Sub-Modal Hypothesis Test and Ambiguity Correlation Classification

Because of the cyclic symmetric structure of rolling bearings, its vibration signals are regular when the rolling bearing is working in a normal state. But when the rolling bearing fails, whether the outer race fault or the inner race fault, the symmetry of the rolling bearing is broken and the fault destroys the rolling bearing’s stable working state. Whenever the bearing passes through the fault point, it will send out vibration signals representing the fault characteristics. These signals are often non-linear, non-stationary, and full of Gaussian noise which are quite different from normal signals. According to this, the sub-modal obtained by empirical wavelet transform (EWT), secondary decomposition is tested by the Gaussian distribution hypothesis test. It is regarded that sub-modal following Gaussian distribution is Gaussian noise which is filtered during signal reconstruction. Then by taking advantage of the ambiguity function superiority in non-stationary signal processing and combining correlation coefficient, an ambiguity correlation classifier is constructed. After training, the classifier can recognize vibration signals of rolling bearings under different working conditions, so that the purpose of identifying rolling bearing faults can be achieved. Finally, the method effect was verified by experiments.

A Prediction Method for the Damping Effect of Ring Dampers Applied to Thin-Walled Gears Based on Energy Method

In turbomachinery applications, thin-walled gears are cyclic symmetric structures and often subject to dynamic meshing loading which may result in high cycle fatigue (HCF) of the thin-walled gear. To avoid HCF failure, ring dampers are designed for gears to increase damping and reduce resonance amplitude. Ring dampers are installed in the groove. They are held in contact with the groove by normal pressure generated by interference or centrifugal force. Vibration energy is attenuated (converted to heat) by frictional force on the contact interface when the relative motion between ring dampers and gears takes place. In this article, a numerical method for the prediction of friction damping in thin-walled gears with ring dampers is proposed. The nonlinear damping due to the friction is expressed as equivalent mechanical damping in the form of vibration stress dependence. This method avoids the forced response analysis of nonlinear structures, thereby significantly reducing the time required for calculation. The validity of this numerical method is examined by a comparison with literature data. The method is applied to a thin-walled gear with a ring damper and the effect of design parameters on friction damping is studied. It is shown that the rotating speed, geometric size of ring dampers and friction coefficient significantly influence the damping performance.

On the Existence of Self-Excited Vibration in Thin Spur Gears: A Theoretical Model for the Estimation of Damping by the Energy Method

The gear is a cyclic symmetric structure, and each tooth is subjected to a periodic mesh force. These mesh forces have the same phase difference tooth by tooth, which can excite gear vibrations. The mechanism of additional axial force caused by gear bending is shown and examined, which can significantly affect the stability of a self-excited thin spur gears vibration. A mechanical model based on energy balance is then developed to predict the contribution of additional axial force, leading to the proposed numerical integration method for vibration stability analysis. By analyzing the change in the system energy, the occurrence of the self-excited vibration is validated. A numerical simulation is carried out to verify the theoretical analysis. The impacts of modal damping, contact ratio, and the number of nodal diameters on the stability boundaries of the self-excited vibration are revealed. The results prove that the backward traveling wave of the driven gear as well as the forward traveling wave of the driving gear encounter self-excited vibration in the absence of sufficient damping. The model can be used to predict the stability of the gear self-excited vibration.

Multistability and localization in forced cyclic symmetric structures modelled by weakly-coupled Duffing oscillators

Many engineering structures are composed of weakly coupled sectors assembled in a cyclic and ideally symmetric configuration, which can be simplified as forced Duffing oscillators. In this paper, we study the emergence of localized states in the weakly nonlinear regime. We show that multiple spatially localized solutions may exist, and the resulting bifurcation diagram strongly resembles the snaking pattern observed in a variety of fields in physics, such as optics and fluid dynamics. Moreover, in the transition from the linear to the nonlinear behaviour isolated branches of solutions are identified. Localization is caused by the hardening effect introduced by the nonlinear stiffness, and occurs at large excitation levels. Contrary to the case of mistuning, the presented localization mechanism is triggered by the nonlinearities and arises in perfectly homogeneous systems.