Component Mode Synthesis Method Research Articles

Vibration of a metaplate made of an acoustic black hole and local resonators

In recent years, acoustic black holes (ABHs) exhibit fascinating damping effects at mid-to-high frequencies. However, flexural waves having long wavelength cannot be trapped in the ABH pit when there is a residual thickness at the ABH tip, therefore below the cut-on frequency the ABH effect fails to work. While locally resonant acoustic metamaterials allows mitigating vibrations in subwavelength scale, usually in a narrow band. In this paper we combine the advantages of the metamaterials and ABHs (MMABH) to control vibrations in the whole frequency band. The designed MMABH plate remains the same weight to the reference uniform plate. To characterize the MMABH performance, an artificial spring component mode synthesis (GECMS) method is suggested, based on the modal coupling between the resonators and the ABH plate via springs. The band gap for the infinite periodic MMABH and the modes for the finite MMABH have been accurately predicted with the GECMS, compared to a reference FEM model. Numerical results show that the low-frequency peaks of the ABH plate can be substantially suppressed when the resonators with proper loss factor are set at its first resonant frequency. The proposed MMABH shows great potentials for light-weight and whole-band vibration reduction.

An artificial spring component mode synthesis method for built-up structures

In this paper, we develop a component mode synthesis method that relies on artificial springs to connect the subsystems of a built-up structure. The modal behaviour of the subsystems can be computed using different approaches, from analytical to numerical methods, depending on their complexity. The advantage of using artificial springs to link the subsystems is that one can model them with the Rayleigh–Ritz method, which is problematic in component mode synthesis. As known, the approximating functions must satisfy the system boundary conditions and the use of artificial springs avoids this difficulty. Moreover, and as in other component mode synthesis approaches, one can build reduced-order models at the subsystem level to significantly diminish the dimensions of the entire built-up structure, resulting in substantial reduction of the computational cost when performing numerical simulations. The method is first presented for the simple case of two axially connected beams. Two more complex cases are then addressed. The first one deals with two beams connected at right angles and the second one deals with an internal floor attached to a cylindrical shell, which reminds of some aeronautical structures. The performance of the method is carefully analysed and validated against finite element simulations.

Nonlinear vibration analysis and stability analysis of rotor systems with multiple localized nonlinearities

This study proposes a methodology to analyze the nonlinear vibration characteristics of rotor systems with multiple localized nonlinearities adopting the Finite Element Method (FEM), free interface Component Mode Synthesis (CMS) method, and modified Incremental Harmonic Balance (IHB) method. The rotor system is supported by squeeze film dampers (SFDs) on both sides, and at the nodes of the SFD arrangement, strong local nonlinearities will appear due to fluid-film forces. The methodology to analyze the nonlinear vibration characteristics of the system by reducing the degree of freedom of the rotating system with multiple local nonlinear factors and combining with the IHB method is proposed for the first time in this paper. The FEM is used to write motion equations in components, and the CMS method is applied to reduce the degrees of freedom of linear components. The IHB method is used to solve the motion equations of the nonlinear system. The system has one linear component and two nonlinear components. For linear components, modal coordinates are used, and for nonlinear components, the original physical coordinate system is used. By synthesizing these three components, the motion equation of the whole system is created. In order to validate the effectiveness of the method, the results obtained by the proposed method are compared with the data in the published literature, and the system responses are considered when specific parameters are changed. The stability analysis of the calculated solutions is carried out using the Floquet theory.

Experimental and numerical dynamic analysis of marine herringbone planetary gearbox supported by journal bearings

Previous studies supposed that planet bearings are springs with isotropic stiffness and damping, whereas the stiffness and damping of journal bearings are asymmetric and interactive. As a result, a coupled dynamic model of the herringbone planetary gear train (HPGT) with journal bearings is established, considering the planetary bearing eccentricities and the asymmetry and interaction of journal bearing dynamic parameters (stiffness and damping). Besides, the gearbox casing condensation model is proposed by the Component Mode Synthesis method, is verified by modal tests, and is incorporated into the established dynamic model. The dynamic performances of the marine gearbox for different operating conditions and planetary bearing parameters are explored under the uncoupled and coupled casing conditions. The result shows that different operating conditions may result in different effects of gearbox casing flexibility on meshing force and load-carrying capacity. Meshing forces and bearing capacities are positively correlated with width-diameter ratios, while they are inversely correlated with radial clearances of planet bearings. A test rig for the herringbone planetary gearbox with journal bearings is constructed for verifying the reliability of the proposed model. Experiments demonstrate the maximum relative error between the measurement and prediction of structural noise of the gearbox casing is 1.72%. This work provides a theoretical basis for the optimal design of the marine gearbox.

Vibroacoustic mitigation for a cylindrical shell coupling with an acoustic black hole plate using Gaussian expansion component mode synthesis

In this paper, we consider a composite cylindrical shell having an internal thin plate. When some ABHs are embedded on the interior floor, the vibration of the cylindrical shell and the sound power radiated from it are expected to be effectively mitigated. To characterize the built-up system, the Gaussian expansion method (GEM) is employed to carry out the modal parameters of the shell and the plate, respectively, where mode truncation is possible for model order reduction. Then the Gaussian expansion component mode synthesis (GECMS) method is developed to describe the final modes of the whole system. The accuracy of the GECMS is validated against reference finite element simulations. To manifest the coupling between the components, the modal participation factors of the shell and the ABH plate are carried out. Furthermore, the sound radiation model for the cylindrical shell is built, together with the non-negative intensity strategy for sound source localization. Results show that the vibration and the sound power level of the shell can be remarkably suppressed, thanks to the exceptional damping effect of the ABHs, especially when the excitation force locates on the ABH plate. The present study is dedicated to pushing the applications of the ABHs.

Multi-mode vibration attenuation of mistuned bladed disks by frictional tuned mass dampers array

An effective method is presented for multi-mode vibration attenuation of mistuned bladed disk (blisk) by employing ring-like array structure composed of multiple frictional tuned mass dampers (FTMDs). The multi-mode vibration attenuation is realized by arranging multiple series of FTMDs. A novel design scheme of the FTMD array is presented. 3D finite element method is employed for quantitative modelling of the blisk-FTMD array system, and the rotordynamic effects of Coriolis forces, spin softening and centrifugal stiffening are taken into account to reach better tuning in natural frequencies in the rotating conditions. The free-interface component mode synthesis method is used to reduce the order of the model. An efficient technique, which combines the harmonic balance method, the analytical formulation of Jacobian matrix and the parameterized Newton-Raphson method, is employed to solve the nonlinear equations. Finally, a blisk, with its first 4ND and 5ND modes being targeted, is employed as an example to illustrate the effectiveness of the presented method. Numerical results show that the method can obviously mitigate the multi-mode vibration of blisk with wide levels of mistuning and is robust against the small tuning deviations of the oscillators.

A reduced-order improved rational polynomial method for viscoelastically damped structures considering ambient temperature effect

This study focuses on the development of a framework for the seismic analysis of viscoelastically damped structures to investigate the ambient temperature effect on seismic responses. The improved rational polynomial model is used to describe the temperature dependence of the complex stiffness of viscoelastic (VE) dampers. The free‐interface component mode synthesis method is employed to improve the computing efficiency. Numerical simulations of a 20-storey benchmark building with added VE dampers under historical earthquakes are presented to illustrate the ambient temperature effect. The results show that the proposed framework is feasible in the seismic analysis of viscoelastically damped structures considering ambient temperature effect. The floor displacement and interstory drift reductions decrease with the increase of the ambient temperature, while fluctuations of the floor acceleration and base shear reductions with the ambient temperature are complicated due to coupled influences of spectrum characteristics of the earthquake, dynamic characteristics of the structure, and temperature dependence of VE dampers. The VE damper with a strong temperature dependence may not be effective in reducing acceleration responses and base shear of structures under specific earthquakes at high and low ambient temperatures.

A free‐interface component mode synthesis of moving contact multibodies and its application to gear contacts

AbstractMoving contact components are common in mechanical systems, and they exhibit a large rigid motion with linear‐elastic deflection under contact boundary conditions. The finite element method can capture local deformation and high gradient stresses, but finely meshed contact interfaces lead to numerous degrees of freedom. Model order reduction techniques for multibodies are effective, but moving contact interfaces also introduce multiple interface degrees of freedom. In the present method, the interface degrees of freedom are retained to ensure the accuracy of the local response induced by contacts, but the computational scale is not further enlarged. The motion of moving contact components is described by the floating frame of reference formulation. A free‐interface component mode synthesis method is proposed for model order reduction. Kept free‐interface normal modes represent global flexibility and residual modes accurately compensate for the local quasi‐static responses of omitted modes induced by contacts. The contribution of residual modes is represented by interface load coordinates, which can be directly evaluated by contact forces. The contacts are calculated with the present contact algorithm. The accuracy and efficiency of the proposed method are fully demonstrated through correlation with the finite element method in gear meshing simulations.

Free-vibration analysis of cracked beam with constant width and linearly varying thickness

This study carries out free-vibration analysis of a cracked tapered beam with a constant width and a linearly varying thickness. The finite-element and component mode synthesis methods are used to determine the dynamic behavior of the tapered cracked beam. In accordance with the Bernoulli–Euler beam theory, the stiffness and mass matrices are obtained to find the natural frequencies and mode shapes of the tapered beam. The beam containing a crack is divided into subcomponents from the crack section. A massless spring model is used to define the local flexibility caused by the crack in the beam. The spring stiffness is calculated by taking the inverse of the compliance matrix found by using proper stress intensity factors and strain energy release rates calculated from linear elastic fracture mechanics. Different boundary conditions are considered such as fixed–free and pinned–pinned ends. Variations of natural frequencies and mode shapes depending on the crack location, crack depth and taper ratio of the beam are found, presented in graphs and examined. The applicability, accuracy and validity of the present method are found to be compatible with the literature.

Model order reduction of thermo-mechanical coupling flexible multibody dynamics via free-interface component mode synthesis method

A systematical procedure of model order reduction for the thermo-mechanical coupling flexible multibody system (FMS) is proposed. The displacement and temperature field are discretized by unified element mesh based on the absolute nodal coordinate formulation (ANCF). Since the expression of the elastic forces is highly nonlinear and the tangential stiffness matrix of ANCF is time varying, the Taylor expansion is used to linearized the dynamic and heat transfer equations to obtain constant tangential stiffness matrix. The thermo-mechanical coupling system is divided into several substructures and the free-interface component mode synthesis method is used at the interface. The modal truncation for the dynamic and thermal coordinates are performed to reduce the system degrees of freedom (DOF). Therefore, the computational efficiency is greatly improved. The system thermo-mechanical equations are reunited and the solving scheme is given. Numerical results show that the proposed method can significantly improve the simulation efficiency without losing accuracy.

Aeroelastic Response of a Z-Shaped Folding Wing During the Morphing Process

This paper presents the development of an aeroelastic analysis approach for the dynamic response of a Z-shaped folding wing. The structural model is established by the finite element method and the component mode synthesis method by accounting for the configuration changing effects on the inertial and stiffness characteristics. The aerodynamic model is directly built using the continuous-time state-space unsteady vortex lattice method. The coupling relationship between structural and aerodynamics models is proposed innovatively with interpolation between structural generalized coordinates and incremental normal vectors on the collocation points of aerodynamic panels. The resulting aeroelastic model with the state-space equation can be used for flutter and dynamic response analysis by considering a rapid folding/unfolding process. The analysis results show that the folding and unfolding processes have opposite effects on both the aerodynamic load and aeroelastic characteristics. Moreover, the effects become more significant with an increasing morphing rate.

Interface reduction in flexible multibody systems using the Floating Frame of Reference Formulation

The influence of interface reduction on the dynamics of flexible multibody systems has not been well investigated and the conventional multipoint constraints still suffer when applied to multibody systems. Here, we propose a novel approach to derive rigid and interpolation multipoint constraints exploiting the role of the reference conditions, used in the Floating Frame of Reference Formulation to cancel rigid body modes from the elastic field. Since the connection of the reference conditions to the multipoint constraints seems to be limited to the free-surface case, the proposed formulation combines the interface reduction problem to create a novel Component Mode Synthesis method able to work with any reference condition. Two case studies are presented to show the differences between models with different combinations of reference conditions and multipoint constraints.

Mobile Interface CMS Method for Vibration Characteristics Prediction of Mistuned Bladed Disk with Different Coupling Degrees

The prediction of vibration characteristics was studied in the mistuned bladed disk by the mobile interface prestressed component mode synthesis (CMS) superelement method. When the strongly, generally, and weakly coupling in the mistuned bladed disk, according to the results of the direct FEM method, the prediction accuracy of this method was verified and compared with the fixed-interface CMS method by using the relative error of dynamic frequency, vibration mode matching function, and dimensionless root mean square error of vibration amplitudes. It is pointed that for mistuned bladed disk in the strong coupling, the prediction accuracy of dynamic frequency and vibration amplitudes are higher by the mobile interface CMS method and the vibration modes are matched with the direct method. In weak coupling, the results of dynamic frequency and vibration modes predicted by the mobile interface CMS method and the fixed-interface CMS method are consistent with the direct method, but the vibration amplitudes’ prediction error of the mobile interface CMS method is lower than that of the fixed-interface CMS method. In general coupling, the mobile interface CMS method has higher dynamic frequency prediction accuracy at low order, and the two methods have comparable dynamic frequency prediction accuracy at high order. The vibration modes predicted by the two methods are matched with the direct FEM method, and the prediction accuracy of vibration amplitude by the mobile interface CMS method is better than that of the fixed-interface CMS method. The results indicate that the mobile interface CMS method could more accurately predict vibration characteristics of the mistuned bladed disk with different coupling degrees and could be an effective measurement for studying the vibration characteristics of the mistuned bladed disk system.

Numerical Methods for Calculating Component Modes for Geometric Mistuning Reduced-Order Models

AbstractGeometric mistuning models formulated from component mode synthesis methods often require the calculation of component modes, particularly constraint and fixed interface normal modes (FINMs), during substructuring. For integrally bladed rotors, these calculations are required for each sector. This paper proposes methods that reuse information garnered from solving the constraint modes of a single sector on the remaining sectors to reduce memory requirements and solution times. A mesh metamorphosis tool is used to ensure finite element models match geometry obtained from a three-dimensional optical scanner. This tool also produces a common mesh pattern from sector-to-sector. This is exploited to produce common permutation matrices and symbolic factorizations of sector stiffness matrices that are proposed for reuse in solving subsequent constraint modes. Furthermore, a drop tolerance is introduced to remove small values during constraint mode calculation to reduce memory requirements. It is proposed to reuse this dropping pattern produced from a single sector on the remaining sectors. Approaches are then extended to a parallel processing scheme to propose effective matrix partitioning methods. Finally, information gathered during the constraint mode calculations is reused during the solution of the FINMs to improve solution time. Results show reusing permutation matrices and symbolic factorizations from sector-to-sector improves solution time and introduces no error. Using a drop tolerance is shown to reduce storage requirements of a constraint mode matrix, while reusing the dropping pattern introduces minimal error. Similarly, reusing constraint mode information in calculating normal modes greatly improves the performance.

Mistuning parameter identification and vibration localization analysis of the integration rotor

This paper deals with the coupling vibration characteristic of the disk-blade-shaft integration rotor. First, a reduced-order model (ROM) based on an improved hybrid interface component mode synthesis method (IHISCMSM) is carried out, which takes the prestress effect into account. The frequency of the disk-blade-shaft integration rotor at different rotating speeds are calculated and the influence of selecting different mode truncation numbers is investigated. In order to quantitatively evaluate the coupling degree of blade and disk, the coupling factor is defined from the perspective of strain energy, and the influence of prestress on system’s dynamic is discussed. Then, an experimental modal analysis is performed on blades to identify the mistuning parameters, and the mode localization of the disk-blade-shaft integration rotor is analyzed with and without blade mistuning. The results indicate that there are several types of coupling modes among blade, disk and shaft of the integration rotor. After considering the prestress, the frequency increases, and the axial coupling vibration degree and radial coupling vibration degree of the integration rotor change. The mode localization of mistuned rotor is more likely to occur in the modes dominated by mistuning stage blades. There also exists a subtle mode localization phenomenon for tuned integration rotor.

A massless boundary component mode synthesis method for elastodynamic contact problems

We propose to combine the ideas of mass redistribution and component mode synthesis. More specifically, we employ the MacNeal method, which readily leads to a singular mass matrix, and an accordingly modified version of the Craig-Bampton method. Besides obtaining a massless boundary, we achieve a drastic reduction of the mathematical model order in this way compared to the parent finite element model. Contact is modeled using set-valued laws and time stepping is carried out with a semi-explit scheme. We assess the method’s computational performance by a series of benchmarks, including both frictionless and frictional contact. The results indicate that the proposed method achieves excellent energy conservation properties and superior convergence behavior. It reduces the spurious oscillations and decreases the computational effort by about 1–2 orders of magnitude compared to the current state of the art (mass-carrying component mode synthesis method). We believe that the computational performance and favorable energy conservation properties will be valuable for the prediction of vibro-impact processes and physical damping.

Optimization of hybrid cable networks with dampers and cross-ties for vibration control via multi-objective genetic algorithm

For long-span cable-stayed bridges, interconnecting adjacent cables using cross-ties and connecting each cable to bridge deck become a promising solution for cable vibration mitigation, leading to hybrid cable networks. However, optimizing the system parameters and configuration for best vibration control effects is still challenging. Therefore, this study proposes an optimization method based on three important ingredients, i.e., a general and efficient numerical model of cable networks, a reasonable definition of the optimization objectives, and a multi-objective optimization method. The cable network is modeled using the component mode synthesis method with modal expansion and quasi-static correction, the optimization objectives consider both multi-mode damping effects and stiffness enhancement, and the multi-objective genetic algorithm is then employed. The proposed method is first verified by comparing with analytical methods when applying to relatively simple cable–damper systems. Subsequently, a five-cable network is studied using the method for optimal design of cross-tie and damper parameters. Furthermore, the method is applied to optimize cross-ties and dampers for the eight longest cables in the Sutong Bridge, considering different numbers, locations and anchoring conditions of the cross-ties. The results have demonstrated the effectiveness of the proposed method in dealing with real-world large-scale cable networks. It is shown that the optimization can ensure that vibrations in local modes of the cable network are dominated by cable segments with a damper installed in the span and hence are sufficiently damped. Besides, damping ratios of hybrid cable networks are shown to be sensitive to the cross-tie location and parameters while they are relatively insensitive to the damper coefficients.

Dynamic characteristics of an aeroengine dual-rotor system with inter-shaft rub-impact

Inter-shaft rub-impact, i.e., the rub-impact between two rotors, may occur in an aeroengine dual-rotor system. This paper represents the first attempt to investigate the nonlinear dynamics of a dual-rotor system with inter-shaft rub-impact. The dynamic model of a complex dual-rotor system is built using the 3D finite solid element method by considering the inter-shaft rubbing forces. An analysis strategy combining the component mode synthesis method and numerical integration method is developed to overcome the problem of solving the large-DOF model. The modes and vibration responses of the dual-rotor system under the effect of inter-shaft rub-impact are clarified. The results show that the inter-shaft rub-impact leads to vibration coupling between LP and HP rotors, leading to the appearance of novel resonance peaks in the vibration response. Three kinds of rubbing types, i.e., continuous rub-impact, intermittent rub-impact and self-excited vibration may be excited when inter-shaft rub-impact occurs in a dual-rotor system. The characteristics and mechanisms of these rubbing motions are examined using the orbit, time and frequency curves as well as the rubbing force and relative velocity curves. By reducing the rubbing stiffness and friction coefficient, the dangerous self-excited rubbing vibrations can be eliminated. Moreover, by comparing the rubbing responses under constant and variable speeds, the key effects of the initial conditions on the inter-shaft rubbing responses are highlighted. Finally, the results of inter-shaft rubbing experiments conducted using a real aeroengine dual-rotor system are discussed and used to verify the correctness of the theoretical results.

A Multiple and Multi-Level Substructure Method for the Dynamics of Complex Structures

Based on the fixed interface component mode synthesis, a multiple and multi-level substructure method for the modeling of complex structures is proposed in this paper. Firstly, the residual structure is selected according to the structural characteristics of the assembled complex structure. Secondly, according to the assembly relationship, the parts assembled with the residual structure are divided into a group of substructures, which are named the first-level substructure, the parts assembled with the first-level substructure are divided into a second-level substructure, and consequently the multi-level substructure model is established. Next, the substructures are dynamically condensed and assembled on the boundary of the residual structure. Finally, the substructure system matrix, which is replicated from the matrix of repeated physical geometry, is obtained by preserving the main modes and the constrained modes and the system matrix of the last level of the substructure is assembled to the upper level of the substructure, one level up, until it is assembled in the residual structure. In this paper, an assembly structure with three panels and a gear box is adopted to verify the method by simulation and a rotor is used to experimentally verify the method. The results show that the proposed multiple and multi-level substructure modeling method is not unique to the selection of residual structures, and different classification methods do not affect the calculation accuracy. The selection of 50% external nodes can further improve the analysis efficiency while ensuring the calculation accuracy.

A fresh look at the dynamics of a flexible body application to substructuring for flexible multibody dynamics

AbstractThe general formulation of the dynamics of an elastic body undergoing arbitrary motion is revisited with emphasis on the orthogonality between rigid body motion and elastic deformation on the one hand, and on the appropriate definition of the relative velocities induced by elastic deformation on the other hand. A mixed displacement–velocity approach is adopted, thus allowing a local frame definition of velocities which results in simplification of the discretization and solution steps. The proposed formulation is applied to model reduction starting from a set of free‐free elastic modes and attachment modes. The Herting–Martinez component mode synthesis method is used to replace the modal intensities by a body‐frame nodal set of elastic degrees of freedom. A body‐to‐global frame transformation is then applied to the system equations in incremental form to avoid the use of Lagrange multipliers for expressing node‐to‐node connections. Examples are presented to assess the methodology proposed and its numerical implementation.