Component Mode Synthesis Method Research Articles

An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation

Abstract In this work, we propose and analyze an extension of the approximate component mode synthesis (ACMS) method to the two-dimensional heterogeneous Helmholtz equation. The ACMS method has originally been introduced by Hetmaniuk and Lehoucq as a multiscale method to solve elliptic partial differential equations. The ACMS method uses a domain decomposition to separate the numerical approximation by splitting the variational problem into two independent parts: local Helmholtz problems and a global interface problem. While the former are naturally local and decoupled such that they can be easily solved in parallel, the latter requires the construction of suitable local basis functions relying on local eigenmodes and suitable extensions. We carry out a full error analysis of this approach focusing on the case where the domain decomposition is kept fixed, but the number of eigenfunctions is increased. The theoretical results in this work are supported by numerical experiments verifying algebraic convergence for the method. In certain, practically relevant cases, even super-algebraic convergence for the local Helmholtz problems can be achieved without oversampling.

Simulation and experimental analysis of traveling wave resonance of flexible spiral bevel gear system

Considering the traveling wave resonance (TWR) phenomenon of the spiral bevel gear system in the aero engine accessory casing, the flexible spiral bevel gear model is established by three-dimensional (3D) shell element and beam element. The dynamic model of flexible bevel gear system considering the change of meshing teeth pairs in the operating process is established by component mode synthesis (CMS) method and rotational modal projection method. The proposed model is verified by finite element (FE) solid model and experiment test results. On this basis, the TWR phenomenon of the bevel gear system is studied in combination with dynamic response and vibration stress experimental test. The results show that the bevel gear system has 3 nodal diameter resonance speeds within the actual operating speed range, which are one 1st nodal diameter resonance of pinion and two 1st nodal diameter resonance of gear. When the bevel gear system operates at TWR state, the amplitude corresponding to fm ± m·fr (fm is meshing frequency, m is nodal diameter number, fr is shaft rotational frequency) is larger than that corresponding to fm in the Fast Fourier transform (FFT) spectrum of the of the gear foundation vibration stress, and the vibration stress cloud map of the gear foundation shows the corresponding nodal diameter vibration shape. The research results provide a theoretical basis for feature recognition and influence law evaluation of TWR of bevel gear system.

Analysis of the Nonlinear Complex Response of Cracked Blades at Variable Rotational Speeds

The operation of an aero-engine involves various non-stationary processes of acceleration and deceleration, with rotational speed varying in response to changing working conditions to meet different power requirements. To investigate the nonlinear dynamic behaviour of cracked blades under variable rotational speed conditions, this study constructed a rotating blade model with edge-penetrating cracks and proposes a component modal synthesis method that accounts for time-varying rotational speed. The nonlinear response behaviours of cracked blades were examined under three distinct operating conditions: spinless, steady speed, and non-constant speed. The findings indicated a competitive relationship between the effects of rotational speed fluctuations and unbalanced excitation on crack nonlinearity. Variations in rotational speed dominated when rotational speed perturbation was minimal; conversely, aerodynamic forces dominated when the effects of rotational speed were pronounced. An increase in rotational speed perturbation enhanced the super-harmonic nonlinearity induced by cracks, elevated the nonlinear damage index (NDI), and accentuated the crack breathing effect. As the perturbation coefficient increased, the super-harmonic nonlinearity of the crack intensified, resulting in a more complex vibration form and phase diagram.

Modeling and vibration analysis of an aero-engine dual-rotor-support-casing system with inter-shaft rub-impact

In order to reveal the nonlinear dynamics of dual-rotor-support-casing system, a numerical procedure is proposed using the finite element method and the component mode synthesis method. Especially, the casing system is subdivided into the casings, struts and housings, which are modeled by shell elements, beam elements and rigid regions with mass elements, respectively. Furthermore, the inter-shaft rub-impact, which may occur in the aero-engine because of imbalance and other faulty, is considered in the dynamic model as a nonlinear excitation. The modal analysis verified the validity of the dual-rotor-support-casing system model. The motion equations are established using the reduced-order model and solved by Newmark-β method with Newton-Raphson iteration. The results show that larger rubbing stiffness will cause the rubbing to enter a self-excited vibration state, and the rotational speed ratio will change the speed region where the self-excited vibration occurs. Finally, the experiment is performed based on a real dual-rotor aeroengine under inter-shaft rub-impact and the results are consistent with the theoretical results, which proves the effectiveness of dual-rotor-support-casing system dynamic model.

A system-level interface sampling and reduction method for component mode synthesis with varying parameters

This paper presents a new interface sampling and reduction method for varying parameters within the framework of the component mode synthesis (CMS) method. In the conventional interpolation-based parametric reduced-order model (IB-PROM) based on CMS, reduced subsystems are synthesized, and interface degrees-of-freedom are reduced via the secondary eigenvalue analysis in the online stage, which is essential to obtain highly reduced system matrices. However, it inhibits rapid parameter updating and requires extra computational resources in the online stage. In the present study, the pre-reduced interface sampling method is proposed by synthesizing substructures and reducing the interface before beginning the online stage. The interface solution is obtained through offline and online adaptation, likewise the procedure of computing the interior solution of previous studies. Additionally, congruence transformations are applied to each reduced interface, which is mandatory for interpolating symmetric positive definite matrices with respect to a new input parameter. As a result, the online performance of IB-PROM is greatly enhanced, showing rapid and near real-time adaptations to the variations in system parameters.

Nonlinear aeroelastic analysis of a twin-tail boom UAV with rudder freeplay nonlinearities

The issue of nonlinear structural freeplays in aircraft has always been a significant concern. This study investigates the aeroelastic characteristics of a twin-tail boom Unmanned Aerial Vehicle (UAV) with simultaneous freeplay nonlinearity in its left and right rudders. A comprehensive Limit Cycle Oscillation (LCO) solution route is proposed for complex aircraft with multiple freeplays, which can consider both accuracy and efficiency. For the first time, this study reveals the unique LCO characteristics exhibited by twin-tail boom UAVs with rudder freeplays and provides simulations and explanations of interesting phenomena observed during actual flight. The governing equations are established using the free-interface component mode synthesis method, and the LCOs of the system are mainly solved through the improved time-domain numerical continuation method and frequency-domain numerical continuation method. Furthermore, the study investigates the influence of the left and right rudder freeplay size ratio on the LCO characteristics. The results demonstrate that the twin-tail boom UAV exhibits two stable LCO types: close and differing left and right rudder amplitudes. The proposed method successfully describes the complete LCO behaviors of the system. Overall, this study makes significant contributions to our understanding of the aeroelastic behavior of twin-tail boom UAVs with rudder freeplays.

Physical modeling and time-domain simulation of a piano

The elaborate method to simulate the sound produced by a piano is presented, using the physical model that considers almost all parts of the piano with wood complex elastic orthotropy and manufacturing stress, but excluding action parts. The purpose of this work is to improve the efficiency of piano development by clarifying the causal relationship between the design and the produced sound. The completed piano is modeled as a 3D coupled system consisting of “nonlinear systems including hammers and strings” and “a linear coupled body-air system.” The hammer felt property are represented by the double layer nonlinear generalized Maxwell model. For the strings with geometrical nonlinearity and their support end anisotropy, the Galerkin method with the component mode synthesis (CMS) method is applied. And also, for the large-scale linear coupled body-air system, the complex CMS method via nonlinear eigenvalue analysis is used. The simulated sounds, including the ringing and body sounds, are so realistic that they help piano designers predict the actual sounds before making heavy prototypes. The method presented here can be applied to other musical instruments that consist of strings and a body, such as guitars and violins.

Dynamic characteristics investigation of a dual rotor-casing system considering the comprehensive stiffness of the intermediate bearing

This paper focuses on the vibrational properties of a dual rotor-casing coupled system with nonlinear clearance in the intermediate bearing, considering both the oil film stiffness and Hertzian contact stiffness between rolling elements and raceways. A comprehensive stiffness model for the intermediate bearing has been developed based on elastohydrodynamic lubrication (EHL) and Hertz contact theory. Subsequently, a dynamical model of the dual rotor-bearing-casing system is established using the finite element method, thereby verifying the feasibility of simulating the casing with beam elements. The dimensionality of the system equation is reduced using the fixed-interface component mode synthesis (CMS) method, and the resulting reduced-dimensionality equation is solved using the improved Newmark-β method. The effects of casing support, rotational speed, and radial clearance on the vibration characteristics of the system are investigated. Then, the comprehensive stiffness and coupled vibration characteristics of the system are analyzed to elucidate its intrinsic mechanism. Finally, the contact characteristics within the intermediate bearing under dynamic load are also studied. The results indicate that the support stiffness of the casing has a more significant impact on the second resonance caused by both the inner and outer rotors. The system is prone to experiencing varying compliance (VC) resonance at low rotational speeds. The excessive bearing clearance leads to complex vibrational characteristics of the system, making it challenging to predict its motion. The impact of minor changes in stiffness on system vibration is small when the comprehensive stiffness is large. At this time, the limited effect of oil film stiffness allows it to be disregarded in terms of comprehensive stiffness. Furthermore, the speed significantly influences the contact characteristics within the intermediate bearing under the unbalanced force of the rotor and gravity.

A novel 3D train–bridge interaction model for monorail system considering nonlinear wheel-track slipping behavior

Variable speed operation of the train cause easily the wheel-track slipping phenomenon, inducing strong nonlinear dynamic behavior of the suspended monorail train and bridge system (SMTBS), especially under an insufficient wheel-track friction coefficient. To investigate the coupled vibration features of the SMTBS under variable speed conditions, a novel 3D train–bridge interaction model for the monorail system considering nonlinear wheel-track slipping behavior is developed. Firstly, based on the D’Alembert principle, the vibration equations of the vehicle subsystem are derived by adequately considering the nonlinear interactive behavior among the vehicle components. Then, a high-efficiency modeling method for the large-scale bridge subsystem is proposed based on the component mode synthesis (CMS) method. The vehicle and bridge subsystems are coupled with a spatial wheel-track interaction model considering the nonlinear wheel-track sliding behavior. Furtherly, by a comprehensive comparison with the field test data, the effectiveness of the proposed method is verified, as well as the reasonable modal truncation frequencies of the bridge subsystem are determined. On this basis, the dynamics performances of the SMTBS are evaluated under different initial braking speeds and wheel-track interfacial adhesion conditions; besides, the nonlinear wheel-track slipping characteristics and their influences on the vehicle–bridge interaction are also revealed. The analysis results indicate that the proposed model is reliable for investigating the time-varying dynamic features of SMTBS under variable train speeds. Both the axle load transfer phenomenon and longitudinal slip of the driving tire would be easy to appear under the braking condition, which would significantly increase the longitudinal vehicle–bridge dynamic responses. To ensure a good vehicle–bridge dynamics performance, it is suggested that the wheel-track interfacial friction coefficient is larger than 0.35.

Topological structures of vibration responses for dual-rotor aeroengine

Many advanced aeroengines are typically designed with coaxial coupling dual-rotor structures. As a result, the vibration responses of the rotor system are characterized by complex multifrequency oscillations. In the paper, the topological structures of the nonlinear vibration responses of the dual-rotor system are revealed, providing a better understanding of the complex structural dynamics of many advanced aeroengines. A nonlinear dynamic model of the structural system of a dual-rotor-aeroengine experimental rig is established using the finite element (FE) method. This model takes into the blade-casing rubbing, misalignment, and the nonlinearities of the supporting rolling element bearings. The component mode synthesis (CMS) method is used to reduce the order of the dynamic model with high-degrees of freedom (DOFs) in order to enhance the computational efficiency. We first discover that long periodic structures exist in the multi-harmonic vibration responses of the dual-rotor system. The structures of Poincare mappings and bifurcation diagrams provide clearer and more concise information when obtained through long periodic sampling. The numerical periodic vibration response structures matched the measurements. These results help uncover the inherent structural characteristics of the complex nonlinear vibrations of a dual-rotor aeroengine.

Extended component mode synthesis method for dynamic analysis of mechanical systems with local nonlinearities

Engineering structures are generally designed based on linear elasticity assumptions. However, it is difficult to describe the connected structures using a simple linear system, due to factors such as sliding and friction at interfaces, and variations in contact areas during vibration processes. Therefore, it is necessary to develop an approach to tackle such systems with interface nonlinearities. In this paper, we proposed an extended free-interface component mode synthesis method. According to the proposed method, the substructures are separated at the nonlinear interfaces. Then the component mode synthesis procedure is carried out. By reasonably neglecting the contributions of higher order modes to the damping and inertia terms, it is demonstrated that the nonlinear interface forces and their time derivatives can be expressed as functions of themselves, as well as modal displacements and velocities. This characteristic facilitates the synthesis of the reduced order models for substructures in the state space. Also owing to this operation, the original properties of nonlinear interface forces can be preserved as much as possible, without any linearization. This method is suitable for non-proportionally damped structures with local nonlinearities such as mechanical structures with nonlinear spring and dashpot connections, rotor-bearing systems and so on. It also may be an alternative for the dynamic analysis of jointed structures with contact nonlinearities. Two numerical examples are presented to demonstrate the capability of the proposed method: (1) an eighteen degrees of freedom spring-dashpot-mass system with cubic spring and cubic dashpot interface connections is studied and (2) a more complex mechanical structure: a dual-rotor system with deep-groove ball bearing inter-shaft support. The numerical simulation results indicate that the dynamic responses of the reduced order model match very well with that of the full model, revealing the high accuracy of the proposed method with low computational costs.

Experimental and Numerical Investigations on the Dynamic Response of Blades with Dual Friction Dampers

Friction dampers are widely employed to reduce blade resonance vibration amplitude in turbomachinery. In this paper, a study was performed on the forced response of two blades with dual friction dampers. Numerical simulation and experimental testing were conducted. Firstly, the dynamics of the blade and dual friction damper system assembly are modeled. A nonlinear code based on the multi-harmonic balance method was developed to calculate the resonance response. In this analysis, both the blade and the damper are modeled with the finite element and the matrices reduced with the component mode synthesis method, while the contact forces are modeled with a one-dimensional variable normal load array element. Secondly, a test rig made of two blades and dual friction dampers, the material of which was steel, was established to measure the nonlinear frequency response function curves of the blade system. The results indicate that when a dual friction damper is applied, superior vibration reduction characteristics are demonstrated, with the system exhibiting an average 21% reduction in the response amplitude levels and an increase of 3% in the frequency shifting range compared to a single damper. Dampers positioned at relatively higher locations contribute significantly to the vibration reduction process. In the end, the numerical predictions match very well with the experimental ones.

Experimental and numerical flutter analysis of a folding fin with multiple asymmetric free-plays

Experimental folding fin models with an adjustable free-play are tested in a wind tunnel. The fin structure is modeled using the free-interface component mode synthesis method, and its free-play is modeled as four independent nonlinear springs with asymmetric stiffness. A nonplanar unsteady vortex-lattice method considering compressibility is employed to address nonlinear deformation and high subsonic flow. Surface spline interpolation is improved through projection and partition. The aeroelastic characteristics of folding fins with different free-play magnitudes, initial conditions and elastic-axis positions are analyzed using an established time-marching method because of its relatively small computation scale and high precision. The results show good consistency among the presented method, the wind tunnel test and the harmonic balance method. There is a negative correlation between the critical speed of divergent motion and the ratio of the initial condition to the free-play magnitude. If either the free-play magnitude or the initial condition is extreme (tiny or vast), the system nonlinearity degenerates to linearity. Generally, the flutter prevention design of a linear model can be applied to a nonlinear model, such as moving the elastic-axis position aftward. The presented fin configuration exhibits an unstable limit cycle oscillation because the orders of coupled flutter modes do not change with variations in equivalent linear stiffness.

A hybrid Bloch mode synthesis method based on the free interface component mode synthesis method

For band-structure computation, the Bloch mode synthesis (BMS) or the generalized Bloch mode synthesis (GBMS) method is a series of model-reduction computational algorithms based on the Craig-Bampton method. A hybrid Bloch mode synthesis (HBMS) method with a hybrid Floquet-Bloch periodic boundary condition is proposed in this paper. A force-based Floquet-Bloch periodic boundary condition is derived and combined with the default displacement-based Floquet-Bloch periodic boundary condition to form the hybrid Floquet-Bloch periodic boundary condition. The hybrid BMS method has been developed by combining the hybrid Floquet-Bloch periodic boundary conditions with the free-interface component mode synthesis method. The HBMS method considers the influence of higher-order modes on periodic boundary structures so that the hybrid Floquet-Bloch periodic boundary conditions can be applied to the reduced-order modal space without introducing errors, which improves the computational efficiency. According to different truncation errors, the HBMS method has been developed into three algorithms, and they each have a different precision-efficiency ratio. Numerical experiments demonstrate that the developed method outperforms in accuracy and computational efficiency compared to the default BMS/GBMS method.

Study on dynamic characteristics of the gear-dual-rotor system with multi-position rubbing

To reveal the mechanism of gear meshing induced rubbing and investigate the nonlinear dynamic response characteristics of a gear-dual-rotor system with multi-position rubbing, a new finite element model (FEM) of the gear-dual-rotor system considering the gear meshing excitation is established based on the in-house beam element code. Then, the fixed interface component mode synthesis method (FICMSM) is used to establish the reduced gear-dual-rotor dynamic model. A rotor-rotor rubbing model between the inner and outer rotors (IORs) and a rotor-stator rubbing model are built through vectors considering whirling velocities. In addition to the general concern about the low-frequency components around rotating frequencies and transversal responses with rubbing, more attention is given to the high-frequency components around the gear meshing frequency and the torsional responses. Finally, some interesting phenomena are found as follows: gear meshing is a new cause of partial rubbing. Some combined frequencies between the rotating frequencies of IORs and modulation frequencies between the meshing frequency and rotating frequencies are found in the transversal and torsional responses.

Nonlinear Dynamic Behavior Analysis of Dual-Rotor-Bearing Systems with Looseness and Rub–Impact Faults

The nonlinear dynamic behaviors of dual-rotor-bearing systems with looseness and rub–impact faults are discussed in this paper. The dual-rotor-bearing system with looseness and rub–impact coupling faults is established by the finite element method. The component mode synthesis (CMS) and proper orthogonal decomposition (POD) methods are introduced. The CMS and POD methods are used to reduce the original rotor system model, and the efficiency of the order reduction method is verified by comparing the dynamic behaviors of the original and reduced systems. The frequency spectrum and amplitude–frequency behaviors of rotor systems are studied. The results can provide qualitative guidance to structural design optimization of large rotating machines and prior information for looseness and rub–impact coupling faults.

Nonlinear vibration analysis and stability analysis of rotor systems supported on SFD by combining DQFEM, CMS and IHB methods

In this paper, the differential quadrature finite element method (DQFEM), component mode synthesis (CMS) method and incremental harmonic balance (IHB) method are combined to propose a methodology to perform vibration analysis and stability analysis of rotor systems supported on fluid-film bearings that exhibit localized nonlinearities such as squeeze-film damper (SFD). In multi-degree of freedom systems with SFD, a method for dynamic analysis of the system by reducing the degrees of freedom is first suggested in this paper. DQFEM are used to create the mass matrix, stiffness matrix, and gyroscopic matrix of the system. Then, the system is divided into linear system and nonlinear system, and the motion equation of the linear system reduces the degrees of freedom using the free interface CMS method, and the motion equation of the nonlinear system is written without reducing the degrees of freedom. By synthesizing the equations of motion of each component, the motion equation of the entire system is created. That is, the motion equation of the whole system uses hybrid coordinates. The motion equation of the system is solved using the modified IHB method. To evaluate the effectiveness of the suggested method, the computed solutions are compared with the results presented in the previous literature, and the results are very consistent. Also, stability analysis of the obtained solutions is performed by the Floquet theory. The method suggested in the paper can be effectively used to analyze the dynamic behavior of rotor systems exhibiting localized nonlinearities such as SFD.

Review on the Theories and Applications of Dynamic Condensation and Component Mode Synthesis Methods in Solving FEM-based Structural Dynamics

Review on the Theories and Applications of Dynamic Condensation and Component Mode Synthesis Methods in Solving FEM-based Structural Dynamics

Dynamical modeling and experimental validation for squeeze film damper in bevel gears

Squeeze film damper (SFD) effectively reduces rotor motion amplitude while crossing critical speeds for vibration and noise reduction in a wide application of aero-engine rotor system. The squirrel cage elastic support (SCES) as an essential component of SFD, accurate modelling of SCES using component modal synthesis (CMS) to account for its flexibility, and calculation of nonlinear oil film forces for SFD using the finite difference method. The CMS method is also used to model the bevel gear wheel body and to establish a finite element nodal dynamics model of the gear-rotor-bearing-SFD system considering time-varying mesh stiffness, tooth clearance, gyroscopic effects, transmission error excitation and SFD oil film forces. The modal frequency of the gear shaft in aero-engine power transmission is exercised to verify the established bevel gear model. Moreover, a bevel gear test platform with SFD is established to study the vibration reduction effect of SFD. It is found that SFD can effectively improve the transmission of the bevel gear system, and the flexibility of the SCES makes the critical speed shift. Actually, the smaller the stiffness of SCES, the more the critical speed is moved to the left. By using CMS to determine the flexibility of bevel gear and SCES, it can show the complete phenomenon of the dynamic behaviors of the bevel gear system with SFD.

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

In recent years, acoustic black holes (ABHs) have been shown very effective to suppress radiation. However, below the cut-on frequency, the waves cannot be trapped into the ABH. In this paper, locally resonant metamaterials are introduced to the ABH plate to improve the low-frequency performance. The suggested material is called MMABH. The total mass of resonators is kept the same to the one when fabricating the ABH, such that the lightweight feature still remains valid for the MMABH. Using the Gaussian expansion component mode synthesis (GECMS) method, the band gap and the coincidences between the MMABH and the air are first characterized, together with the paramagnetic studies of the plate thickness and lattice constant. In what follows, the vibration field of the MMABH is recovered, showing that whole-band reduction can be achieved. The sound pressure, power and radiation efficiency are next characterized via a discretized radiation model. Likewise, the radiated power is substantially mitigated in the whole frequency range being considered. Finally, the implementation of non-negative intensity (NNI) shows that the sound reduction not only associates with the outstanding damping capabilities of the resonators and the ABH, but also involves the wave slowdown phenomenon that breaks the coupling with air.