Mode Synthesis Method Research Articles

A Comparison Study on Co- and Counterrotating Dual-Rotor System with Squeeze Film Dampers and Intermediate Bearing

Nonlinear dynamic model of a coaxial rotor system was established with a method combining the finite element method and the fixed interface modal synthesis method. Then an implicit time domain method was presented to solve the nonlinear equations of motion; thus dynamic characteristics of the rotor system can be obtained. With nonlinear forces of squeeze film damper and intermediate bearing considered, nonlinear dynamic response characteristics of the co- and counterrotating coaxial rotor system under multiple unbalance forces were studied and compared in this work. It was found that the critical speeds of the corotating system were equal to or slightly higher than those of the counterrotating case due to the gyroscopic moments. The results showed that the unbalance excitation frequencies are dominant in the responses of the rotor system. Besides, due to coupling effect of the intermediate bearing some combinations of the unbalance excitation frequencies were also observed in the spectrogram but the combinations were different for co- and counterrotating cases. Stability and periodicity of the rotor system were investigated with bifurcation diagram, Poincare map, and phase diagram. It was found that the rotor system executes four-period quasi-periodic motion around critical speeds.

A reduced interface component mode synthesis method using coarse meshes

Component mode synthesis is a technique to simplify the analysis of complicated finite element models. A structure is split into substructures from which reduced order models can be generated and subsequently assembled. A model reduction performance gain can be limited if the component interfaces contain many degrees of freedom, which is often the case for high resolution models. In this paper a substructuring framework with interface reduction is presented. The method first splits a detailed model into substructures. The substructures’ fine mesh is then coarsened on the internal region, while keeping the boundary mesh intact. Thereafter a Guyan reduction is performed on each coarse mesh substructure. The Guyan computations are cheap due to the reduced size of the linear equation system necessary to solve for the coarse mesh system. After synthesis of the statically reduced systems, a reduction basis for the interface degrees of freedom is computed. Thereafter a Craig-Bampton reduction is performed on each fine mesh substructure using projections with the reduced interface degrees of freedom and fixed interface modes. The method is verified on a dense mesh plate model consisting of two substructures.

A 3D nonlinear finite element method for the dynamic analysis of rotating rotor with a transverse crack

A general and efficient method is presented in this paper for studying the effects of unbalance on the breathing mechanism of crack. Based on 3D finite element models combined with a nonlinear contact approach for crack modeling, the method is free from the assumption of weight-dominance and can be used to gain deep insights into the breathing mechanism of crack. In order to greatly reduce the computational time, a complex free-interface component mode synthesis (CMS) method is employed to reduce the order of the model. Based on the proposed method, the effects of unbalance on the breathing mechanism of crack are discussed. Numerical results show that the unbalance can lead to significant changes in the breathing of crack, even when the unbalance force is about an order of magnitude smaller than the self-weight. Moreover, the level and orientation of the unbalance have also remarkable effects on the breathing behaviors of crack. Besides, a new universal non-steady breathing phenomenon of crack is firstly found in this paper, which denotes that the breathing speed of a crack is fluctuated over one revolution when there exists residual unbalance in the cracked rotor.

Metal Sounds Stiffer than Drums for Ears, but Not Always for Hands: Low-Level Auditory Features Affect Multisensory Stiffness Perception More than High-Level Categorical Information.

Most real-world events stimulate multiple sensory modalities simultaneously. Usually, the stiffness of an object is perceived haptically. However, auditory signals also contain stiffness-related information, and people can form impressions of stiffness from the different impact sounds of metal, wood, or glass. To understand whether there is any interaction between auditory and haptic stiffness perception, and if so, whether the inferred material category is the most relevant auditory information, we conducted experiments using a force-feedback device and the modal synthesis method to present haptic stimuli and impact sound in accordance with participants’ actions, and to modulate low-level acoustic parameters, i.e., frequency and damping, without changing the inferred material categories of sound sources. We found that metal sounds consistently induced an impression of stiffer surfaces than did drum sounds in the audio-only condition, but participants haptically perceived surfaces with modulated metal sounds as significantly softer than the same surfaces with modulated drum sounds, which directly opposes the impression induced by these sounds alone. This result indicates that, although the inferred material category is strongly associated with audio-only stiffness perception, low-level acoustic parameters, especially damping, are more tightly integrated with haptic signals than the material category is. Frequency played an important role in both audio-only and audio-haptic conditions. Our study provides evidence that auditory information influences stiffness perception differently in unisensory and multisensory tasks. Furthermore, the data demonstrated that sounds with higher frequency and/or shorter decay time tended to be judged as stiffer, and contact sounds of stiff objects had no effect on the haptic perception of soft surfaces. We argue that the intrinsic physical relationship between object stiffness and acoustic parameters may be applied as prior knowledge to achieve robust estimation of stiffness in multisensory perception.

Modal reduction of brake squeal systems using complex interface modes

This paper deals with a new efficient reduction method for predicting the stability analysis of a damped nonlinear brake system subjected to friction-induced vibration. The originality of the present work is to propose a generalized double modal synthesis method that combines a classical modal reduction and a condensation at the frictional interface by computing a reduced complex mode basis.Comparisons with the existing double modal synthesis reduction method are performed. It is demonstrated that the new suggested reduction technique is more efficient. It allows to provide satisfactory results with a small number of interfaces modes.

Reduced Order Models for Dynamic Behavior of Elastomer Damping Devices

In the context of passive damping, various mechanical systems from the space industry use elastomer components (shock absorbers, silent blocks, flexible joints...). The material of these devices has frequency, temperature and amplitude dependent characteristics. The associated numerical models, using viscoelastic and hyperelastic constitutive behaviour, may become computationally too expensive during a design process. The aim of this work is to propose efficient reduced viscoelastic models of rubber devices. The first step is to choose an accurate material model that represent the viscoelasticity. The second step is to reduce the rubber device finite element model to a super-element that keeps the frequency dependence. This reduced model is first built by taking into account the fact that the device's interfaces are much more rigid than the rubber core. To make use of this difference, kinematical constraints enforce the rigid body motion of these interfaces reducing the rubber device model to twelve dofs only on the interfaces (three rotations and three translations per face). Then, the superelement is built by using a component mode synthesis method. As an application, the dynamic behavior of a structure supported by four hourglass shaped rubber devices under harmonic loads is analysed to show the efficiency of the proposed approach.

Seismic analysis of the APR 1400 reactor vessel internals using the model reduction method

ABSTRACTTo secure reliability of the seismic design of the reactor vessel internals (RVIs) through the finite element analysis, it is important to develop the accurate analysis model which can represent the geometric complexity of the RVIs. However, the seismic analysis requires too large computation cost to solve the complex equations; thus, it needs to reduce the overall size of the analysis model. Here, we apply a model reduction method based on the fixed-interface component mode synthesis (CMS) method to practical RVIs to solve complex numerical problems efficiently. To verify the model reduction method, several cases of the RVIs with different conditions are analyzed for the static and dynamic problems. Finally, the seismic analysis was performed with the suggested reduced model with the CMS method. The time history analysis is performed to extract important seismic responses at the specified locations, and the stress analysis is also performed to identify that the RVIs satisfy the seismic design. In the last part of the paper, an example of the design modification is suggested to reduce the stress intensity at the support locations.

Optimization of the dynamic behavior of vehicle structures by means of passive interface controls

As a form of passive control, padding rubber layers onto the most heavily deformed zones of a system can improve the dynamic behavior and the acoustic comfort of a vehicle system. This paper proposes an extensive hybrid modal synthesis method in order to study coupled fluid-structure systems, in retaining a few degrees of freedom. Modal criteria, corresponding to noise transmission paths between substructures in the system, have been derived to characterize the dynamic phenomenon from a modal view. These criteria were then substituted by Kriging interpolation models to avoid prohibitive simulation steps during optimization of the complex system. Once the mathematical models of the investigated modal criteria were established and the multi-objective functions for rubber characteristics defined, an approximate optimal solution leading to superior dynamic performance could be obtained based on a genetic algorithm. The analytical results and numerical experiments conducted have also justified the efficiency of our proposed strategy.

Component mode synthesis methods for a body-in-white noise and vibration analysis

In this work the dynamic substructuring approach was applied to a noise, vibration and harshness problem within the automotive engineering field. In particular, a noise, vibration and harshness analysis was carried out on the body-in-white structure of a passenger car. The work focuses on the theory of component mode synthesis. Two component mode synthesis reduction methods, namely the Craig–Bampton method and the Craig–Chang method, were applied to the body-in-white structure of the Volvo V40. The influences of various parameters were investigated. In particular, the effect of the reduction basis on the response accuracy and on the reduction time was studied. Moreover, the effects of the connection properties between different parts of the model were examined. The simulation times of the reduced models and of the full finite element model were compared. The results showed that the Craig–Chang method performs better when the modes are retained for up to one and a half times the maximum frequency response studied. Additionally, the Craig–Chang method gives a very accurate representation of the system dynamics even when connections with a low stiffness are used. Finally, it is possible to reduce the simulation time by up to 90% if component mode synthesis methods are used instead of the full finite element model.

Dynamic modeling and active control of a strap-on launch vehicle

Dynamic modeling and active control of a strap-on launch vehicle are studied in this paper. In the dynamic modeling, the double-compatible free-interface modal synthesis method is used to establish dynamic model of the system, and its model precision is compared with those of finite element method (FEM), fixedinterface modal synthesis method and free-interface modal synthesis method. In the active control, the swing angle of rocket motor is used as design variable, and the control law design based on the model of mass center motion is adopted to validate the system. Simulation results indicate that the double-compatible model synthesis method can properly approximate the FEM which is used as the benchmark solution, and the model precision of the double-compatible modal synthesis method is obviously higher than those of the fixed-interface and freeinterface modal synthesis methods. Based on the control law design, the deflection of mass center of the launch vehicle is very small.

High-Fidelity Flexibility-Based Component Mode Synthesis Method with Interface Degrees of Freedom Reduction

The partitioned structural equations of motion via the method of localized Lagrange multipliers (PALM) consist of internal and interface variables. The model reduction achieved based on PALM is called flexibility-based component mode synthesis (F-CMS), and it thus far has concentrated on reducing the internal degrees of freedom (DOFs) or reducing the internal and interface DOFs at once. In this work, well-defined three-level reduction formulations of the F-CMS method are proposed for better precision: internal DOFs, the localized Lagrange multipliers, and the interface boundary displacement. The proposed formulations include three different reduction techniques for the interface boundary displacement for better computational efficiency. Numerical experiments indicate that the proposed formulations can substantially reduce the overall reduced-order models further without compromising the accuracy, thus achieving a substantial additional reduction of computational cost compared with the previous works. In addition, the performance of a novel error estimator in the overall reduction levels is tested.

A nonlinear component mode synthesis method for the computation of steady-state vibrations in non-conservative systems

This paper presents an extension to classic component mode synthesis methods to compute the steady-state forced response of nonlinear and dissipative structures. The procedure makes use of the nonlinear complex modes of each substructure, computed by means of a modified harmonic balance method, in order to build a reduced-order model easily solved by standard iterative solvers. The proposed method is applied to a mistuned cyclic structure subjected to dry friction forces, and proves particularly suitable for the study of such systems with high modal density and non-conservative nonlinearities.

Study on the Stress-Stiffening Effect and Modal Synthesis Methods for the Dynamics of a Spatial Curved Beam

In this paper, based on the nonlinear strain–deformation relationship, the dynamics equation of a spatial curved beam undergoing large displacement and small deformation is deduced using the finite-element method of floating frame of reference (FEMFFR) and Hamiltonian variation principle. The stress-stiffening effect, which is also called geometric stiffening effect, is accounted for in the dynamics equation, which makes it possible for the dynamics simulation of the spatial curved beam with high rotational speed. A numerical example is carried out by using the deduced dynamics equation to analyze the stress-stiffening effect of the curved beam and then verified by abaqus software. Then, the modal synthesis methods, which result in much fewer numbers of coordinates, are employed to improve the computational efficiency.

A differential quadrature hierarchical finite element method and its applications to vibration and bending of Mindlin plates with curvilinear domains

SummaryA differential quadrature hierarchical finite element method (DQHFEM) is proposed by expressing the hierarchical finite element method matrices in similar form as in the differential quadrature finite element method and introducing interpolation basis on the boundary of hierarchical finite element method elements. The DQHFEM is similar as the fixed interface mode synthesis method but the DQHFEM does not need modal analysis. The DQHFEM with non‐uniform rational B‐splines elements were shown to accomplish similar destination as the isogeometric analysis. Three key points that determine the accuracy, efficiency and convergence of DQHFEM were addressed, namely, (1) the Gauss–Lobatto–Legendre points should be used as nodes, (2) the recursion formula should be used to compute high‐order orthogonal polynomials, and (3) the separation variable feature of the basis should be used to save computational cost. Numerical comparison and convergence studies of the DQHFEM were carried out by comparing the DQHFEM results for vibration and bending of Mindlin plates with available exact or highly accurate approximate results in literatures. The DQHFEM can present highly accurate results using only a few sampling points. Meanwhile, the order of the DQHFEM can be as high as needed for high‐frequency vibration analysis. Copyright © 2016 John Wiley & Sons, Ltd.

In-plane vibration analysis of plates in curvilinear domains by a differential quadrature hierarchical finite element method

Free in-plane vibration analysis of plates is carried out by a differential quadrature hierarchical finite element method (DQHFEM). The NURBS (Non-Uniform Rational B-Splines) patches of geometries were first transformed into differential quadrature hierarchical (DQH) patches, and then the elastic field was discretized by the same DQH basis. The DQHFEM solved the compatibility problem caused by different parametrization of neighbouring patches of isogeometric analysis using NURBS. And mesh refinement in DQHFEM does not propagate from patch to patch. The DQHFEM matrices also have the embedding property as the hierarchical finite element method (HFEM). In-plane vibration analyses of plates of several planforms showed that the DQHFEM is similar as the fixed interface mode synthesis method that can analyse a structure using a few nodes on the boundary of substructure elements and only several clamped modes inside each substructure element, but the DQHFEM does not need modal analysis and is of high accuracy. The accuracy and convergence of the DQHFEM were validated through comparison with exact and approximate results in literatures and computed by the authors.

A Hybrid Coordinates Component Mode Synthesis Method for Dynamic Analysis of Structures With Localized Nonlinearities

This paper reports on the development of the component mode synthesis (CMS) method using hybrid coordinates for a localized nonlinear dynamic system. As is well known, the CMS method is effective in reducing the degrees-of-freedom (DOF) of the system. In contrast to most existing CMS methods, which are usually developed for linear systems, a new CMS method using hybrid coordinates for nonlinear dynamic analysis has been developed in this paper. Generally, the system is divided into two parts, namely, a linear component and a nonlinear component. The equations of the linear component can be transformed into the modal coordinates using its linear vibration modes. To improve the accuracy, the equivalent higher-order matrix of the system is developed to capture the effects of the neglected higher-order modes. Quite different from early works, the flexibility attachment matrix can be obtained without using the inverse of the stiffness matrix by using an equivalent higher-order matrix, thus making it easier to deal with those components that have rigid-body freedom when formulating the residual flexibility attachment matrix. By introducing the residual flexibility attachment matrix and the retained lower-order modes, the dynamic governing equations of the linear component can be converted into the modal space and expressed by a few modal coordinates. To adopt the entire set of nonlinear terms into the final equations, the equations of the nonlinear component are kept in their original form. Compatibility conditions at the interface are used to combine the nonlinear component and the linear component to form the synthesis equations, which are expressed in hybrid coordinates. Finally, the computational efficiency and accuracy of the presented method is demonstrated using numerical examples.

Identification Technology of Modal Parameter of a Strap-on Launch Vehicle

It is generally needed to conduct ground modal tests over strap-on launch vehicle to provide modal parameters for the attitude control design and load calculating. These modal parameters can also provide basis of installing sense organs on the launch vehicle. This paper introduces a modal parameter identification method based on input-output data of the system, which can be used to check calculations of modal parameters after ground modal tests. In this paper, the double-compatible free-interface modal synthesis method is first used for the modeling of the system so as to get the input and output data of the system. Then the identification techniques of observer/Kalman filter identification (OKID) and eigensystem realization algorithm (ERA) are introduced. Finally, numerical simulations are carried out to demonstrate the validity of the presented identification method. Simulation results indicate that the modal parameter of the system can be effectively identified using OKID and ERA.

エンジンのギアケーシングへの制振処理の適用

As the diesel engine used for the forklift and construction machineries is operated at the continuous heavy duty condition, it employs the gear train to drive the fuel injection pump and valve train. When the driving torque acting on the rotating shaft is decreased in the idling condition, separation of gear tooth and rattle noise are induced. Moreover, torsional vibration of crankshaft increases the impact vibration between the gear train. This gear train is commonly installed at the pulley side of the crankshaft and the gear casing is attached to an engine block to cover the gear train. Vibration level of this gear casing is often magnified due to the engine block vibration and consequently the engine noise is increased at the pulley side. The effects of changing the structure of gear casing and noise radiation from the gear casing are evaluated by use of the modal synthesis method which employs the vibratory characteristics of each structure and static deformation caused by the displacement of the connecting points. The rubber pasting on the gear casing is carried out to reduce engine noise. A modal damping ratio of coupled system after pasting rubber on the gear casing evaluated by use of modal analysis technique and its validity is confirmed by vibration experiment of a box model which imitates engine block and gear casing. Change of actual engine noise level by rubber pasting on gear casing is also discussed in detail.

Combination of component mode synthesis method and penalty method for contact-impact of flexible multi-body system

Combination of component mode synthesis method and penalty method for contact-impact of flexible multi-body system

A free interface component mode synthesis method for viscoelastically damped systems

The analysis of large, complex systems can be simplified and more computationally efficient if smaller, simpler subcomponents can be treated and assembled. Component Mode Synthesis (CMS) methods allow one to reduce the order of the model at the subcomponent level. Although CMS methods are widely applied in the field of structure dynamics for undamped or viscously damped systems, their counterparts for viscoelastically damped systems seem not well-developed. Motivated by such points, this paper develops a model-order-reduction approach for viscoelastically damped systems based on a free interface CMS method. Due to the fact that the nonviscous modes and higher-order elastic modes are difficult to be obtained accurately for viscoelastic systems, the mode truncation error is often introduced. To reduce the mode truncation error problem, two residual attachment modes of viscoelastically damped systems are proposed in terms of the lower modes and system matrices by using the first one or two terms of Neumann expansion. Finally, three numerical studies are provided to illustrate the accuracy and effectiveness of the derived methods. It is shown that the proposed methods can receive good results and reduce the modal truncation error significantly over the conventional method.