Component Mode Synthesis Technique Research Articles

A model considering the longitudinal track–bridge interaction in ballasted railway bridges subjected to high-speed trains

AbstractIn this paper, a dynamic interaction model of the coupled system of railway bridge, foundation, subsoil, ballast, track, and high-speed train is presented, with special emphasis on the longitudinal interaction between the track and the bridge structure, taking into account the flexibility of the ballast. After a description of the model of this interaction system, the equations of motion are given separately for each subsystem. The discretization of the bedded rails is performed by two different approaches. In the first approach, the deflection of the rails is expanded into the eigenfunctions of a finitely long bedded beam representing the rails. In the second, simplified approach, the track response is represented by a superposition of the static deflection of the infinitely long bedded beam due to a concentrated load. The coupling of the bridge structure with the track is achieved by a component mode synthesis technique, which in the first approach leads to a representation of the equations of motion in state-space. A discrete substructure technique is used to couple this subsystem with the train model. The two presented strategies are verified by comparison with results of a finite element model of this interaction system. Several application examples reveal the influence of the horizontal track–bridge interaction and other modeling parameters on the dynamic bridge response.

Stability Analysis of an Industrial Blade Accounting for a Blade-Tip/Casing Nonlinear Interface

Abstract This paper investigates the local stability analysis of periodic solutions corresponding to the nonlinear vibration response of an industrial compressor blade, NASA rotor 37, on which are applied different types of nonlinearities. These solutions are obtained using a harmonic balance method-based approach presented in a previous paper. It accounts for unilateral contact and dry friction of the rotating blade against a rigid casing through a regularized penalty law. A Lanczos filtering technique is also employed to mitigate spurious oscillations related to the Gibbs phenomenon thus enhancing the robustness of the solver. In addition, a component mode synthesis technique is used to reduce the dimension of the numerical model. Stability assessment of the computed solutions relies on Floquet theory. It is performed through the computation of the monodromy matrix as well as Hill's method. Both methodologies are applied and thoroughly compared as the severity of the nonlinearity is gradually increased from a cubic spring to three-dimensional contact conditions on a deformed casing. While the presented results underline the applicability of both stability assessment methodologies for all types of nonlinearities, they also put forward the much higher computational effort required when computing the monodromy matrix. Indeed, it is shown that Hill's method yields converged results for significantly lower values of both the number of retained harmonics and the considered number of time steps thus making it a far more efficient method when dealing with industrial models. It is also underlined that the presented results are in excellent agreement with reference solution points obtained with time domain solution methods. Specific implementation tweaks that were found to be of critical importance in order to efficiently assess the stability of computed solutions are also detailed in order to provide a comprehensive view of the challenges inherent to such numerical developments.

Numerical study of bladed structures with geometric and contact nonlinearities

This paper deals with the reduced order modeling of turbomachine bladed structures accounting for both geometric nonlinearities and nonlinear blade-tips/casing contacts in a numerically efficient way. A recently derived methodology, based on the concept of modal derivatives, is first used to study the contact interactions of a single rotating blade impacting a rigid casing. In this reduction procedure, nonlinear internal forces due to large displacements are evaluated using the stiffness evaluation procedure and contact is numerically handled with Lagrange multipliers. In-depth analyses, including clearance consumption computations and frequency analyses with a continuation procedure, are performed to understand and characterize the combined influence of contact and geometric nonlinearities on the blade’s dynamics. The methodology is then generalized to full bladed disks using Component Mode Synthesis techniques with fixed interfaces. Each sector of the cyclically symmetric structure is projected onto a reduction basis composed of Craig–Bampton modes and a selection of their modal derivatives. A second reduction allows reducing the cyclic boundary degrees-of-freedom. The methodology is first assessed using a harmonic excitation at blade-tips, without contact. When applied to the bladed disk subjected to contact interactions, the methodology allows identifying the main interaction speeds that can be detrimental for the engine integrity. Through this work, the numerical strategy is applied on an open industrial compressor bladed disk model based on the NASA rotor 37 in order to promote the reproducibility of results.

Relative cyclic component mode synthesis: A reduced order modeling approach for mistuned bladed disks with friction interfaces

A new reduced order modeling technique for nonlinear dynamics of mistuned bladed disks with friction interfaces is presented. Although various models have been developed to characterize the dynamics of either mistuned or frictionally damped structures, predicting the vibratory response of mistuned bladed disks with frictional interfaces remains challenging due to the:large size of refined industrial models,nonsmooth friction nonlinearities and amplitude dependent dynamics, andlack of cyclic symmetry properties.The Relative Cyclic Component Mode Synthesis (RCCMS) approach benefits from the application of relative coordinates prior to performing a reduction. Based on the new method, the kinematics of contact surfaces are first described in terms of relative displacements between contact node pairs, afterwards the Craig-Bampton Component Mode Synthesis (CB-CMS) technique is employed to reduce the model size. Implementation of relative coordinates halves the number of nonlinear unknowns in the reduced space and enhances the CB-CMS reduction basis by incorporating system-level stick modeshapes.The proposed approach admits introduction of either blade or sector frequency mistuning into the final reduced order model. In this way, the reduction is performed only once, which makes it favorable for statistical analyses. In addition, the mistuned ROM can be developed with minimal computational effort by using the sector model only.The developed RCCMS-based ROM is demonstrated on a finite element model of a mistuned bladed disk with blade-root friction damping. Numerical simulations reveal the excellent accuracy in predicting the nonlinear forced response.

Modal reduction procedures for flexible multibody dynamics

Through a critical review of the various component mode synthesis techniques developed in the past, it is shown that both Craig–Bampton’s and Herting’s methods are particular cases of the mode-acceleration method and furthermore, Rubin’s method is equivalent to Herting’s method. Consequently, the mode-acceleration method is the approach of choice due to its simplicity and because unlike the other methods, it imposes no restriction on the selection of the modes. Next, a general approach to the modal reduction of geometrically nonlinear structures is developed within the framework of the motion formalism, based on the small deformation assumption. The floating frame of reference is defined unequivocally by imposing six linear constraints on the deformation measures, which are defined as the vectorial parameterization of the relative motion tensor that brings the fictitious rigid-body configuration to its deformed counterpart. This approach yields deformation measures that are both objective and tensorial, unlike their classical counterparts that share the first property only. Derivatives are expressed in the material frame, leading to computationally advantageous properties: tangent matrices are functions of the deformation measures only and become nearly constant during the simulation. Numerical examples demonstrate the accuracy, robustness, and numerical efficiency of the proposed approach. With a small number of modal elements, the formulation is able to capture geometrically nonlinear effects accurately, even in the presence of inherently nonlinear phenomena such as buckling.

Nonlinear aeroservoelastic analysis of a supersonic aircraft with control fin free-play by component mode synthesis technique

To investigate the nonlinear aeroservoelastic behaviors of a three-dimensional supersonic aircraft with control fin free-play, a modeling method based on the component-mode synthesis technique is utilized. The most distinctive feature of this method is that the structural nonlinearity can be expressed explicitly in the reduced-order aeroservoelastic model. The unsteady aerodynamic model with discrete gust loads is formulated using the supersonic lifting surface theory and the minimum state approximation. The results validate the feasibility of the reduced-order aeroelastic model for gust response analysis. A hybrid adaptive feedback control algorithm is proposed for the gust load alleviation by integrating the positive position feedback (PPF) and the filtered-x least-mean-square (FxLMS) algorithm. The comparative study demonstrates that the designed PPF-FxLMS algorithm has a better control performance for the alleviation of the pitch attitude angle induced by the gust loads, and it remains effective in the post-flutter regime. Moreover, the effects of free-play nonlinearity on the dynamic behaviors of the aeroservoelastic system are also studied, and numerical results show that the presence of free-play can lead to a larger peak value of gust response. The nonlinear vibration of the control fin induces the aircraft pitch attitude to produce stable limit cycle oscillations with high frequency, and variation in the free-play magnitude significantly influences the gust responses of the nonlinear aeroservoelastic system.

Centrifugal stiffening analysis of gear pair with generalized component mode synthesis and semi-analytic contact technique

Cantilever beam assumption is widely used to estimate gear flexibility for analytical calculation of transmission error of gear pair. Similar to the rotating beam, it is supposed that centrifugal stiffening will also affect dynamic transmission error of gear pair. This work presents an efficient computational procedure for centrifugal stiffening analysis of gear pair. As inertia centrifugal force is distributively applied on gear body, the finite element grid model of gear pair is firstly established. Then, in order to reduce large number of degrees of freedom in finite element grid, the generalized component mode synthesis is employed to establish dynamic equation of flexible gear with constant mass matrix and co-rotated constant stiffness matrix. In order to avoid highly refined grid near contact zone, the semi-analytic contact technique which incorporates advantage of penalty method and Lagrange multiplier method is employed to calculate contact force between non-matching sparse tooth face meshes. After transforming system dynamic differential algebraic equation into ordinary differential equation, dynamic contact analysis of a helical gear pair is carried out, and dynamic transmission error at different rotational speeds are compared. Inflation deformation of gear under inertia centrifugal force is shown as explanation for influence of centrifugal stiffening on dynamic transmission error.

Mixed-Boundary Component Mode Substitution for Nonlinear Dynamics of Mistuned Shrouded Bladed Disks

A new mixed-boundary model reduction technique is presented for nonlinear forced response analysis of mistuned bladed disks with shroud friction contacts. In contrast to conventional fixed- or free-mode component mode synthesis (CMS) techniques, blades are reduced in a mixed-boundary fashion. Accordingly, blade component normal modes with a more realistic mixed boundary condition (fixed at blade-disk interface and free at shroud contacts) are incorporated into the reduction basis. Afterward a modal synthesis based on loaded interface modeshapes of the disk is performed to reduce the disk component and the previously retained blade-disk interface degrees of freedom (DOFs). This results in a compact reduced order model (ROM) in which retained physical DOFs contain only shroud contact nodes. During the reduction steps, sector-level frequency mistuning is efficiently introduced into the ROM. In addition, the reduction basis is invariant of the mistuning pattern, which makes the ROM suitable for statistical characterization of nonlinear response levels. The final ROM is computed using one single sector only and with minimal computational effort. Numerical simulations revealed an improved accuracy of the ROM based on the mixed-boundary model in predicting the nonlinear forced response levels of a mistuned bladed disk with shroud friction contacts.

Interface Reduction with Multilevel Craig–Bampton Substructuring for Component Mode Synthesis

Component mode synthesis is commonly used to simulate the structural behavior of complex systems. Among other component mode synthesis techniques, the Craig–Bampton method stands out for its popularity. However, for finely meshed systems featuring many components, the size of the resulting assembled system is dominated by the interface degrees of freedom. The system-level interface reduction technique aims at reducing the size of the assembled reduced model by extracting a few dominant interface modes. If the size of the interface degrees of freedom is large, the resulting problem is almost as computationally expensive as the one associated to the full model. Conversely, the local-level interface reduction technique reduces the interface of each substructure before assembly. In this case, the computational effort associated to the local eigenvalue problem is moderate, but issues arise when enforcing compatibility between interfaces. In this paper, the computational effort related to the interface reduction is significantly reduced by performing two variants of the multilevel Craig–Bampton reduction when the subsystems are assembled in subsets. This procedure localizes the interface reduction by applying a multilevel static condensation and eigenvalue analysis on each subset in parallel. The different interface reduction techniques are assessed on large-size realistic examples.

Reduction and coupling of substructures via Gram–Schmidt Interface modes

The paper presents a novel reduction method in the class of component mode synthesis techniques. The method is here called Gram–Schmidt Interface (GSI), since it is based on the Gram–Schmidt algorithm. The GSI reduction method addresses problems where the Craig–Bampton reduced order model of a substructure has still a large number of interface degrees of freedom that should be further reduced. Starting from the pre-existing reduction method based on the Characteristic Constraint Modes, the GSI technique goes further, being suitable to reduce a Craig–Bampton model that retains as a master a wide variety of boundary DoFs (interface, active, etc.). The GSI method is here proved to be effective when assembling adjacent substructures having non-compatible meshes at the interfaces. This is possible by performing the interfaces coupling in the modal space of the GSI modes instead in the space of physical degrees of freedom. The technique is here applied to a turbomachinery test case consisting of a blade coupled with a disk’s sector treated in cyclic symmetry conditions. It is also demonstrated how the GSI method appears useful in the reduction of cyclic symmetric structures, representing a valid alternative of the method developed by Tran.

Generalized Bloch mode synthesis for accelerated calculation of elastic band structures

The Bloch mode synthesis (BMS) model-reduction method adapts component mode synthesis techniques to unit-cell problems in order to obtain a reduced-order model that quickly produces band-structure frequencies for any wave vector, or vice versa. Fundamental to BMS is a partitioning of the real-space model into interior and boundary components, and subsequent reduction of the interior via truncated normal mode expansion. In this paper, two enhancements are presented for the BMS method that reduce both computation time and error in band-structure calculations. The first enhancement improves the accuracy of the interior reduction by approximating the participation of the residual modes rather than simply truncating them. The original formulation of BMS includes a modal reduction of the boundary that must be recomputed for every wave vector. This limits computational benefits and prevents the reduced-order model from being useful for the inverse band-structure problem (i.e., the k(ω) calculation). The second enhancement is a local boundary reduction that is independent of wave vector and thus does not suffer from the aforementioned limitations.

A component mode selection method based on a consistent perturbation expansion of interface displacement

A mode selection method is presented for the reduced-order modeling (ROM) of structural systems in conjunction with the Craig–Bampton component mode synthesis technique. The proposed method is derived by using a consistent expansion of the interface displacement in terms of a frequency-dependent small parameter as applied to a Craig–Bampton-like ROM formulation. It is found that this procedure yields a coupling mechanism of the modes of the full model to those of substructures. The present mode selection method employs this coupling mechanism as an indicator, labeled as the CMSσ method, for the substructural modal contributions to the full model. The performance of the proposed method is demonstrated by various numerical examples and compared favorably with existing method such as the CMSχ method.

Multi-level damage identification with response reconstruction

Damage identification through finite element (FE) model updating usually forms an inverse problem. Solving the inverse identification problem for complex civil structures is very challenging since the dimension of potential damage parameters in a complex civil structure is often very large. Aside from enormous computation efforts needed in iterative updating, the ill-condition and non-global identifiability features of the inverse problem probably hinder the realization of model updating based damage identification for large civil structures. Following a divide-and-conquer strategy, a multi-level damage identification method is proposed in this paper. The entire structure is decomposed into several manageable substructures and each substructure is further condensed as a macro element using the component mode synthesis (CMS) technique. The damage identification is performed at two levels: the first is at macro element level to locate the potentially damaged region and the second is over the suspicious substructures to further locate as well as quantify the damage severity. In each level’s identification, the damage searching space over which model updating is performed is notably narrowed down, not only reducing the computation amount but also increasing the damage identifiability. Besides, the Kalman filter-based response reconstruction is performed at the second level to reconstruct the response of the suspicious substructure for exact damage quantification. Numerical studies and laboratory tests are both conducted on a simply supported overhanging steel beam for conceptual verification. The results demonstrate that the proposed multi-level damage identification via response reconstruction does improve the identification accuracy of damage localization and quantization considerably.

Dynamic Response Analysis of Elastic-plastic Structures with Component Mode Synthesis

This paper evaluates the validity of component mode synthesis (CMS) technique for dynamic response of elastic-plastic structures. A method incorporating elastic-plastic material behaviour and the fixed interface mode synthesis technology is presented. To do this, the incremental governing equations of elastic-plastic transient problem derived from nonlinear finite element theory are linearized in each time step, the tangent stiffness matrix is used to generate the basis vector of transformation matrix. Using the proposed method, the response of a simply supported beam loaded impulsively is examined. The Numerical results indicate that the proposed method is accurate, convergent and more efficient compared with the full order finite element model.

The influence of washing machine-leg hardness on its dynamics response within component-mode synthesis techniques

In this paper, we investigate a washing machine's dynamic response. Using an experimental modal analysis, the dependence of the first two natural frequencies on the hardness of leg rubber is demonstrated. In order to model this behaviour, a complete washing-machine numerical model is developed, including a detailed model of the leg. A simple linear leg model is proposed, which accounts for the contact conditions and enables an implicit analysis. The model is validated, based on two measurements with different leg configurations. Additionally, the component-mode synthesis methods are proposed. They allow separate treatment of the washing machine's legs and cabinet, as well as reducing the model order. The four model-reduction techniques are compared with the classic finite-element method. It is shown that the component-mode synthesis methods enable fast recalculation times for the modified substructures, while the remaining structure is calculated only once. This leads to a computationally efficient analysis. A comparison of the results shows good agreement between the component-mode synthesis methods and the classic finite-element method.

Modeling the vibration of spatial flexible mechanisms through an equivalent rigid-link system/component mode synthesis approach

In this paper, a novel formulation for modeling the vibration of spatial flexible mechanisms and robots is introduced. The formulation is based on the concepts of equivalent rigid-link system (ERLS) that allows kinematic equations of motion for the ERLS decoupled from the compatibility equations of the displacement at the joint to be written. With respect to the available literature, in which the ERLS concept has been proposed together with a finite element method (FEM) approach (ERLS-FEM), the formulation is extended in this paper through a modal approach and, in particular, a component mode synthesis technique which allows a reduced-order system of dynamic equations to be maintained even when a fine discretization is needed. The model is validated numerically by comparing it with the results obtained from the Adams-Flex™ software, which implements the well-known floating frame of reference approach for a benchmark L-shaped mechanism. A good agreement between the two models is shown.

Reliability sensitivity analysis of stochastic finite element models

Abstract This contribution presents a scheme for integrating a model reduction technique into a simulation-based method for reliability sensitivity analysis of a class of medium/large nonlinear finite element models under stochastic excitation. The solution of this type of problems requires a large number of finite element model re-analyses to be performed over the space of system parameters. A component mode synthesis technique is implemented to carry out the sensitivity analysis in a reduced space of generalized coordinates. The reliability sensitivity analysis is performed by an approach based on a simple post-processing of an advanced sampling-based reliability analysis. The feasibility and effectiveness of the proposed scheme is demonstrated on a bridge finite element model under stochastic ground excitation.

Prediction of dynamic stresses using flexible multibody system algorithms: Application to tracked vehicle upper structure

The objective of this investigation is to develop a general procedure for predicting the dynamic stresses when the nonlinear floating frame of reference formulation is used to model flexible components in multibody system applications. The procedure utilizes the concept of the floating frame of reference intermediate finite element coordinate system and the general continuum mechanics approach to develop accurate definitions of the floating frame of reference kinematic equations that enter into the formulations of the strains. These general equations are used to design a computational algorithm for the calculation of the dynamic stresses. In order to demonstrate the use of the procedure developed in this investigation for the stress prediction, a tracked vehicle system is used as the study model. Taking into consideration the actual endurance test, a model for a bulldozer upper structure is developed using computer aided engineering solution framework which includes flexible multibody system algorithms. In order to obtain the dynamic stresses, the flexible structure of the tracked vehicle is modeled using Mindlin shell finite elements. The equations of motion are developed using the finite element/floating frame of reference formulation that allows for effectively reducing the number of degrees of freedom by neglecting high-frequency modes of vibration using component mode synthesis techniques. The stresses are obtained using a procedure that takes into consideration the dynamic coupling between the rigid body motion and the elastic deformations. The use of the general continuum mechanics theory to formulate the strain and stress equations allows for applying the procedure developed in this paper to other element types. Kinematic relationships are used to formulate several components and elements of the undercarriage including the idler cushion system and the maximum reverse angle. Numerical results are presented in order to demonstrate the use of the general floating frame of reference procedure in the stress analysis of the tracked vehicle undercarriage system.

Analysis of ballistic transport in nanoscale devices by using an accelerated finite element contact block reduction approach

An accelerated Finite Element Contact Block Reduction (FECBR) approach is presented for computational analysis of ballistic transport in nanoscale electronic devices with arbitrary geometry and unstructured mesh. Finite element formulation is developed for the theoretical CBR/Poisson model. The FECBR approach is accelerated through eigen-pair reduction, lead mode space projection, and component mode synthesis techniques. The accelerated FECBR is applied to perform quantum mechanical ballistic transport analysis of a DG-MOSFET with taper-shaped extensions and a DG-MOSFET with Si/SiO2 interface roughness. The computed electrical transport properties of the devices obtained from the accelerated FECBR approach and associated computational cost as a function of system degrees of freedom are compared with those obtained from the original CBR and direct inversion methods. The performance of the accelerated FECBR in both its accuracy and efficiency is demonstrated.

An Impulse Based Substructuring method for coupling impulse response functions and finite element models

In the field of Dynamic Substructuring (DS), large and complex structures are divided into several smaller and simpler components. The linear substructures are subsequently described in their dominant dynamics and reassembled, allowing one to compute the coupled dynamic behavior. DS methods are often classified into two distinct families, the Frequency Based Substructuring (FBS) methods and Component Mode Synthesis (CMS) techniques. In the former substructures are assembled whose dynamics are described in terms of frequency response functions (FRFs) and the latter are used to reduce and assemble the substructure finite element (FE) models. Lately a new substructuring method has been proposed, one that does not fit the framework of the FBS and CMS methods. The method, named Impulse Based Substructuring (IBS), was first used to obtain the coupled response of a system by assembling its component impulse response functions (IRFs). In this paper the IBS method is extended, thereby allowing one to determine the coupled behavior of structures that are composed of both substructure FE models and substructure IRFs. The method can be regarded as an extension to the normal time integration methods used for obtaining the time responses of FE models. As the linear substructures (described in their IRFs) are fully condensed on the interface of the FE model, one can significantly reduce the computational cost required for time integrating otherwise large FE models. However, as the linear (IRF) domains are exactly accounted for, the IBS method can be seen as a dynamic condensation on the interface, but not as a reduction method in the classical sense. Nonetheless, one can regard IRFs as a sort of “superelements in time” and the IBS method can therefore serve as an attractive alternative to CMS methods in case these are not available in the applied FE modeling programs, or if a high spectral bandwidth of the substructure is required. The method proposed in this work is based on the generalized-α time integration scheme and it is analytically proven that it can be applied in such a way that the simulation results are identical to the responses obtained from a monolithic integration of the full system, thereby guaranteeing its stability and accuracy. The method is demonstrated using a numerical test case, where a wind turbine FE model is coupled with a the IRFs of a marine foundation.