Characteristic Constraint Modes Research Articles

An improved reduced order model for bladed disks including multistage aeroelastic and structural coupling

To assess the influence of mistuning on the vibration amplitudes of turbomachinery rotors, reduced order models (ROMs) are widely used. A variety of methods are available for single-stage configurations and mostly aeroelastic effects can be taken into account. More recent research focusses on extending these methods to include multiple stages. However, due to the significantly increased computational effort of the aeroelastic simulations when adding more stages to the models, these ROMs are rarely applied with the inclusion of multistage aeroelastic effects. It is therefore desirable to develop reduction methods which minimize the number of these simulations to reduce the computational cost and thereby enable analyses of rotors with multiple stages including aeroelastic effects. In this paper, a cyclic Craig-Bampton reduction method with an a priori interface reduction for multistage rotors is extended with an additional a posteriori interface reduction to reduce the number of aeroelastic simulations necessary for a given accuracy level of the ROM. The interface degrees of freedom between stages are reduced using a modified version of Characteristic Constraint Modes, to yield a more efficient representation of their displacements while retaining their monoharmonic nature. The method is applied to a two-stage axial compressor with full aeroelastic coupling between the stages and its reduced computational effort is demonstrated. Additionally, two sorting methods for the degrees of freedom (DOFs) of the ROM are compared.

Refining characteristic constraint modes of component mode synthesis with residual modal flexibility

The Craig–Bampton component mode synthesis uses interface constraint modes and deformed substructural modes for model reduction. In some instances, it may be of interest to realize additional reductions after reducing the substructural degrees of freedom, and this is generally done by the reduction of the interface degrees of freedom. The characteristic constraint modes were developed to achieve the interface reduction of the Craig–Bampton method. However, the interface reduction process can compromise accuracy because using a small number of characteristic constraint modes, focusing on low-frequency range, may interrupt vibration energy transmission between substructures in relatively mid and/or high-frequency ranges. Based on this motivation, an improved reduction technique of the interface regions is introduced for refining characteristic constraint modes. Considering a residual modal effect is the main point of refinement of the characteristic constraint modes. The modified characteristic constraint modes and fixed-interface normal modes computed by a newly derived formulation with residual modal flexibility can represent the characteristics of interface motion better than the conventional constraint modes. Consequently, the improved transformation matrix may facilitate vibration energy transmission between substructures. Numerical experiments indicate that the proposed refinement method can complement the conventional method in terms of accuracy, and the refined constraint and fixed interface normal modes are better representations of the original mode shapes than conventional approaches are.

Nonlinear Interface Reduction for Time-Domain Analysis of Hurty/Craig-Bampton Superelements with Frictional Contact

Virtual prototyping in engineering design rely on modern numerical models of contacting structures with accurate resolution of interface mechanics, which strongly affect the system-level stiffness and energy dissipation due to frictional losses. High-fidelity modeling within the localized interfaces is required to resolve local quantities of interest that may drive design decisions. The high-resolution finite element meshes necessary to resolve inter-component stresses tend to be computationally expensive, particularly when the analyst is interested in response time histories. The Hurty/Craig-Bampton (HCB) transformation is a widely used method in structural dynamics for reducing the interior portion of a finite element model while having the ability to retain all nonlinear contact degrees of freedom (DOF) in physical coordinates. These models may still require many DOF to adequately resolve the kinematics of the interface, leading to inadequate reduction and computational savings. This study proposes a novel interface reduction method to overcome these challenges by means of system-level characteristic constraint (SCC) modes and properly orthogonal interface modal derivatives (POIMDs) for transient dynamic analyses. Both SCC modes and POIMDs are computed using the reduced HCB mass and stiffness matrices, which can be directly computed from many commercial finite element analysis software. Comparison of time history responses to an impulse-type load in a mechanical beam assembly indicate that the interface-reduced model correlates well with the HCB truth model. Localized features like slip and contact area are well-represented in the time domain when the beam assembly is loaded with a broadband excitation. The proposed method also yields reduced-order models with greater critical timestep lengths for explicit integration schemes.

Modal approach to dynamic analysis of structures with attachments at uncertain locations

This paper presents an approach to calculate the natural frequency and frequency response function (FRF) statistics of structures connected to attachments at uncertain locations. The main structure is divided into two regions to which the attachment can, or cannot, be connected. The approach is based on component mode synthesis analysis of finite element models of these regions. Multi-point constraints are used to connect the attachment to the main structure to avoid the need to re-mesh the structure. The use of characteristic constraint modes is used to reduce the number of interface degrees of freedom (DOFs) and hence reduce the computational cost further. One or more baseline systems, for which the attachment is connected at a given position, are analysed and the eigensolution found. A perturbation is then applied to the baseline solution to estimate the solution for other attachment locations. The method can be applied to any problem requiring multiple re-analysis. Here, emphasis is placed on probabilistic approaches using Monte Carlo Simulation (MCS), and the natural frequency and FRF statistics are then estimated through MCS. To illustrate the approach, numerical results are presented for a beam and a plate with a randomly attached mass and a plate with an attachment with a number of internal and attachment DOFs. Numerical results are compared with experimental measurements for the cases of a beam and a plate with an attached mass. The results indicate that the method provides accurate predictions at a significantly reduced computation cost when compared to multiple direct solutions.

Reduced-order modelling using nonlinear modes and triple nonlinear modal synthesis

This paper introduces a novel reduced-order modelling technique well-suited to the study of nonlinear vibrations in large finite element models. The method combines the concepts of nonlinear complex modes and characteristic constraint modes – or interface modes – to build a fully algebraic reduced-order model easily solved by standard iterative solvers. The performance of the method is appraised on a nonlinear finite element model of bladed disk in the presence of structural mistuning, highlighting the potential of this new technique to deal with current industrial problematics.

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.

Modal Substructuring of Geometrically Nonlinear Finite Element Models with Interface Reduction

Substructuring methods have been widely used in structural dynamics to divide large, complicated finite element models into smaller substructures. For linear systems, many methods have been developed to reduce the subcomponents down to a low-order set of equations using a special set of component modes, and these are then assembled to approximate the dynamics of a large-scale model. In this paper, a substructuring approach is developed for coupling geometrically nonlinear structures, where each subcomponent is drastically reduced to a low-order set of nonlinear equations using a truncated set of fixed-interface and characteristic constraint modes. The method used to extract the coefficients of the nonlinear reduced-order model is nonintrusive, in that it does not require any modification to the commercial finite element code but computes the reduced-order model from the results of several nonlinear static analyses. The nonlinear reduced-order models are then assembled to approximate the nonlinear differential equations of the global assembly. The method is demonstrated on the coupling of two geometrically nonlinear plates with simple supports at all edges. The plates are joined at a continuous interface through the rotational degrees of freedom, and the nonlinear normal modes of the assembled equations are computed to validate the models. The proposed substructuring approach reduces a 12,861-degree-of-freedom model down to only 23 degrees of freedom while still accurately reproducing the nonlinear normal modes.

An Efficient Dynamic Substructuring and Application to Reliability Analysis

This work presents an efficient substructure coupling technique that considers interface reduction. The dynamic behavior of the substructures is characterized by dominant and residual fixed interface normal modes. Interface reduction is accomplished by using a reduced number of characteristic constraint modes. The effectiveness of the proposed model reduction technique is demonstrated by means of a numerical example involving the reliability analysis of a structural system under stochastic excitation.

A framework for the analysis of vibrations of structures with uncertain attachments

Attachments affect the dynamic response of an assembled structure. When engineers are modelling structures, small attachments will often not be included in the “bare” model, especially in the initial design stages. The location of these attachments might be poorly known, yet they affect the response of the structure. This paper considers how attachments jointed to the structure at uncertain points, can be included in the dynamic model of a structure. Two approaches are proposed. In the time domain, a combination of component mode synthesis, characteristic constraint modes and modal analysis gives a computationally efficient basis for subsequent analysis using, for example, Monte Carlo simulation. The frequency domain approach is based on assembly of frequency response functions of bare structure and attachment. Numerical examples of a beam and a plate with a point mass added at an uncertain location are considered and predictions compared with experiment results.

Component mode synthesis as a framework for uncertainty analysis

Component mode synthesis (CMS) is a well-established technique for the vibration analysis of built-up structures. It was originally developed as a method for reducing the size of a finite element model, hence reducing computational cost. CMS also offers an appealing framework for the analysis of the dynamics of uncertain structures. The benefits concern the numerical costs, the way uncertainty is included, quantified and propagated. This paper reviews and discusses these issues. The fixed-interface (Craig–Bampton) method is described, while the number of interface degrees of freedom (dofs) can be further reduced using characteristic constraint modes. Quantification and propagation of uncertainty is discussed. Uncertainties in properties can be naturally and straightforwardly introduced at the component level, either in terms of the component physical properties or the component modal properties, while the individual components are typically statistically independent, being made by different manufacturing processes. CMS methods are also amenable to the inclusion of experimentally measured variability data, quantifying it in terms of component modal properties. An example is given. The application of perturbational techniques is considered. The CMS framework is particularly amenable to propagation of uncertainty through one or more of the analysis paths at component or at global level using perturbations. Finally, qualitatively different uncertainty descriptions can be combined, with some components being described probabilistically, some possibilistically, the descriptions then being unified at the global level. Numerical examples are presented. Overall, CMS methods offer a strong physical insight into the analysis of structures with non-deterministic properties.

Power Flow Analysis of Complex Structures Using Characteristic Constraint Modes

Component mode synthesis (CMS) is used as a basis for predicting power flow in complex structures. The power flow formulation is cast in a general framework of multilevel CMS, in which a hierarchy of substructures can be recursively partitioned from a very large finite element model. Additional computational efficiency is achieved by finding characteristic constraint (CC) modes. The CC modes are computed by performing an eigenanalysis on the partitions of the CMS mass and stiffness matrices that correspond to the constraint-mode degrees of freedom. The CC modes describe the vibration, and the exchange of vibration energy, at the interface between connected substructures. Therefore, a relatively small number of CC modes that capture the primary interface motion can be selected to yield a reduced-order model for computing the power flow. The performance and accuracy of the method are illustrated by examples of a two-span beam, a cantilever plate, and the body structure of a military vehicle. For the vehicle structure, multiple levels of substructures are used to map the power flow in the system. It is seen that this multilevel, characteristic-mode-based approach provides a general framework for the efficient calculation of power How in the low- to midfrequency range.

Characteristic Constraint Modes for Component Mode Synthesis

A technique is presented for reducing the size of a model generated by the Craig-Bampton method (Craig, R. R., and Bampton, M. C. C., Coupling of Substructures for Dynamic Analyses, AIAA Journal, Vol. 6, No. 7, 1968, pp. 1313-1319) of component mode synthesis (CMS). An eigenanalysis is performed on the partitions of the CMS mass and stiffness matrices that correspond to the so-called constraint modes. The resultant eigenvectors are referred to as characteristic constraint modes because they represent the characteristic motion of the interface between the component structures. When the characteristic constraint modes are truncated, a CMS model with a highly reduced number of degrees of freedom may be obtained. An example of a cantilever plate is considered. It is shown that relatively few characteristic constraint modes are needed to yield accurate approximations of the lower natural frequencies. Furthermore, this method yields physical insight into the mechanisms of vibration transmission in complex structures, and it provides an excellent framework for the efficient calculation of power flow.

Characteristic constraint modes for component mode synthesis

Characteristic constraint modes for component mode synthesis