Component Mode Synthesis Approach Research Articles

A hybrid formulation for determining the input power in a stiffened plate over a wide frequency range

A hybrid analytical/numerical formulation is presented for determining the input power in a stiffened plate over a wide frequency range. A Component Mode Synthesis (CMS) approach is used for developing the hybrid formulation. The structure is divided into a local substructure in the vicinity of the excitation and non-local substructures representing the remaining system. It is considered that the excitation is applied at a small number of discrete locations, therefore the local substructure is a small portion of the overall system. The computational efficiency originates from using an analytical approach and periodic structure theory for determining the dynamic modes and natural frequencies for the non-local substructures. Finite elements are used for determining the static modes for the non-local substructures and also for both the dynamic and static modes of the local substructure. The CMS matrices for the non-local substructures are condensed to their common interface degrees of freedom with the local substructures. In this manner, the non-local substructures are eventually represented as boundary conditions on the local substructure. Results from the hybrid formulation are compared with results from dense finite element models in order to demonstrate the validity and the efficiency of the hybrid method.

Modeling of flexible bevel gear rotor systems: Modal and dynamic characterization

Bevel gears, a crucial part of aircraft transmission systems, are more susceptible to wheel body vibration issues as lightweight requirements rise. In the past, the gear wheel body was often considered rigid while assessing the dynamic properties of gear systems. Constructing a bevel gear model with a thin-walled structure might not be appropriate for this modeling technique; this research suggests a dynamic model that considers the flexibility of bevel gears. The web and teeth structure of the gear are preserved by realistic modeling using three-dimensional hexahedral components. The component mode synthesis (CMS) approach reduces the model order for increased computational efficiency. To confirm the accuracy of the modeling method, the modal frequencies and vibration shapes of the proposed model single-gear shaft are compared with results from experiments and ABAQUS. By contrasting ABAQUS system-level meshing modal frequencies and vibration shapes, the accuracy of the system model assembly is further confirmed. The model examines the impact of bevel gear flexibility on the modal frequencies, modal vibration shapes, and system dynamic properties. The mode shape diagram at dangerous modal frequencies found using the modal strain energy technique and the vibration displacement cloud maps at resonance speeds computed using the Newmark-β method are mutually verified, and this demonstrates that lightweight bevel gears are more susceptible to excitation due to nodal diameter type vibrations. Proposed and traditional models anticipate substantially different resonance speeds since the traditional model assumes that the gear, being a rigid disk, can only excite the shaft's bending mode of vibration. The proposed model provides a more comprehensive description of the dynamic characteristics of bevel gear systems, providing a theoretical basis and optimization tool for the high-performance design, vibration reduction, and noise control of high-speed lightweight gear transmission systems.

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

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

A metamaterial consisting of an acoustic black hole plate with local resonators for broadband vibration reduction

Acoustic black hole (ABH) indentations on plates are very efficient to reduce high frequency vibrations. However, when the wavelength of the impinging bending waves on the ABH is larger than its diameter, waves cannot be trapped and dissipated within the ABH and that becomes ineffective. Therefore, it would be highly desirable to extend the performance of ABH plates to lower frequencies. In this work, a method is proposed to accomplish that goal. It is suggested to design a metamaterial in which a set of periodic local resonators are attached to an ABH plate. On the one hand, the resonators are tuned to have a bandgap at the plate first eigenmode so as to suppress it. On the other hand, the resonators are also damped which substantially lowers the peaks of the remaining low-order eigenfrequencies. In combination with the ABH effect, such design, hereafter termed the MMABH plate, provides broadband vibration reduction covering the whole frequency range. To characterize the MMABH, the Gaussian expansion method (GEM) for determining the vibrations of the ABH plate is integrated with a component mode synthesis (CMS) approach, which allows one to link the resonators to the plate. That method is validated against finite element simulations. The MMABH is designed so that its overall mass (ABH plate plus resonators) equals that of the uniform plate without ABH indentation, to offer a light-weight solution. Theoretical explanations of the functioning of the MMABH plate are provided based on the analysis of the ABH effect, the dispersion curves and bandgaps of infinite periodic plates with local resonators and finally, the merging of both topics.

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.

Finite Element Model Reduction Applied to Nonlinear Impact Simulation for Squeak and Rattle Prediction

<div class="section abstract"><div class="htmlview paragraph">Increasing demand for simulation accuracy often leads to increased finite element model complexity, which in turn, results in higher computational costs. As a provision, component mode synthesis approaches are employed to approximate the system response by using dynamic substructuring and model reduction techniques in linear systems. However, the use of available model reduction techniques in nonlinear problems containing the contact type of nonlinearities remains an interesting topic. In this paper, the application of a component mode synthesis method in squeak and rattle nonlinear simulation has been investigated. Critical regions for squeak and rattle of the side door model of a passenger car were modelled by nonlinear contact definition in finite element simulation. Craig-Bampton model reduction method was employed to substructure the finite element model while keeping the nonlinear contacts in the model. The model response was evaluated using the modal assurance criterion, frequency response analysis and contact force magnitude in comparison with the baseline model. Results showed that a great reduction in computational time (about 98%) can be achieved while the accuracy of the system response was maintained at an acceptable range for the intended application for squeak and rattle simulation. Although the prediction of impact events in time was done accurately, the contact force magnitude was estimated with average error of 2.5% to 22%, compared with the baseline results. The outcomes of the study show that to empower squeak and rattle prediction by including contact interfaces in finite element simulations, implementation of the model reduction approach can compensate the simulation cost.</div></div>

Probabilistic Response of a Bladed Disk With Uncertain Geometry

AbstractGeometric uncertainties in the blade manufacturing process have important consequences in terms of dynamical properties of bladed disks. In this paper, we address the problem of modeling a full bladed disk composed by blades having uncertain geometry. The geometric imperfection of the blades is represented and analyzed according to a procedure previously presented by the authors, based on the principal component analysis (PCA) and the mesh morphing. The dynamical model of the full disk is constructed following the component mode synthesis (CMS) approach. The blade geometry is represented using a probabilistic model constructed from an experimental dataset. The effect of the geometric uncertainties is assessed using a linear uncertainty propagation approach, leading to a procedure that is fast enough to be embedded into a Monte Carlo simulation (MCS) loop.

Reduced-order thermomechanical modeling of multibody systems using floating frame of reference formulation

In this study, a reduced-order thermomechanical coupling model, which accounts for the inertia coupling of the thermoelastic deformation and the large reference body motion, is proposed using the floating frame of reference formulation for the transient thermomechanical analysis of constrained multibody systems. In this approach, the reduced-order heat equations are fully embedded in the final form of the equations of motion. Accordingly, the transient thermal response as well as the resulting thermoelastic behavior of constrained multibody system can be predicted within the general multibody dynamics computer algorithm. It is demonstrated that appropriate selection of the thermal interface coordinates is crucial for describing the thermal modes (i.e. temperature distribution) induced by external heat sources using the Craig–Bampton component mode synthesis approach generalized for thermomechanical systems. Furthermore, a systematic procedure for imposing prescribed surface temperature given, for example, from thermal-fluid dynamics simulations is proposed for the thermomechanical floating frame of reference formulation. Using several numerical examples, simulation capabilities of the thermomechanical floating frame of reference formulation model are demonstrated for multibody dynamics applications. Numerical results show good agreement with the nonlinear thermomechanical finite element solutions considering the large rotational motion with substantial reduction in the model dimensionality and computational time.

A Free Interface Component Mode Synthesis Approach for Determining the Micro-End Mill Dynamics

The prediction accuracy of the stability boundary in the machining process depends upon accurate estimation of cutting tool-tip dynamics. Note that the experimental modal analysis using direct impact at miniature end mill (typically 50–500 μm in diameter) is not feasible as it can result in tool failure. Consequently, alternative techniques such as experimental modal analysis using reciprocity theory and frequency-based receptance coupling substructure analysis (RCSA) have been used extensively for determining tool-tip dynamics. The experimental approach based on reciprocity theory assumes that the structure is symmetric (cross frequency response functions (FRFs) are same between two points of interest in a structure). RCSA requires a very fine frequency resolution and matrix inversion, which can lead to computational complexities. In addition, RCSA takes into account the FRFs only at the interface and free end, which can induce errors. Owing to these issues with existing approaches, this paper proposes a free-interface component mode synthesis (CMS) approach for estimation of micro-end mill dynamics. The effect of machine tool compliance including the collet–tool interface has been included for estimation of micro-end mill dynamics via a free-interface CMS approach wherein the experimental and analytical mode shapes are coupled. The predicted micro-end mill dynamics have been compared with RCSA and experimental modal analysis using reciprocity theory. Finally, the stability lobe diagrams for high-speed micromilling of Ti6Al4V has been made using the tool-tip dynamics from CMS, RCSA, and experimental technique using reciprocity theory and validated against experimental measurements for onset of instability.

Substructure-based model updating using residual flexibility mixed-boundary method

Substructure method has been widely applied in dynamic analysis of complex structures due to high computational efficiency. On the basis of Residual flexibility mixed-boundary (RFMB) substructure method, a model updating approach is proposed in this paper. Four major steps of the RFMB model updating method are summarized as: 1) Substructuring: Dividing the whole structure into residual part and reduced part according to the junction surface; 2) reduction: Using the RFMB component mode synthesis approach to reduce the order of each substructures; 3) assembly: Residual structure analysis by using the reduced assemble matrix; 4) updating: Model updating by solving the optimization problem. Numerical simulation is conducted to verify the effectiveness by adopting a cantilever plate in case I. In case II, the proposed method is applied to identify the elastic parameter of interface of a bolted joint structure using experimental data. After parameter identification, the maximum error between numerical results and the experimental data decreases to 2.44 %. And three component mode synthesis model updating methods: Craig-Bampton (CB), Mixed-boundary (MB) without considering residual flexibility and RFMB model updating approach, are applied to update the same bolted joint structure for comparing the accuracy. For comparing the computational efficiency, the RFMB model updating approach is applied to the complicated aero-engine casing structure. In case III, the average time-consuming of the Whole finite element model (WFEM) is 5.15 times to the Residual finite element model (RFEM) in the single updating iteration. Results show that the proposed approach has better performance in the finite element model updating.

Component Mode Synthesis Approach for Micro End Mill Dynamics Considering Machine Tool Compliance

Chatter in machining can be avoided by machining at stable process parameters obtained from the stability lobe diagram. The accuracy of the stability lobe diagram depends on the accurate determination of micro-endmill dynamics. It is not trivial to experimentally determine the tool tip dynamics of a micro-endmill due to relatively small diameters (10μm to 500μm). There are different approaches to estimate the micro-endmill dynamics, however, one of the most widely used techniques is the receptance coupling method. It combines the frequency response functions for the two substructures (machine tool with a portion of shank and the micro-endmill) to determine the tool-tip dynamics. There are certain issues with the receptance coupling which uses only two locations for determining the FRF one at the coupling and other at tip unlike the component mode synthesis which allows use of mode shapes at multiple points in the substructures. Another limitation for receptance coupling method is that the FRF for both substructures must be sampled at same frequency for coupling but the first non-rigid modal frequency for free-free substructure of micro-endmill can exceed 50kHz which would require a high bandwidth sensor with a sampling frequency in excess of 100kHz. Consequently, the component mode synthesis approach has been used to couple the substructures. The mode synthesis approach adopted in the present work contains the position dependent modes along the length of substructures. In order to ensure that the component mode synthesis yields accurate results, a test case of a micro-endmill as a fixed cantilever beam has been analyzed. Note that the actual micro-end mill dynamics is affected by the machine tool compliance which needs to be accounted for. Hence, a component mode synthesis of the two substructures, machine-tool/shank and the micro-endmill is carried out. The mode shapes of machine-tool shank substructure are determined experimentally whereas the micro-endmill mode shapes are determined via finite element method (FEM). The predicted micro-endmill dynamics with machine tool compliance has been experimentally validated.

Modeling and simulation of structural components in recursive closed-loop multibody systems

Passenger cars, transit buses, railroad vehicles, off-highway trucks, earth moving equipment and construction machinery contain structural and light-fabrications (SALF) components that are prone to excessive vibration due to rough terrains and work-cycle loads’ excitations. SALF components are typically modeled as flexible components in the multibody system allowing the analysts to predict elastic deformation and hence the stress levels under different loading conditions. Including SALF component in the multibody system typically generates closed-kinematic loops. This paper presents an approach for integrating SALF modeling capabilities as a flexible body in a general-purpose multibody dynamics solver that is based on joint-coordinates formulation with the ability to handle closed-kinematic loops. The spatial algebra notation is employed in deriving the spatial multibody dynamics equations of motion. The system kinematic topology matrix is used to project the Cartesian quantities into the joint subspace, leading to a condensed set of nonlinear equations with minimum number of generalized coordinates. The proposed flexible body formulation utilizes the component mode synthesis approach to reduce the large number of finite element degrees of freedom to a small set of generalized modal coordinates. The resulting reduced flexible body model has two main characteristics: the stiffness matrix is constant while the mass matrix depends on the elastic modal coordinates. A consistent set of pre-computed inertia shape integrals are identified and used to update the modal mass matrix at each time step. The implementation of the component mode synthesis approach in a closed-loop recursive multibody formulation is presented. The kinematic equations are modified to include the effect of the flexible body modal elastic coordinates. Also, modified constraint equations that include the effect of flexibility at the joint connections and the necessary details of the Jacobian matrix are presented. Baumgarte stabilization approach is used to stabilize the constraint equations without using iterative schemes. A sample results for flexible body impeded in a closed system will be presented to demonstrate the above mentioned approach.

Extended bloch mode synthesis: Ultrafast method for the computation of complex band structures in phononic media

In this article an efficient numerical technique, named Extended Bloch Mode Synthesis, is proposed for the fast calculation of the elastic complex band structures in phononic media. The Bloch Mode Synthesis approach, originally developed for reducing the computational cost for the calculation of real band structures, is here extended to evaluate also evanescent/complex near field wave solutions by solving a k(ω) Bloch eigenvalue problem. The k(ω) Bloch eigenvalue problem is built by means of a Wave Finite Element (WFE) discretization of the unit cell combined with a Component Mode Synthesis approach. The Component Mode Synthesis approach is based on the Craig Bampton modal reduction of the interior unit cell degrees of freedom and provides a basis to reduce the Bloch eigenproblem dimension allowing for a fast computation of the complex band structures. The performances of the proposed scheme in terms of band structures accuracy and computational cost saving are demonstrated for a phononic stubbed plate, a case already used in literature as a benchmark for band structures calculation. It is shown that the complex band structures computational time can be reduced by two orders of magnitude with respect to the computational time needed to solve the full model with negligible errors for both real and evanescent/complex solutions.

Low cost metamodel for robust design of periodic nonlinear coupled micro-systems

To achieve robust design, in presence of uncertainty, nonlinearity and structural periodicity, a metamodel combining the Latin Hypercube Sampling (LHS) method for uncertainty propagation and an enriched Craig-Bampton Component Mode Synthesis approach (CB-CMS) for model reduction is proposed. Its application to predict the time responses of a stochastic periodic nonlinear micro-system proves its efficiency in terms of accuracy and reduction of computational cost.

Prediction of dynamic behavior for different configurations in a drilling–milling machine based on substructuring analysis

Dynamic models of machine tool structures are essential instruments to evaluate structural performance in cutting operations. In order to fully exploit these models, they must meet three critical requirements such as accuracy, computational efficiency and multi-configuration machine behavior predictability. These virtues are hard to combine in one single model, as the improvement of one feature can easily involve a negative effect in the others. This paper deals with this problem and presents a robust procedure to develop reliable, efficient and versatile dynamic models. First, the machine tool structure is split up in several components. For each component, a finite element model is developed and the necessary corrections are made comparing the numerical results to the experimental data obtained from a modal analysis test. Once suitable numerical definitions of the components are available, substructures are defined, considering some components individually and grouping those with no relative movement. In this process, the connection between components is accurately modeled, checking the resulting substructures experimentally. Secondly, the order of the substructures is reduced using a Component Mode Synthesis approach based on Craig–Bampton method. Traditionally, when working with movable substructures, the order reduction capability has been limited by the large amount of connection degrees of freedom which must be kept in the reduced representation to cover all possible positions of substructures. This paper presents a procedure to solve this issue, where these degrees of freedom are eliminated from the reduced model as the substructures are assembled sequentially. Finally, in order to evaluate the reliability of the resulting model, the dynamic characteristics of the structure in various configurations are calculated and the results are compared with the experimentally obtained natural frequencies, mode shapes and frequency response functions for the same configurations. Results indicate that the proposed method predicts correctly the position-dependent dynamic behavior of a machine, validating the developed procedure.

Comparison of Common Methods in Dynamic Response Predictions of Rotor Systems with Malfunctions

The efficiency and accuracy of common time and frequency domain methods that are used to simulate the response of a rotor system with malfunctions are compared and analyzed. The Newmark method and the incremental harmonic balance method are selected as typical representatives of time and frequency domain methods, respectively. To improve the simulation efficiency, the fixed interface component mode synthesis approach is combined with the Newmark method and the receptance approach is combined with the incremental harmonic balance method. Numerical simulations are performed for rotor systems with single and double frequency excitations. The inherent characteristic that determines the efficiency of the two methods is analyzed. The results of the analysis indicated that frequency domain methods are suitable single and double frequency excitation rotor systems, whereas time domain methods are more suitable for multifrequency excitation rotor systems.

Simulations and Measurements of the Vibroacoustic Effects of Replacing Rolling Element Bearings With Journal Bearings in a Simple Gearbox

The effects of replacing rolling element bearings with journal bearings on the noise and vibration of a simple gearbox are computationally and experimentally evaluated. A modified component mode synthesis (CMS) approach is used, where the component modes of the shafting and gearbox housing are modeled using finite element analysis (FEA). Instead of using component modes with free boundary conditions, which is typical of CMS, the shafting and gearbox are coupled using nominal impedances computed for the different bearing types, improving convergence of the solution. Methods for computing the actual bearing impedances, including the high damping coefficients in journal bearings, are summarized. The sound radiated by the gearbox is computed using a boundary element (BE) model. The modeling results are validated against measurements made at the NASA Glenn Research Center. Both simulations and measurements reveal that the journal bearings, although highly damped, do not necessarily lead to strong reductions in gearbox vibration and noise.

Component Mode Synthesis Using Undeformed Interface Coupling Modes to Connect Soft and Stiff Substructures

Classical component mode synthesis methods for reduction are usually limited by the size and compatibility of the coupling interfaces. A component mode synthesis approach with constrained coupling interfaces is presented for vibro-acoustic modelling. The coupling interfaces are constrained to six displacement degrees of freedom. These degrees of freedom represent rigid interface translations and rotations respectively, retaining an undeformed interface shape. This formulation is proposed for structures with coupling between softer and stiffer substructures in which the displacement is chiefly governed by the stiffer substructure. Such may be the case for the rubber-bushing/linking arm assembly in a vehicle suspension system. The presented approach has the potential to significantly reduce the modelling size of such structures, compared with classical component mode synthesis which would be limited by the modelling size of the interfaces. The approach also eliminates problems of nonconforming meshes in the interfaces since only translation directions, rotation axes and the rotation point need to be common for the coupled substructures. Simulation results show that the approach can be used for modelling of systems that resemble a vehicle suspension. It is shown for a test case that adequate engineering accuracy can be achieved when the stiffness properties of the connecting parts are within the expected range of rubber connected to steel.

Component Mode Synthesis Using Undeformed Interface Coupling Modes to Connect Soft and Stiff Substructures

Classical component mode synthesis methods for reduction are usually limited by the size and compatibility of the coupling interfaces. A component mode synthesis approach with constrained coupling interfaces is presented for vibro-acoustic modelling. The coupling interfaces are constrained to six displacement degrees of freedom. These degrees of freedom represent rigid interface translations and rotations respectively, retaining an undeformed interface shape. This formulation is proposed for structures with coupling between softer and stiffer substructures in which the displacement is chiefly governed by the stiffer substructure. Such may be the case for the rubber-bushing/linking arm assembly in a vehicle suspension system. The presented approach has the potential to significantly reduce the modelling size of such structures, compared with classical component mode synthesis which would be limited by the modelling size of the interfaces. The approach also eliminates problems of nonconforming meshes in the interfaces since only translation directions, rotation axes and the rotation point need to be common for the coupled substructures. Simulation results show that the approach can be used for modelling of systems that resemble a vehicle suspension. It is shown for a test case that adequate engineering accuracy can be achieved when the stiffness properties of the connecting parts are within the expected range of rubber connected to steel.

Propagation of Uncertainty in Substructured Spacecraft Using Frequency Response

In many situations, it is either impossible to perform a spacecraft system level vibration test, or it is highly desirable to avoid one to conserve time and resources. If modeling and analysis are to replace system tests, it is imperative to have confidence in the results. Therefore, a probabilistic system correlation analysis must be performed by quantifying uncertainty in the substructures, and then propagating it into the system correlation metrics. A systematic procedure is presented for studying the effects of substructure uncertainty on system test-analysis correlation of spacecraft that are validated on a substructure-by-substructure basis. Covariance propagation is used to propagate uncertainty in a free-interface substructure frequency response into the expected frequency response uncertainty for the system using a component mode synthesis approach. The frequency response-based procedure is of special interest in systems with high-modal density, where modal correlation methods do not work. A simple two-substructure example is investigated, and the results of the proposed procedure are substantiated using a full system Monte Carlo analysis. The approach presented is very fast, compared with the Monte Carlo techniques, and it propagates uncertainty in the substructure frequency response directly into the system results.