The wave-based substructuring approach for the efficient description of interface dynamics in substructuring

A flexibility‐based component mode synthesis (CMS) is proposed for reduced‐order modelling of dynamic behaviour of large structures. The approach employs partitioning via the localized Lagrange multiplier method. The use of the localized Lagrange multipliers leads to, unlike the classical Lagrange multipliers, a linearly independent set of interface forces without any redundancies at multiply connected interface nodes. The flexibility‐based CMS method has shown significant advantages over the classical Craig–Bampton method. A key feature of the method is its substructural mode selection criterion that is independent of loading conditions. Unlike the majority of available CMS approaches, where one retains the full dimension of partition boundary degrees of freedom (DOFs), the flexibility‐based method allows to reduce significantly the interface DOFs. The reduction of interface DOFs represents the major contribution of the present communication. The efficiency of the proposed approach is demonstrated on an analysis of a simple plate partitioned and of a more complex 3D structure, both partitioned into several substructures. Copyright © 2006 John Wiley & Sons, Ltd.

The component mode synthesis (CMS) methods are often utilized for modal analysis to investigate the vibration characteristics of the complex structures which are commonly divided into several substructures. However, non-matching finite element meshes may occur at the interfaces between components and virtual gaps are easily produced along the curved interfaces, which limit the application of CMS and lead to larger numerical errors for vibration analysis. To overcome the problem, a novel gap element method (GEM) is employed into a free-interface CMS method in this paper, where both displacements and forces of the nodes on the incompatible interfaces are introduced by two independent Lagrange multipliers to enforce the compatibility conditions. Two-dimensional numerical examples are given to validate the effectiveness of the proposed method. The results of natural frequencies and modal shapes obtained using the proposed method agree very well with the ones obtained using full finite elements model, no matter the gaps along the interface exist or not. The influence of the number of nodes on the non-matching interfaces on the accuracy of frequencies is also discussed.

A classic issue in component mode synthesis methods is the choice for fixed or free boundary conditions at the interface degrees of freedom and the associated vibration modes in the component's reduction basis. Closely related is the aspect of substructure assembly. In this paper a general approach to mixed boundary component mode synthesis problems is proposed, by considering component reduction and assembly separately. First, a general framework for substructure assembly is presented, enabling assembly of all types of reduced component models, regardless of their interface conditions. Thereafter, a general mixed boundary component mode synthesis method called the mixed Craig―Bampton method is introduced. The method is derived by dividing the substructure degrees of freedom into a set of internal degrees of freedom, free interface degrees of freedom, and fixed interface degrees of freedom using a simple but effective selection scheme. Based on this selection a reduction basis is computed consisting of vibration modes, constraint modes, and residual flexibility modes, which leads to relatively sparse reduced matrices. As such, the mixed Craig―Bampton method forms a natural generalization of the classic Craig―Bampton and more recent dual Craig―Bampton methods. Finally, the method is applied to a test problem, revealing that the proposed method has comparable or better accuracy and superior versatility with respect to existing methods.

Noise, Vibration and Harshness (NVH) is the study and modification of the noise and vibration characteristics of vehicles, particularly cars and trucks. NVH parameters which consist of noise (unwanted sound), vibration (mechanical oscillation) and harshness (severity or discomfort of noise or vibration) need to be evaluated in order to improve comfort and safety levels for vehicle occupants. In this paper, NVH study was conducted on Malaysian Armed Forces (MAF) tactical vehicles in order to ensure that the vehicles meet the requirements of the Occupational Safety and Health Administration (OSHA) and Directive 2002/44/EC. The project aim was to identify the preferable solution that can improve the NVH parameters of the vehicles in accordance to the standard and test protocols. A 4x4 Troop Transporter vehicle which carried 8 persons at the rear cabin was used in the NVH testing. The vibration parameters studied are hand arm vibration (HAV), focussing on vibration on the steering wheels of the vehicle, and whole body vibration (WBV), focussing on vibration on the seats. For noise, it was measured at the driver cabin and rear cabin of the vehicles. From the results, the HAV value for driver during idle speed was found to be higher than the others velocity. For WBV, vibration value for passenger 1 in rear cabin was highest compared to the other WBV test points.

Element-by-element model updating of large-scale structures based on component mode synthesis method

An efficient methodology for predicting the nonlinear forced vibration response of a turbine engine rotor with a cracked blade is presented and used to investigate the effects of the damage on the forced response. The effects of small, random blade-to-blade differences (mistuning) and rotation on the forced response are also considered. Starting with a finite element model, a hybrid-interface method of Component Mode Synthesis (CMS) is employed to generate a reduced-order model (ROM). The crack surfaces are retained as physical degrees of freedom in the ROM so that the forces due to contact interaction in three-dimensional space can be properly calculated. The resulting nonlinear equations of steady-state motion are solved by applying an alternating frequency/time-domain method, which is much more computationally efficient than traditional time integration. Using this reduced-order modeling and analysis framework, the effects of the cracked blade on the system response are investigated for various mistuning levels and rotation speeds. First, the advantages of the selected hybrid-interface CMS method are discussed and demonstrated. Then, the resonant frequency shift associated with the stiffness loss due to the crack, as well as vibration localization about the cracked blade are thoroughly investigated. In addition, the results of the nonlinear ROMs are compared to those obtained with linear ROMs as well as blade-alone ROMs. It is shown that several key system vibration characteristics are not captured by the simpler models, but that some insight into the system response can be gained from the blade-alone response predictions. Furthermore, it is demonstrated that while the effects of the crack often appear similar those of mistuning, differences between the effects of mistuning and damage can be discerned by observing and comparing the response across different families of system modes.

The chapter introduces the simulation methods (MBS, FEM, SEA) which are generally used in the vehicle NVH. Alongside also hybrid methods will be introduced. An overview will be given about the usage of the methods depending on the frequency range for the simulation and prediction of the NVH behaviour of full vehicle and its components.

<div class="section abstract"><div class="htmlview paragraph">Road noise caused by road excitation is a critical factor for vehicle NVH (Noise, Vibration, and Harshness) performance. However, assessing the individual contribution of components, particularly bushings, to NVH performance is generally challenging, as automobiles are composed of numerous interconnected parts. This study describes the application of Component Transfer Path Analysis (CTPA) on a full vehicle to provide insights into improving NVH performance. With the aid of Virtual Point Transformation (VPT), blocked forces are determined at the wheel hubs; afterward, a TPA is carried out. As blocked forces at the wheel hub are independent of the vehicle dynamics, these forces can be used in simulations of modified vehicle components. These results allow for the estimation of vehicle road noise. To simulate changes in vehicle components, including wheel/tire and rubber bushings, Frequency-Based Substructuring (FBS) is used to modify the vehicle setup in a simulation model. In this process, analytic springs were added in parallel to the actual rubber mounts, which were also measured using VPT on the actual vehicle. These springs simulate positive or negative changes in stiffness, allowing the study of how changes in the rubber mounts affect the vehicle's NVH performance. In this study, a sensitivity analysis was conducted using the FBS model to observe changes in road noise resulting from variations in bushing stiffness and blocked force. Based on the results of this analysis, the optimal combination of components was proposed for reducing the road noise. This approach provides an efficient and accurate method for analyzing the effects of rubber mount modifications on NVH performance.</div></div>

Vehicle noise, Vibration and Harshness (NVH) has become a key enabler for successful distinction between competitive vehicles because consumers have come to judge vehicle quality by NVH levels. One of many critical areas of NVH is the amount of vibration felt in the steering wheel as the customer drives the vehicle. This paper presents the evaluation of steering column vibration for a rear-wheel drive sports sedan. A significant amount of component level work had previously been done for the column under study, the results of which did not yield a complete understanding of in-vehicle response. For this follow-up study, a varying force input level test was designed to understand how this steering column responded under forces seen in both bedplate and in-vehicle testing. It appears that the column reacts in a non-linear fashion, and the forced response spectra changes with varying input force. The additional insight of the component response yielded greater understanding of the vehicle response. Transfer functions between key chassis attachment points and the response at the wheel were studied to understand the vehicle response and, where possible, to improve the column response. These investigations yielded significant understanding to component and vehicle weaknesses to arrive at improved designs. Significant improvements in tactile response were achieved by altering the vehicle chassis sensitivity to input loads. © 2004 Inderscience Enterprises Ltd.

The driver subjective perception of a vehicle is strongly determined by its Noise, Vibration and Harshness (NVH) behavior. Consequently, the NVH performance has become an important design and marketing criterion for vehicle manufacturers. The low-frequency (0–500Hz) noise and vibration perceived by a passenger are mainly determined by the dynamic behavior of rolling tyres. Besides the increasing awareness for the problems associated with road traffic noise has lead to the demand for more quiet tyres and road surfaces. Computer aided engineering tools play an important role for improving the vehicle vibration behavior at the earliest step of the design. In order to carry out precise modeling of the vehicle and its subsystems, an accurate tyre model is a key element in predicting the performance of the vehicle system with respect to ride comfort, NVH, durability, safety (braking manoeuvre) and tyre/road interaction. To obtain the best possible performance from a tyre model, a number of different measurements are required to support the tyre model parameter identification process. Although tyre vibration and noise behavior has been studied for several decades, there are still some missing links in the process of accurately predicting and controlling the overall tyre/road noise and vibration linked to the difficulty of performing modal testing on a tyre when rotating. This paper deals with the main test benches, experimental activities and modal analysis techniques available nowadays for characterizing the dynamic behavior of a static (unloaded tyre on a fixed hub and loaded tyre on a fixed hub) and a rotating tyre under different boundary conditions. The main results arising from modal analysis of a static and dynamic test on a tyre are highlighted and compared. In particular the effect of rolling speed, inflation pressure, preload, temperature and excitation amplitude on the dynamic response of a rolling tyre will be discussed.

A component mode synthesis method capable of estimating acoustic responses efficiently in a coupled structure–acoustic fluid interaction problem is presented. In order to analyze the response in a high-frequency range, a high- and low-order truncatable mode superposition method, namely, the most general mode superposition method by Ma-Hagiwara, is incorporated in the reduction process included in both the formulation of component mode synthesis and solving the superimposed global equation of motion. By truncating lower modes, it becomes possible to analyze a system with fewer degrees of freedom than the existing modal reduction methods. Frequency response analyses of several test problems of damped structures are performed and compared with the multipurpose versatile code, MSC/NASTRAN, to demonstrate the efficiency and versatility of the proposed method.

A technique is presented for estimating the residual flexibility between nonexcited structural degrees of freedom from experimental structural vibration data. Using this method, one can include the residual flexibility estimated from modal measurements in the computation of measured flexibility for experiments with incomplete reciprocity, i.e., when the response and excitation measurement sensors are not fully collocated. The method can also be used to estimate the unknown entries in the residual flexibility matrix for experimental component mode synthesis when excitations are not provided at all interface degrees of freedom. A general solution is presented that contains an unknown positive semidefinite contribution. The general solution satisfies modal orthogonality in the limit that all of the structural degrees of freedom are instrumented and when the positive semidefinite contribution lies in a nullspace defined by the stiffness matrix and the modal flexibility. With a limited number of measurements, modal orthogonality is shown to be satisfied to the extent that the measured modes are preserved by static condensation. A rank-deficient solution is presented that allows the residual to be used in the computation of the flexibility matrix without further modeling assumptions. Numerical and experimental results that demonstrate the application of the method to both flexibility matrix convergence and experimental component mode synthesis are presented.

High-order dilated nested arrays with increased degrees of freedom and reduced mutual coupling

<p indent="0mm">The railway track, as a long-scale composite structure, exhibits significant non-uniformity or randomness in its material properties and the state of its underlying foundation support. In this paper, based on the theory of vehicle-track coupling dynamics, a model for vehicle-track dynamic interaction is constructed using a generalized Rayleigh-Ritz method and component mode synthesis. This model is particularly suited for conditions where the structural cross-section size and material stiffness properties are non-uniform. The study begins with the track structure as the subject of research. An improved Fourier series is chosen as the displacement basis function for the track structure. The modal analysis of the track substructures is then conducted using the variational method of energy functionals, followed by truncation of these modal forms. The boundary conditions between each substructure of the track are integrated to reduce the overall system matrix of the track. For the first time, the GRRM-CMS is applied to analyze the vehicle-track dynamic interaction model. The model’s accuracy is validated by comparing its dynamic response results with those obtained from a FEM-based vehicle-track coupling model. The impact of the number of truncated terms in the track’s basis functions and the selected modal order on the convergence of the dynamic response is quantified. Finally, several examples are presented to demonstrate the model’s capability to perform vehicle-track coupling dynamics analysis under various non-uniform conditions of track structure parameters. The model proves to have considerable engineering reliability and applicability. This approach allows for a more efficient and accurate analysis of complex railway systems, especially in scenarios where traditional methods might fall short due to the inherent non-uniformity and randomness of track properties.

One of the most popular methods for reducing the complexity of assemblies of finite element models in the field of structural dynamics is component mode synthesis. A main challenge of component mode synthesis is balancing model complexity and model accuracy, because it is difficult to predict how component reduction influences assembly model accuracy. This work introduces an approach that allows for the translation of assembly model accuracy requirements in the frequency domain to the automatic selection of the cut-off frequencies for the model-order reduction (MOR) of components. The approach is based on a mathematical approach for MOR for coupled linear systems in the field of systems and control. We show how this approach is also applicable to structural dynamics models. We demonstrate the use of this approach in the scope of component mode synthesis (CMS) methods with the aim to reduce the complexity of component models while guaranteeing accuracy requirements of the assembly model. The proposed approach is illustrated on a mechanical, three-component structural dynamics system for which reduced-order models are computed that are reduced further compared to reduction using standard methods. This results in lower simulation cost, while maintaining the required accuracy.