Response Of Complex Structures Research Articles

Nonintrusive Structural Dynamic Reduced Order Modeling for Large Deformations: Enhancements for Complex Structures

This paper focuses on the development of nonlinear reduced order modeling techniques for the prediction of the response of complex structures exhibiting “large” deformations, i.e., a geometrically nonlinear behavior, which are nonintrusive, i.e., the structure is originally modeled within a commercial finite element code. The present investigation builds on a general methodology successfully validated in recent years on simpler beam and plate structures by: (i) developing a novel identification strategy of the reduced order model parameters that enables the consideration of the large number of modes (>50 say) that would be needed for complex structures, and (ii) extending a step-by-step strategy for the selection of the basis functions used to represent accurately the displacement field. The above novel developments are successfully validated on the nonlinear static response of a nine-bay panel structure modeled with 96,000 degrees of freedom within Nastran.

Parametric time‐domain identification of multiple‐input systems using decoupled output signals

SUMMARYCivil engineering structures are often subjected to multidirectional actions such as earthquake ground motion, which lead to complex structural responses. The contributions from the latter multidirectional actions to the response are highly coupled, leading to a MIMO system identification problem. Compared with single‐input, multiple‐output (SIMO) system identification, MIMO problems are more computationally complex and error prone. In this paper, a new system identification strategy is proposed for civil engineering structures with multiple inputs that induce strong coupling in the response. The proposed solution comprises converting the MIMO problem into separate SIMO problems, decoupling the outputs by extracting the contribution from the respective input signals to the outputs. To this end, a QR factorization‐based decoupling method is employed, and its performance is examined. Three factors, which affect the accuracy of the decoupling result, including memory length, input correlation, and system damping, are investigated. Additionally, a system identification method that combines the autoregressive model with exogenous input (ARX) and the Eigensystem Realization Algorithm (ERA) is proposed. The associated extended modal amplitude coherence and modal phase collinearity are used to delineate the structural and noise modes in the fitted ARX model. The efficacy of the ARX‐ERA method is then demonstrated through identification of the modal properties of a highway overcrossing bridge. Copyright © 2013 John Wiley & Sons, Ltd.

Retrieval of relevant parameters of natural multilayer systems by means of bio-inspired optimization strategies

Natural photonic structures exhibit remarkable color effects such as metallic appearance and iridescence. A rigorous study of the electromagnetic response of such complex structures requires to accurately determine some of their relevant optical parameters, such as the refractive indices of the materials involved. In this paper, we apply different heuristic optimization strategies to retrieve the real and imaginary parts of the refractive index of the materials comprising natural multilayer systems. Through some examples, we compare the performances of the inversion methods proposed and show that these kinds of algorithms have a great potential as a tool to investigate natural photonic structures.

Next-generation parametric reduced-order models

Novel parametric reduced-order models are proposed for fast reanalysis to predict the dynamic response of complex structures, which suffered thickness variations caused by design changes or damage in one or more substructures. Parametric reduced-order models developed previously have two important challenges to overcome to improve accuracy and performance: (a) the transformation matrix is not mathematically stable and (b) the Taylor series parameterization techniques do not capture thickness variations of the structure modeled with solid-type elements due to the highly nonlinear dependence on thickness changes. Thus, herein, a new transformation matrix and novel parameterization techniques are proposed. Usual reduced-order models have an additional challenge, namely the difficulty in reducing the interface degrees of freedom. Thus a way of reducing the interface degrees of freedom is also proposed. The predicted vibration responses of complex structures are shown to agree very well with results obtained using a much more computationally expensive commercial tool.

Radiation efficiency, impedance, and the acoustics of rattle

Rattle issues consistently rank as one of the top consumer complaints in initial quality surveys for many new products. Predicting the acoustics of rattle is complicated by the need to model the vibro-acoustic response of large complex structures across a broad frequency range. The complexity of the analysis can be reduced by making use of standard methods derived almost 50 years ago. In particular, this paper discusses a computationally efficient method for assessing the propensity for rattle in large complex structures. A finite element model is used to predict the probability that impacts will occur when a product is exposed to a particular low frequency random vibro-acoustic environment. The expected contact forces arising from each impact are then estimated by making use of expressions involving the drive point impedances of infinite structures. Finally, the vibration and acoustic radiation associated with the various impacts are predicted and ranked using a SEA model. A number of examples are presented including one which makes use of a radiation efficiency formula due to G. Maidanik. The results are in very good agreement with analytical reference results.

Simulation of stochastic wind field for large complex structures based on modified Fourier spectrum

Simulation for stochastic wind field is very important in analyzing dynamic responses of large complex structures due to strong wind. The typical simulation method is the spectrum representation method (SRM), but the SRM has drawbacks of inferior precision in lower frequency and slow calculating speed. In view of this, the modified Fourier spectrum method (MFSM) is introduced into the simulation of stochastic wind field in this paper. In this method, phase information of wind velocity time history is determined by cross power spectral density (CPSD) between adjacent points, and the wind velocity time history with time and space correlation is generated by iterative modification for CPSD considering auto power spectral density (APSD). Simulation of the wind field for a long-span bridge is undertaken to verify the effectiveness of the MFSM. Simulation results of the SRM and the MFSM are compared. It can be concluded that the MFSM is more accurate and has higher calculation speed than the SRM.

Dynamic Response of Complex Structure in Half Space

The problems of SH-wave scattering caused by a subsurface circular lining structure and a beeline crack with arbitrary length at an arbitrary position were studied by using the methods of Green's function, complex variables and multi-polar coordinates. A adaptive Green's function, an essential solution to the displacement field for the elastic space possessing circular lining structure while bearing out-plane harmonic lining loads at an arbitrary point, was constructed firstly, and then a crack was created using “crack-division”. Thus the expressions of displacement and stress were established while the crack and the inclusion both existed. Finally, we give some numerical examples to discuss the variety of the horizontal surface displacement in the case of different parameters.

A Least Square Support Vector Machine Approach Based on Uniform Design Method for Structural Reliability Analysis

Since the performance functions of large complex structures can not be expressed explicitly in the process of reliability analysis, support vector machines (SVM) with good ability of generalization are used as the response surface function based on the small training samples. The uniform design method was adopted in selecting the training data. The least support vector machines (LS-SVM) were used to find the support vectors. The limit state function was expressed by the LS-SVM regression. Reliability analysis was then performed by the usual reliability method (e.g., the first-order reliability method, the second-order reliability method or Monte Carlo) on the response surface. The results of calculations of numerical examples and a typical cable-stayed bridge show that LS-SVM using the uniform design method can well approximate the real response of complex structures which has a good efficiency and accuracy and can be applied in complex structures.

Parametric reduced-order models for predicting the vibration response of complex structures with component damage and uncertainties

Modeling and fast reanalysis techniques are proposed for predicting the dynamic response of complex structures with uncertainty represented by parameter variability (in geometric and material properties) at component-level. The novel models allow for accurate reanalyses and are useful in many applications where the model of the pristine structure may not capture the changes in the system-level response due to component-level parameter variations. Herein, such models are obtained by using a novel approach based on a modified concept of component mode synthesis. The novel models, referred to as parametric reduced-order models, are developed for the general case of multiple substructures with parameter variabilities. Three types of parameteric variabilities are considered: (a) geometric (thickness) variability, (b) structural deformations (dents), and (c) cracks. For the first case, a novel parametrization of component-level mass and stiffness matrices is employed to predict the system-level response. For the second case, a novel approximate method based on static mode compensation is implemented. For the third case (cracks), a generalized formulation for the bi-linear frequency approximation is used. The predicted vibration responses of complex structures are shown to agree very well with results obtained using a much more computationally expensive commercial tool.

Shake‐table simulation study of small scale layered models

PurposeThe purpose of this paper is to investigate the correlation between the dynamic behavior of a full‐scale steel prototype and a small‐scale plastic model fabricated using fused deposition modeling (FDM).Design/methodology/approachBased on the use of a known input excitation, the small‐scale model is tested on a shake‐table. Experimental results are compared with results of a full prototype study and with computational models in an effort to assess the feasibility of testing small‐scale FDM models.FindingsTime History Records present strong correlation with prototype data and are reproducible using computational methods. Matching the first natural frequency of the studied structure proved to be a large part of achieving the desired response.Research limitations/implicationsIncluding the direct measurement of floor displacements will potentially highlight different aspects of model behavior not observed by recording accelerations only. Further investigation into the damping properties of acrylonitrile butadiene styrene plastic is recommended towards further understanding the model response.Practical implicationsAlthough this paper is based on a simple structure, the benefits of layered manufacturing (LM) methods include speed and ease of generating geometrically complex solids. The implications of the success of this pilot study include the ease in which the dynamic response of complex structures can be assessed using small‐scale LM models.Originality/valueThis project obtained baseline information on the dynamic behavior of FDM plastic parts. It provides assessment of the value of using small‐scale LM models to accurately predict the dynamic response of structures subjected to earthquake excitation.

Prediction condensed models adapted to the nonlinear structures in time domain

This paper deals with a method to study closely the stationary solution of nonlinear dynamic systems in time domain. This method is based on the exploitation of Karhunen–Loève decomposition with or without parametric modifications as well as on the characteristics of localized nonlinearities. With the application of this method at first on linear models initially condensed by Karhunen–Loève, the predictions of nonlinear responses can be obtained rapidly. This method is adapted to a condensed linear model used in the first optimization procedure of the nonlinear dynamic behaviour. This robust basis will be used as condensation basis of the modified model local per zone, which leads to a prediction of vibratory responses of complex structures modified and affected by localized nonlinearities.

System identification-guided basis selection for reduced-order nonlinear response analysis

Reduced-order nonlinear simulation is often times the only computationally efficient means of calculating the extended time response of large and complex structures under severe dynamic loading. This is because the structure may respond in a geometrically nonlinear manner, making the computational expense of direct numerical integration in physical degrees of freedom prohibitive. As for any type of modal reduction scheme, the quality of the reduced-order solution is dictated by the modal basis selection. The techniques for modal basis selection currently employed for nonlinear simulation are ad hoc and are strongly influenced by the analyst's subjective judgment. This work develops a reliable and rigorous procedure through which an efficient modal basis can be chosen. The method employs proper orthogonal decomposition to identify nonlinear system dynamics, and the modal assurance criterion to relate proper orthogonal modes to the normal modes that are eventually used as the basis functions. The method is successfully applied to the analysis of a planar beam and a shallow arch over a wide range of nonlinear dynamic response regimes. The error associated with the reduced-order simulation is quantified and related to the computational cost.

Structural reliability analysis using Monte Carlo simulation and neural networks

This paper examines a methodology for computing the probability of structural failure by combining neural networks (NN) and Monte Carlo simulation (MCS). MCS is a powerful tool, simple to implement and capable of solving a broad range of reliability problems. However, its use for evaluation of very low probabilities of failure implies a great number of structural analyses, which can become excessively time consuming. The proposed methodology makes use of the capability of a NN to approximate a function for reproducing structural behavior, allowing the computation of performance measures at a much lower cost. This approach seems very attractive, and its main challenge lies in the ability of a NN to approximate accurately complex structural response. In order to assess the validity of this methodology, a test function and two structural examples are presented and discussed. The second example is also used to show how this methodology can be used to perform reliability-based structural optimization.

Orthonormal Vector Fitting: A Robust Macromodeling Tool for Rational Approximation of Frequency Domain Responses

Vector Fitting is widely accepted as a robust macromodeling tool for approximating frequency domain responses of complex physical structures. In this paper, the Orthonormal Vector Fitting technique is presented, which uses orthonormal rational functions to improve the numerical stability of the method. This reduces the numerical sensitivity of the system equations to the choice of starting poles significantly and limits the overall macromodeling time

Spatial statistics of diffuse elastodynamic modes

Spatial statistics of the modes in an elastic body plays a useful role in predicting certain average elastodynamic responses of complex structures, such as transmitted power or nonlinear spectral energy redistribution. For single wave-speed fields, extensive evidence exists that high-frequency modes in the interior of a generic ray-chaotic body resemble a superposition of plane waves with random amplitudes and phases. This implies centered Gaussian statistics for the field amplitude and a specific form for the two-point correlation function. The former can also be linked to the imaginary part of the Green’s function averaged over a frequency band or over a characteristic ensemble of body realizations, which allows extension to the boundaries and inclusion of effects of finite body size. In the present study we analyze and compare theoretical expressions for a number of spatial statistics, such as intensity distribution, wave-type energy equipartition, and two-point correlation function, with numerical results for an ensemble of 2D plane-strain elastic bodies with rough boundaries.

Identification of nonlinear parameters for reduced order models

Constructing nonlinear structural dynamic models is a goal for numerous research and development organizations. Such a predictive capability is required in the development of advanced, high-performance aircraft structures. Specifically, the ability to predict the response of complex structures to engine induced and aero acoustic loading has long been a United States Air Force (USAF) goal. Sonic fatigue has plagued the USAF since the advent and adoption of the turbine engine. While the problem has historically been a maintenance one, predicting the dynamic response is crucial for future aerospace vehicles. Decades have been spent investigating the dynamic response and untimely failure of aircraft structures, yet little work has been accomplished towards developing practical nonlinear prediction methods. Further, the last decade has witnessed an appreciable amount of work in the area of nonlinear parameter identification. This paper outlines a unique and important extension of a recently introduced nonlinear identification method: Nonlinear Identification through Feedback of the Outputs (NIFO). The novel extension allows for a ready means of identifying nonlinear parameters in reduced order space using experimental data. The nonlinear parameters are then used in the assembly of reduced order models, thus providing researchers with a means of conducting predictive studies prior to expensive and questionable experimental efforts. This paper details both an analytical and experimental study conducted on a well-characterized clamped–clamped beam subjected to broadband random loading. Amplitude dependent, constant stiffness parameters were successfully identified for a multiple-degree-of-freedom (MDOF) nonlinear reduced order model. The nonlinear coefficients identified from the analytical scenario compare well with previously published studies of the beam. Nonlinear parameters were also successfully identified from the raw experimental data. Finally, a MDOF nonlinear reduced order model, constructed from experimental data, was used to predict the experimental response of the beam to other loading conditions. Beam response spectra and average displacement values from the prediction model also compare well with the experimental results.

SEA를 이용한 차량 진동 특성 해석

Statistical Energy Analysis(SEA) has been considered as a Possible method for predicting responses of complex structures, especially at higher frequencies. In this paper, an SEA model of a vehicle was built using 138 energy storing subsystems connected together using 1019 Junctions. SEAM software program was used to build and calculate the model. To demonstrate the accuracy of the SEA model, predicted response levels were compared with measured levels. The source Input levels were measured at the engine mounting parts. There is good agreement between the estimated and the experimental results. This paper also identifies some dominant energy flow paths from sources. It is finally presented that the SEA model can optimize the design parameters of vehicles using model parameters and energy flow paths.

Effect of variation of normal force on seismic performance of resilient sliding isolation systems in highway bridges

AbstractIn this study, a series of shaking table tests are carried out on scaled models of two seismically isolated highway bridges to investigate the effect of rocking motion and vertical acceleration on seismic performance of resilient sliding isolators. In addition, performance of RSI is compared with system having solely natural rubber bearings. Test results show that variation of normal force on sliders due to rocking effect and vertical acceleration makes no significant difference in response of RSI systems. In addition, analytical response of prototype isolated bridge and the model used in experiments is obtained analytically by using non‐linear model for isolation systems. It is observed that for seismically isolated bridges, dynamic response of full‐scale complex structures can be predicted with acceptable accuracy by experiments using a simple model of the structure. Copyright © 2005 John Wiley & Sons, Ltd.

Robust component modal synthesis method adapted to the survey of the dynamic behaviour of structures with localised non-linearities

This paper deals with a method to study closely the stationary solution of non-linear dynamic systems at several degrees of freedom (dof) subjected to harmonic excitations with or without parametric modifications. This solution is obtained basically with the utilisation of an iterative algorithm adapted to the first approximation of Newton–Raphson method. This method is based on the exploitation of the eigensolutions of the associated conservative linear system with or without parametric modifications as well as on the characteristics of localised non-linearities and finally on the exploitation of the equivalent linearisation method. With the application of this method at first on linear models initially condensed by modal synthesis, the predictions of non-linear responses can be obtained rapidly. In a second step, this method is adapted to a condensed linear model used in the first optimisation procedure of the non-linear dynamic behaviour. In fact, before the non-linear analysis, the appropriate choice of the basis of reduction, that is referred to as “robust” obtained from the initial linear system is necessary. This robust basis will be used as condensation basis of the modified model local per zone or global by substructure leads to a prediction of vibratory responses of complex structures greatly modified and affected by localised non-linearities.At the end of this article, two examples are given to illustrate the efficiency and the performances of the proposed method.

주파수응답함수를 이용한 감쇠가 있는 유한요소모형의 개선

The finite element analysis is frequently used to prediet dynamic responses of complex structures. Since the predicted responses often differ from experimentally measured ones, updating of the finite element models is performed to make the finite element results agree with the measured ones. Among several model updating methods, one is to use FRF(frequency response funetion) data without a modal analysis. This paper investigates characteristics of the model updating method in order to improve the method. The investigation is focused on hOw to obtain FRfs for unmeasured rotational displacements and how to consider damping. For the investigation simulated data and experimental data for a cantilever beam are used.