Substructuring Technique Research Articles

Dynamic modeling of complex structures in a broad frequency range.

A general method, the so‐called finite substructurE method (FSEM), is presented for the dynamic analysis of complex structural systems. In this method, a complex structure is considered as a collection of a finite number of basic structural components such as beams, plates, and shells. Instead of seeking a numerical solution at a number of discrete or grid points, the current displacement solution is sought, over each component, as a continuous field in the form of Fourier series expansions. Thus, the number of degrees of freedom, which now represent the Fourier expansion coefficients, can be substantially reduced in comparison with a grid‐based solution for the same spatial resolution. Mathematically, the resulting system tends to be better conditioned than those in the finite element methods as the number of DOF’s increases with frequency. The robustness of this model for high frequency applications can be further improved by incorporating statistical processes into the modeling method to properly reflect the means or account for the uncertainties of certain input variables. The proposed substructure method is considerably different from the existing substructure techniques in that no modal data are required for any component. Numerical examples are presented to demonstrate the reliability of this method.

Dynamic substructure model for multiple impact responses of micro/nano piezoelectric precision drive system

A dynamic substructure technique which considers the electromechanical coupling effect of the PZT and the inertial effect of flexible components is presented to study the multiple impact dynamic behavior of micro/nano piezoelectric impact drive systems. It can investigate the step-like motion of object body and the multiple impacts behaviors reasonably by the comparison of the experimental data and the numerical solution of the spring-mass model. It is expected to have higher accuracy in the numerical simulations of the motion and the responses, especially to high frequency pulse voltage excitations. The present dynamic substructure technique has firstly studied reasonably the propagations of piezoelectric-induced transient waves and impact-induced transient waves. It is helpful to the failure analysis and the design of piezoelectric stack and flexible components. The present dynamic substructure technique can be applied to the transient dynamics optimization design and the precision control of the micro/nano piezoelectric impact drive systems.

Finite-Strip Method for the Analysis of Cracked Plates with Application to Plate-Girder Bridges

The finite-strip method (FSM) is one of the most efficient methods for structural analysis of bridges, reducing the time required for analysis without affecting the degree of accuracy. FSM is therefore an ideal platform for traditional time-consuming fracture analysis. This paper presents a new bending crack strip (BCS) that combines the shape functions of the spline finite strip with the eigenfunction solutions of the differential equations that govern the deflection around a crack. When this new BCS is implemented in FSM, complex three-dimensional plate structures with embedded cracks that are perpendicular to the longitudinal axis of the strips can be analyzed efficiently. The stress intensity factors of such structures can be directly computed with BCS. Extra nodal points are used around the cracked area to increase the precision of the numerical method. A substructuring technique is employed to determine the corresponding degrees of freedom of the boundary nodes, resulting in efficient analyses of cracked plates and plate-girder structures. This paper presents two examples to illustrate the accuracy and convergence rate of BCS as well as to demonstrate applications of the proposed model in the analysis of plates and plate girder bridges.

Pseudodynamic testing and nonlinear substructuring of damaging structures under earthquake loading

Pseudodynamic or PSD testing with substructuring technique is a highly relevant approach to capture the dynamic failure mechanisms of a structure. It allows one to realistically test a structure subject to an earthquake with accessible experimental equipment. We propose in this paper to apply such a PSD testing with substructuring to damaging structures, where the nonlinear behaviour is represented by an advanced damage model: an anisotropic damage model, initially proposed for monotonic applications, is upgraded in order to asses cyclic and seismic loading. The keypoint is the macroscopic representation of the microcrack closure (the cracks generated in tension close in compression) and its effect on stiffness recovery. The chain–PSD testing/advanced damage model/experimental measure setup based on image correlations–is successfully used for the study of the failure of a reinforced concrete frame structure.

General Model to Predict Power Flow Transmitted into Laminated Beam Bases in Flexible Isolation Systems

For estimating the vibration transmission accurately and performing vibration control efficiently in isolation systems, a novel general model is presented to predict the power flow transmitted into the complicate flexible bases of laminated beams. In the model, the laminated beam bases are simulated by the first-order shear deformation laminated plate theory, which is relatively simple and economic but accurate in predicting the vibration solutions of flexible isolation systems with laminated beam bases in comparison with classical laminated beam theories and higher order theories. On the basis of the presented model, substructure technique and variational principle are employed to obtain the governing equation of the isolation system and the power flow solution. Then, the vibration characteristics of the flexible isolation systems with laminated bases are investigated. Several numerical examples are given to show the validity and efficiency of the presented model. It is concluded that the presented model is the extension of the classical one and it can obtain more accurate power flow solutions.

A Modal‐Based Substructure Method Applied to Nonlinear Rotordynamic Systems

The discretisation of rotordynamic systems usually results in a high number of coordinates, so the computation of the solution of the equations of motion is very time consuming. An efficient semianalytic time‐integration method combined with a substructure technique is given, which accounts for nonsymmetric matrices and local nonlinearities. The partitioning of the equation of motion into two substructures is performed. Symmetric and linear background systems are defined for each substructure. The excitation of the substructure comes from the given excitation force, the nonlinear restoring force, the induced force due to the gyroscopic and circulatory effects of the substructure under consideration and the coupling force of the substructures. The high effort for the analysis with complex numbers, which is necessary for nonsymmetric systems, is omitted. The solution is computed by means of an integral formulation. A suitable approximation for the unknown coordinates, which are involved in the coupling forces, has to be introduced and the integration results in Green′s functions of the considered substructures. Modal analysis is performed for each linear and symmetric background system of the substructure. Modal reduction can be easily incorporated and the solution is calculated iteratively. The numerical behaviour of the algorithm is discussed and compared to other approximate methods of nonlinear structural dynamics for a benchmark problem and a representative example.

Developing a method for coupling branch modal models

An original method is proposed to reduce diffusion problem characterized by a thermal contact resistance and non-linear boundary conditions. The method use a substructuring technique in which a branch modal basis is obtained for each subdomain. The amalgam reduction method for each basis leads to a global reduced model. A specific flux jump functional allows to optimize reduction. The numerical test case is an electronic component coupled with a radiator and which follows a thermal regulation. For this case a numerical study is made in order to observe the influence of the flux jump functional. Comparison between the detailed model and the reduced one with optimal parameters gives a gain in computation time of 82 for the same precision.

Vibrations of spatially suspended flexible pipe‐beam with moving fluid

AbstractThe nonlinear dynamic model of flexible pipe–beam suspended by spatial system of cables is proposed for vibration analysis of pipeline suspension bridges. The model, based on substructure technique, is considered as an assemblage of the following substructures: cables, hangers and pipe–beam. Equation of motion of pipe–beam is derived by Galerkin's FEM with the original finite element formulated in order to include moving mass of transported fluid. To describe cable vibrations, general continuum approach proposed in Ref.[1] is adopted with application into 3D model. Cable model takes into account initial sag, pre–tension force, large displacements and hangers' point reactions. Equation governing the motion of pipe–beam with cables and hangers is obtained regarding equilibrium conditions of interaction forces and compatibility of displacements at connection points between substructures. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Optimization and static strength test of carbody of light rail vehicle

Preliminary structure of light rail vehicle (LRV) carbody made of steel was designed considering its usage, strength, manufacturing, etc. Based on the finite element analysis, the optimization of design parameters associated with thickness of LRV carbody is carried out to increase the whole strength of the carbody and to reduce its mass. With the aids of the substructure technique and the modified technique with discrete variables in the optimization based on the finite element method, the consumed computing time is reduced dramatically. The optimized LRV carbody is re-analyzed by FEM to obtain its static strength and vibrating mode and is manufactured. The mass of the optimized carbody reduces about 1.3 kg, and the relative reduction ratio is about 10%. Then, the strength test of the real carbody under the static load is executed. It is shown by the numerical and test results that the design requirements of the LRV carbody are satisfying. The newly designed carbody is used in the LRV, which is the first one used commercially developed by China independently. Nowadays, the LRV is running on the transportation circuit in Changchun of China.

Seismic Performance of a Transfer Plate Structure

In regions without seismic provisions, it is common to use transfer plates as part of the structural system. In this paper, pseudodynamic tests with substructure techniques were conducted on an 1:4 scale test specimen representing the first 2 stories of a 18-story high-rise building with a transfer plate under earthquake actions. Columns of the test specimen were strengthened to prevent failure under the pseudodynamic tests. Three types of time–history records were applied, including triangular waves and a series of El-Centro Earthquakes. Based on the experimental results, it is predicted that the bottom of the transfer plate will be severely damaged when subjected to an El-Centro Earthquake with maximum acceleration at 64% g .

Stability analysis of a substructured model of the rotating beam

One of the most well-known situations in which nonlinear effects must be taken into account to obtain realistic results is the rotating beam problem. This problem has been extensively studied in the literature and has even become a benchmark problem for the validation of nonlinear formulations. Among other approaches, the substructuring technique was proven to be a valid strategy to account for this problem. Later, the similarities between the absolute nodal coordinate formulation and the substructuring technique were demonstrated. At the same time, it was found the existence of a critical angular velocity, beyond which the system becomes unstable that was dependent on the number of substructures. Since the dependence of the critical velocity was not so far clear, this paper tries to shed some light on it. Moreover, previous studies were focused on a constant angular velocity analysis where the effects of Coriolis forces were neglected. In this paper, the influence of the Coriolis force term is not neglected. The influence of the reference conditions of the element frame are also investigated in this paper.

Transmissibility of a deformed rotating tyre

The major source of environmental noise exposure is road traffic noise. Of all noise sources, tyre rolling noise is dominant for speeds above 30 km/h for passenger cars. Tyre rolling noise can be subdivided into interior and exterior noise. For the interior noise to which the passengers are exposed to, the tyre transmissibility is essential since it relates the contact forces with the axle forces. These axle forces are responsible for the structure borne interior noise. Here, a Finite Element tyre model, including a fully coupled air column, is used to examine the transmissibility in the frequency domain 0‐300 Hz. It is shown that three aspects are essential in modeling the axle forces resulting from tyre‐road interaction: 1) the tyre deformation since it leads to a set of non‐axisymmetric eigenmodes, 2) the relatively low‐damped non‐axisymmetric acoustic resonance, and 3) rotation. A methodology using substructuring techniques is presented to include rotational effects both in the case of an undeformed and deformed tyre. These effects of rotation on the transmissibility differ in the deformed and undeformed case: frequency loci veering occurs in the deformed case, while in the undeformed case rotation results in a pure split of the eigenfrequencies.

General Framework for Dynamic Substructuring: History, Review and Classification of Techniques

Four decades after the development of the first dynamic substructuring techniques, there is a necessity to classify the different methods in a general framework that outlines the relations between them. In this paper, a certain vision on substructuring methods is proposed, by recalling important historical milestones that allow us to understand substructuring as a domain decomposition concept. Thereafter, based on the dual and primal assembly of substructures, a general framework for the classification of the methods is presented. This framework allows us to indicate how the various classes of methods, proposed along the years, can be derived from a clear mathematical description of substructured problems. Current bottlenecks in experimental dynamic substructuring, as well as solutions found in literature, will also be briefly discussed.

Modification of an interface parameter between sub‐system and vehicle: case of a fan system attached to the front end of a car

The car industry, as well as many others, is constantly undergoing modifications to comply security and comfort regulations. Among the major causes for concern is the integration of sub‐systems on vehicles. In fact, when a sub‐system runs on a vehicle, it can lead to vibrations on the body of the car and thus acoustical radiations, causing trouble for the passengers, and also outside the vehicle. The frequency response functions (FRF)‐based substructuring technique and impedances coupling methods are used to predict the forces entering the vehicle from those measured on a test bed. When a given constraint on the forces entering the vehicle is not satisfied, the car supplier may choose to modify the interface parameters between the substructures. In this paper an analytical approach to filter the entering forces by elastic suspension is proposed. An application is given by way of numerical experiments on a fan system attached to the front end of a car, both subsystems being identified through measurements.

High‐frequency response analysis via algebraic substructuring

AbstractHigh‐frequency response analysis (Hi‐FRA) of large‐scale dynamical systems is critical to predict the resonant behavior of modern micro‐devices and systems operated over MHz or GHz frequency range. Algebraic substructuring (AS) is a powerful technique to extract a large number of natural frequencies. In this work, we extend the AS technique for FRA between two specified cutoff frequencies ωmin and ωmax. The technique is referred to as ASFRA. ASFRA can be efficiently applied to Hi‐FRA, as demonstrated by two examples of microelectromechanical sensors operated at 1–2 and 200–250 MHz ranges. To some extent ASFRA generalizes the underlying numerical algorithm and functionality of commercially viable automated multi‐level substructuring (AMLS) technique. AMLS is designed for FRA up to a specific frequency ωmax, starting from the lowest, and is inefficient for Hi‐FRA. Copyright © 2008 John Wiley & Sons, Ltd.

Development and validation of a strain-based Structural Health Monitoring system

An innovative Structural Health Monitoring (SHM) methodology, based on structural strain measurements, which are processed by a back-propagation feed-forward Artificial Neural Network (ANN), is proposed. The demonstration of the SHM methodology and the identification of its capabilities and drawbacks are performed by applying the method in the prediction of fatigue damage states of a typical aircraft cracked lap-joint structure. An ANN of suitable architecture is developed and trained by numerically generated strain data sets, which have been preprocessed by Fast Fourier Transformation (FFT) for the extraction of the Fourier Descriptors (FDs). The Finite Element (FE) substructuring technique is implemented in the stress and strain analysis of the lap-joint structure, due to its efficiency in the calculation of the numerous strain data, which are necessary for the ANN training. The trained network is successfully validated, as it is proven capable to accurately predict crack positions and lengths of a lap-joint structure, which is damaged by fatigue cracks of unknown location and extent. The proposed methodology is applicable to the identification of more complex types of damage or to other critical structural locations, as its basic concept is generic.

Improvement of substructuring reduction technique for large eigenproblems using an efficient dynamic condensation method

An accelerated substructuring reduction procedure for the iterated improved reduced system (IIRS) method is proposed. The iterated IIRS method can be combined with a substructuring scheme to provide an efficient methodology for large-scale eigenvalue problems. Not only can it reduce eigenvalue analysis errors through successive iterations, but the accuracy of the eigenanalysis is not sensitive to the selected master degrees of freedom. In practical structural eigenproblems, reducing the number of iterations can save a great deal of computation cost. The present substructuring technique modifies the iterative form of the transformation matrices in each substructure to achieve faster convergence. Applications of the present method to two numerical examples demonstrate that the proposed method can obtain lower eigensolutions of structures more accurately and efficiently, as compared with those of the current substructuring technique.

A multidomain integrated‐radial‐basis‐function collocation method for elliptic problems

AbstractThis article is concerned with the use of integrated radial‐basis‐function networks (IRBFNs) and nonoverlapping domain decompositions (DDs) for numerically solving one‐ and two‐dimensional elliptic problems. A substructuring technique is adopted, where subproblems are discretized by means of one‐dimensional IRBFNs. A distinguishing feature of the present DD technique is that the continuity of the RBF solution across the interfaces is enforced with one order higher than with conventional DD techniques. Several test problems governed by second‐ and fourth‐order differential equations are considered to investigate the accuracy of the proposed technique. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008

Identification of equivalent modal damping for a wind turbine at standstill using Fourier and wavelet analysis

The current paper presents approaches to evaluate the equivalent modal viscous damping ratios for a wind turbine tower, using Fourier and wavelet analysis based linear regression. In the Fourier analysis, the estimated damping ratios are constant, but in the wavelet analysis, temporally varying damping ratios using a time-segmented least squares approach can be identified. The estimation of time varying damping is important in assessing the risk of negative aeroelastic damping in wind turbines. In absence of experimental data, the proposed approaches have been illustrated on numerically simulated response obtained from a simple wind turbine model. The model used for generating the response time history data for the wind turbine tower is composed of a flexible tower and rotor blade system, inter-connected using a substructuring technique, which facilitates the blade/tower coupling. A model order reduction technique is first used to model each of the two substructures (tower/nacelle and rotor system) as single-degree-of-freedom systems (DOF), allowing both systems to be later coupled together to form an equivalent two-DOF coupled tower/blade wind turbine tower model with equivalent viscous damping. A wind-induced forced vibration analysis of the coupled system is then carried out using artificially generated wind drag time histories. Two identification procedures based on (a) Fourier and (b) wavelet analysis, and linear regression, are used to solve the inverse problem for evaluating the first- and second-equivalent modal damping ratios of the coupled system. For the numerical example presented in order to demonstrate the applicability of the proposed approaches, good agreement was observed between the originally specified modal damping ratios and the subsequently estimated values. The estimation of damping using wavelet analysis techniques can be applied to situations where damping may vary with time. The proposed approaches can be used without any limitation on response data obtained for more complex models of turbines and loadings or on experimentally measured data from wind turbines.

Numerical simulation of dynamic soil–structure interaction in shaking table testing

This paper provides an insight into the numerical simulation of soil–structure interaction (SSI) phenomena studied in a shaking table facility. The shaking table test is purposely designed to confirm the ability of the numerical substructure technique to simulate the SSI phenomenon. A model foundation–structure system with strong SSI potential is embedded in a dry bed of sand deposited within a purpose designed shaking-table soil container. The experimental system is subjected to a strong ground motion. The numerical simulation of the complete soil–foundation–structure system is conducted in the linear viscoelastic domain using the substructure approach. The matching of the experimental and numerical responses in both frequency and in time domain is satisfying. Many important aspects of SSI that are apparent in the experiment are captured by the numerical simulation. Furthermore, the numerical modelling is shown to be adequate for practical engineering design purposes.