Vahid Yaghoubi

The Modal Observability Correlation as a Modal Correlation Metric

The historical development of the Modal Assurance Criterion (MAC) originated from the need of a correlation metric for comparingexperimental modal vectors, estimated from measured data, to eigenvectors that have been determined from finite element calculation. For systems with well separated eigenvalues with many system degrees-of-freedom (DOF) represented in the eigenvectors it is normally easy to distinguish eigenvectors associated to different eigenvalues by low MAC correlation numbers. However, for eigenvectors with a sparse DOF sampling it may be hard to distinguish between vectors by MAC correlation numbers. To reduce the problem of distinguishing between eigensolutions, this paper advocates the use of a new correlation metric based on the observability matrix of the diagonal state-space realization. This is instead of using a metric based on the eigenvectors only.

Automated Modal Analysis Based on Frequency Response Function Estimates

Given measured data as estimated frequency responses of a quasi-linear system, there is a variety of system identification methods that identify a state-space model that gives good correlation to the data. Such methods are the N4SID and the PolyMAX methods. Using these methods, a key problem is to select the proper model order. In this work we investigate a method for the automatic detection of proper model order. The method is based on the statistical evaluation of an ensemble of state-space models all identified from the same basic set of frequency response functions, but with different realizations based on a bootstrapping scheme. We apply the method to real test data.

A Parallel Solution Method for Structural Dynamic Response Analysis

With the continuous improvements of technology in and around multi-core CPU:s and GPU:s there is a strong desire to exploit this technology in its full potential. For structural dynamics problems, the domain decomposition is a very mature technique that is well adapted to parallel computations in multi-core machines as it is almost trivially parallelizable. However, competing alternatives with model reduction without parallel computation has also reached an extremely high level of maturity and are thus highly competitive. In this paper, a domain decomposition method, in a procedure named the split-stitch-spread (S3) procedure, is proposed to do transient analysis of large finite element models in parallel. In the method, the structure splits into model substructures with elastic interfacial substructures coupling them together. Each of them can be sent to different computer cores to do time discretization. The model substructures stitch to each other by using interfacial forces and as a result, the systems’ state sequence will be obtained. The solution can then be spread into the substructures and response quantities can be evaluated in parallel processing. The method is applied to a multi-story building subjected to earthquake loading and the results are compared with mode displacement method as a model reduction method with focus on computational efficiency.

An efficient simulation method for structures with local nonlinearity

In general, simulating the nonlinear behavior of systems needs a lot of computational effort. Since researchers in different fields are increasingly targeting nonlinear systems, attempts toward fast nonlinear simulation have attracted much interest in recent years. Examples of such fields are system identification and system reliability. In addition to efficiency, the algorithmic stability and accuracy need to be addressed in the development of new simulation procedures. In this paper, we propose a method to treat localized nonlinearity in a structure in an efficient way. The system will be separated by a linearized part and a nonlinear part that is considered as external pseudo forces that act on the linearized system. The response of the system is obtained by iterations in which the pseudo forces are updated. Since the method is presented in linear state space model form, all manipulations that are made on these, like similarity transformations and model reduction, can easily be exploited. To do numerical integration, time-stepping schemes like the triangular hold interpolation can be used to the advantage. We demonstrate the efficiency, stability and accuracy of the method on numerical examples.

Automated modal parameter estimation using correlation analysis and bootstrap sampling

The estimation of modal parameters from a set of noisy measured data is a highly judgmental task, with user expertise playing a significant role in distinguishing between estimated physical and noise modes of a test-piece. Various methods have been developed to automate this procedure. The common approach is to identify models with different orders and cluster similar modes together. However, most proposed methods based on this approach suffer from high-dimensional optimization problems in either the estimation or clustering step. To overcome this problem, this study presents an algorithm for autonomous modal parameter estimation in which the only required optimization is performed in a three-dimensional space. To this end, a subspace-based identification method is employed for the estimation and a non-iterative correlation-based method is used for the clustering. This clustering is at the heart of the paper. The keys to success are correlation metrics that are able to treat the problems of spatial eigenvector aliasing and nonunique eigenvectors of coalescent modes simultaneously. The algorithm commences by the identification of an excessively high-order model from frequency response function test data. The high number of modes of this model provides bases for two subspaces: one for likely physical modes of the tested system and one for its complement dubbed the subspace of noise modes. By employing the bootstrap resampling technique, several subsets are generated from the same basic dataset and for each of them a model is identified to form a set of models. Then, by correlation analysis with the two aforementioned subspaces, highly correlated modes of these models which appear repeatedly are clustered together and the noise modes are collected in a so-called Trashbox cluster. Stray noise modes attracted to the mode clusters are trimmed away in a second step by correlation analysis. The final step of the algorithm is a fuzzy c-means clustering procedure applied to a three-dimensional feature space to assign a degree of physicalness to each cluster. The proposed algorithm is applied to two case studies: one with synthetic data and one with real test data obtained from a hammer impact test. The results indicate that the algorithm successfully clusters similar modes and gives a reasonable quantification of the extent to which each cluster is physical.

Model calibration of a locally non-linear structure utilizing multi harmonic response data

Model correlation and model calibration using test data are natural ingredients in the process of validating computational models. Here, model calibration for the important sub-class of non-linear systems consisting of structures dominated by linear behavior having presence of local non-linear effects is studied. The focus is on the selection of uncertain model parameters together with the forming of the objective function to be used for calibration. To give precise estimation of parameters in the presence of measurement noise, the objective function data have to be informative with respect to the parameters chosen. Also, to get useful data the excitation force is here designed to be multi-harmonic since steady-state responses at the side frequencies are shown to contain valuable information for the calibration process. In this paper, test data from a replica of the Ecole Centrale de Lyon (ECL) nonlinear benchmark together with steady-state solutions stemming from calculations using the Multi-Harmonic Balancing method are used for illustration of the proposed model calibration procedure.

Automated Modal Analysis Based on Statistical Evaluation of Frequency Responses

This paper presents a newly developed method for obtaining the modal model with a proper model order from experimental frequency response functions (FRF). The method is a multi-step procedure which commences with the identification of a high-order state-space model, Exhaustive Model (EM), using the full FRF data set. Then, modal states that give small contribution to the output, quantified by a metric associated to the observability grammian, are rejected from the EM resulting in a Reference Model (RM). Competing models, with the same model order as the RM, are then found by bootstrapping realization using same-size fractions of the full FRF. Eigensolutions of the Bootstrapping Models (BMs) are then paired by the eigensolutions of the RM based on high Modal Observability Correlation (MOC) indices. In a second reduction stage, the modal states with low MOC index are rejected from the BMs. Final model is found by an averaging through BMs. Only one threshold quantity, related to observability grammians need to be set by the user. The method thus requires very little user interaction. The method is applied to experimental data used in a previous IMAC Round Robin exercise for experimental modal analysis evaluation.

Locally Non-linear Model Calibration Using Multi Harmonic Responses: Applied on Ecole de Lyon Non-linear Benchmark Structure

In industry, linear FE-models commonly serve to represent global structural behavior. However, when test data are available they may show evidence of nonlinear dynamic characteristics. In such a case, an initial linear model may be judged being insufficient in representing the dynamics of the structure. The causes of the non-linear characteristics may be local in nature whereas the major part of the structure is satisfactorily represented by linear descriptions. Although the initial model then can serve as a good foundation, the parameters needed to substantially increase the model’s capability of representing the real structure are most likely not included in the initial model. Therefore, a set of candidate parameters controlling nonlinear effects, opposite to what is used within the vast majority of model calibration exercises, have to be added. The selection of the candidates is a delicate task which must be based on engineering insight into the structure at hand.The focus here is on the selection of the model parameters and the data forming the objective function for calibration. An over parameterized model for calibration render in indefinite parameter value estimates. This is coupled to the test data that should be chosen such that the expected estimate variancesof the chosen parameters are made small. Since the amount of information depends on the raw data available and the usage of them, one possibility to increase the estimate precision is to process the test data differently before calibration. A tempting solution may be to simply add more test data but, as shown in this paper, the opposite could be an alternative; disregarding low excessive data may make the remaining data better to discriminate between different parameter settings.Since pure mono-harmonic excitation during test is an abnormality, the excitation force is here designed to contain sub and super harmonics besides the fundamental one. Further, the steady-state responses at the side frequencies are here shown to contain most valuable information for the calibration process of models of locally nonlinear structures.Here, synthetic test data stemming from a model representing the Ecole Centrale de Lyon (ECL) nonlinear benchmark are used for illustration. The nonlinear steady state solutions are found using iterative linear reverse path state space calculations. The model calibration is here based on nonlinear programming utilizing several parametric starting points. Candidates for starting points are found by the Latin Hypercube sampling method. The best candidates are selected as starting points for optimization.

Towards an Automatic Modal Parameter Estimation Framework: Mode Clustering

The estimation of modal parameters from a set of measured data is a highly judgmental task, with user expertise playing a significant role for distinguishing between physical and spurious modes. However, it can be very tedious especially in situations when the data is difficult to analyze. This study presents a new algorithm for mode clustering as a preliminary step in a multi-step algorithm for performing physical mode selection with little or no user interaction. The algorithm commences by identification of a high-order model from estimated frequency response functions to collect all the important characteristics of the structure in a so-called library of modes. This often results in the presence of spurious modes which can be detected on the basis of the hypothesis that spurious modes are estimated with a higher level of uncertainty comparing to physical modes. Therefore, we construct a series of data using a simple random sampling technique in order to obtain a set of linear systems using subspace identification. Then, their similar modes are grouped together using a new correlation criterion, which is called Modal Observability Correlation (MOC). An illustrative example shows the efficiency of the proposed clustering technique and also demonstrates its capability to dealing with inconsistent data.

Stochastic Finite Element Model Updating by Bootstrapping

This paper presents a new stochastic finite element model calibration framework for estimation of the uncertainty in model parameters, which combines the principles of bootstrapping with the technique of FE model calibration with damping equalization. The bootstrapping allows to quantify the uncertainty bounds on the model parameters by constructing a number of resamples, with replacement, of the experimental data and solving the FE model calibration problem on the resampled datasets. To a great extent, the success of the calibration problem depends on the starting value for the parameters. The formulation of FE model calibration with damping equalization gives a smooth metric with a large radius of convergence to the global minimum and its solution can be viewed as the initial estimate for the model parameters. In this study, practical suggestions are made to improve the performance of this algorithm in dealing with noisy measurements. The performance of the proposed stochastic calibration algorithm is illustrated using simulated data for a six degree-of-freedom mass-spring model.