System Equivalent Reduction Expansion Process Research Articles

State Observing Method and Active Vibration Control Based on Virtual Sensing

State feedback is mostly used in the active vibration control of structural systems. However, the most challenging problem is to achieve the whole state of the structure. Another issue in active vibration control is that, in many cases, only some particular modes need to be controlled, while the other modes can remain unchanged in order to save on the costs of the control. To solve the above issues, this paper proposes a state-observing method and conducts a non-spillover partial pole placement control using a multistep method. Using the finite element method, the electromechanical coupling equations of motion of the beam are formulated. The reduced finite element model is then obtained by using the system equivalent reduction expansion process method. Introducing the real measurements by sensors and applying the local corresponding principle, the modes of the structural system are corrected, which are then utilized to predict all the degrees of freedom without measuring by sensors. Then, the non-spillover partial pole placement based on the state feedback is conducted using the multistep method. The optimal locations of actuators are derived by the genetic algorithm to control the specific structural modes effectively. The numerical simulations and experimental studies are also carried out.

Reduced-Order Modeling in Rotordynamics and Its Robustness to Random Matrix Perturbation

Abstract The aim of this paper is to apply and compare four reduced-order modeling strategies to compute the unbalanced vibration response of a rotating machine and evaluate its robustness to random matrix perturbation. The full finite element model of the rotor is built using a rotordynamic open source software (ROSS), and is reduced through different methods, namely: (1) modal reduction, (2) Krylov subspace, (3) Guyan reduction, and (4) system equivalent reduction–expansion process (SEREP). To evaluate the robustness of the obtained results, this paper proposes to perturb the stiffness matrix obtained using the reduced-order models applying the random matrix theory. A simple rotor (three discs) and a more complex gas turbine model (21 discs) are analyzed. Results show that Guyan is not the most appropriate reduction technique for the systems analyzed, but the other three strategies yield good results. In addition, the same random perturbation in the reduced-order stiffness matrices produces a similar level of uncertainty on the stochastic unbalanced responses.

Mechanical characterization of human skin—A non-invasive digital twin approach using vibration-response integrated with numerical methods

This paper proposes an innovative approach to identify elastic material properties and mass density of soft tissues based on interpreting their mechanical vibration response, externally excited by a mechanical indenter or acoustic waves. A vibration test is performed on soft sheets to measure their response to a continuous range of excitation frequencies. The frequency responses are collected with a pair of high-speed cameras in conjunction with 3-D digital image correlation (DIC). Two cases are considered, including suspended/fully-free rectangular neoprene sheets as artificial tissue cutout samples and continuous layered human skin vibrations. An efficient theoretical model is developed to analytically simulate the free vibrations of the neoprene artificial sheet samples as well as the continuous layered human skins. The high accuracy and validity of the presented analytical simulations are demonstrated through comparison with the DIC measurements and the conducted frequency tests, as well as a number of finite element (FE) modeling. The developed analytical approach is implemented into a numerical algorithm to perform an inverse calculation of the soft sheets' elastic properties using the imported experimental vibration results and the predicted system's mass via the system equivalent reduction/expansion process (SEREP) method. It is shown that the proposed frequency-dependent inverse approach is capable of rapidly predicting the material properties of the tested samples with high accuracy.

Two Stage SEREP Method for Condensation of Space Frame Structure for Dynamic Analysis

Analysis of frame structure can be done by the computer analysis, but there is requirement of simple and easy model which can give accurate result or with tolerable accuracy. The process of reducing the size of matrix is called condensation. It needs to be converted into system or model having less number of degrees of freedom as huge computational effort is required to analyze big models. Also, the models which requires analysis multiple times, for example, optimization of structural system, condensation is very useful. The single stiffness value can represent the stiffness of the whole story (elastic, rectangular frame with evenly distributed load), which give the responses accurately or tolerable accuracy. With this we can create a single lumped mass system of the frame structure which come very handy in further analysis of structure. For finding the single stiffness value, there are various approach like by using Approximate Story Stiffness Formulas, Static condensation, System Equivalent Reduction Expansion Process (SEREP) etc. Approximate Story Stiffness formulae don’t give very accurate result. To overcome its shortcoming Static Condensation and SEREP methods are used. Static Condensation gives better result but there is significant difference in the responses. Especially for space frame structures it is difficult to directly condense in to single degrees of freedom at each floor level. Hence, the present study proposes two stage SEREP method for condensation of spaceframe structure. Firstly, space frame is reduced to single degrees of freedom at each node, then it is further reduced to single degree of freedom at each floor. The proposed method gives very less (acceptable) error in comparison of natural frequencies.

A framework for numerical and experimental correlation of mode shapes with low spatial resolution: a catamaran’s case study

This work presents a framework for correlating numerical and experimental mode shapes with low spatial resolution using the Local Correspondence of Modes and Modal Coordinates (LCMC). As a case study, an operational modal analysis of a catamaran was performed. Thirteen global natural frequencies and their respective modes were identified in the range between 1 and 50 Hz, using only six sensors placed on its main deck. Thus, due to experimental modes’ spatial incompleteness, this required its full numerical model to be assembled and reduced using the System Equivalent Reduction Expansion Process to its main deck. Consequently, the reduced numerical modal matrix was used to fill in the gaps in the sparse set of the experimental modes, creating the mixed modal matrix. Then, this matrix was correlated and expanded using LCMC to the degrees of freedom (DOFs) that have not been measured. As a result, the spatial resolution of the catamaran’s experimental mode shapes was successfully increased from 6 to 174 DOFs. In general, the use of LCMC improved the Modal Assurance Criteria (MAC) of the correlated modes matrix compared to the MAC of the mixed modal matrix.

System equivalent model mixing: A modal domain formulation

Hybrid modeling is an increasingly common procedure for predicting the structural dynamics of complex products. System equivalent model mixing (SEMM), a dynamic substructuring-based method, is a recent addition to the field. Its implementation within frequency-based substructuring has great potential. Therefore, it is reasonable to explore the options for implementing an equivalent substructuring framework in any of the other substructuring domains. The objective of this paper is to present M-SEMM, system equivalent model mixing in the modal domain. A theoretical derivation reveals that under certain constrained conditions, the proposed methodology is equivalent to the system equivalent reduction expansion process (SEREP), a well-established reduction/expansion technique. When considering the expansion of spatially sparse models with a high modal density, M-SEMM represents a novel expansion method, which can be seen as a potentially useful extension to SEREP. The proposed implementation offers certain advantages, the most notable being a superior ability to disregard spurious modes in the hybrid model. A study of the proposed methodology on a numerical and an experimental case demonstrates its applicability and provides a comparison with SEREP and the original implementation of SEMM in the frequency domain.

Component model synthesis using model updating with neural networks

A component model synthesis (CMS) has been explored by carrying out model updating with neural networks. Structural system identification was achieved by applying the degree of freedom (DOF)-based reduction method and the inverse perturbation method. Experimental vibration data were restored to a full finite element model to update the numerical model. The experimental data were obtained using the specific sensor location selection method. The system identification was improved using the system equivalent reduction–expansion process by carrying out proper orthogonal decomposition. The proposed CMS was applied for a tank bracket model by simplifying the bolt assembly. Top and bottom parts of the tank bracket were each constructed to have a modally equivalent model. The convolutional neural network was adopted by training the density, Young’s modulus and contact properties to improve the computational efficiency.

Efficient Model Updating Method for System Identification Using a Convolutional Neural Network

Model updating processes are important for improving a model’s accuracy by considering experimental data. Structural system identification was achieved here by applying the degree-of-freedom-based reduction method and the inverse perturbation method. Experimental data were obtained using the specific sensor location selection method. Experimental vibration data were restored to a full finite element model using the reduction method to compare and update the numerical model. Applied iteratively, the improved reduced system method boosts model accuracy during full model restoration; however, iterative processes are time consuming. The computation efficiency was improved using the system equivalent reduction–expansion process in concert with proper orthogonal decomposition. A convolutional neural network was trained and applied to the updating process. We propose the use of an efficient model updating method using a convolutional neural network to reduce computation time. Experimental and numerical examples were adopted to examine the efficiency and accuracy of the model updating method using a convolutional neural network.

Weighted Transmissibility Assurance Criterion for Structural Damage Detection

Transmissibility function (TF) is a useful damage indicator for effective structural damage detection (SDD) under unknown external excitations. However, TF is not sensitive enough to structural damage under measurement noise. To improve the sensitivity of this indicator, a common model reduction technique, that is, the system equivalent reduction expansion process (SEREP) was first adopted to obtain reduced element stiffness matrices in the initial analytical finite-element model of a structure; this was used to weight the TFs in this study. Then, inspired by the cosine similarity measure, a novel damage indicator, called the weighted transmissibility assurance criterion (WTAC), was proposed for evaluating structural states. The effectiveness of the proposed damage indicator, WTAC, was first studied by a six-story frame in numerical simulations. The results showed that the TF weighted by the reduced element stiffness matrices improved the signal-to-noise ratio of the TF, and the proposed WTAC detected changes in structural states under noise. Moreover, further experimental studies on an aluminum alloy frame were conducted in order to assess the proposed WTAC. The results indicated that the proposed WTAC may provide a potential tool for effective SDD with strong robustness and stability for long-term monitoring on site.

Prediction of dynamic responses in a rectangular beam using the modal expansion method

Structures and equipment subject to dynamic loading are prone to a shortened life span due to excessive vibration levels, which can lead to fatigue failure of its components. Continuous monitoring of those systems can be a complicated and expensive task, due to the complexity and difficulty of accessibility to some locations, which makes it difficult to assess the structural integrity. One way to deal with this subject is to use finite element model, operational modal analysis, and the modal expansion method to predict dynamic responses in locations that have not been measured. In this work, a calibrated modal model of a rectangular aluminum beam suspended in air, in free–free boundary condition, was used to predict acceleration responses. The experimental modal matrix was obtained through impact tests and the system equivalent reduction expansion process (SEREP) technique was used to reduce the beam’s Finite Element model, thus ensuring the compatibility between numerical and experimental degrees of freedom. Model calibration was carried out using the local correspondence principle. Finally, by using the modal expansion method, the calibrated modal model was used to predict dynamic responses. Results showed high accuracy between the measured and the predicted acceleration signals.

Damage detection of cylindrical shells based on Sander's theory and model updating using incomplete modal data considering random noises

In this paper, a new system of damage detection equations for damage identification of cylindrical shells is developed based on classic finite element method, incomplete noisy modal data and model updating. Classic finite element technique in conjunction with Sander's thin shell theory is deployed in order to construct the finite element model of cylindrical shells and compute vibration data. To overcome the difficulty of incomplete data, “system equivalent reduction expansion process” (SEREP) method is applied to find a proper complete approximation of incomplete modal data. The proposed system is usually ill-posed due to noisy data and ill-conditioned system derived from the model updating process. To tackle this issue, a constraint as a regularization parameter is employed in model updating problem to differentiate potentially damaged elements and undamaged elements. This constraint is imposed on the model updating process by defining a Reduction Index (RI) and forcing severities of undamaged elements to tend to zero. Furthermore, the presented system is non-square and large depending on the number of structural unknowns. For these reasons, Biconjugate gradient (BCG) method along with an appropriate preconditioner are employed to solve the system of equations. Not only does the preconditioner stabilize the solution against numerical errors and inaccuracies caused by noisy data, but also it alters the system to a square one. Numerical studies are performed to demonstrate the efficiency of the presented method. Moreover, a statistical analysis is carried out to evaluate the effect of different sequences of random noises on the damage detection results. Findings show that the proposed method is capable of detecting the severities and sites of damages for cylindrical shells in the presence of noisy data.

Damage Detection of Truss Structures by Reduction of Degrees of Freedom Using the Serep Method

Damage detection of bridge structures during their operating lifetime is essential. In this paper, two approaches, All Degrees of Freedom and Reduction of the Degrees of Freedom methods, are used to detect the damages in structures. The first method considers All Degrees of Freedom of the structure and the second method, Reduction of the Degrees of Freedom. Since the sensors are installed only on a few degrees of freedom, the responses are available for some of them. The Degrees of Freedom must be reduced and System Equivalent Reduction Expansion Process method is one of the most efficient ways to solve the problem. This research aimed to identify the damage of structures using the Modal Strain Energy method by reducing the structural degree of freedom. Two standard examples are used and the results compared to different damage cases to examine the efficiency of the mentioned method. The results illustrated the proper performance of the Reduction of the Degrees of Freedom method to identify the damage in truss structures. By increasing the number of modes, Reduction of the Degrees of Freedom method detects considerably more accurate the damaged elements, especially when the noise is considered. Also, based on the outcomes to identify damaged elements, it is possible to consider more modes instead of more sensors.

SEREP-based reduced model of higher order viscoelastic propeller shaft considering various asymmetries

The material damping of rotors causes instability by introducing tangential force to whirl orbit. Literature shows that the improvement of stability range is achieved using fibre-reinforced composites or layered rotors which are light yet strong and exhibit better damping properties. In this study, a multi-layered viscoelastic propeller shaft mounted on three journal bearings is considered. Higher order finite-element model is developed after utilizing operator-based constitutive relationship, which includes various asymmetries from several sources like gyroscopic effect, internal damping, and journal bearing support. A large number of states of the higher order model, as well as the presence of asymmetry, make the system complicated for post-processing. Modified system equivalent reduction expansion process (SEREP) is applied for model reduction that uses both right and left eigenvectors of the original propeller-shaft system. Different numerical results are compared to test the effectiveness of reduced model. Though SEREP is found in the literature, its application over most realistic and higher order asymmetric system is completely new.

Data Consistency Assessment Function (DCAF)

A Data Consistency Assessment Function (DCAF) is developed to check the consistency of a measurement or set of measurements to all of the data in the entire data set. The inconsistent data can be precisely spotted and identified, which are recognized as the poorly measured data and can also be modified to be consistent with the rest of the data with an expansion process. The data can be either mode shapes, dynamic time response, frequency response functions or strain fields. Depending on the particular situation, three forms of the data expansion approach can be selected to implement the DCAF: System Equivalent Reduction Expansion Process (SEREP) with finite element model, SEREP with experimental mode shapes, and polynomial expansion. Some academic and industrial structures are used as examples to study the application of DCAF. Experimental data are used to validate the technique.

Model reduction and dynamic matrices extraction from state-space representation applied to rotating machines

Model reduction is a relevant subject within the field of rotordynamics since low order models are fundamental for control strategies design and implementation, health monitoring, behavior prediction, Fault Detection and Identification (FDI) and stochastic analyses. In this context, this article proposes a complete review of three widely used reduction methods: static or Guyan technique, the System Equivalent Reduction Expansion Process (SEREP) and the modified SEREP. Regarding SEREP, a new approach is presented in which right and left eigenvectors from the undamped original system (with mass and stiffness matrices not symmetric) are used to transform all original system dynamic matrices. To modified SEREP (that contemplates all original system characteristics, including frequency dependent damping and gyroscopic effect) an extraction from the reduced state-space representation is achieved to build rotor and bearings reduced dynamic matrices with physical interpretability. A set of practical recommendations is presented, highlighting key aspects to increase reduction success chances. The methods are applied to two different rotors and results show satisfactory agreement between reduced and complete model responses when analyzing Frequency Response Functions (FRFs) and Campbell diagrams. The computational costs of processing each reduced model and running common rotordynamic analyses with reduced and complete models are compared.

SEREP Integrated Control of Flexible Structures

Flexible structures are continuum modeled as infinite degrees of freedom. Most of the time finite element models are made with large number of degrees-of-freedom to analyse flexible structures. Continuous monitoring, analysis and control of dynamics of flexible structures need simulation of large degrees of freedom in real time. This is computationally expensive and realtime control fails. This study focuses on the development of a reduced order framework for dynamics of flexible structures and use the reduced order model for controlling the structure. Essentially, in the proposed framework, full order state space obtained from finite element modeling of the flexible structure has been reduced to lower subspace using a reduced order algorithm keeping the dynamical characteristic intact. The transformation matrix for reduction has been calculated using system equivalent reduction expansion process (SEREP). Traditionally, full order dynamical system is being used to find the gain matrix to suppress the uncontrolled dynamics associated with the dynamical system. Here in this methodology, the reduced dimension of state space of the system is used for estimating the gain matrix using optimal linear quadratic regulator (LQR). The gain obtained through the reduced system is subsequently used as a feedback to the attached actuators which produce the required force to control the system. A numerical example of a flexible cantilever beam has been shown to investigate the effectiveness of the algorithm.

Expansion of experimental mode shape from operational modal analysis and virtual sensing for fatigue analysis using the modal expansion method

Offshore structures are exposed to fatigue damage due to fluctuating environmental and operational forces. The actual stress history of structures in operation is an important parameter for the remaining fatigue lifetime. Unfortunately, the fatigue critical locations are often difficult to reach for inspection of cracks or installation of sensors and these direct locations lead to corruption of sensors due to the environmental effects. Virtual sensing enables full-field stress/strain estimation that facilitates fatigue analyses of offshore structures in operation. In the literature, various virtual sensing methods exist but this paper focuses on modal expansion. This technique uses the mode shapes (the linear subspace) of the structure to transform the measured system response into the full-field stress/strain response. The applied mode shapes either hail from a finite element model or an expansion of experimental mode shapes from an operational modal analysis. This paper compares the use of mode shapes from a finite element model to the use of expanded experimental mode shapes. Modal expansion is applied to a scaled offshore platform in the laboratory to estimate the strain response using the finite element mode shapes and expanded experimental mode shapes using different expansion techniques: System Equivalent Reduction Expansion Process (SEREP), local correspondence principle, and the new leave-p-out cross validation implementation of the local correspondence principle introduced in this paper. The results are analysed with different metrics and this paper introduces a new quality measurement – the normalised error of fatigue damage – for strain estimation intended for fatigue analysis. Based on this study, expansion of experimental mode shapes has the potential to reduce errors in stress/strain estimation. The expansion is, however, a fitting process and, thus, it contains fitting error dependent on the case. In this study, the finite element mode shapes outperform some of the mode shape expansion techniques due to this fitting errors. Therefore, expansion of experimental mode shapes can improve stress/strain estimation but it should be used with care to avoid an overfit.

Model Reduction Methods Applied to A Nonlinear Mechanical System

Modern structures of high flexibility are subject to physical or geometric nonlinearities, and reliable numerical modeling to predict their behavior is essential. The modeling of these systems can be given by the discretization of the problem using the Finite Element Method (FEM), however by using this methodology, it is a very robust model from the computational point of view, making the simulation process difficult. Using reduced models has been an excellent alternative to minimizing this problem. Most model reduction methods are restricted to linear problems, whichmotivated us to maximize the efficiency of these methods considering nonlinear problems. For better accuracy, in this study, adaptations and improvements are suggested in reduction methods such as the Enriched Modal Base (EMB), the System Equivalent Reduction Expansion Process (SEREP), QUASI-SEREP and the Iterated Improved Reduced System (IIRS). The stability of a system is discussed according to the calculation of the Lyapunov exponents and phase space. Numerical simulations showed that the reduced models presented a good performance, according to the commitment of quality and speed of responses (or time saving).

A novel strategy for effectively implementing a typical AVC scheme using finite element model updating

Finite element (FE) model of a square cantilevered plate instrumented with a piezoelectric sensor and an actuator is created using Hamilton’s principle. Rotational degrees of freedom (dofs) of the FE model are eliminated using system equivalent reduction expansion process (SEREP). Experimental mode shapes and natural frequencies are extracted from the structure using high speed cameras and digital image correlation (DIC) technique. Initial FE model is updated using experimental mode shapes and natural frequencies by well-known Berman and Nagy approach. Updated FE model thus derived is further reduced to first three modes using orthonormal modal reduction technique. Modal model of the smart plate is then used to derive state space model of the smart plate. Two Kalman observers are constructed: one using initial FE model and other using updated FE model. Active vibration control experiments are conducted on the cantilevered using these two Kalman observers in the control law. It is observed that much better vibration suppression occurs when Kalman observer based on updated FE model is used in the control law. Strategy suggested in this work to implement a typical active vibration control scheme on a structure is simple and yet very effective.

Non-Model Based Expansion from Limited Points to an Augmented Set of Points Using Chebyshev Polynomials

Expansion of measured dynamic response at a limited set of points is of great interest in the study of structural dynamic systems. Measurements at limited points on a structure can be expanded to unmeasured points by using System Equivalent Reduction Expansion Process (SEREP). However, a finite element model is required to obtain the mode shapes of the system. In this work, a non-model based expansion technique is proposed to expand the vibration characteristics at a sparse set of points to a much larger set of points without the use of a finite element model. Shape functions based on orthogonal polynomials can be used to decompose the deformation of vibrating structures into the summation of a set of shape functions with corresponding weighting coefficients. To obtain the vibration characteristics at an augmented set of points, measurements at a sparse set of points can be related to a set of shape functions which have the same mesh resolution as the sparse configuration. The vibration characteristics at an augmented set of points can then be reconstructed from the shape functions of the high resolution and scaled shape function weighting coefficients of the sparse configuration. Only a sparse set of measurements, geometry and coordinates of measured points are needed. In the work studied here, an analytical plate and a Base-Upright (BU) structure are used as samples to study the application of the proposed method. Both analysis and experiment are used to prove the concept.