Modal Participation Factors Research Articles

Assessment of Effects of Reductions of Lateral Stiffness along Height on Buildings Modeled as Elastic Cantilever Shear Beams

A simplified model useful for assessing economic losses due to moderate seismicity events in urban areas has been developed by studying the behavior of buildings before yielding their structural system, allowing for nonuniform stiffness along their height. In particular, buildings are modeled as cantilever shear beams with uniform mass and parabolic reduction of lateral stiffness. This particular stiffness distribution is relevant, as it could be expected to occur in buildings where earthquake action is a critical structural design criterion. The equation of motion governing the dynamic behavior of the proposed model is solved analytically, finding mode shapes in terms of first and second zero-order Legendre functions. The solution is verified by comparing it with results obtained from fine mesh finite element models. The effect of reducing the lateral stiffness is then studied in the first five modes of vibration. Results include modal periods, mode shapes, modal participation factors, and derivatives of mode shapes. In general, it is found that effects of reduction of lateral stiffness in mode shapes are moderate when the lateral stiffness in the free end is smaller than about seventy percent of the lateral stiffness at the fixed end, but become significant for larger reductions. Effects are particularly important for the derivative of the mode shapes, which could play a significant role in estimating interstory drift demands in buildings. Model usefulness is showcased by analyzing a test case where both acceleration and drift demands are assessed by considering uniform beams and beams with parabolic stiffness variation, finding notable improvements by considering the latter.

Influence of Modal Mass Participation in Damage Detection of Cable Structures

Increasing applications of large diameter and long-span cables as key structural components in cable-supported structures are becoming evident. However, these cables accumulate damage over time during their life cycle and such damage needs to be detected to avoid the detrimental influences on the serviceability and ultimate capacity of the structure. In this context, vibration-based damage detection (VBDD) methods have been used in some structures. Cable structures however exhibit complex vibration patterns, namely with vertical, lateral, torsional and coupled modes which complicate the VBDD procedure. To address this matter, this paper proposes a new approach for detecting and locating damage in cables using component-specificdamage indices (DIs) based on the modal flexibility method considering the modal mass participation. Application of this new procedure is illustrated through two case studies: (a) a suspended cable and (b) a three dimensional (3D) suspension bridge structure. Results verify that the DIs based on the lateral and vertical components of mode shapes, identified through their modal mass participation factors, are effective for detecting and locating damage in the suspended cables and main cables in a suspension bridge respectively, under a range of damage scenarios. The research outcomes of this paper confirm that the modal mass participation factor is an important signature in the damage detection of a structure using VBDD techniques.

Reproduction of vibration patterns of elastic structures by block-wise modal expansion method (BMEM)

The quality of vibration pattern reproduction of elastic structures by the modal expansion method is influenced by the modal expansion method and the sensor placement as well as the accuracy of measured natural modes and the total number of vibration sensors. In this context, this paper presents an improved numerical method for reproducing the vibration patterns by introducing a block-wise modal expansion method (BMEM), together with the genetic algorithm (GA). For a given number of vibration sensors, the sensor positions are determined by an evolutionary optimization using GA and the modal assurance criterion (MAC). Meanwhile, for the proposed block-wise modal expansion, a whole frequency range of interest is divided into several overlapped frequency blocks and the vibration field reproduction is made block by block with different natural modes and different modal participation factors. A hollow cylindrical tank is taken to illustrate the proposed improved modal expansion method. Through the numerical experiments, the proposed method is compared with several conventional methods to justify that the proposed method provides the improved results.

Dynamic displacements-based model updating with motion capture system

Summary In this paper, a dynamic displacements-based model updating method using a motion capture system (MCS) is proposed. The dynamic characteristics from MCS are used to find the parameters that minimize the difference between updated model and direct measurement. Using a multi-objective optimization algorithm of non-dominated sorting genetic algorithm-II, the number of objective functions for model updating is set to the same number of modes under consideration, and all the objective function are simultaneously minimized. To consider the contribution of each mode on model updating and to avoid biased results, a rule for weighting of solutions associated to each mode based on modal participation factors is suggested and tested. Using a free vibration experimental test of a three-story shear model, the performance of model updating method is verified by the comparison of the dynamics characteristics between the updated model and direct measurement by MCS. In addition, time histories of displacements from the updated model are compared with the direct measurement. Copyright © 2016 John Wiley & Sons, Ltd.

Oscillation Energy Analysis of Inter-Area Low-Frequency Oscillations in Power Systems

Low-frequency inter-area oscillation is a potentially dangerous phenomenon in power systems involving exchange of power among generators in different areas. In this paper, we study the characteristics of inter-area oscillation by analyzing the distribution of the oscillation energy. First, we introduce equations for the kinetic energy of generators and the potential energy of network branches, derived from the linearization of system dynamic equations. Then, we study the mode composition and the distribution of the generators' kinetic energy during low-frequency oscillations. A modal kinetic energy participation factor is proposed for evaluating the participation of each generator in the oscillation. The relationships among electromagnetic power oscillation distribution, rotor speed deviation, and generator kinetic energy are analyzed and compared to their modal parameter left eigenvectors, right eigenvectors, and linear participation factors, respectively. The bus voltage angle deviation, active power oscillation, and branch potential energy are calculated using the right eigenvector from modal analysis. The distribution characteristics of voltage angle oscillation and branch potential energy in inter-area oscillation is developed. The simulation results for a four-generator two-area system and the New England 10-machine 39-bus system illustrate the feasibility and validity of the proposed theory and method for analysis of inter-area oscillations.

Tri-axial Seismic Simulation of Solar Inverter – Productivity Improvement

Often, earthquake is a tri-axial event, as in, it excites the system in all three directions. Evaluation of big systems like high capacity solar inverters for such events through testing is quite expensive. Trend these days is to qualify the system using CAE simulation, in contrast to testing. Simulation of big systems is always a challenge, as it becomes computationally expensive due to high node count and large number of modes in the frequency range of interest. In this work, seismic analysis is performed using commercial FEA software ANSYS. Conventional way of seismic simulation in ANSYS is to first excite the system separately in the orthogonal directions and then compute the tri-axial response by superposition of responses in individual directions. In this paper, based on the response spectrum analysis, a tri-axial seismic analysis methodology is proposed and implemented on complicated high capacity solar inverters. Proposed methodology calculates a resultant mode participation factor from mode participation factors due to excitation in individual directions and then performs the mode combination. It was observed that computational time was reduced to 1/3rd without any compromise in the accuracy of response. Also, proposed simulation methodology doesn’t require separate static analysis to account for the effect of missing mass. This further increases the productivity of a seismic engineer performing a finite element analysis.

Experimental validation of the Kalman-type filters for online and real-time state and input estimation

In this study, a novel dual implementation of the Kalman filter proposed by Eftekhar Azam et al. (2014, 2015) is experimentally validated for simultaneous estimation of the states and input of structural systems. By means of numerical simulations, it has been shown that the proposed method outperforms existing techniques in terms of robustness and accuracy for the estimated displacement and velocity time histories. Herein, dynamic response measurements, in the form of displacement and acceleration time histories from a small-scale laboratory building structure excited at the base by a shake table, are considered for evaluating the performance of the proposed Dual Kalman filter and in order to compare this with available alternatives, such as the augmented Kalman filter (Lourens et al., 2012b) and the Gillijn De Moore filter (GDF) (2007b). The suggested Bayesian approach requires the availability of a physical model of the system in addition to output-only measurements from limited degrees of freedom. Two categories of such physical models are herein studied to evaluate the effect of model error on the filter performances; the first, is a model that comprises identified modal parameters, i.e., natural frequencies, mode shapes, modal damping ratios and modal participation factors; the second, is a model that is extracted from a recently developed subspace identification procedure, namely the transformed stochastic subspace identification method. The results are encouraging for the further use of the dual Kalman filter and its available alternatives for addressing the important problems of full response reconstruction and fatigue estimation in the entire body of linear structures, using a limited number of output-only vibration measurements.

Optimal selection of distributed generating units and its placement for voltage stability enhancement and energy loss minimization

Abstract The integration of distributed generation (DG) in distribution network may significantly affect its performance. Transmission networks are no longer accountable solely for voltage security issues in distribution networks with penetration of DGs. The reactive power support from the DG sources greatly varies with the type of DG units and may potentially distress the larger portion of the network from the voltage stability aspects. This paper presents the analysis for the selection of the best type of DG unit among different categories and its optimal location that can enhance the voltage stability of distribution network with simultaneous improvement in voltage profile. Voltage sensitivity index and bus participation factors derived from continuation power flow and Modal Analysis, respectively, are used together for voltage stability assessment and placement of DGs. Changes in mode shapes and participation factors with the placement of DGs are comprehensively analyzed for 33 and 136 nodes radial distribution network.

Uplift of deck or footings in bridges with distributed mass subjected to transverse earthquake

SummaryThe eigenvalue problem is analytically formulated in symmetric bridges with distributed mass and moment of inertia under transverse earthquake. The piers are elastically supported on the ground. The deck is monolithically connected to one or two piers for all degrees of freedom and restrained or transversely free at the abutments. The characteristic equation, symmetric normal modes, modal participation factors, and participating mass ratios are given analytically. The problem is expressed in terms of few dimensionless parameters: (i) the radius of gyration of the deck mass divided by the pier height; (ii) the ratio of the rotational stiffness of a footing to that of the pier at the base; (iii) the ratio of flexural stiffness of the outer spans to those of the pier; (iv) the ratio of torsional stiffness of side spans to the rotational stiffness of the pier top; (v) for two piers, the side‐to‐central‐span ratio. Modal response spectrum analysis gives the moment at the base of the footings and the torque in the deck at its supports on the abutments as ratios to the values at incipient uplifting from the ground or the bearings. The peak ground acceleration of the motion at the onset of either one of these two types of nonlinearity is depicted as a function of the dimensionless parameters and the fundamental period of an elastic deck supported only at the abutments, or of a rigid deck on piers fixed against rotation at top and bottom. Copyright © 2015 John Wiley & Sons, Ltd.

Model Updating Technique Based on Modal Participation Factors for Beam Structures

AbstractThis article proposes a model updating technique based on modal participation factors for a beam structure. In this model updating technique, the error functions of the dynamic characteristic differences between measurement and model are generated as the number of modes under consideration and minimized using the multiobjective optimization techniques. A modal influence factor defined by modal participation factors for each mode is presented for the selection of the best solution from among Pareto solutions. The selection rule represented in this article makes it possible to reflect the contributions of each mode on the behavior of a structure. The model is updated using natural frequencies measured in an impact hammer test of a beam structure and the validity of the updated model is confirmed by the strain responses measured from the test. It is found that the bending stiffness of the beam structure as the parameter for model updating can be identified by the proposed techniques. Furthermore, through comparing the models updated by the simple sum model updating and the technique in this research, it is verified that the proposed technique is more appropriate for the model updating.

Modal Parameters Estimation of Building Structures from Vibration Test Data Using Observability Measurement

The load distribution to each mode of a structure under seismic loading depends on the modal participation factors and mode shapes and thus the exact estimation of modal participation factors and mode shapes is essential to analyze the seismic response of a structure. In this study, an identification procedure for modal participation factors and mode shapes from a vibration test is proposed. The modal participation factors and mode shapes are obtained from the relationship between observability matrices realized from the system identification. Using the observability matrices, it is possible to transform an arbitrarily identified state space model obtained from the experimental data into a state space model which is defined in a domain with physical meaning. Then, the modal participation factor can be estimated based on the transformation matrix between two state space models. The numerical simulation is performed to evaluate the proposed procedure, and the results show that the modal participation factor and mode shapes are estimated from the structural responses accurately. The procedure is also applied to the experimental data obtained from the shaking table test of a three-story shear building model.

Dynamic analysis of uncertain structures using imprecise probability

A new method for dynamic response spectrum analysis of structures with uncertainty in their mechanical properties utilising the notion of imprecise probability is developed. This finite-element-based method is capable of obtaining probabilistic bounds of the dynamic response of the structure with uncertainty defined by enveloping imprecise probability structures. The developed method obtains probabilistic bounds on (1) natural circular frequencies, (2) mode shapes, (3) modal coordinates and (4) modal participation factors leading to the imprecise probability structures of modal responses. Finally, maximum modal responses are combined to obtain the structure's maximum total response with consideration of uncertainty. Numerical examples demonstrating the developed method are presented.

Approximate soil–structure interaction analysis by a perturbation approach: The case of soft soils

An approximate solution of the classical eigenvalue problem governing the vibrations of a relatively stiff structure on a soft elastic soil is derived through the application of a perturbation analysis. The full solution is obtained as the sum of the solution for an unconstrained elastic structure and small perturbing terms related to the ratio of the stiffness of the soil to that of the superstructure. The procedure leads to approximate analytical expressions for the system frequencies, modal damping ratios and participation factors for all system modes that generalize those presented earlier for the case of stiff soils. The resulting approximate expressions for the system modal properties are validated by comparison with the corresponding quantities obtained by numerical solution of the eigenvalue problem for a nine-story building. The accuracy of the proposed approach and of the classical normal mode approach is assessed through comparison with the exact frequency response of the test structure.

Modal Participation Factor-Constrained Optimal PMU Placement in Power System Networks

Post-mortem studies carried out after recent power system blackouts have shown that time-stamped measurements from Phasor Measurement Units (PMUs) would have helped to avert these blackouts if they were available. Consequently, there is the need to optimally locate these PMUs to provide information about the system’s state. This paper proposes a novel method incorporating modal participation factors computed from the eigenvalues and eigenvectors of the reduced Jacobian matrix obtained from the system’s power flow with Binary Integer Programming (BIP), in solving the Optimal PMU Placement (OPP) problem while considering the system’s voltage stability. The proposed method identifies the critical buses prone to voltage instability, determines the minimum number of PMUs required for complete topological observability, and the best locations to site the PMUs. The proposed method was tested on benchmark IEEE 14-bus, 30-bus, and 57-bus test networks. Analyses of the results show that the proposed method provides a better procedure for the strategic placement of PMUs in practical networks for voltage stability monitoring especially for the cases of phased (multi-stage) PMU placement limited by the availability of funds and communication channels. Thereby making it necessary to give priority to critical/voltage weak buses during the phased installation process.

Shear horizontal wave excitation and reception with shear horizontal piezoelectric wafer active sensor (SH-PWAS)

This article discusses shear horizontal (SH) guided-waves that can be excited with shear type piezoelectric wafer active sensor (SH-PWAS). The paper starts with a review of state of the art SH waves modelling and their importance in non-destructive evaluation (NDE) and structural health monitoring (SHM). The basic piezoelectric sensing and actuation equations for the case of shear horizontal piezoelectric wafer active sensor (SH-PWAS) with electro-mechanical coupling coefficient are reviewed. Multiphysics finite element modelling (MP-FEM) was performed on a free SH-PWAS to show its resonance modeshapes. The actuation mechanism of the SH-PWAS is predicted by MP-FEM, and modeshapes of excited structure are presented. The structural resonances are compared with experimental measurements and showed good agreement. Analytical prediction of SH waves was performed. SH wave propagation experimental study was conducted between different combinations of SH-PWAS and regular in-plane PWAS transducers. Experimental results were compared with analytical predictions for aluminium plates and showed good agreement. 2D wave propagation effects were studied by MP-FEM. An analytical model was developed for SH wave power and energy. The normal mode expansion (NME) method was used to account for superpositioning multimodal SH waves. Modal participation factors were presented to show the contribution of every mode. Power and energy transfer between SH-PWAS and the structure was analyzed. Finally, we present simulations of our developed wave power and energy analytical models.

Simultaneous optimization of structural shape and control system of large-scale space frame based on sine wave inputs

This paper proposes a simultaneous optimal design method of asymmetric large-scale space frames with tuned mass dampers (TMDs). The objective function is defined by the maximum absolute acceleration response of the structure to input ground motions of sine waves. Sine waves of periods with the five natural periods having large modal participation factors of the structure are input, and the maximum responses are calculated by time-history response analysis to evaluate the objective function. The shape of the space frame, i.e. nodal coordinates of the space frame's joints, is described by a Bezier surface to reduce the number of design variables. The change from the initial values of the nodal coordinates is constrained to preserve the initial design shape, which is provided by an architect. The method employs a genetic algorithm in optimization. In addition, a case study is conducted for an asymmetric steel space frame of a vault-like shape. The results confirm the reduction of maximum absolute acceleration responses in the optimal shapes not only to the five sine waves but also to four scaled ground motion records. Moreover, the presence of TMDs enables the reduction of the peak response value and maintains similarity to the initial shape. Keywords architectural design, Bezier surface, earthquake engineering, genetic algorithm, optimiza- tion, seismic control, structural engineering, structural shape, space frame, tuned mass damper Language: en

센서최적배치 기법에 의한 원통형 구조물의 진동장 예측

This study is concerned with the estimation of vibration-field of a cylindrical structure by modal expansion method(MEM). MEM is a technique that identifies modal participation factors using some of vibration signals and natural modes of the structure: The selection of sensor locations has a big influence on predicted vibration results. Therefore, this paper deals with four optimal sensor placement(OSP) methods, EFI, EFI-DPR, EVP, AutoMAC, for the estimation of vibration field. It also finds optimal sensor locations of the cylindrical structure by each OSP method and then performs MEMs. Predicted vibration results compared with reference ones obtained by forced response analysis. The standard deviations of errors between reference and predicted results were also calculated. It is utilized to select the most suitable OSP method for estimation of vibration field of the cylindrical structure.

Fatigue of Solder Interconnects in Microelectronic Assemblies under Random Vibration

This study investigates vibration fatigue durability of solder interconnects of surface mount electronic IC packages under harmonic and random excitation, through accelerated stress testing and physics-of-failure (PoF) simulations. The solder material of interest is Sn3.0Ag0.5Cu. The test vehicle consists of several HVQFN (Heat-sink Very-thin Quad Flat-pack No-leads) components soldered onto a Printed Wiring Board (PWB). The fatigue damage accumulation rates under ED random vibration excitation is found to be much higher than under harmonic vibration excitation, for equivalent RMS responses of PWB. To understand the reasons for such discrepancy we use FEA-based modeling. For simplicity of modeling we use a simple PWB with a simple Leadless Ceramic Resistor (LCR) component. We use a multiscale FEA modeling approach where the global analysis provides insights into the PWB modal response under the given excitation and can be conducted in the frequency-domain or in the time-domain. The local FEA model allows us to establish a transfer function between PWB curvature (flexural strain) in the vicinity of the critical component and equivalent strain in the critical region of the critical solder joint. al analysis is conducted to investigate the PWB mode shapes and natural frequencies, and they are compared with the measured strain time history, to understand the modal participation factors under both harmonic and ED random vibration excitation. The failure data is found to correlate well with the modal participations. Multiscale finite-element (FE) simulations are conducted to assess the dynamic response of the critical interconnects. This is done with two steps, first using a FE global analysis in the frequency domain for the global response consisting of all participating modes. Participation factors for individual modes are estimated from the global response, using numerical filtering methods. Inverse FFT is used to convert the frequency-domain response into the time-domain response histories for each dominant mode. In the second step, dynamic FE local analysis is conducted to obtain the transfer function between the PWB flexure and the strain in the critical solder interconnect for each dominant mode. Finally, rainflow cycle counting and Coffin-Manson fatigue model are used to estimate durability model constants and to compare fatigue damage accumulation rates under harmonic versus ED random vibration excitation.

Exact analytical modeling of power and energy for multimode lamb waves excited by piezoelectric wafer active sensors

This article presents an analytical model for power and energy transfer between excited piezoelectric wafer active sensors and host structure. This model is based on exact multimodal Lamb waves, normal mode expansion technique, and orthogonality of Lamb waves. Modal participation factors are presented to show the contribution of every mode to the total energy transfer. The model assumptions include the following: (1) straight-crested multimodal ultrasonic guided wave propagation, (2) propagating waves only, (3) ideal bonding (pin-force) connection between piezoelectric wafer active sensors and structure, and (4) ideal excitation source at the transmitter piezoelectric wafer active sensors. Constrained piezoelectric wafer active sensor admittance is reviewed. Electrical active power, mechanical converted power, and Lamb wave kinetic and potential energies are derived in closed-form formulae. Numerical simulations are performed for the case of symmetric and antisymmetric excitation of thin aluminum structure. The simulation results are compared with axial and flexural approximation for the case of low-frequency Lamb waves. In addition, a thick steel structure example is considered to illustrate the case of multimodal guided waves. A parametric study for different excitation frequencies and different transducer sizes is performed to show the best match of frequency and piezoelectric wafer active sensor size to achieve maximum energy transfer into the excited structure.

PSASP Power Grid Voltage Static Stability Analysis Method -Determine the System Key Node and the Region

In recent years, with the growth of the power grid power load, the power system voltage static stability problems gradually appear,The study found that, often from the system voltage stability the weakest node set off, and gradually into the surrounding weak (nodes) regional spread. How to determine the system key nodes and area is PSASP voltage static stability analysis method, from the essence of voltage stability and principle, to calculate the power of each node Leading voltage instability mode participation factors to determine the size of voltage stability judgment the weakest node or area.