Modal Participation Factors Research Articles

Assessment of floor response spectrum by parametric error estimation and its application to a spring-mounted reactor vessel assembly

For large facilities having several floors or containers, floor response spectra, FRS, other than ground response spectra need to be developed. However, FRS can have error especially when components are not small in their masses. In this paper, error is estimated in order to specify applicability of the FRS by deriving and comparing with analytic results for two degrees of freedom system. An identity regarding modal vectors and participation factors in the modal method is used to measure the FRS error. It is found that FRS is sufficiently accurate if the mass of a component is one hundredth or less than that of the floor. On the contrary, it is shown that fixed frequency of a component does not affect the FRS accuracy considerably. A compact power plant system consists of two main assemblies with spring mounts is applied as an example for the derivation of the FRS and several aspects associated with its modeling and calculation are discussed.

Compact, Generalized Component Mode Mistuning Representation for Modeling Bladed Disk Vibration

New techniques are presented for generating reduced-order models of the vibration of mistuned bladed disks from parent finite element models. A novel component-based modeling framework is developed by partitioning the system into a tuned bladed disk component and virtual blade mistuning components. The mistuning components are defined by the differences between the mistuned and tuned blade mass and stiffness matrices. The mistuned-system model is assembled with a component mode synthesis technique, using a basis of tuned-system normal modes and attachment modes. The formulation developed is general and can be applied to any mistuned bladed disk, including those with large geometric mistuning (e.g., severe blade damage). In the case of small (i.e., blade frequency) mistuning, a compact reduced-order model is derived by neglecting the attachment modes. For this component mode mistuning model, the blade mistuning is projected first onto the component modes of a tuned, cantilevered blade, and then projected again onto the tuned-system normal modes via modal participation factors. In this manner, several natural frequencies of each mistuned blade can be used to capture systematically the effects of the complex physical sources of mistuning. A numerical validation of the developed methods is performed for both large and small mistuning cases using a finite element model of an industrial rotor.

Dynamic Modal Analysis and Stability of Cantilever Shear Buildings: Importance of Moment Equilibrium

The dynamic modal analysis i.e., the natural frequencies, modes of vibration, generalized masses, and modal participation factors and static stability i.e., critical loads and buckling modes of two-dimensional 2D cantilever shear buildings with semirigid flexural restraint and lateral bracing at the base support as well as lumped masses at both ends and subjected to a linearly distributed axial load along its span are presented using an approach that fulfills both the lateral and moment equilibrium conditions along the member. The proposed model includes the simultaneous effects and couplings of shear deformations, translational and rotational inertias of all masses considered, a linearly applied axial load along the span, the shear force component induced by the applied axial force as the member deforms and the cross section rotates, and the rotational and lateral restraints at the base support. The proposed model shows that the stability and dynamic behavior of 2D cantilever shear buildings are highly sensitive to the coupling effects just mentioned, particularly in members with limited rotational restraint and lateral bracing at the base support. Analytical results indicate that except for members with a perfectly clamped base i.e., zero rotation of the cross sections, the stability and dynamic behavior of shear buildings are governed by the flexural moment equation, rather than the second-order differential equation of transverse equilibrium or shear-wave equation. This equation is formulated in the technical literature by simply applying transverse equilibrium ignoring the flexural moment equilibrium equation. This causes erroneous results in the stability and dynamic analyses of shear buildings with base support that is not perfectly clamped. The proposed equations reproduce, as special cases: 1 the nonclassical vibration modes of shear buildings including the inversion of modes of vibration when higher modes cross lower modes in shear buildings with soft conditions at the base, and the phenomena of double frequencies at certain values of beam slenderness L/r; and 2 the phenomena of tension buckling in shear buildings. These phenomena have been discussed recently by the writer 2005 in columns made of elastomeric materials.

On a modal damping identification model of building structures

The simple modal damping identification model of Hart and Vasudevan is generalized. The model works in the frequency domain and provides time-invariant modal damping ratios of building structures under seismic excitations in terms of modal participation factors and the roof-to-basement transfer function. Translational as well as torsional modes of vibration are considered. The performance of the model is assessed through a small, yet indicative, number of numerical examples involving steel plane and space frames under seismic excitations and on the basis of a number of criteria an ideal identification model should satisfy. It is concluded that the presented model, in spite of its simplicity, gives very good results for low-amplitude seismic excitations resulting in linear elastic structural behavior (with damping) even for cases of closely spaced modes, local modes, and very small or large amounts of damping. Some numerical pitfalls regarding the application of the model are mentioned and carefully treated. The limitations of the model when used in conjunction with inelastic structural behavior are determined and discussed. Experimental verification of the model is also provided.

Dynamic characteristics of rectangular twisted beam with fins surrounded with fluid

This study investigates the vibration characteristics of rectangular twisted beam and the safety assessment of the potential for fretting-wear damages caused by foreign object. To get the natural frequency, corresponding mode shape and participation factor, modal analyses are performed for a twisted beam with fins submerged in fluid of hexagonal cylinder. Special emphases are on the effects of fins, boundary conditions at both ends, fluid surrounding the beam and the number of turns on the modal characteristics, which are expressed in terms of the natural frequency and corresponding mode shape. Also, the wear rate caused by foreign object is calculated using the Archard formula and the remaining life of the beam is predicted.

Numerical Simulations of Dynamic Instability of Elastic-Plastic Beams

This paper numerically investigates the dynamic instability of different elastic-plastic beams subjected to transverse pulse load. Two types of beam dynamic instability, i.e., symmetrical and asymmetrical instabilities, are studied. The unstable responses of the elastic-plastic beams are illustrated by investigating the modal participation factors of the lowest nine vibration modes, which are determined by inverse derivation of the numerically simulated beam deflection response. The critical pulse loads for the beam symmetrical and asymmetrical instabilities are obtained with respect to different load durations. Characteristic diagrams for beams with different boundary conditions and subjected to different type pulse loads are given. The present study conforms that not only axial compressive load induces instability of slender members, but also transverse pulse load results in instability of elastic-plastic beams.

Approximate Floor Acceleration Demands in Multistory Buildings. I: Formulation

An approximate method to estimate floor acceleration demands in multistory buildings responding elastically or practically elastic when subjected to earthquake ground motion is presented. The method can be used to estimate floor acceleration demands at any floor level for a given ground motion record. The dynamic characteristics of the building are approximated by using a simplified model based on equivalent continuum structure that consists of a combination of a flexural beam and a shear beam. Closed-form solutions for mode shapes, period ratios, and modal participation factors are presented. The effect of reduction of lateral stiffness along the height is investigated. It is shown that the effect of reduction in lateral stiffness on the dynamic characteristics of the structure is small in buildings that deflect laterally like flexural beams. For other buildings, approximate correction factors to the closed-form solutions of the uniform case are presented to take into account the effects of reduction of lateral stiffness. Approximate dynamic properties of the building are then used to estimate acceleration demands in the building using modal analysis.

구분모드합성에 의한 드럼 브레이크 스퀼 소음 해석 및 저감

Recent studies have dealt with brake squeal in terms of the coupled vibration of brake component parts. In this paper, we assemble the mode models derived from FE analysis of the individual components of the drum brake system into the system model by considering the friction interaction of the lining and drum at the interface. The validity of the component models are backed up by the experimental confirmation work. By scrutinizing the real parts of the complex eigen-values of the system, the unstable modes, which may be strong candidate sources of squeal noise, are identified. Mode participation factors are calculated to examine the modal coupling mechanism. The model predictions for the unstable frequencies pointed well the actual squeal noise frequencies measured through field test. Sensitivity analysis is also performed to identify parametric dependency trend of the unstable modes, which would indicate the direction for the squeal noise reduction design. Finally, reduction of the squeal noise tendency through shape modification is tried.

Estimation of floor acceleration demands in high-rise buildings during earthquakes

A simplified method to estimate lateral acceleration demands in high-rise buildings subjected to earthquakes is presented. In the proposed method acceleration demands are obtained by approximating the dynamic characteristics of the building with those of a continuous model consisting of a combination of a flexural cantilever beam and a shear cantilever beam. Closed-form solutions for mode shapes, period ratios and modal participation factors for the first six modes of vibration are presented. The method is evaluated by comparing peak floor acceleration demands and acceleration time histories computed with the proposed method to those recorded during earthquakes in six instrumented high-rise buildings. A comparison of floor spectra computed with the approximate method and spectra computed with recorded motions is also presented. Results indicate that the proposed method produces relatively good results with a very small computational effort and requires only a small amount of information about the building. Variations of accelerations demands along the height are closely examined in each building for each component. It is shown that the variation of acceleration demands along the height of high-rise buildings can differ significantly from that currently recommended in US seismic provisions for anchoring building nonstructural components. Copyright © 2005 John Wiley & Sons, Ltd.

Dynamic characteristics of steam generator U-tubes with defect

This study investigates the fluid elastic instability characteristics of steam generator (SG) U-tubes with defect and the safety assessment of the potential for fretting-wear damages caused by foreign object in operating nuclear power plants. To get the natural frequency, corresponding mode shape and participation factor, modal analyses are performed for the U-tubes either with axial or circumferential flaw with different sizes. Special emphases are on the effects of flaw orientation and size on the modal and instability characteristics of tubes, which are expressed in terms of the natural frequency, corresponding mode shape and stability ratio. Also, the wear rate of U-tube caused by foreign object is calculated using the Archard formula and the remaining life of the tube is predicted, and discussed in this study is the effect of the flow velocity and vibration of the tube on the remaining life of the tube. In addition, addressed in this study is the effect of the internal pressure on the vibration and fretting-wear characteristics of the tube.

Modal Scheduling and Switching Systems

This paper defines a modal scheduling method for obtaining reduced order systems. In contrast to most order reduction schemes for linear structural systems, the methodology in this paper introduces a switched system to achieve order reduction. The switched system model is based on modal participation factors calculated from an input forcing function and the initial conditions at each switching time. Stability of the reduced order model is discussed, and necessary conditions for stability of the proposed switched model are derived. A numerical example that considers a flexible beam provides a testbed for the study of mode switching. The transient response in the testbed example is constructed such that it oscillates between two distinct subspaces. Numerical experiments show that an error indicator based on modal participation factors provides an effective switching strategy.

Implications of vertical mass modeling errors on 2D dynamic structural analysis

AbstractThe use of diagonal (lumped) mass matrices is common in dynamic structural analysis. The assumptions made when performing the lumping procedure, however, are not always consistent with the underlying physical behavior and can therefore modify the dynamic properties of the modeled structure. The influence of errors in mass modeling for vertical degrees of freedom on dynamic behavior and seismic response is illustrated. We demonstrate that overestimation of the lumped masses associated with vertical displacements in 2D frame models of these structures leads to inaccurate modal periods and associated modal participation factors for dynamic response. The ramifications of these inaccuracies on seismic response are shown and implications in other dynamic analysis situations are discussed. Copyright © 2004 John Wiley & Sons, Ltd.

MW and MVar Management on Supply and Demand Side for Meeting Voltage Stability Margin Criteria

This paper presents a methodology to improve the power system economical dispatch from a voltage stability margin perspective. The time horizon under discussion is the short-term operation planning. The proposed method is based on active/reactive power re-dispatch for normal operation, and also minimum load shedding strategies in case of critical contingencies. The actions are taken in the direction provided by modal participation factors computed for generator and load buses. The generators with negative impact on system margin, which are indicated by the modal index, are penalized with high costs on the objective function of the optimal power flow program used to run the re-dispatch process. Results of this work show a decrease on system losses and significant increase on voltage stability margin as well as on system reactive reserves. In addition, this work presents a study considering critical contingencies, for which is proposed an optimal load shedding strategy also based on modal participation factors to identify the most adequate buses for load shedding purposes. Finally, the proposed methodology is applied considering a typical hour-to-hour daily load curve, and the method presented very good performance since it considerably increases voltage stability margin for the insecure intervals.

Characterization of asymmetrical beam instability to impulsive load

Asymmetrical instability of a 3-DoF Shanley-type beam model subjected to impulsive transverse load is investigated. The effects of viscous damping on the asymmetrical response of the model are discussed. A new energy analysis approach is proposed for identification of the response category when the beam model is subjected to different impulsive loads. Through the energy analysis, three-dimensional Poincaré maps in terms of the displacement variables and the modal participation factors are plotted. Parametric studies for identifying the critical load above which asymmetrical instability occurs are carried out. Influences of the load duration, the material and geometrical parameters on the asymmetrical instability are demonstrated. The characteristic diagrams show clearly when the asymmetrical instability of the model occurs. The curves are also capable of predicting the final response mode if an external impulsive load is given to the beam model.

The influence of stiffened plate modifications on the energy balance

The change in plate stiffness introduced by a system of stiffeners slightly changes the whole system mass and results in a significant change in natural frequencies, distribution and values of vibrational energy. Stiffeners can be treated as an additional subsystem that not only changes the plate stiffness but also accumulates and dissipates a part of vibrational energy. The energy balance analysis of a stiffened plate was carried out for a plate treated as an assembly of a homogeneous plate and a system of beams. On the basis of modal participation factors the energies dissipated in plates and system of stiffeners were determined. The natural frequencies and modal participation factors were computed as a solution of an unconstrained optimization problem by the use of the Nelder Mead optimization algorithm and compared to a twice faster genetic algorithm. Theoretical computations of a homogeneous plate and stiffeners energies were verified by energy distribution carried out on the basis of frequency response functions of FEM models. The influence of geometrical and material properties on natural frequencies and energy dissipated by plate and stiffeners was determined.

증기발생기 나선형 전열관의 프레팅 마모 특성

This study investigates the safety assessment of the potential for fretting-wear damages caused by foreign object in operating nuclear power plants. To get the natural frequency, corresponding mode shape and participation factor, modal analyses are performed for the helical type tubes with various conditions. The wear rate of helical type tube caused by foreign object is calculated using the Archard formula and the remaining life of the tube is predicted, and discussed in this study is the effect of the vibration of the tube on the remaining life of the tube. In addition, addressed is the effect of the external pressure on the vibration and fretting-wear characteristics of the tube.

MVAR management on the pre-dispatch problem for improving voltage stability margin

The paper presents a methodology that provides a solution for the one-day-ahead pre-dispatch problem considering the evaluation and improvement of the voltage stability margin by optimising reactive-power injections by generators and synchronous condensers. Modal participation factors are used to define penalty indices for all generators, which are then added to the optimal power flow objective function. This identifies the most adequate reactive-power injection for each generator or synchronous condenser, to maximise voltage stability margins. The results obtained for the New England test system and a version of the Southwest Brazilian system show that the proposed methodology leads to significant improvement in voltage stability margin for all the critical time intervals of the day. The occurrence of contingencies in the system is also considered. One clear advantage of the proposed methodology is that the optimal (economical) solution for active power injection of generators is maintained unchanged, which means that there will be no impact on the energy targets of the hydroelectric plants, nor on total generation costs. Hence, the voltage stability margin is improved by simply managing reactive-power dispatch.

Chaotic and asymmetrical beam response to impulsive load

Both symmetrical and asymmetrical final displacements are observed for elastic–plastic beams under symmetrical impulsive loading. A three-degree-of-freedom Shanley-type model is developed in this study, which is capable of revealing chaotic and asymmetrical responses of an elastic–plastic beam by introducing initial imperfections. To identify the asymmetrical displacement, the beam response is decomposed into three vibration modes. Corresponding modal participation factors are derived based on the displacement of the three-degree-of-freedom beam model. Phase plane trajectories, Poincaré maps and power spectral density diagrams are derived to illustrate both the symmetrical and asymmetrical chaotic vibrations. Numerical simulations using a general-purpose FE code LS-DYNA are carried out for an elastic–plastic beam subjected to impulsive load. The simulation results indicate that the elastic–plastic beam demonstrates chaotic and asymmetrical vibration when the applied impulsive load exceeds a critical value, which agrees with experimental observations.

Simplified method for the seismic qualification using measured modal data

The seismic response estimation by the response spectrum method using only the experimental modal data are presented here. The modal participation factors (MPFs) used for the response estimation are calculated using the experimental mode shapes only. This response is compared with the response estimated using the conventional MPFs but the experimentally extracted mode shapes are used along with the mass matrix estimated corresponding to the measured degree of freedoms (dofs) using physical dimensions of the structure. The presented study eliminates the uncertainty associated with analytical modelling for evaluating mass matrix.

Optimal location of sensors for reconstruction of seismic responses through spline function interpolation

AbstractIn this paper, a criterion is proposed for the choice of the optimal location of a limited number of recording sensors for reconstruction of seismic responses of multistorey frames. Reconstruction of unknown responses is performed by modelling through a spline shape function the evolution of the relative acceleration along the height of the building. The coefficients of the spline function are determined by interpolating available responses in terms of absolute accelerations taking into account boundary conditions. The optimal location for a given number of recording sensors is defined as the one corresponding to the minimum value of the global error evaluated on the whole set of reconstructed responses. The criterion has been verified by using simulated responses of numerical models of multistorey frames and responses recorded on real buildings during the Northridge earthquake.A physical characterization of the location of available sensors leading to the minimum global error between calculated and real responses is proposed in terms of effective modal participation factors relevant to the principal modes of the structure. This characterization allows for frames of known modal characteristics the definition of a criterion for the choice of the location of a limited number of sensors for monitoring or vibration control purposes. Copyright © 2003 John Wiley & Sons, Ltd.