Domain Modal Analysis Research Articles

In-operation modal analysis of containments using ambient vibration

The importance of preserving the structural integrity of a containment can never be overemphasized. Currently the structural integrity of the containment is evaluated periodically via various nondestructive inspection methods. However, these methods can only provide local integrity data, and cannot deliver continuous monitoring information on overall structural integrity. In this paper, the result of a feasibility study on utilizing ambient vibration measurements to monitor the structural integrity of a containment is presented. The ambient vibration testing was selected, because it can avoid the interruption of normal operation of power plants. For the purpose of the study, the ambient vibration of four different containments in Korea, i.e. Ulchin Unit 5, Wolsong Unit 2, Ulchin Unit 1, and Yeonggwang Unit 4, was measured and modal parameters were extracted via frequency domain modal analysis techniques. The experimental modes were then compared to the analytical modes estimated using finite element analysis to examine the correlation, and highly correlated modes were adopted in identifying structural parameters using a system identification method. The structural parameter selected in this study was the elastic modulus of the concrete, and the identified moduli were compared to design values and inspection results. The result shows the ambient vibration can be utilized in monitoring the structural integrity of a containment, though further extensive study needs to be conducted to identify and exclude the effect of ambient conditions such as temperature and humidity on resonant frequencies.

Structural Nonlinearity Identification Using Perturbed Eigen Problem and ITD Modal Analysis Method

Identification of nonlinear behavior in structural dynamics has been considered here, in this paper. Time domain output data of system are directly used to identify system through Ibrahim Time Domain (ITD) modal analysis method and perturbed eigen problem. Cubic stiffness and Jenkins element, as case studies, are employed to qualify the identification method. Results are compared with Harmonic Balance (HB) estimation of nonlinear dynamic stiffness. Results of ITD based identification are in good agreement with the HB estimation, for stiffness parts of nonlinear dynamic stiffness but for damping parts of nonlinear dynamic stiffness, method needs some additional improvements which are under investigation.

Reduced Order Model of a Multistage Bladed Rotor With Geometric Mistuning via Modal Analyses of Finite Element Sectors

An algorithm to generate a reduced order model of a multistage rotor in which each stage has a different number of blades has been developed. It is shown that a reduced order model can be developed on the basis of tuned modes of certain bladed disks which can be easily obtained via sector analyses. Further, it is shown that the reduced order model can also be obtained when blades are geometrically mistuned. This algorithm is similar to the modified modal domain analysis, which has been recently developed for a single-stage bladed rotor with geometric mistuning. The validity of this algorithm is shown for the finite element model of a two-stage bladed rotor. In addition, the statistical distributions of peak maximum amplitudes and natural frequencies of a two-stage rotor are generated via Monte Carlo simulations for different patterns of geometric mistuning.

Generalized Eigenvalue Decomposition in Time Domain Modal Parameter Identification

This paper is intended to point out the relationship among current time domain modal analysis methods by employing generalized eigenvalue decomposition. Ibrahim time domain (ITD), least-squares complex exponential (LSCE) and eigensystem realization algorithm (ERA) methods are reviewed and chosen to do the comparison. Reformulation to their original forms shows these three methods can all be attributed to a generalized eigenvalue problem with different matrix pairs. With this general format, we can see that single-input multioutput (SIMO) methods can easily be extended to multi-input multioutput (MIMO) cases by taking advantage of a generalized Hankel matrix or a generalized Toeplitz matrix.

An Analysis of Ride Coupling in Automobile Suspensions

The tuning of a pitch-plane model of a passenger car with a ‘coupled’ suspension system was investigated and compared to that of a conventional suspension system. Direct comparison of stiffness matrices for the two models was used as well as modal analysis and frequency domain analysis, to gain insight into the optimal pitch-bounce tuning. Performance was measured in terms of body acceleration, dynamic tyre force, and suspension working space. Optimum tuning of the coupled system was found to be at lower pitch stiffness, but greater pitch damping than that of a popular saloon car. Uncorrelated inputs to the model were found to give tuning suitable for a realistic range of road speeds from 5 to 25 m/s. The Olley suspension tuning criterion was found to represent an optimal conventional suspension stiffness tuning for dynamic tyre force minimization, but not for driver chest acceleration minimization.

Frequency domain modal analysis of earthquake input energy to highly damped passive control structures

AbstractA new complex modal analysis‐based method is developed in the frequency domain for efficient computation of the earthquake input energy to a highly damped linear elastic passive control structure. The input energy to the structure during an earthquake is an important measure of seismic demand. Because of generality and applicability to non‐linear structures, the earthquake input energy has usually been computed in the time domain. It is shown here that the formulation of the earthquake input energy in the frequency domain is essential for deriving a bound on the earthquake input energy for a class of ground motions and for understanding the robustness of passively controlled structures to disturbances with various frequency contents. From the viewpoint of computational efficiency, a modal analysis‐based method is developed. The importance of overdamped modes in the energy computation of specific non‐proportionally damped models is demonstrated by comparing the energy transfer functions and the displacement transfer functions. Through numerical examinations for four recorded ground motions, it is shown that the modal analysis‐based method in the frequency domain is very efficient in the computation of the earthquake input energy. Copyright © 2004 John Wiley & Sons, Ltd.

LOCAL REDUCTION OF FOLD BIFURCATION RELEVANT TO VOLTAGE COLLAPSE IN AN ELECTRIC POWER SYSTEM

In this paper, the reduction approach based on the center manifold theory for parameter-dependent nonlinear dynamical systems is applied to simplify the analysis of the fold bifurcation relevant to voltage collapse in a simple electric power system. This technique enables us to obtain a lower-dimensional and topologically equivalent system in the neighborhood of the bifurcation value, which is a subset in the state-parameter space. Explicit formulas are presented for the computation of quadratic coefficients of the Taylor approximations to the center manifold for the fold bifurcation in parameter-dependent dynamical systems. It is demonstrated that it is the fold bifurcation but not other static bifurcations that occurs in the reduced power system, via validating the nondegenerate conditions for the fold bifurcation. The dynamics of the reduced system are visualized by two- and three-dimensional plots, which show that the reduction method is applicable and accurate to analyze local bifurcations in the power system. Furthermore, time domain simulation and modal analysis technique for linear systems are applied to distinguish the voltage stability from the rotor (angle) stability in the electric power system.

Integration of Pulsed‐Laser ESPI with Spatial Domain Modal Analysis

The present paper discusses the various critical elements of a modal testing system based on pulsed‐laser holographic ESPI measurements. Such system allows making very high spatial resolution measurements on panel‐like structures at frequencies that are of relevance for the vibro‐acoustic behavior.Next to the optical parts, the integration with the modal analysis procedure, including the integration of geometry and response information, is reviewed. The ESPI based Image‐domain frequency response functions are accumulated frequency by frequency. Special attention is paid to the modal parameter extraction of the large DOF number data, for which a special data reduction scheme was developed. The approach has been applied to several case studies related to car panels. The research was conducted in the context of the Brite/Euram project SALOME and the EUREKA project HOLOMODAL.

Torsional vibration of a back-to-back gearbox rig Part 1: Frequency domain modal analysis

The torsional vibration of a back-to-back gearbox system has been investigated experimentally and analytically. Gearboxes are typically part of a larger system, or drivetrain, which commonly includes electric motors, shafts, couplings, ball mills, turbines and generators. The dynamics of the system has been shown significantly to influence the gearbox response and requires inclusion in gearbox modelling. Assumptions are commonly made, however, to reduce the degrees of freedom of the model, and system dynamics is subsequently neglected. Receptance methods were applied to model a back-to-back gearbox system and were shown to be an effective modelling technique for geared system analysis. A detailed experimental modal analysis was performed using the swept-sine technique with an a.c. servomotor torsional exciter. Torsional excitation allowed for the dynamic response to be measured up to 1600 Hz. A multi-degree-of-freedom, frequency domain torsional model of the gearbox system was developed using receptances. The model included a combination of lumped-mass elements, and continuous shafts with distributed inertia and hysteretic damping. The modelled torsional natural frequencies were matched to the measured frequencies by the adjustment of system model parameters to achieve a high level of agreement. Matching of system natural frequencies resulted in the accurate prediction of the first nine associated torsional deflected shapes up to 1548 Hz. This paper presents the detailed results of a full torsional, modal analysis of a gearbox system, demonstrating receptance system modelling and an effective method for torsional excitation on rotating machines.

Variable Modal Parameter Identification for Non-Linear Mdof Systems. Part I: Formulation and Numerical Validation

This paper deals with the formulation of a frequency domain modal analysis technique that is applicable to weakly non-linear multi-degree of freedom (MDOF) systems with well-separated modes. The concept of linear modal superposition is combined with the normal non-linear mode technique, an approach that allows the formulation of a system identification procedure in terms of variable modal parameters. The numerical study was focused on a 4-DOF system with cubic stiffness non-linearity, and the modal parameters were obtained as functions of the modal amplitude. It was shown that the methodology was well suited to the study of practical cases for which the underlying linear model may be approximate. Similarly, the technique was found to be robust in the presence of measurement noise, though some adverse effects were observed for high noise levels. Once the variable modal parameters were extracted at some given force level, the non-linear responses were predicted at other force levels via synthesis of normal non-linear modes. The same responses were also obtained using a harmonic balance approach and very good agreement was obtained between the two sets of results. The procedure is well suited to the study of industrial cases because of its compatibility with existing finite element methods and linear modal analysis techniques.

Time Domain Finite Element Analysis of Dynamic Systems

In this study, dynamic analyses of systems are conducted using the time domain finite element analysis when initial and final conditions are specified. Dynamics of systems can be analyzed using the time-marching procedure based on Hamilton's weak principle. A new method is developed to describe the motion of a system and is extended to accommodate the initial displacement as a state variable. System matrices are constructed to evaluate time histories of the system from initial time to a specified time without the information on the velocity and momentum at each time step. Galerkin's weak principle is used to construct element matrices in the time domain and assemble the whole matrices. The Neumann boundary condition is modified to reflect the effect of initial displacement and velocity that are inherent to dynamic problems through the momentum conservation of the system. The matrices obtained thereafter conserve the symmetricity and enable us to reduce the model order by the systematic approach. Spatial propagation equation is built up, and two-point boundary conditions are used to estimate the unknown initial conditions at one end of the beam. Modal domain analysis is introduced to reduce the size of matrices. It can be seen that the matrices constructed using the time domain finite element analysis are applicable to the model reduction that is analogous to the space-domain model reduction methodology. Several numerical examples show that a consequence of the suggested method can be applied to describe the motion of various dynamic systems successfully and to validate the effect of the model reduction in the time domain.

Active-power stabilizers for multimachine power systems: challenges and prospects

The problem of stabilizing a bulk power system by modulating controllable active loads is investigated. With today's advances in power distribution and home automation, fibre optic communications and networking, such a scheme for improving the reliability of bulk power supply is increasingly attractive. After developing a generic model for active load-modulation studies, time domain modal analysis is applied to a three-machine-nine-bus system in order to assess quantitatively its responsiveness with respect to controller location and observed response signals. This structural analysis shows that proportional control is unsatisfactory and may artificially restrict the economic benefits of active-load modulation. By contrast, using the bus voltage and frequency as input signals, combined with suitable dynamic compensators, yields a fully decentralized, two-loop load stabilizer able to add damping to all grid modes in the sample power system. Finally, placement and sizing issues are considered and preliminary observations based on transient stability studies are made.

Sensitivity analysis for feedforward control system design

A sensitivity analysis for feedforward‐controlled finite domain systems is presented. Using an analytical approach, the sensitivity of a controlled system to variations in the design variables of the control inputs and the error sensors is obtained. Since the physical implementation of the control inputs and the error sensors is directly related to the modal control forces and the modal error sensor components, the analysis is performed in the modal domain. This modal domain analysis offers the advantage that the highest computational effort in obtaining the sensitivities is independent of the physical characteristics of the transducers. From the results in the modal domain, the sensitivities of the controlled system with respect to particular physical parameters are obtained by using the chain rule of differentiation. Therefore, different types of transducers can be investigated within a minimum computational cost. The formulation can be easily incorporated into an optimization procedure for the optimal design of feedforward control systems. A numerical example in which the proposed formulation is compared to results obtained using finite differences is included.

Calculating response spectral moments by complex modal analysis

Response spectral moments are useful for system reliability analysis. Usually, spectral moments are calculated by the frequency domain method. Based on the time domain modal analysis of random vibrations, the authors present a new method for calculating response spectral moments through response correlation functions. The method can be applied to both classical and non-classical damping cases and to three kinds of random excitations, i.e., white noise, band-limited white noise, and filtered white noise.

Nonlinear vibrations of guy cable systems

A nonlinear static and dynamic analysis has been performed with ADINA for the cable system of a 50-kW vertical axis wind turbine. The nonlinear static deformation and tension distribution are compared to the results of a nonlinear catenary cable model. The first natural frequencies and mode shapes, computed with ADINA, are in close agreement with experimental values obtained with the Ibrahim Time Domain (ITD) modal analysis method. The aerodynamic forces acting on the rotor of a vertical axis wind turbine contain many harmonics. These excitation forces located in a rotating coordinates system are transformed into the fixed coordinates of the guy cable system and a solution for the forced nonlinear vibration is obtained with ADINA. The results are compared to the experimental values obtained from field tests on IREQ-50-kW wind turbine.