Search results for: Nonlinear and linear programming

Authors: Carlos Mendez Galindo, Toshiro Hayashikawa, Javier Gil Belda

Abstract:

This paper investigates the effectiveness of the use of seismic isolation devices on the overall 3D seismic response of curved highway viaducts with an emphasis on expansion joints. Furthermore, an evaluation of the effectiveness of the use of cable restrainers is presented. For this purpose, the bridge seismic performance has been evaluated on four different radii of curvature, considering two cases: restrained and unrestrained curved viaducts. Depending on the radius of curvature, three-dimensional non-linear dynamic analysis shows the vulnerability of curved viaducts to pounding and deck unseating damage. In this study, the efficiency of using LRB supports combined with cable restrainers on curved viaducts is demonstrated, not only by reducing in all cases the possible damage, but also by providing a similar behavior in the viaducts despite of curvature radius.

Keywords: Nonlinear dynamic response, seismic design, seismic isolation, unseating prevention system.

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Authors: Zeshan Ahmad, Matteo Zoppi, Rezia Molfino

Abstract:

The objective of positioning the fixture elements in the fixture is to make the workpiece stiff, so that geometric errors in the manufacturing process can be reduced. Most of the work for optimal fixture layout used the minimization of the sum of the nodal deflection normal to the surface as objective function. All deflections in other direction have been neglected. We propose a new method for fixture layout optimization in this paper, which uses the element strain energy. The deformations in all the directions have been considered in this way. The objective function in this method is to minimize the sum of square of element strain energy. Strain energy and stiffness are inversely proportional to each other. The optimization problem is solved by the sequential quadratic programming method. Three different kinds of case studies are presented, and results are compared with the method using nodal deflections as objective function to verify the propose method.

Keywords: Fixture layout, optimization, strain energy, quadratic programming.

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Authors: A. K. Al-Othman, H. A. Al-Mekhaizim

Abstract:

Multilevel inverters supplied from equal and constant dc sources almost don-t exist in practical applications. The variation of the dc sources affects the values of the switching angles required for each specific harmonic profile, as well as increases the difficulty of the harmonic elimination-s equations. This paper presents an extremely fast optimal solution of harmonic elimination of multilevel inverters with non-equal dc sources using Tanaka's fuzzy linear regression formulation. A set of mathematical equations describing the general output waveform of the multilevel inverter with nonequal dc sources is formulated. Fuzzy linear regression is then employed to compute the optimal solution set of switching angles.

Keywords: Multilevel converters, harmonics, pulse widthmodulation (PWM), optimal control.

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Authors: Alexander Wachtel, Michael T. Franzen, Walter F. Tichy

Abstract:

Programming requires years of training. With natural language and end user development methods, programming could become available to everyone. It enables end users to program their own devices and extend the functionality of the existing system without any knowledge of programming languages. In this paper, we describe an Interactive Spreadsheet Processing Module (ISPM), a natural language interface to spreadsheets that allows users to address ranges within the spreadsheet based on inferred table schema. Using the ISPM, end users are able to search for values in the schema of the table and to address the data in spreadsheets implicitly. Furthermore, it enables them to select and sort the spreadsheet data by using natural language. ISPM uses a machine learning technique to automatically infer areas within a spreadsheet, including different kinds of headers and data ranges. Since ranges can be identified from natural language queries, the end users can query the data using natural language. During the evaluation 12 undergraduate students were asked to perform operations (sum, sort, group and select) using the system and also Excel without ISPM interface, and the time taken for task completion was compared across the two systems. Only for the selection task did users take less time in Excel (since they directly selected the cells using the mouse) than in ISPM, by using natural language for end user software engineering, to overcome the present bottleneck of professional developers.

Keywords: Natural language processing, end user development; natural language interfaces, human computer interaction, data recognition, dialog systems, spreadsheet.

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Authors: Ö. Tolga Altınöz

Abstract:

The adaptive backstepping controller for inverted pendulum is designed by using the general motion control model. Backstepping is a novel nonlinear control technique based on the Lyapunov design approach, used when higher derivatives of parameter estimation appear. For easy parameter adaptation, the mathematical model of the inverted pendulum converted into the motion control model. This conversion is performed by taking functions of unknown parameters and dynamics of the system. By using motion control model equations, inverted pendulum is simulated without any information about not only parameters but also measurable dynamics. Also these results are compare with the adaptive backstepping controller which extended with integral action that given from [1].

Keywords: Adaptive backstepping, inverted pendulum, nonlinear adaptive control.

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Authors: Olugbara O.O., Adebiyi A.A.

Abstract:

Most standard software development methodologies are often not applied to software projects in many developing countries of the world. The approach generally practice is close to what eXtreme Programming (XP) is likely promoting, just keep coding and testing as the requirement evolves. XP is an agile software process development methodology that has inherent capability for improving efficiency of Business Software Development (BSD). XP can facilitate Business-to-Development (B2D) relationship due to its customer-oriented advocate. From practitioner point of view, we applied XP to BSD and result shows that customer involvement has positive impact on productivity, but can as well frustrate the success of the project. In an effort to promote software engineering practice in developing countries of Africa, we present the experiment performed, lessons learned, problems encountered and solution adopted in applying XP methodology to BSD.

Keywords: Requirements engineering, Requirements elicitation, Extreme programming, Mobile Work force

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Authors: A. Zare Bazghaleh, M. R. Naghashan, H. Mahmoudimanesh, M. R. Meshkatoddini

Abstract:

Linear induction motors are used in various industries but they have some specific phenomena which are the causes for some problems. The most important phenomenon is called end effect. End effect decreases efficiency, power factor and output force and unbalances the phase currents. This phenomenon is more important in medium and high speeds machines. In this paper a factor, EEF , is obtained by an accurate equivalent circuit model, to determine the end effect intensity. In this way, all of effective design parameters on end effect is described. Accuracy of this equivalent circuit model is evaluated by two dimensional finite-element analysis using ANSYS. The results show the accuracy of the equivalent circuit model.

Keywords: Linear induction motor, end effect, equivalent circuitmodel, finite-element method.

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Authors: Cuneyt Yucelbas, Seral Ozsen, Sule Yucelbas, Gulay Tezel

Abstract:

Due to the fact that there exist only a small number of complex systems in artificial immune system (AIS) that work out nonlinear problems, nonlinear AIS approaches, among the well-known solution techniques, need to be developed. Gaussian function is usually used as similarity estimation in classification problems and pattern recognition. In this study, diagnosis of breast cancer, the second type of the most widespread cancer in women, was performed with different distance calculation functions that euclidean, gaussian and gaussian-euclidean hybrid function in the clonal selection model of classical AIS on Wisconsin Breast Cancer Dataset (WBCD), which was taken from the University of California, Irvine Machine-Learning Repository. We used 3-fold cross validation method to train and test the dataset. According to the results, the maximum test classification accuracy was reported as 97.35% by using of gaussian-euclidean hybrid function for fold-3. Also, mean of test classification accuracies for all of functions were obtained as 94.78%, 94.45% and 95.31% with use of euclidean, gaussian and gaussian-euclidean, respectively. With these results, gaussian-euclidean hybrid function seems to be a potential distance calculation method, and it may be considered as an alternative distance calculation method for hard nonlinear classification problems.

Keywords: Artificial Immune System, Breast Cancer Diagnosis, Euclidean Function, Gaussian Function.

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Authors: Integral square error, Least-squares, Markovparameters, Moment matching, Order reduction.

Abstract:

The concept of order reduction by least-squares moment matching and generalised least-squares methods has been extended about a general point ?a?, to obtain the reduced order models for linear, time-invariant dynamic systems. Some heuristic criteria have been employed for selecting the linear shift point ?a?, based upon the means (arithmetic, harmonic and geometric) of real parts of the poles of high order system. It is shown that the resultant model depends critically on the choice of linear shift point ?a?. The validity of the criteria is illustrated by solving a numerical example and the results are compared with the other existing techniques.

Keywords: Integral square error, Least-squares, Markovparameters, Moment matching, Order reduction.

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Authors: Soyoon Bak, Sunyoung Bu, Philsu Kim

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In this paper, a backward semi-Lagrangian scheme combined with the second-order backward difference formula is designed to calculate the numerical solutions of nonlinear advection-diffusion equations. The primary aims of this paper are to remove any iteration process and to get an efficient algorithm with the convergence order of accuracy 2 in time. In order to achieve these objects, we use the second-order central finite difference and the B-spline approximations of degree 2 and 3 in order to approximate the diffusion term and the spatial discretization, respectively. For the temporal discretization, the second order backward difference formula is applied. To calculate the numerical solution of the starting point of the characteristic curves, we use the error correction methodology developed by the authors recently. The proposed algorithm turns out to be completely iteration free, which resolves the main weakness of the conventional backward semi-Lagrangian method. Also, the adaptability of the proposed method is indicated by numerical simulations for Burgers’ equations. Throughout these numerical simulations, it is shown that the numerical results is in good agreement with the analytic solution and the present scheme offer better accuracy in comparison with other existing numerical schemes.

Keywords: Semi-Lagrangian method, Iteration free method, Nonlinear advection-diffusion equation.

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Authors: Kourosh Parand, Jamal Amani Rad

Abstract:

In this paper, Exp-function method is used for some exact solitary solutions of the generalized Pochhammer-Chree equation. It has been shown that the Exp-function method, with the help of symbolic computation, provides a very effective and powerful mathematical tool for solving nonlinear partial differential equations. As a result, some exact solitary solutions are obtained. It is shown that the Exp-function method is direct, effective, succinct and can be used for many other nonlinear partial differential equations.

Keywords: Exp-function method, generalized Pochhammer- Chree equation, solitary wave solution, ODE's.

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Authors: B. Ghosh, H. Sahoo, K. K. Mondal

Abstract:

Propagation of arbitrary amplitude nonlinear Alfven waves has been investigated in low but finite β electron-positron-ion plasma including full ion dynamics. Using Sagdeev pseudopotential method an energy integral equation has been derived. The Sagdeev potential has been calculated for different plasma parameters and it has been shown that inclusion of ion parallel motion along the magnetic field changes the nature of slow shear Alfven wave solitons from dip type to hump type. The effects of positron concentration, plasma-β and obliqueness of the wave propagation on the solitary wave structure have also been examined.

Keywords: Alfven waves, Sagdeev potential, Solitary waves.

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Authors: E. Assareh, M.A. Behrang, M. Ghalambaz, A.R. Noghrehabadi, A. Ghanbarzadeh

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In this paper, a new approach is introduced to solve Blasius equation using parameter identification of a nonlinear function which is used as approximation function. Bees Algorithm (BA) is applied in order to find the adjustable parameters of approximation function regarding minimizing a fitness function including these parameters (i.e. adjustable parameters). These parameters are determined how the approximation function has to satisfy the boundary conditions. In order to demonstrate the presented method, the obtained results are compared with another numerical method. Present method can be easily extended to solve a wide range of problems.

Keywords: Bees Algorithm (BA); Approximate Solutions; Blasius Differential Equation.

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Authors: Michal Cerny

Abstract:

We consider linear regression models where both input data (the values of independent variables) and output data (the observations of the dependent variable) are interval-censored. We introduce a possibilistic generalization of the least squares estimator, so called OLS-set for the interval model. This set captures the impact of the loss of information on the OLS estimator caused by interval censoring and provides a tool for quantification of this effect. We study complexity-theoretic properties of the OLS-set. We also deal with restricted versions of the general interval linear regression model, in particular the crisp input – interval output model. We give an argument that natural descriptions of the OLS-set in the crisp input – interval output cannot be computed in polynomial time. Then we derive easily computable approximations for the OLS-set which can be used instead of the exact description. We illustrate the approach by an example.

Keywords: Linear regression, interval-censored data, computational complexity.

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Authors: Mehdi Ghayoumi

Abstract:

We have applied new accelerated algorithm for linear discriminate analysis (LDA) in face recognition with support vector machine. The new algorithm has the advantage of optimal selection of the step size. The gradient descent method and new algorithm has been implemented in software and evaluated on the Yale face database B. The eigenfaces of these approaches have been used to training a KNN. Recognition rate with new algorithm is compared with gradient.

Keywords: lda, adaptive, svm, face recognition.

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Authors: A. Nuric, S. Nuric, L. Kricak, I. Lapandic, R. Husagic

Abstract:

This paper describes topic of computer simulation with regard to the ground movement above an underground mine. Simulation made with software package ADINA for nonlinear elastic-plastic analysis with finite elements method. The one of representative profiles from Mine 'Stara Jama' in Zenica has been investigated. A collection and selection of both geo-mechanical data and geometric parameters of the mine was necessary for performing these simulations. Results of estimation have been compared with measured values (vertical displacement of surface), and then simulation performed with assumed dynamic and dimensions of excavation, over a period of time. Results are presented with bitmaps and charts.

Keywords: Computer, finite element method, mine, nonlinear analysis, numerical modeling, simulation, subsidence.

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Authors: Kazuyoshi Mori, Keisuke Hashimoto

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In this paper, we consider the two-stage compensator designs of SISO plants. As an investigation of the characteristics of the two-stage compensator designs, which is not well investigated yet, of SISO plants, we implement three dimensional visualization systems of output signals and optimization system for SISO plants by the parametrization of stabilizing controllers based on the two-stage compensator design. The system runs on Mathematica by using “Three Dimensional Surface Plots,” so that the visualization can be interactively manipulated by users. In this paper, we use the discrete-time LTI system model. Even so, our approach is the factorization approach, so that the result can be applied to many linear models.

Keywords: Linear systems, visualization, optimization, two-Stage compensator design, Mathematica.

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Authors: Saman M. Siddiqui, Fang Jiancheng

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Robust nonlinear integrated navigation of GPS and low cost MEMS is a hot topic of research these days. A robust filter is required to cope up with the problem of unpredictable discontinuities and colored noises associated with low cost sensors. H∞ filter is previously used in Extended Kalman filter and Unscented Kalman filter frame. Unscented Kalman filter has a problem of Cholesky matrix factorization at each step which is a very unstable operation. To avoid this problem in this research H∞ filter is designed in Square root Unscented filter framework and found 50% more robust towards increased level of colored noises.

Keywords: H∞ filter, MEMS, GPS, Nonlinear system, robust system, Square root unscented filter.

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Authors: B. X. Tchomeni, A. A. Alugongo, L. M. Masu

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An analytical 4-DOF nonlinear model of a de Laval rotor-stator system based on Energy Principles has been used theoretically and experimentally to investigate fault symptoms in a rotating system. The faults, namely rotor-stator-rub, crack and unbalance are modeled as excitations on the rotor shaft. Mayes steering function is used to simulate the breathing behaviour of the crack. The fault analysis technique is based on waveform signal, orbits and Fast Fourier Transform (FFT) derived from simulated and real measured signals. Simulated and experimental results manifest considerable mutual resemblance of elliptic-shaped orbits and FFT for a same range of test data.

Keywords: A breathing crack, fault, FFT, nonlinear, orbit, rotorstator rub, vibration analysis.

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Authors: Zixin Liu, Jian Yu, Daoyun Xu

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This paper aims to establish a delayed dynamical relationship between payoffs of players in a zero-sum game. By introducing Markovian chain and time delay in the network model, a delayed game network model with sector bounds and slope bounds restriction nonlinear function is first proposed. As a result, a direct dynamical relationship between payoffs of players in a zero-sum game can be illustrated through a delayed singular system. Combined with Finsler-s Lemma and Lyapunov stable theory, a sufficient condition guaranteeing the unique existence and stability of zero-sum game-s Nash equilibrium is derived. One numerical example is presented to illustrate the validity of the main result.

Keywords: Game networks, zero-sum game, delayed singular system, nonlinear perturbation, time delay.

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Authors: Kahil Amar, Nekmouche Aghiles, Titouche Billal, Hamizi Mohand, Hannachi Naceur Eddine

Abstract:

The ductility is an important parameter in the nonlinear behavior of portal structures reinforced concrete. It may be explained by the ability of the structure to deform in the plastic range, or the geometric characteristics in the map may influence the overall ductility. Our study is based on the influence of geometric slenderness (Lx / Ly) on the overall ductility of these structures, a study is made on a structure has 05 floors with varying the column section of 900 cm², 1600 cm² and 1225 cm². A slight variation in global ductility is noticed as (Lx/Ly) varies; however, column sections can control satisfactorily the plastic behavior of buildings.

Keywords: Ductility, nonlinear behavior, pushover analysis, geometric slenderness, structural behavior.

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Authors: A. Pereira, S. Haffner, L. V. Gasperin

Abstract:

This paper presents two simplified models to determine nodal voltages in power distribution networks. These models allow estimating the impact of the installation of reactive power compensations equipments like fixed or switched capacitor banks. The procedure used to develop the models is similar to the procedure used to develop linear power flow models of transmission lines, which have been widely used in optimization problems of operation planning and system expansion. The steady state non-linear load flow equations are approximated by linear equations relating the voltage amplitude and currents. The approximations of the linear equations are based on the high relationship between line resistance and line reactance (ratio R/X), which is valid for power distribution networks. The performance and accuracy of the models are evaluated through comparisons with the exact results obtained from the solution of the load flow using two test networks: a hypothetical network with 23 nodes and a real network with 217 nodes.

Keywords: Distribution network models, distribution systems, optimization, power system planning.

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Authors: P. Beraldi, M. E. Bruni

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Data Envelopment Analysis (DEA) is one of the most widely used technique for evaluating the relative efficiency of a set of homogeneous decision making units. Traditionally, it assumes that input and output variables are known in advance, ignoring the critical issue of data uncertainty. In this paper, we deal with the problem of efficiency evaluation under uncertain conditions by adopting the general framework of the stochastic programming. We assume that output parameters are represented by discretely distributed random variables and we propose two different models defined according to a neutral and risk-averse perspective. The models have been validated by considering a real case study concerning the evaluation of the technical efficiency of a sample of individual firms operating in the Italian leather manufacturing industry. Our findings show the validity of the proposed approach as ex-ante evaluation technique by providing the decision maker with useful insights depending on his risk aversion degree.

Keywords: DEA, Stochastic Programming, Ex-ante evaluation technique, Conditional Value at Risk.

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Authors: Yang Weiping, Zhang Zhilong, Zhang Yan, Chen Zengping

Abstract:

Infrared focal plane arrays (IRFPA) sensors, due to their high sensitivity, high frame frequency and simple structure, have become the most prominently used detectors in military applications. However, they suffer from a common problem called the fixed pattern noise (FPN), which severely degrades image quality and limits the infrared imaging applications. Therefore, it is necessary to perform non-uniformity correction (NUC) on IR image. The algorithms of non-uniformity correction are classified into two main categories, the calibration-based and scene-based algorithms. There exist some shortcomings in both algorithms, hence a novel non-uniformity correction algorithm based on non-linear fit is proposed, which combines the advantages of the two algorithms. Experimental results show that the proposed algorithm acquires a good effect of NUC with a lower non-uniformity ratio.

Keywords: Non-uniformity correction, non-linear fit, two-point correction, temporal Kalman filter.

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Authors: Ioannis P. Panagopoulos, Christos C. Pavlatos, George K. Papakonstantinou

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Conventional approaches in the implementation of logic programming applications on embedded systems are solely of software nature. As a consequence, a compiler is needed that transforms the initial declarative logic program to its equivalent procedural one, to be programmed to the microprocessor. This approach increases the complexity of the final implementation and reduces the overall system's performance. On the contrary, presenting hardware implementations which are only capable of supporting logic programs prevents their use in applications where logic programs need to be intertwined with traditional procedural ones, for a specific application. We exploit HW/SW codesign methods to present a microprocessor, capable of supporting hybrid applications using both programming approaches. We take advantage of the close relationship between attribute grammar (AG) evaluation and knowledge engineering methods to present a programmable hardware parser that performs logic derivations and combine it with an extension of a conventional RISC microprocessor that performs the unification process to report the success or failure of those derivations. The extended RISC microprocessor is still capable of executing conventional procedural programs, thus hybrid applications can be implemented. The presented implementation is programmable, supports the execution of hybrid applications, increases the performance of logic derivations (experimental analysis yields an approximate 1000% increase in performance) and reduces the complexity of the final implemented code. The proposed hardware design is supported by a proposed extended C-language called C-AG.

Keywords: Attribute Grammars, Logic Programming, RISC microprocessor.

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Authors: Inta Volodko, Valentina Koliskina

Abstract:

Linear stability analysis of wake-shear layers in twophase shallow flows is performed in the present paper. Twodimensional shallow water equations are used in the analysis. It is assumed that the fluid contains uniformly distributed solid particles. No dynamic interaction between the carrier fluid and particles is expected in the initial moment. The stability calculations are performed for different values of the particle loading parameter and two other parameters which characterize the velocity ratio and the velocity deficit. The results show that the particle loading parameter has a stabilizing effect on the flow while the increase in the velocity ratio or in the velocity deficit destabilizes the flow.

Keywords: Linear stability, Shallow flows, Wake-shear flows.

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Authors: Takashi Kaburagi, Takamasa Komura, Yosuke Kurihara

Abstract:

Absolute pitch is the ability to identify a musical note without a reference tone. Training for absolute pitch often occurs in preschool education. It is necessary to clarify how well the trainee can make use of synesthesia in order to evaluate the effect of the training. To the best of our knowledge, there are no existing methods for objectively confirming whether the subject is using synesthesia. Therefore, in this study, we present a method to distinguish the use of color-auditory synesthesia from the separate use of color and audition during absolute pitch training. This method measures blood volume in the prefrontal cortex using functional Near-infrared spectroscopy (fNIRS) and assumes that the cognitive step has two parts, a non-linear step and a linear step. For the linear step, we assume a second order ordinary differential equation. For the non-linear part, it is extremely difficult, if not impossible, to create an inverse filter of such a complex system as the brain. Therefore, we apply a method based on a self-organizing map (SOM) and are guided by the available data. The presented method was tested using 15 subjects, and the estimation accuracy is reported.

Keywords: Absolute pitch, functional near-infrared spectroscopy, prefrontal cortex, synesthesia.

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Authors: K. V. Geetha, N. R. Mangalambal

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Aim. We have introduced the notion of order to multinormed spaces and countable union spaces and their duals. The topology of bounded convergence is assigned to the dual spaces. The aim of this paper is to develop the theory of ordered topological linear spaces La,b, L(w, z), the dual spaces of ordered multinormed spaces La,b, ordered countable union spaces L(w, z), with the topology of bounded convergence assigned to the dual spaces. We apply Laplace transformation to the ordered linear space of Laplace transformable generalized functions. We ultimately aim at finding solutions to nonhomogeneous nth order linear differential equations with constant coefficients in terms of generalized functions and comparing different solutions evolved out of different initial conditions. Method. The above aim is achieved by • Defining the spaces La,b, L(w, z). • Assigning an order relation on these spaces by identifying a positive cone on them and studying the properties of the cone. • Defining an order relation on the dual spaces La,b, L(w, z) of La,b, L(w, z) and assigning a topology to these dual spaces which makes the order dual and the topological dual the same. • Defining the adjoint of a continuous map on these spaces and studying its behaviour when the topology of bounded convergence is assigned to the dual spaces. • Applying the two-sided Laplace Transformation on the ordered linear space of generalized functions W and studying some properties of the transformation which are used in solving differential equations. Result. The above techniques are applied to solve non-homogeneous n-th order linear differential equations with constant coefficients in terms of generalized functions and to compare different solutions of the differential equation.

Keywords: Laplace transformable generalized function, positive cone, topology of bounded convergence

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Authors: M. Ravindra, R. Srinivasa Rao

Abstract:

This paper presents a Generalized Binary Integer Linear Programming (GBILP) method for optimal allocation of Phasor Measurement Units (PMUs) and to generate Dynamic State Estimation (DSE) solution with complete observability. The GBILP method is formulated with Zero Injection Bus (ZIB) constraints to reduce the number of locations for placement of PMUs in the case of normal and single line contingency. The integration of PMU and conventional measurements is modeled in DSE process to estimate accurate states of the system. To estimate the dynamic behavior of the power system with proposed method, load change up to 40% considered at a bus in the power system network. The proposed DSE method is compared with traditional Weighted Least Squares (WLS) state estimation method in presence of load changes to show the impact of PMU measurements. MATLAB simulations are carried out on IEEE 14, 30, 57, and 118 bus systems to prove the validity of the proposed approach.

Keywords: Observability, phasor measurement units, PMU, state estimation, dynamic state estimation, SCADA measurements, zero injection bus.

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Authors: Shuenn-Yih Chang, Chiu-Li Huang

Abstract:

Three structure-dependent integration methods have been developed for solving equations of motion, which are second-order ordinary differential equations, for structural dynamics and earthquake engineering applications. Although they generally have the same numerical properties, such as explicit formulation, unconditional stability and second-order accuracy, a different performance is found in solving the free vibration response to either linear elastic or nonlinear systems with high frequency modes. The root cause of this different performance in the free vibration responses is analytically explored herein. As a result, it is verified that a weak instability is responsible for the different performance of the integration methods. In general, a weak instability will result in an inaccurate solution or even numerical instability in the free vibration responses of high frequency modes. As a result, a weak instability must be prohibited for time integration methods.

Keywords: Dynamic analysis, high frequency, integration method, overshoot, weak instability.

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