• Industrial Engineering and Management Systems
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• v.12 no.1
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• pp.2-8
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• 2013

Uncertainty theory is a branch of mathematics based on normolity, duality, subadditivity and product axioms. Uncertain process is a sequence of uncertain variables indexed by time. Canonical Liu process is an uncertain process with stationary and independent increments. And the increments follow normal uncertainty distributions. Uncertain differential equation is a type of differential equation driven by the canonical Liu process. Stability analysis on uncertain differential equation is to investigate the qualitative properties, which is significant both in theory and application for uncertain differential equations. This paper aims to study stability properties of linear uncertain differential equations. First, the stability concepts are introduced. And then, several sufficient and necessary conditions of stability for linear uncertain differential equations are proposed. Besides, some examples are discussed.

• Structural Engineering and Mechanics
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• v.23 no.6
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• pp.599-613
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• 2006

A method for the dynamical analysis of FE discretized uncertain linear and nonlinear structures is presented. This method is based on the moment equation approach, for which the differential equations governing the response first and second-order statistical moments must be solved. It is shown that they require the cross-moments between the response and the random variables characterizing the structural uncertainties, whose governing equations determine an infinite hierarchy. As a consequence, a closure scheme must be applied even if the structure is linear. In this sense the proposed approach is approximated even for the linear system. For nonlinear systems the closure schemes are also necessary in order to treat the nonlinearities. The complete set of equations obtained by this procedure is shown to be linear if the structure is linear. The application of this procedure to some simple examples has shown its high level of accuracy, if compared with other classical approaches, such as the perturbation method, even for low levels of closures.

• Coupled systems mechanics
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• v.7 no.6
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• pp.707-729
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• 2018

The semi-active optimal vibration control of nonlinear torsion-bar suspension vehicle systems under random road excitations is an important research subject, and the boundedness of MR dampers and the uncertainty of vehicle systems are necessary to consider. In this paper, the differential equations of motion of the coupling torsion-bar suspension vehicle system with MR damper under random road excitation are derived and then transformed into strongly nonlinear stochastic coupling vibration equations. The dynamical programming equation is derived based on the stochastic dynamical programming principle firstly for the nonlinear stochastic system. The semi-active bounded parametric optimal control law is determined by the programming equation and MR damper dynamics. Then for the uncertain nonlinear stochastic system, the minimax dynamical programming equation is derived based on the minimax stochastic dynamical programming principle. The worst-case disturbances and corresponding semi-active bounded parametric optimal control are obtained from the programming equation under the bounded disturbance constraints and MR damper dynamics. The control strategy for the nonlinear stochastic vibration of the uncertain torsion-bar suspension vehicle system is developed. The good effectiveness of the proposed control is illustrated with numerical results. The control performances for the vehicle system with different bounds of MR damper under different vehicle speeds and random road excitations are discussed.

• Journal of the Korean Institute of Intelligent Systems
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• v.5 no.2
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• pp.86-96
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• 1995

In this paper the fuzzy extension for the classical engineering mechanics problems is studied. The governing differential equation is derived for the buckling loads of the columns with uncertain mediums: the their own weight and the flexural rigidity. The columns with one typical end constraint(hinged1 clarnped/free) and the other finite rotational spring with fuzzy constant are considered in numerical examples. The vertex method is used to evaluate the fuzzy functions. The Runge-Kutta method and Determinant Search method are used to solve the differential equation and determine the buckling loads, respectively. The membership functions of the buckling load are calculated. The index of fuzziness to quantitatively describe the propagation of fuzziness is defined. According to the fuzziness of governing factors, the varlation of index of fuzziness for buckling load is investigated, and the sensitivity for the end constraints is analyzed.

• Journal of Advanced Marine Engineering and Technology
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• v.27 no.2
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• pp.218-228
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• 2003

A hydraulic system is modeled as the second order differential equation with uncertain system parameters and disturbance composed of modeling errors. To Position the load of the hydraulic system to a desired point. the servo valve of the hydraulic system is controlled. As a control scheme. a global sliding mode control(GSMC) is Proposed Since the servo valve has a torque limit. the GSMC is designed to coordinate the position of the load along the minimum time trajectory within the torque limit. The Proposed control scheme can be designed with ranges of parametric uncertainties and specified torque limits. By the proposed control scheme, the closed form solution of the arriving time at the desired position can be estimated.

• Structural Engineering and Mechanics
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• v.50 no.5
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• pp.697-708
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• 2014

The intentional buckling design of micro-films has various potential applications in engineering. The buckling amplitude and critical strain of micro-films are the crucial parameters for the buckling design. In the reported studies, the film parameters were regarded as deterministic. However, the geometrical and physical parameters uncertainty of micro-films due to manufacturing becomes prominent and needs to be considered. In the present paper, the probabilistic nonlinear buckling analysis of micro-films with uncertain parameters is proposed for design accuracy and reliability. The nonlinear differential equation and its asymptotic solution for the buckling micro-film with nominal parameters are firstly established. The mean values, standard deviations and variation coefficients of the buckling amplitude and critical strain are calculated by using the probability densities of uncertain parameters such as the film span length, thickness, elastic modulus and compressive force, to reveal the effects of the film parameter uncertainty on the buckling deformation. The results obtained illustrate the probabilistic relation between buckling deformation and uncertain parameters, and are useful for accurate and reliable buckling design in terms of probability.

• Journal of the Korean Institute of Intelligent Systems
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• v.5 no.3
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• pp.11-35
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• 1995

In this paper the fuzzy extension for the classical engineering mechanics problems is studied. The governing differential equation is derived for the buckling loads of the columns with uncertain mediums: the their own weight and the flexural rigidity. The columns with one typical end constraint(hinged1 clarnped/free) and the other finite rotational spring with fuzzy constant are considered in numerical examples. The vertex method is used to evaluate the fuzzy functions. The Runge-Kutta method and Determinant Search method are used to solve the differential equation and determine the buckling loads, respectively. The membership functions of the buckling load are calculated. The index of fuzziness to quantitatively describe the propagation of fuzziness is defined. According to the fuzziness of governing factors, the varlation of index of fuzziness for buckling load is investigated, and the sensitivity for the end constraints is analyzed.

• Transactions on Control, Automation and Systems Engineering
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• v.3 no.1
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• pp.21-26
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• 2001

In this study, we propose a two degree of freedom robust output tracking control method for a class of nonlinear system. We consider hyperbolically nonminimum phase single-input single-output uncertain nonlinear systems. We also consider the case that the nominal input-state equation is differentially flat. Nominal stable state trajectory is obtained in the flat output space via the flat output. Nominal feedforward control input is also computed from the nominal state trajectory. Due to the nature of the method, the generated flat output trajectory and control input are noncausal. Robust feedback control is designed to stabilize the systems around the nominal trajectory. A numerical example is given is given to demonstrate that robust tracking is achieved.

• Structural Engineering and Mechanics
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• v.28 no.4
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• pp.373-386
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• 2008

In this study stochastic analysis of non-linear dynamical systems under ${\alpha}$-stable, multiplicative white noise has been conducted. The analysis has dealt with a special class of ${\alpha}$-stable stochastic processes namely sub-Gaussian white noises. In this setting the governing equation either of the probability density function or of the characteristic function of the dynamical response may be obtained considering the dynamical system forced by a Gaussian white noise with an uncertain factor with ${\alpha}/2$- stable distribution. This consideration yields the probability density function or the characteristic function of the response by means of a simple integral involving the probability density function of the system under Gaussian white noise and the probability density function of the ${\alpha}/2$-stable random parameter. Some numerical applications have been reported assessing the reliability of the proposed formulation. Moreover a proper way to perform digital simulation of the sub-Gaussian ${\alpha}$-stable random process preventing dynamical systems from numerical overflows has been reported and discussed in detail.