| In the engineering structures, the design parameters may be uncertain because of complexity of structures, manufacture errors and inaccuracy in measurement, etc. The uncertainties of structural parameters may lead to large and unexpected excursion of responses that may lead to drastic reduction in accuracy and precision of the operation. In prevenient design and analysis of structures, people generally neglect the effect of the uncertainties and make the deterministic structures as the object of research. But with the development of high speed and efficient computers, people are not satisfied with the results of the deterministic analysis, and more accurate one is required. Therefore, the uncertainties are considered in the analysis of structure in order to describe actual operation.The uncertainties may be described as following three kinds: (1)Physical uncertainty. It is correlated to load, material and geometrical size. Generally speaking, the change of working conditions and the errors in manufacture or installation can bring this kind of uncertainty. (2)Statistical uncertainty. Nowadays, probability statistical methods are used to analysis uncertainty. (3) Model uncertainty. During structural analysis and design, the models are constructed between the inputs, including load, geometric size and plastic modules, and the outputs, including displacement, stress and strain. Besides the physical uncertainty, there are uncertainties in modeling such as the theoretical simplifying and unknown boundary conditions.Several mathematical models have been used for uncertain structural analysis in resent research, such as stochastics model, fuzzy model, covex model and interval one, etc. In the past, the methods used generally in these studies were based on probabilistic simulation approaches. Among the most commonly used techniques are direct Monte Carlo simulation. But it is difficult to use in the following two cases: (1)No more data can be given to obtain the statistical character. (2)The parameters follow the random mechanism. For the fuzzy and convex models, they rely on the subjective information. Proper model should be selected under actual conditions in the uncertain analysis.More attention has been paid to the interval model since the interval mathematics appeared, because it needs only the bounds of the structural parameters, which can be obtained in the engineering. Moore and his co-workers, Alefeld and Herzberger have done the pioneering work. Chen, Qiu, etc. have used interval method in the study of the response and eigenvalue problems of structures with bounded uncertain parameters. In their studies, several important results have been obtained.In recent years, the dynamic vibration control problems have attracted more researchers and the deterministic one has been well developed. But the components of control system may grow old or be damaged and the controlled object may change as the working conditions change. Many factors lead to errors and uncertainties of the mathematical model and the actual object. Researchers looked for a method to analyze and design the control system, it could use the uncertain information to guide the analysis and design, and it makes the closed-loop system stabile or lets it statisfy some performance index. So robustness has been an important problem in the field of control. Robustness analysis of the control systems aims at stability and dynamic characteristic of the systems when there are uncertainties and outer disturbance on the system. And the robustness analysis is the groundwork for synthesis problem. However, few papers can be found about the robustness of structures vibration control with interval parameters in engineering. Hence, it is necessary to develop an effective method to analyze this kind of problems.In the paper, the vibration control problem of structures with uncertain parameters is discussed with interval method. And two aspects of the active control problems of vibration structures with uncertain parameters are given respectively: one is robustness of system stability; the other is robustness of dynamic response. The second-order control problem is approximated by the corresponding deterministic one. The uncertain parameters are modeled to be an interval set. This does not require the probabilistic distribution descriptions of the uncertain parameters. The control system is changed into deterministic part and uncertain one. With the interval extension theory, the method for building the interval mass, stiffness and damping matrices with uncertain physical parameters is presented. In this paper, two modal control methods are presented: one is coupled modal space control method; the other is independent modal space control (IMSC) method. The former requires few sensors to more modal control, it's difficult to work out the equations, there is much work to be done, and the stability of system is not good; the latter needs more sensors (no less than the controlled modes), it needs little work, so it is easy to design, and the stability of system is good. Therefore, the IMSC is more convenient to deal with the eigenvalues and response analysis of the closed-loop system than the coupled modal space control method. We use coupled modal space control method and independent modal space control method respectively to obtain the displacement feedback gain matrix and velocity feedback gain matrix, and we apply them to the uncertain systems.Based on the second-order system, using the matrix perturbation and interval extension of functions, a new method for estimating the upper and lower bounds of eigenvalues and responses of closed-loop system with uncertain parameters was developed. The results can be used to discuss robustness of system stability and robustness of dynamic response. The advantage of these methods is that the present procedure is easy to implement on the computer and incorporate to the large finite element code,and provide information for the problem of the robustness synthesis. |
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